Arbitrary powers of D’Alembertians and the Huygens’ principle

By means of some reasonable rules the operators that can represent arbitrary powers of the D’Alembertian and their corresponding Green’s functions are defined. It is found which powers lead to the validity of Huygens’ principle. The specially interesting case of powers that are half an odd integer in spaces of odd dimensionality, obey Huygens’ principle, and can be expressed as iterated D’Alembertians of the retarded potential are discussed. Arbitrary powers of the Laplacian operator as well as their corresponding Green’s functions are also discussed.


I. INTRODUCTION
The ordinary wave equation, as well as its relation to the Huygens' principle (HP), has received considerable attention, and has also been the object of some beautiful works. We would like to mention the classical book on the subject by Baker and Copson,' and the elegant analytic continuation method of Riesz.2 It is well known that HP is valid for the usual wave equation when the number n of space-time dimensions is even, but not when it is odd.
Nowadays some physicists are not happy living in a world of only four dimensions. Furthermore, second-order wave equations are no longer mandatory for the description of the evolution of physical particles or fields. For example, in gravitational theories, terms quadratic in the curvature tensor are sometimes introduced in the Lagrangian. Then, in some approximation the iterated D'Alembertian (lZ12) is found to operate on the field.3 There are also examples, in particular, for the bosonization in 2 + 1,4 in which the equation of motion involves the square root of the Alembertian ( q "2) .
The observations lead us to consider the general problem of constructing arbitrary powers of the Lorentz invariant differential operator q i, and then of finding, in any number of dimensions, their relation to a general HP that we are going to specify later.
In Sec. II, with the aid of some reasonable rules, we find the general form of Da, which, although dependent somewhat on the boundary conditions, it is almost completely specified.
In Sec. III we define the Green's functions Gca) and find some of their properties. In Sec. IV we introduce the Huygens' principle. In Sec. V we study the analytic distribution Q'+. In Sec. VI, the relations of Gcn) with HP are expressed in terms of the properties found in previous paragraphs. In Sec. VII we study, in particular, the interesting and less-known case of space-time with odd dimensionality (n=odd). Finally, in Sec. VIII, we introduce and discuss arbitrary powers of the Laplacian operator and their Green's functions.
In an appendix we show how to evaluate the Fourier transform of Riesz's classical retarded Green's function.

II. DEFINITION OF q a
We suppose that space-time has d+ 1 =n dimensions, d being the number of Euclidean space dimensions.

A (13)
This is the operator found by Riesz by a generalization of the Rieman-Liouville complex integral (cf. Ref. 2).
Note that the by taking half the sum of the retarded plus the advanced solutions ( 13), we obtain an operator whose Fourier transform is which for a =s= integer coincides with (2) but does not satisfy (6).
We will show below that for cr=s=positive integer ( 12) and ( 13) (15) so that in a convolution El* acts effectively as a differential operator when Q=S: The Green's function for the operator q a is the fundamental solution of the equation q %f=g, i.e., q "*G(")=S(x).
By taking the Fourier transform, we find We then have [cf. (7) and (8)] (12) and (13)], (20) If we take into account ( 10) and ( 1 1 >, we can write Gy) in terms of G (a) as Pi where s*=k-ye( do). (24) So that the GF' Green's functions propagate the positive (negative) frequencies with the retarded Gp) Green's function and the negative (positive) frequencies with the advanced one.
The Fourier transform of (26) can be found from (8) if care is taken with the poles of K", at a= -1 (see below). The result is

IV. THE HUYGENS' PRINCIPLE
The equation corresponding to the pseudodifferential operators introduced in the next paragraph are of the form q *f=g.
The solution f can be found by using the Green's function G '"', defined by ( 17) Note that (29) and (30) are dual to each other, as GCn) is the operator Elma, and (30) can be considered to be an equation for the determination of g, if f is given.
There are several statements that can be considered to represent the principle that Huygens introduced to describe the propagation of light waves (see Ref. 1 for a discussion of this point).
We are going to adopt the following statement.
The solution (30) of Eq. (29) is said to obey Huygens' principle (HP) if the Green's function GCn) has its support on the surface of the light cone.
This HP implies that the signals generated by the source propagate with one sharp velocity, that of the light.
Due to Eq. (23), we see that the properties of GCa) (a) k can be deduced from those of GR . In fact, Gy' propagates the positive frequencies of the source by means of Gp) and the neiative frequencies by means of Gy'. In this sense, we can say that G$$ obeys HP if Gp' and Gp' do so. It is then enough to examine GR (cx) (Gy' is similar) to find out when HP is satisfied.
From (26) we see immediately that GR (a) obey HP in n = 4, as S(Q) has its support on the light cone Q=O. For n =odd number, it follows from (22) for a = 1 that which is well defined and zero outside the light cone [cf.
(3)], but it is different from zero everywhere inside the light cone, and so, as is well known, the solutions of the ordinary wave equation obey HP for n = 4 (n = even), but do not obey HP for n = odd.
In the general case, we have to examine the singularities of the functions on which Gp) depends [cf. (22) We now observe that Q"; has the types of singularities that are present in the product I( 1 +A)r(A+nn/2).
In fact, when n=odd, this product has simple poles at ;1 = -k (k = positive integer), and at A = -n/2-k (k = positive integer or zero), just as in (a). Further, when n =even (as in (b), the product presents simple poles for A = -k (0 < k < n/2), and double poles for A= -k, if k>n/2. For these reasons, if we divide e"; by that product, we obtain where, due to the presence of the factor 6( -t), the contribution of the retarded cone is only a half the quoted value in (37) and (39).
It is now easy to see when the Green's function GR (n) obeys HP. The only cases for which Q'(A) has its support on the light cone are those for which (36) and (38) when n is even the values of k are restricted to be less than n/2 (k < n/2), but for n =odd, k is an unrestricted positive integer. For n=4 we have the usual retarded potential (26). Further, this kind of potential holds in any number of dimensions for k= 1: Equations (44) and (45) are true for any n (even or odd).
The usual wave equation Of =g is the only one whose solution obeys HP in any number of dimensions (n > 2

VII. THE CASE n=ODD
The results found in Sec. VI, Eq. (43), do not seem to be well known for n =odd, and they are interesting enough to deserve explicit mention, at least for low values of n (also see Ref. 8). For any odd n there are an infinite number of convolution operators whose Green's function obeys HP. They are rJ""-1 n/2-2 R ,o, ,w.,@~-~ ,... .
The Green's functions corresponding to the operators (46) can also be expressed in terms of the retarded potential (44) The Green's function corresponding to ha is so that, from (63) and (66), we obtain A"*f=A%(x)*f=A 'f (s=O,1,2,...).
In expression (65), the poles of RaMd" are compensated or neutralized by the poles of T(a).
The residues of GCa) at these poles are proportional to R' (a polynomial in XT), and they are solutions of the homogeneous equation Ad/2+s*Rs,0.
This is trivial for d=even, but it is also true for d=odd, as can be proved by computing RwSmd*R' (see Ref. 5,p. 361).
For this reason we can drop the poles of GCa) and define, for a = d/2 +s, where we have dropped terms proportional to RS (residues). In particular, for d=2, and s=O we have the well-known logarithmic potential: As a matter of fact, the logarithmic potential is the Green's function corresponding to the operator AdI in any number of dimensions: Ad'2*ln R -6 (x ) -.
We may ask, in general, which is the operator that has a potential of the form Rp in a d-dimensional Euclidean space. The answer is given by (63) and (65)  The logarithmic potential corresponds to /3=0. For the Newtonian potential t-1=R-1'2, fl= -f and (73) and (74) For odd-dimensional spaces, (75) is just the Laplacian iterated (d-1)/2 times. In d= 3 it is the usual Laplacian A. In d=5 it is A2=AA, etc.
For even-dimensional spaces (70) gives an exponent that is half an odd integer.