Progress of Theoretical Physics, Vol. 69, No.5, May 1983 On the Inter'action of Quantum Spinor Fields with Extended Objects H. MATSUMOTO, G. SEMEN OFF and H. UMEZAWA Theoretical Physics Institute, The University of Alberta Edmonton, Alberta T6G 2J1 (Received November 17, 1982) 1631 In past studies of scalar quantum field theory with an extended object it has been shown that the space· time c·numbers (x, t) and the quantum mechanical operators associated with the extended object appear only through a particular combination called the generalized coor- dinates. In this paper we study the structure of a spinor field interacting with an extended object. It will be shown that the spinor field becomes a product of a kind of spinor operator and a scalar-like field. This spin or operator turns out to be the Lorentz boost operator with the quantum velocity Q = pi H. The scalar-like field describes the quantum particles in reference to a coordinate system which is attached to the extended object. Thus, the behaviour of these quantum particles attached to the extended object does not manifest Lorentz covariance, although the whole theory is relativistically invariant. This knowledge of the structure of the spinor field if; is useful when we compute the effects of quantum mechanical fluctuations of extended objects. The result will be generalized to quantum fields of arbitrary spin. § 1. Introduction Through recent studies of quantum field theory with extended objects,l) it has become known that the Hilbert space describing a system with extended objects is an extended space which is constructed by two mutually commuting sets of physical operators; the creation and annihilation operators (a, at) of the quantum particles and the quantum mechanical operators ( q, p ).2)_5) The quantum coordinate q and its canonical momentum p appear to describe the quantum motion of the extended object. In this new situation, a key question is to ask how the behaviour of quantum particles is modified by the presence of the new quantum mechanical operators. In a series of papers,2)-5) we studied this question and found that the total momentum P obtained from a Lagrangian becomes p (when it is expressed in terms of the physical operators) and therefore commutes with (a, at). Another significant feature 3 )-5) is that q can be chosen to be the Newton-Wigner position operator. 6 ) In the case of a scalar field theory, the c-number variables (x, t) appear through a special combination with (q, p); this combination is called the generalized coordinate (X, T).3)-5),7) This is a result of the Poincare covariance of the theory. The change (x, t)-+(X, T) corresponds to the coordinate transformation caused by the quantum motion of the extended object. Then a scalar Heisenberg field operator ¢(x, t) is expres- sed as Downloaded from https://academic.oup.com/ptp/article-abstract/69/5/1631/1860619 by guest on 22 May 2020