Wigner's Theorem And Its Extension

We present Wigner's theorem on symmetry transformations and the generalizations of Wigner's theorem for all kinds of conditions. Wigner's theorem plays a fundamental role in the foundations of quantum mechanics. It has deep connections with the theory of projective space and justifies the use of unitary and anti-unitary group representations in physics.Wigner's theorem was first published in 1931. The fact that symmetry transformations which preserve absolute values of inner products led Wigner to a general investigation of such transformations. This led to the fundamental Wigner's theorem.The classical Wigner's theorem states that every symmetry operation of a quantum system is induced by a unitary or an anti-unitary transformation. There are several equivalent formulations of Wigner's theorem in our discussion. Now we introduce one of them.Theorem: Let H be a complex Hilbert space andφa bijective transformation on the set of all one-dimensional linear subspaces of H preserving the angle between every pair of such spaces. Then there exists a unitary or an anti-unitary operator U:H→H such thatφ(L) = {Ux : x∈L}, for every one-dimensional subspace L of H.Theorem: Let H be a Hilbert space. A transformation T maps unit rays in H onto unit rays such thatThen there exists a unitary or an antiunitary operator U : H→H, such that Ux∈T(_|x), (?)x∈T(_|x), when T(_|x) is defined. When Wigner put forward to his theorem, the theorem has been showed. However, in Wigner's book the proof of Wigner's theorem is not rigorous in the view of mathematics. Hence, the rigorous proofs of this theorem were given by many authors. These proofs were published during two periods.In the 1900's , some authors gave the rigorous mathematical proofs of the theorem along the basic mind of Wigner's verifications. Among these proofs, Bargmann's and Lemont and Mendelson's proofs are remarkable. However, it becomes difficult that we construct an analogue of inner product in the general space as the perfect one in Hilbert space, so in the 1990's, scholars gave many new further proofs for the fundamental theorem. By the new proofs, they hoped to generalize Wigner's theorem by a different point of view and to generalize it. In 1998, Molnar presented an operator algebra approach to Wigner's theorem using some classical results from ring theory. Molnar's work is quite effective. The common characteristic of the verifications on Wigner's theorem presented in the other papers is that they are based upon the characteristics and construction of Hilbert space. In Molnar's paper, he pushed the problem toa certain operator algebra over H and applied some well-known results from ring theory to obtain the desired conclusion. His argument provided not only a new point of view for Wigner's theorem but also a broader path for the generalization that we list in our paper are obtained by Molnar's argument.In the last part, we give some generalization of Wigner's theorem. There are six main sections in this part. We introduce the extension of Wigner's theorem in Hilbert spaces, in indefinite inner spaces, in the quaternionic Hilbert spaces, in Banach spaces and in the other spaces. The difference of these results is their background. In 1963, Uhlhorn gets the same conclusion as Wigner's theorem under his weaker assumptions.In his paper, Uhlhorn generalized Wigner's result by requiring only that T preserves the orthogonality between the one dimensional subspaces of H. This is a significant achievement in physics since Uhlhorn's transformation preserves only the logical structure of the quantum mechanical system in question while Wigner's transformation preserves its complete probabilistic structure.The work of the indefinite inner product spaces becomes more and more important in the discussion of quantum physical problems. This has raised the need to study Wigner's theorem in the indefinite setting. L.Bracci, G.Morehio, and F.Strocchi first generalized Wigner's result to indefinite metric spaces. P.M.Van den Broek also discussed this generalization. Later, Molnar presented a generalization of Wigner's theorem for pairs of ray transformations. As a particular case, he got a new Wigner-type theorem for non-Hermitian indefinite inner product spaces.The mathematical theory of quantum mechanics is customarily formulated in terms of complex separable Hilbert spaces. However, in some physical problems, we usually want to describe the states and transformations in quantum system by quaternion Hilbert space instead of common Hilbert space. Hence we present an analogue of Wigner's theorem in quaternion Hilbert space. We also have some Wigner-type results in quaternion indefinite inner product space. The Wigner-type theorem in Banach spaces that would be introduced ,also can be generalized to more general spaces which is vector space about quaternion number field. In our paper we do not list these results, because they are far away from our topics.Make use of the algebra proof of Wigner's theorem, Molnar gets a series of generalizations of Wigner's theorem. One hand, we can generalized the Wigner's theorem and its Uhlhorn's extension for the setting of Banach space, we introduced two articles presented by Molnar on this aspect, on On the other hand, we can get the generalization of Wigner's theorem to Hilbert modules and to Hilber C~* modules.