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Functional Time Evolution, Anomaly Potentials, and the Geometric Phase

A free quantum field in 1+1 dimensions admits unitary Schrodinger picture dynamics along any foliation of spacetime by Cauchy curves. Kuchar showed that the Schrodinger picture state vectors, viewed as functionals of spacelike embeddings, satisfy a functional Schrodinger equation in which the generators of time evolution are the scalar field energy-momentum densities with a particular normal-ordering and with a (non-unique) c-number contribution. The c-number contribution to the Schrodinger equation, called the ``anomaly potential'', is needed to make the equation integrable in light of the Schwinger terms present in the commutators of the normal-ordered energy-momentum densities. Here we give a quantum geometric interpretation of the anomaly potential. In particular, we show the anomaly potential corresponds to the expression in a gauge of the natural connection on the bundle of vacuum states over the space of embeddings of Cauchy curves into the spacetime. The holonomy of this connection is the geometric phase associated with dynamical evolution along a closed path in the space of embeddings generated by the normal-ordered energy-momentum densities. The presence of the anomaly potential in the functional Schrodinger equation provides a dynamical phase which removes this holonomy, so that there is no net phase change for quantum transport around closed loops in the space of embeddings.


I. INTRODUCTION
Time evolution in a field theory is usually specified in terms of the behavior of various dynamical variables as one moves along a given foliation of spacetime by Cauchy surfaces.
One is often interested -particularly in applications to gravitational physics -in the possibility of considering dynamical evolution along any foliation by Cauchy surfaces. In order to consider different foliations it is convenient to view them as different curves in the space of embeddings E of Cauchy surfaces into spacetime. In this setting, dynamical variables can be viewed as functionals on E and this more general type of dynamical evolution is often For a free quantum field in a flat 1+1 dimensional spacetime it is a non-trivial fact that one can use a simple Fock representation to implement functional time evolution in the Schrödinger picture [4]. Put differently, one has unitary Schrödinger picture dynamics along any foliation of spacetime by Cauchy curves, all within a single Fock representation. In this setting the Schrödinger picture state vector of the quantum field can be viewed as a mapping from E into the Fock space. As shown in Ref. 4,6, the state vector then satisfies a functional Schrödinger equation in which the role of the Hamiltonian density is played by the quantized energy-momentum densities of the field. The precise definition of these densities is, however, somewhat intricate. The energy-momentum densities are defined as the sum of (i) the normal-ordered classical expression, using an embedding-dependent set of creation and annihilation operators, and (ii) a non-unique embedding-dependent multiple of the identity operator. The need for contribution (ii) was first recognized by Kuchař, who called this c-number the "anomaly potential" [6]. This terminology arose because the c-number contribution is needed to make the functional Schrödinger equation integrable in light of anomalous terms ("Schwinger terms") present in the commutators of the energymomentum densities. The presence of the anomaly potential can be viewed as an embeddingdependent adjustment to the naive normal-ordering definition of the Schrödinger picture energy-momentum densities. This adjustment guarantees that dynamical evolution of the state vector between two time slices is independent of the choice of interpolating foliation.
In this note we expose the quantum theoretic origins of the anomaly potential in the Schrödinger equation. A hint is given in Ref. 6 where an analogy is drawn between the anomaly potential and the vector potential for a particle moving in an electromagnetic field.
A geometrical interpretation of the latter situation is, of course, that the vector potential corresponds to a connection on a principal bundle over spacetime. Is there a corresponding geometrical interpretation of the anomaly potential? Here we give such an interpretation for the anomaly potential using techniques and results from the extensive investigations into the quantum geometric phase [7].
We consider the Fock representation used in Ref. 4, 6 for a free, massless scalar field on a flat spacetime R × S 1 (Section II). The functional Schrödinger picture is defined by a family of unitary transformations on the Fock space parametrized by the time slice, i.e., the embedding (Section III). The Hilbert space of Fock state vectors is a principal U(1) bundle Q → S over the manifold of Fock states; Q is equipped with a natural connection (Section IV). The Schrödinger picture images of the Heisenberg picture vacuum state define a mapping from E to S. This mapping determines an induced principal U(1) bundle B → E (Section V). There is a connection on B induced from the connection on Q. Its holonomy corresponds to the geometric phase associated with the transport of state vectors along closed curves in E. The expression of this connection in a suitable gauge is precisely the Kuchař anomaly potential (Section VI). The effect of the anomaly potential in the Schrödinger equation is then to provide a dynamical phase which cancels the geometric phase associated with time evolution along closed paths in E, the latter phase being generated by the normalordered energy-momentum densities (Section VII). From this it follows that dynamical evolution of the state vector in the Schrödinger picture is independent of the choice of foliation connecting given initial and final time slices.

II.
THE QUANTUM SCALAR FIELD ON R × S 1 We will consider a free, massless, scalar quantum field on a flat cylindrical spacetime (M, η). The spacetime manifold is M = R × S 1 with coordinates t ∈ (−∞, ∞) and x ∈ (0, 2π). The spacetime metric in these coordinates is We will also use null coordinates The (real symplectic vector) space of smooth solutions to the wave equation is given by where q and p are real and in the usual way (Î (±) are the identity operators on F (±) ). Introduce a Hilbert space L 2 (R) with elements being complex-valued functions ψ = ψ(p) upon which we definê We define the Hilbert space for the field as On F the quantum scalar field is defined as an operator-valued spacetime distribution viâ (2.10) Throughout this paper we use a hat (ˆ) to denote an operator on F .
For each ψ ∈ L 2 (R) there is a state vector, denoted by |ψ, 0 , which we call "the" Heisenberg vacuum state, since it satisfies A basis for F is obtained from a basis for L 2 (R) and from repeated applications of the creation operators to |ψ, 0 .
Throughout this article all kets and bras will denote elements of F and F * .

III. EMBEDDINGS AND FUNCTIONAL TIME EVOLUTION
We denote by X : S 1 → M a smooth, orientation-preserving, spacelike embedding of a circle into the two-dimensional spacetime. Using coordinates x α and σ ∈ (0, 2π) on M and S 1 respectively, the embedding is defined by 2 smooth functions X α : where dx is the 1-form appearing in (2.1). Using null coordinates x ± these two conditions are equivalent to For future use we note that the set of all orientation-preserving spacelike embeddings, denoted by E, is an infinite-dimensional manifold which is contractible. To see this latter feature we consider X 0 , X 1 ∈ E. For any λ ∈ [0, 1] the quantity is an orientation-preserving spacelike embedding as can easily be seen, e.g., by working in null coordinates. It follows that E is contractible to a point.
Although the field operatorφ is defined as an operator-valued distribution on M, the simplicity of two dimensions allows one to restrict it and its derivatives to any embedded circle thus defining operator-valued distributions on S 1 . In particular, we define the canonical field operators associated to an embedding X aŝ where γ is the induced metric (3.2) on the embedded circle and n is the future-pointing unit normal to the embedded circle. In null coordinates x ± on M and using σ as a coordinate on S 1 these definitions arê Such operators are functionals of the embeddings and can be used, in conjunction with a choice of state vector, to compute outcomes of measurements of observables associated to any specific time slice in the Heisenberg picture.
The canonical field operators associated with any two spacelike embeddings X 1 and X 2 are unitarily related [4], that is, for each pair of embeddings there exists a unitary transformation Thus functional time evolution is unitarily implemented in the Heisenberg picture.
Because the Fock representation constructed in II is irreducible,Û is unique up to a (possibly embedding-dependent) phase. See Ref. 7 for an explicit expression ofÛ .
Functional Heisenberg equations can be obtained from (3.9) by viewing X 2 as an infinitesimal deformation of X 1 . We then get [4,6], whereĤ α [X] is the energy-momentum density of the scalar field at the slice embedded by ). In null coordinates we havê Just as the unitary transformation appearing in (3.9) can be redefined by a (possibly embedding-dependent) phase factor, so one can also redefine the energy and momentum densities by the addition of a (possibly embedding-dependent) multiple of the identity without disturbing the validity of (3.10), (3.11).
In the (functional) Schrödinger picture the canonical field operators are defined once and for all, independently of the embedding, and the state vectors are embedding-dependent.
To construct this picture of dynamics we pick once and for all a fiducial embedding X 0the initial slice -upon which we identify the Heisenberg and Schrödinger canonical field operators. Given an initial state vector |Ψ ∈ F characterizing the state of the system at the time defined by X 0 , the state vector |Ψ, X ∈ F on the embedding X is defined by We remark that, asÛ is only determined up to a (possibly embedding-dependent) phase, the state vector on the slice defined by X is determined only up to such a phase.
By considering deformations of the embedding in (3.13), it follows that the Schrödinger picture state vector satisfies the following functional Schrödinger equation [4]: represents the normal-ordered Schrödinger picture energy-momentum densities associated with the slice embedded by X. We note thatĤ α [X] is not normal-ordered in the (â,â † ) operators. Rather,Ĥ α [X] is normal-ordered in the Schrödinger picture image of the creation and annihilation operators:b The c-number contribution to (3.14), A α = A α [X], is Kuchař's anomaly potential [6], and is explicitly computed fromÛ in Ref. 4. Because U[X 0 , X] and hence |Ψ, X is only determined up to a phase, the expression for A α is likewise non-unique. We have . defines a principal fiber bundle with structure group U(1) [7]. The action on Q of an element A tangent vector to Q at |Ψ is any vector |t ∈ F such that Re( Ψ|t ) = 0. (4.5) A vertical tangent vector |v to Q at |Ψ satisfies π * (|v ) = 0 and is of the form |v = iα|Ψ , α ∈ R. (4.6) The bundle π : Q → S has a natural connection [7]. Horizontal tangent vectors |h at |Ψ are defined by this connection to satisfy In light of (4.5), we see that horizontal vectors at |Ψ are elements of F orthogonal to |Ψ .
The associated connection 1-form Ω defines a map from the tangent space to Q at |Ψ to the Lie algebra u(1) (the Lie algebra of imaginary numbers) given by |t → Ω(|t ) = Ψ|t . There is then a canonically defined principal U(1) bundle over E obtained by copying each fiber from Q over P vac [X] to sit over X. This is the induced (or "pull-back") bundle [8].
More precisely, the induced bundle B → E is defined as the subset of points (X, |Ψ ) ∈ E ×Q such that The projection mapping Π : B → E is given by The action of e iβ ∈ U(1) on B is given by Because the set E of orientation-preserving spacelike embeddings is contractible, the bundle B is trivial.
As for any induced bundle, there is a bundle morphism f : B → Q satisfying In our case, this mapping is given by and |T is an element of F of the form where ν ∈ R and |vac, X . In light of the triviality of the induced bundle, this cross-section can be defined globally; for simplicity we assume this in what follows. We define the gauge potential to be the 1-form on E defined by A = iρ * ω. Let V be a tangent vector to E at X, as in (5.9). The push-forward ρ * (V ) is a tangent vector to B at ρ(X) given by At the point X we have , (6.9) then using (3.14) and taking account of the normal-ordering of H α , we have Thus the anomaly potential coincides with the gauge potential of the connection ω, i.e., to the expression of the natural connection on the induced bundle in a particular gauge.
Neither of the connections ω or Ω is flat. In particular, the gauge-invariant field strength dA -which is i times the pull-back to E of the curvature dΩ by f • ρ -is non-vanishing.
The components of dA correspond to the Schwinger terms in the commutator algebra of the operators H α . This can be easily seen from our geometric construction as follows. Let U and V be tangent vectors to E: . (6.12) We have |vac, X . (6.13) Using the functional Schrödinger equation (3.14), we then get 14) The commutators of the normal-ordered operatorsĤ α close back into theĤ α up to embedding-dependent multiples of the identity -the Schwinger terms [6]: See Ref. 6 for explicit expressions. The Schwinger terms are then produced by the vacuum expectation value appearing in (6.14) for dA. VII.

PENDENCE OF TIME EVOLUTION
The integrability of (3.14) implies that dynamical evolution between given initial and final slices is independent of the interpolating foliation [6]. This can be seen directly using the geometric ingredients discussed in the previous sections.
Consider a closed curve γ in E. Physically this corresponds to considering time evolution from an initial slice X 0 to a final slice X 1 along a foliation X(s), s ∈ [0, 1] followed by evolution from X 1 to X 0 along a different foliationX(s), s ∈ [1, 2]. The mapping P vac (see The geometric lift [7] of C, denoted by |Ψ(s) geom , is the horizontal lift defined by the connection Ω. This is not a closed curve; we have where the phase factor is the gauge-invariant holonomy of the connection Ω -it is the geometric phase for this closed path in S. It can be computed by integrating A (i.e., the anomaly potential) around the closed curve γ in E, where γ is represented by the 1-parameter family of embeddings: That this phase is non-trivial follows from the non-triviality of the curvature of A. As we have seen this curvature is determined by the Schwinger terms in the commutator algebra ofĤ α ; see (6.14). Thus the Schwinger terms are responsible for the geometric phase.
The geometric lift obeys a Schrödinger equation with a dynamical phase removed [7]: Now,Û † (1, 2) is the time evolution operator for the geometric lift going from X 0 to X 1 along the slicing determined byX(s). We see then that the time evolution of the initial (vacuum) state vector generated byĤ along two different foliations connecting the same initial and final slices leads to final state vectors which differ by the geometric phase.
The dynamical lift is simply the curve |Ψ(s) in Q solving the Schrödinger equation defined from (3.14) and the foliation X (see (3.22)): Thus the anomaly potential leads to a dynamical phase which cancels the geometric phase caused by the Schwinger terms in the algebra of energy-momentum densities. Reasoning as before, we see that the dynamical evolution generated byĤ is foliation independent.