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Covariant Phase Space Formulation of Parametrized Field Theories C

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Covariant Phase Space Formulation of Parametrized Field Theories C Covariant phase space formulation of parametrized field theories C. G. Torre Department of Physics, Utah State University, Logan, Utah 84322-4415 (Received 8 April 1992; accepted for publication 29 June 1992) Parametrized field theories, which are generally covariant versions of ordinary field theories, are studied from the point of view of the covariant phase space: the space of solutions of the field equations equipped with a canonical (pre)symplectic structure. Motivated by issues arising in general relativity, we focus on phase space representations of the space-time diffeomorphism group, construction of observables, and the relationship between the canonical and covariant phase spaces. 1. INTRODUCTION one could solve some very basic problems’ that are espe- cially troublesome for the program of canonical quanti- One of the central features of Newtonian mechanics zation of the gravitational field. is the presence of an absolute time: a preferred foliation of A relatively unexplored formulation of Hamiltonian Galilean space-time. Despite the presence of a universal gravity is based on the “covariant phase space.“*-” The notion of time, it is still possible to formulate dynamics in covariant phase space is defined as the space of solutions terms of an arbitrary time parameter. This is the “param- to the equations of motion and thus has the virtue of etrized” formulation of mechanics, which is obtained by preserving manifest covariance. Because the space of so- adjoining the Newtonian time to the configuration vari- lutions admits a (pre)symplectic structure, one can still ables of the mechanical system. * The resulting formalism employ sophisticated Hamiltonian methods to formulate is often elegant but, given the existence of a preferred the quantization problem. Thus it is of interest to try to time, never really necessary. apply covariant phase space methods to study the canon- The need for a field-theoretic formalism that includes ical quantum theory of gravity. Given the importance of arbitrary notions of time (and space) becomes apparent parametrized field theory both as a paradigm for general when one studies dynamical theories consistent with Ein- relativity and as an elegant formulation of field theories, it stein’s general theory of relativity. Here there is no pre- is worth examining such theories from the point of view ferred standard of time (or space), and it is usually best of the covariant phase space. In particular, how do the to keep this fact manifest by never selecting such a stan- stubborn problems of time and observables’ appear in the dard. This can be done by including the gravitational covariant phase space formulation of parametrized field field, in the guise of the space-time metric, as a dynamical theory? Can we use the parametrized field theory para- variable and keeping the resulting “general covariance”- digm to better understand the covariant phase space of more precisely: the space-time diffeomorphism general relativity? This latter question is made more covariance-of the theory manifest. However, it is not pressing since it has been shown recently that, strictly necessary to add new physics (gravitational dynamics) in speaking, the canonical (as opposed to covariant) phase order to achieve a generally covariant formulation of a space structure of general relativity cannot be identified field theory. It has been known for a long timez4 that any with that of any parametrized field theory.” As we shall field theory on a fixed background space-time can be see, the covariant and canonical approaches to the phase made generally covariant by adjoining suitable space-time space of parametrized theories are quite different, and variables to the configuration space of the theory in much hence it is plausible that the parametrized field theory the same way as one does in the parametrized formula- paradigm will be more suitable in the context of the co- tion of mechanics. This diffeomorphism covariant formu- variant phase space formulation. lation of field theory is likewise called “parametrized field In this paper we will present the covariant phase theory.” space formulation of a general parametrized field theory. Parametrized field theory allows one to study field In particular we will address the issue of the action of the theory without prejudicing the choice of time (space), diffeomorphism group on the phase space, which is a and for this reason alone it is a useful tool (see, e.g., Ref. delicate problem in the conventional Hamiltonian formu- 5). Because parametrized theories are generally covari- lation,*’ as well as the related issue of how to construct ant, they also serve as an important paradigm for the “observable? in this formalism. In canonical gravity the dynamics of gravitation6 Indeed, general relativity is of- construction of observables has so far proven intractable, ten viewed as an “already parametrized” field theory; if so it is useful to see how the covariant phase space ap- this point of view could be explicitly implemented, then proach handles this question. Most important perhaps, 3802 J. Math. Phys. 33 (1 l), November 1992 0022-2488/92/l 13802-l 1$03.00 @ 1992 American Institute of Physics Downloaded 02 Apr 2010 to 129.123.67.89. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp C. G. Torre: Parametrized field theories 3803 we will spell out in detail the (somewhat complicated) trized field theories; this includes the usual canonical for- relationship between the covariant and canonical phase mulation. In Sec. III we provide a quick tour of the co- space approaches to parametrized field theory by com- variant phase space formalism and apply it to paring the phase spaces, group actions, and observables in parametrized field theories. Next, in Sets. IV and V, we each formulation. In particular, we would like to see how turn to the representations of the diffeomorphism group the reduced phase spaces of the canonical and covariant on phase space and the extraction of the observables; both approaches manage to be the same, which is implied by of these issues are simply and neatly treated using the the general results of Ref. 11. This is a nontrivial issue covariant phase space. The final section, Sec. VI, is in mainly because the covariant phase space admits an ac- many ways the most interesting; in it we spell out the tion of the space-time diffeomorphism group while the relationship between the covariant and canonical phase canonical phase space does not. I3 Presumably, these re- space formulations. sults will at least hint at the corresponding results in general relativity. There are some disadvantages associated with trying II. PARAMETRIZED FIELD THEORIES to give an analysis that includes a “general field theory.” In particular, if one tries to give too broad a coverage of We consider a collection of fields, fl, propagating on possible field theories, then the description becomes quite a globally hyperbolic space-time (A, gcrs) according to opaque if only because of the notational difficulties. Thus, the extrema of the action functional for simplicity, we make some simplifying assumptions about the field theories being studied that, while perhaps violated in some very exceptional cases, are typically wp? = I, J%w%f% (2.1) valid. One important exception to the previous statement is that we will not attempt to include parametrized gauge For simplicity we assume that the fields are nonderiva- theories in our analysis. There are a couple of reasons for tively coupled to the background geometry, and that the this. First, the structure of a parametrized gauge theory is Lagrangian only depends on the fields and their first de- rather different from that of a theory without any gauge rivatives. The equations of motion are invariances. This is because the parametrized formulation of nongauge theories leads to a phase space formulation &T/s@ = 0. (2.2) that is well behaved with respect to the diffeomorphism Note that the solution space of this equation generally group of the space-time manifold, while the parametrized cannot admit an action of the space-time diffeomorphism gauge theory brings in the larger group of bundle auto- group, Diff(d), because the metric is fixed. morphisms. It is an interesting problem to find a globally The parametrization process enlarges the configura- valid formulation of parametrized gauge theory, but we tion space by the space of diffeomorphisms from J? to shall not do it here. At any rate, if one wants to use itself. When dealing with these new dynamical variables, parametrized field theory to understand general relativ- denoted X, it will be convenient to work with two copies ity, then the relevant group is the diffeomorphism group of &, ..M” and J‘, and then view XEDiff(d) as a map and gauge theories can thus be played down in impor- from Mp to .M”, tance (however, see Ref. 14). One of the other main as- sumptions we will make is designed to simplify the task of X:.JtP-bP. relating the covariant and canonical phase spaces. Specif- ically, we will identify the space of Cauchy data for the Thus one can think of X as a field on &p taking values in field theory with the canonical phase space for the theory. Aa. Tensors on Mp will be distinguished by Greek in- Given a spacelike (Cauchy ) hypersurface, the Cauchy dices from the end of the alphabet, likewise tensors on data will be assumed to be the fields and their normal Lie Aa will have Greek indices from the beginning of the derivatives on that surface. (There are, of course, impor- alphabet. As an important example, the metric on &” is tant cases that violate this assumption, e.g., the Dirac g&; given XEDiff(&) this metric can be pulled back to field, but such field theories present no new features in the Ai?? context of the present investigation.) In practice this identification, which is tantamount to identifying the tan- gpv= (x*g),,=x;xfk.lpx, (2.3) gent and cotangent bundles over the space of field con- figurations, is done in terms of metrics, both on space- where XE is the differential of the map X.
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