Generalized Symmetries of Massless Free Fields on Minkowski Space

A complete and explicit classification of generalized, or local, symmetries of massless free fields of spin $s \geq 1/2$ is carried out. Up to equivalence, these are found to consists of the conformal symmetries and their duals, new chiral symmetries of order $2s$, and their higher-order extensions obtained by Lie differentiation with respect to conformal Killing vectors. In particular, the results yield a complete classification of generalized symmetries of the Dirac-Weyl neutrino equation, Maxwell's equations, and the linearized gravity equations.


Introduction
Recent years have seen a growing interest in the study of the symmetry structure of the main field equations originating in mathematical physics. Generalized, or local, symmetries, which arise as vectors that are tangent to the solution jet space and preserve the contact ideal, are important for several reasons. Besides their original application to the construction of conservation laws, they play a central role in various methods, in particular in the classical symmetry reduction [16], Vessiot's method of group foliation [12], and separation of variables, for finding exact solutions to systems of partial differential equations. Generalized symmetries also arise in the study of infinite dimensional Hamiltonian systems [16] and are, moreover, connected with Bäcklund transformations and integrability [11]. In fact, the existence of an infinite number of independent generalized symmetries has been proposed as a test for complete integrability of a system of differential equations [13].
For the important examples of the Einstein gravitational field equations and the Yang-Mills field equations with semi-simple structure group, classifications of their symmetry structures [4,19] have shown that, besides the obvious gauge symmetries, these equations essentially admit no generalized symmetries. In contrast, the linear graviton equations and the linear Abelian Yang-Mills equations possess a rich structure of generalized symmetries, which to-date has yet to be fully determined.
In this paper we present a complete, explicit classification of the generalized symmetries for the massless field equations of any spin s = 1 2 , 1, 3 2 , . . . on Minkowski spacetime, formulated in terms of spinor fields. These equations comprise as special cases Maxwell's equations, i.e., U (1) Yang-Mills equations for s = 1, and graviton equations, i.e., linearized Einstein equations for s = 2. Other field equations of physical interest which are included are the Dirac-Weyl, or massless neutrino equation, and the gravitino equation, corresponding to the spin values s = 1 2 and s = 3 2 , respectively. There are two main results of the classification. First, we obtain spin s generalizations of constant coefficient linear second order symmetries found some years ago by Fushchich and Nikitin for Maxwell's equations [8] and subsequently generalized by Pohjanpelto [18]. The new second order symmetries for Maxwell's equations are especially interesting because, under duality rotations of the electromagnetic spinor, they possess odd parity as opposed to the even parity of the well-known conformal point symmetries. Consequently, we will refer to the new spin s generalizations as chiral symmetries. Furthermore, we show that the spacetime symmetries and chiral symmetries, together with scalings and duality rotations, generate the complete enveloping algebra of all generalized symmetries for the massless spin s field equations. In particular, these equations admit no other generalized symmetries apart from the elementary ones arising from the linearity of the field equations. It is also worth noting that, due to the conformal invariance of the massless field equations, our results provide as a by-product a complete classification of generalized symmetries of spin s fields on any locally conformally flat spacetime, extending earlier results for the electromagnetic field obtained by Kalnins et al. [10].
This classification is a counterpart to our results classifying all local conservation laws for the massless spin s field equations [1,2,3]. We emphasize, however, that there is no immediate Noether correspondence between conservation laws and symmetries in our situation, as the formulation of the massless spin s field equations in terms of spinor fields does not admit a local Lagrangian.
Our paper is organized as follows. First, in Section 2 we cover some background material on symmetries of differential equations and on spinorial formalism, including a factorization property of Killing spinors on Minkowski space that is pivotal in the symmetry analysis carried out in this paper. Then in Section 3 we state and prove our classification theorem for generalized symmetries of arbitrary order for the massless field equations. These, in particular, include novel chiral symmetries of order 2s for spin s = 1 2 , 1, 3 2 , . . . fields. As applications, in Section 4 we transcribe our main result in the spin s = 1 case into tensorial form to derive a complete classification of generalized symmetries for the vacuum Maxwell's equations, and, finally, in Section 5, we employ the methods of Section 3 to carry out a full symmetry analysis of the Weyl system, or the massless Dirac equation, on Minkowski space. Our results for the Weyl system complement those found in [5,7,15], and, in particular, provide a classification of symmetries of arbitrarily high order for the massless neutrino equations.
where φ A 1 A 2 ···A 2s ,j 1 j 2 ···jp stands for the pth order derivative variables. As is customary, we write where σ j BB ′ , 0 ≤ j ≤ 3, are, up to a constant factor, the identity matrix and the Pauli spin matrices. We also write where the bar stands for complex conjugation. Here and in the sequel we employ the Einstein summation convention in both the space-time and spinorial indices, and we lower and raise spinorial indices using the spinor metric ǫ AB and its inverse ǫ AB ; see [17] for further details.
In order to streamline our notation, we will employ boldface capital letters to designate spinorial multi-indices. Thus, for example, we will write and we will combine multi-indices by the rule B p C q = (B 1 B 2 · · · B p C 1 C 2 · · · C q ). We let Here, in accordance with the standard spinorial notation, we have written ǫ C A for the Kronecker delta and we use round brackets to indicate symmetrization in the enclosed indices.
A generalized vector field X on E s in spinor form is a vector field ) are spinor valued functions in x j and the derivative variables φ B ′ q A 2s ,Bq up to some finite order p. An evolutionary vector field Y , in turn, is a generalized vector field of the form where Q A 2s is called the characteristic of Y . Let stand for the spinor representative of the standard total derivative operator, which, as is easily verified, satisfies the commutation formula The infinite prolongation pr X of X in (2.1) to a vector field on J ∞ (E s ) is given by where R A 2s is the characteristic of the evolutionary form The massless free field equation of spin s and its differential consequences determine the infinitely prolonged solution manifold R ∞ (E s ) ⊂ J ∞ (E s ) of the equations. According to [17], the symmetrized derivative variables known as Penrose's exact sets of fields, together with the independent variables x i provide coordinates for R ∞ (E s ). Moreover, as is easily verified, the unsymmetrized and symmetrized variables φ Bp agree on R ∞ (E s ). A generalized, or local, symmetry of massless free fields is a generalized vector field X satisfying pr Xφ Note that any generalized vector field of the form with the prolongation pr T P = P CC ′ D CC ′ automatically satisfies the determining equations (2.4) for a symmetry. Hence we will call a symmetry trivial if its prolongation agrees with a total vector field pr T P = P CC ′ D CC ′ on R ∞ (E s ) and we call two symmetries equivalent if their difference is a trivial symmetry. See, e.g., [6,16] for further details and background material on generalized symmetries. In this paper we explicitly classify all equivalence classes of generalized symmetries of massless free fields of spin s = 1 2 , 1, 3 2 , . . . on Minkowski space. By the above, in our classification we only need to consider symmetries in evolutionary form, and for such a vector field the determining equations (2.4) for the characteristic Q A 2s become Moreover, after replacing X by an equivalent symmetry, we can always assume that the components Q A 2s are functions of only the independent variables x i and the symmetrized derivative for some function k = k(x i ), where η ij stands for the Minkowski metric. As can be verified by a direct computation, a conformal Killing vector field ξ gives rise to the symmetry of massless free fields of spin s, with the characteristic which agrees with the conformally weighted Lie derivative of the spinor field φ A 2s ; see [2,17].
In the course of the present symmetry classification we will repeatedly use the fact that, on account of the linearity of the massless free field equations, the componentwise derivative of an evolutionary symmetry Y with respect to the prolongation of the vector field Z[ξ] is again a symmetry of the equations; see, e.g., [18].
In spinor form the conformal Killing vector equation (2.6) becomes An obvious generalization of equations (2.8) to spinor fields κ and symmetric spinor fields κ satisfying these equations are called Killing spinors of type (k, l). Thus, in particular, a type (1, 1) Killing spinor κ A ′ A corresponds to a complex conformal Killing vector. The following Lemma, which is a special case of the well-known factorization property of Killing spinors on Minkowski space, is pivotal in our classification of symmetries of massless free fields. For more details, see [17].

Main results
Let ξ, ζ 1 , . . . , ζ p be real conformal Killing vectors and let π A ′ 4s be a type (0, 4s) Killing spinor. Let Z[ξ] be the symmetry associated with ξ as in (2.7) and define Z[iξ] by be an evolutionary vector field with components where the coefficients c 2s,p are given by for the repeated componentwise derivatives of the vector fields (2.7), (3.1), (3.2) with respect to conformal symmetries.
Proposition 1. Let ξ, ζ 1 ,. . . ,ζ p be conformal Killing vectors and let π A 4s be a Killing spinor of type (0, 4s). Then the evolutionary vector fields are symmetries of the massless free field equations of spin s of order p+1 and q +2s, respectively. Moreover, when restricted to the solution manifold R ∞ (E s ), the leading order terms in the

respectively.
Proof . We only need to show that W[π] in (3.2), (3.3) satisfies the symmetry equations (2.4). First note that due to the Killing spinor equations (2.9) we have that and, consequently, is a symmetry of massless free field equations provided that We compute where we used (3.9) and (3.10). On the other hand, Now it follows from (3.12), (3.13) that Consequently, by virtue of (3.4), equation (3.11) holds and hence W[π] is a symmetry of the massless free field equations.
The massless free field equations of spin s also admit the obvious scaling symmetry S, its dual symmetry S, and the elementary symmetries E[ϕ] given by and where Theorem 1. Let Q be a generalized symmetry of the massless free field equations of spin s = 1 2 , 1, 3 2 , . . . . If the evolutionary form of Q is of order r, then Q is equivalent to a symmetry Q of order at most r which can be written as In particular, the dimension d r of the real vector space of equivalence classes of spinorial symmetries of order at most r spanned by the symmetries (3.17) is and d r = (r + 1) 2 (r + 2) 2 (r + 3) 2 /18 The above result was originally announced without proof in [3].
Proof . Without loss of generality, we can assume that the components Q A 2s of Q are functions of the independent variables x i and the symmetrized derivative variables φ (3.18) It follows from the determining equations for spinorial symmetries that on R ∞ (E s ). By virtue of the commutation formulas (2.2), the above equations with p = r + 1 show that identically on J ∞ (E s ). Equations (3.20), in turn, combined with (3.18), imply that Now substitute expressions (3.21) into (3.22) to deduce that But it is easy to see that the above equations force S Cr C r−2s to be independent of the symmetrized derivative variables φ B ′ p B p+2s , p ≥ 0, and, consequently, they must satisfy the Killing spinor equations Thus where U A 2s only involves the derivative variables φ so that on R ∞ (E s ), the highest order terms in V r agree with those in Q, and, consequently, the symmetry Q is equivalent to a linear combination of the basic symmetries (3.5)-(3.7) with p = r − 1, q = r − 2s and an evolutionary symmetry of order r − 1. Now proceed inductively in the order of the symmetry. In the last step the symmetry Q is equivalent to a linear combination of the symmetries (3.5)-(3.7) with p ≤ r − 1, q ≤ r − 2s, and an evolutionary symmetry V o of order 0. But it is straightforward to solve the determining equations (2.5) for V o to see that where a ∈ C is a constant and ϕ A 2s = ϕ A 2s (x i ) is a solution of the massless free field equations of spin s. Thus (3.16) holds.
Finally, the above arguments show that the vector space of equivalence classes of symmetries of order r ≥ 1 modulo symmetries of order r − 1 is isomorphic with the real vector space of Killing spinors of type (r, r), if r < 2s and with the direct sum of the real vector spaces of Killing spinors of type (r, r) and (r − 2s, r + 2s) if r ≥ 2s. The dimension of the space spanned by the spinorial symmetries (3.17) now can be computed by adding up the dimensions given in Lemma 1. This concludes the proof of the Theorem.

Symmetries of Maxwell's equations
In this section we transcribe the spinorial symmetries of Theorem 1 for s = 1 to tensorial form in order to classify generalized symmetries of Maxwell's equations on Minkowski space. Here F ij = −F ji are the components of the electromagnetic field tensor F and * stands for the Hodge dual.
We write Λ 2 (T * M) → M for the associated bundle with coordinates F ij , i < j. Then For notational convenience, we write F ij,k 1 ···kp = −F ji,k 1 ···kp for i ≥ j.
In spinor form the electromagnetic field tensor F becomes on solutions of (4.1). If one defines Q AB by of Maxwell's equations, where F is any solution of (4.1) with components F ij = F ij (x k ). Conformal symmetries (2.7) and their duals (3.1) in turn give rise to the symmetries where ξ is a conformal Killing vector on M and where square brackets indicate skew-symmetrization in the enclosed indices. In order to transcribe the second order chiral symmetries W[π] introduced in (3.2) to symmetries of Maxwell's equations in physical form, we first introduce the following polynomial tensors on M. Let x m x n x n , (4.8) and where we raise indices using the inverse η ij of the Minkowski metric. Note that a h ijkl , h = 0, 4, possesses the symmetries of Weyl's conformal curvature tensor. Hence their spinor representatives can be written in the form where α h I ′ J ′ K ′ L ′ is a constant, symmetric spinor. See, for example, [17]. Moreover, it is easy to verify that by (4.11), (4.12), the spinor representatives of a 2 ij and a h ijk , h = 1, 3, are given by where p h ijkl are the polynomials in (4.5)-(4.9), and write where Moreover, on the solution manifold R ∞ (E 1 ), the spinor representatives of the evolutionary vector fields W[F ; p h ] in (4.13) reduce to Proof . The proofs of the equations in the Lemma are based on straightforward albeit lengthy computations. We will therefore, as an example, derive equation (4.14) for h = 2 and equation (4. 16), and omit the proofs of the remaining equations in the Lemma. One can check by a direct computation that the polynomials p h ijkl possess the symmetries of Weyl's conformal curvature tensor, that is, Consequently, we can write the spinor representatives of p h ijkl in the form (4.14); see [17]. Then, in particular, Thus we can find π h I ′ J ′ K ′ L ′ by substituting the spinor representative of the expression on the right-hand side of equations (4.5)-(4.9) into (4.18) and, in each of the resulting terms, symmetrizing over primed indices. The computations can be further simplified by using the observation that given a tensor f ijkl with the spinor representative f II ′ JJ ′ KK ′ LL ′ , then the spinor Now insert the spinor representative of the right-hand side of the equation (4.7) into (4.18). Then by virtue of (4.19), the first term yields the expression The spinor forms of each of the remaining terms on the right-hand side of (4.7) contain an instance of the conjugate of the epsilon tensor, and, consequently, upon symmetrization over primed indices these terms vanish. Hence we have that as required.
In order to derive (4.16), we first write We also have that F Note that by virtue of the Killing spinor equations Consider, for example, the spinor representative of the second term on the right-hand side of equation (4.20). On the solution manifold R ∞ (E 1 ) we have where we used (4.21). Similarly, on R ∞ (E 1 ), we see that Moreover, by virtue of (3.9), which immediately yields equation (4.16).  (4.17), they possess odd parity, i.e., chirality, under the duality transformation interchanging the electric and magnetic fields. Hence, in a marked contrast to the conformal symmetries Z[F ; ξ], which behave equivariantly under the duality transformation, the new second order symmetries, when regarded as differential operators acting on solutions of Maxwell's equations, map self-dual electromagnetic fields to anti-self-dual fields and vice versa.

Applications to the Dirac operator on Minkowski space
Recall that a Dirac spinor consists of a pair of spinor fields of type (0, 1) and (1, 0) (see, e.g., [20]), with the conjugate spinor Ψ * given by A vector v i ∈ M determines a linear transformation / v on Dirac spinors given by Then the Dirac γ-matrices are defined by the condition v i γ i Ψ = / vΨ, so that Moreover, as is customary, we will write The Weyl system, or the massless Dirac equation, is given by is the Dirac operator. As an application of the methods developed in Section 3 we present here a complete classification of generalized symmetries of the Weyl system on Minkowski space. Evidently, the Weyl system is equivalent to the decoupled pair of spin s = 1 2 massless free field equation and its conjugate equation, and, consequently, we will be able to analyze symmetries of (5.1) by the methods employed in the proof of the classification result in Theorem 1.
Now an evolutionary vector field takes the form where the characteristic can be assumed to depend on the spacetime variables x i and the exact sets of fields, that is, the symmetrized derivatives ,Ap , ϕ of the components of the Dirac spinor. As in the case of massless free fields, the Weyl system obviously admits scaling, conformal and chiral symmetries and their duals. The scaling symmetry and its dual possess the characteristics Let ξ CC ′ be a real conformal Killing vector, π B ′ C ′ a type (0, 2) Killing spinor. We write for the characteristics of the basic conformal and chiral symmetries of the Weyl system ( (r + 1) 2 (r + 2) 2 (r + 3) 2 + ((r + 1) 2 − 1)((r + 2) 2 − 1)((r + 3) 2 − 1) , for r ≥ 1.
Proof . The proof of Theorem 3 follows very much the same lines as the proof of Theorem 1, and we can safely omit the details.