Paul, key point is any tetrad map and any GCT is quantum mechanically a Bogoliubov transformation with UIRs (Unitarily Inequivalent Representations) of the quantum vacua/ground states and their quasi-particle and collective mode elementary excitations. In contrast to special relativity where all mappings between geodesic zero g-force inertial frames INDUCE UER (Universally Equivalent Representations) of QUBITS.
This is the lesson of Hawking-Unruh effect. All GCTs are locally approximated by Rindler special conformal hyperbolic boosts with local thermodynamic horizons - seems to be Ted Jacobson's et-al basic intuition?
"unitarily inequivalent representations
From Physics wik Quantum field theory
In ordinary quantum mechanics of systems with finitely many degrees of freedom, the "choice problem", namely choosing a particular representation of the canonical commutation relations, is resolved by the Stone–von Neumann theorem which states that any representation is essentially unique, up to unitary transformations. This is not so for systems with infinitely many degrees of freedom (e.g., Quantum field theories), and the statement is known as Haag's theorem[1]"
http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/unitarily_inequivalent_representations
If spacetime is a world crystal lattice (Hagen Kleinert), then the number of degrees of freedom is finite ~ 10^122 using hologram idea.
On Mar 6, 2011, at 7:19 PM, JACK SARFATTI wrote:
I think the mode transformation plane waves to spherical harmonics with Bessel function radial dependence - is "unitary," i.e. equivalent vacua both in global inertial frames, no Unruh effect (Bogoliubov transformation) - no change in boundary conditions.
However, any switch to off-geodesic/rotating frames does need a Bogoliubov-Unruh change of creation and destruction operators - change of "quanta" identity (non-unitarily equivalent vacua). I could be wrong about that I need to check. If I am right then from a QM POV quantum gravity is inherently non-unitary.
Spontaneous symmetry breakdown (More is different) vacuum condensation's effect is shown in (4.2) - that is Fourier transform (flat spacetime) of the coherent order parameter that obeys a Landau-Ginzburg equation that under some conditions has no "i" in the time derivative, i.e. non-unitary dissipative dynamics with signal nonlocality in contrast to unitary wave dynamics with signal locality.
How to Build Unitarily Inequivalent Representations in Quantum Field Theory
by T Lupher - Related articles
How to Construct Unitarily Inequivalent Representations ... key to generating unitarily inequivalent representations. 1 Introduction ...
https://webspace.utexas.edu/lupher/www/papers/UIRsinQFT.pdf
"Following Haag and Kastler's lead, it was claimed by most proponents of algebraic quantum field theory that all physical content resides in a specific class of observables. It is shown in the dissertation that such claims are exaggerated and misleading. UIRs are used to elucidate the nature of quantum field theory by showing that UIRs have different expectation values for some classical observables of the system, such as temperature and chemical potential, which are not in Haag and Kastler's specific class. It is shown how UIRs may be used to construct classical observables. To capture the physical content of quantum field theory it is shown that a much larger algebra than that of Haag and Kastler is necessary. Finally, the arguments that UIRs are incommensurable theories are shown to be flawed. The lesson of UIRs is that the mathematical structures in both canonical quantum field theory and Haag and Kastler's version of algebraic quantum field theory are not sufficient to capture all of the physical content that UIRs represent. A suitable algebraic structure for quantum field theory is provided in the dissertation."
The philosophical significance of unitarily inequivalent representations (UIR)
by TA Lupher - 2008 - Cited by 1 - Related articles
This dissertation gives a general account of the properties of unitarily inequivalent representations (UIRs) in both canonical quantum field theory and ...
adsabs.harvard.edu/abs/2008PhDT........91L
"Physical (heuristic) point of view
As was already noticed by Haag in his original work, it is the vacuum polarization that lies at the core of Haag's theorem. Any interacting quantum field (including non-interacting fields of different masses) is polarizing the vacuum, and as a consequence its vacuum state lies inside a renormalized Hilbert space HR that differs from the Hilbert space HF of the free field. Although an isomorphism could always be found that maps one Hilbert space into the other, Haag's theorem implies that no such mapping would deliver unitarily equivalent representations of the corresponding CCR, i.e. unambiguous physical results.
Workarounds
Among the assumptions that lead to Haag's theorem is translation invariance of the system. Consequently, systems that can be set up inside a box with periodic boundary conditionsor that interact with suitable external potentials escape the conclusions of the theorem [5]. Haag [6] and Ruelle [7] have presented a modified ('Haag-Ruelle') scattering theory that allows to circumvent the problems posed by Haag's theorem, but this approach is complicated in practical application and so far it has been applied to a limited set of model systems only.
Ignorance on the part of the QFT practitioner
Most practitioners of QFT appear to ignore the implications of Haag's theorem entirely and prefer to go ahead producing numbers. It is currently unknown why, and under which conditions or limitations, QFT produces accurate numbers in real life situations. In fact, within the canonical development of perturbative quantum field theory — which includes quantum electrodynamics, cited as one of the great successes of modern science — the interaction picture is used throughout."
Haag's theorem - Wikipedia, the free encyclopedia
On Mar 6, 2011, at 7:11 PM, Paul Zielinski wrote:
OK.
On 3/6/2011 7:10 PM, JACK SARFATTI wrote:
Not at all. I never wrote that.
On Mar 6, 2011, at 5:56 PM, Paul Zielinski wrote:
"Are you saying that a unitary relationship between spherical harmonic and plane wave modes of the same
quantized EM field is not guaranteed?"
On 3/6/2011 4:37 PM, JACK SARFATTI wrote:
On Mar 6, 2011, at 1:29 PM, Paul Zielinski wrote:
"Do the number and type of constituent quanta depend on the basis function representation of the field?"
Yes and no because there is missing necessary information - your question not well-posed.
For the special case of expansion in plane waves in infinite space
http://en.wikipedia.org/wiki/Canonical_quantization
The c-number mode transformations, e.g. to spherical harmonics should not affect the q-number 2nd quantized operators a & a* - provided there is no change in boundary conditions. As in
http://farside.ph.utexas.edu/teaching/jk1/lectures/node102.html
Clearly, however, the Unruh effect Bogoliubov transformation http://en.wikipedia.org/wiki/Bogoliubov_transformation from inertial to accelerating non-inertial frames is a transformation http://en.wikipedia.org/wiki/Quasiparticle is the kind of change you want - in general unitarily non-equivalent ground states have excitations that are not equivalent to each other.
Are there
"spherical wave" quanta, "plane wave quanta", and so on? And to what extent and in what sense are such
representations mathematically or physically equivalent?
For example, given a single quantum spherical harmonic quantum excitation of an EM field, what is the plane wave
quantum content of the field? Do the numbers of quanta agree in the two different representations?
General answer - agreement only when the ground states are unitarily equivalent.
[math-ph/0609065] Existence and non existence of a ground state ...
by A Panati - 2006 - Cited by 6 - Related articles
Sep 24, 2006 ... We prove that this model does not admit ground state in the Fock ... it does in another not unitarily equivalent coherent representation. ...
arxiv.org › math-ph - Cached
On the Absence of Non-Periodic Ground States for the ...
by T Matsui - 2005 - Cited by 1 - Related articles
Nov 17, 2004 ... non-translationally invariant ground states of XXZ models. Their construction of non ... unitarily equivalent to the following Hamiltonian: ...
www.springerlink.com/index/91WE7W0RBKF2UJU4.pdf
Symmetry breaking - Google Books Result
F. Strocchi - 2008 - Mathematics - 216 pages
Finally, if π is regular, so is πγ by the regularity of γ and therefore the two representations areunitarily equivalent by Von Neumann's uniqueness theorem ...
books.google.com/books?isbn=3540735925...
The dissipative quantum model of brain: how does memory localize ...
by E Alfinito - 2000 - Cited by 3 - Related articles
The density of the DWQ condensed in the ground state represents the .... Remarkably, it is found that the couple of Eqs. (1) is equivalent to the spherical Bessel ... The generator of such a non-unitary time evolution is found to be ...
linkinghub.elsevier.com/retrieve/pii/S0020025500000542
[PDF] Broken Symmetry and Spacetime
File Format: PDF/Adobe Acrobat - Quick View
by DJ Baker - 2010 - Related articles
definition of SSB – the existence of a non-invariant ground state – since ..... the conservative-about-states only if unitarily equivalent, they've also ...
philsci-archive.pitt.edu/8362/1/BrokenSymmetry.pdf