Summary
The infinite tensor product is generalized to a tensor product of certain Hilbert spaces over a topological index set. The criterion for positivity of characteristic functionals of generalized random processes is used to construct two distinct types of continuous tensor products. Representations of the canonical commutation relations (CCR) are constructed in one of these. The other is seen to be useful in the construction of certain quantum fields.
Riassunto
Si generalizza il prodotto tensoriale infinito in un prodotto tensoriale di alcuni spazi di Hilbert su un insieme topologico di indici. Si usa il criterio per la positività dei funzionali caratteristici dei processi casuali generalizzati per costruire due tipi distinti di prodotti tensoriali continui. In uno di questi si costruiscono le rappresentazioni delle relazioni di commutazione canoniche (CCR), l’altro risulta utile per la costruzione di alcuni campi quantizzati.
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References
H. Araki andE. J. Woods:Publ. Res. Inst. Math. Sci. Kyoto, A2, 2 (1966).
I. M. Gelfand andN. Vilenkin:Generalised Functions, vol.4 (New York, 1964).
A. Guichardet:Comm. Math. Phys.,5, 262 (1967).
J. von Neumann:Compt. Math.,6, 1 (1938).
A. Guichardet andA. Wulfsohn:Sur les produits tensoriels continus d’espaces hilbertiens, to appear.
D. Dubin andR. F. Streater:Nuovo Cimento,50 B, 154 (1967).
R. F. Streater:Nuovo Cimento,53 A, 487 (1968).
R. F. Streater:Comm. Math. Phys.,6, 233 (1967).
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The work has been supported in part by the Air Force Office of Scientific Research through the European Office of Aerospace Research, United States Air Force and by the D.G.R.S.T. and C.N.R.S., Paris.
This work was done while one of us (R.F.S.) was Visiting Associate Professor at the University of Maryland (supported by the Science Development Grant NSF GU 2061).
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Streater, R.F., Wulfsohn, A. Continuous tensor products of hilbert spaces and generalized random fields. Nuovo Cimento B (1965-1970) 57, 330–339 (1968). https://doi.org/10.1007/BF02710205
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DOI: https://doi.org/10.1007/BF02710205