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Journal of Mathematical Physics

Properties of states on Weyl algebra with variable multiplication law

Piotr Ługiewicz, Lech Jakóbczyk, Andrzej Frydryszak

Published Feb 1, 2021• 0 citations
Publisher

Abstract

We consider possible quantum effects for infinite systems implied by variations of the multiplication law in the algebra of observables. Using the algebraic formulation of quantum theory, we study the behavior of states ω under changes in the defining relations of the canonical commutation relations (CCR-algebra). These defining relations of the multiplication law depend explicitly on the symplectic form σ, which encodes commutation relations of canonical field operators. We consider the change in this form given by simple rescaling of σ by a positive parameter h. We analyze to what extent changes in h preserve the original state space (this gives restrictions on the admissible changes in the scaling parameter h) and which properties have original quantum states ω as states on the new algebra. We answer such questions for the quasi-free states. We show that any universally invariant state can be interpreted as a convex combination of Fock states with different values of constant h. The second important class of states we study is the KMS-state; here, the rescaling alters in a nontrivial way the relevant dynamics. We also show that it is possible to go beyond the limits restricting the changes in h, but then one has to restrict the CCR-algebra to a subalgebra.

DOI
10.1063/5.0032842

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We consider possible quantum effects for infinite systems implied by variations of the multiplication law in the algebra of observables. Using the algebraic formulation of quantum theory, we study the behavior of states ω under changes in the defining relations of the canonical commutation relations (CCR-algebra). These defining relations of the multiplication law depend explicitly on the symplectic form σ, which encodes commutation relations of canonical field operators. We consider the change in this form given by simple rescaling of σ by a positive parameter h. We analyze to what extent changes in h preserve the original state space (this gives restrictions on the admissible changes in the scaling parameter h) and which properties have original quantum states ω as states on the new algebra. We answer such questions for the quasi-free states. We show that any universally invariant state can be interpreted as a convex combination of Fock states with different values of constant h. The second important class of states we study is the KMS-state; here, the rescaling alters in a nontrivial way the relevant dynamics. We also show that it is possible to go beyond the limits restricting the changes in h, but then one has to restrict the CCR-algebra to a subalgebra.
We consider the possible quantum effect for infinite systems produced by variations of the Planck's constant. Using the algebraic formulation of quantum theory we study behaviour of states ωωω defined as positive, normalized functionals on the canonical commutation relations algebra (CCR-algebra) under the changes of the defining relations of the CCR. These defining relations of the multiplication in the CCR-algebra depend explicitly on the value of the Planck's constant. We analyse to what extend changes of the ℏ\hbarℏ preserve the original state space (this gives restrictions on the admissible changes of the Plank's constant) and what properties have original quantum states ωωω as states on the new algebra. We answer such questions for the quasi-free states. We show that any universally invariant state can be interpreted as a convex combination of Fock states with different values of Planck's constant. The second important class of states we study are the KMS-states, here the rescaling alters in a nontrivial way the relevant dynamics. We also show that it is possible to go beyond the limits restricting the changes of the ℏ\hbarℏ, but then one has to restrict the CCR-algebra to a subalgebra.
The standard C*-formalism of representations of commutations relations for systems of infinitely many degrees of freedom over a real numbers field is extended to the case of a p-adic numbers field. It is proved that any positive function on the Weyl algebra over a p-adic symplectic space is regular and restriction of the generating function of a state of the Weyl algebra on any nongenerate finite-dimensional subspace of p-adic symplectic space is continuous. Some nontrivial class of nonequivalent representations of commutations relations is described and analogies of Fock representations and representations with number operators are constructed. The criteria for the existence of free-field unitary quantization of symplectic transformations is given.
This paper studies a class of representations (called quadratic) of the canonical commutation relations over symplectic spaces of arbitrary dimension, which naturally generalizes coherent and symplectic (i.e. quasifree) representations and which has previously been heuristically employed in the special case of finite degrees of freedom in the physics literature. An explicit characterization of canonical quadratic transformations in terms of a 'standard form' is given, and it is shown that they can be exponentiated to give representations of the Weyl algebra. Necessary and sufficient conditions are presented for the unitary equivalence of these representations with the Fock representation. Possible applications to quantum optics and quantum field theory are briefly indicated.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.