On the motion of particles and strings, presymplectic ... - Springer Link

Gen. Relativ. Gravit. (2005) 37(3): 437–459 DOI 10.1007/s10714-005-0034-y

R E S E A R C H A RT I C L E

Enrique G. Reyes

On the motion of particles and strings, presymplectic mechanics, and the variational bicomplex

Received: 28 January 2004 / Published online: 24 March 2005
C Springer-Verlag 2005

Abstract Examples of equations of motion in classical relativistic mechanics are studied: the equations of motion of a charged spinning particle moving in a space-time (with or without torsion) in the presence of an electromagnetic field are derived via Souriau presymplectic reduction. Then, the extension of Souriau’s ideas to Lagrangian field theory due to Witten, Crnkovi´c, Zuckerman is reviewed using the variational bicomplex, the basic properties of the Lund–Regge equations describing the motion of a string interacting with a scalar field and moving in Minkowski spacetime are recalled, and a symplectic structure for their space of solutions is found. Keywords Presymplectic mechanics · String · Torsion

1 Introduction This paper presents some first steps in the (pre)symplectic approach to Hamiltonian mechanics. Its goal is to introduce some basic facts of symplectic and presymplectic geometry, and to apply them to some mechanical problems in relativistic physics which have relevance even today. The problems considered here are instances of the classical equations of motion theme. As is well-known, equations of motion are usually obtained from conservation laws, action principles or “general covariance”; in general relativity for αβ example, it is standard to start with the covariant conservation law T;β = 0, E. G. Reyes (B) Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA; Departamento de Matem´aticas y Ciencias de la Computaci´on, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile E-mail: [email protected], [email protected]

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where T αβ denotes the energy–momentum density describing the body, and follow a method due to Papapetrou (see papers in [7] and references therein) to find the covariant equations of motion for pole and pole–dipole particles. (A related method has been recently proposed for strings [11]). It will be pointed out here, following classical work by Souriau [24] that one can study this problem in a very general setting using geometrical considerations. The content of this paper is as follows: Section 2 is on symplectic and presymplectic manifolds, and on the relation between Hamiltonian mechanics and what is called here Souriau reduction, that is, the understanding of the equations of motion as a (perhaps local) description of the leaves of a foliation on a presymplectic manifold [24]. Of course, no claim can be made as to the novelty of the results stated in this part of the paper: in one way or another, they are bound to appear in [24] and/or in the treatises by Abraham, Marsden [3], Arnold [5], Guillemin, Sternberg [12], Marsden and Ratiu [19] and Woodhouse [26]. The examples alluded to before are in Sects. 3–5: Section 3 summarizes some results of K¨unzle [15] on the presymplectic description of a spinning particle in a gravitational field. K¨unzle’s work is by now classical, but it is relevant to review it briefly here as it appears to be the first deep application of Souriau reduction to general relativity (see also the second reference of [24]). As an easy corollary of K¨unzle’s analysis, it is also observed in this section that if the set of time-like geodesics of a spacetime is a manifold, it carries a symplectic structure. Section 4 generalizes K¨unzle’s construction. Equations of motion which can be interpreted as the equations of motion of a charged spinning particle interacting with an electromagnetic field, and moving in a spacetime with torsion, are derived using Souriau reduction. Even though the Einstein–Cartan theory of gravity has been examined for a long time (see [13] and Trautman’s reports [25]) torsion has recently reappeared in several contexts. For the purposes of this paper, it is noteworthy to mention the work by R.T. Hammond on the motion of strings in spacetime [11], and the study of pseudo-classical particles by Rietdijk and van Holten, and by Vaman and Visinescu [23]. A short review of other interesting developments, as well as a classification of the torsion tensor of spacetime, appears in the work of Capozzielo, Lambiase, Stornaiolo [9]. Section 5 is on the motion of a string in interaction with a scalar field in Minkowski spacetime. In this case, the relevant equations of motion have been found by Lund and Regge [17] and studied further by Lund [16]. In particular, Lund has shown in the second reference of [16] that the Lund–Regge equation is Hamiltonian by using scattering/inverse scattering techniques. In this paper one is interested in finding a coordinate-free description of the phase space for the Lund– Regge equation, as an introduction to work by Witten, Crnkovi´c and Zuckerman [10, 27] on the extension of Souriau’s ideas to field theory. Following Zuckerman [27], the main tools used here are the formal geometric theory of differential equations and the variational bicomplex [2, 22]. As [1, 10, 27] testify, this is an exciting area of research, and much remains to be done! The Einstein summation convention will be used throughout.

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2 Hamiltonian systems and presymplectic manifolds In this section are all manifolds are finite-dimensional, and all maps, vector fields and tensors are assumed to be of class C ∞ . The main references for the differential geometric facts used in this section are the books [3, 4].

2.1 Presymplectic and symplectic manifolds Let ω be a two-form on a manifold M . For each q ∈ M one defines the linear mapping ω  (q) : Tq M → Tq∗ M by ω  (q)(Xq ) = iXq ω for all Xq ∈ Tq M , in which iXq ω(Yq ) = ω(q)(Xq , Yq ) for all Yq ∈ Tq M . The kernel of ω  (q) is called the kernel of ω(q), and its rank, the rank of ω(q). If the rank of ω(q) is independent of q , the two-form ω is called a two-form of constant rank. If ω is a two-form of constant rank and rank(ω) = dim(M ), ω is said to be nondegenerate. It is not hard to see that the rank of a two-form of constant rank is an even number. In particular, if ω is non-degenerate, the dimension of M must be even. A presymplectic manifold is a pair (M, ω ) in which M is a manifold and ω is a presymplectic form on M , that is, a closed two-form of constant rank on M . If ω is non-degenerate, then (M, ω ) is a symplectic manifold and ω is a symplectic form on M . From now on, the adjective “presymplectic” will be applied exclusively to closed two-forms of constant rank strictly less than dim(M ). If (M1 , ω1 ) and (M2 , ω2 ) are symplectic manifolds, φ : M1 → M2 is a smooth mapping, and the pull-back φ∗ ω2 is precisely ω1 , then φ is called a symplectic mapping. If, in addition, φ is a diffeomorphism, then (M1 , ω1 ) and (M2 , ω2 ) are said to be symplectomorphic. Locally, (pre)symplectic manifolds all look alike. More precisely, the following result hold: Theorem 1 (a) Suppose that ω is a non-degenerate 2-form on a 2n-dimensional manifold M . Then dω = 0 if and only if for any m ∈ M there exists a chart (U, φ) about m such that φ(m) = 0 and ω|U = dxi ∧ dyi , (1) in which φ|U = (x1 , . . . , xn , y1 , . . . , yn ). (b) Let (M, ω) be a (2n + k)-dimensional presymplectic manifold with rank(ω) = 2n. For each m ∈ M there is a chart (U, φ) about m such that, in this chart, ω |U = dq i ∧ dri ,

where coordinates in the chart are written (q 1 , . . . , q n , r1 ,. . . , rn , w1 , . . . , wk ). This is the classical Darboux theorem. The proofs of (a) and b appear in [3, p. 175] and [3, p. 371] respectively.

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Remark 1 Darboux’s theorem may be thought of as stating that symplectic manifolds are, in a sense, “flat”. Indeed, assume that the symplectic manifold (M, ω) is equipped with a symplectic connection, that is, with a linear connection ∇ which is torsion-free and satisfies the compatibility condition X[ω(Y, Z)] = ω(∇X Y, Z) + ω(Y, ∇X , Z)

for all vector fields X, Y, Z on M , so that the parallel transport operator defined by ∇ determines symplectomorphisms between the tangent spaces to M . The connection ∇ on M determines a normal coordinate system (U, φ) about m ∈ M , with φ|U = (x1 , . . . , xn , y 1 , . . . , y n ), as follows [14, p. 148]: the coordinates of a point q ∈ U are found by projecting, via the exponential map, the unique geodesic joining m and q into Tm M equipped with a fixed basis {ej }, so that in particular, φ(m) = 0. If in addition the connection ∇ is flat, one can define coordinate vec∂ ∂ tor fields ( ∂x i , ∂y i ) on U by parallel transporting the basis {ej } of Tm M . Now, choose the fixed basis {ej } such that ω(m) has the form (1). Since, as previously said, parallel transport is a symplectic isomorphism, it follows that ω in the coordinate system (U, φ) has the form required by Darboux’s theorem. Conversely, Darboux’s theorem implies that one can locally identify the symplectic manifold (M, ω) with R2n equipped with its canonical symplectic structure (1), and the standard connection of R2n is symplectic relative to (1). Another important observation is that if M is an arbitrary manifold, there exists a canonical symplectic form ω0 on T ∗ M : One takes αq ∈ Tq∗ M and lets π : T ∗ M → M be the canonical projection. The equation θ0 (αq ) · Xαq = αq · T π(Xαq )

for all

Xαq ∈ Tαq (T ∗ M ),

defines a one-form on T ∗ M . The symplectic form ω0 is ω0 = −dθ0 . If (q i ) is a coordinate chart on M , and αq = (q 1 , . . . , q n , p1 , . . . , pn ) is an element of T ∗ M , then θ0 (αq ) = pi dq i , and therefore ω0 = dq i ∧ dpi . The symplectic manifold (T ∗ M, ω0 ) is called the canonical phase space of the configuration space M . 2.2 Hamiltonian systems Let (M, ω ) be a symplectic manifold and let H : M → R be a smooth function on M . The triplet (M, ω, H) is called the Hamiltonian system on (M, ω) with Hamiltonian function H and phase space (M, ω ). The evolution of the system is determined by the flow of the unique vector field XH satisfying the equation iXH ω = dH.

(2)

Note that a solution XH to Eq. (2) exists and is unique because ω is nondegenerate: the function ω  :T M → T ∗ M given by ω  |Tq M = ω  (q) is an isomorphism, and therefore Eq. (2) implies that XH = (ω  )−1 (dH). That Eq. (2) does encode Hamilton’s equations is a consequence of Darboux’s result reviewed in Theorem 1: Proposition 1 Let (M, ω) be a symplectic manifold and let (q 1 , . . . , q n , p1 , . . . , pn ) be canonical coordinates (i.e. given by Darboux’s theorem) on M . Then, the equation iXH ω = dH , in which H: M → R is a smooth function

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∂H on M , implies that XH = ( ∂p , − ∂H ∂q i ). Thus (q(t), p(t)) is an integral curve of i XH if and only if Hamilton‘s equations

∂H dq i = , dt ∂pi

∂H dpi =− i, dt ∂q

i = 1, . . . , n

(3)

are satisfied.

2.3 The space of motions It is not always straightforward to find a symplectic description of a mechanical system [3, 5, 12]: it is not uncommon to begin with a (singular) Lagrangian, and find a canonical formulation of the system at hand by means of the Dirac constraint algorithm [3, 8, 10, 26]. The final result of this algorithm is a presymplectic manifold (M, ω). In general, if a dynamical system is described by a presymplectic manifold (M, ω), one says that (M, ω) is the evolution space of the system. The corresponding phase space is constructed as follows: For each v ∈ M set kerv ω = kernel ω  (v) = {Zv ∈ Tv M : iZv ω = 0}, and define the distribution of vector spaces
ker ω = kerv ω. v∈M

Since ω is of constant rank, the dimension of kerv ω is independent of v and therefore ker ω is a sub-bundle of the tangent bundle T M . Moreover, if Z , Y are vector fields on M such that Z(v) and Y (v) belong to kerv ω for all v ∈ M , then, i[Z,Y ] ω = LZ (iY ω) − iY (LZ ω) = 0 − iY (d(iZ ω) + iZ dω) = 0,

and so [Z, Y ](v) ∈ kerv ω for all v ∈ M . Frobenius’ theorem [4, p. 333] then implies that the distribution ker ω is integrable, that is, there exists a foliation Φω = {Lα }α∈A of M satisfying ker ω = T (M, Φω ), in which

T (M, Φω ) = Tm Lα α∈A m∈Lα

is the tangent bundle of Φω . Definition 1 Let (M, ω) be a presymplectic manifold. The space of motions UM of (M, ω) is the set of leaves of the foliation Φω , that is, UM = M/ker ω . The procedure of constructing the space of motions UM will be called Souriau reduction, after the fundamental contributions to the subject made by Souriau [24]. The space UM is a manifold if and only if [4, p. 334] for every v ∈ M there exists a local submanifold Σv of M such that Σv intersects every leaf in at most one point (or nowhere), and Tv Σv ⊕ Tv (M, Φω ) = Tv M . The submanifold Σv is called a slice or local cross section for Φω . A corollary of this theorem is that if UM is a manifold, it is Hausdorff and its dimension is equal to dim(M ) − dim(ker ω).

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Theorem 2 Let (M, ω) be a presymplectic manifold, and assume that the space of motions UM is a manifold. Then, UM can be equipped with a symplectic structure. Proof Let π: M → M/ker ω be the canonical projection from M onto UM . Since π is an onto submersion, the tangent space T[v] UM , in which [v] = π(v), is isomorphic to the quotient vector space Tv M/kerv ω . Define a two-form ω ˜ on UM by the equation π ∗ ω ˜ = ω , that is, ω ˜ (π(v)) (Tv πZv , Tv πWv ) = (π ∗ ω ˜ )(v)(Zv , Wv ) = ω(v)(Zv , Wv )

for v ∈ M and Zv , Wv ∈ Tv M . Now, if Zv , Wv ∈ Tv M are such that (Tv π)Zv = (Tv π)Zv , and (Tv π)Wv = (Tv π)Wv , then Zv − Zv and Wv − Wv ∈ kerv ω , and therefore ω(v)(Zv , Wv ) = ω(v)(Zv , Wv ) + ω(v)(Zv − Zv , Wv ) + ω(v)(Zv , Wv − Wv ) = ω(v)(Zv , Wv ),

so that ω ˜ is well-defined. That ω ˜ is closed follows from the equations 0 = dω = ˜ = π ∗ (d˜ ω ) and the fact that the projection π is an onto submersion. dπ ∗ ω Finally, ω ˜ is non-degenerate, for, if iZ ω ˜ (π(v))((T π)Yv ) = 0 for all Yv ∈   ˜ (π(v))((T π)Zv , (T π)Yv ) = 0 for all Yv ∈ Tv M , in which Zv ∈ Tv M , then ω ˜ )(v)(Zv , Yv ) = ω(v)(Zv , Yv ) = 0 Tv M is such that (T π)Zv = Z . But then, (π ∗ ω   for all Yv ∈ Tv M , and therefore Zv ∈ kerv ω , so that Z = (T π)Zv = 0.
Remark 2 In the 1980’s, Sternberg and his coworkers formulated a program to reduce the classical theory of particle motion to the construction of presymplectic manifolds and the corresponding spaces of motion. Of interest for this paper are the articles cited in [20]; the general principles of this program are spelled out in Guillemin and Sternberg’s book [12] and references therein. 2.4 From Hamiltonian systems to presymplectic manifolds and back If ω is a symplectic form, the foliation Φω is zero-dimensional and UM is diffeomorphic to the manifold M , but this is not always the case, of course. It is important then to connect Souriau’s space of motions with the Hamiltonian systems discussed in Sect. 2.2. Souriau’s original discussion is in [24, pp. 128–132]; here one begins with the following lemma due essentially to Cartan, see [3, p. 376]. Lemma 1 Let (M, ω, H) be a Hamiltonian system on the symplectic manifold (M, ω). Define N = M × R, and set Ω = p∗1 ω + (p∗1 dH) ∧ (p∗2 dt), in which p1 : N → M and p2 : N → R are the canonical projection maps. Then, (N, Ω) is a presymplectic manifold. Proof The two-form Ω is closed. Moreover, for each (m, t) ∈ N , ker(m,t) Ω = ∂ {αXH (m) + α ∂t : α ∈ R}, so that the dimension of ker(m,t) Ω is equal to 1 for all (m, t) ∈ N .
Note that if m(t) is an integral curve of XH and one sets n(t) = (m(t), t), then n (t) ∈ kern(t) Ω for all t. Following Souriau [24], one identifies the motions of the system described by (M, ω, H) with the leaves of the foliation induced by the integrable distribution ker Ω. Conversely, given a leaf one recovers, up to parameterizations, an integral curve of XH :

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Lemma 2 Let (M, ω, H) be a Hamiltonian system on (M, ω), and let (N, Ω) be the presymplectic manifold defined in Lemma 1. The integral curves of the Hamiltonian system (M, ω, XH ) can be obtained, up to parametrization, by projecting the leaves of the foliation ΦΩ into M . Proof If n(s) = (m(s), γ(s)) satisfies n (s) = 0 for all s and n (s) ∈ kern(s) Ω, then n(s) can be reparametrized to be of the form (m(t), t), where m(t) is an integral curve of XH . In fact, n (s) = (m (s), γ  (s)) ∈ kern(s) Ω implies m (s) = γ  (s)XH (m(s)), and since by hypothesis n (s) = 0 for all s, γ  (s) = 0 for all s; thus, if one sets t = γ(s), then n(t) = ((m ◦ γ −1 )(t), t), and n (t) = (XH (m ◦ γ −1 (t)), 1).
These two lemmas imply the following characterization result: Proposition 2 The space of motions UN = N/ker Ω, in which N and Ω are defined in Lemma 1, is a manifold. Moreover, the original symplectic manifold (M, ω) and UN are symplectomorphic. Proof The foregoing discussion implies that L ∈ UN can be described as a curve (m(t), t), in which m(t) is an integral curve of XH . Define the map λ : UN → M by λ(L) = m(0). Then, the fact that two different leaves have to determine two different initial conditions [10], imply that λ is a bijection. In order to see that UN is a manifold, and that λ is smooth and symplectic, one can proceed as follows: the flow box theorem [3, p. 66] says that for each m ∈ M there exists an open set m ⊆ M and a smooth map F : Um × I → M , in which I = (−a, a) with a > 0 or a = ∞, such that for each u ∈ Um , the curve cu : I → M given by cu (s) = F (u, s) is the integral curve of XH at u. Now, since the leaves of N through u ∈ Um are precisely the integral curves of XH , the submanifold Σ = Um × {0} is a slice for the foliation ΦΩ . Thus, UN is a manifold, and for (u, 0) ∈ Σ, the function λ is simply the projection λ(u, 0) = u. This is of course a bijective smooth symplectic map.
This proposition means that one is justified in considering the phase space as the space of classical solutions of the system at hand. This observation is at the core of the generalization of Souriau’s point of view to Lagrangian field theory by Crnkovi´c, Witten and Zuckerman [10, 27], as it will be shown in Sect. 5. Partial converses to Lemmas 1 and 2 are given by the next two results. Proposition 3 Let (M, ω) be a presymplectic manifold. There exists a symplectic manifold (P, Ω) and an embedding j: M → P such that j ∗ Ω = ω . Proof Set P = T ∗ M , and let the embedding j: M → P be the identification of M with the zero-section of P , that is, j(q) = 0q for all q ∈ M . Define a two-form Ω on P by Ω = ω0 + π ∗ ω , where ω0 is the canonical symplectic form of P and π: P → M is the canonical projection. One easily shows that for q ∈ M and Xq , Yq ∈ Tq M , j ∗ Ω(q)(Xq , Yq ) = ω(q)(Xq , Yq ). The form Ω is obviously closed. Non-degeneracy is checked thus: By the generalized Darboux theorem, there exists a coordinate chart (q i , ri , wj ) on M such that ω = dq i ∧ dri , where i = 1, . . . , n, rank ω = 2n, and j = 1, . . . , dim M − 2n. The two-form ω0 on P can be written as ω0 = dq i ∧ dpi + dri ∧ dsi + dwj ∧ dvj ,

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where pi , si , vj are the conjugate variables of q i , ri , wj respectively, and Ω becomes Ω = ω0 + π ∗ ω = dq i ∧ dpi + dri ∧ dsi + dwj ∧ dvj + dq i ∧ dri . ∂ ∂ ∂ j ∂ Write Xαq = Qi ∂q∂ i +Pi ∂p +Ri ∂r∂ i +S i ∂s i +W ∂w j +Vj ∂v . Then, Ω(Xαq , ·) = i j i i i i i i j Q dpi − Pi dq + Ri ds − S dri + Q dri − Ri dq + W dvj − Vj dwj , and so Ω(Xαq , ·) = 0 clearly implies Qi = Ri = Pi = S i = 0 for all i = 1, . . . , n, and W j = Vj = 0 for all j = 1, . . . , dim M − 2n.


Remark 3 A more precise result has been proven by Marle [18], who showed that if (M, ω) is a presymplectic manifold of dimension m = 2p + k , where 2p = rank(ω), then (M, ω) can be symplectically embedded into a symplectic manifold (P, Ω) with dim P = m + k , which is obviously less than 2m = dim P . Marle’s result is relevant for geometric quantization [26]. Proposition 4 Let (M, ω) be a presymplectic manifold, (P, Ω) be the symplectic manifold defined in Proposition 3, and suppose that the space of motions UM = M/ker ω possesses a manifold structure. Assume further that the distribution ker ω is of dimension one. Then, there exists an open set U ∗ ⊆ P and a function H: U ∗ → R such that the flow of the (local) Hamiltonian vector field determined by H projects, via π: P → M , onto the leaves of the foliation Φω of M . Proof Let m ∈ M , and consider the leaf φ of Φω to which m belongs. If m ∈ φ and m = m , there is a vector field X defined on an open subset U of M , and everywhere in ker ω , such that m and m lie on the same integral curve of X , c(t), say, and furthermore, one can assume that c (t) = 0 for all t [4, p. 333]. As φ is one-dimensional, φ (locally) is the integral curve. By the straightening  of U , with m ∈ U  , and a out theorem [4, p. 247] there exists an open subset U i  coordinate system (q ) on U such that  ∂ d  i X(q ) = 1 = (t + q 1 , q 2 , . . . , q n ), ∂q dt t=0  , and consider the vector field X ∗ on U ∗ where n = dim M . Set U ∗ = π −1 U d 1 2 given by X ∗ (q i , pi ) = dt (t + q , q , . . . , q n , pi ), where (q i , pi ) are the fibre t=0 coordinates on U ∗ constructed from the coordinates (q i ) on U . A trivial computation shows that (T π)X ∗ (q i , pi ) = X(q i ) and therefore the projection via π of the flow of X ∗ is the flow of X . Moreover, if Y ∈ T(qi ,pi ) T ∗ M then (iX ∗ Ω)(q i , pi )Y = (iX ∗ ω0 )(q i , pi )Y + (π ∗ ω)(q i , pi )(X ∗ (q i , pi ), Y ) = (iX ∗ ω0 )(q i , pi )Y + ω(q i )((T π)X ∗ (q i , pi ), (T π)Y ) = (iX ∗ ω0 )(q i , pi )Y

because X(q i ) ∈ ker(qi ) ω . Thus, since ω0 (q i , pi ) = dq i ∧ dpi , (iX ∗ Ω)(q i , pi ) = (dq i ∧dpi )(X ∗ (q i , pi ), ·) = dp1 (q i , pi ),

and therefore, defining H: U ∗ → R by H(q i , pi ) = p1 (q i , pi ), one obtains iX ∗ Ω = dH .


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3 A spinning particle in a gravitational field The goal of this section is to find the equations of motion of a spinning particle in an arbitrary space–time (M, g) by using Souriau reduction ab initio. Latin indices will be lowered/raised by using the Minkowski metric ηab = diag(−1, 1, 1, 1); greek indices will be lowered/raised by using the spacetime metric gαβ . Hereafter (M, g) will denote an arbitrary spacetime. A local moving frame {fa } on M is a set of local vector fields defined on an open set U ⊆ M such that for all m ∈ U , {fa (m)} ⊂ Tm M is a basis of Tm M . The set of local one-forms {θa } given by θa (fb ) = δba is called the dual local moving coframe. If (U, xα ) is a chart in M , one can write fb = fbα ∂x∂α and θa = θβa dxβ . It is then trivial to check that δba = fbα θαa and that δβα = faα θβa . Let LMm be the set of all the bases {fa (m)} of Tm M satisfying 

g(m)(fa (m), fb (m)) = ηab ,

and define LM = m∈M LMm . The set LM equipped with the Lorentz group O(1, 3) as structure group is called the Lorentz bundle of M [14]. A local moving frame {fa } defined on an open set U ⊆ M and satisfying {fa (m)} ∈ LM for all m ∈ U is called a tetrad or orthonormal moving frame. Assuming that M is connected, time orientable and orientable, one restricts LM to LM, the set of all oriented and future-pointing tetrads on M . The proper homogeneous orthochronous Lorentz group O++ (1, 3) acts on LM by the restriction of the action of O(1, 3) on LM . The 10-dimensional fibre bundle LM will be called the restricted Lorentz bundle over (M, g). Orthonormal moving frames {ea } will have the following interpretation: the vector field e0 is the direction of the 4-momentum P of the spinning particle, and the vector field e1 the direction of its 4-spin S . In local coordinates xα , P = ∂ α ∂ α ∂ pα ∂x∂α = meα 0 ∂xα , and S = s ∂xα = se1 ∂xα , in which the numbers m > 0 and s ∈ R represent the mass and spin magnitude of the particle respectively. One [α β] also defines the contravariant spin tensor as S αβ = 2se2 e3 , so that the following constraints hold: pα pα = −m2 ,

sα sα = s2 ,

S αβ pα = pα sα = 0.

(4)

In the classical general relativistic approach (see [7] and references therein) a test particle (or “extended test body”) is described by an energy–momentum density T αβ , as stated in Sect. 1, and the motion of the body is described by a set of time-like curves which form a “world tube” W ⊂ M such that T αβ = 0 only inside W . The total momentum P α and angular momentum S αβ of the test body are defined by integrals depending on T αβ , where the integration is carried out over a compact space-like section Σ of W . A time-like curve l with unit tangent vector U α representing the body is chosen along W , and one then finds equations of motion for P α and S αβ . In order to completely determine the motion of a spinning body, one needs to add supplementary conditions and a natural one is precisely the third equation of (4), Pα S αβ = 0, see [7]. The equations of motion and this assumption then imply that Pα P α and Sα S α , where S α = − 12 αβγδ Uβ Sγδ , are constant along the curve l. Thus, the presymplectic viewpoint reviewed here abstracts the essential characteristics of the physical approach to the problem of motion.

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3.1 The evolution space Let {Xab } be a fixed basis of the Lorentz Lie algebra. Denote by {θa } the local coframe dual to a local orthonormal moving frame {ea }, and let ω  = ωba Xab be the unique torsion-free metric connection form of LM. The evolution space modeling a spinning particle moving in a gravitational field is (LM, ω ), in which ω = dθ, and θ is given by [15] θ({ea }) = m θ0 + s ω23 .

(5)

K¨unzle [15] then proves the following technical lemma: Lemma 3 The two-form ω = dθ, in which θ is given by (5), is a presymplectic form on LM. More explicitly, whenever ∆ = 1 + s2 m−2 R2323 is different from zero, the dimension of ker(ω) is either two (if s = 0) or four (if s = 0). 3.2 The space of motions It will be assumed, by restricting consideration to some open submanifold of LM if needed, that ∆ = 0 on LM. One would like to describe the leaves of the foliation induced by the distribution ker ω , that is, one would like to α α find tensorial equations for pα = m eα 0 and s = s e1 along the integral curves α α α ∂ c(λ) = (x (λ), ea (λ)) of a vector field Z = v ∂xα + Eaα ∂e∂α everywhere in a ker ω . Assume that an integral curve c(λ) = (xα (λ), eα a (λ)) of Z satisfies dxα = vα , dλ

and

deα a = Eaα . eλ

Then, v a = θa (Z) = θαa dxα (Z) = θαa v α , and [14, p. 142]  α α   deb α γ β α γ dx + Γβγ e˙ab = ωba (Z) = θαa Ebα + Γβγ eb v = θαa eb = θαa v β ∇β eα b. dλ dλ 1 The scalar ∆ of Lemma 3 becomes ∆ = 1 + m−2 Rαβγδ S αβ S γδ and, as long 4 as ∆ = 0, one finds the equations [15] 1 α λ µν p˙α = − Rλµν v S , 2  α  β δβ + eα 0 e0β s˙ = 0,  1 −2 −1 µ αβ α 0 α γδ mv = v p + m ∆ p S Rβµγδ S , 2

(6) (7) (8)

in which the dots indicate covariant differentiation along the vector v α . These equations are called the Dixon–Souriau equations of motion in the first paper cited in [7]. Note that (8) shows that in the general relativistic case the 4velocity and the 4-momentum of the particle are not necessarily parallel. In the physical literature, an equation analog to (8) first appeared in the last reference of [7]: it is an algebraic consequence of (6), (7) and the constraints (4). Also noteworthy is the fact that an equation formally identical to Eq. (6) can be derived for

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pseudo-classical spinning particles [23], but that, in contradistinction with Eq. (7), the theory of [23] implies that the spin tensor is covariantly constant. It appears to be unknown whether one can reproduce the theory reviewed in this section from the point of view of spinning spaces. Applications of Eqs. (6)–(8), as well as explicit solutions to the equations of motion appear, for example, in Ref. [7]. Remark 4 Consider the special case s = 0. As pointed out in the proof of Lemma 3, in this case the dimension of ker(ω) is 4, so that the space of motions M/ker ω is of dimension 6. Equations (6)–(8) describing the leaves the foliation induced by ker ω become simply p˙α = 0,

mv α = v 0 pα ,

so that the momentum is conserved along the motion, and the velocity is parallel to it. Thus, v˙ α = 0, that is, the leaves in LM project onto unique (time-like) geodesics on M . It follows that the set of time-like geodesics can be represented by LM/ker ω , and one concludes that if LM/ker ω is a manifold (a nontrivial restriction, see for example Zuckerman [27]) it can be equipped with a symplectic structure. Related results were proven by Cari˜nena and L´opez [8] in 1991 by using Lagrangian methods. 4 Charged spinning particles in an electromagnetic and gravitational field with torsion The goal of this section is to extend K¨unzle’s analysis to the case of a charged spinning particle interacting with an electromagnetic field and moving in a space– time whose Lorentz bundle is equipped with a connection compatible with the space–time metric, but not necessarily torsion-free. The work [20] by Rapoport and Sternberg is related to the present paper, but no explicit calculations appear in [20], in contradistinction with the treatment in this article. Torsion is usually associated with spin [9, 13, 25] and therefore the problem considered in this section is a natural one. It should be mentioned, though, that the interaction between spin and torsion is not the only aspect of modern research in which torsion appears: it is also of importance in the study of twistors, string theory, and supergravity, see [9, 11] and references therein. K¨unzle’s equations reviewed in Sect. 3 are recovered when the torsion is identically zero and, as in the torsion-free case, one finds that the velocity and momentum of the particle are not necessarily parallel. Another interesting consequence of the equations found here is that if the particle is spinless and the electromagnetic field vanishes, the particle moves along a geodesic of the unique torsion-free metric connection of the space–time, a result which coincides with observations made in [20, 25]. 4.1 The evolution space Let (Q, gαβ ) be a space–time, and assume that ω is a metric connection form on LQ whose torsion tensor does not necessarily vanish. As in Sect. 3, the arena for

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this system will be the restricted Lorentz bundle LQ. Also as before, latin indices will be raised/lowered by means of ηab = diag(−1, 1, 1, 1), while greek indices will be raised/lowered by means of gαβ . Let {θa } be the dual basis corresponding to {ea }, and let ω = ωba Xab , in which b {Xa } is a fixed basis of the Lorentz Lie algebra, be a metric connection form on LQ not necessarily torsion-free. For a given local moving frame {ea } define M =m+

1 µ S αβ Fαβ = m + µ s F23 2

and

θ = M θ0 + s ω23 + eA,

(9)

[α β]

in which S αβ = 2 s e2 e3 , the number e is the charge of the particle, µ ∈ R, µs is the magnetic dipole moment, and F = dA = 12 Frs θr ∧ θs . In analogy with α α Sect. 3, the vectors pα = M eα 0 and s = s e1 will represent the 4-momentum and 4-spin of the particle respectively. Lemma 4 The two-form Ω = dθ, in which θ is given by (5), is a presymplectic form on the restricted Lorentz bundle LQ. Proof The structure equations for a moving coframe in the presence of torsion are [14, p. 121] 1 dθa = −ωra ∧θr + Qars θr ∧θs , 2

1 a r s dωba = −ωra ∧ωbr + Rbrs θ ∧θ , 2

(10)

and moreover, ωba = −ωab and ωac = ηac ωbc . One computes dM = µsdF23 as follows:  ∂Fαβ β β β β α α dF23 = d Fαβ eα dxγ eα 2 e3 = 2 e3 + Fαβ e3 de2 + Fαβ e2 de3 . ∂xγ α γ β Now, recall that [14, p. 142] ωba = θαa (deα b + Γβγ eb dx ), and so r α α γ δ deα 2 = ω2 er − Γδγ e2 dx ,

and

β γ deβ3 = ω3r eβr − Γδγ e3 dxδ .

Thus, ∂Fαβ α β γ λ β α λ α β dF23 = Fr3 ω2r +F2r ω3r + e e dx −Fλβ Γγα e3 e2 dxγ − Fαλ Γγβ e2 e3 dxγ γ 2 3 ∂x  ∂Fαβ λ λ α β γ r r = − Γ F − Γ F γα λβ γβ αλ e2 e3 dx + Fr3 ω2 + F2r ω3 ∂xγ β γ = Fr3 ω2r + F2r ω3r + (∇γ Fαβ )eα 2 e3 dx β r γ = Fr3 ω2r + F2r ω3r + (∇γ Fαβ )eα 2 e3 θ er = θr ∇r F23 + Fr3 ω2r + F2r ω3r ,

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and therefore dM = µ s (∇r F23 θr + Fr3 ω2r + F2r ω3r ). Now expand the two-form Ω: Ω = d(M θ0 + sω23 + eA) = dM ∧θ0 + M dθ0 + s dω32 + e dA   0  1 0 r s r r r 0 r = µs ∇r F23 θ + Fr3 ω2 + F2r ω3 ∧θ + M −ωr ∧θ + Qrs θ ∧θ 2  1 2 r s 1 + s −ωr2 ∧ω3r + R3rs θ ∧θ + eFrs θr ∧θs 2 2  1 1 1 0 0 M Qrs + sR23rs + eFrs − µ sδr ∇s F23 θr ∧θs = 2 2 2   r 0 − M θ ∧ω0r − µsθ ∧ F2r ω3r + Fr3 ω2r − sω2r ∧ω3r . (11)

The next step is to determine the kernel of Ω. Let Z be a vector field on LQ, define v a = θa (Z) and e˙ ab = ωba (Z), and note that ηac e˙cb + ηbc e˙ca = 0 for all a, b = 0, . . . , 3. Substituting in (11) one finds  1 1 1 0 0 M Qrs + sR23rs + eFrs − µsδr ∇s F23 (v r θs − v s θr ) iZ Ω = 2 2 2   r   r r ˙ 2 ˙ −s er ω3 − e3 ω2r − M v ω0r + e˙0r θr     − µsv 0 F2r ω3r + Fr3 ω2r + µs F2r e˙ r3 + Fr3 e˙ r2   = M v s Q0sr + sv s Rsr23 + eFsr v s − µs∇r F23 v 0 − M e˙0r θr   + µs v r ∇r F23 + F2r e˙ r3 + Fr3 e˙ r2 θ0 − M v r ω0r     + s e˙r2 − µv 0 F2r ωr3 − s e˙r3 + µv 0 F3r ωr2 , and therefore, iZ Ω = 0 if and only if M Q0sr v s + sRsr23 v s + eFsr v s − µs∇r F23 v 0 = M e˙0r , (12) v r ∇r F23 + F2r e˙ r3 + Fr3 e˙ r2 = 0, (13)   −M v 1 ω01 − M v 2 ω02 − M v 3 ω03 + s e˙02 − µv 0 F20 ω03     ˙1   +s e2 − µv 0 F21 ω13 − s e˙03 + µv 0 F30 ω02 − s e˙13 + µv 0 F31 ω12 = 0. (14)

Equation (14) implies, assuming M and s different from 0, v 1 = 0,

e˙12 = µv 0 F21 ,

e˙13 = µv 0 F31 ,

−M v 2 − se˙03 − µsv 0 F30 = 0, −M v 3 + se˙0 − µsv 0 F 0 = 0. 2

2

(15) (16)

On the other hand, let ∆ := 1 + s2 M −2 R2323 + esM −2 F23 + M −1 sQ023 .

(17)

Expanding (12) and using v 1 = 0 one finds M e˙0r = M Q00r v 0 + M Q02r v 2 + M Q03r v 3 − sRr023 v 0 − sRr223 v 2 − sRr323 v 3 + eF0r v 0 + eF2r v 2 + eF3r v 3 − µsv 0 ∇r F23 ,

(18)

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E. G. Reyes

and therefore (15) and (16) imply   M e˙02 = M Q002 v 0 + Q032 se˙ 02 − µsv 0 F20 − sR2023 v 0     − sM −1 R2323 se˙ 02 − µsv 0 F20 + eF32 M −1 se˙ 02 − µsv 0 F20 + eF02 v 0 − µsv 0 ∇2 F23 ,

that is,



 sM −1 Q023 + s2 M −2 R2323 + esF23 M −2 + 1 M e˙ 02

= M Q002 v 0 + Q023 µsv 0 F20 − sR2023 v 0 + s2 µM −1 R2323 v 0 F20 + eF02 v 0 − eF32 M −1 µsv 0 F20 − µsv 0 ∇2 F23 ,

and it follows that M e˙ 02 = v 0 ∆−1 (sR0223 + [e − µM (1 − ∆)]F02 + M Q002 − µs∇2 F23 ).

(19)

Analogous computations yield M e˙ 03 = v 0 ∆−1 (sR0323 + [e − µM (1 − ∆)]F03 + M Q003 − µs∇3 F23 ) and    0 M e˙ 1 = v 0 M Q001 + sR0123 + eF01 − µs∇1 F23 − v 2 M Q012    + sR1223 + eF12 − v 3 M Q013 + sR1323 + eF13 .

(20) (21)

Substituting (19) and (20) into (15) and (16) one finds   v 2 = −sv 0 ∆−1 M −2 sR0323 + [e − µM (1 − ∆)]F03 + M Q003 − sµ∇3 F23  − µsv 0 M −1 F30 = −sv 0 ∆−1 M −2 sR0323  + (e − µM )F03 − sµ∇3 F23 + M Q003 , (22) and

  v 3 = sv 0 ∆−1 M −2 sR0223 + (e − µM )F02 − sµ∇2 F23 + M Q002 .

(23)

Thus, if ∆ = 0 and M = 0, all of the v a and e˙ cb are either zero or multiples of v 0 , except for e˙ 23 which remains arbitrary. The dimension of ker Ω is therefore 2 for this generic case. On the other hand, if s = 0, then M = m, ∆ = 1, v 1 = v 2 = v 3 = 0, me˙ 0r = mQ00r v 0 + eF0r v 0 for r = 1, 2, 3, and v 0 , e˙ 12 , e˙ 13 , and e˙ 23 are arbitrary, and so the dimension of ker Ω is 4.
4.2 The space of motions One now forms the space of motions ULQ := LQ/ker Ω and projects the leaves of the foliation induced by ker Ω into Q. Tensorial equations for the integral curves of a vector field everywhere in ker Ω are found as follows. Suppose that Z = v α ∂x∂α + Eaα ∂e∂α is a (local) vector field on LQ everywhere a in ker Ω, and let c(λ) = (xα (λ), eα a (λ)) be an integral curve of Z . As in Sect. 3, set d α d α x = v α , and e = Eaα , dλ dλ a

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and recall that v a = θa (Z) = θαa dxα (Z) = θαa v α and e˙ab = θαa v β ∇β eα b . One obtains, 1 1 1 ∆ = 1 + M −2 Rαβγδ S αβ S γδ + M −2 Fαβ S αβ − M −2 Qαβγ pα S βγ , (24) 4 2 2

and also vα =

 α dxα 0 = v a eα e0 − ∆−1 M −3 Qµβγ pµ pβ S γα a =v dλ 1 − ∆−1 M −3 pµ Rµβγδ S γδ S αβ +M −3 ∆−1 (e−µM )pγ Fγβ S βα 2 1 −2 −1 βγ αδ + M ∆ µ∇δ Fβγ S S , 2

in which Eqs. (22) and (23) have been used. Thus, 

 1 µ 1 p Rβµγδ + µM ∇β Fγδ S γδ M v α = v 0 pα + 2 S αβ M ∆ 2  + (e − µM )pλ Fβλ + Qµγβ pµ pγ .

(25)

This equation reduces to the one found in Sect. 3 if Qαβγ and Fγδ are identically zero. The presence of non-vanishing torsion, therefore, does not alter in general the fact that for spinning particles, the velocity and momentum are not parallel. From Eq. (12) one finds 1 M e˙ r0 eµr = −Qβαδ pβ erδ eµr v γ − Rβαγδ v α erβ eµr S γδ 2 1 0 rβ µ α + eFαβ e er v − µv ∇α Fβγ S βγ erα eµr , 2 µ λα and using erα eµr = η rk eα = η ab eλa eα k er and g b , one obtains

1 µ α γδ 1 M e˙ r0 eµr = −Qβα µ pβ v α − Rαγδ v S + eFαµ v α − µv 0 ∇µ Fβγ S βγ . 2 2

The left hand side of this equation is equal to v α ∇α v α ∇α

pµ , and therefore M

pµ 1 µ α γδ 1 = −Qβα µ pβ v α − Rαγδ v S + eFαµ v α − µv 0 ∇µ Fβγ S βγ . (26) M 2 2

An analogous equation has been obtained by Trautman (second reference of [25]) for the special case Fαβ ≡ 0. Finally, recall that e˙ a1 = θβa v α ∇α eβ1 = θβa v α ∇α (sβ /s). Using Eq. (14) one 0 ν λ α λ α obtains s˙ α − (gγβ eγ0 s˙ β )eα 0 = µv Fνλ se1 (e3 e3 + e2 e2 ), and therefore      λα  0 ν λ α λ α 0 ν λ α s˙ β δβα + e0β eα + eλ0 eα 0 = µv Fνλ s e3 e3 + e2 e2 = µv Fνλ s g 0 − e1 e1     0 β ν α α = µv 0 Fνλ sν g λα + eλ0 eα 0 = µv Fν s δβ + e0 e0β ,

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E. G. Reyes

that is,  α  β   α  β  0 β ν 0 β ν δβ + eα = δβ + eα = 0. 0 e0β s˙ − µv Fν s 0 e0β s˙ + µv Fν s

(27)

Equations (25)–(27) are the required equations of motion. The following is an interesting consequence of (25)–(27): Proposition 5 If the tensor fields S αβ and Fαβ are identically zero, then the particle moves along a geodesic of the unique torsion free, metric-compatible connection of the space–time (Q, g), that is, the presence of a torsion field does not alter the motion of spinless particles. β γ α 0 α Proof From Eqs. (25) and (26), p˙α = Qα βγ p v and mv = v p . Thus,

m α m β γ v˙ = 0 Qα βγ v v = 0 0 v v

as Q is antisymmetric, and therefore v˙ α = 0. Now one can write, for arbitrary vector fields X, Y , ∇X Y =

1 1 (∇X Y + ∇Y X + [X, Y ]) + T (X, Y ), 2 2

in which T (X, Y ) is the torsion tensor associated with ∇. The derivation ∇ = (1/2)(∇X Y + ∇Y X + [X, Y ]) is a covariant derivative with Christoffel symbols  k Γijk = (1/2)(Γijk + Γji ), where Γijk are the Christoffel symbols of ∇. It is metric preserving and it has vanishing torsion, and therefore it is the unique metric and torsion free connection on the space–time Q. It follows from [14, p. 146] that the geodesics of ∇ and ∇ coincide.
5 Motion of a string in interaction with a scalar field 5.1 The Lund–Regge equations The equations of motion for a string in interaction with a scalar field moving in Minkowski spacetime reduce to the Lund–Regge equations [17, 16]   2 2 ∂λ ∂λ ∂2θ ∂2θ cos θ − 2 − sin θ cos θ + − =0 ∂x2 ∂t ∂x ∂t sin3 θ   ∂ ∂ 2 ∂λ 2 ∂λ cot θ − cot θ = 0. ∂x ∂x ∂t ∂t

(28) (29)

Here, θ(x, t) determines the intrinsic Riemannian metric ds2 = cos2 θ dx2 + sin2 θ dt2 on the world-surface described by the string, and λ(x, t) determines its extrinsic curvature Lµν as follows: L12 = cot θ

∂λ , ∂t

1 ∂λ (L11 + L22 ) = cot θ , 2 ∂x

1 (L11 − L22 ) = sin θ cos θ. 2

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A most interesting characteristic of Eqs. (28) and (29) is that they are the integrability condition of a linear problem with spectral parameter [16], and that one can use this linear problem to find exact soliton solutions to (28) and (29) by scattering/inverse scattering techniques (see the second and third papers of [16]). More recently, an infinite-parameter family of exact, explicit multivortex solutions to (28) and (29) has been obtained by Barashenkov, Shchesnovich and Adams [6] using a different approach. Another important consequence of the existence of an associated linear problem for (28) and (29) is that these equations possess an infinite number of local conservation laws. They were found by Lund [16] and have been re-derived using a purely geometrical approach in [21]. Crucial for the present paper is the fact that Eqs. (28) and (29) are the Euler– Lagrange equations corresponding to the Lagrangian 1 L= 2

 

∂θ ∂t



2 −

∂θ ∂x



2 − sin θ + cot θ 2

2

∂λ ∂t



2 −

∂λ ∂x

2  dx,

(30) as it can be readily checked. As stated in Sect. 1, the goal of this section is to construct a phase space for (28) and (29) as in the work by Witten, Crnkovi´c and Zuckerman [10, 27]. The following subsection is a review of these important developments after Zuckerman’s rigorous exposition [27] using the variational bicomplex: the main references used are [2] and the recent paper [22]. Its application to the Lund–Regge equation is carried out in Sect. 5.3.

5.2 Zuckerman’s universal current Consider a fixed fiber bundle π: E → M and let J ∞ E be the infinite jet bundle of π: E → M , see [2, 27]. M is the space of independent variables and the typical fiber the space of dependent variables. Local fibered coordinates on E will be denoted by (xi , uα ), 1 ≤ i ≤ n and 1 ≤ α ≤ m. In these coordinates the projection map π: E → M is simply π: (xi , uα ) → (xi ). For any p ≥ 0, a differential p-form ω on J ∞ E may be written in canonical α coordinates (xi , uα , uα i , . . . , ul1 ,...,lq , . . .) as a finite linear combination of terms of the form  i1  α1 αs α ir A xi , uα , uα i , . . . , ui1 ,...,ik dx ∧· · ·∧dx ∧duj1 ,...,jq ∧· · ·∧dul1 ,...,lq , 1

s

(31)

in which r + s = p. The space of p-differential forms on J ∞ E will be denoted by Ωp (J ∞ E). E) admits a bigrading by horizontal and vertical degree. One The space Ωp (J ∞ writes Ωp (J ∞ E) = r+s=p Ωr,s (J ∞ E), in which Ωr,s (J ∞ E) is the space of all r-horizontal (i.e. r-forms generated by the differentials dxi ) and s-contact (i.e. si α α forms generated by the one-forms θα = duα − uα i dx and θi1 ,...,ij = dui1 ,...,ij − α i ∞ p ui1 ,...,ij i dx ) differential forms on J E . The exterior derivative d: Ω (J ∞ E) →

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E. G. Reyes

Ωp+1 (J ∞ E) can then be decomposed into the sum d = dH + dV of horizontal and vertical differentials. In coordinates, one sets Dxj =

∂ ∂ ∂ ∂ + uα + ui1 j α + ui1 i2 j α + · · · . j ∂xj ∂uα ∂ui1 ∂ui1 i2

and defines dH and dV by means of  dH f = (Dxi f ) dxi ,

(32)

(33)

i

∂f α ∂f α ∂f α θ + θi + α θij + · · · , ∂uα ∂uα ∂u i ij  i α dH (dx ) = 0, dH θi1 ,...,ik = dxj ∧ θiα1 ,...,ik j , dV f =

(34) (35)

j

dV (dxi ) = 0,

dV θiα1 ,...,ik = 0,

(36)

α so that, for example, dV uα i1 ...ik = θi1 ,...,ik . The variational bicomplex for the fiber ∗,∗ bundle E is the double complex (Ω (J ∞ E), dH , dV ) of differential forms on the infinite jet bundle J ∞ E . Writing Ω∗,∗ for Ω∗,∗ (J ∞ E), this important bicomplex looks like follows:

.. .. . . ↑ dV ↑ dV ↑ dV ↑ dV dH dH dH dH 0 → Ω0,2 → Ω1,2 → ··· → Ωn−1,2 → Ωn,2 ↑ dV ↑ dV ↑ dV ↑ dV dH dH dH dH 0 → Ω0,1 → Ω1,1 → ··· → Ωn−1,1 → Ωn,1 ↑ dV ↑ dV ↑ dV ↑ dV dH 0,0 dH 1,0 dH n−1,0 dH 0 → R → Ω → Ω → ··· → Ω → Ωn,0

(37)

The space Ωn,1 (J ∞ E) possesses a distinguished subspace E n+1 (E) of all source forms on J ∞ E [2]: one says that ω ∈ Ωn,1 (J ∞ E) is a source form if in any local system of coordinates (xi , uα ) on E the form ω looks like   β α 1 n ω = Pβ xi , uα , uα i , . . . , ui1 ,...,ik du ∧ dx ∧ · · · ∧ dx , α β 1 n or, equivalently, ω = Pβ (xi , uα , uα i , . . . , ui1 ,...,ik ) θ ∧ dx ∧ · · · ∧ dx . Their importance is due to the following lemma [2, 27].

Lemma 5 Assume that ω ∈ Ωn,1 (J ∞ E). Then, ω can be uniquely written as (38)

ω = ω1 + dH η,

in which ω1 ∈ E

n+1

(E) is a source form and η ∈ Ω

n−1,1

∞

(J E).

1 Suppose now that one fixes a Lagrangian density λ = L(xi , uα , uα i , . . .) dx ∧ n n,0 ∞ · · · ∧ dx in Ω (J E). The construction of Zuckerman’s universal current (denoted hereafter by U (λ)) proceeds as follows: the vertical exterior derivative dV λ belongs to Ωn,1 (J ∞ E), and the last lemma implies that one can write

dV λ = E(λ) + dH η

(39)

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uniquely, in which E(λ) is a source form and η ∈ Ωn−1,1 (J ∞ E). The differential form E(λ) is the Euler–Lagrange operator evaluated at L (see [2]) that is, E(λ) = Eα (L) duα ∧ dx1 ∧ · · · ∧ dxn , in which    ∂L ∂L ∂L Eα (L) = − Dxi − ··· . + Dxi Dxj ∂uα ∂uα ∂uα i ij One now defines U (λ) ∈ Ωn−1,2 (J ∞ E) by (40)

U (λ) = dV η.

Then, dV U (λ) = 0, and moreover, dH U (λ) ∈ Ωn,2 (J ∞ E) vanishes on solutions uα (xi ) to the Euler–Lagrange equations Eα (L) = 0. Indeed, on solutions to Eα (L) = 0, Eq. (39) becomes dV λ = dH η , and therefore 0 = dV dV λ = dV dH η = −dH dV η = −dH U (λ).

(41)

After Zuckerman [27], one says that the differential form U (λ) is a conserved current for the Euler–Lagrange equations Eα (L) = 0 : Definition 2 Fix a form λ ∈ Ωn,0 (J ∞ E) as above. A differential form K ∈ Ωn−1,q (J ∞ E), q = 0, 1, 2, . . . , is a conserved current for λ (or, for the Euler– Lagrange equations Eα (L) = 0) if dH K = 0

whenever uα (xi ) is a solution to the equations Eα (L) = 0. The conserved currents of Definition 2 are a generalization of the standard conservation laws in field theory: usually one thinks of a conservation law as a differential form K ∈ Ωn−1,0 (J ∞ E) which is closed on solutions, see [2, 3, 15, 21] and references therein. The importance of these new conserved currents, also called higher-degree or form-valued conservation laws, has been recognized only recently [2]. Of course, Definition 2 extends mutatis mutandis to arbitrary systems of partial differential equations. Thus, for example, it is straightforward to check that the canonical symplectic form of Hamiltonian mechanics (see Sect. 2) is a formvalued conservation law for Hamilton’s equations: Proposition 6 The symplectic form ω = dq i ∧ dpi is a conserved current for Hamilton’s equations ∂H dq i = , dt ∂pi

∂H dpi =− i, dt ∂q

i = 1, . . . , n.

(42)

Proof Consider the trivial fiber bundle E given by (t, q i , pi ) → (t). Then ω = dq i ∧ dpi is a (0, 2)-valued differential form on J ∞ E . The total derivative Dt is given by ∂ ∂ ∂ ∂ ∂ + q˙i i + p˙i Dt = + q¨i + p¨i + ··· , ˙ i ∂t ∂q ∂pi ∂ p˙i ∂q

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E. G. Reyes

and therefore one computes dH ω as follows: dH ω = d(Dt q i ) ∧ dt ∧ dpi − dq i ∧ d(Dt pi ) ∧ dt = dq˙i ∧ dt ∧ dpi − dq i ∧ p˙i ∧ dt.

Now, whenever (q(t), p(t)) is a solution to Hamilton’s equations one has dH ω = −d(∂H/∂pi ) ∧ dpi ∧ dt + dq i ∧ d(∂H/∂q i ) ∧ dt  2 ∂ H ∂2H j =− dq + dp j ∧ dpi ∧ dt ∂pi ∂q j ∂pi ∂pj  2 ∂ H ∂2H j + dq i ∧ dq + dp j ∧ dt ∂q i ∂q j ∂q i ∂pj =−

∂2H ∂2H dq j ∧ dpi ∧ dt + i dq i ∧ dpj ∧ dt = 0. j ∂pi ∂q ∂q ∂pj




Definition 3 (a) The solution variety SL associated with a Lagrangian density λ = Ldx1 ∧ · · · ∧ dxn is the set of all local smooth sections ψ : (xi ) → (xi , uα (xi ))

of the bundle E such that uα (xi ) is a solution to the Euler–Lagrange equations Eα (L) = 0. (b) For each ψ ∈ SL , the tangent space Tψ SL at ψ is the set of all vector fields δψ = Gα

∂ ∂ ∂ + Di1 Gα α + Di1 Di2 Gα α + · · · α ∂u ∂ui1 ∂ui1 i2

(43)

on the infinite jet bundle J ∞ E such that Gα (ψ) (the function Gα evaluated on the solution ψ ) satisfy the Jacobi equations, that is, the linearization of the Euler–Lagrange equations at ψ . Zuckerman’s main result [27] is the following: Theorem 3 For any Lagrangian density λ = L dx1 ∧ dx2 ∧ · · · ∧ dxn ∈ Ωn,0 (J ∞ E), consider the associated differential forms η ∈ Ωn−1,1 (J ∞ E) and U (λ) ∈ Ωn−1,2 (J ∞ E) defined in (39) and (40) respectively. Then, U (λ) is a conserved current for the Euler–Lagrange equations Eα (L) = 0. Moreover, (a) Suppose that C is a compact, oriented (n − 1)-dimensional submanifold of M . Define   θC =

η C

and

ωC =

U (λ), C

that is, for any solution ψ ∈ SL and any δ1 ψ, δ2 ψ ∈ Tψ SL ,  θC (ψ)δ1 ψ = ψ ∗ (iδ1 ψ η) and ωC (ψ)(δ1 ψ, δ2 ψ) C    ψ ∗ iδ2 ψ iδ1 ψ U (λ) . = C

Then, ωC = dθC and dωC = 0.

On the motion of particles and strings

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(b) The two-form ωC does not depend on the submanifold C . The two-form ωC is the (pre)symplectic form on the space of solutions SL one was trying to obtain. Special cases of the differential form U (λ) and the corresponding two-form ωC on SL have appeared several times in the literature. For example, by the time [27] appeared, instances of ωC had already been used to describe a covariant Hamiltonian formalism for Yang–Mills theory and general relativity (Crnkovi´c and Witten [10], Ashtekar et al. [1], Woodhouse [26]). Some subsequent work on the subject is listed in [1]. The important question of when one can assure that ωC is in fact symplectic is briefly considered in the last work of [10], in [27] and in the papers by Lee and Wald, and Torre cited in [1]. A recent short discussion on this issue also appears in [22]. One now proceeds in analogy with Sect. 2 [10, 1, 27]: Assume that the kernel of the two-form ωC determines a foliation on SL . Then, the covariant phase space of the Lagrangian theory at hand is SL /ker (ωC ). 5.3 A symplectic structure for the Lund–Regge equations Since the Lund–Regge Lagrangian is of first order, the theory of Sect. 5.2 will be 1 n applied to a Lagrangian density of the form λ = L(xi , uα , uα i ) dx ∧ · · · ∧ dx . It was checked in [22] (and also in [10, 26] using some formal computations) that in this case dV λ = Eα (L)θα ∧ ν + dH η , in which η=

∂L α θ ∧ νi ∂uα i

(44)

and νi = (−1)i dx1 ∧· · ·∧dxi−1 ∧dxi+1 ∧· · ·∧dxn . It follows that the differential forms θC and ωC on SL read [22, 26]   ∂L α θC (ψ)δψ = ψ ∗ (iδψ η) = (45) α G νi , C C ∂ui in which δψ is given by (43) and ωC (ψ)(δ1 ψ, δ2 ψ)    ∂2L ∂2L β α β α β β α α [G1 G2 − G2 G1 ] + [(Dj G1 )G2 − (Dj G2 )G1 ] νi = ∂uβ ∂uα ∂uβj ∂uα C i i

(46) respectively, in which δ1 ψ and δ2 ψ are vectors in Tψ SL as in (43). In the special case of the Lund–Regge Lagrangian (30), consider the corresponding space SL of local solutions to (28) and (29). One sets ∂ ∂ ∂ ∂ ∂ ∂ + Gλ + Dx Gθ + Dt Gθ + Dx Gλ + Dt Gλ +··· , ∂θ ∂λ ∂θx ∂θt ∂λx ∂λt (47) and obtains the formulae  θC (ψ)δψ = (λt cot2 θ Gλ + θt Gθ ) dx + (λx cot2 θ Gλ + θx Gθ ) dt (48)

δψ = Gθ

C

458

and ωC (ψ)(δ1 ψ, δ2 ψ) =

E. G. Reyes

 

 cos θ   2 Gθ2 Gλ1 − Gλ2 Gθ1 λt 3 + Gθ2 Dt Gθ1 sin θ C    − Gθ1 Dt Gθ2 − Gλ1 Dt Gλ2 − Gλ2 Dt Gλ1 cot2 θ dx (49)   cos θ    + 2 Gθ2 Gλ1 − Gλ2 Gθ1 λx 3 + Gθ2 Dx Gθ1 − Gθ1 Dx Gθ2 sin θ   λ − G1 Dx Gλ2 − Gλ2 Dx Gλ1 cot2 θ dt, (50)

in which ψ is a local solution to the Lund–Regge equations and δψ , δ1 ψ and δ2 ψ are vectors in Tψ SL . The symplectic manifold (SL , ωC ) is the covariant phase space for the Lund–Regge equations. Acknowledgements The author first learned some symplectic geometry from his teachers at the University of Saskatchewan during his M.Sc. studies: He is particularly grateful to his advisor, Prof. J.A. Brooke for his guidance and friendship. The computations of Sect. 4 first appeared in his unpublished 1993 M.Sc. thesis.

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