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A densely defined, symmetric sesquilinear form is a prima facie candidate to represent a physical quantity or an observable. Since t is symmetric, the corresponding quadratic form is real-valued. Hence, for each unit vector ψ ∈ D(t), we might say that the “expectation value” of t in state ψ is t(ψ). Indeed, at first glance, the expectation value mapping t → t(ψ) seems to have all the same properties as the corresponding expectation mapping for operators. 17 Theorem. Let Φ(·) be a Wightman field on the Hilbert space H.
Field quantization: the Hilbert space of the quantum field is is space L2 (H, d) of ‘square integrable’ functions from H into C relative to the isonormal distribution d on H. (In a rigorous treatment, elements of L2 (H, d) are not really functions. ) The free field theories constructed by these two methods are known to be unitarily equivalent. However, the field quantization approach lends itself more naturally to a field interpretation. Indeed, in a recent survey of the foundations of QFT ([Huggett, 2000]), one finds the suggestive notation: Ψ(φ), φ ∈ L2 (R3n ), for a function on the space H := L2 (R3n ) of wavefunctions.
10 Fact. Net automorphisms correspond to automorphisms of the quasilocal algebra that leave each local subalgebra globally invariant. To state this precisely, let A denote the functor from K into C∗ , and let B denote the inductive limit of A. We identify A(O) with its image in B. Then α is a net automorphism of A iff there is an automorphism β of B such that β|A(O) = αO . Now, given a net A with inductive limit B, what should we consider as a symmetry of B? Proposal 1. e. a natural transformation of A.