In 1894 Pierre Curie announced what has come to be known as Curie's Principle: the asymmetry of effects must be found in their causes. In the same publication Curie discussed a key feature of what later came to be known as spontaneous symmetry breaking: the phenomena generally do not exhibit the symmetries of the laws that govern them. Philosophers have long been interested in the meaning and status of Curie's Principle. Only comparatively recently have they begun to delve into the mysteries of spontaneous symmetry breaking. The present paper aims to advance the discussion of both of these twin topics by tracing their interaction in classical physics, ordinary quantum mechanics and quantum field theory. The features of spontaneous symmetry that are peculiar to quantum field theory have received scant attention in the philosophical literature. These features are highlighted here, along with an explanation of why Curie's Principle, though valid in quantum field theory, is nearly vacuous in that context.

Curie's Principle and spontaneous symmetry breaking

Hilbert space and the C*-algebra approach are outlined. There follows sections devoted to Wightman's axiomatic treatment with recent generalisations. The review surveys the work covered in the period 1954-74.

https://iopscience.iop.org/article/10.1088/0034-4885/38/7/001/pdf

Outline of axiomatic relativistic quantum field theory

Geometrical and topological aspects of chiral anomalies in gauge theories are reviewed. Geometrical and topological concepts and results for chiral anomalies in gauge theories are considered, including differential forms, Lie groups, homotopy, homology, cohomology, Riemannian manifolds, fibre bundles, characteristic classes, index theorems and spectral flow. Gauge theories and their formulation in terms of differential forms and fibre bundles are described. The quantisation of gauge theories is performed using path integrals, and the orbit space, BRST symmetries and ϑ vacuum are discussed. Gauge theories with fermions are formulated, including realistic models of strong and weak interactions. Chiral anomalies and related issues such as the existence of Schwinger terms, their origins in terms of differential forms, cohomology, the orbit space, BRST identities, Hamiltonian systems and relations to index theorems are analysed. Constraints on models for particle physics from chiral anomalies and theo...

Geometry and topology of chiral anomalies in gauge theories

This lecture intended for graduate students is devoted to the concept of chiral symmetry in field theory and its spontaneous lowering (breaking) to ordinary isosymmetry. The aim is to state the conditions under which chiral symmetry can be realized in particle physics and to formulate the signatures announcing the different modes of realization.

Elements of chiral symmetry

Some physicists and philosophers argue that unitarily inequivalent representations (UIRs) in quantum field theory (QFT) are mathematical surplus structure. Support for that view, sometimes ...

The Limits of Physical Equivalence in Algebraic Quantum Field Theory

Approximate or «broken» symmetries are discussed in the framework of axiomatic field theory. A previous result is generalized to space-time symmetries:i.e., that a nonexact symmetry cannot be implemented by a time-dependent unitary operator. It is shown that this has nothing to do with the well-known difficulties of defining a charge, which exist also for exact symmetries. A further results is that if a symmetry is not exact, there is at least one observable whose VEV is maximally noninvariant.

L. E. Picasso

Papers

Broken symmetries in quantum field theory.

The theorem of carruthers and the decay of positronium