Mathematical characterization of the physical vacuum for a linear Bose-Einstein field

Abstract

representation for the canonical variables used in conventional theory, but its utilization of "state" as defined by its expectation value form, rather than as a vector in a Hilbert space, means that, a priori, the energy of a state is a somewhat vague mathematical object.The problem is thus that of dealing, in a fashion which is effective from a quite broad point of view, with the vacuum for free fields, the ultimate aim being to arrive at a simple characteri- zation which may be adapted with a reasonable degree of confidence to the case of interacting fields.From this point of view the main result is essentially that, although con- trary to common intuitive belief, Lorentz-invariance in itself is materially insufficient to characterize the vacuum for any free field (this remarkable fact is due to David Shale; it should perhaps be emphasized 'that this lack of uniqueness holds even in such a simple case as the conventional scalar meson field; in particular, there exist euclidean-invariant states satisfying the Haag-Coester cluster decomposition property which differ from the conventional vacuum, despite the existence of heuristic indications of uniqueness as pre- sented in [2]), none of the Lorentz-invariant states other than the conventional vacuum is consistent with the postulate of the positivity of the energy, when suitably and simply formulated.More specifically, in physical terms, what is done here is first to investigate the possibility of characterizing the vacuum for free fields as the unique state invariant under all classical unitary transformations, or more exactly, their induced action on the quantum field. (Physi- cally, this is similar to attempting to characterize the free-field representation as that admitting occupation-number operators with the usual formal proper-

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Affiliated Institutions

• Massachusetts Institute of Technology US

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