A. A. Logunov

Relativistic Invariance in Quantum Theory

By the general Poincare group we mean the set ρ of all transformations x → x’ of Minkowski space M that leaves the interval between any pair of points of M invariant, that is, transformations such that (s’ − y’)2 = (x − y)2 for all x,y ∈ M. Each transformation of ρ automatically turns out to be an inhomogeneous linear (that is, affine) transformation; more precisely, it has the form n n

Fields in an Indefinite Metric

x = Lambda x + a,

Lehmann-Symanzik-Zimmermann Formalism

n n(7.1) n nwhere a is a fixed vector of M and Λ is a transformation of the general Lorentz group. Hence it is clear that the general Poincare group can also be defined as the set of pairs (a, Λ), where a ∈ M, Λ ∈ L, with the multiplication law n n

Analyticity with respect to Momentum Transfer and Dispersion Relations

left( {{a_1},{Lambda _1}} right)left( {{a_2},{Lambda _2}} right) = left( {{a_1} + {Lambda _1}{a_2},{Lambda _1}{Lambda _2}} right).

Preliminaries on Functional Analysis

n n(7.2)

Haag-Ruelle Scattering Theory

We saw (Theorem 8.5) that by virtue of the spectrum condition, the Wightman functions w(xl,…, x n ) = W(ξ1,…,ξn−1) are boundary values of functions W(ζ1,…,ζn−1) of ζ1,…, ζn−1 that are holomorphic in the past tubes T n−1 − (8.41). It turns out that the combination of the spectrum property and Lorentz-invariance provides extra information about analyticity, in particular, it allows one to continue the functions analytically to a wider domain (called the extended tube). This theorem, due to Bargmann, Hall and Wightman, lies at the basis of the TCP theorem and the theorem on the connection between spin and statistics, which we shall become acquainted with in this chapter.