Quantum Yang-Mills Theory, section 3 (Quantum Fields)

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A quantum field, or local quantum field operator, is an operator-valued generalized function on space-time obeying certain axioms. The properties required of the quantum fields are described at a physical level of precision in many textbooks, see, for example, [27]. Garding and Wightman gave mathematically precise axioms for quantum field theories on R4 with a Minkowski signature, see [45], and Haag and Kastler introduced a related scheme for local functions of the field, see [24]. Basically, one requires that the Hilbert space H of the quantum field carry a representation of the Poincare group (or inhomogeneous Lorentz group). The Hamiltonian H and momentum P are the self-adjoint elements of the Lie algebra of the group that generate translations in time and space. A vacuum vector is an element of H that is invariant under the (representation of the) Poincare group. One assumes that the representation has positive energy, 0 <= H, and a vacuum vector Omega in H that is unique up to a phase. Gauge-invariant functions of the quantum fields also act as linear transformations on H and transform covariantly under the Poincare group. Quantum fields in space-time regions that cannot be connected by a light signal should be independent; Garding and Wightman formulate independence as the commuting of the field operators (anti-commuting for two fermionic fields).

Referencing norms: WightmanAxiomsClaim, WightmanAxiomsSource, bridge-locality, bridge-poincare-covariance, bridge-positive-energy, bridge-unique-vacuum, corresponds_to_gauge_theory_relation, states-locality, states-poincare-covariance, states-positive-energy, states-unique-vacuum, states_axiom_requirement, wightman-source-asserts-axioms

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[45] R. Streater and A. Wightman, PCT, Spin and Statistics and all That, W. A. Benjamin, New York, 1964.