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Jul
06

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In a strong measurement, the results are eigenvalues of eigenfunctions. The eigenfunctions describe Bohr's "total experimental arrangement." Therefore, Zielinski's false premise below is that a transformation of basis eigenvectors is a purely formal matter. That is false. Every basis transformation is a physical change of the total experimental arrangement, e.g. changing relative orientations of Stern-Gerlach magnets in testing Bell's inequality in entangled pairs of magnetic spins.

From: Paul Zielinski <iksnileiz@gmail.com>

To: Jack Sarfatti <sarfatti@pacbell.net>

Sent: Tue, July 5, 2011 2:18:24 PM

Subject: Re: Creon Levit's Nano-electronics & The issue of small n - FTL signals via over-complete non-orthogonal coherent states?

On 7/4/2011 5:23 PM, Jack Sarfatti wrote:

From: Paul Zielinski <iksnileiz@gmail.com>

To: Ruth Elinor Kastner <rkastner@umd.edu>

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".

PZ: Not at all. The point here is that you have to apply a non-orthogonal transformation to get from an orthogonal basis to a non-orthogonal basis, and the inner product defined by the Euclidean metric is not preserved.

JS: You mean a "non-unitary" transformation. I never denied that. But my argument for entanglement signals never explicitly used a non-unitary transformation. The physical idea is that the non-orthogonal basis is there objectively physically ab-initio for dynamical reasons, e.g. stimulated emission population inversion, some spontaneous symmetry breakdown with a Glauber state condensate of Goldstone bosons etc.

Given that non-orthogonal basis, unitary transformations on it give a new non-orthogonal basis.

OQT does not permit non-unitary dynamics - a weakness of OQT admittedly. It does not apply to open systems. Dissipation introduces non-Hermitian observables for example, e.g. imaginary part of the effective Hamiltonian.

You apply a transformation to a basis.

PZ: Right.

JS: Second, when you move to a uniformly accelerating non-inertial Rindler frame for example, there are cross terms in the metric

gtz =/= 0

PZ: OK...

JS: 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.

PZ: I was making a general mathematical point about inner products and non-orthogonal/non-unitary transformations.

JS: That's not good physics. The job of the theoretical physicist is more physics with less math not the other way round. Any fundamental mathematical process must describe a physical change of some kind. If not, if it's redundant then it is a gauge transformation that must be "gauged away" in some sense. Of course, in GR the gauge transformations have direct physical operational meaning unlike the internal U1, SU2, SU3 gauge transformations.

The tetrad map connecting the two of them is a real physical change i.e. switching a rocket engine on or off for example.

PZ: My point is that any changes in lengths that result purely from the application of non-orthogonal transformations in ordinary 3D Euclidean space, for example, are not actual geometric changes. I am saying that the same applies to non-unitary transformations and inner products in the Hilbert space formalism of QM.

JS: Your remarks are too vague to pin down to my actual toy model. I am saying that stimulated emission with population creates an objectively real Glauber state non-orthogonal basis whose partial traces leaves an entanglement signal when we can entangle two back to back laser beams without orthogonal polarizations destroying the effect. 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.

PZ: I think this is a different issue.

JS: No that IS my issue here. You have introduced extraneous side issues.

PZ: Yes one could for example modify the Schrodinger equation such that time evolution of the wave

function is no longer unitary, but that would represent a change in the physics. I suppose one could interpret such non-unitary time evolution in terms of creation/annihilation of particle density (sources and sinks), but that would constitute a change of physical interpretation, which goes well beyond the purely mathematical application of a non-unitary transformation.

JS: That's what happens in a laser where the macro-quantum coherence dynamics is no longer the linear unitary Schrodinger equation entangled in configuration space, but morphs to a nonlinear nonunitary emergent Landau-Ginzburg equation for the order parameter local in physical space! - but with some residual entanglement and still a coupling to micro-quantum noise (Goldstone bosons popping into and out of the Goldstone boson condensate order parameter).

PZ: I'm making a mathematical point about the invariance of lengths and angles (inner products) under a certain class of transformations.

JS: Completely irrelevant to the physics. I am not interested in redundant formal transformations that must be factored out of the actual physics of testable measurements. Please I am not interested side issues of formalism - unless it's an error in my own use of formal symbol strings.

PZ: Non-orthogonal transformations in Euclidean space don't preserve lengths and angles, but in standard Euclidean theory this has no geometric meaning. I'm suggesting that you can make the same argument in a Hilbert space. I'm arguing that changes in the computed inner products that result purely from the mathematical application of a non-unitary transformation do not have any physical meaning in the standard Hilbert space formulation of QM, any more than the analogous changes in lengths and angles resulting from the application of non-orthogonal transformations in Euclidean theory have actual geometric meaning.

JS: Not relevant to the physics I am proposing.

PZ: As you like to say, the map is not the territory.

JS: 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.

PZ: Not in 3D Euclidean space, which is what I was referring to. There we are talking about ordinary orthogonal transformations. Very simple and intuitive. You can construct orthogonal and non-orthogonal bases, and apply orthogonal and non-orthogonal transformations, quite generally in Riemannian geometry.

JS You are losing me. This is a tangent. A distraction.

Adding in the Rindler observers is adding the special conformal transformations I suppose. That is, going to a bigger group.

PZ: Rindler transformations do not change the intrinsic (coordinate invariant) value of the Riemann metric, they only change the representation of the metric. I'm saying an analogous distinction applies in the Hilbert space formalism.

JS: The real measurements of the Rindler observer differ qualitatively from those of the Minkowski observer. True the classical metric structure is the same, but the quantum properties are different. You fail to make the difference that makes a difference. Classical measurements e.g. curvature are the same. On the other hand the Rindler observer feels g-force whilst the Minkowski observer does not. The Rindler observer feels heat, the Minkowski observer does not. Therefore, including quantum measurements, the special conformal boost describes a physical change in the total experimental arrangement in the sense of Bohr. Your mistake Paul is to think too classically. These are new quantum gravity effects.

PZ: The map is not the territory.

JS: 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: Well you thought I was confusing non-orthogonal transformations with non-orthogonal basis vectors. Which I wasn't.

JS: Whatever you thought you were doing was not helpful to the real problem at hand which is

1) Do entanglement signals exist in complex biological systems ubiquitously? e.g. Josephson, Bem et-al

2) Do they exist in non-living quantum mechanical systems? - Moddell's problem.

e.g. do they exist whenever LCAO non-orthogonal bases are needed for a correct computation of observables in quantum chemistry for example?

PZ: It seems obvious to me that changes in computed quantities that result purely from the merely mathematical application of a non-unitary transformation to an orthonormal basis spanning a QM Hilbert space in itself has no physical meaning under the standard interpretation of the QM formalism.

JS: Of no relevance to what I am interested in. Different ball park.

PZ: Of course, if the physical interpretation changes, then this may not apply. But that is a different kettle of fish.

JS: It's the kettle I am stirring.

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.

PZ: Why are you so sure that such correlations do not result purely from the non-zero projections of the non-orthogonal basis vectors onto each other? Why are you so sure that such "entanglement" has the same physical meaning as when the observables are represented in an orthonormal basis? Couldn't such "correlations" simply reflect the non-orthogonal character of the basis?

JS: I have no idea what the above string of words means. There is evidence of signal nonlocality in living matter. I am trying to model how that can happen. The basic idea is that the inner products of base states with each other are not formal redundancies, but are objectively real control parameters.

Non-Hermitian Hamiltonians for open systems with dissipation will have non-orthogonal over-complete energy eigenfunctions with complex energy eigenvalues - I think. True they will evolve with a non-unitary dynamics for the emergent order parameter.

PZ: What happens to analogous geometric quantities when we transform to a non-orthogonal vector basis, in plain vanilla Euclidean geometry?

JS: Who cares? Not an interesting question for what I am trying to do.

Your red herring of a non-unitary transformation is nowhere in the actual calculation I gave.

PZ: I feel you are missing the point, which is that in order to get from an orthogonal basis to a non-orthogonal basis you have to apply a non-unitary transformation. I'm not saying that you did that explicitly.

JS: Show how that changes the actual equations in the model I proposed? I don't care in this initial phase what specific physically real non-unitary map may have created the non-orthogonal basis to begin with. I assume it is there as my starting point to get the entanglement signal. The details of the non-unitary physics that forces the non-orthogonal basis is a separate problem of importance obviously.

Now the LCAO basis in the many electron problem may be an example of what I am looking for, but I am not sure.

Well they are definitely using non-orthogonal basis representations of correlated density matrices. There is no

question in my mind at least that technically one can do this.

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.

PZ: As I said, I think you may be missing the point, which is that in order to get from an orthogonal basis to a

non-orthogonal basis you have apply a non-unitary transformation, which "stretches" the Hilbert space inner products.

That is not a true geometric stretch -- it's a Coney Island fun house optic!

Of course we've already been through all this with objective spacetime warps vs. mere coordinate artifacts in GR. For

example, black hole event horizons are not coordinate-dependent artifacts. Also there is no such thing as a purely formal non-unitary transformation in a real physics theory. Well there is in Euclidean geometry. In Euclidean geometry a non-orthogonal transformation is a fun house mirror. A passive diffeomorphism is a fun house optic.

I'm not saying that one cannot change to physical interpretation of QM such that the effects of non-unitary transformations on inner products can acquire objective physical meaning.

Of course here I'm talking about the standard Hilbert space formulation of orthodox QM.

JS: 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: Exactly.

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.

PZ; I meant to write, "... such changes in the computed values of inner products..."

JS: 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: What I'm saying here is not very subtle Jack. Obviously non-orthogonal vectors are "correlated" since they have non-zero projections onto each other. But it has no objective geometric meaning in Euclidean geometry, and I'm saying that something similar applies in QM Hilbert spaces.

I agree that there is a close analogy here with some possible misconceptions in the interpretation of GR. You seem to be implicitly applying a kind of "equivalence principle" to basis-dependent artifacts in QM, rendering them physically indistinguishable from actual changes in physical correlations predicted by QM. Am I wrong?

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.

PZ: OK.

JS: In other words you will get the wrong answer if you used an orthogonal basis.

PZ: But how do you think can you get a different physical answer simply by changing the basis representation of a QM observable? That's what I don't get.

JS: 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: But that is the result of a real physical interaction of the moving detectors with the vacuum. You are confusing a mere coordinate transformation with a physical interaction.

JS: Not at all. Switching on a rocket engine is a real physical process that transforms a Minkowski to a Rindler observer.

PZ: OK then I guess I was right -- you are applying an analog of Einstein's equivalence principle to basis dependent artifacts.

Heuristically speaking this might be OK, and might produce interesting results, but it's kind of crazy from a purely analytical perspective.

There is no such "equivalence principle" in orthodox QM. I think you would have to modify the standard interpretation of the Hilbert space formulation of QM for this even to be relevant.

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.

PZ: Don't we have to distinguish carefully here between the mathematical choice of basis, on the one hand, and the physical

motivation for the choice of state vectors, on the other? Clearly we want our state vectors to represent the states we actually

prepare, but I think the choice of basis in which to represent Hilbert space operators is another issue.

I say it's a mathematical question. Just as the choice of coordinates in Euclidean space need not have any relationship to the behavior of "observers" -- unless one establishes a convention to that effect.

JS: In OQT you can only use unitary transformations of basis.

PZ: OK.

JS: The inner products are invariant in OQT for a closed system.

PS: OK.

JS: Non-unitary transformations are forbidden in OQT.

Nonunitary transformations require new physics beyond OQT.

PZ: OK.

JS: However, I do agree we need non-unitary transformations in physics. But they are not merely formal crutches.

PZ: Well I would say that mathematically speaking we already have them in any Hilbert space, but they don't have physical meaning in the orthodox interpretation of the QM formalism. Any more than non-orhtogonal transformations have geometric meaning in ordinary Euclidean geometry.

JS: 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: Which raises questions about exactly what the physical chemists think they're doing with those "strongly correlated" density matrices expressed in non-orthogonal bases! I don't think they are offering any modifications of the standard interpretation of the QM formalism.

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.

Didn't you say or at least imply above that the analogy is good in the standard interpretation of the von Neumann formalism?

PZ: You said non-unitary transformations are not physically admissible in orthodox QM. In order to get from an orthogonal basis to a non-orthogonal basis in QM, you have to apply a non-unitary transformation, which "stretches" the inner products. So any "observable" derived from such quantities will similarly be "stretched". You want to say that an artifact of such a transformation is in certain cases a real physical effect. I'm saying that this would require a fundamental modification of OQT.

JS: Of course it does. However, lasers do that. Spontaneous broken symmetries do it.

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.

Well exactly. And isn't that my point?

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.

-RK

From: Paul Zielinski <iksnileiz@gmail.com>

To: Jack Sarfatti <sarfatti@pacbell.net>

Sent: Tue, July 5, 2011 2:18:24 PM

Subject: Re: Creon Levit's Nano-electronics & The issue of small n - FTL signals via over-complete non-orthogonal coherent states?

On 7/4/2011 5:23 PM, Jack Sarfatti wrote:

From: Paul Zielinski <iksnileiz@gmail.com>

To: Ruth Elinor Kastner <rkastner@umd.edu>

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".

PZ: Not at all. The point here is that you have to apply a non-orthogonal transformation to get from an orthogonal basis to a non-orthogonal basis, and the inner product defined by the Euclidean metric is not preserved.

JS: You mean a "non-unitary" transformation. I never denied that. But my argument for entanglement signals never explicitly used a non-unitary transformation. The physical idea is that the non-orthogonal basis is there objectively physically ab-initio for dynamical reasons, e.g. stimulated emission population inversion, some spontaneous symmetry breakdown with a Glauber state condensate of Goldstone bosons etc.

Given that non-orthogonal basis, unitary transformations on it give a new non-orthogonal basis.

OQT does not permit non-unitary dynamics - a weakness of OQT admittedly. It does not apply to open systems. Dissipation introduces non-Hermitian observables for example, e.g. imaginary part of the effective Hamiltonian.

You apply a transformation to a basis.

PZ: Right.

JS: Second, when you move to a uniformly accelerating non-inertial Rindler frame for example, there are cross terms in the metric

gtz =/= 0

PZ: OK...

JS: 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.

PZ: I was making a general mathematical point about inner products and non-orthogonal/non-unitary transformations.

JS: That's not good physics. The job of the theoretical physicist is more physics with less math not the other way round. Any fundamental mathematical process must describe a physical change of some kind. If not, if it's redundant then it is a gauge transformation that must be "gauged away" in some sense. Of course, in GR the gauge transformations have direct physical operational meaning unlike the internal U1, SU2, SU3 gauge transformations.

The tetrad map connecting the two of them is a real physical change i.e. switching a rocket engine on or off for example.

PZ: My point is that any changes in lengths that result purely from the application of non-orthogonal transformations in ordinary 3D Euclidean space, for example, are not actual geometric changes. I am saying that the same applies to non-unitary transformations and inner products in the Hilbert space formalism of QM.

JS: Your remarks are too vague to pin down to my actual toy model. I am saying that stimulated emission with population creates an objectively real Glauber state non-orthogonal basis whose partial traces leaves an entanglement signal when we can entangle two back to back laser beams without orthogonal polarizations destroying the effect. 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.

PZ: I think this is a different issue.

JS: No that IS my issue here. You have introduced extraneous side issues.

PZ: Yes one could for example modify the Schrodinger equation such that time evolution of the wave

function is no longer unitary, but that would represent a change in the physics. I suppose one could interpret such non-unitary time evolution in terms of creation/annihilation of particle density (sources and sinks), but that would constitute a change of physical interpretation, which goes well beyond the purely mathematical application of a non-unitary transformation.

JS: That's what happens in a laser where the macro-quantum coherence dynamics is no longer the linear unitary Schrodinger equation entangled in configuration space, but morphs to a nonlinear nonunitary emergent Landau-Ginzburg equation for the order parameter local in physical space! - but with some residual entanglement and still a coupling to micro-quantum noise (Goldstone bosons popping into and out of the Goldstone boson condensate order parameter).

PZ: I'm making a mathematical point about the invariance of lengths and angles (inner products) under a certain class of transformations.

JS: Completely irrelevant to the physics. I am not interested in redundant formal transformations that must be factored out of the actual physics of testable measurements. Please I am not interested side issues of formalism - unless it's an error in my own use of formal symbol strings.

PZ: Non-orthogonal transformations in Euclidean space don't preserve lengths and angles, but in standard Euclidean theory this has no geometric meaning. I'm suggesting that you can make the same argument in a Hilbert space. I'm arguing that changes in the computed inner products that result purely from the mathematical application of a non-unitary transformation do not have any physical meaning in the standard Hilbert space formulation of QM, any more than the analogous changes in lengths and angles resulting from the application of non-orthogonal transformations in Euclidean theory have actual geometric meaning.

JS: Not relevant to the physics I am proposing.

PZ: As you like to say, the map is not the territory.

JS: 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.

PZ: Not in 3D Euclidean space, which is what I was referring to. There we are talking about ordinary orthogonal transformations. Very simple and intuitive. You can construct orthogonal and non-orthogonal bases, and apply orthogonal and non-orthogonal transformations, quite generally in Riemannian geometry.

JS You are losing me. This is a tangent. A distraction.

Adding in the Rindler observers is adding the special conformal transformations I suppose. That is, going to a bigger group.

PZ: Rindler transformations do not change the intrinsic (coordinate invariant) value of the Riemann metric, they only change the representation of the metric. I'm saying an analogous distinction applies in the Hilbert space formalism.

JS: The real measurements of the Rindler observer differ qualitatively from those of the Minkowski observer. True the classical metric structure is the same, but the quantum properties are different. You fail to make the difference that makes a difference. Classical measurements e.g. curvature are the same. On the other hand the Rindler observer feels g-force whilst the Minkowski observer does not. The Rindler observer feels heat, the Minkowski observer does not. Therefore, including quantum measurements, the special conformal boost describes a physical change in the total experimental arrangement in the sense of Bohr. Your mistake Paul is to think too classically. These are new quantum gravity effects.

PZ: The map is not the territory.

JS: 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: Well you thought I was confusing non-orthogonal transformations with non-orthogonal basis vectors. Which I wasn't.

JS: Whatever you thought you were doing was not helpful to the real problem at hand which is

1) Do entanglement signals exist in complex biological systems ubiquitously? e.g. Josephson, Bem et-al

2) Do they exist in non-living quantum mechanical systems? - Moddell's problem.

e.g. do they exist whenever LCAO non-orthogonal bases are needed for a correct computation of observables in quantum chemistry for example?

PZ: It seems obvious to me that changes in computed quantities that result purely from the merely mathematical application of a non-unitary transformation to an orthonormal basis spanning a QM Hilbert space in itself has no physical meaning under the standard interpretation of the QM formalism.

JS: Of no relevance to what I am interested in. Different ball park.

PZ: Of course, if the physical interpretation changes, then this may not apply. But that is a different kettle of fish.

JS: It's the kettle I am stirring.

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.

PZ: Why are you so sure that such correlations do not result purely from the non-zero projections of the non-orthogonal basis vectors onto each other? Why are you so sure that such "entanglement" has the same physical meaning as when the observables are represented in an orthonormal basis? Couldn't such "correlations" simply reflect the non-orthogonal character of the basis?

JS: I have no idea what the above string of words means. There is evidence of signal nonlocality in living matter. I am trying to model how that can happen. The basic idea is that the inner products of base states with each other are not formal redundancies, but are objectively real control parameters.

Non-Hermitian Hamiltonians for open systems with dissipation will have non-orthogonal over-complete energy eigenfunctions with complex energy eigenvalues - I think. True they will evolve with a non-unitary dynamics for the emergent order parameter.

PZ: What happens to analogous geometric quantities when we transform to a non-orthogonal vector basis, in plain vanilla Euclidean geometry?

JS: Who cares? Not an interesting question for what I am trying to do.

Your red herring of a non-unitary transformation is nowhere in the actual calculation I gave.

PZ: I feel you are missing the point, which is that in order to get from an orthogonal basis to a non-orthogonal basis you have to apply a non-unitary transformation. I'm not saying that you did that explicitly.

JS: Show how that changes the actual equations in the model I proposed? I don't care in this initial phase what specific physically real non-unitary map may have created the non-orthogonal basis to begin with. I assume it is there as my starting point to get the entanglement signal. The details of the non-unitary physics that forces the non-orthogonal basis is a separate problem of importance obviously.

Now the LCAO basis in the many electron problem may be an example of what I am looking for, but I am not sure.

Well they are definitely using non-orthogonal basis representations of correlated density matrices. There is no

question in my mind at least that technically one can do this.

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.

PZ: As I said, I think you may be missing the point, which is that in order to get from an orthogonal basis to a

non-orthogonal basis you have apply a non-unitary transformation, which "stretches" the Hilbert space inner products.

That is not a true geometric stretch -- it's a Coney Island fun house optic!

Of course we've already been through all this with objective spacetime warps vs. mere coordinate artifacts in GR. For

example, black hole event horizons are not coordinate-dependent artifacts. Also there is no such thing as a purely formal non-unitary transformation in a real physics theory. Well there is in Euclidean geometry. In Euclidean geometry a non-orthogonal transformation is a fun house mirror. A passive diffeomorphism is a fun house optic.

I'm not saying that one cannot change to physical interpretation of QM such that the effects of non-unitary transformations on inner products can acquire objective physical meaning.

Of course here I'm talking about the standard Hilbert space formulation of orthodox QM.

JS: 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: Exactly.

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.

PZ; I meant to write, "... such changes in the computed values of inner products..."

JS: 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: What I'm saying here is not very subtle Jack. Obviously non-orthogonal vectors are "correlated" since they have non-zero projections onto each other. But it has no objective geometric meaning in Euclidean geometry, and I'm saying that something similar applies in QM Hilbert spaces.

I agree that there is a close analogy here with some possible misconceptions in the interpretation of GR. You seem to be implicitly applying a kind of "equivalence principle" to basis-dependent artifacts in QM, rendering them physically indistinguishable from actual changes in physical correlations predicted by QM. Am I wrong?

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.

PZ: OK.

JS: In other words you will get the wrong answer if you used an orthogonal basis.

PZ: But how do you think can you get a different physical answer simply by changing the basis representation of a QM observable? That's what I don't get.

JS: 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: But that is the result of a real physical interaction of the moving detectors with the vacuum. You are confusing a mere coordinate transformation with a physical interaction.

JS: Not at all. Switching on a rocket engine is a real physical process that transforms a Minkowski to a Rindler observer.

PZ: OK then I guess I was right -- you are applying an analog of Einstein's equivalence principle to basis dependent artifacts.

Heuristically speaking this might be OK, and might produce interesting results, but it's kind of crazy from a purely analytical perspective.

There is no such "equivalence principle" in orthodox QM. I think you would have to modify the standard interpretation of the Hilbert space formulation of QM for this even to be relevant.

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.

PZ: Don't we have to distinguish carefully here between the mathematical choice of basis, on the one hand, and the physical

motivation for the choice of state vectors, on the other? Clearly we want our state vectors to represent the states we actually

prepare, but I think the choice of basis in which to represent Hilbert space operators is another issue.

I say it's a mathematical question. Just as the choice of coordinates in Euclidean space need not have any relationship to the behavior of "observers" -- unless one establishes a convention to that effect.

JS: In OQT you can only use unitary transformations of basis.

PZ: OK.

JS: The inner products are invariant in OQT for a closed system.

PS: OK.

JS: Non-unitary transformations are forbidden in OQT.

Nonunitary transformations require new physics beyond OQT.

PZ: OK.

JS: However, I do agree we need non-unitary transformations in physics. But they are not merely formal crutches.

PZ: Well I would say that mathematically speaking we already have them in any Hilbert space, but they don't have physical meaning in the orthodox interpretation of the QM formalism. Any more than non-orhtogonal transformations have geometric meaning in ordinary Euclidean geometry.

JS: 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: Which raises questions about exactly what the physical chemists think they're doing with those "strongly correlated" density matrices expressed in non-orthogonal bases! I don't think they are offering any modifications of the standard interpretation of the QM formalism.

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.

Didn't you say or at least imply above that the analogy is good in the standard interpretation of the von Neumann formalism?

PZ: You said non-unitary transformations are not physically admissible in orthodox QM. In order to get from an orthogonal basis to a non-orthogonal basis in QM, you have to apply a non-unitary transformation, which "stretches" the inner products. So any "observable" derived from such quantities will similarly be "stretched". You want to say that an artifact of such a transformation is in certain cases a real physical effect. I'm saying that this would require a fundamental modification of OQT.

JS: Of course it does. However, lasers do that. Spontaneous broken symmetries do it.

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.

Well exactly. And isn't that my point?

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.

-RK