Initial conditions and degrees of freedom of non-local gravity

Initial conditions and degrees of freedom of non-local gravity


We prove the equivalence between non-local gravity with an arbitrary form factor and a non-local gravitational system with an extra rank-2 symmetric tensor. Thanks to this reformulation, we use the diffusion-equation method to transform the dynamics of renormalizable non-local gravity with exponential operators into a higher-dimensional system local in spacetime coordinates. This method, first illustrated with a scalar field theory and then applied to gravity, allows one to solve the Cauchy problem and count the number of initial conditions and of non-perturbative degrees of freedom, which is finite. In particular, the non-local scalar and gravitational theories with exponential operators are both characterized by four initial conditions in any dimension and, respectively, by one and eight degrees of freedom in four dimensions. The fully covariant equations of motion are written in a form convenient to find analytic non-perturbative solutions.

\emailAdd \affiliationInstituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain

\emailAdd \affiliationDepartment of Physics, Southern University of Science and Technology, Shenzhen 518055, China

\emailAdd \affiliationDipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore,
via Musei 41, 25121 Brescia, Italy \affiliationTIFPA – INFN c/o Dipartimento di Fisica, Università di Trento,
38123 Povo (Trento), Italy


Classical Theories of Gravity, Models of Quantum Gravity, Nonperturbative Effects \preprintJHEP 05 (2018) 087 [arXiv:1803.00561]

1 Introduction

There is cumulative evidence that theories with exponential non-local operators of the form


have interesting renormalization properties. After early studies of quantum scalar field theories [1, 2, 3, 4] and gauge and gravitational theories [5, 6, 7, 8, 9, 10, 11], in recent years there has been a surge of interest in non-local classical and quantum gravity [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. A non-local theory of gravity aims to fulfill a synthesis of minimal requirements: (i) spacetime is a continuum where Lorentz invariance is preserved at all scales; (ii) classical local (super-)gravity should be a good approximation at low energy; (iii) the quantum theory must be perturbatively super-renormalizable or finite; (iv) the quantum theory must be unitary and ghost free, without extra pathological degrees of freedom in addition to those present in the classical theory; (v) typical classical solutions must be singularity-free.

The typical structure of the gravitational action in topological dimensions is

where is the gravitational constant and are form factors dependent on the dimensionless ratio , where is the characteristic energy scale of the system, is the Laplace–Beltrami or d’Alembertian operator and is the covariant derivative of a vector . Our conventions for the curvature invariants are


The particular choice of form factors

leads to the action [21, 23, 24, 25, 28]


where is the Einstein tensor (107) and


This model is dictated by the above program (i)–(v) and may be also regarded as a phenomenological non-local limit of M-theory [28]. The role of the non-local operator is to compensate the second-order derivatives in curvature invariants. Its definition is presented in appendix A. To date, the perturbative renormalizability of the theory with (6) has been proven only with the use of the resummed propagator [30], while infinities have not been tamed yet in the orthodox expansion with the bare propagator. Nevertheless, this theory encodes all the main features of those non-local quantum gravities that have been shown to be renormalizable and its dynamics is simpler to deal with.

Even without considering gravity and the quantum limit, there is a general conceptual issue usually characterizing non-local physics. Namely, the Cauchy problem can be ill defined or highly non-standard in non-local theories [35, 36, 37, 38]. In fact, while there is a time-honored tradition on linear differential equations with infinitely many derivatives that admit a fair mathematical treatment [35, 36], non-linear non-local equations such as those appearing in non-local field theories are a very different and much trickier business. For any tensorial field , it entails an infinite number of initial conditions , , , …, representing an infinite number of degrees of freedom. As the Taylor expansion of around is given by the full set of initial conditions, specifying the Cauchy problem would be tantamount to knowing the solution itself, if analytic [39]. This makes it very difficult to find analytic solutions to the equations of motion, even on Minkowski spacetime. Fortunately, the exponential operator (1) is under much greater control than other non-local operators, since (at least for finite ) the diffusion-equation method is available to find analytic solutions [41, 40, 42, 43, 44, 45, 46, 47] which are well defined when perturbative expansions are not [40]. The Cauchy problem can be rendered meaningful, both in the free theory [36, 37, 48] and in the presence of interactions [42]. Consider a real scalar field dependent on spacetime coordinates . According to the diffusion-equation method, one promotes to a field living in an extended spacetime with a fictitious extra coordinate . This field is assumed to obey the diffusion equation , implemented at the level of the -dimensional action by introducing an auxiliary scalar field (dynamically constrained to be ). Since the diffusion equation is linear in (and , consequently), the Laplace–Beltrami operator commutes with the diffusion operator and exponential operators act as translations on the extra coordinate, . One can then show that, from the point of view of spacetime coordinates, the -dimensional system is fully localized and that the only initial conditions to be specified are , , , [42]. The infinite number of initial conditions , , , …have been transferred into two initial conditions, which are actually boundary conditions in , for an auxiliary field. When interactions are turned off, vanishes and one obtains the single degree of freedom, represented by and , of the free local theory.1 The original system is recovered when acquires a specific, fixed value proportional to the scale . This value depends on the solution and is determined by solving the localized equations at , where is a constant. The resulting solutions are not exact in general but they satisfy the equations of motion to a very good level of approximation [41, 45, 47].

For non-local gravity, one would like to apply the same method to the metric itself or to curvature invariants , but this is not possible in a direct way. Calling the curvature invariants of a putative localized theory, since the diffusion equation would be non-linear in the metric , one would have


and one would be unable to trade non-local operators for shifts in the extra direction. Moreover, the diffusion method applies for exponential operators, while in the actual quantum-gravity action (5) non-locality is more complicated.

In this paper, we address this problem. First, we will use a field redefinition (already employed in other non-local gravities, although not for (5) [16, 27], and similar to those used in scalar-tensor theories and modified gravity models) to transfer all non-locality to an auxiliary field . Next, we impose the diffusion equation on : the linearity problem is thus immediately solved and one can proceed to localize the non-local system, count the initial conditions and identify the degrees of freedom, which are finite in number. From there, one can begin the study of the dynamical solutions of the classical Einstein equations, but this goes beyond the scope of the present work. Counting non-local degrees of freedom is a subject surrounded by a certain halo of mystery and confusion in the literature. To make it hopefully clearer, we will make a long due comparison of the counting procedure and of its outcome in the methods proposed to date: the one based on the diffusion equation and the delocalization approach by Tomboulis [49].

1.1 Plan of the paper

In preparation for the study of non-local gravity, the diffusion-equation method is reviewed in section 2 for a scalar field. This example is very useful because it contains virtually all the main ingredients we will need to localize non-local gravity and rewrite it in a user-friendly way: localized action, auxiliary fields, slicing choice, matching of the non-local and localized equations of motion, counting of degrees of freedom, solution of the Cauchy problem, and so on. The non-local scalar is introduced in section 2.1, while the localization procedure is described in section 2.2. The counting of initial conditions and degrees of freedom is carried out in section 2.3, where we find that this number is, respectively, 4 and 1 for the real non-local scalar with non-linear interactions. Section 2.4 reviews another practical use of the diffusion-equation method, the construction of analytic solutions of the equations of motion. In section 2.5, we compare the diffusion-equation method with the results obtained in other approaches, mainly the delocalization method by Tomboulis [49]. A generalization of the method to non-local operators with polynomial exponents is proposed in section 2.6, while non-polynomial profiles require some extra input which is discussed in a companion paper [50].

The non-local gravitational action (5) is studied in section 3, where we find the background-independent covariant Einstein equations for any form factor and recast the system in terms of an auxiliary field. Contrary to other calculations in the literature [27, 51, 52], we find the equations of motion for an exponential-type form factor (6) in terms of parametric integrals rather than from the series expansion of the non-local operators. This new form is crucial both to solve the initial-value problem and to find explicit solutions with the diffusion-equation method.

The localized system corresponding to the non-local gravitational action (5) is introduced and discussed in section 4. After defining the localized action in section 4.1, we obtain the equations of motion in section 4.2, which agree with the non-local ones. The counting of initial conditions and degrees of freedom is done in section 4.3, where we find that they amount to, respectively, 4 and . Appendices contain several technical details and the full derivation of the equations of motion.

Therefore, although in sections 2.2 and 4 we will concentrate on the form factor (6) for which renormalization is likely but still under debate, our results with auxiliary fields (section 3.2) will be valid for an arbitrary form factor, while in section 2.6 and in [50] we will generalize the diffusion-equation approach to form factors associated with finite quantum theories.

1.2 Summary of main equations and claims

To orient the reader, we summarize here the key formulæ:

  • Scalar field theory.

    • Non-local action: (8).

    • Non-local equation of motion: (9).

    • Localized action: (11).

    • Localized equations of motion: (16), (21), (22).

    • Constraints on localized dynamics: (19), (23).

    • Number of field degrees of freedom: (40).

    • Number of initial conditions: (41).

  • Gravity.

    • Non-local action: (46).

    • Non-local equations of motion: (49).

    • Non-local action with auxiliary field: (57).

    • Non-local equations of motion with auxiliary field: (58).

    • Localized action: (67).

    • Localized equations of motion: (75), (76), (77), (LABEL:lasto).

    • Constraints on localized dynamics: (83), (84).

    • Number of field degrees of freedom: (85).

    • Number of initial conditions: (86).

2 Diffusion-equation method: scalar field

Before considering gravity, it will be useful to illustrate the main philosophy beyond and advantages of the diffusion-equation method. To this purpose, we review its application to a classical scalar field theory [42], expanding the discussion therein to cover all important points that will help us to understand the results for non-local gravitational theories. We present a simplified version of the scalar system, with no nested integrals, no free parameters in the diffusion equation, and fewer assumptions than in [42]. The original version of [42] can be found in appendix B. A comparison between the scalar and gravitational systems will be done in section 4.2.

2.1 Non-local system: traditional approach and problems

Consider the scalar-field action in -dimensional Minkowski spacetime (with signature )


where is a constant of mass dimension and is a potential. We chose the exponential operator as the simplest example where the diffusion method works, but we will relax this assumption later to include operators of the form not contemplated in the original treatment in [42]. Applying the variational principle to , the equation of motion is


where a prime denotes a derivative with respect to . The action (8) and the dynamical equation (9) are a prototype of, respectively, a non-local system and a non-local equation of motion.

The initial-condition problem associated with (9) suffers from the conceptual issues outlined in the introduction. Rather than repeating the same mantra again, we recast the Cauchy problem as a problem of representation of the non-local operator . To find a solution of (9), one must first define the left-hand side. The most obvious way to represent the exponential is via its series,


To find solutions, one can use different strategies. One of the oldest and most disastrous is to truncate the non-local operator up to some finite order . In doing so, one introduces instabilities corresponding to the Ostrogradski modes of a higher-derivative theory which has little or nothing to do with the starting theory [39, 40]. Exact procedures such as the root method exist for linear equations of motion [37, 53, 48, 54] but they have the disadvantage of being applicable only to non-interacting systems. Another possibility is to choose a profile and apply the operator (10), but the series does not converge in general [40]. This does not necessarily mean that the chosen profile is not a solution of the equations of motion. Rather, the series representation (10) is ill defined for a portion of the space of solutions. Even in the case where an exact solution is found, however, this may be non-unique for a given set of initial conditions [53, 39, 49].

2.2 Localized system

The diffusion-equation method [41, 42, 44, 47], some elements of which can be found already in [37] (section III.B.3), bypasses the above-mentioned issues by converting the Cauchy problem into a boundary problem.2 All the non-locality is transferred into a fictitious extra direction and infinite initial conditions for the scalar field are converted to a finite number of field conditions on the slice along the extra direction, where is a positive dimensionless constant (i.e., it is the physical value of measured in units). In other words, the rectangle can be spanned either along the (time) direction, as done when trying to solve the problem of initial conditions by brute force at , or along the direction, as done in the boundary-value problem with the diffusion method; see Fig. 1 here and Fig. 1 of [44].

Figure 1: Diffusion-equation method describing the dynamics of the scalar field theory (8) as the dynamics of the localized system (11) on the slice .

We will also be able to find the exact number of conditions required and to compare these results with those from other methods [49].

Lagrangian formalism

The main idea is to exploit the fact that the exponential operator in (8) acts as a translation operator if obeys a diffusion equation. Using this property, we can convert the non-local system into a localized one where the diffusion equation is part of the dynamical equations, the field is evaluated at different points in an extra direction (along which the system is thus non-local), and only second-order derivative operators appear in the action and in the equations of motion. In this way, one can make sense of the Cauchy problem in the localized system and also in the non-local one, after establishing the conditions for which the two systems are equivalent [42]. This construction goes through some initial guesswork about the form of the correct localized system, especially regarding the integration domain of certain parts of the action, but this is not difficult in general. Both the scalar case (8) and the gravitational action (5) are simple enough to create no big trouble.

Let us therefore forget temporarily about the non-local system (8) and consider the -dimensional local system


where is an extra direction, is a specific value of , and are -dimensional scalar fields and


hence . The action (11) is second-order (hence local) in spacetime derivatives and non-local in (because the fields take different arguments). The integration range of in (11) is arbitrary, it can be set equal to or any other interval containing (the slices and play a special role: the former is the value where to specify the initial condition in of the diffusion equation, while the latter will be the physical value of the parameter , for a given ).

The equations of motion are calculated from the infinitesimal variations of the action, using the functional derivative for a field . Since and are arbitrary, one can always assume the support of these delta distributions to lie within the integration domains in (11), so that integrations in , and are removed and the fields evaluated at and . Bars will be removed in the final equations of motion.

The first variation we calculate is with respect to . To keep notation light, let us ignore the trivially local -dependence from now on. Doing it step by step,


The integration of the Dirac distribution in gives the prescription , hence such that the support of the lies in both - and -integration ranges. After a reparametrization , one gets an integral of the form . Since is arbitrary, the integrand must be identically zero on shell for any integration range:


Another way to obtain the same result is to restrict from the very beginning the integration range in (11) from 0 to or from to . In the first case, the integration range in (15) is reduced to , since . In the second case, the range in (15) is reduced to , since . In both cases, due to the arbitrariness of the width of the integration domain is arbitrary, which implies that the integrand is zero.

The diffusion equation (16) is the first equation of motion. The second equation of motion is more complicated but very instructive, so that we report it in full. We integrate (13) by parts, in order to load all derivatives onto :


Therefore, varying with respect to gives


From this, we conclude that equation (18) reproduces (9) if


where is a real constant, and


In fact, in this case obeys the same diffusion equation (16) as , so that the two contributions in (18) must vanish separately, thus yielding the two equations of motion (restoring -dependence)


Then, when evaluating (21) at the first term yields , the second term vanishes and (21) reproduces (9) exactly. See Fig. 2 for a toy example. Note that imposing (19) only at ,


or at any given instead of for all would again yield (19), provided obeyed (22). In fact, parametrizing with , .

The introduction of the parameter in (19) reflects the fact that the choice of the slice where the -dimensional scalar field coincides with the -dimensional field does not affect the final result. For instance, one could have chosen and identified (the “initial” condition in of the diffusion equation), . However, in section 2.4 we will argue that equation (19) is far better suited than for the task of finding dynamical solutions. This is why we introduced a strictly positive in the first place.

Figure 2: In flat Euclidean space, the solution of the diffusion equation (16) with initial condition is . This solution is represented in the plane as an orange surface (concavity upwards) in the left plot, together with (blue surface, concavity downwards). The section of these surfaces at (black thick line) are shown in the right plot.

To summarize the logic here, given the non-local system (8) one can always write down the system (11)–(13) localizing it. This localized system is not in one-to-one correspondence with the non-local system but it always admits, among its solutions, the solutions of the non-local system. These solutions are defined by the boundary condition (19) together with the local condition (23). The sub-set of solutions of the localized system obeying these conditions are solutions to the original non-local one, since the above conditions are valid on shell (i.e., applying (16) and (22) to (21)). In other words, (19) and (23) define the sub-set of solutions of the localized system that recover the equations of motion and solutions of the original non-local system. Recalling that the localized system (11) must be reducible to the non-local one (8) only at a certain slice in the extra direction, it is clear that we do not need to study the most general -dimensional evolution of the localized dynamics, which is obtained by dropping (23).

Notice that it is not possible, while keeping the diffusing structure unaltered, to change the status of (23) from a condition imposed by hand to a consequence of the dynamics. For instance, one could try to add an extra term to the action (11), which would give (23) when varying with respect to the Lagrange multiplier . However, equations (15) and (18) would become, respectively, and , where the extra terms would vanish separately if, again, we imposed by hand


This condition, replacing (23), amounts to forbid source terms in the diffusion equation (16). Indeed, the infinitely many degrees of freedom of the original non-local system are encoded in equation (23) or in the alternative equation (24), both of which are a condition on the infinitely many -values of the fields and . Thus, demanding to get a fully self-determined diffusing localized system equivalent to the non-local one is not only impossible,3 but also meaningless, since the equivalence between the localized and the non-local system on one hand and the statement of the initial-value problem for the non-local system on the other hand must both go through the setting of an infinite number of conditions external to the dynamics.

For future use, we highlight three important features of the localization procedure which will apply, in their essence, also to the non-local gravity action (5).

  1. By the diffusion-equation method, one does not establish a one-to-one correspondence between the localized system (11) and the non-local system (8). Rather, we showed that there exist field conditions on the slice such that the localized system has the same spacetime dynamics as the non-local system. This correspondence on a slice is depicted in Fig. 1.

  2. To get the correct result, it was crucial to make a careful choice of the arguments in the diffusion-equation term (13) and a careful treatment of the boundary terms when integrating (13) by parts as in (17). Without such boundary terms, (21) would have been unable to reproduce (9) on the slice with the correct numerical factors.

  3. The localized system is second-order in spacetime derivatives, for both and . Therefore, the Cauchy problem for this system, when restricted to spacetime directions , is solved by four initial conditions at some :


    In particular, these conditions are valid at , so that also the Cauchy problem of the non-local system (8) is solved by four initial conditions, corresponding (via (23)) to and its first three time derivatives. We will find a similar result also in non-local gravity gravity, with four initial conditions for the metric. In general, given a non-local action with exponential non-locality for tensorial field representing physical degrees of freedom, the diffusion-equation method relies on a second-order localized system for a field and an auxiliary field with the same symmetry properties as , thus leading to initial conditions.

Ghost mode

In this subsection, we analyze a hidden ghost mode which, however, does not influence the non-local dynamics. To understand this aspect, we will employ a reformulation of the localized dynamics (equation (LABEL:tildeL)), physically equivalent to (11)–(13), which is convenient to study the degrees of freedom of the theory but is unsuitable for the practical treatment (Cauchy problem, solutions, and so on) of the dynamics, due to problems we will comment on in due course.

It is very well known that the kinetic term in (8) can be symmetrized after integrating by part, so that the Lagrangian becomes


From here, one can make the field redefinition so often used in -adic and string field theory. We will do something similar by considering the localized version of (26), which is given by (11) with (-dependence omitted everywhere)


replacing (12). We note that the integral in (13) is pleonastic for the Laplace–Beltrami term, since both and obey the diffusion equation:


However, replacing (13) with a mixed term

would not give the correct equations of motion, as we will see shortly. The reason is that is originated by an on-shell condition, a trick that invalidates the variational principle. To find the correct Lagrangian, we generalize this term with a generic functional of the fields, . A last step we take (not necessary, but useful to simplify the physical interpretation) is to consider the field redefinitions


so that


and the total Lagrangian on Minkowski spacetime is



The function is determined in appendix C by requiring the recovery of the non-local dynamics on the slice.

Observing (LABEL:tildeL), one sees that the canonical scalar propagates with a kinetic term of the correct sign, while the canonical scalar (hence ) is a ghost. This detail went unnoticed in [42].

There are two issues affecting (LABEL:tildeL) and described in appendix C, but we should not lose sight of the reason why we introduced this Lagrangian. One may choose either (11)–(13) or (LABEL:tildeL) depending on what one wants to study. For the analysis of the Cauchy problem and of the dynamical solutions, the action (11)–(13) is to be preferred, and in fact we will analyze non-local gravity under the same scheme. On the other hand, for the characterization (ghost-like or not) of the localized degrees of freedom the Lagrangian (LABEL:tildeL), or the Hamiltonian (36) we will derive from it in the next subsection, is more indicated. The counting of the localized degrees of freedom (section 2.3) can be performed indifferently in the original system (11)–(13), in the Lagrangian (LABEL:tildeL), or in the Hamiltonian formalism derived from (LABEL:tildeL).

Hamiltonian formalism

To count the number of degrees of freedom in a non-local theory, we must first count the number of localized degrees of freedom in the associated localized -dimensional theory. In the case of the scalar field, this information is already available in Lagrangian formalism, but for completeness we can obtain the same result from Hamiltonian formalism. The example presented in this subsection will illustrate the general method and its caveats. Its application in the localization of the scalar field was sketched in [42], but here we will fill several gaps in that discussion. The actual counting of localized degrees of freedom will be done in section 2.3.

Although we do not write the non-local system (8) in Hamiltonian formalism, we can reach a lesser but still instructive goal, namely, the formulation of the Hamiltonian approach for the associated localized system. However, if we take the localized system (11)–(13) as a starting point we soon meet several problems, all of which stem from the non-locality with respect to the direction. Momenta acquire a rather obscure non-invertible form and one cannot write down a Hamiltonian in phase space. However, the system is not constrained. We can avoid all the trouble by acting directly on (LABEL:tildeL). Calling the Lagrangian, we can define the phase space and the Hamiltonian. The momenta are


Notice that, if we had calculated the momenta directly from (12) and (27), we would have obtained and , which are not invertible locally with respect to and .

The non-vanishing equal-time Poisson brackets in terms of the spatial -vectors are


while the Hamiltonian of the system is (-dependence omitted again)


where it is understood that . Since is shifted in , is non-local in due to the terms in the last line of (36). Nevertheless, the Hamiltonian is written solely in terms of phase-space variables and the phase-space fields are completely local in spacetime coordinates.

The evolution equations for the fields and trivially gives the momenta, , , while the Hamiltonian evolution of the momenta give the localized equations of motion (98) and (99):


2.3 Initial conditions and degrees of freedom

The question about how many initial conditions we should specify for the non-local scalar system is related to another one: How many degrees of freedom are hidden in equation (9)? In higher-derivative theories, the presence of many degrees of freedom (Ostrogradski modes) is well known. For a system with derivatives, the Cauchy problem is uniquely solved by initial conditions. However, there is an uncrossable divide between higher-derivative and non-local theories, and one cannot conclude that non-local theories need initial conditions; conversely, truncating a non-local theory to finite order leads to a physically different model [53, 40].

To understand the problem, we review its root and also some confusion surrounding it. First of all, there is agreement in the literature about the fact that the free system with constant, linear or quadratic has two initial conditions. In the absence of interactions, the Cauchy problem associated with (9) is specified only by and . The entire functional introduces no new poles in the spectrum of and the system is equivalent to the local one with , as is obvious from the field redefinition .4 This was first recognized as early as 1950 in the seminal paper by Pais and Uhlenbeck [37] (section III.B.3) and reiterated more recently, sometimes using very different terminology and techniques, in other works [53, 56, 48, 42].

More contrived is the case with interactions. The reader unfamiliar with non-local theories may wonder why interactions should make any difference when counting the number of initial conditions. The reason is that, in this case, there is no field redefinition absorbing the non-local operator of the kinetic term. Any other rewriting will not work, either. For instance, a non-local kinetic term can always be expressed as a convolution with a kernel [37]. Consider the scalar-field Lagrangian with generic form factor . In momentum space, calling the Fourier transform of ,


Specifying the form factor determines the spectrum of the field. In general, the poles of the propagator correspond to the zeros of and to the poles of . This correspondence is straightforward for a massless dispersion relation , where and denotes the derivative of order of the delta. The derivative order of the delta is the order of the pole. Polynomial dispersion relations have a similar structure, e.g., gives . For , this dispersion relation corresponds to one massive and one massless scalar mode, for a total of two double poles.5 Furthermore, when is non-local the field spectrum depends on whether the form factor is entire or not. In the case of (9), the propagator


has a massless double pole, while has a double massive pole. In the last two cases, the order and nature (massless or massive) of the particle poles and the poles of is less transparent, although their counting agrees.

From this exercise, it should become clear that hiding infinitely many derivatives into integrals with non-trivial kernels such as (38), or to transfer part of these derivatives onto the scalar potential and then converting them into integral operators, does not help in solving the Cauchy problem, since the two formulations are equivalent (on the space of real analytic functions [39]). In [50], we complement this no-go result with its way out: If the kernel can be found by solving some finite-order differential equation extra with respect to the dynamical equations, then its contribution to the Cauchy problem becomes under full control.

The novelty brought in by the diffusion-equation method is that it allows one to go beyond the free theory and count the extra number of initial conditions. Surprisingly, in the scalar-field case this number is two, which sums to the two initial conditions of the free theory for a total of four. This result goes against the belief, implicitly endorsed in some literature, that the information from the free theory is complete. In particular, when one says that the number of initial conditions for solving the Cauchy problem of the theory (8) is two, one should specify that this is true only for the free, perturbative case.

We reach this conclusion in three steps: (i) counting the number of field degrees of freedom of the localized theory; (ii) specifying the number of initial conditions (in time) for each localized field; (iii) restricting our attention to the slice where the non-local dynamics is recovered, and proceeding with the counting thereon. In Lagrangian formalism, we saw that there are two independent localized fields, either the pair and or the pair and . Consistently, the same result is obtained in Hamiltonian formalism, where there are two non-vanishing independent momenta and . Since the dynamics is second-order in spacetime derivatives, there are two initial conditions per field, for a total of four.

Number of degrees of freedom: scalar field. The localized real scalar field theory (11)–(13) in dimensions has two scalar degrees of freedom and . On the -slice where the system is equivalent to the non-local real scalar field theory (8) in dimensions, the degree of freedom is no longer independent. Consequently, the non-local theory has one non-perturbative scalar degree of freedom . (40)
Number of initial conditions: scalar field. The Cauchy problem on spacetime slices of the localized real scalar field theory (11)–(13) in dimensions is specified by four initial conditions , , , . As a consequence, the Cauchy problem of the non-local non-perturbative real scalar field theory (8) in dimensions is specified by four initial conditions , , , . (41)

The nature of the new degree of freedom is quite peculiar. As we saw above with a diagonalization trick (used, for instance, also in [57]), this field is a ghost and, in fact, the Hamiltonian (36) is unbounded from below. From the point of view of the -dimensional localized system (11)–(13), arises as a Lagrange multiplier introduced to enforce the diffusion equation of ; itself does not appear in its own equation of motion (16). Its -dimensional dynamics, given by the equation of motion of , is non-trivial (in Hamiltonian formalism, the momentum does not vanish) but it only amounts to diffusion, equation (22). Eventually, it turned up that it is associated with by the second-order derivative relation (23). From the point of view of the -dimensional non-local system, disappears because its diffusion is frozen at a given slice, and the dynamics is written solely in terms of , its derivatives and its potential. At this point, there is only one degree of freedom whose perturbative classical propagator (39) describes a non-ghost massive scalar mode. The potentially dangerous ghost mode in the -dimensional system turns out to be non-dynamical in -dimensions and in the free theory.

In the interacting non-local theory, does play a part in the dynamics, but in the form of the potential for . Combined with equation (9), the local condition (23) explains in part the finite proliferation of degrees of freedom in the interacting case. Since (23) implies for all , then from (9) one has


If , then and there is no extra degree of freedom with respect to the case. For a cubic or higher-order polynomial, is not linearly equivalent to . Non-linearities can generate new degrees of freedom (a typical example is gravity, which contains a hidden scalar mode apart from the graviton) but not in this case, since the field is not dynamical on the slice where (42) holds.

2.4 Solutions

Solutions of non-local theories can be categorized into perturbative and non-perturbative. Perturbative solutions can have two meanings, either as the solutions obtained when truncating the non-local operators to a finite order (a procedure we will not discuss here [53, 39, 40]) or as the solutions obtained, order by order, starting from the free theory and modeling interactions as a perturbative series [39, 53, 58]. When all non-locality acts on interactions, the two meanings coincide. Non-perturbative solutions are all those solutions that cannot be reached in these ways and, in general, they constitute the great majority of all possible solutions of the system. The diffusion-equation method permits to get access precisely to these solutions with generic non-perturbative potential [41, 40, 43, 45, 46, 18].

When introducing the condition (19), we commented on the fact that the identification of the localized dynamics with the non-local one could take place at any slice, including at where . However, for the sake of the construction of actual solutions this choice is not fortunate, since it corresponds to the initial condition of the heat kernel. In other words, setting the initial condition (in ) of the -dimensional system to be the solution of the non-local system would take us back to the usual paradox with non-local dynamics, namely, that knowing all the infinite number of initial conditions (in time) is tantamount to already knowing the Taylor expansion of the full solution around . It is more logical, then, to impose (19) (the non-local solution is the outcome of the diffusion from to rather than of anti-diffusion from to ) and to set the initial condition in as something else. This “something else” can be most naturally recognized as the solution of the local system obtained by setting in equations (8) and (9):


Then, the solution of the diffusion equation (16) can be found in integral form in momentum space. Calling the eigenvalue of the Laplace–Beltrami operator and writing


one has


Since we know , we also know its Fourier transform and we can obtain the full non-local solution . Examples of solutions of the scalar-field equation of motion (9) using the diffusion-equation method can be found in [40] (on a Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological background), [41, 47] (Minkowski background, rolling tachyon of open string field theory), [42, 46] (Minkowski and FLRW backgrounds,