# Analysis over infinite dimensional spaces and applications to quantum field theory

Séminaire Jean Leray (1973-1974)

- Issue: 2, page 1-8

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topGlimm, James. "Analysis over infinite dimensional spaces and applications to quantum field theory." Séminaire Jean Leray (1973-1974): 1-8. <http://eudml.org/doc/112582>.

@article{Glimm1973-1974,

author = {Glimm, James},

journal = {Séminaire Jean Leray},

language = {eng},

number = {2},

pages = {1-8},

publisher = {Collège de France},

title = {Analysis over infinite dimensional spaces and applications to quantum field theory},

url = {http://eudml.org/doc/112582},

year = {1973-1974},

}

TY - JOUR

AU - Glimm, James

TI - Analysis over infinite dimensional spaces and applications to quantum field theory

JO - Séminaire Jean Leray

PY - 1973-1974

PB - Collège de France

IS - 2

SP - 1

EP - 8

LA - eng

UR - http://eudml.org/doc/112582

ER -

## References

top- (1) See for example Gross, J. Funct. Analysis1 (1961), p. 123. Zbl0165.16403
- (2) Jürgens, Math. Ann.138 (1959) p. 179 and Math Z.17 (1961), p. 265. MR144073
- (3) Segal, Ann. of Math.18 (1963) p. 339.
- (4) Morawetz and Strauss, Comm. Pure Appl. Math.25 (1972) p. 1. Zbl0228.35055MR303097
- (5) Glimm and Joffe, Acta Math.125 (1970) p. 203. MR269234
- (6) Glimm, Joffe and Spencer, Ann. of Math., to appear and a contribution in: Constructive Quantum Field Theory, Ed. by Velsand Wightman, Springer Verlag, Berlin (1973). MR395513
- (7) Nelson, in: Constructive Quantum Field Theory, Ed. by Velsand and Wightman, Springer Verlag, Berlin (1973). Zbl0325.00006MR395513
- (8) Guerra, Rosen and Simon, Ann. of Math., to appear.
- (9) Dobrushya and Minlos, J. of Functional Analysis and its applications (Russian).
- (11) This approach has been basic to many papers on constructive quantum field theory, starting with Symanzik, NYU, 1964 (see also earlier papers of Schwinger) as well as Nelson, in: Mathematical theory of elementary particles, Ed. by Goodman and Segal, MIT press, 1966. It was central to the approach of Joffe and the author ; see for example Glimm-Joffe, Ann. of Math.91 (1970), or Glimm, Comm. Math. Physics8 p. 12 (1968).
- In 1968, a covariant form of the Feynman-Kac formula was known to Symanzin, as part of his formal program for a covariant construction of Euclidean quantum fields. This paper appears in: ffocal quantum field theory, proceedings of the International School of Physics "Enrier Ferme" Course45, Ed. by Jost. Academic Press, New York (1969).
- Three years later, Symanzik's covariant Feynman-Kac formula was used by Nelson as part of a simplified bound on the vacuum energy per unit volume, in: Proceedings of the Summer Institute of Partial Differential Equations, Berkeley1971, Amer. Math. Soc.Providence R.I.1973. Symanzik' s Feynman-Kac formula was subsequently used by Guerra, Phys. Rev. Lett.28 p. 1213 (1972) to obtain new results on the vacuum energy per unit volume. Following the Nelson and Guerra papers, Symanzik's covariant Feynman-Kac formula came into wide usage ; see footnote references 6, 7, 8 and 12.
- (12) The passage from the Euclidean field theory, defined on L2(S'(Rd+1)) to the Hilbert space H = L2(S'(Rd)) is given by the theory of Euclidean axioms of Nelson, J. Funct. Anal.12, p. 91 (1973) and Osterwalder-Schroder, Comm. Math. Phys.31 p. 83 (1973) and in Constructive Quantum Field Theory, Ed. Velo and Wightman, Springer Verlag, Berlin (1973).

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