Master Dissertation

4.2 The Wightman Axioms
We will treat a quantum field theory as a tuple
(H, U, φ, D, |0〉),
where H is a separable Hilbert space, U a unitary representation of the
proper Poincaré group P ↑
+ , φ are field operators, D a dense subspace of H
and |0〉 the vacuum state.
We expect the tuple to satisfy the Wightman axioms listed on the next
page. 5
FIXME: make sure the syntax is right
We will not go in to the deeper meaning of the axioms. Instead we will
note the most important properties for our use. We should note that the
fields φ are assumed to be operator valued distributions, which are
well-defined on a dense domain D ⊂ H.
FIXME: note that D0 is contained in axiom 4
In the construction of the scattering matrix, however, it suffices to regard
the dense domain D0 ⊂ D ⊂ H given by,
D0 = span{φ(g1) · · · φ(gn)|0〉|g1, . . . , gn ∈ S, n ∈ N}.
5 We cite them from [9] page 103 − 104.
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