Master Dissertation

The Wightman Axioms.
Axiom 1 (quantum field) The operators φ1(f), . . . , φn(f) are given
for each C ∞ -function f with compact support on the Minkowski space
R 4 . Each φj(f) and its Hermitian conjugate operator φj(f) ∗ are defined
at least on a common dense linear subset D of the Hilbert space H and
D satisfies
φj(f)D ⊂ D, φj(f) ∗ D ⊂ D,
for any f and j = 1, . . . , n. For any Φ, Ψ ∈ D,
is a complex valued distribution.
f ↦→ (Φ, φj(f)Ψ),
Axiom 2 (relativistic symmetry) On H there exists a unitary repre-
sentation U(a, A) of ˜ P ↑
+ (a ∈ R4 , A ∈ SL(2, C)), satisfying U(a, A)D =
D (invariance of the common domain of the fields).
U(a, A)φj(f)U(a, A) ∗ = S(A −1 )jkφk(f (a,A))
f (a,A)(x) = f(Λ(A) −1 (x − a)),
where the matrix (S(A)j,k) is an n-dimensional representation of A ∈
SL(2C).
Axiom 3 (local commutativity) If the support of f and g is spacelike
separated, then for any vector Φ ∈ D
[φj(f) (∗) , φk(g) (∗) ]±Φ = 0,
where (∗) indicates that the equation holds for any choice.
Axiom 4 (vacuum state) There exists a vector |0〉 in D satisfying the
following conditions:
(i) (Ua, A)|0〉 = |0〉 (invariance).
(ii) The set of all vectors obtained by acting an arbitrary polynomial
P of the fields on |0〉 is dense in H (cyclicity).
(iii) The Spectrum of the translation group U(a, 1) on |0〉 ⊥ is contained
in
V m = {p|(p, p) ≥ m 2 , p 0 > 0} (m > 0)
(Spectrum condition).
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