Related papers: Wigner's theorem on Grassmann spaces

As established by Sol\`er, Quantum Theories may be formulated in real, complex or quaternionic Hilbert spaces only. St\"uckelberg provided physical reasons for ruling out real Hilbert spaces relying on Heisenberg principle. Focusing on this…

We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for…

We consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar Cartesian case are systematically…

This paper deals with the Newton--Wigner position observable for Poincar\'e-invariant classical systems. We prove an existence and uniqueness theorem for elementary systems that parallels the well-known Newton--Wigner theorem in the quantum…

We generalise Wigner's theorem to its most general form possible for B(h) in the sense of completely characterising those vector state transformations of B(h) that appear as restrictions of duals of linear operators on B(h). We then use…

We establish a noncommutative generalisation of the Borel-Weil theorem for the Heckenberger-Kolb calculi of the quantum Grassmannians. The result is formulated in the framework of quantum principal bundles and noncommutative complex…

We study a generalization of the Wigner function to arbitrary tuples of hermitian operators, which is a distribution uniquely characterized by the property that the marginals for all linear combinations of the given operators agree with the…

Let $H$ be an infinite-dimensional complex Hilbert space and let ${\mathcal G}_{\infty}(H)$ be the set of all closed subspaces of $H$ whose dimension and codimension both are infinite. We investigate (not necessarily surjective)…

A simple proof is provided to show that any bounded normal operator on a real Hilbert space is orthogonally equivalent to its transpose(adjoint). A structure theorem for invertible skew-symmetric operators, which is analogous to the finite…

The quantum measurement problem can be regarded as the tension between the two alternative dynamics prescribed by quantum mechanics: the unitary evolution of the wave function and the state-update rule (or "collapse") at the instant a…

In this paper we generalize the concept of Wigner function in the case of quantum mechanics with a minimum length scale arising due to the application of a generalized uncertainty principle (GUP). We present the phase space formulation of…

Let $V$ be an $n$-dimensional left vector space over a division ring $R$ and $n\ge 3$. Denote by ${\mathcal G}_{k}$ the Grassmann space of $k$-dimensional subspaces of $V$ and put ${\mathfrak G}_{k}$ for the set of all pairs $(S,U)\in…

We study the behaviour of the geometric phase under isometries of the ray space. This leads to a better understanding of a theorem first proved by Wigner: isometries of the ray space can always be realised as projections of unitary or…

We give a complete characterization of the (non)classicality of all stabilizer subtheories. First, we prove that there is a unique nonnegative and diagram-preserving quasiprobability representation of the stabilizer subtheory in all odd…

Gleason's theorem [A. Gleason, J. Math. Mech., \textbf{6}, 885 (1957)] is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it…

The classical Mazur-Ulam theorem establishes that every surjective isometry between normed real vector spaces is an affine transformation. In various applied mathematical settings, however, one encounters maps that preserve distances not…

Using a key observation due to Thiemann, a generalized Wick transform is introduced to map the constraint functionals of Riemannian general relativity to those of the Lorentzian theory, including matter sources. This opens up a new avenue…

We prove the theorem stated in the title. This answers a question of John Cobb (1994). We also consider the case of the Hilbert space $\ell _2$.

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…

Wigner's irreducible positive energy representations of the Poincare group are often used to give additional justifications for the Lagrangian quantization formalism of standard QFT. Here we study another more recent aspect. We explain in…