Sunday, January 22, 2006

Russian Torsion Field Weapons from String Theory?

Lecture 5

Lee Smolin wrote:

"5.4 Background independent approaches to string and M theory
It has been often argued that string theory requires a background independent formulation. This is required, not just because any quantum theory of gravity must be background independent, but because there is a need to unify all the different perturbative string theories into one theory. As this must combine theories defined on different backgrounds, it must not be restricted by the choice of a particular background.

There are some claims that string theory does not need a background independent formulation, and can be instead defined for fixed boundary or asymptotic conditions as dual to a field theory on a fixed background, as in the AdS/CFT correspondence. To respond to this, it first should be emphasized that the considerable evidence in favor of some form of an AdS/CFT correspondence [AntideSitter/Conformal Field Theory] falls short of a proof of actual equivalence, which would be needed to say that a full quantum theory of gravity, rather than just limits of correlation functions taken to the boundary, is coded in the dual conformal field theory [33, 2]). But even granting the full Maldacena conjecture it is hard to see how a theory defined only in the presence of boundary or asymptotic conditions, as interesting as that would be, could be taken as a candidate for a complete formulation of a fundamental theory of spacetime. This is because the boundary or asymptotic conditions can only be interpreted physically as standing for the presence of physical degrees of freedom outside the theory. For example, the timelike or null killing fields at the boundary stand for the reading of a clock which is not part of the physical systems. Such a formulation cannot be applied to cosmological problems, where the problem is precisely to formulate a consistent theory of the entire universe as a closed system. General relativity with spatially compact boundary conditions is such a theory. Hence, it seems reasonable to require that a quantum theory of gravity, which is supposed to reproduce general relativity, must also make sense as a theory of a whole universe, as a closed system.

Some string theorists have also claimed that string theory does not need a background independent formulation, because the fact that string perturbation theory is, in principle, defined on many different backgrounds is sufficient. This assertion rests on exaggeration and misunderstanding. First, string perturbation theory is so far only defined on stationary backgrounds that have timelike killing fields. But this is a measure zero of solutions to the Einstein equations. It is, however, difficult to believe that a consistent string perturbation theory can be defined on generic solutions to the Einstein equations because, in the absence of timelike Killing fields, one cannot have spacetime supersymmetry, without which the spectrum will generally contain a tachyon 21.”

Comment 10: A real on-mass-shell tachyon makes pocket universes outside our horizon directly observable and smashes the event horizon of a black hole. That may not be a bad thing even though it demolishes retarded signal causality and quantum cryptography. It also violates the spin-statistics connection. We need the extra space dimensions in order to have the cosmic landscape diversity as explained by Lenny Susskind. Without it, life is effectively not possible.

Back to Smolin:

“More generally, this assertion misses completely the key point that general relativity is itself a background independent theory. Although we sometimes use the Einstein’s equations as if they were a machine for generating solutions, within which we then study the motion of particles and fields, this way of seeing the theory is inadequate as soon as we want to ask questions about the gravitational degrees of freedom, themselves. Once we ask about the actual local dynamics of the gravitational field, we have to adopt the viewpoint which understands general relativity to be a background independent theory within which the geometry is completely dynamical, on an equal footing with the other degrees of freedom. The correct arena for this physics is not a particular spacetime, or even the linearized perturbations of a particular spacetime. It is the infinite dimensional phase space of gravitational degrees of freedom. From this viewpoint, individual spacetimes are just trajectories in the infinite dimensional phase or configuration space; they can play no more of a role in a quantization of spacetime than a particular classical orbit can play in the quantization of an electron.”

Comment 11: On the other hand in my emergent gravity theory we do not want to re-quantize the emergent dynamical c-number buv metric. There is the small q-number huv spin 2 quantum field relative to buv that is a source and sink for the huv gravitons.

Back to Smolin:
“To ask for a background independent formulation of string theory is to ask only that it conserve the fact that the dynamics of the Einstein equations does not require, indeed does not allow, the specification of a fixed background metric. For, if one means anything at all by a quantum theory of gravity, one certainly means a theory by which the degrees of freedom of the classical theory emerge from a suitable limit of a Hilbert space description. This does not commit oneself to the belief that the elementary degrees of freedom are classical metrics or connections, nor does it commit oneself to a belief that the correct microscopic dynamics have to do with the Einstein equations. But it does imply that a quantum theory must have a limit in which it reproduces the correct formulation of general relativity as a dynamical system, which is to say in the background independent language of the classical phase space. It would seem very unlikely that such a background independent formulation can emerge as a classical limit of a theory defined only on individual backgrounds, which are just trajectories in the exact phase space.

21 Note that for none of the theories in the landscape is it known how to construct the free string world-sheet theory.

In fact, there have been a few attempts to develop a background independent approach to string and M theory [35, 36].These have been based on two lessons from loop quantum gravity: i)Background independent quantum theories of gravity can be based on matrix models, so long as their formulation depends on no background metric. Such a model can be based on matrices valued in a group, as in certain formulations of spin foam models. ii) The dynamics of all known gravitational theories can be understood by beginning with a topological field theory and then extending the theory so as to minimally introduce local degrees of freedom. This can be extended to supergravity, including the 11-dimensional theory [34]. By combining these, a strategy was explored in which a background independent formulation of string or M theory was to be made which is an extension of a matrix Chern-Simons theory[35]. The Chern-Simons theory provides a starting point, which may be considered a membrane dynamics, but without embedding in any background manifold. The background manifold and embedding coordinates then arise from classical solutions to the background independent membrane model. It was then found that background dependent matrix models of string and M theory emerged by expanding around these classical solutions.”
A recent development in this direction is a proposal for how to quantize a certain reduction of M theorynon-perturbatively [36].”

Comment 12: Anyons in thin films are also Chern-Simons models with fractional quantum statistics.

Smolin continues:

“These few, preliminary, results, indicate that it is not difficult to invent and study hypotheses for background independent formulations of string theory.

6 Relationalism and reductionism
I would now like to broaden the discussion by asking: Does the relational view have implications broader than the nature of space and time? I will argue that it does 22. A starting point for explaining why is to begin with a discussion of reductionism.

To a certain degree, reductionism is common sense. When a system has parts, it makes sense to base an understanding of it on the laws that the parts satisfy, as well as on patterns that emerge from the exchanges of energy and information among the parts. In recent years we have learned that very complex patterns can emerge when simple laws act on the parts of a system, and this has led to the development of the study of complex systems. These studies have shown that there are useful principles that apply to such complex systems and these may help us to understand an array of systems from living cells to ecosystems to economic systems. But this is not in contradiction to reductionism, it is rather a deepening of it. But there is a built in limit to reductionism. If the properties of a complex system can be understood in terms of their parts, then we can keep going and understanding the parts in terms of their parts, and so on. We can keep looking at parts of parts until we reach particles that we believe are elementary, which means they cannot be further divided into parts. These still have properties, for example, we believe that the elementary particles have masses, positions, momenta, spin, and charges. When we reach this point we have to ask what methodology we can follow to explain the properties of the elementary particles? As they have no parts, reductionism will not help us. At this point we need a new methodology. Most thinking about elementary particle physics has taken place in the context of quantum field theory and its descendants such as string theory, which are background dependent theories. Let us start by asking how well these background dependent theories have done resolving the problem of how to attribute properties to particles thought to be elementary. After this we will see if background independent theories can do better.

Comment 13, the passage from classical special relativistic field theory to general relativity is the local gauging of the global Poincare group. This results in several kinds of compensating gauge fields that include both curvature and torsion when all 10 parameters are localized. Therefore, the Poincare symmetries are now no longer fixed but are dynamical i.e. relational.
Locally gauging the T4 subgroup of the Poincare group gives the curved tetrad 1-form field out of which the usual spin connection and its associated symmetric Levi-Civita connection field can be constructed. Going beyond 1915 GR by locally gauging the Lorentz group introduces an additional anti-symmetric torsion field connection in 4D. One can also think of the additional 6 local phase variables as the Calabi-Yau internal coordinates in a 10D manifold including the 4 local phases from T4.

For example, the curved tetrad 1-form (base space) components can be written as

Lu^a = P^a(&x)u/hbar

Where a is in the tangent fiber and u is in the base space.

{Pa} is the commutative Lie algebra of T4

(&x)u are 4 local parameters (local displacements in base space)

L^a = Lu^adx^u

{dx^u} is a form frame in base space

L = L^a&a = invariant 1-form

&a is a co-form basis in the tangent fiber.

L = (hG/c^3)[(Theta)(dPhi) – (dTheta)(Phi)]

V = G/H = S2 vacuum manifold defined by Goldstone phases Theta, Phi

Similarly for the antisymmetric torsion connection 1-form the components are

S^a^b = S(ab)(chi)(^a^b) = S^a^budx^u

Note that ( ) means suspend summation convention

With 6 Calabi-Yau “extra dimensions” (12) (13) (23) (01) (02) (03)

{Sab} is the Lie algebra of the Lorentz group.

If we include a conformal dilation we get the 11 dimensional manifold.

There are still 4 more conformal boosts in (3 + 1) space-time.

The 10D configuration space is Gennady Shipov’s “oriented point” in his low-energy torsion effective field theory.

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