Euclidean Fermi fields

Abstract

This paper is intended to clarify the relationships among different approaches to Euclidean Fermi fields. The previous work being considered can be found in: Osterwalder and Schrader [7], Gross [5], Schrader and Uhlenbrock [IS] and Osterwalder [6]. The construction of the Osterwalder-Schrader fields is carried out in the first part of this paper. The smeared fields are defined directly in a standard FockCook representation. Thus no estimates are required to establish the existence of the smeared fields, and the consistency and rotational covariance of the construction are immediate. The free Dirac field is then defined. In the standard constructions of the free Dirac field [7] a certain amount of dedication is required to check consistency, Lorentz covariance, and locality. In the construction presented here some effort has been made to make the labor involved in checking the details as insignificant as possible. Once the Schwinger functions for the free Dirac field are defined their representation in terms of the Osterwalder-Schrader fields is demonstrated. This result is proved in much the same manner that Gross [5] used to prove a more complicated result for the Yukawa model. The proof is included partly because it is naturally simpler than that in [5] and partly because the different definitions adopted here make translations from [S] and (71 difficult. Incident]? the complications in this proof are absent from [7] because they use “Wick’s theorem.” the proof of which would necessarily involve similar considerations. ‘I’he next part of the paper is devoted to a comparison of the unitary dilation analysis in [ 151 and the map W constructed in [7]. It is shown that W =:. T(w) where ZL’ is a complex linear map from the ‘one particle’ Euclidean Hilbert space to the ‘one particle’ Dirac space which compresses the unitary Euclidean time translations into the contraction semi-group associated with the Dirac equation. The term ‘one particle’ is qualified since the Hilbert spaces involved are each double the size of the usual one particle spaces. The connection between this doubling and Euclidean rotational covariance is especially transparent.