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Physical Meaning of a Quantum Field
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[QUOTE="DarMM, post: 6234712, member: 140662"] [USER=666861]@Bobjoesmith[/USER] I just went over those notes and they're sort of an odd presentation of the Wightman-[I]Gårding[/I] axioms. I much prefer the presentation in Streater and Wightman's classic "PCT, Spin and Statistics and all that" Chapter 3. The Wightman axioms there conceptually are: [LIST=1] [*]There is a Hilbert space carrying a (projective) representation of the Poincaré group. The generator of translations ##P^{\mu}## has eigenvalues lying in or on the positive light cone. There is one invariant state, the vacuum ##\Omega## These are basically the conditions of a Lorentz covariant quantum theory with sensible spectral properties[*]For each Schwartz function ##f## there is a set of self-adjoint operators ##\phi_{i}\left(f\right)## that are endomorphisms of a dense subspace ##\mathcal{D}## of the Hilbert space. ##\Omega## the vacuum is in this subspace. The unitary (projective) representations of the Poincaré group map ##\mathcal{D}## into itself. Whenever ##\Psi , \Phi \in \mathcal{D}## then ##\left(\Psi, \phi\left(f\right)\Phi\right)## is a tempered distribution. This basically states the existence of a set of operator valued distributions. The second part, together with the fact that they and the Poincaré transformations are endomorphisms, establishes the minimal conditions necessary for them to have n-point functions in any given reference frame.[*]The action of the Poincaré group on these operators is equivalent to transforming their components as if they were tensor or spinor fields on Minkowski spacetime: $$U\left(a,A\right)\phi_{j}\left(f\right)U\left(a, A\right)^{-1} = \sum S_{jk}\left(A^{-1}\right)\phi_{k}\left(\left\{a, A\right\}f\right)$$ This just establishes that these operators are in fact fields[*]The fields commute or anti-commute at spacelike distances. This just implements locality. [/LIST] [/QUOTE]
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