An algebraic approach to Wigner's unitary-antiunitary theorem

Generalization of Wigner's unitary-antiunitary theorem for indefinite inner product spaces

We present a generalization of Wigner's unitary-antiunitary theorem for pairs of ray transformations. As a particular case, we get a new Wigner-type theorem for non-Hermitian indefinite inner product spaces.

A generalization of Wigner's unitary-antiunitary theorem to Hilbert modules

Let H be a Hilbert $C^*$-module over a matrix algebra A. It is proved that any function $T:H\to H$ which preserves the absolute value of the (generalized) inner product is of the form $Tf=ϕ(f)Uf$ $(f\in H)$, where $ϕ$ is a phase-function and U is an A-linear isometry. The result gives a natural extension of Wigner's classical unitary-antiunitary theorem for Hilbert modules.

将反酉对称变换化为标准形

Transforming antiunitary symmetries to a normal form

本文通过酉共轭将反酉对称化为标准形，重现Wigner构造性证明并给出算法，进而证明有限维十倍路分类，为Altland-Zirnbauer对称类的数值标准化实现提供方法。

Antiunitary representations and modular theory

Antiunitary representations of Lie groups take values in the group of unitary and antiunitary operators on a Hilbert space H. In quantum physics, antiunitary operators implement time inversion or a PCT symmetry, and in the modular theory of operator algebras they arise as modular conjugations from cyclic separating vectors of von Neumann algebras. We survey some of the key concepts at the borderline between the theory of local observables (Quantum Field Theory (QFT) in the sense of Araki--Haag--Kastler) and modular theory of operator algebras from the perspective of antiunitary group represent…

Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Let $V$ be a standard subspace in the complex Hilbert space $H$ and $G$ be a finite dimensional Lie group of unitary and antiunitary operators on $H$ containing the modular group $(Δ_V^{it})_{t \in R}$ of $V$ and the corresponding modular conjugation~$J_V$. We study the semigroup \[ S_V = \{ g\in G \cap U(H) : gV \subseteq V\} \] and determine its Lie wedge $L(S_V) = \{ x \in L(G) : exp(R_+ x) \subseteq S_V\}$, i.e., the generators of its one-parameter subsemigroups in the Lie algebra $L(G)$ of~$G$. The semigroup $S_V$ is analyzed in terms of antiunitary representations and their analytic exte…

Transformations on the set of all n-dimensional subspaces of a Hilbert space preserving principal angles

Wigner's classical theorem on symmetry transformations plays a fundamental role in quantum mechanics. It can be formulated, for example, in the following way: Every bijective transformation on the set L of all 1-dimensional subspaces of a Hilbert space H which preserves the angle between the elements of L is induced by either a unitary or an antiunitary operator on H. The aim of this paper is to extend Wigner's result from the 1-dimensional case to the case of n-dimensional subspaces of H with n fixed.