arXiv:gr-qc/9607032v1 14 Jul 1996 SWAT 96/124 gr-qc/9607032 14 July, 1996. The Schr¨odinger Wave Functional and Vacuum States in Curved Spacetime II – Boundaries and Foliations D.V. Long1andG.M. Shore2 Department of Physics University of Wales Swansea Singleton Park Swansea, SA2 8PP, U.K. Abstract In a recent paper, general solutions for the vacuum wave functionals in the Schr¨odinger picture were given for a variety of classes of curved spacetimes. Here, we describe a number of simple examples which illustrate how the presence of spacetime boundaries influences the vacuum wave functional and how physical quantities are independent of the choice of spacetime foliation used in the Schr¨odinger approach despite the foliation dependence of the wave functionals themselves. PACS numbers: 04.62+v, 04.70.Dy and 98.80.Cq 1D.V.Long@swansea.ac.uk 2G.M.Shore@swansea.ac.uk
1Introduction The Schr¨odinger wave functional provides a simple and intuitive description of vacuum states in quantum field theory in curved spacetimes.It is particularly useful in situations where the background metric is time-dependent or in the presence of boundaries. This is the second paper in a series where we develop the Schr¨odinger picture formalism in curved spacetime.In the first paper [1], we reviewed and devel- oped techniques for solving the Schr¨odinger wave functional equation for broad classes of spacetimes, viz. static (where the metric depends only on the spacelike coordinates), dynamic or Bianchi type I (where the metric depends only on the timelike coordinates) and a certain class of conformally static metrics including the Robertson-Walker spacetimes. Here, we continue this development by study- ing examples of spacetimes with boundaries, in particular regions described by coordinate patches which can be analytically extended to a larger spacetime. We describe how the presence of boundaries influences the choice of foliation in the Schr¨odinger formulation and determines the nature of possible vacuum states. The main advantage of the Schr¨odinger picture over other ways to characterise vacuum states is that it describes states explicitly by a simple wave functional specified by a single, possibly time-dependent, kernel function satisfying a differ- ential equation with the prescribed boundary conditions. This makes no reference to the assumed spectrum of excited states and so circumvents the difficulties of the conventional canonical description of a vacuum as a ‘no-particle’ state with respect to the creation and annihilation operators defined by a particular mode decomposition of the field, an approach which is not well suited to time-dependent problems. Unlike the alternative of specifying a vacuum state implicitly by giving a prescription for determining the Green functions, the Schr¨odinger wave func- tional is an explicit description, and this simplifies the interpretation of the nature of the states.In the end, of course, the same fundamental ambiguities appear in very similar guises in all these formalisms, but while the Green function ap- proach is perhaps better suited to more elaborate issues such as renormalisation and higher-order perturbative calculations, the Schr¨odinger picture frequently gives the clearest insight into the nature of the vacuum state. So far, we have spoken loosely about ‘the vacuum state.’ In fact, it is only for the very special class of static spacetimes that an essentially unique state exists which possesses most of the defining attributes of the Minkowski vacuum.In the general case, there may be no distinguished candidate at all for a vacuum state with the usual properties.For example, in a dynamic spacetime, there is a one-parameter family of ‘vacuum’ solutions to the Schr¨odinger wave functional equation and the selection of one of these requires a physically motivated initial condition on the first-order time-dependent equation for the kernel.Although 1
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