Volume 73A, number 4 PHYSICS LETTERS 1 October 1979 NONGRASSMANN QUANTIZATION OF THE DIRAC SYSTEM Piotr GARBACZEWSKI Institute of Theoretical Physics, University of Wroctaw, 50-205 Wroctaw, Poland Received 3 July 1979 We develop a path integral formalism which allows understanding of the Dirac equation in terms of the conventional canonical (phase space) variables: the internal, which are constrained and the external. 1. We denote ~ an 8-dimensional euclidean mani- = (1/\[2)(p~ hi, 1) , a~ = (l/\/~)(p~+ iir~).(5) fold parametrized by the canonical coordinates fp~., ~ ~ with/I being an euclidean label. In ternis ofa~, a~ an operator 1’~~ appears in the form The following Poisson bracket structure is imposed on — •~ * * F—itg~a0--a0a~), so that [F NJ = 0 with N= ~ a*a and {p~,p0}=0= ~~,n0} , (I) — p p p so that the antisymmetric second rank tensor Ak = i(a~a4 a4ak) = icllkalak (7) As a consequence the 0(4) Lie algebra commutation F pir—pir pv P V V /L ‘ ‘~‘ relations are immediately satisfied. can be used to define the two three-vectors: A~ = F~4 By defining L~ = —- ~, A~ = s~ + we find further- = ~i~4 ~4~i’ L~ = ~eiik F/k, i,j, k = I, 2, 3 which more: satisfy the 0(4) group Lie algebra commutation rela- 1 * * * * tionson~: s1 =~i(a2a3 --a3a2+a1a4 —a4a1), {L~, L1} = eiikLk , {L~,AJ}= ci/kAk , = ~i(aa1— a~a3 + aa4 s-i(aa —aa +aa~—a~,a). ~Aj,A/}=eiJkLk (3) 1. * * * * () = ~i(a1a4 — a4a1 -~ a2a3 + a3a2) - and set on ~J in a linear way, according to 1. * * * * = ~i(a2a4 -- a4a2 - a3a1 + a1a3) () ~3~i(aa4 —a~a3—a~a2+aa1), where [s~,NJ_ =0= [~~,N]_ =0= [N,s 2j s2 =~2 2. From now on, we will only use the natural sys- - d tern of units h = c = 1. Let us make a conventional an canonical quantization step, by introducing a [se, ~] — = 0 , [si, s 1] = i ijkSk Schrodinger representation of the canonical commuta- — tion relations L~i’ s11 — — ~ijk ~k ~ ~] = i~0 , [pu, P01 = = [np’ ~pJ — ‘ (5) 3. By virtue of the above commutation relations, we [an, a~J = , [a,~, a0]_ = 0 = [a~, a~] , can represent the 0(4) group Lie algebra in the N = 280