PCT, spin and statistics, and all that

by R. F. Streater and A. S. Wightman.

CONTENTS

Chapter I

• Super-selection Rules
• Symmetry Operations
• The Lorentz and Poincaré Groups
• Relativistic Transformation Laws of States

Chapter II

• Definition of Distribution
• Fourier Transforms
• Laplace transforms and Holomorphic Functions
• Tubes and Extended Tubes
• The Edge of the Wedge Theorem
• Hilbert Space

Chapter III

• Axioms for the Notions of Field and Field Theory
• Independence and Compatibility of the Axioms
• Properties of the Expectation Values
• The Reconstruction Theorem: Recovery of a Theory from its Expectation Values
• Symmetries in a Field Theory

Chapter IV

• The Global Nature of Local Commutativity
• Properties of the Polynomial Algebra of an Open Set
• The PCT Theorem
• Spin and Statistics
• Haag's Theorem and its Generalizations
• Equivalence Classes of Local Fields (Borchers Classes)

• Appendix: Some More Recent Developments in Quantum Field Theory (second edition, 1978 and later editions)

A more natural starting point than the Wightman axioms for the proof of the connection between spin and statistics, and PCT, is the algebraic approach of Haag. This avoids the assumption, made in Wightman theory, that unobservable fields either commute or anticommute at space-like separation. Unfortunately, there are counter examples to these theorems for fields with infintely many components, as shown by myself and I. Todorov. A natural mathematical axiom, wedge-duality, excludes these examples. See An algebraic spin and statistics theorem I, by Daniele Guido and Roberto Longo, 1994. This is a natural axiom, since (1) it leads to a unique action of the Poincaré group on the observables and (2) it leads to the split property, that an algebra of type I lies between the local algebra of a double cone and that of any properly larger cone. This means that there exist some states on the smaller algebra given by density operators of finite entropy.

For a sharp analysis of the consequences of positive energy and locality on the observability of particles see Hans P. Halvorson's thesis, by going to "search the archives", and then entering his name in the search box.

For some experimental results on PC and T violation, see the article by Ladd and Helliwell.

For a good account of antimatter, and you have a pdf-reader, see the article by Casadei

The correct connection between spin and statistics has been put to the test by De Mille, Budker, Derr and Deveney, in "Search for exchange-antisymmetric two-photon states"; this can be read here. It is found that any violation of bose statistics is less than 12 parts in 100 million.

A long list of papers discussing the symmetrization postulate of quantum mechanics is to be found here. Dr. Gillaspy's own work is listed here.

A. Pasquinucci and K. Roland have presented a proof that in the first-quantised version of string theory in Lorentzian space-time, PCT symmetry holds.

The fifth printing of the book, PCT, Spin and Statistics, and All That, with corrections, is now available from Princeton University Press, 2000. It is published in their series, Landmarks in Mathematics and Physics.