Symmetry Groups in Quantum Mechanics and the Theorem of Wigner on the Symmetry Transformations

Cassinelli, Gianni; Vito, Ernesto De; Lahti, Pekka; Levrero, Alberto

doi:10.1142/s0129055x97000324

Cited by 50 publications

(69 citation statements)

References 6 publications

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“…The symmetry property allows us to apply Wigner's theorem, and in doing so the following implication is obtained [5,6,25,26].…”

Section: (C) Symmetry Transformationsmentioning

confidence: 99%

“…Different equivalent formulations of Wigner's theorem [6,30] have been demonstrated in the literature. The following version can be applied for the the mapping S of propostion 3.1.…”

Section: Corollary 32 From Proposition 31 Immediately Follows Thatmentioning

confidence: 99%

“…Put simply, these approaches proceed according to the following scheme. First, they enforce the condition that Galilei's group G (or Poincaré's group P, for a relativistic theory) is a group of symmetry transformations for an isolated particle, so that Wigner's theorem [5,6] on the representation of symmetries implies the existence of a projective representation of that group. Then, since the position observables (Q 1 , Q 2 , Q 3 ) = Q determine the existence of an imprimitivity system with respect to the restriction of the projective representation to the Euclidean group, Mackey's imprimitivity theory [7,8] can be applied to derive the explicit Quantum Theory of a free particle [9].…”

Section: Introductionmentioning

confidence: 99%

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Group theoretical derivation of the minimal coupling principle

Nisticò

2017

Proc. R. Soc. A.

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The group theoretical methods worked out by Bargmann, Mackey and Wigner, which deductively establish the Quantum Theory of a free particle for which Galileian transformations form a symmetry group, are extended to the case of an interacting particle. In doing so, the obstacles caused by loss of symmetry are overcome. In this approach, specific forms of the wave equation of an interacting particle, including the equation derived from the minimal coupling principle, are implied by particular firstorder invariance properties that characterize the interaction with respect to specific subgroups of Galileian transformations; moreover, the possibility of yet unknown forms of the wave equation is left open.

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“…The symmetry property allows us to apply Wigner's theorem, and in doing so the following implication is obtained [5,6,25,26].…”

Section: (C) Symmetry Transformationsmentioning

confidence: 99%

“…Different equivalent formulations of Wigner's theorem [6,30] have been demonstrated in the literature. The following version can be applied for the the mapping S of propostion 3.1.…”

Section: Corollary 32 From Proposition 31 Immediately Follows Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Group theoretical derivation of the minimal coupling principle

Nisticò

2017

Proc. R. Soc. A.

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“…The first concerns representations of symmetry groups and algebras. In the Hilbert space formulation, elements of a symmetry group or more generally, of any group of automorphisms of the set of quantum states, are represented by unitary or antiunitary operators that define ray representations of the group as a whole, in accordance with Wigner's celebrated theorem [15,16,17]. In contrast, in the phase-space formulation, elements of a symmetry group are represented by real unitary operators that define a true representation.…”

Section: And Dirac: So If One Asks What Is the Main Feature Of Quantumentioning

confidence: 99%

Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for wigner functions

Bracken

2006

Reports on Mathematical Physics

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Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.1

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“…These sets of operators, that is, the set of all states and the set of all effects, play important role in the mathematical description of quantum mechanics (see, for example, [3,7] and the references therein as well). Just as with any algebraic structure, the study of the automorphisms of these sets when equipped with certain algebraic structures is of considerable importance.…”

Section: S = {T ∈ B(h) : T ≥ 0 Tr T = 1} Where Tr Denotes the Usualmentioning

confidence: 99%

Local automorphisms of the sets of states and effects on a Hilbert space

Barczy

Tóth

2001

Reports on Mathematical Physics

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Symmetry Groups in Quantum Mechanics and the Theorem of Wigner on the Symmetry Transformations

Abstract: Various mathematical formulations of the symmetry group in quantum mechanics are investigated and shown to be mutually equivalent. A new proof of the theorem of Wigner on the symmetry transformations is worked out.

Cited by 50 publications

References 6 publications

Group theoretical derivation of the minimal coupling principle

Group theoretical derivation of the minimal coupling principle

Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for wigner functions

Local automorphisms of the sets of states and effects on a Hilbert space

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