Position Operators for Extended Objects in Quantum Field Theory

Abstract

The operator structure of the collective coordinate associated with extended objects in quantum field theory is discussed in the context of renormalized perturbation theory. The analysis of a quantum field theory with extended objects must take into account the presence of these position operators. Two methods have been proposed so far. One is the collective coordinate method,4) in which the Heisenberg field operator and its canonical conjugate are decomposed into a new set of Heisenberg operators, namely the collective coordinate operator X( t) and field operator x(x, t) and their conjugates p( t) and Jr(x, t), respectively. This decomposition is accompanied by certain constraints which define X( t) and p( t). It is required that X( t)--> X( t) + a induces the space translation of the Heisenberg operators and that X( t) and x(x, t) are independent as Heisenberg operators. The other method expresses the Heisenberg operators in terms of the physical operators or asymptotic fields which construct the physical Hilbert space. 3 ) This expression is called the dynamical map. In this method it has been shown that the set of physical operators consists of two mutually commuting sets (q, p) and ((l, (l t), where q is the quantum mechanical position-operator (quantum coordinate) and p is its canonical con­ jugate while (l and (l t stand for the annihilation and creation operators of particle-like modes respectively.3),5) Thus, the Hilbert space is found to be a