Infinite derivatives vs integral operators. The Moeller-Zwiebach puzzle

We study the relationship between integral and infinite-derivative operators. In particular, we examine the operator p12∂t2 that appears in the theory of p-adic string fields, as well as the Moyal product that arises in non-commutative theories. We also attempt to clarify the apparent paradox presented by Moeller and Zwiebach, which highlights the discrepancy between them.


Introduction: A short chronological review
Theoretical physics has resorted to nonlocal Lagrangians and infinite-order Lagrangians in several instances.As far as we know, a chronological list should include Fokker-Wheeler-Feynman electrodynamics [1,2], an attempt to set up electrodynamics without an intermediate field.The Lagrangian has two contributions, namely a part for free point charges L 0 plus an interaction term L int resulting from the addition of the symmetric Liénard-Wiechert potentials -halfadvanced and half-retarded-of each pair of charges.
Then, in the fifties, there were several attempts, such as that of Pais and Uhlenbeck [3], who proposed higher-order field equations -even equations of infinite order-"to eliminate the divergent features of the present theory", or the proposals by Yukawa [4] or Kristensen and Møller [5] to avoid divergent perturbation expansions in nucleon-meson interaction.
Later, in the seventies, we find Marnelius' significant contribution [6] to formulating a variational principle and conserved currents for nonlocal field theories.He starts with an action made of a local free part plus a nonlocal interaction part S int (ϕ) = dx 1 . . .dx N K AB...D (x 1 , . . .x N )ϕ A (x 1 ) . . .ϕ D (x N ) , i.e., already shaped in the "integral operator" (IO) fashion, which he turns into an infinite derivative (ID) form by means of the formal Taylor expansion where we have used the multi-index notation, namely, x ∈ R n , (α 1 , . . ., α n ) ∈ N n 0 , and |α| := α 1 + . . .+ α n .This procedure transforms the action into which is (pseudo)local in that it contains field derivatives of any order.
Once the action is written in this form, the variational principle and Noether's theorem can be applied in the standard way for a Lagrangian of order n -but replacing n with ∞to derive: (i) the field equations whose interaction term depends on the field derivatives of any order, and (ii) the conserved current associated to any continuous symmetry of the Lagrangian (again the interaction contribution contains a series with all field derivatives).Finally, Marnelius manages to sum those ID series and write the interaction terms, both in the field equations and conserved currents, in the "integral operator" shape.
The method relies on the formal Taylor expansion (2), which is abusive and leads to a glaring inconsistency.Indeed, not all functions on which the action is defined are real analytic functions with an infinite radius of convergence, not even if we limit ourselves to the solutions of the field equations.The inconsistency is that, whereas in finite order (n) Lagrangians, the variational principle reads δS = 0 for B meaning the boundary.When we replace n with ∞, it becomes δS = 0 for δ∂ α ϕ(B) = 0 for all α = (α 1 , α 2 , . ..) , which, due to the Taylor expansion assumption, implies that δS = 0 for δϕ(x) = 0 for all x , which is a triviality.One may reply to this objection by saying that Marnelius' method is merely an intermediate heuristic trick which, despite its inconsistencies, allows to derive useful field equations and conserved currents, whose local conservation on-shell can be checked a posteriori.In our view, this fact indicates that there must be a consistent reasoning leading to the same results.
In the eighties, nonlocality arises again as ". . .a fundamental property of string field theory and not just as an artifact . . ." [7].They deal with it in the "higher derivative representation" (in their terminology), consisting in truncating the infinite derivative series to work it as a higher (finite) derivative theory and then replacing n with infinity.
More recently, nonlocality -either as IO or ID-appears in modified gravity theories.Inspired by string theory's ultraviolet (UV) finiteness, they are being proposed to solve both cosmological and black hole singularities (see, for instance, [8][9][10][11] and references therein).Likewise, they are also being used to explain the cosmic expansion of the Universe without relying on a contribution from dark energy [12], inflationary models [13], and gravitational waves [14].
The central question is: are both approaches, ID and IO, equivalent?In what follows, we will examine the operator p 1 2 ∂ 2 t that appears in the p-adic particle, i.e the p-adic string field independent of spatial coordinates [15,16], as well as the Moyal product, which arises in noncommutative theories [17,18].
where p is a prime integer, and Rescaling the variable τ = t/ √ r , both operators keep the same form replacing the parameter r with 1. Hereon, we will analyze the case r = 1 because it presents the same features as the general case.Furthermore, it is worth mentioning that the connection between operators (4) and ( 5) is derived in Section 2.2 below, and it relies on the use of Fourier transforms [7,19].
In mathematical literature, representation (5) is known as the Weierstrass transform, which was extensively studied around the 1960s.For instance, the necessary and sufficient conditions for this representation to be well-defined were discussed in ref. [20].Additionally, the inverse of this transformation, denoted as e ∂ 2 t , was examined in ref. [21], shedding light on the ineffectiveness of replacing the inverse with a Taylor series, given that many functions diverge under such an approach1 .It should also be mentioned that Hermite polynomials have been used to study the interpretability and summability of the inverse Weierstrass transform when interpreted as infinite series [23].For more recent literature, it is worth mentioning alternative derivations [24,25] that are based on the heat kernel.

The puzzle
Moeller and Zwiebach [26] pointed out that K1 ψ and K2 ψ are actually different for a smooth, compact support function such that because This said, the following questions arise: (a) Which operator, K1 ψ or K2 ψ, is "the good one"?(b) do they act on the same domain of functions?, and (c) is there any subdomain in which both operators coincide?
We must always bear in mind that the operators K1 ψ and K2 ψ usually occur in a Lagrangian (and an action integral), and therefore, for Ka to be a "valid" operator, Ka ψ(t) must be welldefined for any t ∈ R , in some sense.
As for the domains of definition, it is evident that K1 acts on smooth functions ψ ∈ C ∞ (R) , and moreover, the series defining K1 ψ must be "summable" in some sense, namely pointwise, uniformly, or the like2 .

How are the operators K1 and K2 connected?
As pointed out in ref. [7], provided that ψ(t) has a Fourier transform both operators are connected through the following steps: (1 st ) Substituting and differentiating under the integral sign, we have that (2 nd ) commuting the series and the integral signs, we can write and, since ) by the convolution theorem, we arrive at The soundness of the above chain of reasoning relies on the validity of the three highlighted steps: (1 st ) differentiation under the integral sign, (2 nd ) the commutability of the series and the integral, and (3 rd ) the convolution theorem.Some mathematical theorems provide sufficient conditions for each one to be allowed.While the conditions for steps 1 st and 3 rd to be valid are reasonably mild, the validity of step 2 nd might rely either on the termwise integration theorem for uniformly convergent series [27] or on Lebesgue's dominated convergence theorem [28], namely

Theorem [Lebesgue]
Let {f n } be a sequence of Lebesgue integrable functions in an interval I .Assume that: (i) the sequence converges to f pointwise almost everywhere (a.e.w.) in I , and (ii) it exists a non-negative Lebesgue integrable function g such that a.e.w. in I , ∀n .
Then f is Lebesgue integrable and (In our case, {f n (z)} is the sequence of partial sums of a series.) The present work aims to address the apparent paradox raised in ref. [26] and to understand why the action of the operator e r∂ 2 t on a bounded support function yields a different result, depending on whether the ID form K1 or the IO form K2 is used.
We will prove that K1 η is not globally defined for the particular finite smooth function η(t) considered in ref. [26], i.e., the series diverges for a wide range of values of t .
Carlson et al. [19] did already prove that, when acting on a finite smooth function, a certain class of operators, which are defined by an infinite series containing derivatives of any order, lead to divergent series, and therefore they are not globally defined.The operator e r∂ 2 t belongs to this class.However, instead of invoking that result in ref. [19], which is based on a rather powerful and sophisticated mathematical formalism, we think that our contribution -however clumsy it may be-might be helpful in that, on the one hand, it is more accessible to a wider community and, on the other, we emphasize pointwise convergence that, in our view, is more suitable than L 1 convergence for a classical physics context.

The domains of the operators K1 and K2
There is a common domain where both operators coincide, but apparently, the domain of K2 is wider than that of K1 , as illustrated by the following examples.

Entire functions
K2 and K1 coincide on the space of entire functions, i.e., real functions that result from restricting entire complex functions to R. Indeed, let Applying the operator K2 , we have and since uniformly on any interval [−K, K] , ∀K ∈ R + , the series may be termwise integrated [28].The individual integrals are easily performed and by substituting them in (10), we finally obtain

An apparently "harmless" function
Let us consider the function Although it is C ∞ (R), bounded, and also L p (R) , its analytic extension does not yield an entire function because it presents two single poles at t = ±i .It is a familiar function because, apart from normalization, it is Cauchy probability density and its Fourier transform is It belongs to the domain of K2 ; indeed, is well-defined and finite for any t ∈ R .However, despite ψ being so little pathological, it presents serious difficulties for the operator K1 .We must first obtain all derivatives of even order and, to make it simpler, we include that whence it quickly follows that ψ (2n) 2n+1) .Thus, from (14), we have that which, using the new variable can be written as and, therefore, We will prove that the series is divergent for all t ∈ R .Our proof relies on a corollary of Cauchy's condition [29] that provides a necessary condition for pointwise convergence, namely lim n→∞ |a n (t)| = 0 , and therefore the sequence A := {|a n (t)|} must be bounded for the series to converge.
It is obvious that for t = 0 and α = 0, the sequence {|a n (0)|} is unbounded.For other values of t, we restrict to positive ones and distinguish two cases: If α/π is rational, then α/π is an irreducible fraction k/l < 1/2 and where D n and s n < l denote the integer quotient and remainder, respectively.Therefore, and, as n runs over N, it repeats periodically, yielding a finite set of positive numbers (some may be null but, as a rule, not all of them vanish), namely Now, pick a p ̸ = 0 among these and consider the subsequence F p ⊂ A corresponding to those m such that |cos(2m + 1)α| = p , namely This sequence is not bounded as it easily follows from Stirling's formula Therefore, A is unbounded, and the series ( 17) is divergent for those t = tg α , where α is a rational multiple of π .
If α/π is irrational, then e i(2n+1)α does not repeat periodically as a function of n, and the sequence E := e i(2n+1)α , n ∈ N ⊂ C is infinite and bounded.
Then, the Bolzano-Weierstrass theorem [30] implies that E has at least an accumulation point different from ±i .Indeed, assume that i was the only accumulation point, then there would exist a subsequence, E 1 := e i(2n j +1)α , j ∈ N , such that lim j→∞ e i(2n j +1)α = i .
Consider now the subsequence resulting from choosing the next of each term in E 1 , that is, E 2 := e i(2[n j +1]+1)α , j ∈ N .It is obvious that lim j→∞ e i(2[n j +1]+1)α = e i 2α lim j→∞ e i(2n j +1)α i e i 2α , and, as 2α < π , we have proved that ie i 2α ̸ = ±i is an accumulation point of E as well.
For the n j + 1 term in the subsequence E 2 , we have that If we then consider the subsequence F 1 ⊂ A corresponding to those m such that |cos(2m + 1)α| > p/2 , we quickly see that F 1 is unbounded.Therefore, the series ( 17) is also divergent for those t = tg α , such that α is an irrational multiple of π .□

Can Lebesgue's dominated convergence theorem be applied to convert K1 into K2 for this function?
A consequence of the results so far is that Lebesgue's theorem does not apply in the case of the function (12).Indeed, whereas K2 ψ is well defined, K1 ψ runs into an everywhere divergent series.Let us examine in detail how the hypothesis of the theorem does fail.For this particular function, the conversion of K1 ψ into K2 ψ requires swapping the integral and the series in (8), which is easily written as For Lebesgue's theorem to apply, we need that a non-negative dominant function g ∈ L([0, ∞[) exists such that and therefore Thus, for a dominant function g to exist, the quantities ∞ 0 f n must be bounded.Let us now introduce that, using the variable ( 14), becomes Now, the sequence {I n (α) , n ∈ N} is unbounded.Were it bounded, so would be the sequence {I n (α) that, as proved before, is unbounded and, therefore, the dominant function g does not exist.
In summarizing, a simple smooth function as (12), which is square summable but not an entire function, belongs to the domain of K2 but not to that of K1 .Therefore, the 2 nd step in the conversion of K1 into K2 must fail, and we have shown why the dominated convergence theorem does not apply in this case.

The anomalous function of Moeller and Zwiebach
We will now study the action of the ID operator K1 on the particular finite smooth function considered in ref. [26].We will see that it is not globally defined, i.e., it fails for a wide range of values of the variable.
We consider the finite smooth function defined by [31] with where ω is a smooth finite function, i. e. its support [−1, 1] is bounded.Notice that the parity of its derivatives is ω Since ω(t) is even, so is η(t), and it suffices to study η(t) for t ≥ 0 .Thus, including that ω(τ ) vanishes for |τ | > 1 and that t + c ≥ c > 1 , we have that whence it follows that where the facts that ω(τ ) is even and It is evident that η(t) is a particular case of the functions considered in ref. [26].The action of the operator K2 yields which is well-defined for any finite smooth ω .
To study the action of K2 , we need the derivatives of η for n ≥ 1 , which are better obtained directly from (18), and we have For t positive, c + t > 1 and ω(c + t) = 0 .Therefore, or where we have included that c + t > 1, and therefore ω (n−1) (c + t) = 0 .Some particular values are Then, using (24) for even values of n, we obtain that and, introducing the variable τ = c − t , we have It is obvious that Thus, we need the derivatives of ω(τ ) at all odd orders.For practical reasons, we introduce the variable and have that We will prove that the derivatives of ω(τ ) are: odd order where P n and Q n are polynomials in x.Indeed, from ( 27), we have that ω ′ (τ ) = −2 τ x 2 ω(τ ) , whence the proposition holds for n = 0 with Deriving now ω (2n) (τ ), we have that and, therefore, Similarly, by deriving ω (2n−1) (τ ), we have that and, using that 2τ 2 x 2 = 2x(x − 1) , we arrive at Combining equations ( 31) and ( 32), we finally obtain which provides an iterative formula to derive Q n (x) from Q 1 (x) .The relations (28-29) are followed by the induction principle.Back to equation (26), including ( 27) and (28)(29), we arrive at therefore, K1 η(c−τ ) is well-defined if, and only if, the series on the right-hand side converges.
A necessary condition for convergence is that lim To prove that the series diverges for all |τ | < 1 , i. e. x > 1 , we will show that the general term in the series, q n (x , is unbounded, but, as the expressions are not so simple as in Section 2.3.2,we have to resort to a numerical simulation with Mathematica3 .Figure 1 is a plot of the general term q n (x) for several values of x as a function of n and it shows that the larger is n, the more q n (x) grows, which clearly indicates that the general term is unbounded.We have tested it using different values of x = {1.1, 2, 10, 50, 100, 500} and n = 1, 2, . . ., 350 and found that even for n = 350 and x = 500, the value of q n (x) exceeds 10 2800 .We have also tried even larger values of n and x, and the result remains the same.
Figure 2 is a logarithmic plot of the quotient q n+1 (x) q n (x) .Notice that the result is strictly positive, indicating that the term q n+1 (x) is larger than q n (x).We have used the same values of x and n as in the previous figure and also tested larger values of n, leading to the same conclusion.Therefore, these numerical simulations confirm that the series is not convergent since the general term

The Moyal star product
In the non-commutative U (1) theory [17,18], we can find another intriguing example featuring operators with infinite derivatives.This theory revolves around the Moyal star product, which plays a central role.Given any two (smooth) functions, f and g, the star product is defined as where θ a j b j is a skewsymmetric constant matrix that we assume to be non-degenerate.Consider now that f and g are entire (real) functions on R 4 , then the ID operator (35) is equivalent to the following IO form where |θ| = det (θ ab ), v θu = v b θbc u c , and θ ab θbc = δ a c .Indeed, we can write f and g as formal Taylor series (2), namely, and similarly for g(x − v).Plugging them into (36), we arrive at  where M a 1 ...an b 1 ...bm is M a 1 ...an b 1 ...bm := R 8 du dv u a 1 . . .u an v b 1 . . .v bm e 2iv θu , and we have permuted the integrals by the infinite series because these Taylor series converge uniformly on any interval [−K, K], K ∈ R + .Consequently, the series can be termwise integrated [28].The result of these integrals is Therefore, substituting it in (38), we get . □ From now on, we shall restrict to 1+1 dimensions for the sake of simplicity, where Consider the functions f (x a ) = f (t) and g(x a ) = g(x) , where x a = (x, t) , then the Moyal product (35) reduces to Now, we will prove that, for two particular "harmless" smooth functions like the series diverges.Indeed, as in Section 2.3.2,write f ] and similarly for g(x) to obtain .
Substituting these derivatives into (39), we have that f * g(x, t) = ∞ n=0 a n (x, t) , where Now, a similar analysis as in Section 2.3.2 leads to conclude that {|a n (x, t)| , n ∈ N} is unbounded, and the series diverges for almost all values of t and x .

Conclusions
Two approaches to the operators arise in nonlocal mechanics and nonlocal field theories, namely, one based on integral operators (IO) and another based on infinite derivatives (ID) that usually involve an infinite series.We have started by asking whether both approaches are equivalent.It is a rhetorical question because, as Moeller and Zwiebach [26] have demonstrated for the operator p 1 2 ∂ 2 t , each approach gives different results when applied to some compact support smooth function, which answers the question in the negative.
It is well-known [7] that both approaches can be connected through some manipulation that includes Fourier transforms.The crucial weakness of this procedure consists of the exchange of an integral and a series, which requires that some conditions are fulfilled to be legitimate.
Another point to consider is that each form of the operator, ID or IO, may act on a different functional space.We have analyzed the domains of both approaches and found that there is a common domain, but the domain of IO is broader than the one for ID.The common domain includes at least all entire real functions, i.e., real functions that result from restricting entire complex functions to R. In this case, the series and the integral may be exchanged on the basis of uniform convergence of the series on compact sets.We have concentrated on the particular operator p 1 2 ∂ 2 t and the Moyal product but the proof for a wider class of operators can be found in [19].
We have then examined both forms of the operator p 1 2 ∂ 2 t and the Moyal product acting on a function that a priori seems "harmless" since it is smooth, bounded, and L p summable.We have seen that, although the result of applying the IO form of the operators is well-defined, the infinite series arising when applying the ID form is divergent.
Furthermore, we have considered both forms of the operator p 1 2 ∂ 2 t acting on a smooth compact support function, an example of the kind of functions proposed by Moeller and Zwiebach.Again, the function clearly belongs to the operator's domain in IO form, but, as we have illustrated with a numerical simulation, it does not belong in the domain of the operator's ID form because the series is not convergent.It is worth mentioning here the work of Carlson et al. [19] where, by using a powerful and more sophisticated mathematical formalism, it is proved that any smooth compact support function acted by the ID form of p 1 2 ∂ 2 t yields a divergent series.All this leads us to conjecture that the domain of the IO form of the operator is wider than the domain of the ID form.For this reason, it seems more advisable to use integral operators whenever possible.
One of the reasons for using the ID form is its practicality for setting up the variational principle, Noether's theorem and a Hamiltonian formalism by merely mimicking what is done in standard local mechanics and field theory.This was the way chosen by Marnelius [6], who should first convert the action integral from IO form into the ID form before proceeding.However, as previously stated, the value of the proofs and derivations included in the procedure is only heuristic unless the convergence of all series is proven.For this reason, methods [32][33][34] have been recently developed to work with the same subjects, such as the variational principle, Noether's theorem, and Hamiltonian formalism, directly in the IO form.

Figure 1 :
Figure 1: The general term q n (x) increases with n.