On Mode Regularization of the Configuration Space Path Integral in Curved Space

The path integral representation of the transition amplitude for a particle moving in curved space has presented unexpected challenges since the introduction of path integrals by Feynman fifty years ago. In this paper we discuss and review mode regularization of the configuration space path integral, and present a three loop computation of the transition amplitude to test with success the consistency of such a regularization. Key features of the method are the use of the Lee-Yang ghost fields, which guarantee a consistent treatment of the non-trivial path integral measure at higher loops, and an effective potential specific to mode regularization which arises at two loops. We also perform the computation of the transition amplitude using the regularization of the path integral by time discretization, which also makes use of Lee-Yang ghost fields and needs its own specific effective potential. This computation is shown to reproduce the same final result as the one performed in mode regularization.


Introduction
The Schrödinger equation for a particle moving in a curved space with metric g µν (x) has many applications ranging from non-relativistic diffusion problems (described by a Wick rotated version of the Schrödinger equation) to the relativistic description of particles moving in a curved space-time. However it cannot be solved exactly for an arbitrary background metric g µν (x), and one has to resort to some kind of perturbation theory. A very useful perturbative solution can be obtained by employing a well-known ansatz introduced by De Witt [1], also known as the heat kernel ansatz. This ansatz makes use of a power series expansion in the time of propagation of the particle. The coefficients of the power series are then determined iteratively by requiring that the Schrödinger equation be satisfied perturbatively.
Equivalently, the solution of the Schrödinger equation can be represented by a path integral, as shown by Feynman fifty years ago [2]. One can formally write down the path integral for the particle moving in curved space and check that the standard loop expansion reproduces the structure of the heat kernel ansatz of De Witt. However the proper definition of the path integral in curved space is not straightforward. In fact it has presented many challenges due to complications arising from: i) the non-trivial path integral measure [3], ii) the proper discretization of the action necessary to regulate the path integral. A quite extensive literature has been produced over the years addressing especially the latter point [4].
In this paper we short cut most of the literature and discuss a method of defining the path integral by employing mode regularization as it is by now standard in many calculations done in quantum field theory. The methods extends the one employed by Feynman and Hibbs in discussing mode regularization of the path integral in flat space [5]. It has been introduced and successively refined in [6], [7] and [8] where quantum mechanics was used to compute one loop trace anomalies of certain quantum field theories. The key feature is to employ ghost fields to treat the non-trivial path integral measure as part of the action, in the spirit of Lee and Yang [3]. These ghost fields have been named "Lee-Yang" ghosts and allow to take care of the non-trivial path integral measure at higher loops in a consistent manner. The path integral is then defined by expanding all fields, including the ghosts, in a sine expansion about the classical trajectories and integrating over the corresponding Fourier coefficients. The necessary regularization is obtained by integrating all Fourier coefficients up to a fixed mode M, which is eventually taken to infinity. A drawback of mode regularization is that it doesn't respect general coordinate invariance in target space: a particular non-covariant counterterm has to be used in order to restore that symmetry [8]. General arguments based on power counting (quantum mechanics can be thought as a super-renormalizable quantum field theory) plus the fact that the correct trace anomalies are obtained by the use of this path integral suggest that the mode regularization described above is consistent to any loop order without any additional input.
As usual when dealing with formal constructions, it is a good practice to check with explicit calculations the proposed scheme. It is the purpose of this paper to present a full three loop computation of the transition amplitude. The result is found to be correct since it solves the correct Schrödinger equation at the required loop order. This gives a powerful check on the method of mode regularization for quantum mechanical path integrals on curved space. In addition, we present our computation in such a way that it can be easily extended and compared to the time discretization method developed in refs. [9], which is also based on the use of the Lee-Yang ghosts. This method requires its own specific counterterm (also called effective potential) to restore general coordinate invariance. As expected both schemes give the same answer.
The paper is structured as follows. In section 2 we review the method of mode regularization and discuss the effective potential specific to this regularization. In section 3 we present a three loop computation of the transition amplitude. Here we make use of general coordinate invariance to select Riemann normal coordinates to simplify an otherwise gigantic computation. We check that the result satisfies the Schrödinger equation at the correct loop order. In section 4 we extend our computation to the time discretization scheme. This is found to compare successfully with the results previously obtained in section 3. Finally, in section 5 we present our conclusions and perspectives. In appendix A we present a technical section with a list of loop integrals employed in the text. In particular, we discuss how to compute them in mode regularization as well as in time discretization regularization.

Mode regularization
The Schrödinger equation for a particle of mass m moving in a D-dimensional curved space with metric g µν (x) and coupled to a scalar potential V (x) is given by where with ∇ 2 the covariant laplacian acting on scalars. It can be obtained by canonical quantization of the model described by the classical action when ordering ambiguities are fixed by requiring general coordinate invariance in target space and requiring in addition that no scalar curvature term be generated by the orderings in the quantum potential 1 . For convenience we will Wick rotate the time variable t → −it and set m =h = 1 to obtain the following heat equation and corresponding euclidean action As mentioned in the introduction the heat equation can be solved by the heat kernel ansatz of De Witt [1]: a n (x, y)t n (6) which depends parametrically on the point y µ that specifies the boundary condition . Here σ(x, y) is the so-called Synge world function and corresponds to half the squared geodesic distance. The coefficients a n (x, y) are sometimes called Seeley-De Witt coefficients 2 and are determined by plugging the ansatz (6) into (4) and matching powers of t. Now we want to describe in detail how to get the solution of eq. (4) by the use of a path integral which employs the classical action in (5). Following refs. [6,7,8] we write the transition amplitude for the particle to propagate from the initial point x µ i at time t i to the final point x µ f at time t f as follows For commodity we have shifted and rescaled the time parameter in the action, Note that the total time of propagation β plays the role of the Planck constanth (which we have already set to one) and counts the number of loops. In the loop expansion generated by β the potentials V and V M R start contributing only at two loops 3 . The full action S includes terms proportional to the Lee-Yang ghosts, namely the commuting ghosts a µ and the anticommuting ghosts b µ and c µ . Their effect is to reproduce a formally covariant measure: integrating them out producesDx = (det g µν (x(τ ))) 1/2 d D x(τ ). As we will discuss, mode regularization destroys this formal covariance. Nevertheless reparametrization covariance is recovered thanks to the effects of the potential V M R directly included in the action (8). With precisely this counterterm the mode regulated path integral in (7) solves the equation in (4) in both sets of variables (x µ f , t f ) and (x µ i , t i ) and with the boundary condition .
For an arbitrary metric g µν (x) one is able to calculate the path integral only in a perturbative expansion in β and in the coordinate displacements ξ µ about the final point 2 It is also customary to redefine the a n (x, y) by extracting a common factor ∆ 1 2 (x, y), where ∆(x, y) is a scalar version of the so-called Van Vleck-Morette determinant. 3 Reintroducingh one can see that the classical potential V must be of orderh 0 while the counterterm V MR is a truly two loop effect of orderh 2 .
The actual computation starts by parametrizing where x µ bg (τ ) is a background trajectory and q µ (τ ) the quantum fluctuations. The background trajectory is taken to satisfy the free equations of motion and is a function linear in τ connecting x µ i to x µ f in the chosen coordinate system, thus enforcing the proper boundary conditions Note that by free equations of motion we mean the ones arising from (8)  The quantum fields q µ (τ ) in (11) should vanish at the time boundaries since the boundary conditions are already included in x µ bg (τ ). Therefore they can be expanded in a sine series. For the Lee-Yang ghosts we use the same Fourier expansion since the classical solutions of their field equations are where φ stands for all the quantum fields q µ , a µ , b µ , c µ . The measureDx in (10) is now properly defined in terms of integration over the Fourier coefficients φ µ m as follows where A is a constant. Note that this fixes the path integral for a free particle to It is well-known that A = (2πβ) − D 2 , however this value can also be deduced later on from a consistency requirement.
The way to implement mode regularization is now quite clear: limiting the integration over the number of modes for each field to a finite mode number M gives the natural regularization of the path integral. This regularization resolves the ambiguities that show up in the continuum limit.
The perturbative expansion is generated by splitting the action into a quadratic part S 2 , which defines the propagators, and an interacting part S int , which gives the vertices. We do this splitting by expanding the action about the final point x µ f and obtain where In this expansion all geometrical quantities, like g µν and ∂ α g µν , as well as V and V M R , are evaluated at the final point x µ f , but for notational simplicity we do not exhibit this dependence. S 2 is taken as the free part and defines the propagators which are easily obtained from the path integral where ∆ is regulated by the mode cut-off and has the following limiting value for M → ∞ Note that we indicate • ∆(τ, σ) = ∂ ∂τ ∆(τ, σ), ∆ • (τ, σ) = ∂ ∂σ ∆(τ, σ) and so on. Details on the properties of these functions are given in appendix A. Now, the quantum perturbative expansion reads: where the brackets · · · denote the averaging with the free action S 2 , and amount to use the propagators given in (21) in the perturbative expansion. Note that in the last line of the above equation we have kept only those terms contributing up to two loops, i.e. up to O(β), by taking into account that ξ µ ∼ O(β 1 2 ), as follows from the exponential appearing in the last line of (24) after one averages over ξ µ . Note also that having extracted the coefficient A together with the exponential of the quadratic action S 2 evaluated on the background trajectory implies that the normalization of the left over path integral is such that 1 = 1.
To test its consistency one can use it to evolve in time an arbitrary wave function Ψ(x, t) and the terms of order β give This last equation means that the wave function Ψ satisfies the correct Schrödinger equation (4) at the final point (x µ f , t f ).
It is interesting to note that the counterterm V M R appears only in the last line of eq. (26). Actually the value of the counterterm reported in eq. (9) has been deduced in [8] by imposing that the transition amplitude would solve eq. (30). General arguments can then be used to show that this counterterm should be left unmodified at higher loops. In fact one can consider quantum mechanics on curved spaces as a super-renormalizable one-dimensional quantum field theory, and check by power counting that all possible divergences can only appear at loop order 2 or less in β. In the next section we are going to check that it is so indeed, expelling doubts which have sometimes been raised that mode regularization would be inconsistent at higher loops. Thus one can consider mode regularization as a viable way of correctly defining the path integral in curved spaces.

The transition amplitude at three loops
In this section we want to check eq. (28) at the next order in β, which is equivalent to showing that the transition amplitude computed by the path integral satisfies the Schrödinger equation not only at the point (x µ f , t f ) but in a small neighbourhood of it. This computation can be quite lengthy if done in arbitrary coordinates. To make it feasible we select a useful set of coordinates: the Riemann normal coordinates centred at the point x µ f . In such a frame of reference the coordinates of an arbitrary point x µ contained in a neighbourhood of the origin are given by a vector z µ (x) belonging to the tangent space at the origin. This vector specifies the unique geodesic connecting the origin to the given point x µ in a unit time. In such a frame of reference the coordinates of the origin are obviously given by z µ (x f ) = 0. In what follows we will use Riemann normal coordinates which we keep denoting by x µ since no confusion can arise.
The expansion of the metric around the origin is given by (see e.g. [7] for a derivation) Note that the coefficients in this expansion are tensors belonging to the tangent space at the origin. This is a property of Riemann normal coordinates. In general, the terms contributing to the transition amplitude up to three loops are given by Clearly the computation would be quite complex in arbitrary coordinates. Fortunately, in Riemann normal coordinates many terms are absent since we obtain Note that all structures like R µναβ , V , V M R and derivatives thereof are evaluated at the origin of the Riemann coordinate system, but for notational simplicity we do not indicate so explicitly. The computation is still quite lengthy and we get where the integrals I n are listed and evaluated using mode regularization in appendix A. Inserting the specific values of the terms arising from the effective potential V M R when evaluated at the origin leads us to the following expression for the transition amplitude at the third loop order R αµ R β µ + 1 12 This is the complete expression which should be used to test eq. (28) at order β 2 . A straightforward calculation shows that one indeed obtains an identity after making use of eq. (30). The mode regulated path integral described in the previous section passes this consistency check. Therefore it can be considered as a well defined way of computing path integrals in curved spaces. Before closing this section it may be useful to cast the transition amplitude in a more compact form which can be made manifestly symmetric under the exchange of the initial and final point. Keeping on using the Riemann normal coordinates (in which we recall x µ f = 0 and ξ µ ≡ x µ i − x µ f = x µ i ) and defining symmetrized quantities as we can write From this expression one can extract (by re-expanding part of the exponential) the leading terms of the Seeley-De Witt coefficients a 0 , a 1 , a 2 for non-coinciding points and obtain, in particular, the one loop trace anomalies for the operator H = − 1 2 ∇ 2 + V (x) in two and four dimensions.

Time discretization
The computation performed in the previous section was cast in such a way that can be easily extended to a different regularization scheme: the time discretization method developed in refs. [9]. Such a regularization was obtained by deriving directly from operatorial methods a discretized version of the path integral. Taking the continuum limit one recognizes the action with the proper counterterm, and the rules how to compute Feynman graphs. These rules differ in general from the one required by mode regularization. The counterterm V W arising in time discretization differs from V M R , too.
The time discretization method leads to the following path integral expression of the transition amplitude [9] x µ where The propagators to be used in the perturbative expansion implied by the brackets on the right hand side of eq. (49) are the same as in (21). The only difference is in the prescription how to resolve the ambiguities arising when distributions are multiplied together. The prescription imposed by time discretization consists in integrating the Dirac delta functions coming form the velocities and the ghosts propagators (thanks to the Lee-Yang ghosts they never appear multiplied together) and using consistently the value θ(0) = 1 2 for the step function. Note also the presence of the factor [ g(x f ) g(x i ) ] 1/4 appearing in this scheme.
The result of the calculation has the same structure as the one reported in eqs. (37), (38), (39), (40), (41) with the difference that V M R should be substituted by V W , leading to and with the following different values of the integrals computed in time discretization regularization I 1 = 0, I 3 = 0, I 5 = 0, I 10 = 0, I 13 = 1 12 , The other integrals are as in mode regularization. Inserting all these values back in (48) and expanding the coefficient [ g(x i ) ] 1/4 at the required loop order give the same transition amplitude as in (45) or, equivalently, in (47). Thus this result constitutes a successful test on the method developed in [9].

Conclusions
In this paper we have discussed a proper definition of the configuration space path integral for a particle moving in curved spaces. By performing a three loop computation we have tested its consistency and checked that one can equally well obtain the perturbative solution of the Schrödinger equation by path integrals. This fills a conceptual gap, showing that the perturbative description of a quantum particle moving in a curved space obtained by De Witt by solving the Schrödinger equation (i.e. using the canonical formulation of quantum mechanics [1]) can equally well be obtained in the path integral approach introduced by Feynman fifty years ago. This approach may also have practical applications in quantum field theoretical computations when carried out in curved background using the world line formalism [10].
We have mainly described the mode regulated path integral. Its definition was obtained in [6], [7] and [8] by using a pragmatic approach to identify its key elements, and needed a strong check to test its foundations. This we have provided in this paper. We find that the method of mode regularization is also quite appealing for aesthetic reasons, since it is close to the spirit of path integrals that are meant to give a global picture of the quantum phenomena.
On the other hand we have also extended our computation to the time discretization method of defining the path integrals [9]. This method is in some sense closer to the local picture given by the differential Schrödinger equation, since one imagines the particle propagating by small time steps. It is nevertheless a consistent way of defining the path integral, maybe superior at this stage, since one obtains its properties directly from canonical methods. As we have seen also this scheme gives the correct result for the transition amplitude.
An annoying property of the two regularization schemes we have been discussing is that they both do not respect general coordinate invariance in target space, and require specific non-covariant counterterms to restore that symmetry. It would be interesting to find a reliable covariant regularization scheme or, at least, a scheme which while breaking covariance (e.g. in the decomposition of the action into free and interacting parts) does not necessitates non-covariant counterterms.
being included in square brackets. .