The structure of maps on the space of all quantum pure states that preserve a fixed quantum angle

Let $H$ be a Hilbert space and $P(H)$ be the projective space of all quantum pure states. Wigner’s theorem states that every bijection $\phi \colon P(H)\to P(H)$ that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator $U\colon H\to H$. Uhlhorn’s theorem generalizes this result for bijective maps $\phi $ that are only assumed to preserve the quantum angle $\frac{\pi }{2}$ (orthogonality) in both directions. Recently, two papers, written by Li–Plevnik–Šemrl and Gehér, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle $\alpha $ in both directions, provided that $0 < \alpha \leq \frac{\pi }{4}$ holds. In this paper we solve the remaining structural problem for quantum angles $\alpha $ that satisfy $\frac{\pi }{4} < \alpha < \frac{\pi }{2}$, hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner’s.

Единственность симметричной структуры в идеалах компактных операторов

Let Hbe a separable infinite-dimensional complex Hilbert space, let L(H) be the C∗-algebra of bounded linear operators acting in H, and let K(H) be the two-sided ideal of compact linear operators in L(H). Let (E,∥⋅∥E) be a symmetric sequence space, and let CE:={x∈K(H):{sn(x)}∞n=1∈E} be the proper two-sided ideal in L(H), where {sn(x)}∞n=1 are the singular values of a compact operator x. It is known that CE is a Banach symmetric ideal with respect to the norm ∥x∥CE=∥{sn(x)}∞n=1∥E. A symmetric ideal CE is said to have a unique symmetric structure if CE=CF, that is E=F, modulo norm equivalence, whenever (CE,∥⋅∥CE) is isomorphic to another symmetric ideal (CF,∥⋅∥CF). At the Kent international conference on Banach space theory and its applications (Kent, Ohio, August 1979), A. Pelczynsky posted the following problem: (P) Does every symmetric ideal have a unique symmetric structure? This problem has positive solution for Schatten ideals Cp, 1≤p<∞ (J. Arazy and J. Lindenstrauss, 1975). For arbitrary symmetric ideals problem (P) has not yet been solved. We consider a version of problem (P) replacing an isomorphism U:(CE,∥⋅∥CE)→(CF,∥⋅∥CF) by a positive linear surjective isometry. We show that if F is a strongly symmetric sequence space, then every positive linear surjective isometry U:(CE,∥⋅∥CE)→(CF,∥⋅∥CF) is of the form U(x)=u∗xu, x∈CE, where u∈L(H) is a unitary or antiunitary operator. Using this description of positive linear surjective isometries, it is established that existence of such an isometry U:CE→CF implies that (E,∥⋅∥E)=(F,∥⋅∥F).

Generalized Jordan Semitriple Maps on Hilbert Space Effect Algebras

Letℰ(H)be the Hilbert space effect algebra on a Hilbert spaceHwithdim⁡H≥3,α,βtwo positive numbers with2α+β≠1andΦ:ℰ(H)→ℰ(H)a bijective map. We show that ifΦ(AαBβAα)=Φ(A)αΦ(B)βΦ(A)αholds for allA,B∈ℰ(H), then there exists a unitary or an antiunitary operatorUonHsuch thatΦ(A)=UAU*for everyA∈ℰ(H).

REPRESENTATION OF SPACE INVERSION, TIME REVERSAL, AND PARTICLE CONJUGATION IN QUANTUM FIELD THEORY

The unitary operators of space inversion and particle conjugation and the unitary factor of the antiunitary operator of time reversal can each be written in the form eiΩ, where Ω is the direct sum of two terms, Q = Ω1θ1 + Ω2θ2, with Ω1, Ω2 Hermitean bilinear forms in the creation and annihilation operators of the boson or fermion field under consideration, and θ1 θ2 singular operators which separate the appropriate half spaces needed for the formulation of the symmetry operations. Explicit expressions are given for the generators Ω in case of a non-Hermitean boson field of spin 0, and in case of a four-component fermion field of spin [Formula: see text].