[Chapter] Introduction to Some Aspects of the Relativistic by Arthur S. Wightman

By Arthur S. Wightman

Cargèse Lectures in Theoretical Physics

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0 for all p and q . 60) where again both signs of p 0 are allowed. The usual transformation law U(a, A ) tf;(x ) U (a, A )- 1 S(A- 1 ) tf;(A(A ) x + a) for a spinor field is valid for this if;. '0 in contrast to the situation in a theory satisfying the spectral condition. Bilinear quantities such as = J d 2 xf(x) tf;+(x) iy/J. tf;(x) make sense without Wick ordering in this non-physical world, as one easily sees from their definition in ( P v .

12 x ) ( l + ;2 i. {y + n m-V2-x2}) = = -4 l (4. 10) so the two-point distribution has a singularity like RELATIVISTIC DYNAMICS OF QUANTIZED FIELDS 201 on the light cone x2 = 0. 1 1 ) oo (This is what is usually meant by the statement that the field strength renormalization is infinite. ) The proof is simply that integrals of the form J dp(a)H&1>(vamz) 00 0 J dp(a) = 0 as does H&1>(vmz). with 00 < 0 oo will have the same singularity at z To see how solutions of coupled field equations can be constructed from this exponential, consider the two-component field.

Pn ) � ( - l )i+1 J dQ0(q)f( - (p� - q)) u+ ( pj) iy/J. u(q) x i I x p (q , p1 , . · Pi• . . ,. ) ( 4. (x ) kl'- ( x ) 1T-1 12 a/J. p ( x ) , 1T-11 2 a,,. a (x ) = = R ELATIVISTIC D YNAMICS OF QUAN'l'IZED FIELDS 223 and one has similar infra-red troubles, which can be overcome by similar techniques. 62) and it turns out to be zero in strong contrast to the situation in the theory satisfying the spectral condition. 50). 81 ) below. 57) does indeed define a local field, at least in a formal sense, even though p and a do not commute with ip<0> at space-like separations.

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