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Basic Sciences Journal of Textile Universities

Some remarks on Wigner′s theorem

Cao Huai-xin

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

The relations between several different forms of Wigner′s theorem are investigated,the descriptions of the theorem in terms of physics and geometry are given.By using operator theory and operator algebra,the equivalence between these different forms of propositions is proved.The results show that if the surjection T:R1(H)→R1(K) preserves the inner product between unit rays,the surjection S:R(H)→R(K) preserves the inner product between rays,the surjection Φ:P1(H)→P1(K) preserves the trace of the product of rank-one projections,and the surjection W:H→K preserves the inner product between vectors,then there exists unit or anti-unitary operators U:H→K,such that Uy∈Tx,r(Uy)=Sx,Φ(Px)=UPx U* and W(x)=φ(x)U(x),in which φ:H→C satisfies |φ(x)|=1.

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Symmetry in quantum mechanics

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The relations between several different forms of Wigner′s theorem are investigated,the descriptions of the theorem in terms of physics and geometry are given.By using operator theory and operator algebra,the equivalence between these different forms of propositions is proved.The results show that if the surjection T:R1(H)→R1(K) preserves the inner product between unit rays,the surjection S:R(H)→R(K) preserves the inner product between rays,the surjection Φ:P1(H)→P1(K) preserves the trace of the product of rank-one projections,and the surjection W:H→K preserves the inner product between vectors,then there exists unit or anti-unitary operators U:H→K,such that Uy∈Tx,r(Uy)=Sx,Φ(Px)=UPx U* and W(x)=φ(x)U(x),in which φ:H→C satisfies |φ(x)|=1.
Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.
Abstract It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces A of a Hilbert space H H{\mathcal H}H correspond to direct product decompositions H ≃ A × A ⊥ H≃A×A⊥\mathcal{H}\simeq A\times A^\perpH≃A×A⊥ of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X, the automorphism group of the orthomodular poset Fact(X) of direct product decompositions of X is isomorphic to the permutation group of X. The structure Fact(X) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact(X) are determined in proving our version of Wigner’s theorem, namely that Fact(X) is atomistic in a very strong way.
Symmetry in quantum mechanicsRoughly speaking, a symmetry of some mathematical object is an invertible transformation that leaves all relevant structure as it is.Thus a symmetry of a set is just a bijection (as sets have no further structure, whence invertibility is the only demand on a symmetry), a symmetry of a topological space is a homeomorphism, a symmetry of a Banach space is a linear isometric isomorphism, and, crucially important for this chapter, a symmetry of a Hilbert space H is a unitary operator, i.e., a linear map u : H → H satisfying one and hence all of the following equivalent conditions:• u is invertible and preserves the inner product, i.e., uϕ, uψ = ϕ, ψ (ϕ, ψ ∈ H).The discussion of symmetries in quantum physics is based on the above idea, but the mathematically obvious choices need not be the physically relevant ones.Even in elementary quantum mechanics, where A = B(H), i.e., the C*-algebra of all bounded operators on some Hilbert space H, the concept of a symmetry is already diverse.The main structures whose symmetries we shall study in this chapter are:1.The normal pure state space P 1 (H), i.e., the set of one-dimensional projections on H, with transition probability τ : P 1 (H) × P 1 (H) → [0, 1] defined by (2.44).2. The normal state space D(H), i.e. the convex set of density operators ρ on H. The self-adjoint operators B(H)sa on H, seen as a Jordan algebra (see below).4. The effects E (H) = [0, 1] B(H) , seen as a convex partially ordered set (poset). 5.The projections P(H) on H, seen as an orthocomplemented lattice.6.The unital commutative C*-subalgebras C (B(H)) of B(H), seen as a poset.Each of these structures comes with its own notion of a symmetry, but the main point of this chapter will be to show these notions are equivalent, corresponding in all cases to either unitary or-surprisingly-anti-unitary operators, both merely defined up to a phase.The latter subtlety will open the world of projective unitary group representation to quantum mechanics (without which the existence of spin-1 2 particles such as electrons, and therewith also of ourselves, would be impossible).