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Mathematical Description of Infinitely Extended Quantum Systems

Franco Strocchi

From the discussion of the previous Chapter, it appears that the description of infinite systems looks much more difficult than in the finite-dimensional case, above all because of the existence of (too) many possible representations of the algebra of canonical variables.

Theoretical and mathematical physicsJan 1, 20210 citations

Some remarks on classical representations of quantum mechanics

Werner Stulpe

Foundations of PhysicsJul 1, 19945 citations

Choices of Quantum Representations of Classical Dynamical Systems

C. Dewitt-Morette

Jan 1, 19880 citations

Quantum mechanics, functional integrals

Lowell S. Brown

Quantum field theories are quantum mechanical systems with an infinite number of degrees of freedom. We start this book with a review of quantum mechanics in a form that is particularly suited for the field theory generalization. This form utilizes the Heisenberg picture and the functional integral representation of probability amplitudes — the transformation functions of Dirac. Some time will be spent developing and discussing the functional or path integral representation of quantum mechanics since this will enable us to learn a new formalism in a simple and familiar physical context — we won't have to learn too many new things all at once. The functional formulation will be illustrated by applying it at some length to the harmonic oscillator which is not only the simplest (and exactly soluble) non-trivial quantum system but which is also the dynamics of a single mode of a free field theory. The purpose of this review of quantum mechanics is to introduce in the simplest form much of the notation and methods used later in the book. Throughout this chapter we will explicitly work only with systems that have a single degree of freedom (one coordinate variable). The extension to systems with several degrees of freedom is straightforward.Quantum field theories are quantum mechanical systems with an infinite number of degrees of freedom. We start this book with a review of quantum mechanics in a form that is particularly suited for the field theory generalization. This form utilizes the Heisenberg picture and the functional integral representation of probability amplitudes — the transformation functions of Dirac. Some time will be spent developing and discussing the functional or path integral representation of quantum mechanics since this will enable us to learn a new formalism in a simple and familiar physical context — we won't have to learn too many new things all at once. The functional formulation will be illustrated by applying it at some length to the harmonic oscillator which is not only the simplest (and exactly soluble) non-trivial quantum system but which is also the dynamics of a single mode of a free field theory. The purpose of this review of quantum mechanics is to introduce in the simplest form much of the notation and methods used later in the book. Throughout this chapter we will explicitly work only with systems that have a single degree of freedom (one coordinate variable). The extension to systems with several degrees of freedom is straightforward.

Cambridge University Press eBooksSep 10, 19920 citations

Non-standard quantum algebras and infinite-dimensional PT-symmetric systems

Ángel Ballesteros, Romina Ramírez, M. Reboiro

Abstract In this work, we introduce a PT -symmetric infinite-dimensional representation of the U z ( s l ( 2 , R ) ) Hopf algebra, and we analyse a multiparametric family of Hamiltonians constructed from such representation of the generators of this non-standard quantum algebra. It is shown that all these Hamiltonians can be mapped to equivalent systems endowed with a position-dependent mass. From the latter presentation, it is shown how appropriate point canonical transformations can be further defined in order to transform them into Hamiltonians with constant-mass over suitable domains. By following this approach, the bound-state spectrum and the corresponding eigenfunctions of the initial PT -symmetric Hamiltonians can be determined. It is worth stressing that a relevant feature of some of the new U z ( s l ( 2 , R ) ) systems here presented is found to be their connection with double-well and Pöschl-Teller potentials. In fact, as an application we present a particular Hamiltonian that can be expressed as an effective double-well trigonometric potential, which is commonly used to model several relevant systems in molecular physics.Abstract In this work, we introduce a PT -symmetric infinite-dimensional representation of the U z ( s l ( 2 , R ) ) Hopf algebra, and we analyse a multiparametric family of Hamiltonians constructed from such representation of the generators of this non-standard quantum algebra. It is shown that all these Hamiltonians can be mapped to equivalent systems endowed with a position-dependent mass. From the latter presentation, it is shown how appropriate point canonical transformations can be further defined in order to transform them into Hamiltonians with constant-mass over suitable domains. By following this approach, the bound-state spectrum and the corresponding eigenfunctions of the initial PT -symmetric Hamiltonians can be determined. It is worth stressing that a relevant feature of some of the new U z ( s l ( 2 , R ) ) systems here presented is found to be their connection with double-well and Pöschl-Teller potentials. In fact, as an application we present a particular Hamiltonian that can be expressed as an effective double-well trigonometric potential, which is commonly used to model several relevant systems in molecular physics.

arXiv:2504.21833Journal of Physics A Mathematical and TheoreticalOct 24, 20252 citations

Non linear canonical transformations and their representations in quantum mechanics

M. Moshińsky, Pier A. Mello

Springer eBooksApr 6, 20086 citations

Physically Relevant Representations

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Physically Relevant Representations

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