We describe how geometrical methods can be applied to a system with
explicitly time-dependent second-class constraints so as to cast it in
Hamiltonian form on its physical phase space. Examples of particular interest
are systems which require time-dependent gauge fixing conditions in order to
reduce them to their physical degrees of freedom. To illustrate our results we
discuss the gauge-fixing of relativistic particles and strings moving in
arbitrary background electromagnetic and antisymmetric tensor fields.Comment: 8 pages, Plain TeX, CERN-TH.7392/94 and MPI-PhT/94-4

Evans, Jonathan M.

Tuckey, Philip A.

English

arXiv

We describe how geometrical methods can be applied to a system with explicitly time-dependent second-class constraints so as to cast it in Hamiltonian form on its physical phase space. Examples of particular interest are systems which require time-dependent gauge fixing conditions in order to reduce them to their physical degrees of freedom. To illustrate our results we discuss the gauge-fixing of relativistic particles and strings moving in arbitrary background electromagnetic and antisymmetric tensor fields.We describe how geometrical methods can be applied to a system with explicitly time-dependent second-class constraints so as to cast it in Hamiltonian form on its physical phase space. Examples of particular interest are systems which require time-dependent gauge fixing conditions in order to reduce them to their physical degrees of freedom. To illustrate our results we discuss the gauge-fixing of relativistic particles and strings moving in arbitrary background electromagnetic and antisymmetric tensor fields

CERN Document Server

arXiv:hep-th/9408055v1  9 Aug 1994CERN-TH.7392/94MPI-PhT/94–45hep-th/9408055GEOMETRY AND DYNAMICS WITH TIME-DEPENDENT CONSTRAINTSJonathan M. Evans1* and Philip A. Tuckey2**1 Theoretical Physics Division, CERNCH-1211 Geneva 23, Switzerlande-mail: evansjm@surya11.cern.ch2 Max-Planck-Institut fu¨r PhysikWerner-Heisenberg-InstitutFo¨hringer Ring 6D-80805 Munich, Germanye-mail: pht@mppmu.mpg.deAbstractWe describe how geometrical methods can be applied to a system with explicitly time-dependent second-class constraints so as to cast it in Hamiltonian form on its physicalphase space. Examples of particular interest are systems which require time-dependentgauge fixing conditions in order to reduce them to their physical degrees of freedom. Toillustrate our results we discuss the gauge-fixing of relativistic particles and strings movingin arbitrary background electromagnetic and antisymmetric tensor fields.To appear in the Proceedings of the Conference on Geometry of Constrained DynamicalSystems, Isaac Newton Institute for Mathematical Sciences, Cambridge, U.K.,15th-18th June 1994, ed J.M. Charap (CUP)CERN-TH.7392/94MPI-PhT/94–45August 1994* Supported by a fellowship from the EU Human Capital and Mobility programme.** Supported by a fellowship from the Alexander von Humboldt Foundation.11. IntroductionA Hamiltonian dynamical system can be described geometrically by a phase space manifoldΓ (of dimension 2d say) equipped with a symplectic form ω and a Hamiltonian function H(see Abraham and Marsden (1978) and Arnol’d (1978)). The condition that ω is symplecticmeans that it is a non-degenerate closed two-form, so it can be used to introduce a Poissonbracket { , } on Γ. The evolution of the system in time t is given by a particular set oftrajectories on Γ, parametrized by t, such that Hamilton’s equationdfdt=∂f∂t+ {f,H} (1)holds for any time-dependent function f on Γ.Since the seminal work of Dirac (1950,1958,1964) there has been intensive study ofsystems of this type which can be consistently constrained to some physical phase spacemanifold Γ∗ (of dimension 2n say) which is embedded in Γ in a manner we now describe. Inthe most general case the embedding of Γ∗ in Γ can depend on time and it must thereforebe defined by a family of mapsϕt : Γ∗ → Γ , (2)depending smoothly on t, each of which is a diffeomorphism onto its image Xt ⊂ Γ. Weassume that each of the trajectories on Γ for which (1) holds has the property that italways lies in the subspaces Xt for each t, or else that it always lies in the complements ofthese spaces. It is clear that trajectories of the former type correspond exactly under theembedding (2) to trajectories on Γ∗, and one can attempt to reformulate the dynamics forthis subclass of trajectories in a manner which is intrinsic to Γ∗.We define ω∗ on Γ∗ at time t by pulling back ω using ϕt and we assume that thisis also a symplectic form (albeit a time-dependent one in general). We can then use ω∗to introduce the Dirac bracket { , }∗ on Γ∗. The key issue which we shall address here iswhether, for a given choice of embeddings (2), one can find a Hamiltonian function H∗such that Hamilton’s equationdfdt=∂f∂t+ {f,H∗}∗ (3)holds for any time-dependent function f on Γ∗. When the embeddings (2) are independentof time, (3) follows easily from (1) with H∗ = H. In the general case, however, the dynam-ics on Γ∗ is determined not just by the dynamics on Γ but also by the time-dependence ofthe embeddings ϕt, and under these circumstances it is non-trivial to determine whether(3) holds for some function H∗.The most compelling reason for studying this general situation is the fact that gaugechoices with explicit time dependence are essential in order to restrict systems which areinvariant under time-reparametrizations, such as the relativistic particle, string or generalrelativity, to their physical degrees of freedom (but see Henneaux et al (1992) for possiblemodifications of the action to allow other gauge choices). Here we summarize the solutionof this problem given in Evans and Tuckey (1993) and we clarify some related issues.(We have recently learned that Mukunda (1980) has previously obtained results which are2locally equivalent to ours using an algebraic approach. Related work from the Lagrangianpoint of view appears in (Ran˜ada 1994).) We then give some new examples, extending thetreatment of the relativistic particle in a background field (Evans 1993) to the case of astring in an arbitrary antisymmetric tensor background.2. Extended phase space and constrained dynamicsWe define extended phase space to be Γ¯ = Γ × R, where the second factor is time. Wecan, in a natural way, regard H and ω as living on Γ¯ (by pulling back using the projectionmap) and we define the contact form on Γ¯ to beΩ = ω + dH ∧ dt . (4)(In Evans and Tuckey (1993) Ω was called the Poincare´-Cartan two-form; in Abraham andMarsden (1978) Ω is introduced as an example of a contact structure.) Any trajectory onΓ parametrized by t is clearly equivalent to a trajectory on Γ¯ with parameter s chosensuch that dt/ds is nowhere zero. Let V be the tangent vector to the trajectory on Γ¯. ThenHamilton’s equation (1) is precisely the conditioni(V )Ω = 0 (5)(where i(V ) denotes interior multiplication of a form by the vector field V ).When the system is constrained we can similarly define extended physical phase spaceto be Γ¯∗ = Γ∗×R. The family of embeddings (2) is equivalent to the single embeddingϕ¯ : Γ¯∗ → Γ¯ , ϕ¯(x, t) = (ϕt(x), t) , (6)which is a diffeomorphism onto its image X¯ = {(x, t) : x ∈ Xt, t ∈ R} ⊂ Γ¯ (assuming, asstated earlier, that ϕt varies smoothly with t). Define the form Ω∗ on Γ¯∗ to be the pullback of Ω using ϕ¯. In general this has the structureΩ∗ = ω∗ + (dH + Y ) ∧ dt (7)for some one-form Y . (Here we use the fact that any time-dependent form on Γ∗ canbe regarded as a smooth form on Γ¯∗; when the form is time-independent this reduces topulling back using the projection map.)Any solution of Hamilton’s equation (1) which lies in X¯ clearly corresponds to atrajectory in Γ¯∗ with tangent vector V ∗ which satisfiesi(V ∗) Ω∗ = 0 . (8)By comparison with (4) and (5) we see that Hamilton’s equation (3) holds on Γ∗ if andonly ifY = dK mod dt and then H∗ = H +K (9)for some function K on Γ¯∗. One can show that this holds locally (ie. in any contractibleregion on Γ∗) if and only if ω∗ is independent of time on Γ∗, a fact which will prove usefullater.3To discuss specific examples it is convenient to introduce on Γ local coordinates zM,M = 1, . . . , 2d. The subsets Xt ⊂ Γ are defined by a set of time-dependent constraint func-tions ψI(zM, t), I = 1, . . . , 2(d−n), which are, in the language of Dirac (1950,1958,1964),second-class. The fact that these constraint functions are preserved in time is equivalentto our initial assumption that there exists a subset of trajectories confined to the sub-spaces Xt. The condition that the constraint functions are second-class is equivalent toour assumption that the form ω∗ is symplectic.If ξA, A = 1, . . . , 2n, are local coordinates on Γ∗ then the embeddings ϕt or ϕ¯ allowus to regard the zM as time-dependent functions of these variables on X¯, and we have theexplicit expressionY = − ∂zM∂t∂zN∂ξAωMN dξA = −ωMN ∂zM∂tdzN mod dt (10)for the one-form appearing in (9). It is convenient in practice to specify ϕt or ϕ¯ by givingexplicit expressions for a set of functions ξA(zM, t), which we call physical variables; onrestriction to X¯ these functions define (the inverses of) these embeddings in terms of thelocal coordinates.3. RemarksOur result (9) establishes necessary and sufficient conditions for a family of embeddingsϕt, or a choice of physical variables ξA(zM, t), to result in a Hamiltonian time-evolutionequation (3) on Γ∗. For a given set of constraint subspaces Xt, or equivalently a set ofconstraint functions ψI(zM, t), a family of embeddings or physical variables having thisproperty always exists locally. This follows from Darboux’s Theorem, which tells us thatlocally we can find embeddings ϕt or choose coordinates (ξA) = (qα, pα) on Γ∗ such thatω∗ = dqα ∧ dpα. Since this expression is manifestly independent of time on Γ∗, it satisfiesthe criterion which we gave following (9). On the other hand, there are clearly manyembeddings or choices of physical variables for which (3) will not hold, as can be seenby performing an arbitrary time-dependent coordinate transformation to make ω∗ time-dependent.Our result does not tell us how to explicitly construct a set of embeddings or physicalvariables with the required property, and in general this remains an open problem. Apartial result in this direction is case (B) of Evans (1991). This applies to a systemwith time-independent gauge symmetry generators which has imposed on it a set of time-dependent gauge-fixing conditions involving some subset of canonical variables which allcommute under the Poisson bracket. It is worth pointing out that if these canonicalvariables are regarded as configuration space coordinates in some equivalent Lagrangiandescription, then the result in question can also be obtained by first gauge-fixing theLagrangian and then passing to the Hamiltonian formalism. The new examples we shallpresent below lie outside the scope of case (B) of Evans (1991). Thus the result (9) is stilluseful for finding good sets of physical variables, even though it offers no general methodfor doing so.Finally we emphasize that our main motivation for the work summarized here is thereduced phase space approach to the canonical quantisation of systems which require time-dependent gauge choices. Even at the classical level, a system whose time evolution is not4described by an equation of the form of (3) falls outside the realm of conventional Hamil-tonian mechanics. In passing to the quantum theory, (3) becomes the Heisenberg equationof motion, which guarantees the existence of a unitary time evolution operator. In theabsence of a classical evolution equation of the form of (3), Gitman and Tyutin (1990a)have given an alternative prescription for the Heisenberg quantum evolution equation, inwhich extra terms appearing on the right hand side of (3) are taken over. This approachis complicated by the difficulty in obtaining an explicitly unitary time evolution. Theconsideration of simple examples such as the relativistic particle in an arbitrary back-ground electromagnetic field (Evans 1993) reveals that our approach can be much simpler– compare with Gitman and Tyutin (1990b), Gavrilov and Gitman (1993). Batalin andLyakovich (1991) have also considered the quantization of systems with time-dependentHamiltonian and constraints.4. ExamplesWe shall now apply our result (9) to discuss the gauge-fixing of relativistic particles andstrings moving in d-dimensional Minkowski space-time with background gauge fields. Weshall take coordinates xµ on Minkowski space-time which are either ‘orthonormal’ withµ = 0, . . . , d−1, or of ‘light-cone’ type with µ = +,−, 1, . . . , d−2, so that the flat metrichas components −g00 = g11 = . . . = gd−1d−1 = g+− = g−+ = 1 and all others vanishing.(This means that x± = (xd−1 ± x0)/√2 agreeing with the conventions of Evans (1993)but not Evans (1991).) It is useful to set up some conventions regarding indices which willallow us to deal with temporal and light-cone gauge conditions in a uniform way. Thuswe shall let the single index n on any vector denote either 0 or +, and we shall label theremaining components by a = 1, . . . , d−1 or a = −, 1, . . . , d−2 respectively. We shall alsofind it useful to denote the ‘transverse’ components by i = 1, . . . , d−2. In what follows theranges of these indices will always be understood.4.1 Relativistic particle in an electromagnetic fieldA particle of mass m and charge e moving in an arbitrary electromagnetic field Aµ(xν)can be described by the LagrangianL = −[m√−x˙2 + eAµ(xν) x˙µ]. (11)Here xµ(t) is the particle’s trajectory, t is a parameter along the worldline, and x˙ = dx/dt.Introducing the canonical momentum pµ = ∂L/∂x˙µ conjugate to xµ, we have coordinates(xµ, pµ) on phase space, with Poisson bracket {xµ, pν} = δµν . There is a single, first-classconstraintφ = (p+ eA)2 +m2 = 0 . (12)The Hamiltonian is H = λφ, where λ is an arbitrary (time-dependent) function on phasespace.Consider the class of gauge-fixing conditions of the formxn = f(pµ, t) , (13)5where f can be any function of its arguments which defines a good gauge choice (we shallmake no attempt to be more precise concerning this last point). We define a set of physicalvariables(ξA) = (xa∗, pa) where xa∗ = xa −∫dpn∂f∂pa(14)(partial derivatives and integrals of f are to be understood in terms of the functionaldependence given by (13)) and it is clear that in principle the equations (12), (13) and(14) allow us to express all quantities as functions of (ξA, t). We claim that the systemcan then be described by these physical variables together with a HamiltonianH∗ = −∫dpn∂f∂t, (15)which is valid for any background gauge field Aµ and any function f .The explicit restriction to physical phase space is of course very involved for a generalbackground field. In principle we can substitute from (13) and (14) into (12) to find pn asa function of the physical variables and time, and substitution of this result back into (13)and (14) then determines all the xµ as functions of (ξA, t). Fortunately, it is not necessaryto carry out this elimination explicitly in order to verify that our chosen physical variablesdo indeed satisfy the criterion (9) leading to the general expression for the Hamiltoniangiven above. This is because the explicit time dependence of xµ enters only through pn andf ; and by using this fact it is easy to calculate from (10) that Y = −d(∫ dpn ∂f/∂t) mod dt.Since the original Hamiltonian H vanishes when φ = 0, the result follows.Examples are:xn = t giving xa∗ = xa , H∗ = −pn , (16)which reproduces the temporal and light-cone results of Evans (1993);x+= p+t giving x−∗ = x−− p−t , xi∗ = xi , H∗ = −p+p− ;x0 = p0t giving xa∗ = xa , H∗ = 12p20 .(17)These expressions are deceptively simple in appearance because they represent very com-plicated functions of the physical variables in the case of a general background. It isinteresting that the Hamiltonians have universal forms in terms of the original momenta,in the sense that the dependence on the background field enters only through these par-ticular functions.4.2 Relativistic closed string in an antisymmetric tensor fieldA closed string moving in an arbitrary background antisymmetric tensor field Bµν(xρ) canbe described by the LagrangianL = −∫ 2pi0dσ[((x˙.x′)2 − (x˙)2 (x′)2)1/2+Bµν(xρ) x˙µx′ν]. (18)Here xµ(t, σ) describes the string’s trajectory, t and 0 ≤ σ ≤ 2pi parametrize the world-sheet, and x˙ = ∂x/∂t, x′ = ∂x/∂σ. Introducing the momentum pµ(σ) = δL/δx˙µ(σ)6conjugate to xµ(σ) as usual, we have coordinates (xµ(σ), pµ(σ)) on phase space, withPoisson bracket {xµ(σ), pν(σ′)} = δµν δ(σ−σ′). The only constraints are first-class and aregiven by(pµ +Bµνx′ν)2+ (x′)2= 0 , (19)x′. p = 0 . (20)Again the Hamiltonian H is proportional to the constraints.It is useful to introduce the position zero-mode and total momentum of the string byXµ(t) =12pi∫ 2pi0dσ xµ(t, σ) , Pµ(t) =∫ 2pi0dσ pµ(t, σ) . (21)The factor of 2pi ensures Xµ and Pµ are conjugate variables. One can then write a decom-position of the string fieldsxµ = Xµ + x˜µ , pµ =12piPµ + p˜µ , (22)where the tilded variables represent the oscillator degrees of freedom.We consider the class of gauge-fixing conditions of the formxn = f(Pµ, t) , (23)pn = 12piPn , (24)where f is, as before, any function of its arguments which provides a good gauge-fixingcondition. The gauge conditions and constraints allow for the complete elimination of onepair of string position and momentum variables corresponding to the direction in space-time labelled by n, and they allow also for the elimination of one additional set of stringoscillators. We introduce a set of physical variables(ξA) = (Xa∗, Pa, x˜i(σ), p˜i(σ)) where Xa∗ = Xa −∫dPn∂f∂Pa(25)(comments similar to those following (14) apply) and with a little thought one can seethat the equations (19), (20), (23), (24) indeed allow us in principle to express all theoriginal variables in terms of (ξA, t). At this stage the analysis looks very like that forthe particle, at least as far as the string zero mode and total momentum variables areconcerned. For the case of a light-cone gauge (n = +) this comparison is accurate: for thephysical variables given above one can again calculate Y from (10) and deduce that (9)holds, yielding a HamiltonianH∗ = −∫dPn∂f∂t, (26)which is valid for any background field Bµν and any gauge-fixing function f . For the caseof a temporal gauge condition (n = 0), however, the physical variables written above donot satisfy (9) in general, unless the background has some special symmetry. The previousarguments break down because the solutions for the redundant oscillator variables x˜d−17and p˜d−1 in terms of (ξA, t) can depend explicitly on time for a general background field.This difficulty is absent if, for example, the background vanishes, Bµν = 0, and then thephysical variables and Hamiltonian written above hold for any function f .Examples are:xn = t giving Xa∗ = Xa , H∗ = −Pn ; (27)andx+= P+t giving X−∗ = X−− P−t , X i∗ = X i , H∗ = −P+P− ;x0 = P 0t giving Xa∗ = Xa , H∗ = 12P 20 .(28)These last examples generalise previous light-cone and temporal gauge-fixing constructions(Goddard et al 1973, Scherk 1975, Goddard et al 1975).AcknowledgmentsWe would like to thank Prof. J.M. Charap for organising this very stimulating conference.ReferencesAbraham R and Marsden J E, 1978, Foundations of mechanics Benjamin/Cummings, 1978Arnol’d V I, 1978, Mathematical methods of classical mechanics Springer, 1978Batalin I A and Lyakovich S L, 1991, in Group theoretical methods in physics Nova Science,1991, vol. 2, p. 57Dirac P A M, 1950, Canad. J. Math. 2 129Dirac P A M, 1958, Proc. Roy. Soc. A246 326Dirac P A M, 1964, Lectures on quantum mechanics Academic, 1964Evans J M, 1991, Phys. Lett. B256 245Evans J M, 1993, Class. Quantum Grav. 10 L221Evans J M and Tuckey P A, 1993, Int. J. Mod. Phys. A8 4055Gavrilov S P and Gitman D M, 1993, Class. Quantum Grav. 10 57Gitman D M and Tyutin I V, 1990a, Quantization of fields with constraints Springer, 1990Gitman D M and Tyutin I V, 1990b, Class. Quantum Grav. 7 2131Goddard P, Goldstone J, Rebbi C and Thorn C B, 1973, Nucl. Phys. B56 109Goddard P, Hanson A J and Ponzano G, 1975, Nucl. Phys. B89 76Henneaux M, Teitelboim C and Vergara J, 1992, Nucl. Phys. B387 391Mukunda N, 1980, Phys. Script. 21 801Ran˜ada M F, 1994, J. Math. Phys. 35 748Scherk J, 1975, Rev. Mod. Phys. 47 1238

Geometry and Dynamics with Time-Dependent Constraints

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