SUMMING OVER SPACETIME DIMENSIONS IN QUANTUM GRAVITY

Erik Curiel, Felix Finster and J.M. Isidro Munich Center for Mathematical Philosophy, Ludwig-Maximilians-Universität, Geschwister-Scholl-Platz 1, D-80539 München, Germany Fakultät für Mathematik, Universität Regensburg, D-93040 Regensburg, Germany Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Valencia 46022, Spain Erik.Curiel@lrz.uni-muenchen.de, finster@ur.de, joissan@mat.upv.es


Introduction
Feynman propagators encode more information than meets the eye. Let us for simplicity consider a massive scalar particle on a D-dimensional manifold M. When the latter is Minkowskian flat space R D , it has been found [10,11,12] that inertial scalar propagators suffice in order to derive the thermal properties of the Rindler horizon. A slight modification of Schwinger's proper-time representation of the scalar propagator turns out to provide an ultraviolet completion of the scalar theory, both in flat spaces [8,9] and in spaces of constant nonzero curvature [6]; this ultraviolet completion amounts to the existence of a quantum of length L. These are just a few examples of (in principle unexpected) quantum-gravity properties of Feynman scalar propagators, the latter considered in the absence of gravity.
In the naive path-integral approach to quantum gravity one has to sum over all metrics on the given manifold M. In the presence of several inequivalent differentiable structures and/or topologies one has to sum over them: one integrates over more than one manifold structure. This raises the question of summing over all possible dimensions as well.
In this letter we consider a free scalar particle in the spacetime M = R D , the latter endowed with its standard topology, differentiable structure, and Euclidean or Minkowskian metric, and perform a sum of scalar Feynman propagators over all dimensions D ′ ≥ D. Each one of these summands is free of quantum-gravity effects; but the sum of all summands will turn out to produce quantum-gravitational effects on the scalar particle. This will provide yet another unexpected example of quantum-gravity effects that are encoded within flat-spacetime, inertial Feynman propagators. As will be explained in more detail below, the quantum-gravity effects under consideration are those arising from the existence of a quantum of length L.
We present our analysis first in Euclidean R D in section 2. In section 3 we Wickrotate back to Minkowskian R D , the latter with signature (+, −, . . . , −). In section 4 we report an alternative derivation of the same results due to Padmanabhan (private communication). In our use of Bessel functions we follow the conventions and notations of ref. [5].
2 Quantum-gravitational properties of inertial propagators: Euclidean space 2.1 An ultraviolet completion A massive scalar particle in Euclidean momentum space R D has the Feynman propagator (p 2 + m 2 ) −1 . Fourier transforming and inserting Schwinger's proper-time integral one arrives at Above, K ν (z) is a modified Bessel function, a solution to the modified Bessel equation The latter possesses two linearly independent solutions, conventionally denoted I ν (z) and K ν (z). For later use we recall two wellknown properties [13]. First, the K ν (z) are singular at z = 0 while the I ν (z) are everywhere regular. Second, the I n (z) have the following generating function: Now the right-hand side of (3) can be regarded as a certain ultraviolet completion of the standard integral kernel for the Feynman propagator, the latter expressed as a path integral. Indeed, in refs. [8,9] one modifies the standard path integral where the action integral S(x) = x ds equals the proper length, to become The superindex QG in the corrected Feynman propagator (5) stands for quantumgravity. It has been argued in refs. [8,9] that the propagator (5) includes the effects of the presence of a quantum of length as given by the minimal length L. In this sense, the modified path integral (5) can be regarded as an ultraviolet completion of the standard scalar theory on Euclidean space. The right-hand side of the modified path integral (5) closely resembles the generating function (3), while reducing to the usual case (4) in the limit L → 0. Indeed the result of computing the path integral (5), correctly reduces to the propagator (2) when L = 0. Moreover, when L = 0, this ultraviolet completion of the scalar Euclidean propagator has the property that it no longer diverges as r → 0, because the presence of the quantum of length L prevents it. So quantum-gravity effects mollify the Feynman propagator at short distances, while at the same time ensuring invariance under the transformation [8,9] S −→ L 2 S .
We close this section with two observations. i) A related, though different, ultraviolet completion of the Euclidean scalar propagator has been considered in ref. [1]. It is based on the replacement of the proper time variable s within the exponential in Eq. (1) with the function s + s −1 .
ii) The ultraviolet completion of the Euclidean scalar propagator summarised above (as originally presented in refs. [8,9]) was not based on the generating-function-for-Bessel-functions approach followed here. Rather, it was based on the requirement of invariance under the duality (7). However, we have found it useful to reword it in the language of generating functions, with an eye on what comes next.

An identity satisfied by the K ν (z)
Do the Bessel functions K n (z) also satisfy an identity of the type (3)? To the best of our knowledge, no such expression has been published in the standard literature [5,13]. In what follows we derive a new identity satisfied by the K n (z). It reads In order to prove Eq. (8) we start from the integral representation [5] K µ (z) = 1 2 and consider the series Substituting (9) into the above and rearranging terms we find which establishes (8).

Summing over dimensions
As was already the case with the known identity (3), our new identity (8) will turn out to have an interesting application to quantum gravity. We first evaluate (8) at z = y/2 and t = −1/(2y) to obtain valid whenever y = 0 and | arg(y)| < π/4. Without loss of generality it will be convenient to set m = 1, when solving Eq. (2) for the Bessel functions K µ in terms of Euclidean Feynman propagators G D : Next we set y on the right-hand side of (11) equal to r = 2ν+2 j=1 x 2 j 1/2 , bearing in mind that the same y will appear on the left-hand side. We remark that the natural radial variable that the summand K n+ν (y) on the left-hand side of (11) should depend on is not 2ν+2 j=1 x 2 j 1/2 but 2n+2ν+2 j=1 x 2 j 1/2 ; we will return to this point presently.

Substitution of Eq. (13) into (11) finally gives
We know that quantum-gravity effects on the Euclidean scalar propagator in D = 2ν + 2 dimensions cause the appearance of a quantum of length L. These effects have been taken into account in Eq. (14), as shown by the right-hand side. The lefthand side expresses this quantum-gravitationally corrected propagator as an infinite sum of gravity-free propagators. Each summand corresponds to one higher value of the dimension 2n+2ν +2, one for each n ∈ N. Within each (2n+2ν +2)-dimensional space that contributes to the above sum, however, only a (2ν + 1)-dimensional sphere S 2ν+1 ⊂ R 2n+2ν+2 is swept out by the equation r = const. With increasing values of the dimension 2n + 2ν + 2 it is always this same sphere S 2ν+1 that is swept out, i.e., the sphere does not lie along the additional dimensions. Hence the higher dimensions being summed over in Eq. (14) play the role of a virtual spacetime for propagation in the actual dimension 2ν + 2. Being virtual, however, does not imply that they are unphysical, as they add up to a nonvanishing quantum-gravitational correction to the Feynman propagator in 2ν + 2 dimensions.
the superindices M, E referring to Minkowski and Euclidean space, respectively. By Eq.
(2) one thus finds where H (2) µ = J µ − iY µ is a Hankel function of the second kind [5]. With the understanding that we will henceforth work in Minkowski space, we will drop the superindex M from our notations. Also, for simplicity we will restrict our attention to timelike vectors and denote so r > 0. In the units of Eq. (12), the gravity-free propagator (16) simplifies to The quantum-gravity corrected Feynman propagator in Minkowski space R D is readily obtained from the above: following refs. [8,9], it suffices to make the replacement r = √ r 2 → √ r 2 + 4L 2 . This yields in (16) which, in the units of Eq. (12), becomes did the work in path integral duality by an alternative approach. His derivation is as follows: The Schwinger propertime representation of the QG-corrected propagator, in D-dimensional Euclidean sector, is given by the integral: with r 2 as in Eq. (2). Note that the dependence on the dimension arises only from the factor s −D/2 . We now expand the factor e −L 2 /4s in the integrand in a Taylor series. The n-th term in the series will introduce the factor s −n , which will change the original s −D/2 factor to s −(D+2n)/2 , which occurs in the standard (uncorrected) propagator G D+2n (r) for the dimension D + 2n. So we immediately get: The QG corrected propagator can therefore be expressed as the sum of standard propagators for the dimensions (D + 2n). The simplicity of this derivation is noteworthy. It is obvious that the approach also works in the Lorentzian sector.

Conclusions
Our conclusions are summarised by Eqs. (14) and (24), again collected below for convenience: in Euclidean R D we have In both cases we find that a quantum-gravity corrected propagator in D dimensions can be expressed as an infinite sum over all gravity-free propagators in dimension D ′ ≥ D. Under quantum-gravity corrections we understand, as already explained, the ultraviolet completion obtained by the inclusion of a minimal length [8,9]. In the latter papers, the existence of a quantum of length L has been shown to be equivalent to the requirement of invariance under the duality (7), the dimensionality of spacetime being kept fixed. The equivalent viewpoint that emerges from our analysis is that a smallest distance L results from summing over an infinite number of higher dimensions.
Ultraviolet completions of standard theories have been the subject of many analyses, too numerous to quote here; see however ref. [4] for a sample of different standpoints. In this letter we have concentrated on the case of a massive scalar in flat R D , either Euclidean or Minkowski. The particular ultraviolet completion studied here, Eq. (5), is usually regarded as enforcing the duality (7). Our approach in this letter interprets this same completion on the basis of the generating function (3) for the modified Bessel functions I n (z) (and similar identities satisfied by their close cousins the Macdonald functions K n (z), the Hankel functions H (2) (z), etc.) Alternatively but equivalently, these identities (Eqs. (3), (8) and (21)) amount to a sum over an infinite number of virtual dimensions. On the basis of the aforementioned identities we have derived the expansions (28) and (29) for Feynman propagators.
Altogether, our analysis somehow places the dimensionality of spacetime on an equal footing with other variables that are integrated over in quantum gravity, such as the metric and the topology of the spacetime manifold M. That the notion of dimensionality might not be as fundamental as believed has also been hinted at in various settings such as string theory [3] and others. It is an intriguing question to ask, if our expansions (28) and (29) could possibly find a thermal analogue in the fact that thermal scalar Green functions can be written as an infinite, imaginary-time sum of the corresponding zero-temperature Green functions [2,11]. We hope to report in the future.