The Regular Ground States of the Linear Boson Field in Terms of Soft Modes

Abstract

We supplement the characterization of the regular ground states of the linear boson field, as detailed in Segal (Illinois J. Math. 6 (1962), 500–523), Weinless (J. Funct. Anal. 4 (1969), 350–379) and Baez, Segal & Zhou (Introduction to algebraic and constructive quantum field theory, Princeton University Press, 1992, Princeton), by using more developed operator algebraic methods for general Weyl systems and by continuing ideas of Honegger & Rieckers (Photons in Fock space and beyond I, World Scientific, 2015, Singapore). For a linear boson field in the sense of Segal (Illinois J. Math. 6 (1962), 500–523) 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 specifies 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 Segal (Illinois J. Math. 6 (1962), 500–523). 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.