## Monday, February 07, 2005

You're the Top!

Memorandum For The Record

Subject: Fallacy of The Fermi Paradox

http://stardrive.org/cartoon/MagicBean.html

re: http://www.bizspirit.com/science/Sarfatti,%20Jack%20workshop.html

To track swiftly emerging developments by "super-empowered individuals" and small groups bearing 5-th Generation WMD (Ch 11 of Sir Martin Rees's "Our Final Hour") in Russia and elsewhere monitor http://stardrive.org/ daily.

Peter Jennings gets US & World Public ready for UFO Disclosure on ABC PRIME TIME Feb 24, 2004.

We are already here Tootsy! ;-)

On Feb 7, 2005, at 4:08 PM, iksnileiz@earthlink.net wrote:

Z: It's very basic and very important.

J: Yes, it's basic and important, but still trivial. You are quibbling. In GR, Lorentz invariance is a local symmetry. In SR it is a global symmetry.

Z: This is conceptually orthogonal to the question of *universality* of the contraction/dilatation phenomena, i.e.,
the hypothesis that all physical objects are affected in exactly the same way when they are in relative uniform
motion.

J: I don't know what this means operationally or mathematically. What are you talking about? How can you decide this? Give a procedure.
The point is that these symmetries are left intact at the dynamical level.

Z: But the only "warrant" for this that I am aware of is the Einsteinian hypothesis of *universal* contraction. Of course you can still insist formally on Lorentz invariance as a regulative principle for the formulation of all physical laws, but without guaranteed universality of the contraction and dilatation phenomena, the original motivation for such regulative imposition of "Lorentz invariance" on the fundamental dynamical equations is absent.

J: Meaningless. The Lagrangian for every tested theory is precise. It's O(1,3) covariant. It works. I don't know what you mean scientifically.

Z: The point is that in a Lorentzian model, you could in principle recover the Lorentz transformations at the
*phenomenological* level, within a limited domain of validity, from underlying dynamics that is not itself Lorentz invariant.

J: Give examples. This is meaningless to me as it stands.

Z: For example, it is known that while the equilibrium v^3 "black body" spectrum is Lorentz invariant, non-equilibrium distributions may not be Lorentz invariant. That suggests that the property of Lorentz invariance is contingent, among other things, on the thermodynamic equilibrium condition, and thus not fundamental to the underlying physics, but instead is only *emergent* at the phenomenological level.

J: It's not the black body spectrum that is invariant, its the zero point spectrum (1/2)hf^3 per unit frequency per unit volume as I recall. Ask Puthoff he does know about that. The temperature T of the BB is NOT Lorentz invariant! Also we are talking about the Lagrangian of different theories. The Euler-Lagrange equations are O(1,3) form invariant for ALL tested theories when GR effects are not important and when vacuum is not broken. So you are not making any sense at all.

Z: If this is true, then the traditional regulative imposition of Lorentz invariance on the physics of the underlying quantum fields begins to look problematic (at least to me). Of course, that is not to say that you cannot do it.

J: Give examples. There is no evidence at all for any O(1,3) violating terms in any Lagrangian of any tested theory. So much for meta-theoretics. The kind of "preferred frame" Cahill reports is merely a vacuum property that does not affect the dynamics.

Z: He is not questioning the phenomenological validity of the Lorentz transformations, I agree -- in fact he is assuming the phenomenological validity of these transformations in his analysis of the MM experiments.

J: Yes.

Z: But the mere existence of a locally detectable preferred frame that is intrinsic to the physical vacuum completely undermines the classic Einstein (special) relativity model, and thus radically changes the *physical meaning* of the phenomenological Lorentz transformation formulas. It removes the entire Einsteinian basis for the hypothesis of universality of the contraction and dilatation phenomena.

J: You still do not understand the meaning of spontaneous broken vacuum symmetry. What you say is exactly what I deny!

Z: That means that the traditional Einsteinian warrant for the assumption of universality evaporates, and we are then effectively back to a *Lorentzian* model of the so-called "Lorentz transformations".

J: Lorentzian relativity in modern terms MEANS spontaneous broken O(1,3) vacuum symmetry, but the dynamical Euler-Lagrange equations of all correct theories are still O(1,3) form-invariant - mod GR corrections for curvature, possible torsion, possible conformal-gravity and dilation fields. The principle of relativity and covariance refer to the structure of the dynamical action and its Euler-Lagrange equations not to the vacuum solutions!

Z: You seem to be arguing that a locally detectable preferred frame doesn't disturb the (special) principle of relativity, even while the assumed absence of such a frame was a cornerstone of Einstein's 1905 theory.

J: Yes of course! The principle of special relativity ONLY means that the Euler-Lagrange equations from the extremum dynamical action are O(1,3) form-invariant. It says nothing about a preferred inertial frame. That's an additional independent axiom - that is violated if Cahill is right - just like 5th postulate leading to non-Euclidean geometry.

Z: As I understand the 1905 theory, the "principle of relativity" is built into the kinematics, and not just the dynamics. In fact, that is precisely what distinguishes Einstein's theory from Lorentz's.

J: Vague. All it means is that the Euler-Lagrange equations are form-invariant (covariant) under O(1,3) inertial frame transformations. Also it means that the quantum states transform under unitary representations of O(1,3). That's all it means. What decides Lorentz vs Einstein beyond that is breaking the vacuum symmetry with a macro-quantum order parameter like in a superfluid.

Z: What you seem to be suggesting here is that you can start with a fundamental theory of the vacuum for which the dynamics is Lorentz invariant (based on what?)

J: Based on every experiment in physics designed to test for a violation of Lorentz-covariant DYNAMICS (locally in GR situations).

Z; and you can then recover a preferred frame at the phenomenological level in certain solutions of the fundamental dynamical laws of the vacuum that have less symmetry than the Lorentz-invariant dynamical laws themselves,

J: Yes

Z: yielding a non-Einsteinian kinematics.

J: Wrong inference. Starting from the preferred GIF (ignore GR for now) use O(1,3) to get to any other inertial frame whose velocity is v relative to the preferred frame. The field equations are the same as always! Nothing changes for them!

Z: I suppose this is OK, but then what is the physical motivation for requiring the underlying dynamics to be Lorentz invariant?

J: Read Cliff Will & Wheeler & Ciufolini on experimental tests of relativity. Look at relativistic quantum scattering theory etc. Look at QED calculations of cross-sections, of atomic spectra with the Dirac equation etc. etc. Your point here is completely ludicrous!

Z: Isn't this then an arbitrarily imposed formal condition? Who needs it? Why? Since you could in principle also get phenomenological Lorentz invariance from underlying dynamics that is not itself Lorentz invariant?

J: How would you decide if what you say was wrong? I am talking about much more than that.

Z: Without strict relativity, there is no basis for assuming universality of length contraction; with no basis for
assuming universality of contraction; there is no known reason to *assume* that the fundamental laws are Lorentz covariant.
That is not to say that they cannot be.

J: All the experimental evidence requires Lorentz covariance (subject to GR corrections) in the dynamics.

Z: At the phenomenological level, yes. What direct evidence do you have in mind for Lorentz invariance of the underlying dynamics of the vacuum?

J: That is NOT a well-posed problem. You have field equations and you have solutions to them depending on boundary and initial conditions. Dynamics means the structure of the field equations. "Underlying dynamics of the vacuum" beyond what I just said is meaningless to me. Give a counter example!

Z: That is, if you agree that you could in principle get phenomenological Lorentz invariance from an underlying fundamental theory of the vacuum whose dynamics is not itself Lorentz invariant?

J: Ditto. Meaningless string. I cannot compute what you say. There is no evidence to the contrary. Cahill's allegations show spontaneous broken vacuum symmetry - that's all!

Z: No, I agree. That is not what I'm saying.

J: I don't know what you are saying and I am not sure if you do either. ;-)

Z; I am asking, What is the motivation for imposing Lorentz invariance on the underlying dynamics of the vacuum?

J: WRONG QUESTION!

"The Question is: What is The Question?" (Wheeler)

Z: What empirical evidence can you cite that supports imposing this condition on the underlying dynamics?

J: It's all around you. Also, do you not yet understand that Cahill's results, if corroborated, DO NOT VIOLATE LORENTZ INVARIANCE at all where it matters, i.e. in the DYNAMICS? You don't seem to get that? Also I am talking about ALL groups in ALL physical situations not only the Lorentz group O(1,3).

Z: Cahill's argument does not violate Lorentz covariance. It only violates the *relativity hypothesis* and thus undermines the *Einstein model for the Lorentz transformations*, which is different. Clearly, without Einstein relativity there is no need to build the Lorentz transformations into the kinematics. They can be treated as a dynamical effect, as in the classic Lorentzian model.

J: This is too vague. Show the alleged distinction as a mathematical difference.

Z; Jack, this is all well known. As you mentioned, you should have another look at Bell's essay in "Unspeakable".

J: I did. So what? Cite relevant parts. The kinematics is there when you perform a O(1,3) transformation on a physical quantity like the local Lagrangian density, or the local Euler-Lagrange equations of motion.

Z: Yes, and on the phenomenological level, I agree that this is not in question here from an empirical standpoint.

Z: My question is about why, in the absence of Einsteinian strict relativity of uniform motion, we continue to insist on imposing universal Lorentz invariance on the underlying dynamics of the vacuum.

J: Don't speak.

Z: Again, there is the v^3 spectrum argument against universality and against the *a priori* imposition of such invariance on *all* laws of physics.

J: No relevance. Or, if GR is not important, when you perform a GLOBAL O(1,3) transformation on the entire global classical action. Similarly working with states in qubit Hilbert space using unitary representations of O(1,3).

Z: Yes, fine, but I am talking about the *physical meaning* of formal invariance under the transformations included in O(1,3).

J: Your *physical meaning* has no physical meaning.

Z: That is the entire basis for the distinction between Einstein's and Lorentz's models for the Lorentz transformations.

J: I draw a blank. Your message is not coming through.

Z: Of course, either way, Lorentz or Einstein, you are dealing with transformation properties under the group O(1,3).

J: I have told you operationally what the difference means. When the vacuum breaks O(1,3) symmetry then Lorentz wins over Einstein, but in BOTH cases the field dynamics, the laws of nature are THE SAME! Beyond that there is nothing. Your head is in the clouds and your feet are not on the ground.

Z: There is even an ambiguity in the mathematical meaning of O(1,3) in this context, since the transformations can be interpreted as coordinate transformations, or as active transformations of the physical system.

J: What's the operational difference? The Lorentz boosts connect two different classes of inertial observers Alice and Bob relative to preferred frame of Eve in the case of broken O(1,3) vacuum symmetry. v(Alice) and v(Bob) are both relative to Eve. Keep things in 1 space dimension now to make the math simpler.

v(Alice - Bob) = [v(Alice) - v(Bob)]/[1 - v(Alice)v(Bob)/c^2]

Note when v(Alice) = v(Bob) relative to Eve, we still get ZERO!

The boost directly from Alice to Bob depends on v(Alice - Bob) = v(Bob - Alice).

Note on Earth from the Catanian's He-Ne laser beat frequency data on 90 deg rotation of the 2-laser apparatus,

v(Alice) ~ v(Bob) ~ 200 km/sec

c = 3 x 10 ^5 km/sec

So when v(Alice) =/= v(Bob), but roughly the same

v/c ~ 10^-4

(v/c)^2 ~ 10^-8

But since the effect, in Cahill's case also goes as n(n^2 - 1) and only in a gas with index of refraction n, one has to look carefully to a precision better than 1 part in 10^8. This should not be so hard.

Z: Such a purely mathematical distinction has no physical correlate in the Einsteinian model, based on the special relativity principle; but in a Lorentzian model, it does.

J: So, frankly, Paul your words above are meaningless until you show examples.

Z: The physical meaning of O(1,3) invariance depends on the model that is adopted. If you ignore the deep differences between the Einsteinian and Lorentzian models, then of course I can see how my words would appear to be "meaningless" to you.

J: You lost me. You are too vague for me. You are content with a vegetable love that certainly does not suit me. (Patience, Gilbert & Sullivan) In your meta-theoretics the scientific meaning doesn't matter for it's only idle chatter of a transcendental kind.

Z: Once we go to a Lorentzian model, then this guarantee no longer holds, and we are deprived of a theoretical basis for imposing Lorentz covariance *a priori* on the fundamental physical laws; and so we then need only recover length contraction as an *emergent dynamical effect*, which could be based on underlying *Galilean* laws.

J: No you have this all wrong. The dynamical laws (the structure of the action) remain Lorentz covariant even when there is a preferred Lorentz frame

Z: Sure. I'm not disputing that.

J: Then you have no point. Where's the beef? Anything beyond that is of no interest. It's meta-theoretic.

Z: See above. I am saying that the original Einsteinian motivation for the *a priori* imposition of Lorentz invariance evaporates once you admit the existence of locally detectable preferred frames.

J: Huh? You lost me.

Z: Of course I am not saying that Lorentz invariance of the underlying dynamics is incompatible with Lorentz invariance at the phenomenological level.

J: Ditto. Show me where what you say makes a significant difference.
- it's exactly like broken gauge symmetry in a superconductor when the U(1) phases all line up, e.g. see pictures in Frank Wilzcek's Jan 20, 2005 Nature paper. Instead of U(1) ~ O(2) think of O(1,1) in the simplest case.

Z: That's a model for the vacuum in which there are preferred frames, which is OK.

J: It's a model that really works! It's battle-tested. What Cahill is allegedly seeing is what P.W. Anderson calls "generalized phase rigidity" a long-range coherence selecting out a preferred Lorentz inertial frame in a limited space-time region where we can ignore GR effects - another story.

Z: OK, I have no problem with this as a tentative model for a physical vacuum with a locally detectable preferred frame.

J: Well that is what the problem is. I mean that's the only part of the problem that is important. What are you worrying about beyond that?

Z: See above.

J: You lost me again. My eyes glaze over. BTW I suspect that Cahill does not know about the above way of looking at what he is allegedly seeing.

Z: He is selling his own "inward flow" model.

J: I think that's wrong, but at least it's definite falsifiable like Hal's "PV without PV". In fact, my theory is the REAL PV! Hal's PV is the illusory Pot of Fool's Gold at the End of Finian's Rainbow. ;-)
People are still looking for dynamical violations i.e. putting terms into the action that violate O(1,3) covariance like in the magnetic Zeeman effect for O(3) violation. This is a very wrong way to go.

Z: OK, that is certainly a POV. But what direct empirical evidence can you cite for Lorentz invariance of the underlying dynamics of the physical vacuum?

J: Ill-posed, That's not a legitimate question.

Z: I suppose an example might be Maxwell's theory, which is definitely Lorentz invariant -- but I understand that from the standpoint of QED, Maxwell's theory is only statistical in character.

J: The Maxwell field theory as a second quantized field theory is still Lorentz form-invariant just like in classical theory. The 4-potential Au is now an operator in Fock space. U is a unitary operator representation of O(1,3)

U = e^iA^a^bSab

{Sab} is a matrix representation of the Lie algebra of O(1,3)

A^a^b are "phases" i.e. Euler angles of space rotation and rapidities of Lorentz boosts

Sab is anti-symmetric so there are only 6 independent elements to the Lie algebra

(Note in General Relativity we have the Ricci rotation coefficients A^a^bc where the "torsion field" potential is

Tu = eu^cA^a^bcSab )

Back to micro-quantum Maxwell EM free field theory, without the j.A interaction from minimal U(1) local gauge covariant coupling to electrically charged Dirac 4-spinors,

U* = U^-1

Under a O(1,3) inertial frame shift in real classical space-time

Au -> Au' = UAuU^-1

Micro-quantum states obey

| >' = U| >

'< | = < |U^-1

Therefore, the micro-quantum transition amplitudes are LORENTZ FRAME INVARIANT, i.e. scalars like ds^2 in classical space-time

'< |Au'| >' = < |Au| >

The active/passive distinction makes no physical difference here as far as I recall off top of my head.

Z: Which brings us back to the question of the Lorentz invariance of non-equilibrium vacuum EM spectra.

J: Don't know what you mean. Hidden variables? A. Valentini's stuff? Then write the math. Pin it down. As soon as you have non-equilibrium in the micro-quantum substratum you have SIGNAL NON-LOCALITY. You also get it going the other way in emergent "More is different" MACRO-quantum spontaneous breaking of vacuum & ground state (for the living brain)) symmetry. This is why Cahill's speed is not the same as the motion of Earth relative to GR Hubble flow as seen in CMB anisotropy. Two different groups.

Z: OK.

J: Cahill's effect is O(1,3) broken vacuum symmetry,

Z: Well, obviously a preferred inertial frame will violate this symmetry, since it establishes a preferred direction in Minkowski spacetime.

J: Yes, that's what I am saying. However, I am not aware of anyone else saying this?

Z: But isn't this obvious?

Not to anyone but me so far. Do you have written evidence to the contrary? Published papers online prior to my Aha? If so, let me know.
- and the Hubble flow is broken T4 vacuum symmetry!

Z: OK. Of course there can be more than one physically preferred frame of reference, which can show up in different kinds of measurements.

J: Yes, I have said that.

Z: OK.

J: Now if I can also relate the Pioneer anomaly to O(1,3)/O(3) ~ S2? as a hedgehog defect in the "multi-layered multi-colored" (Wilzcek) vacuum order parameter cosmic superconducting field - that would clinch it. I have not shown that yet mathematically, but I smell it's true. I could be wrong - or not even wrong. I think it's actually an easy elementary problem.

Z: So you seem to be confusing Cahill's acceptance of the Lorentz transformation formulas as empirically valid with an Einsteinian insistence on imposing Lorentz covariance *a priori* on *all* physical laws, as a kinematical condition -- which I think is a serious misreading of Cahill's position.

J: No Paul. You obviously do not understand the actual math. I don't care about Cahill's theory position.

Z: OK.

J: I don't think he understands what he is seeing BTW. I am only interested in his empirics, not his interpretation of them.

Z: OK. So you have your own theory to account for the appearance of these residual shifts and their evident connection with the earth's motion through space.

J: Right - all based on only two battle-tested ideas

1. Local gauging of symmetry groups (both internal and space-time)

Z: OK.

J: OK? OK? It's more than OK! It's elegant. It's beautiful. It's parsimonious. It's The Tops! It's the Louvre Museum better than The Colosseum ... http://www.pbs.org/wnet/americanmasters/database/porter_c.html It's MORE WITH LESS and it's BATTLE-TESTED! We do not need to look for anything new. It's already right under our noses!

2. Spontaneous breaking of selected symmetry groups in the vacuum (ground state) is essentially the DEFINITION of what the term "preferred frame" means as "generalized phase rigidity" (PW Anderson).

Z: OK.

J: Example 1: ferromagnetism is a preferred spatial orientation over a finite domain in the ground state even though the dynamics is still rotationally invariant the ground state is not.

Example 2: alleged Cahill effect is a preferred space-time orientation, i.e. a definite hyperbolic "rapidity" (rather than Euclidean "angle of orientation") is selected out in a finite space-time domain over which the vacuum coherence condensate has a coherent phase rigidity in the appropriate internal order parameter space. I will explain that in more detail later. This is a global topology effect.

He is backed up by the Catanians though they get a different number using He-Ne lasers rather than interferometers. I think they claim Cahill made a minor error?

Z: The dust hasn't had time to settle yet.

J: The point is that the dynamical equations remain, for example, Lorentz form-invariant (at least locally when GR is important).

Z: Certainly the phenomenological equations do. I wasn't suggesting otherwise.

J: You weren't? I thought you were. OK. I don't know then what you were suggesting.

Z: See above.

J: Duh! Still don't know. Folks are dumb where I come from ...

http://www.physics.rutgers.edu/~motl/

http://www.bitfurnace.com/TheCuddlyMenace/target0.html

The observed fringe shifts are vacuum effects - a kind of Higgs field effect!

Z: Yes, they are vacuum effects, even while you need a propagation medium in the MM light path to detect them.

J: Maxwell waves propagate in classically empty space. So what's your point?

Z: Only that a locally preferred frame that is an intrinsic feature of the physical vacuum completely blows the 1905 Einstein model, which fundamentally changes the physical meaning of the Lorentz transformations, and at the same time raises serious questions about the theoretic and empirical basis for the traditionally supposed universality of Lorentz invariance.

J: *NO IT DOES NOTHING OF THE SORT! You still don't get it. It's only a minor change in some of the interpretation. It does not change E = mc^2 or any of the effects of fields of moving charges and the many battle-tested successes of relativity!

Quantum mechanically the vacuum is primarily (at low energy) a "normal fluid" micro-quantum random plasma of ionized unbound virtual electron-positron pairs and virtual photons. I have added in the NEW macro-quantum "superfluid" vacuum condensate of bound virtual electron-positron pairs out of which Einstein's 1916 GR local field equation emerge via

Z: OK.

J: Bu = (Goldstone Phase of Virtual Electron-Positron Vacuum Condensate),u from spontaneous breaking of EM U(1) in the physical vacuum out of which the geometrodynamic fabric of warped space-time emerges "More is different" (PW Anderson)

Z: OK.

J: Bu = Bu^aPa/h = compensating gauge potential connection field from locally gauging T4

Z: OK.

J: Bu^a = non-trivial part of Cartan tetrad eu^a = (Kronecker Delta)u^a + Bu^a

{Pa} = Lie algebra of T4 subgroup of the Conformal Group.

Note I use both local gauging of T4 and spontaneous symmetry breaking of U(1) in the vacuum i.e. low energy part of Frank Wilczek's "multi-layered multi-colored" cosmic field of superconductivity. I use BOTH 1 & 2 for different groups.

guv(LNIF) = eu^aev^b(Minkowski)ab is EEP

guv(LNIF) = (Minkowski)uv + Lp^2[Bu,v + Bv,u]

Lp^2 = hG/c^3

Gauge transformations of Bu <---> Diff(4) GCT

Z: OK, this is beginning to look really good.

J: You can say that again Bhubba! In fact you should go to the black board and write it 100 times! You do not understand the different between what's happening in the laws of nature and what's happening in the vacuum!

Z; I am simply pointing out that the fundamental theory of the vacuum doesn't *have* to be kinematically Lorentz covariant in order to recover Lorentz contractions at the phenomenological level -- although it could be.

J: That seems to be a very vague way of saying what I have been saying without any of the all-important details. God is in the details. I never heard you once use "spontaneous symmetry breaking" in this context - a battle-tested idea with a large literature since at least 1967.

Z: Because I don't see a problem with it. I only complain about things that look problematic.

J: Folks are dumb where I come from. Maybe Nick Herbert understands you? He is from The Universe Next Door.

You do not at all AS YET understand what I mean in:

There are only two important ideas in theoretical physics.

1. Locally gauging a symmetry group G (either external or internal).

2. Spontaneously breaking the vacuum (or ground state for real
quanta) to make a "preferred frame" relative to the group G.

Z: Of course I understand that you can recover a certain symmetry by locally gauging the relevant symmetry group and then adding back the resulting gauge field.

J: That's 1. What about 2.?

Z: I also understand in a quantum theory how you could lower the symmetry of the physical vacuum by spontaneous symmetry breaking.
I am simply pointing out the the physical meaning of Lorentz covariance shifts dramatically when you have locally detectable preferred frames.

J: Too vague to be useful. I am being very specific here.

And he's "clear as a Bell". :-)

J: What pages? Until you read P.W. Anderson you will not grok it. It's sort of subtle. Even Eugene Wigner had trouble with this and P.W. Anderson recounts. They were both at Princeton of course.

Z: Yes. I'll have to look at all that.

J: It's important not to confuse the dynamical Lagrange-Euler equations for stationary action with their lowest energy solutions in a given well-posed problem.

Z: OK. Of course I understand that the solutions of a set of dynamical equations can and typically do have lower symmetry than the equations themselves.

J: What is important is that it is the lowest energy state that has a lower symmetry.

Z: OK, but isn't that a little odd?

J: THAT'S THE WHOLE POINT! IT'S COUNTER-INTUITIVE BUT IT'S TRUE. ALL OF SOFT-CONDENSED MATTER PHYSICS IS PROOF. Super fluids is proof. No one really understood it properly until P. W. Anderson in 1967 although Lars Onsager and Oliver Penrose first got the essential idea in "ODLRO". Feynman did not understand it when I met him in 1963 and he was working on superfluid helium. He did know about ODLRO I think, but did not have the idea that the ground state of superfluid helium broke a U(1) gauge symmetry. My PhD thesis came out of my private discussions with Feynman in 1963 and later in 1968 both at Cal Tech. PW Anderson only announced the modern POV at UCSD in 1967. I was there.

That the excited states have lower symmetry is not the point. Apparently up until 1967 no one understood that! It was Eugene Wigner's greatest blunder! (As I read PW Anderson). I think everyone assumed that the lowest energy state had to have ALL the symmetries of the dynamics! This led to false super-selection rules like the one on total electric charge that is violated in a superconductor. There was a lot of confusion on this in the 1960's.

Z: OK, as soon as I have time I'll have a look at Anderson's work, which looks very interesting.