Mathematical Aspects of the Quantum Theory of Fields

Abstract

Part 111. Boson Field in Interaction with a Given Source Distribution Expectation Values of the Energy a d the Number of BosonsThe simplest specific problem that may be posed in field quantum theory is"to determine what happens in a field under the influence of a given source distribution. In the subsequent discussion of this problem we shall employ the Heisenberg picture, unless otherwise stated. As before, the operator which corresponds to the "field quantity" will be denoted by ," and considered as a function Z(s, t ) of position z and time t. In case no external forces are present this function was supposed to satisfy the differential equationin which V, = a/at, V, = grad, and p is the "rest mass", see Section 3. In case the field is under the influence of a given distribution of charges with the density j = j(z, t), the function E(s, 2) should satisfy the differential equationThe term j on the right side is of course to be understood as the identity operator multiplied by the function j (~, t).The solution of this differential equation must satisfy initial conditions.In the present section we shall assume that the operators ," and V,Z are prescribed at a finite time, specifically, at t = 0. Thus we require Z to satisfy conditionin which Zn(z) and no(z) are two-improper-operators which obey the commutation ruleIn Section 16 we shall impose initial conditions which characterize the behavior of the operator E(z, t) as t + -a.The source distribution j will be assumed independent of the time,As, 0 = .w in the present and in the two subsequent sections. In Section 16 we shall show that, the results obtained can be used to deal with a general, time dependent, source distribution.

A Combinatorial approach to the quantification of Lie algebras

“…In this section we apply the above general results to the infinite series A n , B n , C n , D n of nilpotent Lie algebras defined by the Serre relations (11) or, equivalently, (16). Let g be any such Lie algebra.…”

“…In line with the main idea of our approach, the skew primitive elements must play the same role in quantum enveloping algebras as the primitive elements do in the classical case. By the Friedrichs criteria [13], [16], [32], [34], [35], the primitive elements form the ground Lie algebra in the classical case. For this reason we consider the space spanned by the skew primitive elements and equipped with the quantum Lie operations as a quantum analogue of a Lie algebra.…”

Finite-particle representations and states of the canonical commutation relations

“…(See [8] for a reprint.) Among the representations which he studied, he found only the standard zero-interaction (Fock) representation to have a number operator.…”

On bogoliubov transformations. II. The general case

“…condition is well known. It dates back to semirigorous work of Friedrichs in the early fifties [7]. The first rigorous proof of an equivalent condition in the boson case is due to Shale [l I], in the fermion case to Shale and Stinespring [12].…”

“…In this form the transformation has been applied by many authors to solid-state physics, statistical mechanics, and relativistic quantum field theory, in particular the external field problem. Purely mathematical studies using this formalism can be found in the books by Friedrichs [7] and Berezin [3], and in [8]. One introduces new annihilation and creation operators where Tl and T, are complex-linear operators on Z+ , such the CAR (CCR), i.e., such that (2.24) that the ?…”