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L ! 54) 52 III. FREE FIELDS AND FEYNMAN DIAGRAMS Unfortunately, the products of Feynman propagators occuring in this formula are, in general, ill defined. This is in remarkable contrast to the corresponding formulas in the graphical presentation of Wightman functions and is the origin of ultraviolet divergences in quantum field theory. The arising graphs can be classified according to the number of independent loops. Graphs without loops are called trees. The corresponding terms in the expansion of time ordered products are well defined, since the arguments of the Feynman propagators are independent differences yjm = xj − xm , and no higher powers of propagators occur, ljm = 0, 1.

21) 1. THE SCALAR FIELD 47 where the position x plays the role of the component k and ϕ˙ is the canonical conjugated momentum to ϕ. The fact that measurements at spacelike separated points have to be compatible does not at all mean that they have to be uncorrelated. Correlations are present, in contrast to the non-relativistic case, already in the vacuum, Ω, ϕ(x)ϕ(y)Ω = Ω, (a(x) + a∗ (x))(a(y) + a∗ (y))Ω = [a(x), a∗ (y)] = ∆+ (x − y) . 22) The vacuum expectation values of products of fields are called Wightman functions, Wn (x1 , .

33) U (A)(Φ + NpΛ(A) ) = U (A)Φ + Np . 34) by + Hm For each momentum p ∈ there is a subgroup Gp of SL(2, C), the so-called little group, whose elements do not change p, Gp = {A ∈ SL(2, C), pΛ(A) = p} . 35) Then U Gp is an irreducible representation of the little group on Hp . 36) ´ GROUP 3. THE REPRESENTATIONS OF THE POINCARE 37 and the representations of the little groups on the respective Hilbert spaces with sharp momentum satisfy the equation U (ARA−1 ) = U (A)U (R)U (A)−1 . 37) with R ∈ Gq and q = pΛ(A).

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