Comment on "Quantum entropy and special relativity" [by A. Peres, P. F. Scudo, and D. R. Terno, Phys. Rev. Lett. 88, 230402 (2002)]
Two subtleties of this paper are discussed.
quant-ph↗
arXiv subjects
Marek Czachor
Publications and source records attributed to Marek Czachor.
Quantum Aspects of Semantic Analysis and Symbolic Artificial Intelligence
Modern approaches to semanic analysis if reformulated as Hilbert-space problems reveal formal structures known from quantum mechanics. Similar situation is found in distributed representations of cognitive structures developed for the purposes of neural networks. We take a closer look at similarites and differences between the above two fields and quantum information theory.
quant-ph↗
Einstein-Podolsky-Rosen-Bohm experiment with relativistic massive particles
The EPRB experiment with massive partcles can be formulated if one defines spin in a relativistic way. Two versions are discussed: The one using the spin operator defined via the relativistic center-of-mass operator, and the one using the Pauli-Lubanski vector. Both are shown to lead to the SAME prediction for the EPRB experiment: The degree of violation of the Bell inequality DECREASES with growing velocity of the EPR pair of spin-1/2 particles. The phenomenon can be physically understood as a combined effect of the Lorentz contraction and the Moller shift of the relativistic center of mass. The effect is therefore stronger than standard relativistic phenomena such as the Lorentz contraction or time dilatation. The fact that the Bell inequality is in general less violated than in the nonrelativistic case will have to be taken into account in tests for eavesdropping if massive particles will be used for a key transfer.
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Quantum circa-rhythms
A class of solutions of nonlinear von Neumann equations exhibits rhythmic properties analogous to those found for nonlinear kinetic equations employed in phenomenological description of biochemical oscillations. As opposed to kinetic equations, typically solved numerically and involving Hill-type nonlinearities, our exact solutions are derived by means of soliton techniques and involve a simple quadratic nonlinearity.
physics.bio-ph↗
Reducible representations of CAR and CCR with possible applications to field quantization
Reducible representations of CAR and CCR are applied to second quantization of Dirac and Maxwell fields. The resulting field operators are indeed operators and not operator-valued distributions. Examples show that the formalism may lead to a finite quantum field theory.
hep-th↗
States of light via reducible quantization
Multi-photon and coherent states of light are formulated in terms of a reducible representation of canonical commutation relations. Standard properties of such states are recovered as certain limiting cases. The new formalism leads to field operators and not operator-valued distributions. The example of radiation fields produced by a classical current shows an automatic regularization of the infrared divergence.
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Relativistic BB84, relativistic errors, and how to correct them
The Bennett-Brassard cryptographic scheme (BB84) needs two bases, at least one of them linearly polarized. The problem is that linear polarization formulated in terms of helicities is not a relativistically covariant notion: State which is linearly polarized in one reference frame becomes depolarized in another one. We show that a relativistically moving receiver of information should define linear polarization with respect to projection of Pauli-Lubanski's vector in a principal null direction of the Lorentz transformation which defines the motion, and not with respect to the helicity basis. Such qubits do not depolarize.
quant-ph↗
Darboux Covariant Equations of von Neumann Type and their Generalizations
Lax pairs with operator valued coefficients, which are explicitly connected by means of an additional condition, are considered. This condition is proved to be covariant with respect to the Darboux transformation of a general form. Nonlinear equations arising from the compatibility condition of the Lax pairs in the matrix case include, in particular, Nahm equations, Volterra, Bogoyavlenskii and Toda lattices. The examples of another one-, two- and multi-field lattice equations are also presented.
nlin.SI↗
Reducible field quantization (II): Electrons
Quantization of free Dirac fields is formulated in terms of a reducible representation of CAR. Similarly to the bosonic case we arrive at field operators which are indeed operators and not operator valued distributions. Observables such as four-momentum and charge can be defined without any need of normal ordering.
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Darboux-integrable equations with non-Abelian nonlinearities
We introduce a new class of nonlinear equations admitting a representation in terms of Darboux-covariant compatibility conditions. Their special cases are, in particular, (i) the "general" von Neumann equation $i\dotρ=[H,f(ρ)]$, with $[f(ρ),ρ]=0$, (ii) its generalization involving certain functions $f(ρ)$ which are non-Abelian in the sense that $[f(ρ),ρ]\neq0$, and (iii) the Nahm equations.
nlin.SI↗
Correlation experiments in nonlinear quantum mechanics
We show how one can compute multiple-time multi-particle correlation functions in nonlinear quantum mechanics in a way which guarantees locality of the formalism.
quant-ph↗
Relativistic corrections to the Ekert test for eavesdropping
A degree of violation of the Bell inequality depends on momenta of massive particles with respect to a laboratory if spin plays a role af a "yes--no" observable. For ultra-relativistic particles the standard Ekert test has to take into accont this velocity dependent suppression of the degree of violation of the inequality. Otherwise Alice and Bob may "discover" a nonexisting eavesdropper.
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Non-canonical quantum optics (II): Poincaré covariant formalism and thermodynamic limit
The paper contains further development of the idea of field quantization introduced in M. Czachor, J. Phys. A: Math. Gen. {\bf 33} (2000) 8081-8103. The formalism is extended to the relativistic domain. The link to the standard theory is obtained via a thermodynamic limit. Unitary representations of the Poincaré group at the level of fields and states are explicitly given. Non-canonical multi-photon and coherent states are introduced. In the thermodynamic limit the statistics of photons in a coherent state is Poissonian. The $S$ matrix of radiation fields produced by a classical current is given by a non-canonical coherent-state displacement operator, a fact automatically eliminating the infrared catastrophe. Field operators are shown to be operators and not operator-valued distributions, and can be multiplied at the same point in configuration space. An exactly solvable example is used to compare predictions of the standard theory with those of non-canonical quantum optics, and explicitly shows the mechanism of automatic ultraviolet regularization occuring in the non-canonical theory. Similar conclusions are obtained in perturbation theory, where one finds the standard Feynman diagrams, but the Feynman rules are modified. A comparison with the Dicke-Hepp-Lieb model allows to identify the physical structure behind the non-canonical algebra as corresponding to an ensemble of indefinite-frequency oscillators with constant density $N/V$.
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On classical models of spin
The reason for recalling this old paper is the ongoing discussion on the attempts of circumventing certain assumptions leading to the Bell theorem (Hess-Philipp, Accardi). If I correctly understand the intentions of these Authors, the idea is to make use of the following logical loophole inherent in the proof of the Bell theorem: Probabilities of counterfactual events A and A' do not have to coincide with actually measured probabilities if measurements of A and A' disturb each other, or for any other fundamental reason cannot be performed simulaneously. It is generally believed that in the context of classical probability theory (i.e. realistic hidden variables) probabilities of counterfactual events can be identified with those of actually measured events. In the paper I give an explicit counterexample to this belief. The "first variation" on the Aerts model shows that counterfactual and actual problems formulated for the same classical system may be unrelated. In the model the first probability does not violate any classical inequality whereas the second does. Pecularity of the Bell inequality is that on the basis of an in principle unobservable probability one derives probabilities of jointly measurable random variables, the fact additionally obscuring the logical meaning of the construction. The existence of the loophole does not change the fact that I was not able to construct a local model violating the inequality with all the other loopholes eliminated.
quant-ph↗
Non-canonical quantization of electromagnetic fields and the meaning of $Z_3$
Non-canonical quantization is based on certain reducible representations of canonical commutation relations. Relativistic formalism for electromagnetic non-canonical quantum fields is introduced. Unitary representations of the Poincaré group at the level of fields and states are explicitly given. Multi-photon and coherent states are introduced. Statistics of photons in a coherent state is Poissonian if an appropriately defined thermodynamic limit is performed. Radiation fields having a correct $S$ matrix are constructed. The $S$ matrix is given by a non-canonical coherent-state displacement operator, a fact automatically eliminating the infrared catastrophe. This, together with earlier results on elimination of vacuum and ultraviolet infinities, suggests that non-canonical quantization leads to finite field theories. Renormalization constant $Z_3$ is found as a parameter related to wave functions of non-canonical vacua.
hep-th↗
A toy model of bosonic non-canonical quantum field
A harmonic oscillator is an indefinite-frequency one if the parameter $ω$ is replaced by an operator. An ensemble of $N$ such oscillators may be regarded as a toy model of a bosonic quantum field. All the possible frequencies associated with a given problem are present already in a single oscillator and $N$ can be finite. Due to the operator character of $ω$ the resulting algebra of creation-annihilation operators is non-canonical. In the limit of large $N$ one recovers perturbation theory formulas of the canonical quantum field theory but with form factors automatically built in. Vacuum energy of the ensemble is finite, a fact discussed in the context of the cosmological constant problem. Space of states is given by a vector bundle with Fock-type fibers. Interactions of the field with 2-level systems, including Rabi oscillations and spontaneous emission, are discussed in detail.
hep-th↗
Generalized thermostatistics and Kolmogorov-Nagumo averages
We introduce a generalized thermostatistics based on Kolmogorov-Nagumo averages and appropriately selected information measures. The formalism includes Tsallis non-extensive thermostatistics, but also extensive thermostatistics based on Renyi entropy. The Curie-Weiss model is discussed as an example.
cond-mat.stat-mech↗
Thermodynamic origin of universal fluctuations and two-power laws
We discuss universality of response functions in systems with excited degrees of freedom. We propose a unification of two existing phenomenologies, two-power law decay and deviation from power law due to non-extensivity. A universal curve is derived by maximizing entropy with a non-linear constraint. The same formalism can explain the universal fluctuation curves which have been discovered recently by Bramwell, Holdsworth, and Pinton.
cond-mat.stat-mech↗