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Classical History Theory of Vector Fields

We consider the extension of classical history theory to the massive vector field and electromagnetism. It is argued that the action of the two Poincare groups introduced by Savvidou suggests that the history fields should have five components. The extra degrees of freedom introduced to make the fields five-dimensional result in an extra pair of second class constraints in the case of the massive vector field, and in an extended gauge group in the case of electromagnetism. The total gauge transformations depend on two arbitrary parameters, and contain `internal' and `external' U(1) gauge transformations as subgroups.


Motivation
The Hamiltonian formalism provides a strong link between classical and quantum theories and is mathematically well-developed in both cases. However, a major drawback of the standard Hamiltonian approach to relativistic field theories is that it requires a splitting of space-time into space and time, thus breaking the manifest covariance of the theory. This problem becomes particularly acute when canonical methods are applied to generally covariant theories such as general relativity, and this is one aspect of the 'problem of time' in canonical quantum gravity.
Recently a Hamiltonian formalism has been proposed by Isham and coworkers [1,2,3], in which the fundamental physical entities are entire histories of the system under consideration, as opposed to Cauchy data at an instant of time. For a thorough introduction see [5]. The histories formalism was originally developed in the quantum case, but there is a well-defined classical history formalism. The central object in a classical history theory is the space of histories, Π, which is defined to be a one-parameter family of singletime phase spaces. An element of Π is called a history, and the space of histories carries a symplectic structure which provides the crucial link to the corresponding quantum history theory.
A particularly striking fact about history theories is that there are two notions of time evolution [6]. External time corresponds to the parameter labeling the copies of state space, and internal time is generated by Hamiltonian evolution. In this way the notion of time plays two different roles in a history theory, one corresponding to the causal ordering of logical propositions, and the second corresponding to dynamical evolution. The two times are linked together in the action principle.
The histories formalism has been applied to scalar field theory on flat [7] and curved [9,10] space-times, and in the following we discuss the extension of the classical theory to the case of vector fields. In particular we examine the massive vector field and the electromagnetic field from a histories perspective. We shall argue that, as a consequence of the two time directions, a history field should be regarded as a certain type of vector field on a five-dimensional space-time. We also give an analysis of the gauge transformations of histories electromagnetism.
These results are relevant to the ultimate goal of the histories programme: the formulation of a histories version of general relativity [8]. Firstly, they suggest the possibility that a covariant history theory of gravity should be concerned with the metric of a five-dimensional extended space-time. Secondly, gauge symmetry is an important feature of general relativity, particularly when formulated in terms of tetrad fields. The study of the extended gauge transformations of histories electromagnetism is relevant in this context.

Scalar Field Theory
In this section we give a brief account of the histories description of the classical scalar field on Minkowski space-time, M , as given by Savvidou [7]. We define the Minkowski metric η µν to have signature (+, −, −, −).

The history algebra
The canonical configuration space of the scalar field is Q n,t = C ∞ (Σ n,t ) where Σ n,t is a Cauchy surface in M . The Cauchy surfaces are labeled by a future pointing timelike unit vector n normal to Σ n,t , and a real number t. Each Cauchy surface represents an instant of time in a particular inertial frame. The state space P n,t is the cotangent bundle of Q n,t which we identify with C ∞ (Σ n,t ) × C ∞ (Σ n,t ) 1 . The construction of the corresponding history theory begins by defining a trivial vector bundle 2 ξ n : P × R → R such that ξ −1 n (t) = P n,t . Here P is an abstract copy of the state space. The space of histories of the scalar field corresponds to the space of sections of this bundle, Π n = Γ(ξ n ), and the history fields satisfy the following Poisson algebra: 1 The analytical subtleties regarding these infinite dimensional spaces will not concern us here. We shall only be interested in the resulting Poisson algebra.
2 In this paper we will not make use of this bundle structure, and we could just have defined Π n to be the space of paths R → P n . However, the bundle picture is useful in more general situations (eg., on curved space-times), and also gives a motivation for the 'internal' / 'external' nomenclature in history theory: internal transformations act internally to the fibres of ξ n , while external transformations act across the fibres.
where δ (3) n,t (x − x ′ ) is the delta function on Σ n,t . The right hand side of this equation is a space-time scalar density of weight one. In Minkowski space-time, there is no difference between a scalar density and a scalar (if we restrict attention to transformations under the connected part of the Poincare group), which suggests that the fields can be thought of as spacetime scalars. A pair (t, x) ∈ R ×Σ can be identified with a unique four-vector X ∈ M as X = tn + x n , where the three-vector x has been associated with a corresponding four-vector x n that is n-spatial (i.e., n · x n := η µν n µ x ν n = 0). Hence we can write φ n (t; x) as φ n (X). The history algebra can then be written in the more covariant looking form Now it is tempting to drop the n label from the fields since the right hand side of this algebra does not depend on n. However, this would be somewhat misleading in the sense that the field π(X) has no physical meaning. This is because the conjugate momentum corresponds to the field momentum along a certain timelike direction, and so must be written as π n (X). Although the algebra (4), (5), (6) is independent of n, Savvidou [7] has shown that in the quantum theory n labels the physically relevant, inequivalent representations of the algebra on a particular Fock space. So in the analysis of the classical theory the n-labels remain on the fields on the understanding that they are necessary for the physical interpretation of the theory and arise naturally in quantisation. However, we note that it is rather unsatisfactory to have an n label on the φ field because, physically, the value of the field at a point in space-time is independent of the foliation. In this histories formulation of classical scalar field theory all propositions about the field are made in the context of a particular inertial reference frame 3 .

Time translations
For each n, a 'Louville' operator can be constructed from the fields, which generates external time translations in the n direction. In coordinates adapted to n these transformations take the form φ n (t, x) → φ n (t + λ, x), and similarly for π n (t, x). There is another notion of time translation in the history theory. Intuitively the Hamiltonian at each instant of external time generates dynamical evolution internal to each fibre of ξ n . More precisely, the time-averaged Hamiltonian generates transformations φ n (X) → φ n (X, s).
The action operator is made up of the Louville and Hamiltonian operators as follows: S n := V n − H n (9) and the equations of motion can be written in the form {S n , φ n (X)} = 0 (10) {S n , π n (X)} = 0 (11)

Poincare covariance
Savvidou [7] has shown the existence of two Poincare groups in the histories formulation of the scalar field. The n-spatial components of the two groups are identical, but the time translations of the internal Poincare group are generated by the Hamiltonian while the time translations of the external Poincare group are generated by the Louville operator. External boosts correspond to the following automorphism of the history algebra: where we have used adapted coordinates to write φ n (X, 0) = φ n (t, x, 0) and Λ(t, x) denotes the usual Lorentz transformations acting on inertial coordinates (t, x). We note that, in the classical case, these automorphisms cannot be generated by canonical transformations. This is because there is no momentum conjugate to the foliation vector n, and thus no way to generate changes in n. The 'multisymplectic' approach [4] offers a solution to this problem in the classical theory, and in the quantum theory changes in n correspond to mapping between inequivalent representations of the history algebra.
Internal boosts act on the fields as follows: π n (0, x, s) → π n (0, Λ(x, s)) As the internal boosts leave the foliation vector fixed they can be implemented by canonical transformations. The generator of internal boosts on the hyperplane s = const is where H n (X) is the Hamiltonian density and the integral is over the surface s = const.
The fields φ n (X, s) are defined on an extended space-time N = M × R . However, the theory is not invariant under the full SO(2, 3) isometry group of this space-time. This is evident from the fact that the algebra (4), (5), (6) is defined on external space-time, that is the surface in N defined by s = 0, and not on internal space-time (defined by n · X = 0). Also, the generators of the symmetries are defined as integrals over external space-time. This indicates that the fields are not true scalar fields on N . However, the fields are invariant under the internal and external SO(1, 3) subgroups of SO (2,3).
In the case of the scalar field these subtleties can be overlooked, but the construction of a history theory of vector fields acutely illustrates this issue. A particularly relevant question is whether the history vector field should have 4 indices, or 5 as it must to be a vector field on N .

Massive vector field 2.1 State space theory
In this section we give a brief overview of the standard state space theory of the massive vector field on M .
We begin with the covariant theory. The covariant configuration space is X (M ), the space of vector fields on M . The massive vector field is described by the following Lagrangian [11]: The resulting field equations are The first of these equations shows that each component of the field behaves like a massive scalar field. The second equation is known as the Fierz-Pauli equation and it is the first indication of the presence of constraints in the theory.
To pass to the canonical theory we choose a Cauchy surface in M and consider the fields on this Cauchy surface. More precisely, we choose a spacelike embedding ι : Σ → M , where Σ ≃ R 3 , and take the corresponding configuration space to be the space of fields φ µ ι (x) where x ∈ ι(Σ). However the fields φ µ ι (x) are not geometric objects on either Σ or M . The geometrical interpretation of the fields is clarified by considering Emb(Σ, M ), the space of embeddings of Σ into M . Then φ ι can be thought of as an element of T ι Emb(Σ, M ) where the tangent space to Emb(Σ, M ) at the embedding ι is defined as The configuration space of the canonical theory is then defined as Q ι := T ι Emb(Σ, M ) for some fixed ι. The cotangent space of Emb(Σ, M ) at ι is defined similarly: and the pairing between these two spaces is given by where dθ x is an arbitrary volume element on Σ. The state space P ι is the cotangent bundle of Q ι , and can be identified with the Cartesian product Q ι × T * ι Emb(Σ, M ). As we are considering flat space-time, there exists a family of preferred embeddings; those which correspond to inertial frames. The space of preferred embeddings can be parametrised by pairs (n, t) where n is a future pointing unit vector in M and t ∈ R . As we are considering a deterministic system, we choose t = 0 without loss. We denote the configuration space and the state space corresponding to the embedding labeled by (n, 0) as Q n and P n respectively. The state space P n carries the following Poisson algebra; and we have introduced the n-spatial projection tensor defined by It follows that n µ n P µ ν = 0 and n ν n P µ ν = 0. The fields φ t n (x) and n φ µ (x) are defined on the space of embeddings, but φ t n (x) pulls back to give a scalar field on Σ n . We can use the metric on M to lower the index on n φ µ (x). The resulting one-form can be pulled back to Σ n , and then the index can be raised using the metric on Σ n to give a vector field on Σ n 4 . In a similar way, objects defined by n π µ (x) := n P ν µ π n ν (x), and n φ µν (x) := n P µ σ n P ν ρ φ σρ n (x) can be thought of as a one-form and a covariant tensor on Σ n .
The canonical momenta are computed from the Lagrangian and turn out to be [12] π n t (x) = 0 , n π µ (x) = n P ρ µ n ν φ νρ (x) (29) The first of these equations is a primary constraint. The canonical Hamiltonian is computed to be where the n-spatial derivative is defined as n ∂ µ = n P µ ν ∂ ν . For the primary constraint π n t (x) = 0 to be preserved by the dynamical evolution, it is necessary and sufficient that {H n , π n t (x)} = 0. This implies the secondary constraint A point in the constraint surface determines and is fully determined by the pair ( n π µ (x), n φ ν (x)). The constraint surface is therefore isomorphic to the space The constraints do not commute under the Poisson bracket and so they form a second class pair. This implies that the pull-back of the symplectic 2-form on P n to the constraint surface is non-degenerate. Therefore there is a well-defined Poisson algebra on the constraint surface which is given by the Dirac brackets:

Classical History theory
We begin with the abstract state space P , and follow the usual procedure of taking a one-parameter family of copies of P . This results in a trivial vector bundle ξ n : P × R → R where the fibre ξ −1 n (t) = P n,t for each t. Sections of ξ n correspond to histories of the vector field with respect to the foliation labelled by n. Thus a history is a map If we choose a volume element on R then a symplectic structure is induced on the space of maps R → P because P is a symplectic manifold. The symplectic structure defines the following algebra In the case of the scalar field a one-parameter family of functions on Σ n : was identified with a function φ n (X) on M . In a similar way, for a fixed foliation, a one-parameter family of elements of Q n is equivalent to a unique vector field φ µ n (X) ∈ X (M ). The one-parameter family t → π n µ (t; x) can be identified with a one-form π n µ (X) ∈ Λ 1 (M ) in the same way, and these fields satisfy the covariant looking algebra In this way, the history space, Π n , can be identified with the space and we use these fields to define the generators of internal and external time translations. External time translations are generated by the 'Louville' operator, Internal time translations are generated by the time-averaged Hamiltonian The internal time translations generated by H n take the form In the next subsection we will consider the geometric meaning of these curious objects which have four components, but depend on five space-time variables.
In the remainder of this subsection we discuss the history constraint surface C n ⊂ Π n . An arbitrary element of Π n will not be compatible with the constraints. The constraint submanifold, C n , contains all elements of Π which satisfy the constraints: π n t (X) = 0 , m 2 φ t n (X) + n ∂ µ n π µ (X) = 0 Thus a point in C n is equivalent to a pair ( n φ µ (X), n π ν (X)). The Poisson algebra induced on C n by pulling back along the natural inclusion map C n ֒→ Π n is given by and we note that C n is diffeomorphic to the space of sections of the bundle C n × R → R .

Poincare covariance
As discussed in the previous section for the case of the scalar field, the notion of two times leads naturally to the definition of two Poincare groups. The external Poincare group mixes the external time with the n-spatial variables x and the internal Poincare group mixes internal time with x. The crucial new feature of the generators for the vector field is the mixing of the time-like and space-like components of the field.

External Poincare group
The generators of external space-time translations can be written in covariant looking form as Next we define where the 'spin tensor' is The generators of n-spatial rotations 5 can be parametrised by two vectors m 1 and m 2 satisfying m 1 · n = m 2 · n = 0 in the following way: As in the case of the scalar field, the external boosts cannot be implemented by canonical transformations because of the change in foliation. The natural definition of the automorphisms generated by the action of the external boosts is So the external boosts mix t with x, and the external time component φ t Λn (t, x, 0) with the space-like components Λn φ µ (t, x, 0).

Internal Poincare group
The rotation and spatial translation generators of the internal Poincare group coincide with those for the external Poincare group. However, internal time translations are generated by the Hamiltonian, and therefore act internally as φ µ n (X) → φ µ n (X, s). We tentatively define the generator of internal Lorentz transformations on the s = const hyperplane in the 'obvious' way: where H n (X) is the Hamiltonian density, m is a vector satisfying m · n = 0, and the integral is over the surface s = const. This functional generates the automorphisms int K n (m) mixes s with x, and φ t n (0, x, s) with n φ µ n (0, x, s). So in this transformation, the function φ t n (0, x, s) is associated with the internal time direction, whereas it was associated with the external time direction by the external boosts.

Alternative interpretations of the history vector field
The above discussion suggests that it is misleading to think of the history vector fields as a family of 4-vectors on external space-time. We propose two alternative interpretations of the history vector fields.

Four-component fields
One way of thinking of the history fields is as a family of 4-vectors, but with the temporal component in the ∂ τ := ∂ s + ∂ t direction; where φ τ n (X, s) = φ t n (X, s) and we have used coordinates adapted to n. From this perspective it is natural to look for a representation of the Poincare group in which the boosts act in this direction. The orbital part of the internal boost generator on s = 0 would be where V n (X) is the 'Louville' density. However, it can be shown that Thus it is not possible to eliminate the two times in favour of one 'physical' time direction ∂ τ , in a covariant way. Nevertheless, τ does have a special significance in the theory. This is indicated by the equations of motion. Returning to the example of the scalar field for a moment; if φ n (X) is a solution then {S n , φ n (X)} = 0 implies that and so all the temporal change in such histories occurs in the τ direction.

Five-component fields
We can augment φ µ n (X, s) with a new degree of freedom φ s n (X, s) to form whereH n is the Hamiltonian on extended history space, and is defined in the next section. The history algebra can be extended to these fields in a natural way: This defines the algebra of the fields on the surface s = 0, which is a submanifold of N . Hamiltonian evolution can be used to extend this definition to the rest of N . The fact that the algebra is naturally defined on the hyperplane s = 0 and not on the hyperplane n · X = 0 reflects the underlying asymmetry between the two 'modes' of time. As a consequence of this asymmetry, the theory constructed from the fieldsφ n will not be covariant under the full SO(2, 3) isometry group associated with N . In particular it will not be covariant under the action of the SO(2) subgroup acting in the (s, t) plane.
In order to discuss the Poincare transformations of the five-component fields, we make the definition: where M (e) ⊂ N is external space-time, defined as the surface s = 0, and the extended spin tensor is defined as The foliation vector n is an element of the external space-time which is a subspace of N . Using the canonical embedding, n can be considered as a vector in N , which we denote byñ and is given in coordinates (X, s) by (n, 0). Using this coordinate system the internal future pointing unit vectorẽ can be written as (0, 1). We can use these vectors to decompose an extended fieldφ M n (X, s) into its external and internal time components as follows: It remains to be shown that the extra degrees of freedom can be included in the action in a way that is consistent with the symmetries and equations of motion of the theory.

The action
In the case of scalar field theory the physical action functional is written as S n = V n − H n . The Louville operator is associated with external time in the sense that it generates translations in the external time direction. In the same way, the Hamiltonian is associated with internal time, and the action functional mixes the two 'modes' of time.
First we will need the following definition: The n-spatial part ofφ M n (X, s) is defined as where the extended n-spatial projection tensor is Similarly we define n ∂ M := nP M N ∂ N . LetṼ n denote the extension of the Louville operator to the extended fields.Ṽ n is defined in the following natural way, where ∂ t n =ñ M ∂ M is the derivative in the external time direction defined bỹ n. The extended Hamiltonian,H n , is defined as The important thing about this Hamiltonian is thatφ s n andφ t n both appear in the mass term, but onlyφ t n appears as the coefficient of n ∂ M nπ M . Due to this asymmetry betweenφ s n andφ t n , the Hamiltonian is not invariant under SO(2, 3). The action is defined to beS n :=Ṽ n −H n . The resulting field equations are, The physical action has not been derived in the usual way from a Lagrangian, so we do not have the usual identification of primary constraints. The field equations do not determine the time-like components of theπ field, so we augment the equations of motion with the following equations which are interpreted as the primary constraints of the theory.
π n s (X) = 0 ,π n t (X) = 0 (95) We require these constraints to be conserved in internal time which implies the following secondary constraints φ s n (X) = 0 , m 2φt n (X) + n ∂ M nπ M (X) = 0 (96) so in the history theory of the massive vector field there are two pairs of second class constraints. In the state space theory a single pair of constraints allow the theory to be written in a Lorentz covariant way. In the history theory where there are two SO(1, 3) symmetry groups, we have to introduce two pairs of constraints in order to have a covariant theory.
The three functionals,Ṽ n ,H n andS n are all invariant under the action of the internal Poincare group. However, the external boosts change the foliation with respect to whichṼ n andH n are defined, giving the transformations V n →Ṽ Λn andH n →H Λn . These transformations implyS n →S Λn , and the history theory is covariant under both Poincare groups if we include the internal foliation dependence.

State space theory
In this section we consider vacuum electromagnetism on M . The covariant configuration space for electromagnetism is Q = Λ 1 (M ), and the Lagrangian is The covariant equations of motion which follows from this Lagrangian are F µν , ν (X) = 0 (98) Given an embedding ι : Σ → M , the canonical configuration space is Q ι = T * ι Emb(Σ, M ) and the state space, P ι , can be identified with Q ι × T ι Emb(Σ, M ). The Poisson algebra on the state space associated with the embedding labeled by (n, 0) is The canonical momenta are computed from the Lagrangian; where we again use the decomposition into time-like and space-like parts. The canonical Hamiltonian is The equation of motion for E t n implies the secondary constraint which can be recognised as Gauss' law. The constraints form a first class pair. The corresponding gauge freedom is manifested in the fact that the equations of motion do not determine A n t or A n L where A n L is the longitudinal part of A n . The pull back of the symplectic two-form on P n to the constraint surface C n is degenerate because of the first class nature of the constraints. The reduced state space is obtained after gauge fixing and contains only four of the original eight degrees of freedom.

History theory
We follow the same procedure as before and consider sections of the trivial vector bundle P × R → R . So a history is a map and again, there is a unique pair (A n µ (X), E ν n (X)) corresponding to such a history so we fix a n and identify Π n with the abstract space Λ 1 (M )×X (M ). These history fields satisfy the covariant looking algebra, and the Louville, Hamiltonian and action operators are defined as At this point we note that, unlike in the case of the scalar field, the foliation dependence of these fields is empirically verified. It is a well-known fact that observations of magnetic and electric fields do depend on the state of motion of the observer.

Constraints
We follow the procedure detailed for the vector field and work in the extended history spaceΠ n ⊂ Λ 1 (N ) × X (N ). The extended history fields are written in the formÃ and using a basis we have the decomposition ofÃ n into time-like and spacelike components (Ã n s ,Ã n t ,Ã n i ). The action of the Poincare group is very similar to the corresponding definitions for the vector field. From now on we fix an n, and work in coordinates adapted to n, dropping the n-label for typographical convenience. The new feature of electromagnetism is, of course, gauge invariance. We extend the Louville, Hamiltonian and action functionals to the extended history space as

External local symmetries
To investigate the external local symmetries we define the extended action [13].S The transformations are generated by the functional The extended action is invariant under these transformations if The symmetry of the total action, and therefore of the underlying Lagrangian theory is found by setting λ 2 = 0 (and δλ 2 = 0), thus eliminating the secondary constraint. The resulting transformations are δÃ s (X) = ǫ 0 (X) (126) δÃ µ (X) = ∂ µ ǫ 2 (X) where µ = t, 1, 2, 3. These transformations contain two arbitrary real functions on external space-time. Setting ǫ 0 = 0 we obtain 'external' U(1) gauge transformations: which correspond to the symmetries of the external Maxwell tensorF µν = ∂ [ µÃν ] .

Internal local symmetries
The mapÃ M (X) →Ã M (X, s) is one-to-many for gauge systems. This introduces an extra ambiguity into the theory which is not contained in equations (126) and (127). We make this extra ambiguity explicit by introducing Lagrange multipliers to make the mapÃ M (X) →Ã M (X, s) one-to-one. To accomplish this we define the extended Hamiltonian; At each moment of internal time s, the integral is over M Gauge-equivalent histories correspond to different choices of the Lagrange multipliers in the extended Hamiltonian. The functional defined by and these transformations take the form In order that equation (130) is preserved by these transformations up to a change in the Lagrange multipliers associated with the primary constraints, the transformations must satisfy These are the internal local symmetry transformations of histories electromagnetism. They contain two arbitrary real valued functions on extended space-time, but are not just a trivial extension of the external local transformations to each moment of internal time. The transformation ofÃ t contains a derivative with respect to internal time rather than external time. This is because the internal gauge transformations correspond to the symmetries of the internal field equations rather than the symmetries of the external field equations.

Internal symmetries vs. external symmetries
An arbitrary history satisfies the internal equations of motion. Therefore all histories that are related by internal gauge transformations should be regarded as physically equivalent. However, most histories will not satisfy the external equations of motion, and so need not respect the external symmetry

Summary and Conclusion
We have shown that the global symmetry transformations of geometric objects in a history theory suggests the introduction of an extra pair of fields. The extended history fields can then be interpreted as fields on N = M ×R that are covariant under the action of two Poincare groups, but not the SO(2, 3) group associated with N . The history fields can be decomposed into two SO(1, 3)-vectors at each point in the extended space-time, where the spatial components of these two vectors are the same.
In the case of the massive vector field, the theory contains two pairs of second class constraints, and in the case of electromagnetism, we obtain three first class constraints. In both these examples, the extra degrees of freedom can be set to zero, and eliminated from the theory by taking Dirac brackets. It seems reasonable to expect this pattern to continue in the extension to other constrained field theories as there is no physical information in the extra degrees of freedom. So although Lorentz covariance suggests that the fields should have five components, the extra constraints allow the extra variables to be eliminated.
In conclusion, the geometry of classical history theories is not fully understood. In the state space approach the solutions to the field equations are sections of tensor bundles associated to an SO(1, 3) principal bundle over M . This formulation elegantly characterises the way that the fields transform under Lorentz transformations. If the history fields were covariant under SO(2, 3) then we would have a description of history fields in terms of sections of bundles associated to an SO(2, 3) principal bundle over N . However the reality of the situation appears to be more complicated. There is an SO(1, 3) group associated to each surface of constant t, and to each surface of constant s, but the action of these two groups is intertwined in a non-trivial way. Similar remarks apply to the local symmetries in a history theory: they cannot be interpreted as the transformations of a U(1) connection on a principal bundle over N .
In a subsequent paper we will discuss the quantisation of histories electromagnetism using the BRST formalism.

Acknowledgements
I would like to thank Chris Isham for his constant encouragement and support, and PPARC for a studentship.