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Hypercontractivity for local states of the quantized electromagnetic field.
Leonard Gross
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 2, p. 381-399

The quantized free electromagnetic field provides a good example of the structures that arise in the theory of quantized fields. There is a Gaussian measure on an infinite dimensional linear space along with a Dirichlet form on this space. Both are uniquely determined by special relativity. These will be described, along with the operators that represent the quantized electromagnetic field. Hypercontractivity of the operator associated to the Dirichlet form will be proved under the condition that observations made of the field take place in a bounded region of space.

Le champ électromagnétique libre quantifié est un bon exemple des structures qui apparaissent dans la théorie des champs quantifiés. On considère un espace vectoriel de dimension infinie équipé d’une mesure gaussienne et d’une forme de Dirichlet qui sont determinées par la théorie de la relativité restreinte. Nous décrivons ces objets ainsi que l’opérateur qui représente le champ électromagnétique quantifié. L’hypercontractivité de l’opérateur associé à cette forme de Dirichlet est obtenue sous la condition que les observations du champ sont effectuées dans une region bornée de l’espace.

Published online : 2017-04-13
DOI : https://doi.org/10.5802/afst.1537
@article{AFST_2017_6_26_2_381_0,
     author = {Leonard Gross},
     title = {Hypercontractivity for local states of the quantized electromagnetic field.},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {2},
     year = {2017},
     pages = {381-399},
     doi = {10.5802/afst.1537},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_2_381_0}
}
Gross, Leonard. Hypercontractivity for local states of the quantized electromagnetic field.. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 2, pp. 381-399. doi : 10.5802/afst.1537. afst.centre-mersenne.org/item/AFST_2017_6_26_2_381_0/

[1] V. Bargmann; Eugene P. Wigner Group theoretical discussion of relativistic wave equations, Proc. Natl. Acad. Sci. USA, Tome 34 (1948), pp. 211-223 | Article

[2] James D. Bjorken; Sidney D. Drell Relativistic quantum fields, McGraw - Hill Book Co., 1965, xiv+396 pages

[3] Nelia Charalambous; Leonard Gross The Yang–Mills heat semigroup on three-manifolds with boundary, Commun. Math. Phys., Tome 317 (2013) no. 3, pp. 727-785 | Article

[4] Nelia Charalambous; Leonard Gross Neumann domination for the Yang–Mills heat equation, J. Math. Phys., Tome 56 (2015) no. 7 (ID 073505, 21 pages, electronic only) | Article

[5] Leonard Gross The configuration space for Yang–Mills fields (in preparation)

[6] Leonard Gross Some physics for mathematicians, Spring, 2011 (200 pages, http://www.math.cornell.edu/~gross/pubs/newt32y.pdf)

[7] Leonard Gross Stability for the Yang–Mills heat equation (in preparation)

[8] Leonard Gross The Yang–Mills heat equation with finite action (2016) (http://arxiv.org/abs/1606.04151)

[9] Irving Segal The Cauchy problem for the Yang–Mills equations, J. Funct. Anal., Tome 33 (1979) no. 2, pp. 175-194 | Article