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From: Paul Zielinski To: Ruth Elinor Kastner Sent: Mon, July 4, 2011 3:54:52 PM
Subject: Re: Creon's Nano-electronics & The issue of small n - FTL signals via over-complete non-orthogonal coherent states?
PZ: "Orthogonal transformations in ordinary Euclidean 3D space preserve lengths and angles. Non-orthogonal transformations don't.  If we apply a non-orthogonal transformation in a Euclidean space, of course we will see formal changes in lengths and angles, but it will have no objective geometric meaning. "
JS: You are wrong. For one thing, you have confused "transformation" with "basis". You apply a transformation to a basis.

Second, when you move to a uniformly accelerating non-inertial Rindler frame for example, there are cross terms in the metric
gtz =/= 0
Also the non-geodesic Rindler guv-detector sees real photons in the form of thermal black body Unruh radiation. In contrast, the geodesic detector nab sees only virtual zero point photons.
The tetrad map connecting the two of them is a real physical change, i.e. switching a rocket engine on or off for example.
Now you can in the sense of an analogy compare that to a non-unitary transformation in Hilbert space that indeed does violate OQT in Henry Stapp's sense.
The quantum unitary transformations are then analogs of the global Lorentz boosts, the global rotations and translations of Special Relativity restricted to the global 10-parameter Poincare group.
Adding in the Rindler observers is adding the special conformal transformations I suppose. That is, going to a bigger group.
Finally GR involves localizing only the 4-parameter translation sub-group of the Poincare group.
Then e.g. cosmology has spontaneous symmetry breakdown of the T1 time translation group i.e. Hubble flow of accelerating expanding universe in which total global energy is not conserved. Thus, in the dS solution we have a constant / vacuum energy density so that the total vacuum energy density is not conserved. However it does appear to have a finite upper bound because of the finite distance to our future event horizon.
Our actual universe is not dS at the beginning, so that / is not a constant. It is large at inflation and then decreases approaching a small asymptote.
PZ: So how is the situation any different in QM, if we consider the corresponding unitary and non-unitary transformations?
JS: I just showed you. Your analogy is false.
PZ: Surely inner  products computed from state vectors subject to non-unitary transformations have a very different meaning in the Hilbert space formalism of QM from those computed under unitary transformations?
JS: Agreed, but that has nothing to do with my original point which is that GIVEN a physically realizable non-orthogonal basis for an entangled system, partially tracing over a part of the system will allow an entangled signal to the other part. You red herring of a non-unitary transformation is nowhere in the actual calculation I gave.  Now the LCAO basis in the many electron problem may be an example of what I am looking for, but I am not sure.
PZ: My point here is that any change in computed inner products that purely arises from the formal application of a non-unitary transformation  to an orthonormal basis cannot have the same meaning as an inner product that is computed in the usual manner.
JS: Irrelevant as I never did that in this case. Also there is no such thing as a purely formal non-unitary transformation in a real physics theory. Any such transformation must describe an objective change - some operation like firing a rocket engine, or adding dissipation to a complex system, pumping it off thermal equilibrium etc.
PZ: Changes in the computed values of such inner products can only be purely formal in character.
JS: I don't think that is the case. Sure if, it's only formal you are correct. However, I don't think that is what is happening in the situations Brian Josephson had in mind and in the Daryl Bem data.
PZ: Since entanglement correlations for multiparticle systems ultimately have to be computed from such inner products, it is hard to see how such correlations that arise purely from non-unitary  transformations of an orthonormal basis to a non-orthonormal basis can have objective physical meaning within the framework of QM.
JS: My idea is that the non-orthogonal basis are actually there in the complex system in a physical sense and are not simply a formal alternative. In other words you will get the wrong answer if you used an orthogonal basis. To make an analogy, you will not measure Unruh temperature unless you switch on your rocket engine - but when you do, your temperature gauge measures a real temperature that drops to zero when you switch off the engine.
PZ: Of course one can create non-vanishing non-diagonal matrix elements in any "observable" by going to a non-orthogonal basis, but does  this mean that we now have non-vanishing transition probabilities between eigenstates of the observable? It should be obvious that the non-vanishing off-diagonal elements merely reflect the fact that non-orthogonal basis vectors have non-zero projections onto each other.
JS: False. Either the physics gives you an orthogonal basis or not. In OQT you can only use unitary transformations of basis. The inner products are invariant in OQT for a closed system. Non-unitary transformations are forbidden in OQT. Nonunitary transformations require new physics beyond OQT. However, I do agree we need non-unitary transformations in physics. But they are not merely formal crutches.  Gerardus 't Hooft does not think so. He thinks the unitary S-Matrix describes black holes for example if you wait long enough.  On the other hand, applied physicists do use dissipative non-unitary transformations, but I guess the lip service is that when you include the traced over environment everything is unitary.
PZ: So Jack, how is the multiparticle case in essence any different with respect to the entanglement correlations considered in your thought  experiment? Of course one can prepare non-orthogonal coherent photon states as you say, but if the effects you predict arise purely  from the use of a non-orthogonal basis for the representation of quantum "observables", what makes you think they have any more  objective physical meaning than changes in lengths and angles induced in ordinary Euclidean space by non-orthogonal transformations?
JS: Your formal analogy is not correct physically. Unlike pure mathematics, the transformations in real physics must describe differences that make a physical difference, otherwise the physical theory is ill-posed, wrong or worse not even wrong.
On 7/4/2011 9:45 AM, Ruth Elinor Kastner wrote:
I still think the issue is just that the entanglement is with respect to which-slit (which is an observable whose eigenstates are orthogonal) and not with respect to the nonorthogonal eigenstates of the annihilation operators (coherent states).  So I think what needs to be questioned (as you suspected peviously) is your eqn 1-18. Where does this come from? It seems to me that the A1 and A2 have to be just which-slit eigenstates here.
JS: Maybe so, but I am using the Feynman path history picture.
represent the complete complex number Feynman amplitudes for a Glauber state in the entangled pair to be emitted at the source, take all possible paths passing through slits 1' and 2' and be detected on the screen at x' (from the central peak in Young's set up).  
= exp{-[1 - cos@(x')]}
is an intuitive Ansatz I inferred from Glauber's work - but it extends it admittedly and may be wrong.
Remember Feynman played with his intuitive pictures before he really had much of the formal algebra.
@(x') is the net relative phase weighted by all possible paths with the same emission at the laser source landing at same x' through both A1' and A2'.
The RHS is an interesting function as ---> infinity basically it's a comb of Dirac delta functions at @ = integer x 2pi
The physical observable of interest is then the integral over x' of
e^i@(x') = Integral over x' of exp{i@(x') - (1 - cos@(x')}
This is sort of pretty.
However, even if this particular model is wrong, it does not change the general fact that a non-orthogonal partial trace gives an entanglement signal. The use of LCAO non-orthogonal base functions is certainly an alarm bell.
Also, my above model may be correct in the end. It must be tested with entangled laser beams.
Nick's attitude of don't even look in that part of the room is not wise in my opinion.
As I said at the beginning I am Devil's Advocate - no telling where these attempts will lead. The prize is great if in fact there is entanglement signaling ubiquitous in nature especially in biological systems as Brian Josephson was the first to suggest.
That's why I am not content with Henry Stapp's general philosophical arguments though he may be correct in the end - there is also some ambiguity in the boundary of "Orthodox Quantum Theory" - does it include spontaneous symmetry breakdown with the emergence of non-unitary local Landau-Ginzberg dynamics for macro-quantum coherent order parameters?

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