New paper on emergent gravity

http://qedcorp.com/book/EmergentGravity.pdf

## Tuesday, January 31, 2006

## Saturday, January 28, 2006

Civil War in Stanford's Physics Department?

I stand in the cross fire between Susskind and Laughlin. Do they have adjoining offices like Gell-Mann and Feynman? :-)

Laughlin hauls out his heavy artillery against Susskind in his final chapter

"Exploding things, such as dynamite or the big bang are unstable. Theories of explosions, including the first picoseconds of the big bang ... are inherently unfalsifiable, notwithstanding widely cited supportive 'evidence' such as ... the cosmic microwave anisotropy ... Beyond the big bang we have really unfalsifiable concepts of budding little baby universes with different properties that must have been created before the inflationary epoch, but which are now fundamentally undetectable due to being beyond the light horizon."

Not so fast Professor Laughlin. Antony Valentini has shown how quantum theory may be slightly extended so as to permit "signal nonlocality" violating the "no cloning theorem." Indeed your very idea of emergence will force signal nonlocality in open non-equilibrium living matter I suspect. If so, then the pocket universes of eternal inflation beyond the Hubble horizon may not be fundamentally unobservable at all. This would mean that entanglement could be used as a stand alone C^3 channel. There is no reason to require macro-quantum ground states to be linear and unitary subject to the Born probability interpretation simply because their elementary excitations are. Contrary to popular opinion Steven Weinberg did not hammer the last nail into the coffin of signal nonlocality. He only chose experiments in which one would not expect signal nonlocality in the first place.

So that is my first shot across Laughlin's bow, although I do agree with a lot of his ideas about emergence.

On Jan 28, 2006, at 12:40 PM, Jack Sarfatti wrote:

I am significantly revising my book Super Cosmos in a 2nd revised edition to include the creative tension between Robert Laughlin and Lenny Susskind, both physics professors at Stanford, seen at the endings of their respective books "A Different Universe" and "Cosmic Landscape". I will also include Richard Gott's book "Time Travel" on the self-creating megaverse of pocket universes.

I also will be improving the accuracy of the current edition regarding the nature of the zero point energy cores of the charge clusters of Ken Shoulders and the microgeon hidden variable model of the spatially extended J.P. Vigier electron/quark that shrinks to a point from space-warping when hit hard. I will clean up the Cartan form equations for emergent tetrad gravity using two Goldstone phases for the vacuum coherent manifold instead of one. This seems to fit the facts in a much more natural way. The cosmological constant "Mother of all physics problems" is solved by realizing that it is the emergent coherent non-random Higgs vacuum fields Goldstone phase variations that provide the "marble" of Einstein's smooth c-number geometrodynamics Guv, with only a little bit of locally random, but nonlocally EPR correlated, zero point vacuum fluctuation in the "wood" source /\guv.

Lambda = (Quantum of Area Flux Density)^-1(1 - |Higgs Field|^2)

The curved locally invariant gravity tetrad field is the intrinsic Cartan 1-form

B ~ (dtheta)(phi) - (theta)(dphi)

dB ~ 2(dtheta)/\(dphi)

/\ = antisymmetric Cartan exterior multiplication

Einstein's Curved Spacetime local invariant is by the Equivalence Principle

ds^2 = (1 + B)(Flat Spacetime)(1 + B)

No /\ there of course. The metric field is a "bosonic" tetrad pair, the "tetrad" field is a "fermion".

The renormalizable Higgs ODLRO field potential is

V(Higgs) = a|Psi|^2 + b|Psi|^4

Spontaneous symmetry breakdown of the false vacuum to a local minimum in the landscape means

a < 0, b > 0

In my new model

|Psi|^2 = |Psi1|^2 + |Psi2|^2 + |Psi3|^2

The degenerate vacuum manifold is

G/H = S2

i.e.

1 = (|Psi1|^2 + |Psi2|^2 + |Psi3|^2)/|Psi|^2

= cos^2(Theta) + cos^2(Phi) + cos^2(Chi)

Theta & Phi are the two independent Goldstone phases of the Vacuum Manifold.

Note that this is like a simple spherical Fermi surface. The Calabi-Yau parameters would be additional structure on the Vacuum Manifold "surface" - things like wormhole handles et-al to get the 10^500 worlds in the FRW de Sitter limit where Lambda is constant.

I stand in the cross fire between Susskind and Laughlin. Do they have adjoining offices like Gell-Mann and Feynman? :-)

Laughlin hauls out his heavy artillery against Susskind in his final chapter

"Exploding things, such as dynamite or the big bang are unstable. Theories of explosions, including the first picoseconds of the big bang ... are inherently unfalsifiable, notwithstanding widely cited supportive 'evidence' such as ... the cosmic microwave anisotropy ... Beyond the big bang we have really unfalsifiable concepts of budding little baby universes with different properties that must have been created before the inflationary epoch, but which are now fundamentally undetectable due to being beyond the light horizon."

Not so fast Professor Laughlin. Antony Valentini has shown how quantum theory may be slightly extended so as to permit "signal nonlocality" violating the "no cloning theorem." Indeed your very idea of emergence will force signal nonlocality in open non-equilibrium living matter I suspect. If so, then the pocket universes of eternal inflation beyond the Hubble horizon may not be fundamentally unobservable at all. This would mean that entanglement could be used as a stand alone C^3 channel. There is no reason to require macro-quantum ground states to be linear and unitary subject to the Born probability interpretation simply because their elementary excitations are. Contrary to popular opinion Steven Weinberg did not hammer the last nail into the coffin of signal nonlocality. He only chose experiments in which one would not expect signal nonlocality in the first place.

So that is my first shot across Laughlin's bow, although I do agree with a lot of his ideas about emergence.

On Jan 28, 2006, at 12:40 PM, Jack Sarfatti wrote:

I am significantly revising my book Super Cosmos in a 2nd revised edition to include the creative tension between Robert Laughlin and Lenny Susskind, both physics professors at Stanford, seen at the endings of their respective books "A Different Universe" and "Cosmic Landscape". I will also include Richard Gott's book "Time Travel" on the self-creating megaverse of pocket universes.

I also will be improving the accuracy of the current edition regarding the nature of the zero point energy cores of the charge clusters of Ken Shoulders and the microgeon hidden variable model of the spatially extended J.P. Vigier electron/quark that shrinks to a point from space-warping when hit hard. I will clean up the Cartan form equations for emergent tetrad gravity using two Goldstone phases for the vacuum coherent manifold instead of one. This seems to fit the facts in a much more natural way. The cosmological constant "Mother of all physics problems" is solved by realizing that it is the emergent coherent non-random Higgs vacuum fields Goldstone phase variations that provide the "marble" of Einstein's smooth c-number geometrodynamics Guv, with only a little bit of locally random, but nonlocally EPR correlated, zero point vacuum fluctuation in the "wood" source /\guv.

Lambda = (Quantum of Area Flux Density)^-1(1 - |Higgs Field|^2)

The curved locally invariant gravity tetrad field is the intrinsic Cartan 1-form

B ~ (dtheta)(phi) - (theta)(dphi)

dB ~ 2(dtheta)/\(dphi)

/\ = antisymmetric Cartan exterior multiplication

Einstein's Curved Spacetime local invariant is by the Equivalence Principle

ds^2 = (1 + B)(Flat Spacetime)(1 + B)

No /\ there of course. The metric field is a "bosonic" tetrad pair, the "tetrad" field is a "fermion".

The renormalizable Higgs ODLRO field potential is

V(Higgs) = a|Psi|^2 + b|Psi|^4

Spontaneous symmetry breakdown of the false vacuum to a local minimum in the landscape means

a < 0, b > 0

In my new model

|Psi|^2 = |Psi1|^2 + |Psi2|^2 + |Psi3|^2

The degenerate vacuum manifold is

G/H = S2

i.e.

1 = (|Psi1|^2 + |Psi2|^2 + |Psi3|^2)/|Psi|^2

= cos^2(Theta) + cos^2(Phi) + cos^2(Chi)

Theta & Phi are the two independent Goldstone phases of the Vacuum Manifold.

Note that this is like a simple spherical Fermi surface. The Calabi-Yau parameters would be additional structure on the Vacuum Manifold "surface" - things like wormhole handles et-al to get the 10^500 worlds in the FRW de Sitter limit where Lambda is constant.

Lisa Randall's "Warped Passages"

I will review the following books in the revised Super Cosmos:

Warped Passages (Randall) on extra space dimensions

Time Travel (Gott) on time travel to the past and the self-creating megaverse

Cosmic Landscape (Susskind) on why the cosmic energy is so small and parallel pocket universes next door in eternal chaotic inflation.

A Different Universe (Laughlin) - on the physicality of BIT as the emergent organizational principle. No measurements of fundamental constants at the micro-quantum level are possible without spontaneous broken macro-quantum emergence - with different rules.

It's not only

IT FROM BIT

we also need

BIT FROM IT

On Jan 28, 2006, at 6:33 PM, Figaro wrote:

Lisa Randall does a very good job of explaining complex ideas in lay terms, in fact, she over explains certain ideas which might confuse some readers. That's why you need to take time reading the book, you can't speed read it.......Rmc

Jack Sarfatti wrote:

Buy a copy of Robert Laughlin's book "A Different Universe." YOU WILL LOVE IT as well as learn a lot of good stuff. I might be able to get George Chapline to introduce you. He is an experienced outdoorsman and sets his book in a kind of desert trek - native Californian. I wonder how Laughlin would react to the Skinwalker experiences! Now that would be something!

On Jan 28, 2006, at 6:01 PM, Kim Burrafato wrote:

Just finished Exempt from Disclosure. Very sophisticated disinfo? Actual disclosure -- albeit carefully spun? At the very least, it makes for a good read, like Skinwalker. I heard similar stories from naval flight officers on board the Ranger when I was in WestPac in '72-73. What's amazing to me, is how effective those charged with The Keeping Of The Secret have been! Makes one wonder of the ones keeping the secret actually live here... It's a shame that a constructive SF meeting couldn't have come off with Doty, Collins, Knapp, et al in attendance. Damn religious fanatics...they always manage to mess up a good thing.

Dan needs to awaken from his anti-theoretical slumber.

On Jan 28, 2006, at 5:37 PM, Jack Sarfatti wrote:

On Jan 28, 2006, at 5:30 PM, Hank Harrison wrote:

Jack I just finished Lisa Randall's book on coincidence and the matrix aka string theory etc, Seems to me she is spinning her wheels and never really gets anyplace, but maybe I am not sophisticated enough to grasp it all, its a thick sucker, i didn't read every word, What do you think?

I only read pieces of it, then got involved with Richard Gott's, Lenny Susskind's and Robert Laughlin's books. I will get back to hers for my revision of Super Cosmos. Will tell you then. :-)

Laughlin's "A Different Universe" you and Kim will like. Laughlin is an experienced outdoors-man and his book reads like a Zane Gray Western. But it has to be read with Lenny Susskind's "Cosmic Landscape."

I will review the following books in the revised Super Cosmos:

Warped Passages (Randall) on extra space dimensions

Time Travel (Gott) on time travel to the past and the self-creating megaverse

Cosmic Landscape (Susskind) on why the cosmic energy is so small and parallel pocket universes next door in eternal chaotic inflation.

A Different Universe (Laughlin) - on the physicality of BIT as the emergent organizational principle. No measurements of fundamental constants at the micro-quantum level are possible without spontaneous broken macro-quantum emergence - with different rules.

It's not only

IT FROM BIT

we also need

BIT FROM IT

On Jan 28, 2006, at 6:33 PM, Figaro wrote:

Lisa Randall does a very good job of explaining complex ideas in lay terms, in fact, she over explains certain ideas which might confuse some readers. That's why you need to take time reading the book, you can't speed read it.......Rmc

Jack Sarfatti wrote:

Buy a copy of Robert Laughlin's book "A Different Universe." YOU WILL LOVE IT as well as learn a lot of good stuff. I might be able to get George Chapline to introduce you. He is an experienced outdoorsman and sets his book in a kind of desert trek - native Californian. I wonder how Laughlin would react to the Skinwalker experiences! Now that would be something!

On Jan 28, 2006, at 6:01 PM, Kim Burrafato wrote:

Just finished Exempt from Disclosure. Very sophisticated disinfo? Actual disclosure -- albeit carefully spun? At the very least, it makes for a good read, like Skinwalker. I heard similar stories from naval flight officers on board the Ranger when I was in WestPac in '72-73. What's amazing to me, is how effective those charged with The Keeping Of The Secret have been! Makes one wonder of the ones keeping the secret actually live here... It's a shame that a constructive SF meeting couldn't have come off with Doty, Collins, Knapp, et al in attendance. Damn religious fanatics...they always manage to mess up a good thing.

Dan needs to awaken from his anti-theoretical slumber.

On Jan 28, 2006, at 5:37 PM, Jack Sarfatti wrote:

On Jan 28, 2006, at 5:30 PM, Hank Harrison wrote:

Jack I just finished Lisa Randall's book on coincidence and the matrix aka string theory etc, Seems to me she is spinning her wheels and never really gets anyplace, but maybe I am not sophisticated enough to grasp it all, its a thick sucker, i didn't read every word, What do you think?

I only read pieces of it, then got involved with Richard Gott's, Lenny Susskind's and Robert Laughlin's books. I will get back to hers for my revision of Super Cosmos. Will tell you then. :-)

Laughlin's "A Different Universe" you and Kim will like. Laughlin is an experienced outdoors-man and his book reads like a Zane Gray Western. But it has to be read with Lenny Susskind's "Cosmic Landscape."

## Friday, January 27, 2006

Edge of the Ledge as Electroweak Splitting

"the distinction between electromagnetism and the weak nuclear force is central to modern cosmology, because the energy released when it occurs is the ostensible power source for inflation of the hypothesized brief period of rapid expansion following the big bang." p. 154 Robert Laughlin "A Different Universe".

Gravity must start at the Planck scale 10^-33 cm in order for Lambda to be defined as a parameter in say a DeSitter space metric to begin with.

That is

B ~ (dtheta)(phi) - (theta)(dphi)

must come into being at 10^-33 cm

So this must give a slow roll down the variable Lambda landscape to the edge of the ledge picture given by Lenny Susskind in "Cosmic Landscape." Falling over the edge at ~ 10^-16 cm. is when electromagnetism splits from the weak force. The released zero point vacuum energy then is the "reheating" described by the old big bang theory in Steven Weinberg's "The First Three Minutes" leading to "surface of last scattering" seen in WMAP data.

Since /\ is changing in "time" in vacuum obviously torsion fields are needed, i.e. "string theory".

"the distinction between electromagnetism and the weak nuclear force is central to modern cosmology, because the energy released when it occurs is the ostensible power source for inflation of the hypothesized brief period of rapid expansion following the big bang." p. 154 Robert Laughlin "A Different Universe".

Gravity must start at the Planck scale 10^-33 cm in order for Lambda to be defined as a parameter in say a DeSitter space metric to begin with.

That is

B ~ (dtheta)(phi) - (theta)(dphi)

must come into being at 10^-33 cm

So this must give a slow roll down the variable Lambda landscape to the edge of the ledge picture given by Lenny Susskind in "Cosmic Landscape." Falling over the edge at ~ 10^-16 cm. is when electromagnetism splits from the weak force. The released zero point vacuum energy then is the "reheating" described by the old big bang theory in Steven Weinberg's "The First Three Minutes" leading to "surface of last scattering" seen in WMAP data.

Since /\ is changing in "time" in vacuum obviously torsion fields are needed, i.e. "string theory".

Weightless Warp Drive with Curvature and Torsion Fields

math typo - final equation on previous message on emergent gravity should have been

R/\e + Lambda e/\e/\e = 0 vacuum equation

Memorandum for the Record

Military Applications of the Emergent Gravity Theory

Bottom Line:

*Military Applications:

Stealth cloaking of attack fighter-bombers and cruise missiles

Silent running, e.g. "Black Triangle" UFO craft

Tapping of zero point energy for power as in the "energy device" allegedly delivering up to 100,000 watts CW at 100 amps continuously for past 60 years reported by Robert Collins (USAF Intelligence ret) in "Exempt from Disclosure."

All of the above are without torsion fields. Torsion fields give more capabilities of course.

I. Plain Vanilla 1915 Einstein gravity "The Cake"

The vacuum equation is the 1-form equation

*[R/\e + Lambda e/\e/\e] = 0

The 1-form matter source equation is

*[R/\e + Lambda e/\e/\e] = *J

*J is the stress-energy source current density 1-form, so that J is a 3-form.

The torsion 2-form is

S = De

First consider only 1915 GR i.e.

S = De = 0

e = 1 + B

D = d + W/

Therefore the torsion-free spin connection 1-form is

W = -*[dB/\(1 - B)]

Note the subspace nonlinearity ~ dB/\B not found in electromagnetism.

Look at Maxwell's E&M gauge force theory

The U(1) internal symmetry gauge invariant exterior derivative on source fields is

D = d + A/

Compare this to the Diff(4) (AKA locally gauged T4) "background independent" space-time covariant exterior derivative

D = d + W/

So that A is analogous to W not to B. B lives in the antisymmetric "square root" tetrad subspace of the symmetric geometrodynamic space. The boson elements of space-time are pairs of the tetrads. That's one way of thinking of the equivalence principle, AKA EEP.

In abelian U(1) electrodynamics

F = dA

is the 2-form EM field tensor

dF = 0 is the "Bianchi identity"

i.e. Faraday's law & no magnetic monopoles law

from d^2 = 0 locally ignoring global EMERGENT "More is different" topological defects for now.

* is Hodge dual, i.e. 4D antisymmetric tensor depends on vacuum (or material) structure.

d*F = *J this is a 3-form since *F is a 2-form in 4D space-time

(no extra dimensions here)

i.e., Ampere's law with displacement (transverse radiation) current) + Gauss's law

Again from d^2 = 0 locally NOT generically GLOBALLY in DeRham integration

d^2*F = d*J = 0

i.e. local conservation of electric current source density.

One can generalize this to non-Abelian SU(2) & SU(3) Yang-Mills gauge force theories using

D = d + A/

where

F = DA = dA + A/\A

A/\A = 0 in U(1) EM

A/\A =/= 0 in SU(2) weak force and in SU(3) strong force because of non-trivial Lie algebra

[A^a,A^b] = fc^a^bA^c

Note that

A/\A means A^a^c/\Ac^b = (A/\A)^a^b

DF = 0 Bianchi identities

D*F = *J (source equations)

D^2 = 0 locally

D*J = 0

local conservation of Yang-Mills source current densities for weak and strong internal gauge forces of the standard model without gravity.

What about gravity?

Obviously, the geodesic deviation tidal curvature 2-form is

R = DW = dW + W/\W

W = -*[dB/\(1 - B)]

Note Einstein's local invariant is the symmetric bilinear expression (indices suppressed)

ds^2(curved) = guvdx^udx^v = (1 + B)(Minkowski)(1 + B)

= 1(Minkowski)1 + B(Minkowski)1 + 1(Minkowski)B + B(Minkowski)B

Note the nonlinear PLASTIC FRACTURE term B(Minkowski)B that gives "geon" spontaneous self-organization to the fabric of space-time even in the absence of "sources".

B = 0 globally means NO REAL GRAVITY FIELDS in sense of Landau & Lifshitz "Classical Theory of Fields".

Rovelli shows that the Einstein-Hilbert action gives the field vacuum equation

R/\e + (Lambda)e/\e/\e = 0

This is a 3-form equation

Note that

DR = 0 and De = 0

imply that in VACUUM

D(Lambda) = d(Lambda) ---> 4D gradient of 0-form Lambda = 0

i.e. Lambda is a uniform constant in this limiting case, though not in general. This is so in the FRW cosmology of "dark energy" from Type 1a supernovae anomalous redshifts.

The source equation is

R/\e + (Lambda)e/\e/\e = J

Still if no torsion

d(Lambda)e/\e/\e = DJ =/= 0

That is the locally conserved stress-energy density current is

J' = J - (Lambda)e/\e/\e

DJ' = 0

d(Lambda) =/= 0 locally coincident with DJ =/= 0

* This is a sloshing back and forth between off-mass-shell virtual zero point vacuum stress-energy current densities and on-mass-shell real matter-EM field stress-energy current densities.

*Military Applications:

Stealth cloaking of attack fighter-bombers and cruise missiles

Silent running, e.g. "Black Triangle" UFO craft

Tapping of zero point energy for power as in the "energy device" allegedly delivering up to 100,000 watts CW at 100 amps continuously for past 60 years reported by Robert Collins (USAF Intelligence ret) in "Exempt from Disclosure."

II. Add the Russian Torsion Field from local gauging of O(1,3). This is a poor man's string theory with a 10D manifold. The Whipped Cream with the Cherry on Top.

S = De =/= 0

Obviously then we have WWD i.e. Weightless Warp Drive ("G-Engine" of George Trimble, "acceleration field" of Paul Hill)

In vacuum

R/\e + Lambda e/\e/\e = 0

Therefore,

DR = 0 still true, but

R/\De + d(Lambda)e/\e/\e + (Lambda)d(e/\e/\e) = 0

Note the curvature-torsion coupling

R/\De = R/\S =/= 0

and the pure zero point energy torsion propellantless propulsion term

(Lambda)d(e/\e/\e) =/= 0

as well as the WWD term

d(Lambda)e/\e/\e =/= 0

One gets additional effect when J =/= 0

"Nuff said"

On Jan 26, 2006, at 9:29 PM, Jack Sarfatti wrote:

Let (hG/c^3) = 1 for now.

I have derived at least Einstein's classical GR vacuum equation as an emergent Goldstone phase modulation.

I assume a Planck-scale emergence where the degenerate vacuum manifold

G/H ~ S2

So there are 2 independent local Goldstone phase coherent fields theta & phi

I define the invariant 1-form

B = (dtheta)/\(phi) - (theta)/\(dphi)

so that

dB = 2(dtheta)/\(dphi) =/= 0

d^2B = 0 locally

The invariant Einstein-Cartan 1-form is

e = 1 + B

where

e = eudx^u

eu = eu^a&a

Einstein's symmetric metric tensor is by the equivalence principle

guv = eu^a(Minkowski)abev^b

Note the nonlinear terms ~ B^2.

The spin connection is the 1-form W^a^b = Wu^a^bdx^u

The exterior covariant derivative is

D = d + W/

Zero torsion means

De = 0

Therefore

W = -*[dB/\(1 - B)]

* is the Hodge dual

The curvature 2-form is

R = DW = dW + W/\W

i.e.

R^a^b = DW^a^b + Wa^c/\Wc^b

The Bianchi identities are

DR = 0

Define

The vacuum equation is the 1-form equation

*[R/\e + Lambda e/\e/\e] = 0

math typo - final equation on previous message on emergent gravity should have been

R/\e + Lambda e/\e/\e = 0 vacuum equation

Memorandum for the Record

Military Applications of the Emergent Gravity Theory

Bottom Line:

*Military Applications:

Stealth cloaking of attack fighter-bombers and cruise missiles

Silent running, e.g. "Black Triangle" UFO craft

Tapping of zero point energy for power as in the "energy device" allegedly delivering up to 100,000 watts CW at 100 amps continuously for past 60 years reported by Robert Collins (USAF Intelligence ret) in "Exempt from Disclosure."

All of the above are without torsion fields. Torsion fields give more capabilities of course.

I. Plain Vanilla 1915 Einstein gravity "The Cake"

The vacuum equation is the 1-form equation

*[R/\e + Lambda e/\e/\e] = 0

The 1-form matter source equation is

*[R/\e + Lambda e/\e/\e] = *J

*J is the stress-energy source current density 1-form, so that J is a 3-form.

The torsion 2-form is

S = De

First consider only 1915 GR i.e.

S = De = 0

e = 1 + B

D = d + W/

Therefore the torsion-free spin connection 1-form is

W = -*[dB/\(1 - B)]

Note the subspace nonlinearity ~ dB/\B not found in electromagnetism.

Look at Maxwell's E&M gauge force theory

The U(1) internal symmetry gauge invariant exterior derivative on source fields is

D = d + A/

Compare this to the Diff(4) (AKA locally gauged T4) "background independent" space-time covariant exterior derivative

D = d + W/

So that A is analogous to W not to B. B lives in the antisymmetric "square root" tetrad subspace of the symmetric geometrodynamic space. The boson elements of space-time are pairs of the tetrads. That's one way of thinking of the equivalence principle, AKA EEP.

In abelian U(1) electrodynamics

F = dA

is the 2-form EM field tensor

dF = 0 is the "Bianchi identity"

i.e. Faraday's law & no magnetic monopoles law

from d^2 = 0 locally ignoring global EMERGENT "More is different" topological defects for now.

* is Hodge dual, i.e. 4D antisymmetric tensor depends on vacuum (or material) structure.

d*F = *J this is a 3-form since *F is a 2-form in 4D space-time

(no extra dimensions here)

i.e., Ampere's law with displacement (transverse radiation) current) + Gauss's law

Again from d^2 = 0 locally NOT generically GLOBALLY in DeRham integration

d^2*F = d*J = 0

i.e. local conservation of electric current source density.

One can generalize this to non-Abelian SU(2) & SU(3) Yang-Mills gauge force theories using

D = d + A/

where

F = DA = dA + A/\A

A/\A = 0 in U(1) EM

A/\A =/= 0 in SU(2) weak force and in SU(3) strong force because of non-trivial Lie algebra

[A^a,A^b] = fc^a^bA^c

Note that

A/\A means A^a^c/\Ac^b = (A/\A)^a^b

DF = 0 Bianchi identities

D*F = *J (source equations)

D^2 = 0 locally

D*J = 0

local conservation of Yang-Mills source current densities for weak and strong internal gauge forces of the standard model without gravity.

What about gravity?

Obviously, the geodesic deviation tidal curvature 2-form is

R = DW = dW + W/\W

W = -*[dB/\(1 - B)]

Note Einstein's local invariant is the symmetric bilinear expression (indices suppressed)

ds^2(curved) = guvdx^udx^v = (1 + B)(Minkowski)(1 + B)

= 1(Minkowski)1 + B(Minkowski)1 + 1(Minkowski)B + B(Minkowski)B

Note the nonlinear PLASTIC FRACTURE term B(Minkowski)B that gives "geon" spontaneous self-organization to the fabric of space-time even in the absence of "sources".

B = 0 globally means NO REAL GRAVITY FIELDS in sense of Landau & Lifshitz "Classical Theory of Fields".

Rovelli shows that the Einstein-Hilbert action gives the field vacuum equation

R/\e + (Lambda)e/\e/\e = 0

This is a 3-form equation

Note that

DR = 0 and De = 0

imply that in VACUUM

D(Lambda) = d(Lambda) ---> 4D gradient of 0-form Lambda = 0

i.e. Lambda is a uniform constant in this limiting case, though not in general. This is so in the FRW cosmology of "dark energy" from Type 1a supernovae anomalous redshifts.

The source equation is

R/\e + (Lambda)e/\e/\e = J

Still if no torsion

d(Lambda)e/\e/\e = DJ =/= 0

That is the locally conserved stress-energy density current is

J' = J - (Lambda)e/\e/\e

DJ' = 0

d(Lambda) =/= 0 locally coincident with DJ =/= 0

* This is a sloshing back and forth between off-mass-shell virtual zero point vacuum stress-energy current densities and on-mass-shell real matter-EM field stress-energy current densities.

*Military Applications:

Stealth cloaking of attack fighter-bombers and cruise missiles

Silent running, e.g. "Black Triangle" UFO craft

Tapping of zero point energy for power as in the "energy device" allegedly delivering up to 100,000 watts CW at 100 amps continuously for past 60 years reported by Robert Collins (USAF Intelligence ret) in "Exempt from Disclosure."

II. Add the Russian Torsion Field from local gauging of O(1,3). This is a poor man's string theory with a 10D manifold. The Whipped Cream with the Cherry on Top.

S = De =/= 0

Obviously then we have WWD i.e. Weightless Warp Drive ("G-Engine" of George Trimble, "acceleration field" of Paul Hill)

In vacuum

R/\e + Lambda e/\e/\e = 0

Therefore,

DR = 0 still true, but

R/\De + d(Lambda)e/\e/\e + (Lambda)d(e/\e/\e) = 0

Note the curvature-torsion coupling

R/\De = R/\S =/= 0

and the pure zero point energy torsion propellantless propulsion term

(Lambda)d(e/\e/\e) =/= 0

as well as the WWD term

d(Lambda)e/\e/\e =/= 0

One gets additional effect when J =/= 0

"Nuff said"

On Jan 26, 2006, at 9:29 PM, Jack Sarfatti wrote:

Let (hG/c^3) = 1 for now.

I have derived at least Einstein's classical GR vacuum equation as an emergent Goldstone phase modulation.

I assume a Planck-scale emergence where the degenerate vacuum manifold

G/H ~ S2

So there are 2 independent local Goldstone phase coherent fields theta & phi

I define the invariant 1-form

B = (dtheta)/\(phi) - (theta)/\(dphi)

so that

dB = 2(dtheta)/\(dphi) =/= 0

d^2B = 0 locally

The invariant Einstein-Cartan 1-form is

e = 1 + B

where

e = eudx^u

eu = eu^a&a

Einstein's symmetric metric tensor is by the equivalence principle

guv = eu^a(Minkowski)abev^b

Note the nonlinear terms ~ B^2.

The spin connection is the 1-form W^a^b = Wu^a^bdx^u

The exterior covariant derivative is

D = d + W/

Zero torsion means

De = 0

Therefore

W = -*[dB/\(1 - B)]

* is the Hodge dual

The curvature 2-form is

R = DW = dW + W/\W

i.e.

R^a^b = DW^a^b + Wa^c/\Wc^b

The Bianchi identities are

DR = 0

Define

The vacuum equation is the 1-form equation

*[R/\e + Lambda e/\e/\e] = 0

## Thursday, January 26, 2006

I think I have derived at least Einstein's classical GR vacuum equation as an emergent Goldstone phase modulation.

I assume a Planck-scale emergence where the degenerate vacuum manifold

G/H ~ S2

So there are 2 independent local Goldstone phase coherent fields theta & phi

I define the invariant 1-form

B = (dtheta)/\(phi) - (theta)/\(dphi)

so that

dB = 2(dtheta)/\(dphi) =/= 0

d^2B = 0 locally

The invariant Einstein-Cartan 1-form is

e = 1 + B

where

e = eudx^u

eu = eu^a&a

Einstein's symmetric metric tensor is by the equivalence principle

guv = eu^a(Minkowski)abev^b

Note the nonlinear terms ~ B^2.

The spin connection is the 1-form W^a^b = Wu^a^bdx^u

The exterior covariant derivative is

D = d + W/

Zero torsion means

De = 0

Therefore

W = -*[dB/\(1 - B)]

* is the Hodge dual

The curvature 2-form is

R = DW = dW + W/\W

i.e.

R^a^b = DW^a^b + Wa^c/\Wc^b

The Bianchi identities are

DR = 0

Define

The vacuum equation is the 1-form equation

*(R/\e) + *(Lambda/\e/\e/\e) = 0

## Wednesday, January 25, 2006

The Illusion of Quantum Gravity

Lecture 11: The Failure of Loop Quantum Gravity. The Bottom Line is Red.

It’s overly complicated and leads to nothing much it appears. It cannot even derive Einstein’s classical relativity as a suitable limiting case. It’s a big buck for very little bang. This is what happens when pure mathematicians try to do theoretical physics. Angels dancing on the heads of pins? What Feynman called “philofawzy” with “rigor mortis”? ;-)

In contrast I claim to have solved the quantum gravity decisively by showing it is a pseudo-problem asking the wrong question from Nature.

“The Question is: What is The Question?” John A. Wheeler

Q. First what is the wrong question posed by “quantum gravity”?

A. “What about finiteness properties of spin foam models? …. From this point of view, the finiteness properties established so far say nothing about the UV properties of quantum gravity, which should instead follow from some kind of refinement limit, or from an averaging procedure where one sums over all foams, as discussed above. The question of convergence or non-convergence of such limits has so far not received a great deal of attention in the literature. This then, in a sense, brings us back to square one, namely the true problem of quantum gravity, which lies in the ambiguities associated with an infinite number of non-renormalizable UV divergences. As is well known this problem was originally revealed in a perturbative expansion of Einstein gravity around a fixed background, which requires an infinite series of counterterms, … The need to fix an infinite number of couplings in order to make the theory predictive renders perturbatively quantised Einstein gravity useless as a physical theory. What we would like to emphasize here is that any approach to quantum gravity must eventually confront this question, and that the need to fix infinitely many couplings in the perturbative ap•proach, and the appearance of infinitely many ambiguities in non-perturbative approaches are really just different sides of the same coin. In other words, non-perturbative approaches, even if they do not ‘see’ any UV divergences, cannot be relieved of the duty to explain in detail how the above divergences ‘disappear’, be it through cancellations or some other mechanism.”

“Whereas there is a rather direct link between (perturbative) string theory and classical space-time concepts, and string theory can therefore rely on familiar notions and concepts, such as the notion of a particle and the S-matrix, LQG must face up right away to the question of what an observable quantity is in the absence of a proper semiclassical space-time with fixed asymptotics.”

“Hence, for (3+1) gravity a decisive proof of the connection between spin foam models and the full Einstein theory and its canonical formulation appears to be lacking, and it is by no means excluded that such a link does not even exist.”

“Because the space of quantum states used in LQG is very different from the one used in Fock space quantisation, it becomes non-trivial to see how semiclassical ‘coherent’ states can be constructed, and how a smooth classical spacetime might emerge. In simple toy examples, such as the harmonic oscillator, it has been shown that the LQG Hilbert space indeed admits states (complicated linear superpositions) whose properties are close to those of the usual Fock space coherent states [17]. In full (3+1)-dimensional LQG, the classical limit is, however, far from understood (so far only kinematical coherent states are known [18, 19, 20, 21, 22, 23], i.e. states which do not satisfy the quantum constraints). In particular, it is not known how to describe or approximate classical spacetimes in this framework that ‘look’ like, say, Minkowski space, or how to properly derive the classical Einstein equations and their quantum corrections. … A proper understanding of the semi-classical limit is also indispensable to clarify the connection (or lack thereof) between conventional perturbation theory in terms of Feynman diagrams, and the non-perturbative quantisation proposed by LQG. However, the truly relevant question here concerns the structure (and definition!) of physical space and time. This, and not the kinematical ‘discretuum’ on which holonomies and spin networks ‘float’, is the arena where one should try to recover familiar and well-established concepts like the Wilsonian renormalisation group, with its continuous ‘flows’.”

Also, Loop Quantum Gravity does not really deliver on its promise of 4D covariance and it is apparently not able to properly handle virtual zero point energy i.e. “off-mass-shell” processes:

“Quantum space-time covariance?

Spacetime covariance is a central property of Einstein’s theory. Although the Hamiltonian formulation is not manifestly covariant, full covariance is still present in the classical theory, albeit in a hidden form, via the classical (Poisson or Dirac) algebra of constraints acting on phase space. However, this is not necessarily so for the quantised theory. As we explained, LQG treats the diffeomorphism constraint and the Hamiltonian constraint in a very different manner. Why and how then should one expect such a theory to recover full spacetime (as opposed to purely spatial) covariance? The crucial issue here is clearly what LQG has to say about the quantum algebra of constraints. Unfortunately, to the best of our knowledge, the ‘off-shell’ calculation of the commutator of two Hamiltonian constraints in LQG – with an explicit operatorial expression as the final result – has never been fully carried out. Instead, a survey of the possible terms arising in this computation has led to the conclusion that the commutator vanishes on a certain restricted ‘habitat’ of states [30, 34, 35], and that therefore the LQG constraint algebra closes without anomalies. By contrast, we have argued in [7] that this ‘on shell closure’ is not sufficient for a full proof of quantum spacetime covariance, but that a proper theory of quantum gravity requires a constraint algebra that closes ‘off shell’, i.e. without prior imposition of a subset of the constraints. The fallacies that may ensue if one does not insist on off-shell closure can be illustrated with simple examples. In our opinion, this requirement may well provide the 9 acid test on which any proposed theory of canonical quantum gravity will stand or fail.”

“In other words, if we only demand on-shell closure as in LQG, there is no way of telling whether or not the vanishing of a commutator is merely accidental, that is, not really due to the diffeomorphism invariance of the state, but caused by some other circumstance. By weakening the requirements on the constraint algebra and by no longer insisting on off-shell clo•sure, crucial information gets lost. This loss of information is reflected in the ambiguities inherent in the construction of the LQG Hamiltonian. It is quite possible that the LQG Hamiltonian admits many further modi•cations on top of the ones we have already discussed, for which the commutator continues to vanish on a suitably restricted habitat of states — in which case neither (23) nor (24) would amount to much of a consistency test.”

“Let us also mention that, as an alternative to the Euclidean spin foam models, one can try to set up Lorentzian spin foam models, as has been done in [49, 50]. In this case, the (compact) group SO(4)is replaced by the non-compact Lorentz group SO(1,3) [or SL(2, C)]. In the absence of an Osterwalder-Schrader type equivalence, it appears unlikely that there is any relation between these models and the Euclidean ones. Furthermore, the analysis of the corresponding state sums is much more complicated due to the fact that the relevant (i.e. unitary) representations are now infinite-dimensional. The above considerations show very clearly that there is no unique path from canonical gravity to spin foam models, and thus no unique model either (even if there was a unique canonical Hamiltonian); in fact, the number of possibilities for spin foam models appears to be even larger than the number of possible Hamiltonians in canonical LQG. A further problematic issue in the relation between spin foams and the canonical formalism comes from covariance requirements. While tetrahedral symmetry (or the generalisation thereof in four dimensions) is natural in the spin foam picture, the action of the Hamiltonian constraint, depicted in figure 7, does not reflect this symmetry. The Hamiltonian constraint only leads to so-called 1-3 moves, in which a single vertex in the initial spin network is mapped to three vertices in the final spin network. In the spin foam picture, the restriction to only these moves seems to be in conflict with the idea that the slicing of space-time into a space+time decomposition can be chosen arbitrarily.”

“Obviously, a brief introductory text such as this cannot do justice to the numerous recent developments in a very active field of current research. For this reason, we would like to conclude this introduction by referring readers to several ‘inside’ reviews for recent advances and alternative points of view, namely [9, 10, 11] for the canonical formulation, [12, 13, 14] for spin foams, and [15] for both. Readers are also invited to have a look at [16] for an update on the very latest developments in the subject.

2. The kinematical Hilbert space of LQG

There is a general expectation (not only in the LQG community) that at the very shortest distances, the smooth geometry of Einstein’s theory will be replaced by some quantum space or spacetime, and hence the continuum will be replaced by some ‘discretuum’. Canonical LQG does not do away with conventional spacetime concepts entirely, in that it still relies on a spatial continuum •as its ‘substrate’, on which holonomies and spin networks live (or ‘float’) — of course, with the idea of eventually ‘forgetting about it’ by considering ‘abstract spin networks’ and only the combinatorial relations between them. On this substrate, it takes as the classical phase space variables the holonomies of the Ashtekar connection,

…

Here, •a are the standard generators of SU(2) (Pauli matrices), but one can also replace the basic representation by a representation of arbitrary spin, de•noted … The Ashtekar connection A is thus a particular linear combination of the spin

Figure 1: LQG employs holonomies and fluxes as elementary conjugate variables.

…

Instead of building a Hilbert space as the space of functions over configurations of the Ashtekar connection, i.e. instead of constructing wave-functionals … LQG uses a Hilbert space of wave functionals which “probe” the geometry only on one-dimensional submanifolds, so-called spin networks. The latter are (not necessarily connected) graphs • … consisting of •finitely many edges (links). The wave functionals are functionals over the space of holonomies. In order to make them C-valued, the SU(2) indices of the holonomies have to be contracted using invariant tensors (i.e.vClebsch-Gordan coeffcients). … The spin network wave functions are thus labeled by (the spin network graph), by the spins {j} attached to the edges, and the intertwiners associated to the vertices. At this point, we have merely defined a space of wave functions in terms of rather unusual variables, and it now remains to define a proper Hilbert space structure on them. The discrete kinematical structure which LQG imposes does, accordingly, not come from the description in terms of holonomies and fluxes. After all, this very language can also be used to describe ordinary Yang-Mills theory. The discrete structure which LQG imposes is also entirely different from the discreteness of a lattice or naive discretisation of space (i.e. of a finite or countable set). Namely, it arises by ‘polymerising’ the continuum via an unusual scalar product. For any two spin network states, one defines this scalar product to be … where the integrals … are to be performed with the SU(2) Haar measure. The spin network wave functions depend on the Ashtekar connection only through the holonomies.

Figure 2: A simple spin network, embedded in the spatial hypersurface. The hypersurface is only present in order to provide coordinates which labe lthe positions ofthe vertices and edges. Spin network wave functions only probe the geometry along the one-dimensional edges and are insensitive to the geometry …

The kinematical Hilbert space … is then defined as the completion of the space of spin network wave functions w.r.t. this scalar product (5). The topology induced by the latter is similar to the discrete topology (‘pulverisation’) of the real line with countable unions of points as the open sets. Because the only notion of ‘closeness’ between two points in this topology is whether or not they are coincident, whence any function is continuous in this topology, this raises the question as to how one can recover conventional notions of continuity in this scheme.

The very special choice of the scalar product (5) leads to representations of operators which need not be weakly continuous: this means that expectation values of operators depending on some parameter do not vary continuously as these parameters are varied. Consequently, the Hilbert space does not admit a countable basis, hence is non-separable, because the set of all spin network graphs in is uncountable, and non-coincident spin networks are orthogonal w.r.t. (5). Therefore, any operation (such as a diffeomorphism) which moves around graphs continuously corresponds to an uncountable sequence of mutually orthogonal states in Hkin. That is, no matter how ‘small’ the deformation of the graph in •, the associated elements of Hkin always remain a finite distance apart, and consequently, the continuous motion in ‘real space’ gets mapped to a highly discontinuous one in Hkin. Although unusual, and perhaps counter-intuitive, as they are, these properties constitute a cornerstone for the hopes that LQG can overcome the seemingly unsurmountable problems of conventional geometrodynamics: if the representations used in LQG were equivalent to the ones of geometrodynamics, there would be no reason to expect LQG not to end up in the same quandary.

Because the space of quantum states used in LQG is very different from the one used in Fock space quantisation, it becomes non-trivial to see how semiclassical ‘coherent’ states can be constructed, and how a smooth classical spacetime might emerge. In simple toy examples, such as the harmonic oscillator, it has been shown that the LQG Hilbert space indeed admits states (complicated linear superpositions) whose properties are close to those of the usual Fock space coherent states [17]. In full (3+1)-dimensional LQG, the classical limit is, however, far from understood (so far only kinematical coherent states are known [18, 19, 20, 21, 22, 23], i.e. states which do not satisfy the quantum constraints). In particular, it is not known how to describe or approximate classical spacetimes in this framework that ‘look’ like, say, Minkowski space, or how to properly derive the classical Einstein equations and their quantum corrections.

Figure 3: The computation of the spectrum of the area operator involves the division of the surface into cells, such that at most one edge of the spin network intersects each given cell.

A proper understanding of the semi-classical limit is also indispensable to clarify the connection (or lack thereof) between conventional perturbation theory in terms of Feynman diagrams, and the non-perturbative quantisation proposed by LQG. However, the truly relevant question here concerns the structure (and definition!) of physical space and time. This, and not the kinematical ‘discretuum’ on which holonomies and spin networks ‘float’, is the arena where one should try to recover familiar and well-established concepts like the Wilsonian renormalisation group, with its continuous ‘flows’. Because the measurement of lengths and distances ultimately requires an operational definition in terms of appropriate matter fields and states obeying the physical state constraints, ‘dynamical’ discreteness is expected to manifest itself in the spectra of the relevant physical observables. Therefore, let us now turn to a discussion of the spectra of three important operators and to the discussion of physical states.

3. Area, volume and the Hamiltonian

In the current setup of LQG, an important role is played by two relatively simple operators: the ‘area operator’ measuring the area of a two-dimensional surface S •, and the ‘volume operator’ measuringthe volume of a three-dimensional subset V ••. The latter enters the definition of the Hamiltonian constraint in an essential way. Nevertheless, it must be emphasized that the area and volume operators are not observables in the Dirac sense, as they do not commute with the Hamiltonian. To construct physical operators corresponding to area and volume is more difficult and would require the inclusion of matter (in the form of ‘measuring rod fields’). … These spin network states are thus eigenstates of the area operator. The situation becomes considerably more complicated for wave functions, which contain a spin network vertex which lies in the surface S; in this case the area operator does not necessarily act diagonally anymore (see figure 4). Expression (9) lies at the core of the statement tha tareas are quantised in LQG.

The construction of the volume operator follows similar logic, although it is substantially more involved. … [There is also an] operator which replaces the Hamiltonian evolution operator of ordinary quantum mechanics, and encodes all the important dynamical information of the theory (whereas the Gauss and diffeomorphism constraints are merely ‘kinematical’). More specically, together with the kinematical constraints, it defines the physical states of the theory, and thereby the physical Hilbert space Hphys (which may be separable, even if Hkin is not). To motivate the form of the quantum Hamiltonian one starts with the classical expression, written in loop variables. To this aim one re-writes the Hamiltonian in terms of Ashtekar variables, with the result … The key problem in canonical gravity is the definition and implementation of the Hamiltonian (scalar) constraint operator, and the verification that this operator possesses all the requisite properties. The latter include (quantum) space-time covariance as well as the existence of a proper semi-classical limit, in which the classical Einstein equations are supposed 6 For the special values … the last term drops out, and the Hamiltonian simplifies considerably. This was indeed the value originally proposed by Ashtekar, and it would also appear to be the natural one required by local Lorentz invariance (as the Ashtekar variable is, in this case, just the pullback of the four-dimensional spin connection). However, imaginary … obviously implies that the phase space of general relativity in terms of these variables would have to be complexified, such that the original phase space could be recovered only after imposing a reality constraint. In order to avoid the difficulties related to quantising this reality constraint, … is now usually taken to be real. With this choice, it becomes much more involved to rewrite (12) in terms of loop and flux variables.

4. Implementation of the constraints

In canonical gravity, the simplest constraint is the Gauss constraint. In the setting of LQG, it simply requires that the SU(2)representation indices entering a given vertex of a spin network enter in an SU(2)invariant manner. More complicated are the diffeomorphism and Hamiltonian constraint. In LQG these are implemented in two entirely different ways. Moreover, the implementation of the Hamiltonian constraint is not completely independent, as its very definition relies on the existence of a subspace of diffeomorphism invariant states.

Let us start with the diffeomorphism constraint. Unlike in geometrodynamics, one cannot immediately write down formal states, which are manifestly diffeomorphism invariant, because the spin network functions are not supported on all of •, but only on one-dimensional links, which ‘move around’ under the action of a diffeomorphism. A formally diffeomorphism invariant state is obtained by ‘averaging’ over the diffeomorphism group, and more speci•cally by considering the formal sum … Although this is a continuous sum which might seem to be ill-defined, it can be given a mathematically precise meaning because the unusual scalar product (5) ensures that the inner product between a state and a … On the space of diffeomorphism averaged spin network states (regarded as a subspace of a distribution space) one can now again introduce a Hilbert space structure ‘by dividing out’ spatial diffeomorphisms, … As we said above, however, it is the Hamiltonian constraint which plays the key role in canonical gravity, as it this operator which encodes the 7 dynamics. Implementing this constraint on Hdi• or some other space is fraught with numerous choices and ambiguities, inherent in the construction of the quantum Hamiltonian as well as the extraordinary complexity of the resulting expression for the constraint operator [26]. The number of ambiguities can be reduced by invoking independence of the spatial background [10], and indeed, without making such choices, one would not even obtain sensible expressions, as we shall see very explicitly. In other words, the formalism is partly ‘on-shell’ in that the very existence of the (unregulated) Hamiltonian constraint operator depends very delicately on its ‘diffeomorphism covariance’, and the choice of a proper ‘habitat’, on which it is supposed to act in a well defined manner. A further source of ambiguities, which, for all we know, has not been considered in the literature so far, consists in possible dependent ‘higher order’ modifications of the Hamiltonian, which might still be compatible with all consistency requirements of LQG.

In order to write the constraint in terms of only holonomies and fluxes, one has to eliminate the inverse square root E-1/2 in (12) as well as the extrinsic curvature factors. This can be done through a number of tricks found by Thiemann [27… The attitude often expressed with regard to the ambiguities in the construction of the Hamiltonian 3 is that they correspond to different physics, and therefore the choice of the correct Hamiltonian is ultimately a matter of physics (experiment?), and not mathematics. However, it appears unlikely to us that Nature will allow such a great degree of arbitrariness at its most fundamental level: in fact, our main point here is that the infinitely many ambiguities which killed perturbative quantum gravity, are also a problem that other (to wit, non-perturbative) approaches must address and solve. …

3 The abundance of ‘consistent’ Hamiltonians and spin foam models (see below) is sometimes compared to the vacuum degeneracy problem of string theory, but the latter concerns different solutions of the same theory, as there is no dispute as to what (perturbative) string theory is. However, the concomitant lack of predictivity is obviously a problem for both approaches.

…

5. Quantum space-time covariance?

Spacetime covariance is a central property of Einstein’s theory. Although the Hamiltonian formulation is not manifestly covariant, full covariance is still present in the classical theory, albeit in a hidden form, via the classical (Poisson or Dirac) algebra of constraints acting on phase space. However, this is not necessarily so for the quantised theory. As we explained, LQG treats the diffeomorphism constraint and the Hamiltonian constraint in a very different manner. Why and how then should one expect such a theory to recover full spacetime (as opposed to purely spatial) covariance? The crucial issue here is clearly what LQG has to say about the quantum algebra of constraints. Unfortunately, to the best of our knowledge, the ‘off-shell’ calculation of the commutator of two Hamiltonian constraints in LQG – with an explicit operatorial expression as the final result – has never been fully carried out. Instead, a survey of the possible terms arising in this computation has led to the conclusion that the commutator vanishes on a certain restricted ‘habitat’ of states [30, 34, 35], and that therefore the LQG constraint algebra closes without anomalies. By contrast, we have argued in [7] that this ‘on shell closure’ is not sufficient for a full proof of quantum spacetime covariance, but that a proper theory of quantum gravity requires a constraint algebra that closes ‘off shell’, i.e. without prior imposition of a subset of the constraints. The fallacies that may ensue if one does not insist on off-shell closure can be illustrated with simple examples. In our opinion, this requirement may well provide the 9 acid test on which any proposed theory of canonical quantum gravity will stand or fail.

While there is general agreement as to what one means when one speaks of ‘closure of the constraint algebra’ in classical gravity (or any other classical constrained system [36]), this notion is more subtle in the quantized theory. 4 Letus therefore clarify first the various notions of closure that can arise: we see at least three different possibilities. The strongest notion is ‘off-shell closure’(or ‘strong closure’), where one seeks to calculate the commutator of two Hamiltonians … Although on-shell closure may perhaps look like a sufficient condition on the quantum Hamiltonian constraint, it is easy to see, at the level of simple examples, that this is not true. … In other words, if we only demand on-shell closure as in LQG, there is no way of telling whether or not the vanishing of a commutator is merely accidental, that is, not really due to the diffeomorphism invariance of the state, but caused by some other circumstance.

By weakening the requirements on the constraint algebra and by no longer insisting on off-shell closure, crucial information gets lost. This loss of information is reflected in the ambiguities inherent in the construction of the LQG Hamiltonian. It is quite possible that the LQG Hamiltonian admits many further modifications on top of the ones we have already discussed, for which the commutator continues to vanish on a suitably restricted habitat of states — in which case neither (23) nor (24) would amount to much of a consistency test.

6. Canonical gravity and spin foams

Attempts to overcome the difficulties with the Hamiltonian constraint have led to another development, spin foam models [37, 38, 39]. These were originally proposed as space-time versions of spin networks, to wit, evolutions of spin networks in ‘time’, but have since developed into a class of models of their own, disconnected from the canonical formalism. Mathematically, spin foam models represent a generalisation of spin networks, in the sense that group theoretical objects (holonomies, representations, intertwiners, etc.) are attached not only to vertices and edges (links), but also to higher dimensional faces in a simplicial decomposition of space-time.

The relation between spin foam models and the canonical formalism is based on a few general features of the action of the Hamiltonian constraint operator on a spin network (for a review on the connection, see [40]). As we have discussed above, the Hamiltonian constraint acts, schematically, by adding a small plaquette close to an existing vertex of the spin network (as in figure 5). In terms of a space-time picture, we see that the edges of the spin network sweep out surfaces, and the Hamiltonian constraint generates new surfaces, as in figure 7; but note that this graphical representation does not capture the details of how the action of the Hamiltonian affects the intertwiners at the vertices. Instead of associating spin labels to the edges of the spin network, one now associates the spin labels to the surfaces, in such a way that the label of the surface is determined by the label of the edge which lies in either the initial or final surface.

In analogy with proper-time transition amplitudes for a relativistic particle, it is tempting to define the transition amplitude between an initial spin network state and a final one … There are many questions one could ask about the physical meaning of this expression, but one important property is that (just as with the relativistic particle), the transition amplitude will project onto physical states (formally, this projection is effected in the original path integral by integrating over the lapse function multiplying the Hamiltonian density). One might thus consider (25) as a way of defining a physical inner product. In order to make contact with statistical partition sums, and because path integrals with oscillatory measures are difficult to handle, one next applies a formal Wick rotation to (25), replacing the Feynman weight with a Boltzmann weight, as is usually done in Euclidean quantum field theory. However, in making these steps one should always remember that there is no Osterwalder-Schrader type reconstruction theorem in quantum gravity, and therefore the derivation remains formal. Alternatively, one can adopt Hawking’s point of view that the world really is Euclidean, and simply take the Euclidean analog of (25) as the basic definition of the theory.

…

The simplest context in which to study these ideas is (2+1)gravity, because it is a topological (‘BF-type’) theory, that is, without local degrees of freedom, which can be solved exactly (see e.g. [41, 42, 43] and [44] for a more recent analysis of the model within the spin foam picture). … When one tries to formulate spin foam models in four dimensions, the relation to the canonical quantisation approach becomes less clear … Let us also mention that, as an alternative to the Euclidean spin foam models, one can try to set up Lorentzian spin foam models, as has been done in [49, 50]. In this case, the (compact) group SO(4)is replaced by the non-compact Lorentz group SO(1,3) [or SL(2, C)]. In the absence of an Osterwalder-Schrader type equivalence, it appears unlikely that there is any relation between these models and the Euclidean ones. Furthermore, the analysis of the corresponding state sums is much more complicated due to the fact that the relevant (i.e. unitary) representations are now infinite-dimensional. The above considerations show very clearly that there is no unique path from canonical gravity to spin foam models, and thus no unique model either (even if there was a unique canonical Hamiltonian); in fact, the number of possibilities for spin foam models appears to be even larger than the number of possible Hamiltonians in canonical LQG. A further problematic issue in the relation between spin foams and the canonical formalism comes from covariance requirements. While tetrahedral symmetry (or the generalisation thereof in four dimensions) is natural in the spin foam picture, the action of the Hamiltonian constraint, depicted in figure 7, does not reflect this symmetry. The Hamiltonian constraint only leads to so-called 1-3 moves, in which a single vertex in the initial spin network is mapped to three vertices in the final spin network. In the spin foam picture, the restriction to only these moves seems to be in conflict with the idea that the slicing of space-time into a space+time decomposition can be chosen arbitrarily. For space-time covariance, one expects 22 and 04 moves (and their time-reversed partners) as well, see figure 9. It has been argued [38] that these missing moves can be obtained from the Hamiltonian formalism by a suitable choice of operator ordering. In section 4 we have used an ordering, symbolically denoted by FEE, in which the Hamiltonian first opens up a spin network and subsequently glues in a plaquette. If one chooses the ordering to be EEF , then the inverse densitised vielbeine can open the plaquette, thereby potentially inducing a 22 or 04 move. However, ref. [27] has argued strongly against this operator ordering, claiming that in such a form the Hamiltonian operator cannot even be densely defined. In addition, the derivation sketched here is rather symbolic and hampered by the complexity of the Hamiltonian constraint [51]. Hence, for (3+1) gravity a decisive proof of the connection between spin foam models and the full Einstein theory and its canonical formulation appears to be lacking, and it is by no means excluded that such a link does not even exist.

7. Spin foams and discrete gravity

In view of the discussion above, it is thus perhaps best to view spin foam models as models in their own right, and, in fact, as a novel way of defining a (regularised) path integral in quantum gravity. Even without a clear-cut link to the canonical spin network quantisation programme, it is conceivable that spin foam models can be constructed which possess a proper semi-classical limit in which the relation to classical gravitational physics becomes clear. For this reason, it has even been suggested that spin foam models may provide a possible ‘way out’ if the difficulties with the conventional Hamiltonian approach should really prove insurmountable. To clarify the relation between spin foam models and earlier attempts to define a discretised path integral in quantum gravity, we recall that the latter can be roughly divided into two classes, namely:

•

Quantum Regge Calculus (see e.g. [52]), where one approximates space-time by a triangulation consisting of a fixed number of simplices, and integrates over all edge lengths, keeping the ‘shape’ of the triangulation fixed;

•

Dynamical Triangulations (see e.g. [53, 54, 55]), where the simplices are assigned •fixed edge lengths, and one sums instead over different triangulations, but keeping the number of simplices fixed (thus changing only the ‘shape’, but not the ‘volume’ of the triangulation).

Both approaches are usually based on a positive signature (Euclidean) metric, where the Boltzmann factor is derived from, or at least motivated by, some discrete approximation to the Einstein-Hilbert action, possibly with a cosmological constant [but see [56, 57] for some recent progress with a (Wick•rotated) ‘Lorentzian’ dynamical triangulation approach which introduces and exploits a notion of causality on the space-time lattice]. In both approaches, the ultimate aim is then to recover continuum space-time via a refinement limit in which the number of simplices is sent to infinity. Establishing the existence of such a limit is a notoriously difficult problem that has not been solved for four-dimensional gravity. In fact, for quantum Regge models in two dimensions such a continuum limit does not seem to agree with known continuum results [58, 59, 60, 61] (see however [62]).

From the point of view of the above classifica•tion, spin foam models belong to the first, ‘quantum Regge’, type, as one sums over all spins for a given spin foam, but does not add, remove or replace edges, faces or vertices, at least not in a first step. Indeed, for the spin foams discussed in the foregoing section, we have so far focused on the partition sum for a single given spin foam. An obvious question then concerns the next step, or more specically the question how spin foam models can recover (or even only define) a continuum limit. The canonical setup, where one sums over all spin network states in expressions like (25), would suggest that one should sum over all foams … where Z• denotes the partition function for a given spin foam, and where we have allowed for the possibility of a non-trivial weight w• depending only on the topological structure (‘shape’) of the foam. The reason for this sum would be to achieve formal independence of the triangulations. In a certain sense this would mimic the dynamical triangulation approach (except that one now would also sum over foams with a different number of simplices and different edge lengths), and thus turn the model into a hybrid version of the above approaches. However, this prescription is far from universally accepted, and several other ideas on how to extract classical, continuum physics from the partition sum Z•have been proposed. … The key issue is then to ensure that the final result does not depend on the way in which the triangulations are performed and refined. The refinement limit is motivated by the fact that it does appear to work in three space-time dimensions: more specifically, for large spins, the 6j symbol which appears in the Ponzano-Regge model approximates the Feynman weight for Regge gravity [63, 64]. … At present, there is little evidence that triangulation independence can be realised in non-topological theories, or that the problems related to the continuum limit will not reappear in a different disguise.

8. Predictive (finite) quantum gravity?

What about finiteness properties of spin foam models? …. From this point of view, the finiteness properties established so far say nothing about the UV properties of quantum gravity, which should instead follow from some kind of refinement limit, or from an averaging procedure where one sums over all foams, as discussed above. The question of convergence or non-convergence of such limits has so far not received a great deal of attention in the literature.

This then, in a sense, brings us back to square one, namely the true problem of quantum gravity, which lies in the ambiguities associated with an infinite number of non-renormalizable UV divergences. As is well known this problem was originally revealed in a perturbative expansion of Einstein gravity around a fixed background, which requires an infinite series of counterterms, … The need to fix an infinite number of couplings in order to make the theory predictive renders perturbatively quantised Einstein gravity useless as a physical theory. What we would like to emphasize here is that any approach to quantum gravity must eventually confront this question, and that the need to fix infinitely many couplings in the perturbative ap•proach, and the appearance of infinitely many ambiguities in non-perturbative approaches are really just different sides of the same coin. In other words, non-perturbative approaches, even if they do not ‘see’ any UV divergences, cannot be relieved of the duty to explain in detail how the above divergences ‘disappear’, be it through cancellations or some other mechanism.

At least in its present incarnation, the canonical formulation of LQG does not encounter any UV divergences, but the problem reappears through the lack of uniqueness of the canonical Hamiltonian. For spin foams (or, more generally, discrete quantum gravity) the problem is no less virulent. The known finiteness proofs all deal with the behaviour of a single foam, but, as we argued, these proofs concern the infrared rather than the ultraviolet. Just like canonical LQG, spin foams thus show no signs of ultraviolet divergences so far, but, as we saw, there is an embarras de richesse of physically distinct models, again reflecting the non-uniqueness that manifests itself in the infinite number of couplings associated with the perturbative counterterms. Indeed, fixing the ambiguities of the non-perturbative models by ad hoc, albeit well-motivated, assumptions is not much different from defining the perturbatively quantised theory by •fixing infinitely many coupling constants ‘by hand’.

Finally, let us remark that in lattice gauge theories, the classical limit and the UV limit can be considered and treated as separate issues. As for quantum gravity, this also appears to be the prevailing view in the LQG community.

However, the continuing failure to construct viable physical semi-classical states, solving the constraints even in only an approximate fashion, seems to suggest (at least to us) that in gravity the two problems cannot be solved separately, but are inextricably linked —also in view of the fact that the question as to the precise fate of the two-loop divergence (37) can then no longer be avoided.

…

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Lecture 10

Comment 19: Is Smolin’s paper the end of the story?

“Establishing the existence of a semi-classical limit, in which classical spacetime and the Einstein field equations are supposed to emerge, is widely regarded as the main open problem of this approach.”

Comment 19: In other words the Loop Quantum Gravity theorists are not really able to derive Einstein’s 1915 equation

Guv + kTuv = 0

As a limiting case.

I am able to do this in my emergent gravity theory from

B ~ (dTheta)(Phi) – (Theta)(dPhi)

Where Theta and Phi are the two Goldstone phases of the vacuum Planck Higgs field whose bare vacuum manifold is S2 excluding Calabi-Yau parameters. B is the curved piece of the Einstein-Cartan invariant 1-form tetrad field. Einstein’s metric field is obtained trivially from the mathematical form the equivalence principle.

Excerpts & my commentaries on:

AEI-2006-004 hep-th/0601129 January 18, 2006

arXiv:hep-th/0601129 v1 18 Jan 2006

Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners

Hermann Nicolai and Kasper Peeters

Max-Planck-Institut f¨

ur Gravitationsphysik Albert-Einstein-Institut Am M¨

uhlenberg 1 14476 Golm, GERMANY

hermann.nicolai, kasper.peeters@aei.mpg.de

“Abstract:

We review aspects of loop quantum gravity and spin foam models at an introductory level, with special attention to questions frequently asked by non-specialists.

Contributed article to “An assessment of current paradigms in theoretical physics”.

1. Quantum Einstein gravity

The assumption that Einstein’s classical theory of gravity can be quantised non-perturbatively is at the root of a wide variety of approaches to quantum gravity. The assumption constitutes the basis of several discrete methods [1], such as dynamical triangulations and Regge calculus, but it also implicitly underlies the older Euclidean path integral approach [2, 3] and the somewhat more indirect arguments which suggest that there may exist a non-trivial fixed point of the renormalisation group [4, 5, 6]. Finally, it is the key assumption which underlies loop and spin foam quantum gravity. Although the assumption is certainly far-reaching, there is to date no proof that Einstein gravity cannot be quantized non-perturbatively, either along the lines of one of the programs listed above or perhaps in an entirely different way.

In contrast to string theory, which posits that the Einstein-Hilbert action is only an effective low energy approximation to some other, more fundamental, underlying theory, loop and spin foam gravity take Einstein’s theory in four spacetime dimensions as the basic starting point, either with the conventional or with a (constrained) ‘BF-type’ formulation. 1 These approaches are background independent in the sense that they do not presuppose the existence of a given background metric. In comparison to the older geometrodynamics approach (which is also formally background independent) they make use of many new conceptual and technical ingredients. A key role is played by the reformulation of gravity in terms of connections and holonomies. A related feature is the use of spin networks in three (for canonical formulations) and four (for spin foams) dimensions. These, in turn, require other mathematical ingredients, such as non-separable (‘polymer’) Hilbert spaces and representations of operators which are not weakly continuous. Undoubtedly, novel concepts and ingredients such as these will be necessary in order to circumvent the problems of perturbatively quantised gravity (that novel ingredients are necessary is, in any case, not just the point of view of LQG but also of most other approaches to quantum gravity). However, it is important not to lose track of the physical questions that one is trying to answer.

The present text, which is based in part on the companion review [7], is intended as a brief introductory and critical survey of loop and spin foam quantum gravity2, with special focus on some of the questions that are frequently asked by non-experts, but not always adequately emphasized (for our taste, at least) in the pertinent literature.

[1 In the remainder, we will follow established (though perhaps slightly misleading) custom and summarily refer to this frame•work of ideas simply as “Loop Quantum Gravity”, or LQG for short.

2 while our review [7] is focused on the ‘orthodox’ approach] to loop quantum gravity — to wit the Hamiltonian framework — the present text also addresses the more recent spin foam developments. Even though the connections between these approaches are not as strong as one might expect, they do share some historical background and philosophy.

These concern in particular the definition and implementation of the Hamiltonian (scalar) constraint and its lack of uniqueness. Another important question (which we will not even touch on here) concerns the consistent incorporation of matter couplings, and especially the question as to whether the consistent quantisation of gravity imposes any kind of restrictions on them. Establishing the existence of a semi-classical limit, in which classical spacetime and the Einstein field equations are supposed to emerge, is widely regarded as the main open problem of this approach. Explaining the emergence of classical space-times is also a prerequisite for understanding the ultimate fate of the non-renormalisable UV divergences that arise in the conventional perturbative treatment. The latter question also arises in the ‘covariant’ spin foam approach in the form of the existence (or nonexistence) of a proper ‘continuum limit’.

A further question in any canonical approach to quantum gravity is whether one has succeeded in achieving (a quantum version of) full space-time covariance, rather than merely covariance under the diffeomorphisms of the three-dimensional slices. For someone unfamiliar with the concepts of LQG, it is not easy to see whether and how this requirement is met. In the presently known canonical set-up of LQG, it is only possible to establish on-shell closure of the constraint algebra, which means that partial use of the (diffeomorphism) constraint must be made in checking the commutator of two Hamiltonian constraint operators. In [7] we have argued that this is not enough, and that it is rather the off-shell closure of the constraint algebra that should be made the crucial requirement in establishing quantum space-time covariance. Space-time covariance is also an issue in discrete approaches, and thus spin foam quantum gravity, although the problem appears in a different guise. Whereas in conventional lattice discretisation the main question was whether and in which sense it is possible to ‘approximate’ general coordinate transformations on discrete sets of points and links, the key question in modern approaches which work with reparametrisation invariant quantities (proper lengths, etc.), as in Regge calculus, is now whether it is possible to obtain results which do not depend on the way in which the discretisation and the continuum limit are performed.

In view of our continuing ignorance about the ‘true theory’ of quantum gravity, the best strategy is surely to explore all possible avenues, including non-string approaches to quantum gravity. LQG, just like the older geometrodynamics approach [8], addresses several aspects ofthe problem that are currently outside the main focus of string theory, in particular the question of background independence and the quantisation of geometry.”

Comment 20 In my alternative theory the curved space-time geometry should not be quantized top -> down because it is a “More is different” bottom -> up emergent structure from the “organizational order” macro-quantum vacuum ODLRO Goldstone phases of the first Higgs field that forms at 10^-33 inflation to the edge of the ledge on the cosmic landscape before the final plunge off the ledge to make the hot big bang. Quantizing geometry is a silly as quantizing temperature, or quantizing elasticity. The ODLRO is already non-perturbative and background independent. There are still spin 2 quanta relative to the dynamical curved ODLRO spacetime of course. This explains why no giant Schrodinger Cats, why macro-spacetime physics is local and what the correspondence principle really is.

Lecture 11: The Failure of Loop Quantum Gravity. The Bottom Line is Red.

It’s overly complicated and leads to nothing much it appears. It cannot even derive Einstein’s classical relativity as a suitable limiting case. It’s a big buck for very little bang. This is what happens when pure mathematicians try to do theoretical physics. Angels dancing on the heads of pins? What Feynman called “philofawzy” with “rigor mortis”? ;-)

In contrast I claim to have solved the quantum gravity decisively by showing it is a pseudo-problem asking the wrong question from Nature.

“The Question is: What is The Question?” John A. Wheeler

Q. First what is the wrong question posed by “quantum gravity”?

A. “What about finiteness properties of spin foam models? …. From this point of view, the finiteness properties established so far say nothing about the UV properties of quantum gravity, which should instead follow from some kind of refinement limit, or from an averaging procedure where one sums over all foams, as discussed above. The question of convergence or non-convergence of such limits has so far not received a great deal of attention in the literature. This then, in a sense, brings us back to square one, namely the true problem of quantum gravity, which lies in the ambiguities associated with an infinite number of non-renormalizable UV divergences. As is well known this problem was originally revealed in a perturbative expansion of Einstein gravity around a fixed background, which requires an infinite series of counterterms, … The need to fix an infinite number of couplings in order to make the theory predictive renders perturbatively quantised Einstein gravity useless as a physical theory. What we would like to emphasize here is that any approach to quantum gravity must eventually confront this question, and that the need to fix infinitely many couplings in the perturbative ap•proach, and the appearance of infinitely many ambiguities in non-perturbative approaches are really just different sides of the same coin. In other words, non-perturbative approaches, even if they do not ‘see’ any UV divergences, cannot be relieved of the duty to explain in detail how the above divergences ‘disappear’, be it through cancellations or some other mechanism.”

“Whereas there is a rather direct link between (perturbative) string theory and classical space-time concepts, and string theory can therefore rely on familiar notions and concepts, such as the notion of a particle and the S-matrix, LQG must face up right away to the question of what an observable quantity is in the absence of a proper semiclassical space-time with fixed asymptotics.”

“Hence, for (3+1) gravity a decisive proof of the connection between spin foam models and the full Einstein theory and its canonical formulation appears to be lacking, and it is by no means excluded that such a link does not even exist.”

“Because the space of quantum states used in LQG is very different from the one used in Fock space quantisation, it becomes non-trivial to see how semiclassical ‘coherent’ states can be constructed, and how a smooth classical spacetime might emerge. In simple toy examples, such as the harmonic oscillator, it has been shown that the LQG Hilbert space indeed admits states (complicated linear superpositions) whose properties are close to those of the usual Fock space coherent states [17]. In full (3+1)-dimensional LQG, the classical limit is, however, far from understood (so far only kinematical coherent states are known [18, 19, 20, 21, 22, 23], i.e. states which do not satisfy the quantum constraints). In particular, it is not known how to describe or approximate classical spacetimes in this framework that ‘look’ like, say, Minkowski space, or how to properly derive the classical Einstein equations and their quantum corrections. … A proper understanding of the semi-classical limit is also indispensable to clarify the connection (or lack thereof) between conventional perturbation theory in terms of Feynman diagrams, and the non-perturbative quantisation proposed by LQG. However, the truly relevant question here concerns the structure (and definition!) of physical space and time. This, and not the kinematical ‘discretuum’ on which holonomies and spin networks ‘float’, is the arena where one should try to recover familiar and well-established concepts like the Wilsonian renormalisation group, with its continuous ‘flows’.”

Also, Loop Quantum Gravity does not really deliver on its promise of 4D covariance and it is apparently not able to properly handle virtual zero point energy i.e. “off-mass-shell” processes:

“Quantum space-time covariance?

Spacetime covariance is a central property of Einstein’s theory. Although the Hamiltonian formulation is not manifestly covariant, full covariance is still present in the classical theory, albeit in a hidden form, via the classical (Poisson or Dirac) algebra of constraints acting on phase space. However, this is not necessarily so for the quantised theory. As we explained, LQG treats the diffeomorphism constraint and the Hamiltonian constraint in a very different manner. Why and how then should one expect such a theory to recover full spacetime (as opposed to purely spatial) covariance? The crucial issue here is clearly what LQG has to say about the quantum algebra of constraints. Unfortunately, to the best of our knowledge, the ‘off-shell’ calculation of the commutator of two Hamiltonian constraints in LQG – with an explicit operatorial expression as the final result – has never been fully carried out. Instead, a survey of the possible terms arising in this computation has led to the conclusion that the commutator vanishes on a certain restricted ‘habitat’ of states [30, 34, 35], and that therefore the LQG constraint algebra closes without anomalies. By contrast, we have argued in [7] that this ‘on shell closure’ is not sufficient for a full proof of quantum spacetime covariance, but that a proper theory of quantum gravity requires a constraint algebra that closes ‘off shell’, i.e. without prior imposition of a subset of the constraints. The fallacies that may ensue if one does not insist on off-shell closure can be illustrated with simple examples. In our opinion, this requirement may well provide the 9 acid test on which any proposed theory of canonical quantum gravity will stand or fail.”

“In other words, if we only demand on-shell closure as in LQG, there is no way of telling whether or not the vanishing of a commutator is merely accidental, that is, not really due to the diffeomorphism invariance of the state, but caused by some other circumstance. By weakening the requirements on the constraint algebra and by no longer insisting on off-shell clo•sure, crucial information gets lost. This loss of information is reflected in the ambiguities inherent in the construction of the LQG Hamiltonian. It is quite possible that the LQG Hamiltonian admits many further modi•cations on top of the ones we have already discussed, for which the commutator continues to vanish on a suitably restricted habitat of states — in which case neither (23) nor (24) would amount to much of a consistency test.”

“Let us also mention that, as an alternative to the Euclidean spin foam models, one can try to set up Lorentzian spin foam models, as has been done in [49, 50]. In this case, the (compact) group SO(4)is replaced by the non-compact Lorentz group SO(1,3) [or SL(2, C)]. In the absence of an Osterwalder-Schrader type equivalence, it appears unlikely that there is any relation between these models and the Euclidean ones. Furthermore, the analysis of the corresponding state sums is much more complicated due to the fact that the relevant (i.e. unitary) representations are now infinite-dimensional. The above considerations show very clearly that there is no unique path from canonical gravity to spin foam models, and thus no unique model either (even if there was a unique canonical Hamiltonian); in fact, the number of possibilities for spin foam models appears to be even larger than the number of possible Hamiltonians in canonical LQG. A further problematic issue in the relation between spin foams and the canonical formalism comes from covariance requirements. While tetrahedral symmetry (or the generalisation thereof in four dimensions) is natural in the spin foam picture, the action of the Hamiltonian constraint, depicted in figure 7, does not reflect this symmetry. The Hamiltonian constraint only leads to so-called 1-3 moves, in which a single vertex in the initial spin network is mapped to three vertices in the final spin network. In the spin foam picture, the restriction to only these moves seems to be in conflict with the idea that the slicing of space-time into a space+time decomposition can be chosen arbitrarily.”

“Obviously, a brief introductory text such as this cannot do justice to the numerous recent developments in a very active field of current research. For this reason, we would like to conclude this introduction by referring readers to several ‘inside’ reviews for recent advances and alternative points of view, namely [9, 10, 11] for the canonical formulation, [12, 13, 14] for spin foams, and [15] for both. Readers are also invited to have a look at [16] for an update on the very latest developments in the subject.

2. The kinematical Hilbert space of LQG

There is a general expectation (not only in the LQG community) that at the very shortest distances, the smooth geometry of Einstein’s theory will be replaced by some quantum space or spacetime, and hence the continuum will be replaced by some ‘discretuum’. Canonical LQG does not do away with conventional spacetime concepts entirely, in that it still relies on a spatial continuum •as its ‘substrate’, on which holonomies and spin networks live (or ‘float’) — of course, with the idea of eventually ‘forgetting about it’ by considering ‘abstract spin networks’ and only the combinatorial relations between them. On this substrate, it takes as the classical phase space variables the holonomies of the Ashtekar connection,

…

Here, •a are the standard generators of SU(2) (Pauli matrices), but one can also replace the basic representation by a representation of arbitrary spin, de•noted … The Ashtekar connection A is thus a particular linear combination of the spin

Figure 1: LQG employs holonomies and fluxes as elementary conjugate variables.

…

Instead of building a Hilbert space as the space of functions over configurations of the Ashtekar connection, i.e. instead of constructing wave-functionals … LQG uses a Hilbert space of wave functionals which “probe” the geometry only on one-dimensional submanifolds, so-called spin networks. The latter are (not necessarily connected) graphs • … consisting of •finitely many edges (links). The wave functionals are functionals over the space of holonomies. In order to make them C-valued, the SU(2) indices of the holonomies have to be contracted using invariant tensors (i.e.vClebsch-Gordan coeffcients). … The spin network wave functions are thus labeled by (the spin network graph), by the spins {j} attached to the edges, and the intertwiners associated to the vertices. At this point, we have merely defined a space of wave functions in terms of rather unusual variables, and it now remains to define a proper Hilbert space structure on them. The discrete kinematical structure which LQG imposes does, accordingly, not come from the description in terms of holonomies and fluxes. After all, this very language can also be used to describe ordinary Yang-Mills theory. The discrete structure which LQG imposes is also entirely different from the discreteness of a lattice or naive discretisation of space (i.e. of a finite or countable set). Namely, it arises by ‘polymerising’ the continuum via an unusual scalar product. For any two spin network states, one defines this scalar product to be … where the integrals … are to be performed with the SU(2) Haar measure. The spin network wave functions depend on the Ashtekar connection only through the holonomies.

Figure 2: A simple spin network, embedded in the spatial hypersurface. The hypersurface is only present in order to provide coordinates which labe lthe positions ofthe vertices and edges. Spin network wave functions only probe the geometry along the one-dimensional edges and are insensitive to the geometry …

The kinematical Hilbert space … is then defined as the completion of the space of spin network wave functions w.r.t. this scalar product (5). The topology induced by the latter is similar to the discrete topology (‘pulverisation’) of the real line with countable unions of points as the open sets. Because the only notion of ‘closeness’ between two points in this topology is whether or not they are coincident, whence any function is continuous in this topology, this raises the question as to how one can recover conventional notions of continuity in this scheme.

The very special choice of the scalar product (5) leads to representations of operators which need not be weakly continuous: this means that expectation values of operators depending on some parameter do not vary continuously as these parameters are varied. Consequently, the Hilbert space does not admit a countable basis, hence is non-separable, because the set of all spin network graphs in is uncountable, and non-coincident spin networks are orthogonal w.r.t. (5). Therefore, any operation (such as a diffeomorphism) which moves around graphs continuously corresponds to an uncountable sequence of mutually orthogonal states in Hkin. That is, no matter how ‘small’ the deformation of the graph in •, the associated elements of Hkin always remain a finite distance apart, and consequently, the continuous motion in ‘real space’ gets mapped to a highly discontinuous one in Hkin. Although unusual, and perhaps counter-intuitive, as they are, these properties constitute a cornerstone for the hopes that LQG can overcome the seemingly unsurmountable problems of conventional geometrodynamics: if the representations used in LQG were equivalent to the ones of geometrodynamics, there would be no reason to expect LQG not to end up in the same quandary.

Because the space of quantum states used in LQG is very different from the one used in Fock space quantisation, it becomes non-trivial to see how semiclassical ‘coherent’ states can be constructed, and how a smooth classical spacetime might emerge. In simple toy examples, such as the harmonic oscillator, it has been shown that the LQG Hilbert space indeed admits states (complicated linear superpositions) whose properties are close to those of the usual Fock space coherent states [17]. In full (3+1)-dimensional LQG, the classical limit is, however, far from understood (so far only kinematical coherent states are known [18, 19, 20, 21, 22, 23], i.e. states which do not satisfy the quantum constraints). In particular, it is not known how to describe or approximate classical spacetimes in this framework that ‘look’ like, say, Minkowski space, or how to properly derive the classical Einstein equations and their quantum corrections.

Figure 3: The computation of the spectrum of the area operator involves the division of the surface into cells, such that at most one edge of the spin network intersects each given cell.

A proper understanding of the semi-classical limit is also indispensable to clarify the connection (or lack thereof) between conventional perturbation theory in terms of Feynman diagrams, and the non-perturbative quantisation proposed by LQG. However, the truly relevant question here concerns the structure (and definition!) of physical space and time. This, and not the kinematical ‘discretuum’ on which holonomies and spin networks ‘float’, is the arena where one should try to recover familiar and well-established concepts like the Wilsonian renormalisation group, with its continuous ‘flows’. Because the measurement of lengths and distances ultimately requires an operational definition in terms of appropriate matter fields and states obeying the physical state constraints, ‘dynamical’ discreteness is expected to manifest itself in the spectra of the relevant physical observables. Therefore, let us now turn to a discussion of the spectra of three important operators and to the discussion of physical states.

3. Area, volume and the Hamiltonian

In the current setup of LQG, an important role is played by two relatively simple operators: the ‘area operator’ measuring the area of a two-dimensional surface S •, and the ‘volume operator’ measuringthe volume of a three-dimensional subset V ••. The latter enters the definition of the Hamiltonian constraint in an essential way. Nevertheless, it must be emphasized that the area and volume operators are not observables in the Dirac sense, as they do not commute with the Hamiltonian. To construct physical operators corresponding to area and volume is more difficult and would require the inclusion of matter (in the form of ‘measuring rod fields’). … These spin network states are thus eigenstates of the area operator. The situation becomes considerably more complicated for wave functions, which contain a spin network vertex which lies in the surface S; in this case the area operator does not necessarily act diagonally anymore (see figure 4). Expression (9) lies at the core of the statement tha tareas are quantised in LQG.

The construction of the volume operator follows similar logic, although it is substantially more involved. … [There is also an] operator which replaces the Hamiltonian evolution operator of ordinary quantum mechanics, and encodes all the important dynamical information of the theory (whereas the Gauss and diffeomorphism constraints are merely ‘kinematical’). More specically, together with the kinematical constraints, it defines the physical states of the theory, and thereby the physical Hilbert space Hphys (which may be separable, even if Hkin is not). To motivate the form of the quantum Hamiltonian one starts with the classical expression, written in loop variables. To this aim one re-writes the Hamiltonian in terms of Ashtekar variables, with the result … The key problem in canonical gravity is the definition and implementation of the Hamiltonian (scalar) constraint operator, and the verification that this operator possesses all the requisite properties. The latter include (quantum) space-time covariance as well as the existence of a proper semi-classical limit, in which the classical Einstein equations are supposed 6 For the special values … the last term drops out, and the Hamiltonian simplifies considerably. This was indeed the value originally proposed by Ashtekar, and it would also appear to be the natural one required by local Lorentz invariance (as the Ashtekar variable is, in this case, just the pullback of the four-dimensional spin connection). However, imaginary … obviously implies that the phase space of general relativity in terms of these variables would have to be complexified, such that the original phase space could be recovered only after imposing a reality constraint. In order to avoid the difficulties related to quantising this reality constraint, … is now usually taken to be real. With this choice, it becomes much more involved to rewrite (12) in terms of loop and flux variables.

4. Implementation of the constraints

In canonical gravity, the simplest constraint is the Gauss constraint. In the setting of LQG, it simply requires that the SU(2)representation indices entering a given vertex of a spin network enter in an SU(2)invariant manner. More complicated are the diffeomorphism and Hamiltonian constraint. In LQG these are implemented in two entirely different ways. Moreover, the implementation of the Hamiltonian constraint is not completely independent, as its very definition relies on the existence of a subspace of diffeomorphism invariant states.

Let us start with the diffeomorphism constraint. Unlike in geometrodynamics, one cannot immediately write down formal states, which are manifestly diffeomorphism invariant, because the spin network functions are not supported on all of •, but only on one-dimensional links, which ‘move around’ under the action of a diffeomorphism. A formally diffeomorphism invariant state is obtained by ‘averaging’ over the diffeomorphism group, and more speci•cally by considering the formal sum … Although this is a continuous sum which might seem to be ill-defined, it can be given a mathematically precise meaning because the unusual scalar product (5) ensures that the inner product between a state and a … On the space of diffeomorphism averaged spin network states (regarded as a subspace of a distribution space) one can now again introduce a Hilbert space structure ‘by dividing out’ spatial diffeomorphisms, … As we said above, however, it is the Hamiltonian constraint which plays the key role in canonical gravity, as it this operator which encodes the 7 dynamics. Implementing this constraint on Hdi• or some other space is fraught with numerous choices and ambiguities, inherent in the construction of the quantum Hamiltonian as well as the extraordinary complexity of the resulting expression for the constraint operator [26]. The number of ambiguities can be reduced by invoking independence of the spatial background [10], and indeed, without making such choices, one would not even obtain sensible expressions, as we shall see very explicitly. In other words, the formalism is partly ‘on-shell’ in that the very existence of the (unregulated) Hamiltonian constraint operator depends very delicately on its ‘diffeomorphism covariance’, and the choice of a proper ‘habitat’, on which it is supposed to act in a well defined manner. A further source of ambiguities, which, for all we know, has not been considered in the literature so far, consists in possible dependent ‘higher order’ modifications of the Hamiltonian, which might still be compatible with all consistency requirements of LQG.

In order to write the constraint in terms of only holonomies and fluxes, one has to eliminate the inverse square root E-1/2 in (12) as well as the extrinsic curvature factors. This can be done through a number of tricks found by Thiemann [27… The attitude often expressed with regard to the ambiguities in the construction of the Hamiltonian 3 is that they correspond to different physics, and therefore the choice of the correct Hamiltonian is ultimately a matter of physics (experiment?), and not mathematics. However, it appears unlikely to us that Nature will allow such a great degree of arbitrariness at its most fundamental level: in fact, our main point here is that the infinitely many ambiguities which killed perturbative quantum gravity, are also a problem that other (to wit, non-perturbative) approaches must address and solve. …

3 The abundance of ‘consistent’ Hamiltonians and spin foam models (see below) is sometimes compared to the vacuum degeneracy problem of string theory, but the latter concerns different solutions of the same theory, as there is no dispute as to what (perturbative) string theory is. However, the concomitant lack of predictivity is obviously a problem for both approaches.

…

5. Quantum space-time covariance?

Spacetime covariance is a central property of Einstein’s theory. Although the Hamiltonian formulation is not manifestly covariant, full covariance is still present in the classical theory, albeit in a hidden form, via the classical (Poisson or Dirac) algebra of constraints acting on phase space. However, this is not necessarily so for the quantised theory. As we explained, LQG treats the diffeomorphism constraint and the Hamiltonian constraint in a very different manner. Why and how then should one expect such a theory to recover full spacetime (as opposed to purely spatial) covariance? The crucial issue here is clearly what LQG has to say about the quantum algebra of constraints. Unfortunately, to the best of our knowledge, the ‘off-shell’ calculation of the commutator of two Hamiltonian constraints in LQG – with an explicit operatorial expression as the final result – has never been fully carried out. Instead, a survey of the possible terms arising in this computation has led to the conclusion that the commutator vanishes on a certain restricted ‘habitat’ of states [30, 34, 35], and that therefore the LQG constraint algebra closes without anomalies. By contrast, we have argued in [7] that this ‘on shell closure’ is not sufficient for a full proof of quantum spacetime covariance, but that a proper theory of quantum gravity requires a constraint algebra that closes ‘off shell’, i.e. without prior imposition of a subset of the constraints. The fallacies that may ensue if one does not insist on off-shell closure can be illustrated with simple examples. In our opinion, this requirement may well provide the 9 acid test on which any proposed theory of canonical quantum gravity will stand or fail.

While there is general agreement as to what one means when one speaks of ‘closure of the constraint algebra’ in classical gravity (or any other classical constrained system [36]), this notion is more subtle in the quantized theory. 4 Letus therefore clarify first the various notions of closure that can arise: we see at least three different possibilities. The strongest notion is ‘off-shell closure’(or ‘strong closure’), where one seeks to calculate the commutator of two Hamiltonians … Although on-shell closure may perhaps look like a sufficient condition on the quantum Hamiltonian constraint, it is easy to see, at the level of simple examples, that this is not true. … In other words, if we only demand on-shell closure as in LQG, there is no way of telling whether or not the vanishing of a commutator is merely accidental, that is, not really due to the diffeomorphism invariance of the state, but caused by some other circumstance.

By weakening the requirements on the constraint algebra and by no longer insisting on off-shell closure, crucial information gets lost. This loss of information is reflected in the ambiguities inherent in the construction of the LQG Hamiltonian. It is quite possible that the LQG Hamiltonian admits many further modifications on top of the ones we have already discussed, for which the commutator continues to vanish on a suitably restricted habitat of states — in which case neither (23) nor (24) would amount to much of a consistency test.

6. Canonical gravity and spin foams

Attempts to overcome the difficulties with the Hamiltonian constraint have led to another development, spin foam models [37, 38, 39]. These were originally proposed as space-time versions of spin networks, to wit, evolutions of spin networks in ‘time’, but have since developed into a class of models of their own, disconnected from the canonical formalism. Mathematically, spin foam models represent a generalisation of spin networks, in the sense that group theoretical objects (holonomies, representations, intertwiners, etc.) are attached not only to vertices and edges (links), but also to higher dimensional faces in a simplicial decomposition of space-time.

The relation between spin foam models and the canonical formalism is based on a few general features of the action of the Hamiltonian constraint operator on a spin network (for a review on the connection, see [40]). As we have discussed above, the Hamiltonian constraint acts, schematically, by adding a small plaquette close to an existing vertex of the spin network (as in figure 5). In terms of a space-time picture, we see that the edges of the spin network sweep out surfaces, and the Hamiltonian constraint generates new surfaces, as in figure 7; but note that this graphical representation does not capture the details of how the action of the Hamiltonian affects the intertwiners at the vertices. Instead of associating spin labels to the edges of the spin network, one now associates the spin labels to the surfaces, in such a way that the label of the surface is determined by the label of the edge which lies in either the initial or final surface.

In analogy with proper-time transition amplitudes for a relativistic particle, it is tempting to define the transition amplitude between an initial spin network state and a final one … There are many questions one could ask about the physical meaning of this expression, but one important property is that (just as with the relativistic particle), the transition amplitude will project onto physical states (formally, this projection is effected in the original path integral by integrating over the lapse function multiplying the Hamiltonian density). One might thus consider (25) as a way of defining a physical inner product. In order to make contact with statistical partition sums, and because path integrals with oscillatory measures are difficult to handle, one next applies a formal Wick rotation to (25), replacing the Feynman weight with a Boltzmann weight, as is usually done in Euclidean quantum field theory. However, in making these steps one should always remember that there is no Osterwalder-Schrader type reconstruction theorem in quantum gravity, and therefore the derivation remains formal. Alternatively, one can adopt Hawking’s point of view that the world really is Euclidean, and simply take the Euclidean analog of (25) as the basic definition of the theory.

…

The simplest context in which to study these ideas is (2+1)gravity, because it is a topological (‘BF-type’) theory, that is, without local degrees of freedom, which can be solved exactly (see e.g. [41, 42, 43] and [44] for a more recent analysis of the model within the spin foam picture). … When one tries to formulate spin foam models in four dimensions, the relation to the canonical quantisation approach becomes less clear … Let us also mention that, as an alternative to the Euclidean spin foam models, one can try to set up Lorentzian spin foam models, as has been done in [49, 50]. In this case, the (compact) group SO(4)is replaced by the non-compact Lorentz group SO(1,3) [or SL(2, C)]. In the absence of an Osterwalder-Schrader type equivalence, it appears unlikely that there is any relation between these models and the Euclidean ones. Furthermore, the analysis of the corresponding state sums is much more complicated due to the fact that the relevant (i.e. unitary) representations are now infinite-dimensional. The above considerations show very clearly that there is no unique path from canonical gravity to spin foam models, and thus no unique model either (even if there was a unique canonical Hamiltonian); in fact, the number of possibilities for spin foam models appears to be even larger than the number of possible Hamiltonians in canonical LQG. A further problematic issue in the relation between spin foams and the canonical formalism comes from covariance requirements. While tetrahedral symmetry (or the generalisation thereof in four dimensions) is natural in the spin foam picture, the action of the Hamiltonian constraint, depicted in figure 7, does not reflect this symmetry. The Hamiltonian constraint only leads to so-called 1-3 moves, in which a single vertex in the initial spin network is mapped to three vertices in the final spin network. In the spin foam picture, the restriction to only these moves seems to be in conflict with the idea that the slicing of space-time into a space+time decomposition can be chosen arbitrarily. For space-time covariance, one expects 22 and 04 moves (and their time-reversed partners) as well, see figure 9. It has been argued [38] that these missing moves can be obtained from the Hamiltonian formalism by a suitable choice of operator ordering. In section 4 we have used an ordering, symbolically denoted by FEE, in which the Hamiltonian first opens up a spin network and subsequently glues in a plaquette. If one chooses the ordering to be EEF , then the inverse densitised vielbeine can open the plaquette, thereby potentially inducing a 22 or 04 move. However, ref. [27] has argued strongly against this operator ordering, claiming that in such a form the Hamiltonian operator cannot even be densely defined. In addition, the derivation sketched here is rather symbolic and hampered by the complexity of the Hamiltonian constraint [51]. Hence, for (3+1) gravity a decisive proof of the connection between spin foam models and the full Einstein theory and its canonical formulation appears to be lacking, and it is by no means excluded that such a link does not even exist.

7. Spin foams and discrete gravity

In view of the discussion above, it is thus perhaps best to view spin foam models as models in their own right, and, in fact, as a novel way of defining a (regularised) path integral in quantum gravity. Even without a clear-cut link to the canonical spin network quantisation programme, it is conceivable that spin foam models can be constructed which possess a proper semi-classical limit in which the relation to classical gravitational physics becomes clear. For this reason, it has even been suggested that spin foam models may provide a possible ‘way out’ if the difficulties with the conventional Hamiltonian approach should really prove insurmountable. To clarify the relation between spin foam models and earlier attempts to define a discretised path integral in quantum gravity, we recall that the latter can be roughly divided into two classes, namely:

•

Quantum Regge Calculus (see e.g. [52]), where one approximates space-time by a triangulation consisting of a fixed number of simplices, and integrates over all edge lengths, keeping the ‘shape’ of the triangulation fixed;

•

Dynamical Triangulations (see e.g. [53, 54, 55]), where the simplices are assigned •fixed edge lengths, and one sums instead over different triangulations, but keeping the number of simplices fixed (thus changing only the ‘shape’, but not the ‘volume’ of the triangulation).

Both approaches are usually based on a positive signature (Euclidean) metric, where the Boltzmann factor is derived from, or at least motivated by, some discrete approximation to the Einstein-Hilbert action, possibly with a cosmological constant [but see [56, 57] for some recent progress with a (Wick•rotated) ‘Lorentzian’ dynamical triangulation approach which introduces and exploits a notion of causality on the space-time lattice]. In both approaches, the ultimate aim is then to recover continuum space-time via a refinement limit in which the number of simplices is sent to infinity. Establishing the existence of such a limit is a notoriously difficult problem that has not been solved for four-dimensional gravity. In fact, for quantum Regge models in two dimensions such a continuum limit does not seem to agree with known continuum results [58, 59, 60, 61] (see however [62]).

From the point of view of the above classifica•tion, spin foam models belong to the first, ‘quantum Regge’, type, as one sums over all spins for a given spin foam, but does not add, remove or replace edges, faces or vertices, at least not in a first step. Indeed, for the spin foams discussed in the foregoing section, we have so far focused on the partition sum for a single given spin foam. An obvious question then concerns the next step, or more specically the question how spin foam models can recover (or even only define) a continuum limit. The canonical setup, where one sums over all spin network states in expressions like (25), would suggest that one should sum over all foams … where Z• denotes the partition function for a given spin foam, and where we have allowed for the possibility of a non-trivial weight w• depending only on the topological structure (‘shape’) of the foam. The reason for this sum would be to achieve formal independence of the triangulations. In a certain sense this would mimic the dynamical triangulation approach (except that one now would also sum over foams with a different number of simplices and different edge lengths), and thus turn the model into a hybrid version of the above approaches. However, this prescription is far from universally accepted, and several other ideas on how to extract classical, continuum physics from the partition sum Z•have been proposed. … The key issue is then to ensure that the final result does not depend on the way in which the triangulations are performed and refined. The refinement limit is motivated by the fact that it does appear to work in three space-time dimensions: more specifically, for large spins, the 6j symbol which appears in the Ponzano-Regge model approximates the Feynman weight for Regge gravity [63, 64]. … At present, there is little evidence that triangulation independence can be realised in non-topological theories, or that the problems related to the continuum limit will not reappear in a different disguise.

8. Predictive (finite) quantum gravity?

What about finiteness properties of spin foam models? …. From this point of view, the finiteness properties established so far say nothing about the UV properties of quantum gravity, which should instead follow from some kind of refinement limit, or from an averaging procedure where one sums over all foams, as discussed above. The question of convergence or non-convergence of such limits has so far not received a great deal of attention in the literature.

This then, in a sense, brings us back to square one, namely the true problem of quantum gravity, which lies in the ambiguities associated with an infinite number of non-renormalizable UV divergences. As is well known this problem was originally revealed in a perturbative expansion of Einstein gravity around a fixed background, which requires an infinite series of counterterms, … The need to fix an infinite number of couplings in order to make the theory predictive renders perturbatively quantised Einstein gravity useless as a physical theory. What we would like to emphasize here is that any approach to quantum gravity must eventually confront this question, and that the need to fix infinitely many couplings in the perturbative ap•proach, and the appearance of infinitely many ambiguities in non-perturbative approaches are really just different sides of the same coin. In other words, non-perturbative approaches, even if they do not ‘see’ any UV divergences, cannot be relieved of the duty to explain in detail how the above divergences ‘disappear’, be it through cancellations or some other mechanism.

At least in its present incarnation, the canonical formulation of LQG does not encounter any UV divergences, but the problem reappears through the lack of uniqueness of the canonical Hamiltonian. For spin foams (or, more generally, discrete quantum gravity) the problem is no less virulent. The known finiteness proofs all deal with the behaviour of a single foam, but, as we argued, these proofs concern the infrared rather than the ultraviolet. Just like canonical LQG, spin foams thus show no signs of ultraviolet divergences so far, but, as we saw, there is an embarras de richesse of physically distinct models, again reflecting the non-uniqueness that manifests itself in the infinite number of couplings associated with the perturbative counterterms. Indeed, fixing the ambiguities of the non-perturbative models by ad hoc, albeit well-motivated, assumptions is not much different from defining the perturbatively quantised theory by •fixing infinitely many coupling constants ‘by hand’.

Finally, let us remark that in lattice gauge theories, the classical limit and the UV limit can be considered and treated as separate issues. As for quantum gravity, this also appears to be the prevailing view in the LQG community.

However, the continuing failure to construct viable physical semi-classical states, solving the constraints even in only an approximate fashion, seems to suggest (at least to us) that in gravity the two problems cannot be solved separately, but are inextricably linked —also in view of the fact that the question as to the precise fate of the two-loop divergence (37) can then no longer be avoided.

…

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Lecture 10

Comment 19: Is Smolin’s paper the end of the story?

“Establishing the existence of a semi-classical limit, in which classical spacetime and the Einstein field equations are supposed to emerge, is widely regarded as the main open problem of this approach.”

Comment 19: In other words the Loop Quantum Gravity theorists are not really able to derive Einstein’s 1915 equation

Guv + kTuv = 0

As a limiting case.

I am able to do this in my emergent gravity theory from

B ~ (dTheta)(Phi) – (Theta)(dPhi)

Where Theta and Phi are the two Goldstone phases of the vacuum Planck Higgs field whose bare vacuum manifold is S2 excluding Calabi-Yau parameters. B is the curved piece of the Einstein-Cartan invariant 1-form tetrad field. Einstein’s metric field is obtained trivially from the mathematical form the equivalence principle.

Excerpts & my commentaries on:

AEI-2006-004 hep-th/0601129 January 18, 2006

arXiv:hep-th/0601129 v1 18 Jan 2006

Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners

Hermann Nicolai and Kasper Peeters

Max-Planck-Institut f¨

ur Gravitationsphysik Albert-Einstein-Institut Am M¨

uhlenberg 1 14476 Golm, GERMANY

hermann.nicolai, kasper.peeters@aei.mpg.de

“Abstract:

We review aspects of loop quantum gravity and spin foam models at an introductory level, with special attention to questions frequently asked by non-specialists.

Contributed article to “An assessment of current paradigms in theoretical physics”.

1. Quantum Einstein gravity

The assumption that Einstein’s classical theory of gravity can be quantised non-perturbatively is at the root of a wide variety of approaches to quantum gravity. The assumption constitutes the basis of several discrete methods [1], such as dynamical triangulations and Regge calculus, but it also implicitly underlies the older Euclidean path integral approach [2, 3] and the somewhat more indirect arguments which suggest that there may exist a non-trivial fixed point of the renormalisation group [4, 5, 6]. Finally, it is the key assumption which underlies loop and spin foam quantum gravity. Although the assumption is certainly far-reaching, there is to date no proof that Einstein gravity cannot be quantized non-perturbatively, either along the lines of one of the programs listed above or perhaps in an entirely different way.

In contrast to string theory, which posits that the Einstein-Hilbert action is only an effective low energy approximation to some other, more fundamental, underlying theory, loop and spin foam gravity take Einstein’s theory in four spacetime dimensions as the basic starting point, either with the conventional or with a (constrained) ‘BF-type’ formulation. 1 These approaches are background independent in the sense that they do not presuppose the existence of a given background metric. In comparison to the older geometrodynamics approach (which is also formally background independent) they make use of many new conceptual and technical ingredients. A key role is played by the reformulation of gravity in terms of connections and holonomies. A related feature is the use of spin networks in three (for canonical formulations) and four (for spin foams) dimensions. These, in turn, require other mathematical ingredients, such as non-separable (‘polymer’) Hilbert spaces and representations of operators which are not weakly continuous. Undoubtedly, novel concepts and ingredients such as these will be necessary in order to circumvent the problems of perturbatively quantised gravity (that novel ingredients are necessary is, in any case, not just the point of view of LQG but also of most other approaches to quantum gravity). However, it is important not to lose track of the physical questions that one is trying to answer.

The present text, which is based in part on the companion review [7], is intended as a brief introductory and critical survey of loop and spin foam quantum gravity2, with special focus on some of the questions that are frequently asked by non-experts, but not always adequately emphasized (for our taste, at least) in the pertinent literature.

[1 In the remainder, we will follow established (though perhaps slightly misleading) custom and summarily refer to this frame•work of ideas simply as “Loop Quantum Gravity”, or LQG for short.

2 while our review [7] is focused on the ‘orthodox’ approach] to loop quantum gravity — to wit the Hamiltonian framework — the present text also addresses the more recent spin foam developments. Even though the connections between these approaches are not as strong as one might expect, they do share some historical background and philosophy.

These concern in particular the definition and implementation of the Hamiltonian (scalar) constraint and its lack of uniqueness. Another important question (which we will not even touch on here) concerns the consistent incorporation of matter couplings, and especially the question as to whether the consistent quantisation of gravity imposes any kind of restrictions on them. Establishing the existence of a semi-classical limit, in which classical spacetime and the Einstein field equations are supposed to emerge, is widely regarded as the main open problem of this approach. Explaining the emergence of classical space-times is also a prerequisite for understanding the ultimate fate of the non-renormalisable UV divergences that arise in the conventional perturbative treatment. The latter question also arises in the ‘covariant’ spin foam approach in the form of the existence (or nonexistence) of a proper ‘continuum limit’.

A further question in any canonical approach to quantum gravity is whether one has succeeded in achieving (a quantum version of) full space-time covariance, rather than merely covariance under the diffeomorphisms of the three-dimensional slices. For someone unfamiliar with the concepts of LQG, it is not easy to see whether and how this requirement is met. In the presently known canonical set-up of LQG, it is only possible to establish on-shell closure of the constraint algebra, which means that partial use of the (diffeomorphism) constraint must be made in checking the commutator of two Hamiltonian constraint operators. In [7] we have argued that this is not enough, and that it is rather the off-shell closure of the constraint algebra that should be made the crucial requirement in establishing quantum space-time covariance. Space-time covariance is also an issue in discrete approaches, and thus spin foam quantum gravity, although the problem appears in a different guise. Whereas in conventional lattice discretisation the main question was whether and in which sense it is possible to ‘approximate’ general coordinate transformations on discrete sets of points and links, the key question in modern approaches which work with reparametrisation invariant quantities (proper lengths, etc.), as in Regge calculus, is now whether it is possible to obtain results which do not depend on the way in which the discretisation and the continuum limit are performed.

In view of our continuing ignorance about the ‘true theory’ of quantum gravity, the best strategy is surely to explore all possible avenues, including non-string approaches to quantum gravity. LQG, just like the older geometrodynamics approach [8], addresses several aspects ofthe problem that are currently outside the main focus of string theory, in particular the question of background independence and the quantisation of geometry.”

Comment 20 In my alternative theory the curved space-time geometry should not be quantized top -> down because it is a “More is different” bottom -> up emergent structure from the “organizational order” macro-quantum vacuum ODLRO Goldstone phases of the first Higgs field that forms at 10^-33 inflation to the edge of the ledge on the cosmic landscape before the final plunge off the ledge to make the hot big bang. Quantizing geometry is a silly as quantizing temperature, or quantizing elasticity. The ODLRO is already non-perturbative and background independent. There are still spin 2 quanta relative to the dynamical curved ODLRO spacetime of course. This explains why no giant Schrodinger Cats, why macro-spacetime physics is local and what the correspondence principle really is.

String Theory vs Loop Quantum Gravity

Lecture 10

Comment 19: Is Smolin’s paper the end of the story?

“Establishing the existence of a semi-classical limit, in which classical spacetime and the Einstein field equations are supposed to emerge, is widely regarded as the main open problem of this approach.”

In other words the Loop Quantum Gravity theorists are not really able to derive Einstein’s 1915 equation

Guv + kTuv = 0

As a limiting case.

I am able to do this in my emergent gravity theory from

L ~ (dTheta)(Phi) – (Theta)(dPhi)

Where Theta and Phi are the two Goldstone phases of the vacuum Planck Higgs field whose bare vacuum manifold is S2 excluding Calabi-Yau parameters. B is the curved piece of the Einstein-Cartan invariant 1-form tetrad field. Einstein’s metric field is obtained trivially from the mathematical form the equivalence principle.

Excerpts & my commentaries on:

AEI-2006-004 hep-th/0601129 January 18, 2006

arXiv:hep-th/0601129 v1 18 Jan 2006

Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners

Hermann Nicolai and Kasper Peeters

Max-Planck-Institut f¨

ur Gravitationsphysik Albert-Einstein-Institut Am M¨

uhlenberg 1 14476 Golm, GERMANY

hermann.nicolai, kasper.peeters@aei.mpg.de

“Abstract:

We review aspects of loop quantum gravity and spin foam models at an introductory level, with special attention to questions frequently asked by non-specialists.

Contributed article to “An assessment of current paradigms in theoretical physics”.

1. Quantum Einstein gravity

The assumption that Einstein’s classical theory of gravity can be quantised non-perturbatively is at the root of a wide variety of approaches to quantum gravity. The assumption constitutes the basis of several discrete methods [1], such as dynamical triangulations and Regge calculus, but it also implicitly underlies the older Euclidean path integral approach [2, 3] and the somewhat more indirect arguments which suggest that there may exist a non-trivial fixed point of the renormalisation group [4, 5, 6]. Finally, it is the key assumption which underlies loop and spin foam quantum gravity. Although the assumption is certainly far-reaching, there is to date no proof that Einstein gravity cannot be quantized non-perturbatively, either along the lines of one of the programs listed above or perhaps in an entirely different way.

In contrast to string theory, which posits that the Einstein-Hilbert action is only an effective low energy approximation to some other, more fundamental, underlying theory, loop and spin foam gravity take Einstein’s theory in four spacetime dimensions as the basic starting point, either with the conventional or with a (constrained) ‘BF-type’ formulation. 1 These approaches are background independent in the sense that they do not presuppose the existence of a given background metric. In comparison to the older geometrodynamics approach (which is also formally background independent) they make use of many new conceptual and technical ingredients. A key role is played by the reformulation of gravity in terms of connections and holonomies. A related feature is the use of spin networks in three (for canonical formulations) and four (for spin foams) dimensions. These, in turn, require other mathematical ingredients, such as non-separable (‘polymer’) Hilbert spaces and representations of operators which are not weakly continuous. Undoubtedly, novel concepts and ingredients such as these will be necessary in order to circumvent the problems of perturbatively quantised gravity (that novel ingredients are necessary is, in any case, not just the point of view of LQG but also of most other approaches to quantum gravity). However, it is important not to lose track of the physical questions that one is trying to answer.

The present text, which is based in part on the companion review [7], is intended as a brief introductory and critical survey of loop and spin foam quantum gravity2, with special focus on some of the questions that are frequently asked by non-experts, but not always adequately emphasized (for our taste, at least) in the pertinent literature.

[1 In the remainder, we will follow established (though perhaps slightly misleading) custom and summarily refer to this frame•work of ideas simply as “Loop Quantum Gravity”, or LQG for short.

2 while our review [7] is focused on the ‘orthodox’ approach] to loop quantum gravity — to wit the Hamiltonian framework — the present text also addresses the more recent spin foam developments. Even though the connections between these approaches are not as strong as one might expect, they do share some historical background and philosophy.

These concern in particular the definition and implementation of the Hamiltonian (scalar) constraint and its lack of uniqueness. Another important question (which we will not even touch on here) concerns the consistent incorporation of matter couplings, and especially the question as to whether the consistent quantisation of gravity imposes any kind of restrictions on them. Establishing the existence of a semi-classical limit, in which classical spacetime and the Einstein field equations are supposed to emerge, is widely regarded as the main open problem of this approach. Explaining the emergence of classical space-times is also a prerequisite for understanding the ultimate fate of the non-renormalisable UV divergences that arise in the conventional perturbative treatment. The latter question also arises in the ‘covariant’ spin foam approach in the form of the existence (or nonexistence) of a proper ‘continuum limit’.

A further question in any canonical approach to quantum gravity is whether one has succeeded in achieving (a quantum version of) full space-time covariance, rather than merely covariance under the diffeomorphisms of the three-dimensional slices. For someone unfamiliar with the concepts of LQG, it is not easy to see whether and how this requirement is met. In the presently known canonical set-up of LQG, it is only possible to establish on-shell closure of the constraint algebra, which means that partial use of the (diffeomorphism) constraint must be made in checking the commutator of two Hamiltonian constraint operators. In [7] we have argued that this is not enough, and that it is rather the off-shell closure of the constraint algebra that should be made the crucial requirement in establishing quantum space-time covariance. Space-time covariance is also an issue in discrete approaches, and thus spin foam quantum gravity, although the problem appears in a different guise. Whereas in conventional lattice discretisation the main question was whether and in which sense it is possible to ‘approximate’ general coordinate transformations on discrete sets of points and links, the key question in modern approaches which work with reparametrisation invariant quantities (proper lengths, etc.), as in Regge calculus, is now whether it is possible to obtain results which do not depend on the way in which the discretisation and the continuum limit are performed.

In view of our continuing ignorance about the ‘true theory’ of quantum gravity, the best strategy is surely to explore all possible avenues, including non-string approaches to quantum gravity. LQG, just like the older geometrodynamics approach [8], addresses several aspects ofthe problem that are currently outside the main focus of string theory, in particular the question of background independence and the quantisation of geometry.”

Comment 20: In my alternative theory the curved space-time geometry should not be quantized top -> down because it is a “More is different” bottom -> up emergent structure from the “organizational order” macro-quantum vacuum ODLRO Goldstone phases of the first Higgs field that forms at 10^-33 inflation to the edge of the ledge on the cosmic landscape before the final plunge off the ledge to make the hot big bang. Quantizing geometry is a silly as quantizing temperature, or quantizing elasticity. The ODLRO is already non-perturbative and background independent. There are still spin 2 quanta relative to the dynamical curved ODLRO spacetime of course. This explains why no giant Schrodinger Cats, why macro-spacetime physics is local and what the correspondence principle really is.

Appendix: Measuring dark energy and dark matter as virtual vacuum stuff locally in the laboratory with the Casimir force.

On Jan 24, 2006, at 9:28 PM, Jack Sarfatti wrote:

PS I trivially proved below that the electromechanical Casimir force in a closed spherical conducting cavity should be repulsive not attractive as for parallel plates! This is in fact observed!

This only happens of course because of the controversial mechanism that causes

/\ ~ 0.7 critical density to make our pocket universe spatially flat

Given that, my proof goes through when the ZPF energy density's effective micro /\ inside is more negative than the cosmological constant is positive. Therefore, in principle, we can use spherical Casimir cavities as TINY LOCAL DARK ENERGY DETECTORS in the lab! (In principle) so far this is a Gedankenexperiment. This is my DARK ENERGY GRAVITOMETER! It also LOCALLY detects DARK MATTER if it is simply negative /\.

That is, inside the spherical shell we have /\(cosmic) - /\(Casimir) when we exclude enough long wave spherical modes. The fact that repulsion is actually seen shows that this is not hard to do.

On Jan 24, 2006, at 9:04 PM, Jack Sarfatti wrote:

On Jan 24, 2006, at 8:12 PM, Figaro wrote:

The ZPE/ZPF fields are real as Hal mentioned many times not virtual

That's why every one of the top-rank physicists in the field reject the Haisch-Puthoff ZPE theory because it's based on the "SED" theory of Trevor Marshall that has been completely discredited. Also that SED theory is only for electromagnetic waves it says nothing about vacuum polarization of virtual fermion-antifermion pairs. Says nothing about the weak and strong forces. Hal's theory not only rejects Einstein's general relativity but he rejects quantum field theory! Therefore, it is completely inconsistent for Hal to cite Bryce De Witt's quantum field theory calculation of the ZPF tensor! Hal is not able to derive that result in his theory. So Hal is simply hiding the pea in a shell game.

but some still like to say virtual.

"Virtual" means "off-mass-shell," i.e. not the poles in Feynman propagators. It means stuff inside the vacuum. "Real" means stuff excited outside the vacuum.

E^2 = (pc)^2 + (mc^2)^2

dpdx > h

for real quanta excited outside the vacuum

E^2 =/= (pc)^2 + (mc^2)^2

dpdx < h

for virtual quanta inside the vacuum

As for w = -1 that's for no boundaries.

That's correct.

Then there are places out there in space where w = + 1/3..

No!

What you have for EM virtual quanta between the parallel plates where rotational and translational symmetry is broken is diagonal terms in the tensor that are

- 1, +1, +1, -3

The - 3 is the space direction perpendicular to the parallel uncharged conducting plates separated by tiny distance a

In this case the Casimir energy density is negative between the plates because long wave modes between the plates are excluded and in QED they take the space outside the plates to be "zero energy".

Now in fact for gravity purposes you cannot do that.

The naive energy density outside the plates is ~ hc/Lp^4 > 0 and the energy density inside the plates is ~ hc(1/Lp^4 -1/a^4)

Note if there were no plates at all, or for a small spherical cavity, the same tensor would be for virtual photons

+1 - 1 -1 - 1

In general the Newtonian 00 limit of GR is the Poisson equation

Laplacian of potential energy per unit test mass of any source ~ (G/c^2)(Source Energy Density)(Trace of Matrix)

In the case of virtual photons without any boundaries this is

(G/c^2)(Source Energy Density)(1 - 1 -1 -1) = -2(G/c^2)(Source Energy Density)

Therefore, a positive energy density isotropic ZPF distribution will repulsively anti-gravitate from the negative pressure!

In contrast, it will electrically attract from that same negative pressure.

It will do so at different coupling strengths of course.

If, instead, the isotropic ZPF energy density is negative, the positive pressure causes attractive gravity and electrical repulsion on a thin spherical shell (like ordinary gas in a piston)

Next consider the case of the anisotropic Casimir plates.

The negative pressure perpendicular to the plates will electro-mechanically attract (suction), but gravitationally repel. Note the negative longitudinal pressure - 3 acts with the negative energy density in that direction of space with rotationally broken symmetry.

The 2 positive transverse pressures will electro-mechanically repel, but will have no gravity effect because of the cancelation from the equal and opposite negative energy density.

Defining Trace of Matrix = 1 + 3

is not a useful measure when boundary conditions dominate. It is only useful in the homogeneous isotropic cosmological situation.

Jack Sarfatti wrote:

Yes, I am obviously not the only physicist putting 1 + 1 + 1 together to make 1. ;-)

The cosmic energy and the Lamb shift and Casimir force energy are all basically the same virtual zero point vacuum energy. w, of course, need not always be -1 when boundary conditions break isotropy and homogeneity which is assumed in the large-scale cosmological limit.

On Jan 24, 2006, at 7:39 PM, Gary S. Bekkum wrote:

How about:

"... we discuss the relevance of experiments at the interface of astrophysics and quantum field theory, focusing on the Casimir effect in gravitational and cosmological contexts. We conclude that challenging some of the assumptions underlying the cosmological constant problems and putting them to the test may prove useful and necessary to make progress on these questions..."

" C. Vacuum energy and Casimir effect in gravitational and cosmological contexts

Another approach is to think of a situation or an experiment where the validity of the cosmological constant-vacuum energy identification can be put to the test. A geometrical cosmological constant has no quantum properties while vacuum energy has both gravitational and quantum properties. Also, is it possible to learn more on how vacuum energy contributes to the cosmological constant? Some of these questions started to be addressed in the literature as we cite further.

It is perhaps relevant at this point to recall the Casimir effect which is a purely quantum field theory phenomena...

The Casimir effect results from a change in the zero-point oscillations spectrum of a quantized field when the quantization domain is restricted or when the topology of the space is non- trivial. For example, a Casimir force appears as the result of the alteration of the vacuum energy by some boundaries...

BTW topological defects enter here where the Higgs field vanishes and the possibly several Goldstone phases are undefined.

Now, related to our question on how the vacuum energy may fit within the EFE, it has been argued in some papers, see for example [42, 43, 44], that as the measured Casimir effect is related to vacuum energy differences, the vacuum energy may not contribute to the cosmological dynamics via some fixed cutoff energy but rather via energy differences as in Casimir energy. This Casimir energy can be produced from some compact extra dimensions [42, 43, 44] or non-trivial topology of the spacetime..."

Those papers are suspect. They violate the equivalence principle.

SEE:

http://www.arxiv.org/abs/astro-ph/0504416

Remarks on the formulation of the cosmological constant/dark energy problems

Authors: Mustapha Ishak (Princeton University)

Comments: 10 pages, 5 figures

Associated with the cosmic acceleration are the old and new cosmological constant problems, recently put into the more general context of the dark energy problem. In broad terms, the old problem is related to an unexpected order of magnitude of this component while the new problem is related to this magnitude being of the same order of the matter energy density during the present epoch of cosmic evolution. Current plans to measure the equation of state certainly constitute an important approach; however, as we discuss, this approach is faced with serious feasibility challenges and is limited in the type of conclusive answers it could provide. Therefore, is it really too early to seek actively for new tests and approaches to these problems? In view of the difficulty of this endeavor, we argue in this work that a good place to start is by questioning some of the assumptions underlying the formulation of these problems and finding new ways to put this questioning to the test. Motivated by some theorems, we discuss if the full identification of the cosmological constant with vacuum energy is unquestionable. Next, we evaluate how much fine tuning the cosmic coincidence problem represents. We discuss some implications of the simplest solution for the principles of General Relativity. We stress the potential of some cosmological probes such as weak gravitational lensing to identify novel tests to probe dark energy questions and assumptions. Also, we discuss the relevance of experiments at the interface of astrophysics and quantum field theory, focusing on the Casimir effect in gravitational and cosmological contexts. We conclude that challenging some of the assumptions underlying the cosmological constant problems and putting them to the test may prove useful and necessary to make progress on these questions

Sent: Tuesday, January 24, 2006 8:16 PM

Subject: Re: On the true relation of zero point energy to gravity (Robert Laughlin)

Jack Sarfatti wrote:

Robert Laughlin, a Nobel Prize physicist, also a professor of physics at Stanford like Lenny Susskind, makes it clear that all mainstream physicists essentially think that the vacuum energy of the cosmological constant and the vacuum energy of the Lamb shift and the Casimir force are the SAME PHENOMENON at vastly different scales. Thus on the relation of the Lamb shift's "/\zpf" to Einstein's cosmological constant "/\(Dark Energy)" - both are the same vacuum virtual "stuff" mentioned below by Laughlin (off-mass- shell).

Lecture 10

Comment 19: Is Smolin’s paper the end of the story?

“Establishing the existence of a semi-classical limit, in which classical spacetime and the Einstein field equations are supposed to emerge, is widely regarded as the main open problem of this approach.”

In other words the Loop Quantum Gravity theorists are not really able to derive Einstein’s 1915 equation

Guv + kTuv = 0

As a limiting case.

I am able to do this in my emergent gravity theory from

L ~ (dTheta)(Phi) – (Theta)(dPhi)

Where Theta and Phi are the two Goldstone phases of the vacuum Planck Higgs field whose bare vacuum manifold is S2 excluding Calabi-Yau parameters. B is the curved piece of the Einstein-Cartan invariant 1-form tetrad field. Einstein’s metric field is obtained trivially from the mathematical form the equivalence principle.

Excerpts & my commentaries on:

AEI-2006-004 hep-th/0601129 January 18, 2006

arXiv:hep-th/0601129 v1 18 Jan 2006

Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners

Hermann Nicolai and Kasper Peeters

Max-Planck-Institut f¨

ur Gravitationsphysik Albert-Einstein-Institut Am M¨

uhlenberg 1 14476 Golm, GERMANY

hermann.nicolai, kasper.peeters@aei.mpg.de

“Abstract:

We review aspects of loop quantum gravity and spin foam models at an introductory level, with special attention to questions frequently asked by non-specialists.

Contributed article to “An assessment of current paradigms in theoretical physics”.

1. Quantum Einstein gravity

The assumption that Einstein’s classical theory of gravity can be quantised non-perturbatively is at the root of a wide variety of approaches to quantum gravity. The assumption constitutes the basis of several discrete methods [1], such as dynamical triangulations and Regge calculus, but it also implicitly underlies the older Euclidean path integral approach [2, 3] and the somewhat more indirect arguments which suggest that there may exist a non-trivial fixed point of the renormalisation group [4, 5, 6]. Finally, it is the key assumption which underlies loop and spin foam quantum gravity. Although the assumption is certainly far-reaching, there is to date no proof that Einstein gravity cannot be quantized non-perturbatively, either along the lines of one of the programs listed above or perhaps in an entirely different way.

In contrast to string theory, which posits that the Einstein-Hilbert action is only an effective low energy approximation to some other, more fundamental, underlying theory, loop and spin foam gravity take Einstein’s theory in four spacetime dimensions as the basic starting point, either with the conventional or with a (constrained) ‘BF-type’ formulation. 1 These approaches are background independent in the sense that they do not presuppose the existence of a given background metric. In comparison to the older geometrodynamics approach (which is also formally background independent) they make use of many new conceptual and technical ingredients. A key role is played by the reformulation of gravity in terms of connections and holonomies. A related feature is the use of spin networks in three (for canonical formulations) and four (for spin foams) dimensions. These, in turn, require other mathematical ingredients, such as non-separable (‘polymer’) Hilbert spaces and representations of operators which are not weakly continuous. Undoubtedly, novel concepts and ingredients such as these will be necessary in order to circumvent the problems of perturbatively quantised gravity (that novel ingredients are necessary is, in any case, not just the point of view of LQG but also of most other approaches to quantum gravity). However, it is important not to lose track of the physical questions that one is trying to answer.

The present text, which is based in part on the companion review [7], is intended as a brief introductory and critical survey of loop and spin foam quantum gravity2, with special focus on some of the questions that are frequently asked by non-experts, but not always adequately emphasized (for our taste, at least) in the pertinent literature.

[1 In the remainder, we will follow established (though perhaps slightly misleading) custom and summarily refer to this frame•work of ideas simply as “Loop Quantum Gravity”, or LQG for short.

2 while our review [7] is focused on the ‘orthodox’ approach] to loop quantum gravity — to wit the Hamiltonian framework — the present text also addresses the more recent spin foam developments. Even though the connections between these approaches are not as strong as one might expect, they do share some historical background and philosophy.

These concern in particular the definition and implementation of the Hamiltonian (scalar) constraint and its lack of uniqueness. Another important question (which we will not even touch on here) concerns the consistent incorporation of matter couplings, and especially the question as to whether the consistent quantisation of gravity imposes any kind of restrictions on them. Establishing the existence of a semi-classical limit, in which classical spacetime and the Einstein field equations are supposed to emerge, is widely regarded as the main open problem of this approach. Explaining the emergence of classical space-times is also a prerequisite for understanding the ultimate fate of the non-renormalisable UV divergences that arise in the conventional perturbative treatment. The latter question also arises in the ‘covariant’ spin foam approach in the form of the existence (or nonexistence) of a proper ‘continuum limit’.

A further question in any canonical approach to quantum gravity is whether one has succeeded in achieving (a quantum version of) full space-time covariance, rather than merely covariance under the diffeomorphisms of the three-dimensional slices. For someone unfamiliar with the concepts of LQG, it is not easy to see whether and how this requirement is met. In the presently known canonical set-up of LQG, it is only possible to establish on-shell closure of the constraint algebra, which means that partial use of the (diffeomorphism) constraint must be made in checking the commutator of two Hamiltonian constraint operators. In [7] we have argued that this is not enough, and that it is rather the off-shell closure of the constraint algebra that should be made the crucial requirement in establishing quantum space-time covariance. Space-time covariance is also an issue in discrete approaches, and thus spin foam quantum gravity, although the problem appears in a different guise. Whereas in conventional lattice discretisation the main question was whether and in which sense it is possible to ‘approximate’ general coordinate transformations on discrete sets of points and links, the key question in modern approaches which work with reparametrisation invariant quantities (proper lengths, etc.), as in Regge calculus, is now whether it is possible to obtain results which do not depend on the way in which the discretisation and the continuum limit are performed.

In view of our continuing ignorance about the ‘true theory’ of quantum gravity, the best strategy is surely to explore all possible avenues, including non-string approaches to quantum gravity. LQG, just like the older geometrodynamics approach [8], addresses several aspects ofthe problem that are currently outside the main focus of string theory, in particular the question of background independence and the quantisation of geometry.”

Comment 20: In my alternative theory the curved space-time geometry should not be quantized top -> down because it is a “More is different” bottom -> up emergent structure from the “organizational order” macro-quantum vacuum ODLRO Goldstone phases of the first Higgs field that forms at 10^-33 inflation to the edge of the ledge on the cosmic landscape before the final plunge off the ledge to make the hot big bang. Quantizing geometry is a silly as quantizing temperature, or quantizing elasticity. The ODLRO is already non-perturbative and background independent. There are still spin 2 quanta relative to the dynamical curved ODLRO spacetime of course. This explains why no giant Schrodinger Cats, why macro-spacetime physics is local and what the correspondence principle really is.

Appendix: Measuring dark energy and dark matter as virtual vacuum stuff locally in the laboratory with the Casimir force.

On Jan 24, 2006, at 9:28 PM, Jack Sarfatti wrote:

PS I trivially proved below that the electromechanical Casimir force in a closed spherical conducting cavity should be repulsive not attractive as for parallel plates! This is in fact observed!

This only happens of course because of the controversial mechanism that causes

/\ ~ 0.7 critical density to make our pocket universe spatially flat

Given that, my proof goes through when the ZPF energy density's effective micro /\ inside is more negative than the cosmological constant is positive. Therefore, in principle, we can use spherical Casimir cavities as TINY LOCAL DARK ENERGY DETECTORS in the lab! (In principle) so far this is a Gedankenexperiment. This is my DARK ENERGY GRAVITOMETER! It also LOCALLY detects DARK MATTER if it is simply negative /\.

That is, inside the spherical shell we have /\(cosmic) - /\(Casimir) when we exclude enough long wave spherical modes. The fact that repulsion is actually seen shows that this is not hard to do.

On Jan 24, 2006, at 9:04 PM, Jack Sarfatti wrote:

On Jan 24, 2006, at 8:12 PM, Figaro wrote:

The ZPE/ZPF fields are real as Hal mentioned many times not virtual

That's why every one of the top-rank physicists in the field reject the Haisch-Puthoff ZPE theory because it's based on the "SED" theory of Trevor Marshall that has been completely discredited. Also that SED theory is only for electromagnetic waves it says nothing about vacuum polarization of virtual fermion-antifermion pairs. Says nothing about the weak and strong forces. Hal's theory not only rejects Einstein's general relativity but he rejects quantum field theory! Therefore, it is completely inconsistent for Hal to cite Bryce De Witt's quantum field theory calculation of the ZPF tensor! Hal is not able to derive that result in his theory. So Hal is simply hiding the pea in a shell game.

but some still like to say virtual.

"Virtual" means "off-mass-shell," i.e. not the poles in Feynman propagators. It means stuff inside the vacuum. "Real" means stuff excited outside the vacuum.

E^2 = (pc)^2 + (mc^2)^2

dpdx > h

for real quanta excited outside the vacuum

E^2 =/= (pc)^2 + (mc^2)^2

dpdx < h

for virtual quanta inside the vacuum

As for w = -1 that's for no boundaries.

That's correct.

Then there are places out there in space where w = + 1/3..

No!

What you have for EM virtual quanta between the parallel plates where rotational and translational symmetry is broken is diagonal terms in the tensor that are

- 1, +1, +1, -3

The - 3 is the space direction perpendicular to the parallel uncharged conducting plates separated by tiny distance a

In this case the Casimir energy density is negative between the plates because long wave modes between the plates are excluded and in QED they take the space outside the plates to be "zero energy".

Now in fact for gravity purposes you cannot do that.

The naive energy density outside the plates is ~ hc/Lp^4 > 0 and the energy density inside the plates is ~ hc(1/Lp^4 -1/a^4)

Note if there were no plates at all, or for a small spherical cavity, the same tensor would be for virtual photons

+1 - 1 -1 - 1

In general the Newtonian 00 limit of GR is the Poisson equation

Laplacian of potential energy per unit test mass of any source ~ (G/c^2)(Source Energy Density)(Trace of Matrix)

In the case of virtual photons without any boundaries this is

(G/c^2)(Source Energy Density)(1 - 1 -1 -1) = -2(G/c^2)(Source Energy Density)

Therefore, a positive energy density isotropic ZPF distribution will repulsively anti-gravitate from the negative pressure!

In contrast, it will electrically attract from that same negative pressure.

It will do so at different coupling strengths of course.

If, instead, the isotropic ZPF energy density is negative, the positive pressure causes attractive gravity and electrical repulsion on a thin spherical shell (like ordinary gas in a piston)

Next consider the case of the anisotropic Casimir plates.

The negative pressure perpendicular to the plates will electro-mechanically attract (suction), but gravitationally repel. Note the negative longitudinal pressure - 3 acts with the negative energy density in that direction of space with rotationally broken symmetry.

The 2 positive transverse pressures will electro-mechanically repel, but will have no gravity effect because of the cancelation from the equal and opposite negative energy density.

Defining Trace of Matrix = 1 + 3

is not a useful measure when boundary conditions dominate. It is only useful in the homogeneous isotropic cosmological situation.

Jack Sarfatti wrote:

Yes, I am obviously not the only physicist putting 1 + 1 + 1 together to make 1. ;-)

The cosmic energy and the Lamb shift and Casimir force energy are all basically the same virtual zero point vacuum energy. w, of course, need not always be -1 when boundary conditions break isotropy and homogeneity which is assumed in the large-scale cosmological limit.

On Jan 24, 2006, at 7:39 PM, Gary S. Bekkum wrote:

How about:

"... we discuss the relevance of experiments at the interface of astrophysics and quantum field theory, focusing on the Casimir effect in gravitational and cosmological contexts. We conclude that challenging some of the assumptions underlying the cosmological constant problems and putting them to the test may prove useful and necessary to make progress on these questions..."

" C. Vacuum energy and Casimir effect in gravitational and cosmological contexts

Another approach is to think of a situation or an experiment where the validity of the cosmological constant-vacuum energy identification can be put to the test. A geometrical cosmological constant has no quantum properties while vacuum energy has both gravitational and quantum properties. Also, is it possible to learn more on how vacuum energy contributes to the cosmological constant? Some of these questions started to be addressed in the literature as we cite further.

It is perhaps relevant at this point to recall the Casimir effect which is a purely quantum field theory phenomena...

The Casimir effect results from a change in the zero-point oscillations spectrum of a quantized field when the quantization domain is restricted or when the topology of the space is non- trivial. For example, a Casimir force appears as the result of the alteration of the vacuum energy by some boundaries...

BTW topological defects enter here where the Higgs field vanishes and the possibly several Goldstone phases are undefined.

Now, related to our question on how the vacuum energy may fit within the EFE, it has been argued in some papers, see for example [42, 43, 44], that as the measured Casimir effect is related to vacuum energy differences, the vacuum energy may not contribute to the cosmological dynamics via some fixed cutoff energy but rather via energy differences as in Casimir energy. This Casimir energy can be produced from some compact extra dimensions [42, 43, 44] or non-trivial topology of the spacetime..."

Those papers are suspect. They violate the equivalence principle.

SEE:

http://www.arxiv.org/abs/astro-ph/0504416

Remarks on the formulation of the cosmological constant/dark energy problems

Authors: Mustapha Ishak (Princeton University)

Comments: 10 pages, 5 figures

Associated with the cosmic acceleration are the old and new cosmological constant problems, recently put into the more general context of the dark energy problem. In broad terms, the old problem is related to an unexpected order of magnitude of this component while the new problem is related to this magnitude being of the same order of the matter energy density during the present epoch of cosmic evolution. Current plans to measure the equation of state certainly constitute an important approach; however, as we discuss, this approach is faced with serious feasibility challenges and is limited in the type of conclusive answers it could provide. Therefore, is it really too early to seek actively for new tests and approaches to these problems? In view of the difficulty of this endeavor, we argue in this work that a good place to start is by questioning some of the assumptions underlying the formulation of these problems and finding new ways to put this questioning to the test. Motivated by some theorems, we discuss if the full identification of the cosmological constant with vacuum energy is unquestionable. Next, we evaluate how much fine tuning the cosmic coincidence problem represents. We discuss some implications of the simplest solution for the principles of General Relativity. We stress the potential of some cosmological probes such as weak gravitational lensing to identify novel tests to probe dark energy questions and assumptions. Also, we discuss the relevance of experiments at the interface of astrophysics and quantum field theory, focusing on the Casimir effect in gravitational and cosmological contexts. We conclude that challenging some of the assumptions underlying the cosmological constant problems and putting them to the test may prove useful and necessary to make progress on these questions

Sent: Tuesday, January 24, 2006 8:16 PM

Subject: Re: On the true relation of zero point energy to gravity (Robert Laughlin)

Jack Sarfatti wrote:

Robert Laughlin, a Nobel Prize physicist, also a professor of physics at Stanford like Lenny Susskind, makes it clear that all mainstream physicists essentially think that the vacuum energy of the cosmological constant and the vacuum energy of the Lamb shift and the Casimir force are the SAME PHENOMENON at vastly different scales. Thus on the relation of the Lamb shift's "/\zpf" to Einstein's cosmological constant "/\(Dark Energy)" - both are the same vacuum virtual "stuff" mentioned below by Laughlin (off-mass- shell).

## Monday, January 23, 2006

On Jan 23, 2006, at 11:57 AM, Figaro wrote:

"I'm told Jack that you are the only one who tries to explain the Casimir effect in terms of Lambda, the Cosmological Constant."

Nonsense. That is another Straw Man. I use "/\" as a generic term. At the large scale > 300 Megaparsecs it is Einstein's cosmological constant - a coarse grained quantity in the FRW metric.

At smaller scales it is simply a local scalar contribution to the zero point stress-energy tensor. For example if you use the Bryce Dewitt formula beloved by Hal,

tuv(Casimir) ~ (pi^2/720)(hcbar/a^4)Matrix

Here then BY DEFINITION

(pi^2/720)(hcbar/a^4) = (c^4/8piG)/\(Casimir) = (Superstring Tension)/

/\(Casimir) = (8piG/c^4)(pi^2/720)(hcbar/a^4)

Where [/\] = 1/Area

If you can't do the math, then be silent. This all has to do with the math. Your understanding based on "words" is not enough for these technical details.

"No other Physicist is trying to accomplish such a feat I'm told. No wonder I don't have a clue as to what you're saying but yet Hal's et al papers are clear, everything is explained.......Robert C"

That's not why you have not a clue I am afraid. ;-)

On Jan 23, 2006, at 12:42 PM, Jack Sarfatti wrote:

Of course it is not there in Hal's paper.

1. He does not understand it yet as I just explained.

2. It's tiny and unimportant in those experiments so OK to neglect it there.

On Jan 23, 2006, at 9:12 AM, Figaro wrote:

"Hal just sent me his paper on condensed charges see attached. There is no anti-gravity between the plates or with condensed charges. It's simply a "mode" or "vacuum pressure" effect...Rmc"

Hal's model is completely wrong. It's based on his wrong idea that UNIFORM ZERO POINT ENERGY DENSITY is WEIGHTLESS i.e. no direct gravity influence.

Hal has a uniform positive pressure ZPF /\ ~ (mc/h)^2 filling all of space OUTSIDE the shells of charge with /\ = 0 inside.

This is IMPOSSIBLE if you read Lenny Susskind's Cosmic Landscape and actually understand what you are reading.

The pressure in that case is enormous, it is

(c^4/8piG)/\ = 10^19Gev/(10^-33 cm)(10^22)cm^-2

with /\ < 0.

We could not exist in such a universe! It is a DEAD REGION of the COSMIC LANDSCAPE.

Jack Sarfatti wrote:

On Jan 23, 2006, at 1:00 AM, Gary G. Ford wrote:

jack,

"N I take it (in your equations) is the Number

of charge carriers, m their individual mass,

M their total mass, ..."

Yes

"Are their photo's available of the charge clusters

in Puthoff's lab (I presume they are Ken Shoulder's

discover). Has Puthoff or Shoulders improved the

apparatus for generation recently?"

Shoulders has yes.

On Jan 23, 2006, at 7:20 AM, Figaro wrote:

"Well, by the definition you cited then it would be negative pressure that holds the condensed charges together or the force of the vacuum pressure not gravity."

No, you are confused, but it is easy to get confused on this issue. Indeed I first made this mistake, which I am correcting in the revised version of Super Cosmos I am still reworking.

Example 1: Ordinary Casimir Plate effect

The virtual photon pressure is negative in the longitudinal direction perpendicular to the plates. There this negative pressure electrically SUCKS the plates together causing the observed attractive force. You can picture it as excluded modes in between the plates. At the SAME TIME that same negative virtual photon pressure causes a DARK ENERGY anti-gravity repulsion between those plates BUT IT IS VERY SMALL and not noticeable. In principle it might be measurable as a small correction to the QED Casimir force.

Example 2: Thin spherical shell cavity of charge Q at radius r of total mass M, angular momentum J. ZPF pressure outside the shell is assumed to be negligible relative to the ZPF pressure inside the shell.

If the ZPF pressure inside the shell were negative it would add to the Coulomb repulsion and the shell explodes like a huge bomb. If the ZPF pressure inside the shell were positive it will gravitationally attract and hold the shell of charge together. Of course the electrical effect of this same positive pressure repels like the charge itself (that is the hc/r term), but the whole idea is that the gravity attraction induced by the interior ZPF compensates for that to find a stable configuration on the energy "landscape".

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