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The regular ground states of the linear boson field in terms of soft modes. (English) Zbl 1433.81114

Summary: We supplement the characterization of the regular ground states of the linear boson field, as detailed in [I. E. Segal, Ill. J. Math. 6, 500–523 (1962; Zbl 0106.42804)], [M. Weinless, J. Funct. Anal. 4, 350–379 (1969; Zbl 0205.57504)] and [J. C. Baez et al., Introduction to algebraic and constructive quantum field theory. Princeton, NJ: Princeton University Press (1992; Zbl 0760.46061)], by using more developed operator algebraic methods for general Weyl systems and by continuing ideas of R. Honegger and the author [Photons in Fock space and beyond. Vol. I: From classical to quantized radiation systems. Vol. II: Quantized mesoscopic radiation models. Vol. III: Mathematics for photon fields. Hackensack, NJ: World Scientific (2015; Zbl 1381.81005)]. For a linear boson field in the sense of Segal [loc. cit.] the originally real symplectic test function dynamics is given in terms of a continuous unitary group. The quantized field dynamics is realized via a continuous unitary group in a representation space of the Weyl commutation relations and speci es a quasifree automorphism group, which does not act strongly continuous on the C*-Weyl algebra.
If the generator of the test function dynamics is strictly positive, we show that the quasifree dynamical automorphism group is \(R\)-central. Under this condition, we determine the form of all “partially regular” ground states, which are regular only over a subspace of test functions, introduced in [Sega, loc. cit.]. All regular ground states are then functional integrals, in some weakened sense, over the bare vacua dressed by singular classical modes. If the ground states are nuclear continuous these functional integrals are proper ones.
If there are also time-invariant test functions then the related invariant quantum modes intervene into these superpositions of the pure regular ground states and complicate particularly the fully regular case.
Global features of the sets of (partially) regular ground states are detailed and compared with the ground state sets of C*-dynamical systems. By way of example, the vacuum dressing is interpreted as a soft-boson cloud, forming collective classical soft modes.

MSC:

81T05 Axiomatic quantum field theory; operator algebras
22D45 Automorphism groups of locally compact groups
82B10 Quantum equilibrium statistical mechanics (general)
46L55 Noncommutative dynamical systems
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
43A35 Positive definite functions on groups, semigroups, etc.
81V73 Bosonic systems in quantum theory
46L05 General theory of \(C^*\)-algebras
22E70 Applications of Lie groups to the sciences; explicit representations
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