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The family of mathematical problems discussed here has emerged in recent years as a result of efforts to put a small chapter of quantum field theory,… - Arthur Wightman
" "The family of mathematical problems discussed here has emerged in recent years as a result of efforts to put a small chapter of quantum field theory, the so-called external field problem, on a sound mathematical footing. The external field problems is special because the partial differential equations for the unknown field is linear, but the coefficients are allowed to vary in space and time and that gives rise to some surprises, which seem to be of general interest. There is a vast and in large part turgid mathematical physics literature on the subject. To make the general wisdom which has accumulated there more readily available to a mathematical audience I have, in the following, tried to place the problems in their physical context, and still to bring out the essential mathematical questions many of which remain to be answered.
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About Arthur Wightman
(March 30, 1922 – January 13, 2013) was an American mathematical physicist, known for the .
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Alternative Names:
Arthur Strong Wightman
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From the very beginning of quantum mechanics, the notion of the position of a particle has been much discussed. In the nonrelativistic case, the proof of the equivalence of matrix and wave mechanics, the discovery of the uncertainty relations, and the development of the statistical interpretation of the theory led to an understanding which, within the inevitable limitations of the nonrelativistic theory, may be regarded as completely satisfactory.
… there are other things wrong with these models but the fundamental trouble is the non-uniqueness of the vacuum as was first shown by , , and STEINMANN … Actually, … has shown that the cluster decomposition property is not only necessary but sufficient for the uniqueness of the vacuum, if there is at least one cyclic vacuum. …
HEPP, K., JOST, R., RUELLE, D. and STEINMANN, O., Necessary condition on Wightman functions, Helv. Phys. Acta 34 (1961) 542.
BORCHERS, H.J., On the structure of the algebra of field observables, Nuovo Cim. 24 (1962) 214
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Vacuum expectation values of products of neutral scalar field operators are discussed. The properties of these distributions arising from , the absence of negative energy states and the positive definiteness of the scalar product are determined. The vacuum expectation values are shown to be boundary values of analytic functions. Local commutativity of the field is shown to be equivalent to a symmetry property of the analytic functions. The problem of determining a theory of a neutral scalar field given its vacuum expectation values is posed and solved.
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