Date of Award
Spring 2025
Project Type
Dissertation
Program or Major
Mathematics
Degree Name
Doctor of Philosophy
First Advisor
Junhao Shen
Second Advisor
Donald Hadwin
Third Advisor
Rita Hibschweiler
Abstract
Wigner's Theorem, an influential result in quantum mechanics established by Eugene Wigner in 1931, states that every symmetry transformation on a ray space is induced by either a unitary or anti-unitary transformation. In this work, we generalize Wigner's Theorem in two directions: first, to maps between different Hilbert spaces, and second, to maps between factors of type II. In both cases, we focus on weakening the assumptions on the map, specifically by utilizing spectrum shrinking conditions. For the first generalization, let $\mathcal{H}_1$ and $\mathcal{H}_2$ be Hilbert spaces, and let $k$ be a positive integer such that $2\leq k < \frac{1}{2} \dim(\mathcal{H}_1)$. We show that if an injective map $\psi: \mathcal{P}_k(\mathcal{H}_1) \to \mathcal{P}_k(\mathcal{H}_2)$ satisfies two spectrum shrinking conditions, then it is induced by a linear or conjugate-linear isometry. For the second generalization, let $\mathcal{M}$ be a type II$_1$ or type II$_\infty$ factor, let $\tau_\mathcal{M}$ be a faithful normal semi-finite tracial weight of $\mathcal M$, and let $0
Recommended Citation
Allen, Meaghan, "Wigner's Theorem and Spectrum Shrinking Maps" (2025). Doctoral Dissertations. 2894.
https://scholars.unh.edu/dissertation/2894