Master Dissertation

Expected properties of the S-matrix.
1 Unitarity i.e. S(g) −1 = S(g) ∗ . That is
˜Tn(x1, . . . , xn) = Tn(x1, . . . , xn) ∗ .
Fixme: this is not enough according to page 162 probably not important for
this project.
2 Translational invariance. Let U(a, 1) be the unitary translation operator
in the Fock space F, that is
Then we require
(U(a, 1)Φ)j(x) = Φj(x1 + a, . . . , xj + a).
U(a, 1)S(g)U(a, 1) −1 = S(ga), where ga(x) = g(x − a).
Hence for the Tn’s (and of course the ˜ Tn’s as well) we get:
U(a, 1)Tn(x1, . . . , xn)U(a, 1) −1 = Tn(x1 + a, . . . , xn + a).
3 Lorentz covariance Letting U(0, Λ) be the representation of L ↑
+ .
U(0, Λ)S(g)U(0, Λ) −1 = S(gΛ), where gΛ = g(Λ −1 x).
Note that 2. and 3. together form a condition of Poincaré invariance.
4 Causality. Suppose there exists a reference frame in which the
test-functions g1 and g2 have disjoint supports in time. Assuming
supp g1 < supp g2. That is, for some r ∈ R,
supp g1 ⊂ {x ∈ M|x 0 ∈ (−∞, r)} and supp g2 ⊂ {x ∈ M|x 0 ∈ (r, ∞)}.
We require that
S(g1 + g2) = S(g2)S(g1). (5.4)
This is a statement about the fact that what happens at time t < s is not
influenced by what happens at some later time t > s.
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