Initial conditions and degrees of freedom of nonlocal gravity
Abstract
We prove the equivalence between nonlocal gravity with an arbitrary form factor and a nonlocal gravitational system with an extra rank2 symmetric tensor. Thanks to this reformulation, we use the diffusionequation method to transform the dynamics of renormalizable nonlocal gravity with exponential operators into a higherdimensional 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 nonperturbative degrees of freedom, which is finite. In particular, the nonlocal 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 nonperturbative solutions.
g.calcagni@csic.es \affiliationInstituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain
lmodesto@sustc.edu.cn \affiliationDepartment of Physics, Southern University of Science and Technology, Shenzhen 518055, China
giuseppe.nardelli@unicatt.it
\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]
Contents
 1 Introduction
 2 Diffusionequation method: scalar field
 3 Nonlocal gravity: equations of motion
 4 Localization of nonlocal gravity
 A The operator
 B Original version of the localized scalar system
 C Comments on the system ()
 D Variations of curvature invariants and form factors
 E Derivation of the Einstein equations (49)
 F Derivation of equation (58)
 G Equivalence of equations (49) and (58)
 H Derivation of equation ()
1 Introduction
There is cumulative evidence that theories with exponential nonlocal operators of the form
(1) 
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 nonlocal 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 nonlocal 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 superrenormalizable 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 singularityfree.
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
(2)  
(3)  
(4) 
The particular choice of form factors
leads to the action [21, 23, 24, 25, 28]
(5) 
where is the Einstein tensor (107) and
(6) 
This model is dictated by the above program (i)–(v) and may be also regarded as a phenomenological nonlocal limit of Mtheory [28]. The role of the nonlocal operator is to compensate the secondorder 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 nonlocal 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 nonlocal physics. Namely, the Cauchy problem can be ill defined or highly nonstandard in nonlocal theories [35, 36, 37, 38]. In fact, while there is a timehonored tradition on linear differential equations with infinitely many derivatives that admit a fair mathematical treatment [35, 36], nonlinear nonlocal equations such as those appearing in nonlocal 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 nonlocal operators, since (at least for finite ) the diffusionequation 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 diffusionequation 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.
For nonlocal 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 nonlinear in the metric , one would have
(7) 
and one would be unable to trade nonlocal operators for shifts in the extra direction. Moreover, the diffusion method applies for exponential operators, while in the actual quantumgravity action (5) nonlocality is more complicated.
In this paper, we address this problem. First, we will use a field redefinition (already employed in other nonlocal gravities, although not for (5) [16, 27], and similar to those used in scalartensor theories and modified gravity models) to transfer all nonlocality 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 nonlocal 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 nonlocal 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 nonlocal gravity, the diffusionequation 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 nonlocal gravity and rewrite it in a userfriendly way: localized action, auxiliary fields, slicing choice, matching of the nonlocal and localized equations of motion, counting of degrees of freedom, solution of the Cauchy problem, and so on. The nonlocal 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 nonlocal scalar with nonlinear interactions. Section 2.4 reviews another practical use of the diffusionequation method, the construction of analytic solutions of the equations of motion. In section 2.5, we compare the diffusionequation method with the results obtained in other approaches, mainly the delocalization method by Tomboulis [49]. A generalization of the method to nonlocal operators with polynomial exponents is proposed in section 2.6, while nonpolynomial profiles require some extra input which is discussed in a companion paper [50].
The nonlocal gravitational action (5) is studied in section 3, where we find the backgroundindependent 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 exponentialtype form factor (6) in terms of parametric integrals rather than from the series expansion of the nonlocal operators. This new form is crucial both to solve the initialvalue problem and to find explicit solutions with the diffusionequation method.
The localized system corresponding to the nonlocal 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 nonlocal 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 diffusionequation 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:
2 Diffusionequation method: scalar field
Before considering gravity, it will be useful to illustrate the main philosophy beyond and advantages of the diffusionequation 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 nonlocal 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 Nonlocal system: traditional approach and problems
Consider the scalarfield action in dimensional Minkowski spacetime (with signature )
(8) 
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
(9) 
where a prime denotes a derivative with respect to . The action (8) and the dynamical equation (9) are a prototype of, respectively, a nonlocal system and a nonlocal equation of motion.
The initialcondition 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 nonlocal operator . To find a solution of (9), one must first define the lefthand side. The most obvious way to represent the exponential is via its series,
(10) 
To find solutions, one can use different strategies. One of the oldest and most disastrous is to truncate the nonlocal operator up to some finite order . In doing so, one introduces instabilities corresponding to the Ostrogradski modes of a higherderivative 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 noninteracting 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 nonunique for a given set of initial conditions [53, 39, 49].
2.2 Localized system
The diffusionequation method [41, 42, 44, 47], some elements of which can be found already in [37] (section III.B.3), bypasses the abovementioned issues by converting the Cauchy problem into a boundary problem.
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 nonlocal 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 nonlocal), and only secondorder 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 nonlocal 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 nonlocal system (8) and consider the dimensional local system
(11)  
(12)  
(13) 
where is an extra direction, is a specific value of , and are dimensional scalar fields and
(14) 
hence . The action (11) is secondorder (hence local) in spacetime derivatives and nonlocal 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,
(15)  
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:
(16) 
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 :
(17)  
Therefore, varying with respect to gives
(18)  
From this, we conclude that equation (18) reproduces (9) if
(19) 
where is a real constant, and
(20) 
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)
(21)  
(22) 
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 ,
(23) 
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.
To summarize the logic here, given the nonlocal system (8) one can always write down the system (11)–(13) localizing it. This localized system is not in onetoone correspondence with the nonlocal system but it always admits, among its solutions, the solutions of the nonlocal system. These solutions are defined by the boundary condition (19) together with the local condition (23). The subset of solutions of the localized system obeying these conditions are solutions to the original nonlocal 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 subset of solutions of the localized system that recover the equations of motion and solutions of the original nonlocal system. Recalling that the localized system (11) must be reducible to the nonlocal 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
(24) 
This condition, replacing (23), amounts to forbid source terms in the diffusion equation (16). Indeed, the infinitely many degrees of freedom of the original nonlocal 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 selfdetermined diffusing localized system equivalent to the nonlocal one is not only impossible,
For future use, we highlight three important features of the localization procedure which will apply, in their essence, also to the nonlocal gravity action (5).

By the diffusionequation method, one does not establish a onetoone correspondence between the localized system (11) and the nonlocal 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 nonlocal system. This correspondence on a slice is depicted in Fig. 1.

To get the correct result, it was crucial to make a careful choice of the arguments in the diffusionequation 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.

The localized system is secondorder 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 :
(25) In particular, these conditions are valid at , so that also the Cauchy problem of the nonlocal 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 nonlocal gravity gravity, with four initial conditions for the metric. In general, given a nonlocal action with exponential nonlocality for tensorial field representing physical degrees of freedom, the diffusionequation method relies on a secondorder 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 nonlocal 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
(26) 
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)
(27) 
replacing (12). We note that the integral in (13) is pleonastic for the Laplace–Beltrami term, since both and obey the diffusion equation:
(28)  
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 onshell 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
(29) 
so that
(30) 
and the total Lagrangian on Minkowski spacetime is
where
(32)  
The function is determined in appendix C by requiring the recovery of the nonlocal 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 nonlocal gravity under the same scheme. On the other hand, for the characterization (ghostlike 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 nonlocal 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 nonlocal 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 nonlocality with respect to the direction. Momenta acquire a rather obscure noninvertible 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
(33) 
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 nonvanishing equaltime Poisson brackets in terms of the spatial vectors are
(34)  
(35) 
while the Hamiltonian of the system is (dependence omitted again)
(36)  
where it is understood that . Since is shifted in , is nonlocal in due to the terms in the last line of (36). Nevertheless, the Hamiltonian is written solely in terms of phasespace variables and the phasespace fields are completely local in spacetime coordinates.
2.3 Initial conditions and degrees of freedom
The question about how many initial conditions we should specify for the nonlocal scalar system is related to another one: How many degrees of freedom are hidden in equation (9)? In higherderivative 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 higherderivative and nonlocal theories, and one cannot conclude that nonlocal theories need initial conditions; conversely, truncating a nonlocal 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 .
More contrived is the case with interactions. The reader unfamiliar with nonlocal 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 nonlocal operator of the kinetic term. Any other rewriting will not work, either. For instance, a nonlocal kinetic term can always be expressed as a convolution with a kernel [37]. Consider the scalarfield Lagrangian with generic form factor . In momentum space, calling the Fourier transform of ,
(38)  
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.
(39) 
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 nontrivial 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 nogo result with its way out: If the kernel can be found by solving some finiteorder 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 diffusionequation method is that it allows one to go beyond the free theory and count the extra number of initial conditions. Surprisingly, in the scalarfield 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 nonlocal 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 nonvanishing independent momenta and . Since the dynamics is secondorder 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 nonlocal real scalar field theory (8) in dimensions, the degree of freedom is no longer independent. Consequently, the nonlocal theory has one nonperturbative 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 nonlocal nonperturbative 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 nontrivial (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 secondorder derivative relation (23). From the point of view of the dimensional nonlocal 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 nonghost massive scalar mode. The potentially dangerous ghost mode in the dimensional system turns out to be nondynamical in dimensions and in the free theory.
In the interacting nonlocal 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
(42) 
If , then and there is no extra degree of freedom with respect to the case. For a cubic or higherorder polynomial, is not linearly equivalent to . Nonlinearities 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 nonlocal theories can be categorized into perturbative and nonperturbative. Perturbative solutions can have two meanings, either as the solutions obtained when truncating the nonlocal 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 nonlocality acts on interactions, the two meanings coincide. Nonperturbative 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 diffusionequation method permits to get access precisely to these solutions with generic nonperturbative 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 nonlocal 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 nonlocal system would take us back to the usual paradox with nonlocal 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 nonlocal solution is the outcome of the diffusion from to rather than of antidiffusion 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):
(43) 
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
(44) 
one has
(45) 
Since we know , we also know its Fourier transform and we can obtain the full nonlocal solution . Examples of solutions of the scalarfield equation of motion (9) using the diffusionequation 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,