Lee Model and Quantisation of Non Linear Field Equations

Abstract

In a model for the theory of elementary particles an unconventional method of quantisation had been applied which makes use of an indefinite metric in Hilbert space. This method is analysed with the help of the Lee model. The constants of the Lee model can be adjusted in such a manner that, after the renormalisation has been carried out, the mathematical structure is in many ways analogous to that of the method of quantisation mentioned above (§ 2). The total Hilbert space can be divided into two parts, the one of which has a positive metric and comprises the physical states of the system, while the other one contains ‘dipole-ghosts’ and is needed for the commutation relations and the convergence of the theory. The S-matrix is unitary for the states of Hilbert space I, since the states of Hilbert space II are separated from those of Hilbert space I in a manner similar to the case of “non-combining term systems” (§§3 and 5). Some general results concerning the Hilbert space of indefinite metric are discussed in § 4. The questions of causality are briefly raised in § 6. The results seem to show that a Hilbert space with indefinite metric can be used for the description of real phenomena.

Essential parts of this paper are the result of an extensive correspondence with W. Pauli to whom the author is indebted for many valuable suggestions. R. Haag has contributed to the sections 2 and 4.