Quadratic expressions in a free boson field

Klein, Abel

doi:10.1090/S0002-9947-1973-0406213-5

Quadratic expressions in a free boson field
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by Abel Klein PDF

Trans. Amer. Math. Soc. 181 (1973), 439-456 Request permission

Abstract:

Quadratic expressions in a massive spinless free Boson field are treated by an appropriate extension of the method of second quantization. A certain class of these second quantized operators is shown to generate semigroups that act on a suitable scale of ${L_p}$-spaces, obtained through the diagonalization of the field at a fixed time, in a particularly regular fashion. The techniques are developed first in an abstract setting, and then applied to the neutral scalar free field. The locally correct generator of Lorentz transformations for $P{(\varphi )_2}$ is studied in detail, and essential selfadjointness is shown. These techniques are also used to solve explicitly the ${({\varphi ^2})_n}$ model.

References

John T. Cannon and Arthur M. Jaffe, Lorentz covariance of the $\lambda (\varphi ^{4})_{2}$ quantum field theory, Comm. Math. Phys. 17 (1970), 261–321. MR 272301, DOI 10.1007/BF01646027
J. Ginibre and G. Velo, Renormalization of a quadratic interaction in the Hamiltonian formalism, Comm. Math. Phys. 18 (1970), 65–81. MR 266533, DOI 10.1007/BF01649639
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Lon M. Rosen, The $(\phi ^{2n})_{2}$ quantum field theory. Lorentz covariance, J. Math. Anal. Appl. 38 (1972), 276–311. MR 411462, DOI 10.1016/0022-247X(72)90088-1
Lon Rosen, Renormalization of the Hilbert space in the mass shift model, J. Mathematical Phys. 13 (1972), 918–927. MR 299127, DOI 10.1063/1.1666077
I. E. Segal, Tensor algebras over Hilbert spaces. I, Trans. Amer. Math. Soc. 81 (1956), 106–134. MR 76317, DOI 10.1090/S0002-9947-1956-0076317-8

Conjugacy to unitary groups within the infinite dimensional symplectic group

Irving Segal, Nonlinear functions of weak processes. II, J. Functional Analysis 6 (1970), 29–75. MR 0263369, DOI 10.1016/0022-1236(70)90046-7
Irving Segal, Transformations in Wiener space and squares of quantum fields, Advances in Math. 4 (1970), 91–108 (1970). MR 273950, DOI 10.1016/0001-8708(70)90017-4
Irving Segal, Construction of non-linear local quantum processes. I, Ann. of Math. (2) 92 (1970), 462–481. MR 272306, DOI 10.2307/1970628
Irving Segal, Construction of non-linear local quantum processes. II, Invent. Math. 14 (1971), 211–241. MR 295695, DOI 10.1007/BF01418890
David Shale, Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103 (1962), 149–167. MR 137504, DOI 10.1090/S0002-9947-1962-0137504-6
Michael Weinless, Existence and uniqueness of the vacuum for linear quantized fields. , J. Functional Analysis 4 (1969), 350–379. MR 0253687, DOI 10.1016/0022-1236(69)90004-4