Charges and Symmetries in Quantum Theories without Locality

Abstract

We review some rigorous results (and include some new ones) on charges, symmetry breaking and related concepts in quantum theories without locality (micro‐causality), relevant examples of which are quantum lattice systems, (nonrelativistic) many‐body and lattice gauge theories. In particular, Goldstone's theorem and its generalizations (involving long‐range forces) and Swieca's theorem on the connection between the absence of charged states and the existence of a mass gap are discussed.

Locality in quantum systems

“…We do not require the states in the ground state sector to be degenerate with each other, simply the existence of a gap between that sector and the rest of the spectrum. This result is stronger than that in [3] as it is valid in arbitrary dimension; it is also stronger than other previous results [20] which either required a unique ground state or else assumed an ergodic property which is equivalent to requiring the vanishing of the matrix elements in the ground state sector in which case the decay or correlations becomes equivalent to clustering.…”

“…We now present an application to a non-relativistic Goldstone theorem. This theorem is perhaps not that surprising, but the results here (originally in [18]) are more general and simpler than previous nonrelativistic results [20]. Above, we have described the clustering of correlation functions, proving that the connected correlated function, AB − AP 0 B , of two operators A, B with support on sets X, Y is exponentially small in the distance dist(X, Y ).…”

Goldstone Boson Normal Coordinates

“…This is the content of the Goldstone Theorem (see section 4 and references [19,20] for more details). Therefore these systems do not have normal fluctuations as defined in this section, i.e.…”

“…For a more complete discussion of the phenomenon of spontaneous symmetry breaking, see [20]. The local Hamiltonians are determined by an interaction Φ H n = X⊆Λn Φ(X) and the infinite volume Hamiltonian H is defined such that for A ∈ A Λ 0 ,…”

A Goldstone theorem in thermal relativistic quantum field theory

“…In the present paper, we study the spontaneous symmetry breaking (ssb) of continuous (internal) symmetries in relativistic thermal quantum field theory, and prove a version of Goldstone's theorem (see, e.g., [5] for a review and references) -Theorem III.3 of Section III -which relates ssb to the asymptotic decay of (truncated) correlation functions for large space-like distances. In this respect the theorem follows the lines of [6] and [7], the latter having been proved to be an optimal version, generalizing the well-known Mermin-Wagner theorem of quantum statistical mechanics [8].…”