Related papers: Enlargement of symmetry groups in physics: a pract…

A symmetry in quantum mechanics is described by the projective representations of a Lie symmetry group that transforms between physical quantum states such that the square of the modulus of the states is invariant. The Heisenberg…

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary…

We present a pedagogical review of projective representations of finite groups and their physical applications in quantum many-body systems. Some of our physical results are new. We begin with a self-contained introduction to projective…

Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary --- or, in general, projective unitary --- representations implement the action of an abstract symmetry group on physical states…

We give the full representation theory of the gravitational extended corner symmetry group in two-dimensions. This includes projective representations, which correspond to representations of the quantum corner symmetry group. We find that…

The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is…

Hilbert--Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. Here we study…

This is mainly a lecture note taken by myself following Weinberg's book, but also contains some corrections to the abuse of mathematical treatment. This article discusses projective unitary representations of Poincare group on the single…

There is a unique finite group that lies inside the 2-dimensional unitary group but not in the special unitary group, and maps by the symmetric square to an irreducible subgroup of the 3-dimensional real special orthogonal group. In an…

Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…

We study projective unitary (co)representations of compact quantum groups and the associated second cohomology theory. We introduce left/right/bi/strongly projective corepresentations and study them in details. In particular, we prove that…

In quantum mechanics, symmetry groups can be realized by projective, as well as by ordinary unitary, representations. For the permutation symmetry relevant to quantum statistics of N indistinguishable particles, the simplest properly…

We develop the theory of Wigner representations for general probabilistic theories (GPTs), a large class of operational theories that include both classical and quantum theory. The Wigner representations that we introduce are a natural way…

In this paper we present a new procedure to obtain unitary and irreducible representations of Lie groups starting from the cotangent bundle of the group (the cotangent group). We discuss some applications of the construction in…

We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle. We also…

A new representation of Quantum Gravity is developed. This formulation is based on an extension of the group of loops. The enlarged group, that we call the Extended Loop Group, behaves locally as an infinite dimensional Lie group. Quantum…

We examine the notion of symmetry in quantum field theory from a fundamental representation theoretic point of view. This leads us to a generalization expressed in terms of quantum groups and braided categories. It also unifies the…

Quantum symmetries that leave invariant physical transition probabilities are described by projective representations of Lie groups. The mathematical theory of projected representations for topologically connected Lie groups is reviewed and…

Recent advances in classical machine learning have shown that creating models with inductive biases encoding the symmetries of a problem can greatly improve performance. Importation of these ideas, combined with an existing rich body of…

We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the…