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 ≥ 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 obtaine...
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| Цитувати: | Generalized Symmetries of Massless Free Fields on Minkowski Space / J. Pohjanpelto, S.C. Anco // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 20 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859932430764343296 |
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| author | Pohjanpelto, J. Anco, S.C. |
| author_facet | Pohjanpelto, J. Anco, S.C. |
| citation_txt | Generalized Symmetries of Massless Free Fields on Minkowski Space / J. Pohjanpelto, S.C. Anco // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 20 назв. — англ. |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A complete and explicit classification of generalized, or local, symmetries of massless free fields of spin s ≥ 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.
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| first_indexed | 2025-12-07T16:08:58Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 004, 17 pages
Generalized Symmetries of Massless Free Fields
on Minkowski Space?
Juha POHJANPELTO † and Stephen C. ANCO ‡
† Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4605, USA
E-mail: juha@math.oregonstate.edu
URL: http://oregonstate.edu/∼pohjanpp/
‡ Department of Mathematics, Brock University, St. Catharines ON L2S 3A1 Canada
E-mail: sanco@brocku.ca
URL: http://www.brocku.ca/mathematics/people/anco/
Received November 01, 2007; Published online January 12, 2008
Original article is available at http://www.emis.de/journals/SIGMA/2008/004/
Abstract. A complete and explicit classification of generalized, or local, symmetries of
massless free fields of spin s ≥ 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 confor-
mal 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.
Key words: generalized symmetries; massless free field; spinor field
2000 Mathematics Subject Classification: 58J70; 70S10
1 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 struc-
tures [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.
?This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in
Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). The full collection is available at
http://www.emis.de/journals/SIGMA/symmetry2007.html
mailto:juha@math.oregonstate.edu
http://oregonstate.edu/~pohjanpp/
mailto:sanco@brocku.ca
http://www.brocku.ca/mathematics/people/anco/
http://www.emis.de/journals/SIGMA/2008/004/
http://www.emis.de/journals/SIGMA/symmetry2007.html
2 J. Pohjanpelto and S.C. Anco
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.
2 Preliminaries
Let M be Minkowski space with coordinates xi, 0 ≤ i ≤ 3, and let Es, s = 1
2 , 1, 3
2 , . . . , stand
for the coordinate bundle
π : Es = {(xi, φA1A2···A2s)} → {(xi)},
where φA1A2···A2s is a type (2s, 0) spinor. We denote the kth order jet bundle of local sections
of Es by Jk(Es), 0 ≤ k ≤ ∞. Recall that the infinite jet bundle J∞(Es) is the coordinate space
J∞(Es) = {(xi, φA1A2···A2s , φA1A2···A2s,j1 , . . . , φA1A2···A2s,j1j2···jp , . . . )},
Generalized Symmetries of Massless Free Fields on Minkowski Space 3
where φA1A2···A2s,j1j2···jp stands for the pth order derivative variables. As is customary, we write
φ
B′
1B
′
2···B′
p
A1A2···A2s,B1B2···Bp
= σj1
B′
1
B1
σj2
B′
2
B2
· · ·σjpB
′
p
Bp
φA1A2···A2s,j1j2···jp ,
where σjBB′ , 0 ≤ j ≤ 3, are, up to a constant factor, the identity matrix and the Pauli spin
matrices. We also write
φ
B1···Bp
A′1A
′
2···A′2s,B
′
1···B′
p
= φ
B′
1···B′
p
A1A2···A2s,B1···Bp
,
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 εAB and its inverse εAB; 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
φ
B′
p
A2s,Bp
= φ
B′
1B
′
2···B′
p
A1A2···A2s,B1B2···Bp
,
and we will combine multi-indices by the rule BpCq = (B1B2 · · ·BpC1C2 · · ·Cq).
We let
∂CC′ = σiCC′∂/∂xi
denote the spinor representative of the coordinate derivative ∂/∂xi. Moreover, we define partial
derivative operators ∂A2s,Bp
φ B′
p
by
∂
A2s,Bp
φ B′
p
φ
D′
r
C2s,Dr
=
{
ε(C1
A1 · · · εC2s)
A2sε(D1
B1 · · · εDp)
Bpε(B′
1
D′
1 · · · εB′
p)
D′
p , if p = r,
0, if p 6= r,
∂
A2s,Bp
φ B′
p
φ
Dr
C′
2s,D
′
r
= 0,
and write
∂
A′
2s,B
′
p
φ Bp
= ∂
A2s,Bp
φ B′
p
.
Here, in accordance with the standard spinorial notation, we have written εCA for the Kronecker
delta and we use round brackets to indicate symmetrization in the enclosed indices.
A generalized vector field X on Es in spinor form is a vector field
X = PCC
′
∂CC′ +QA2s∂
A2s
φ +QA′
2s
∂
A′
2s
φ , (2.1)
where the coefficients PCC
′
= PCC
′
(xj , φ[p]), QA2s = QA2s(x
j , φ[p]) are spinor valued functions
in xj and the derivative variables φ
B′
q
A2s,Bq
up to some finite order p. An evolutionary vector
field Y , in turn, is a generalized vector field of the form
Y = QA2s∂
A2s
φ +QA′
2s
∂
A′
2s
φ ,
where QA2s is called the characteristic of Y .
Let
DC′
C = ∂C
′
C +
∑
p≥0
(
φ
B′
pC
′
A2s,BpC
∂
A2s,Bp
φ B′
p
+ φ
BpC′
A′
2s,B
′
pC
∂
A′
2s,B
′
p
φ Bp
)
4 J. Pohjanpelto and S.C. Anco
stand for the spinor representative of the standard total derivative operator, which, as is easily
verified, satisfies the commutation formula
[∂A2s,Bp
φ B′
p
, DC′
C ] = ε(B′
p|
C′
εC
(Bp|∂
A2s,|Bp−1)
φ |B′
p−1)
, p ≥ 1. (2.2)
The infinite prolongation prX of X in (2.1) to a vector field on J∞(Es) is given by
prX = PCC
′
DCC′ +
∑
p≥0
(
(DB′
1
B1
· · ·DB′
p
Bp
RA2s)∂
A2s,Bp
φ B′
p
+ (DB′
1
B1
· · ·DB′
p
Bp
RA′
2s
)∂A′
2s,Bp
φ B′
p
)
,
where RA2s is the characteristic of the evolutionary form
Xev = (QA2s − PCC
′
φA2s,CC′)∂A2s
φ + (QA′
2s
− PCC
′
φA′
2s,CC
′)∂
A′
2s
φ
of X.
The massless free field equation of spin s and its differential consequences
φ
A2sB′
p
A2s,A′ Bp
= 0, p ≥ 0, (2.3)
determine the infinitely prolonged solution manifold R∞(Es) ⊂ J∞(Es) of the equations. Ac-
cording to [17], the symmetrized derivative variables
φ
B′
p
A2sBp
= φ
B′
p
(A2s,Bp), p ≥ 0,
known as Penrose’s exact sets of fields, together with the independent variables xi provide
coordinates for R∞(Es). Moreover, as is easily verified, the unsymmetrized and symmetrized
variables φ
B′
p
A2s,Bp
, φ
B′
p
A2sBp
agree on R∞(Es).
A generalized, or local, symmetry of massless free fields is a generalized vector field X
satisfying
prXφ A2s
A2s,A′
= 0 on R∞(Es). (2.4)
Note that any generalized vector field of the form
TP = PCC
′
(∂CC′ + φA2s,CC′∂A2s
φ + φA′
2s,CC
′∂
A′
2s
φ )
with the prolongation
prTP = PCC
′
DCC′
automatically satisfies the determining equations (2.4) for a symmetry. Hence we will call a sym-
metry trivial if its prolongation agrees with a total vector field prTP = PCC
′
DCC′ on R∞(Es)
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
Y = QA2s∂
A2s
φ +QA′
2s
∂
A′
2s
φ ,
the determining equations (2.4) for the characteristic QA2s become
DA2s
A′ QA2s = 0 on R∞(Es). (2.5)
Generalized Symmetries of Massless Free Fields on Minkowski Space 5
Moreover, after replacing X by an equivalent symmetry, we can always assume that the compo-
nents QA2s are functions of only the independent variables xi and the symmetrized derivative
variables φ
B′
q
B2s+q
, 0 ≤ q ≤ p, for some p.
Recall that a vector field ξ = ξi(xj)∂i on M is conformal Killing provided that
∂(iξj) = kηij (2.6)
for some function k = k(xi), where ηij stands for the Minkowski metric. As can be verified by
a direct computation, a conformal Killing vector field ξ gives rise to the symmetry
Z[ξ] = ZA2s [ξ]∂
A2s
φ + ZA′
2s
[ξ]∂A′
2s
φ (2.7)
of massless free fields of spin s, with the characteristic
ZA2s [ξ] = ξCC
′
φA2sCC′ + s∂C′(A2s
ξCC
′
φA2s−1)C +
1− s
4
(∂CC′ξCC
′
)φA2s ,
which agrees with the conformally weighted Lie derivative
L(−1)
ξ φA2s = LξφA2s +
1
4
(∂CC′ξCC
′
)φA2s
of the spinor field φA2s ; 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
prZ[ξ]Y
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
∂
(B′
(B ξ
C′)
C) = 0. (2.8)
An obvious generalization of equations (2.8) to spinor fields κA′
l
Ak
= κ
A′
l
Ak
(xCC
′
) of type (k, l) is
∂
(A′l+1
(Ak+1
κ
A′
l)
Ak) = 0, (2.9)
and symmetric spinor fields κA′
l
Ak
= κ
(A′
l)
(Ak)(x
CC′
) satisfying these equations are called Killing
spinors of type (k, l). Thus, in particular, a type (1, 1) Killing spinor κA
′
A corresponds to a com-
plex 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].
Lemma 1. Let ξA
′
k
Ak
, κ
A′
k+2s
Ak
be Killing spinors of type (k, k) and (k, k + 2s). Then ξ
A′
k
Ak
can
be expressed as a sum of symmetrized products of k Killing spinors of type (1, 1), and κ
A′
k+2s
Ak
can be expressed as a sum of symmetrized products of Killing spinors of type (0, 2s) and k
Killing spinors of type (1, 1). The dimensions of the complex vector spaces of Killing spinors of
type (k, k) and (k, k + 2s) are
(k + 1)2(k + 2)2(2k + 3)/12 and
(k + 1)(k + 2)(k + 2s+ 1)(k + 2s+ 2)(2k + 2s+ 3)/12,
respectively.
6 J. Pohjanpelto and S.C. Anco
3 Main results
Let ξ, ζ1, . . . , ζp be real conformal Killing vectors and let πA′
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
Z[iξ] = iZA2s [ξ]∂
A2s
φ − iZA′
2s
[ξ]∂A′
2s
φ . (3.1)
Furthermore, let
W[π] = WA2s [π]∂A2s
φ +WA′
2s
[π]∂A′
2s
φ (3.2)
be an evolutionary vector field with components
WA2s [π] =
2s∑
p=0
c2s,p∂B′
1(A2s−p+1|∂B′
2|A2s−p+2| · · ·| ∂B′
p|A2s
πB′
pC
′
4s−pφA2s−p)C′
4s−p
, (3.3)
where the coefficients c2s,p are given by
c2s,p =
4s− p+ 1
4s+ 1
(
2s
p
)
, 0 ≤ p ≤ 2s. (3.4)
Moreover, write
Z[ξ; ζ1, . . . , ζp] = prZ[ζ1] · · ·prZ[ζp]Z[ξ], (3.5)
Z[iξ; ζ1, . . . , ζp] = prZ[ζ1] · · ·prZ[ζp]Z[iξ], (3.6)
W[π; ζ1, . . . , ζq] = prZ[ζ1] · · ·prZ[ζq]W[π], (3.7)
for the repeated componentwise derivatives of the vector fields (2.7), (3.1), (3.2) with respect
to conformal symmetries.
Proposition 1. Let ξ, ζ1,. . . ,ζp be conformal Killing vectors and let πA4s be a Killing spinor
of type (0, 4s). Then the evolutionary vector fields
Z[ξ; ζ1, . . . , ζp], Z[iξ; ζ1, . . . , ζp], W[π; ζ1, . . . , ζq], p, q ≥ 0, (3.8)
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∞(Es), the leading order terms in the
components ZA2s [ξ; ζ1, . . . , ζp], ZA2s [iξ; ζ1, . . . , ζp], WA2s [π; ζ1, . . . , ζq] of the symmetries (3.8)
reduce to
(−1)p+1ξ
(C1
(C′
1
ζ1
C2
C′
2
· · · ζp
Cp+1)
C′
p+1)
φ
C′
p+1
A2sCp+1
,
(−1)p+1iξ(C1
(C′
1
ζ1
C2
C′
2
· · · ζp
Cp+1)
C′
p+1)
φ
C′
p+1
A2sCp+1
,
(−1)qζ1
(C1
(C′
1
ζ2
C2
C′
2
· · · ζq
Cq)
C′
q
πB′
4s)
φ
C′
qB
′
4s
A2sCq
,
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
∂CC′∂CD
′
πB′
4s = 0, (3.9)
Generalized Symmetries of Massless Free Fields on Minkowski Space 7
and, consequently,
∂CC′∂DD′πB′
4s = ∂CD′∂DC′πB′
4s . (3.10)
Write
Π1
p,A2s−1A′ = ∂A2s
A′ ∂B′
1(A2s−p+1|∂B′
2|A2s−p+2| · · ·| ∂B′
p|A2s
πB′
pC
′
4s−pφA2s−p)C′
4s−p
,
Π2
p,A2s−1A′ = ∂B′
1(A2s−p+1|∂B′
2|A2s−p+2| · · ·| ∂B′
p|A2s
πB′
pC
′
4s−pφ
A2s
A2s−p)C′
4s−pA
′ ,
0 ≤ p ≤ 2s, so that W[π] is a symmetry of massless free field equations provided that
2s∑
p=0
c2s,p(Π1
p,A2s−1A′ + Π2
p,A2s−1A′) = 0. (3.11)
We compute
Π1
p,A2s−1A′
=
4s
4s+ 1
∂B′
1(A2s−p+1|∂B′
2|A2s−p+2| · · ·| ∂B′
p|A2s
(εA′ (B
′
p∂
|A2s|
|D′| π
B′
p−1C
′
4s−p)D′
)φA2s−p)C′
4s−p
=
p
4s+ 1
∂A′(A2s−p+1|∂B′
1|A2s−p+2| · · ·| ∂B′
p−1|A2s
∂A2s
|D′|π
B′
p−1C
′
4s−pD
′
φA2s−p)C′
4s−p
+
4s− p
4s+ 1
∂B′
1(A2s−p+1|∂B′
2|A2s−p+2| · · ·| ∂B′
p|A2s
∂A2s
|D′|π
B′
pC
′
4s−p−1D
′
φA2s−p)C′
4s−p−1A
′
=
p
4s+ 1
Π1
p,A2s−1A′ +
4s− p
4s+ 1
2s− p
2s
(3.12)
× ∂B′
1(A2s−p|∂B′
2|A2s−p+1| · · ·| ∂B′
p|A2s−1
∂B|D′|π
B′
pC
′
4s−p−1D
′
φA2s−p−1)C′
4s−p−1A
′B,
where we used (3.9) and (3.10). On the other hand,
Π2
p,A2s−1A′ =
p
2s
∂D′B∂B′
1(A2s−p+1|∂B′
2|A2s−p+2| · · ·| ∂B′
p|A2s−1
πB′
pC
′
4s−pD
′
φ
B
A2s−p)C′
4s−pA
′ . (3.13)
Now it follows from (3.12), (3.13) that
Π1
p,A2s−1A′ = − (4s− p)(2s− p)
(4s− p+ 1)(p+ 1)
Π2
p+1,A2s−1A′ .
Clearly
Π1
2s,A2s−1A′ = 0, Π2
0,A2s−1A′ = 0.
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 S̃, and the elementary symmetries E [ϕ] given by
S = φA2s∂
A2s + φA′
2s
∂
A′
2s , S̃ = iφA2s∂
A2s − iφA′
2s
∂
A′
2s , (3.14)
and
E [ϕ] = ϕA2s(x
i)∂A2s + ϕA′
2s
(xi)∂A′
2s , (3.15)
where ϕA2s = ϕA2s(x
i) is any solution of (2.3).
8 J. Pohjanpelto and S.C. Anco
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 Q̂
of order at most r which can be written as
Q̂ = V + E [ϕ], (3.16)
where ϕA2s = ϕA2s(x
i) is a solution of the massless free field equations of spin s, and where V
is equivalent to a spinorial symmetry that is a linear combination of the symmetries
S, S̃, Z[ξ; ζ1, . . . , ζp], Z[iξ; ζ1, . . . , ζp], W[π; ζ1, . . . , ζq] (3.17)
with p ≤ r − 1, q ≤ r − 2s.
In particular, the dimension dr of the real vector space of equivalence classes of spinorial
symmetries of order at most r spanned by the symmetries (3.17) is
dr = (r + 1)2(r + 2)2(r + 3)2/18, if r < 2s,
and
dr = (r + 1)2(r + 2)2(r + 3)2/18
+ ((r + 1)2 − 4s2)((r + 2)2 − 4s2)((r + 3)2 − 4s2)/18, if r ≥ 2s.
The above result was originally announced without proof in [3].
Proof. Without loss of generality, we can assume that the components QA2s of Q are functions
of the independent variables xi and the symmetrized derivative variables φ
A′
p
A2s+p
, 0 ≤ p ≤ r.
Consequently,
∂
B2s,Cp
φ C′
p
QA2s = ∂
(B2s,Cp)
φ C′
p
QA2s , ∂
B′
2s,C
′
p
φ Cp
QA2s = ∂
(B′
2s,C
′
p)
φ Cp
QA2s . (3.18)
It follows from the determining equations for spinorial symmetries that
∂
(B2s,Cp)
φ C′
p
DA2s
A′ QA2s = 0, ∂
(B′
2s,C
′
p)
φ Cp
DA2s
A′ QA2s = 0 (3.19)
on R∞(Es). By virtue of the commutation formulas (2.2), the above equations with p = r + 1
show that
∂
(B2s,Cr
φ C′
r
QA2s)
A2s−1
= 0, ∂
(B′
2s,C
′
r)
φ (Cr
QA2s)A2s−1
= 0 (3.20)
identically on J∞(Es). Equations (3.20), in turn, combined with (3.18), imply that
∂ B2s,Cr
φ C′
r
QA2s = εA1
(B1εA2
B2 · · · εA2s
B2sSCr)
C′
r
,
∂
B′
2s,C
′
r
φ Cr
QA2s = ε(Cr−2s+1
A1εCr−2s+2
A2 · · · εCr
A2sT B′
2sC
′
r
Cr−2s)
, (3.21)
for some spinor valued functions SCr
C′
r
, T Cr+2s′
Cr−2s
on J∞(Es) symmetric in their indices. Note in
particular that T Cr+2s′
Cr−2s
vanishes if r < 2s.
Next use equations (3.19) with p = r together with the commutation formulas (2.2) to
conclude that
DA2s
(A′ ∂
B2s,Cr
φ C′
r)QA2s = 0, DA2s(A′∂
B′
2s,C
′
r)
φ Cr
QA2s = 0 on R∞(Es). (3.22)
Generalized Symmetries of Massless Free Fields on Minkowski Space 9
Now substitute expressions (3.21) into (3.22) to deduce that
D
(Cr+1
(C′
r+1
SCr)
C′
r) = 0, D
(C′
r+2s+1
(Cr−2s+1
T C′
r+2s)
Cr−2s)
= 0 on R∞(Es).
But it is easy to see that the above equations force SCr
C′
r
, T C′
r+2s
Cr−2s
to be independent of the
symmetrized derivative variables φ
B′
p
Bp+2s
, p ≥ 0, and, consequently, they must satisfy the Killing
spinor equations
∂
(Cr+1
(C′
r+1
SCr)
C′
r) = 0, ∂
(C′
r+2s+1
(Cr−2s+1
T C′
r+2s)
Cr−2s)
= 0.
Thus
QA2s = SBr
B′
r
φ
B′
r
A2sBr
+ T B′
r+2s
Br−2s
φ
Br−2s
A2sB′
r+2s
+ UA2s ,
where UA2s only involves the derivative variables φ
A′
p
A2s+p
up to order r − 1.
Now by the factorization property of Lemma 1, the Killing spinors SBr
B′
r
, T Br+2s
Br−2s
can be
expressed as a sum of symmetrized products of r Killing spinors of type (1, 1), and as a sum of
symmetrized products of a Killing spinor of type (0, 4s) and r− 2s Killing spinors of type (1, 1),
respectively. Thus by Proposition 1 there is a linear combination Vr of the basic symmetries
Z[ξ; ζ1, . . . , ζr−1], Z[iξ; ζ1, . . . , ζr−1], W [π; ζ1, . . . , ζr−2s]
so that on R∞(Es), the highest order terms in Vr 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 Vo of order 0. But it is straightforward to solve the determining
equations (2.5) for Vo to see that
Vo,A2s = aφA2s + ϕA2s ,
where a ∈ C is a constant and ϕA2s = ϕA2s(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. �
4 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
Fij,
j = 0, ∗Fij,j = 0 (4.1)
on Minkowski space. Here Fij = −Fji are the components of the electromagnetic field tensor F
and ∗ stands for the Hodge dual.
10 J. Pohjanpelto and S.C. Anco
We write Λ2(T ∗M) → M for the associated bundle with coordinates Fij , i < j. Then
J∞Λ2(T ∗M) is the coordinate bundle
{(xi, Fij , Fij,k1 , Fij,k1k2 , . . . )} → {(xi)}.
For notational convenience, we write Fij,k1···kp = −Fji,k1···kp for i ≥ j.
In spinor form the electromagnetic field tensor F becomes
σiAA′σ
j
BB′Fij = εA′B′φAB + εABφA′B′ ,
while Maxwell’s equations (4.1) correspond to the spin s = 1 massless free field equations (2.3)
for the electromagnetic spinor φAB = φ(AB).
A generalized symmetry of Maxwell’s equations in evolutionary form is a vector field Y =
Qij∂
ij
F satisfying
DjQij = 0, Dj∗Qij = 0 (4.2)
on solutions of (4.1). If one defines QAB by
σiAA′σ
j
BB′Qij = εA′B′QAB + εABQA′B′ , (4.3)
then it easily follows from (4.2), (4.3) that QAB are the components of a generalized symmetry
of the massless field equations of spin s = 1. Thus, by employing the correspondence (4.3), we
can obtain a complete classification of generalized symmetries of Maxwell’s equations from the
classification result in Theorem 1.
Symmetries (3.14), (3.15) clearly correspond to the symmetries
S = Fij∂
ij
F , S̃ = ∗Fij∂ijF , E(F) = Fij∂
ij
F
of Maxwell’s equations, where F is any solution of (4.1) with components Fij = Fij(xk). Con-
formal symmetries (2.7) and their duals (3.1) in turn give rise to the symmetries
Z[F ; ξ] = Zij [F ; ξ]∂ijF , Z[∗F ; ξ] = Zij [∗F ; ξ]∂ijF , (4.4)
with components
Zij [F ; ξ] = ξkFij,k − 2∂[iξ
kFj]k,
where ξ is a conformal Killing vector on M and where square brackets indicate skew-symmetri-
zation in the enclosed indices.
In order to transcribe the second order chiral symmetries W[π] introduced in (3.2) to sym-
metries of Maxwell’s equations in physical form, we first introduce the following polynomial
tensors on M. Let
p0
ijkl = a0
ijkl, (4.5)
p1
ijkl = x[ia
1
j]kl + x[ka
1
l]ij + (η[i|[ka
1
l]|j]n + η[k|[ia
1
j]|l]n)x
n, (4.6)
p2
ijkl = a2
[i|[kxl]|xj] −
1
2
η[i|[ka
2
l]|j]xmx
m
+
1
2
(η[i|[ka
2
l]n|xj] + η[k|[ia
2
j]n|xl])x
n − 1
6
ηi[kηl]ja
2
mnx
mxn, (4.7)
p3
ijkl = (x[ia
3
j]n[kxl] + x[ka
3
l]n[ixj])x
n +
1
4
(x[ia
3
j]kl + x[ka
3
l]ij)xnx
n
+
1
2
(a3
mn[iηj][kxl] + a3
mn[kηl][ixj])x
mxn +
1
4
(a3
[i|m[kηl]|j] + a3
[k|m[iηj]|l])x
mxnx
n, (4.8)
p4
ijkl = (a4
m[i|n[kxl]|xj] −
1
2
a4
m[i|n[kηl]|j]xpx
p)xmxn − 1
16
a4
ijklxmx
mxnx
n, (4.9)
Generalized Symmetries of Massless Free Fields on Minkowski Space 11
where a0
ijkl, a
1
ijk, a
2
ij , a
3
ijk, a
4
ijkl are real constants satisfying
ahijkl = ah[kl][ij], ah[ijkl] = 0, ahijk
j = 0, h = 0, 4, (4.10)
ahijk = ahi[jk], ah[ijk] = 0, ahji
j = 0, h = 1, 3, (4.11)
a2
ij = a2
(ij), a2
i
i = 0, (4.12)
and where we raise indices using the inverse ηij of the Minkowski metric.
Note that ahijkl, h = 0, 4, possesses the symmetries of Weyl’s conformal curvature tensor.
Hence their spinor representatives can be written in the form
ahII′JJ ′KK′LL′ = εIJεKLα
h
I′J ′K′L′ + εI′J ′εK′L′α
h
IJKL, h = 0, 4,
where αhI′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 a2
ij and ahijk, h = 1, 3, are given by
a2
II′JJ ′ = α2
IJI′J ′ and
ahII′JJ ′KK′ = εJKα
h
II′J ′K′ + εJ ′K′αhI′IJK , h = 1, 3,
where α1
II′J ′K′ , α2
IJI′J ′ = α2
IJI′J ′ , α
3
II′J ′K′ , are symmetric constant spinors.
For 0 ≤ h ≤ 4, let W[F ; ph] denote the evolutionary vector field with components
Wij [F ; ph] = phklm[iF
kl
,
m
j] + ∂[ip
h
j]mklF
kl
,
m +
3
5
∂mphklm[iF
kl
,j] +
3
5
∂m∂[ip
h
j]mklF
kl, (4.13)
where phijkl are the polynomials in (4.5)–(4.9), and write
Z[F ; ξ, ζ1, . . . , ζq] = prZ[F ; ζ1] · · ·prZ[F ; ζq]Z[F ; ξ],
W[F ; ph; ζ1, . . . , ζq] = prZ[F ; ζ1] · · ·prZ[F ; ζq]W [F ; ph]
for the componentwise Lie derivatives, where Z[F ; ζi] stands for the conformal symmetry (4.4)
associated with the conformal Killing vector ζi.
Lemma 2. Let phijkl, 0 ≤ h ≤ 4, be the polynomials (4.5)–(4.9). Then the spinor representative
of phijkl is of the form
phII′JJ ′KK′LL′ = εIJεKLπ
h
I′J ′K′L′ + εI′J ′εK′L′π
h
IJKL, (4.14)
where
π0
I′J ′K′L′ = α0
I′J ′K′L′ , π1
I′J ′K′L′ = α1
L(I′J ′K′xLL′),
π2
I′J ′K′L′ =
1
4
α2
KL(I′J ′x
K
K′xLL′), π3
I′J ′K′L′ = −1
2
α3
JKL(I′x
J
J ′x
K
K′xLL′), (4.15)
π4
I′J ′K′L′ =
1
4
α4
IJKLx
I
I′x
J
J ′x
K
K′xLL′ .
Moreover, on the solution manifold R∞(E1), the spinor representatives of the evolutionary vector
fields W[F ; ph] in (4.13) reduce to
WII′JJ ′ [F ; ph] = εI′J ′WIJ [πh] + εIJWI′J ′ [πh], (4.16)
where WIJ [πh] are the components (3.3) of chiral symmetries for spin s = 1 fields. Thus
Wij [F ; ph], 0 ≤ h ≤ 4, is a symmetry of Maxwell’s equations. Moreover,
∗Wij [F ; ph] = −Wij [∗F ; ph]. (4.17)
12 J. Pohjanpelto and S.C. Anco
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 equa-
tion (4.16), and omit the proofs of the remaining equations in the Lemma.
One can check by a direct computation that the polynomials phijkl possess the symmetries of
Weyl’s conformal curvature tensor, that is,
phijkl = ph[kl][ij], ph[ijkl] = 0, phijk
j = 0, 0 ≤ h ≤ 4.
Consequently, we can write the spinor representatives of phijkl in the form (4.14); see [17]. Then,
in particular,
πhI′J ′K′L′ =
1
4
phP (I′
P
J ′|Q|K′
Q
L′). (4.18)
Thus we can find πhI′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, sym-
metrizing over primed indices. The computations can be further simplified by using the ob-
servation that given a tensor fijkl with the spinor representative fII′JJ ′KK′LL′ , then the spinor
representative hII′JJ ′KK′LL′ of hijkl = f[ij][kl] satisfies
hPI′
P
J ′QK′QL′ = fP (I′J ′)
P
Q(K′L′)
Q. (4.19)
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
1
4
α2
KL(I′J ′x
K
K′xLL′).
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
π2
I′J ′K′L′ =
1
4
αKL(I′J ′x
K
K′xLL′),
as required.
In order to derive (4.16), we first write
F+
kl =
1
2
(Fkl − i∗Fkl), W +
ij [F, ph] =
1
2
(Wij [F, ph]− iWij [∗F, ph]),
so that
W +
ij [F, ph] = phklm[iF
+kl
,
m
j] + ∂[ip
h
j]mklF
+kl
,
m
+
3
5
∂mphklm[iF
+kl
,j] +
3
5
∂m∂[ip
h
j]mklF
+kl
. (4.20)
We also have that F+
KK′LL′ = εKLφK′L′ .
Note that by virtue of the Killing spinor equations
∂PI′π
h
J ′K′L′M ′ = −4
5
εI′(J ′∂
PP ′
πhK′L′M ′)P ′ .
Hence
∂P (I′π
h
J ′)K′L′M ′φ
K′L′M ′P =
3
5
∂P
′
P πhK′L′P ′(I′φJ ′)
K′L′P . (4.21)
Generalized Symmetries of Massless Free Fields on Minkowski Space 13
Consider, for example, the spinor representative of the second term on the right-hand side of
equation (4.20). On the solution manifold R∞(E1) we have
σiII′σ
j
JJ ′∂[ip
h
j]mklF
+kl
,
m
= −∂II′πhJ ′K′L′M ′φJ
K′L′M ′
+ ∂JJ ′π
h
I′K′L′M ′φI
K′L′M ′
= εI′J ′∂P ′(Iπ
hK′L′M ′P ′
φJ)K′L′M ′ − εIJ∂P (I′π
h
J ′)K′L′M ′φ
K′L′M ′P (4.22)
= εI′J ′∂P ′(Iπ
hK′L′M ′P ′
φJ)K′L′M ′ −
3
5
εIJ∂
P ′
P πhK′L′P ′(I′φJ ′)
K′L′P ,
where we used (4.21). Similarly, on R∞(E1), we see that
σiII′σ
j
JJ ′p
h
klm[iF
+kl
,
m
j] = εI′J ′π
h
K′L′M ′P ′φIJ
K′L′M ′P ′
, and (4.23)
σiII′σ
j
JJ ′∂
mphklm[iF
+kl
,j] = εI′J ′∂P ′(Iπ
hK′L′M ′P ′
φJ)K′L′M ′ (4.24)
− εIJ∂PP ′πhK
′L′P ′
(I′φJ ′)K′L′
P .
Moreover, by virtue of (3.9),
σiII′σ
j
JJ ′∂
m∂[ip
h
j]mklF
+kl
= εI′J ′∂M ′I∂JP ′πhK
′L′M ′P ′
φK′L′ − εIJ∂PM ′∂P(I′π
h
J ′)
K′L′M ′
φK′L′ (4.25)
= εI′J ′∂M ′I∂JP ′πhK
′L′M ′P ′
φK′L′
on R∞(E1).
Now equations (4.22)–(4.25) together show that the spinor form of W +
ij [F, ph] is
W +
II′JJ ′ [F, ph] = εI′J ′
(
πhK
′L′M ′P ′
φIJK′L′M ′P ′
+
8
5
∂P ′(Iπ
hK′L′M ′P ′
φJ)K′L′M ′ +
3
5
∂M ′I∂JP ′πhK
′L′M ′P ′
φK′L′
)
= εI′J ′WIJ [πh],
which immediately yields equation (4.16). �
Theorem 2. The space of equivalence classes of symmetries of Maxwell’s equations of order r,
r ≥ 2, is spanned by the symmetries
E(F), S, S̃,
Z[F ; ξ, ζ1, . . . , ζq], Z[∗F ; ξ, ζ1, . . . , ζq], q ≤ r − 1,
W[F ; ph; ζ1 . . . , ζq], W[∗F ; p2; ζ1, . . . , ζq], 0 ≤ h ≤ 4, q ≤ r − 2,
where F is an arbitrary solution of Maxwell’s equations, ξ, ζ1,. . . ,ζr, are conformal Killing
vectors and phijkl, 0 ≤ h ≤ 4, are the polynomials given in (4.5)–(4.9). Apart from the trivial
symmetries E(F), there are
dr = (r + 1)(r + 3)(r4 + 8r3 + 17r2 + 4r + 6)/9
independent symmetries of Maxwell’s equations of order r ≥ 2.
Proof. First note that if Y = Qij∂
ij
F is a symmetry of Maxwell’s equations in evolutionary
form with the spinor representative Q determined by (4.3) then the spinor representative of the
symmetry prZ[F ; ζ]Y is prZ[ζ]Q. Thus, in light of Theorem 1 and by linearity, it suffices to
14 J. Pohjanpelto and S.C. Anco
show that the spinorial symmetries E(ϕ), S, S̃, Z[ξ], Z[iξ], W[π], where ξ is a real conformal
Killing vector and π is a type (0, 4) Killing spinor, are contained in the span of the spinorial
symmetries corresponding to
E(F), S, S̃, Z[F ; ξ], ∗Z[F ; ξ], ∗W[F ; p2], W[F ; ph], 0 ≤ h ≤ 4,
under the identification (4.3).
Obviously the symmetries E(F), S, S̃, Z[F ; ξ], ∗Z[F ; ξ] correspond via (4.3) to the spinorial
symmetries E(ϕ), S, S̃ Z[ξ], Z[iξ], where ϕ is the spinorial counterpart of the solution F of
Maxwell’s equations.
Next, by (4.15), the spinor fields πhI′J ′K′L′ corresponding to the polynomials phijkl span the
space of Killing spinors of type (0, 4) provided that we allow α2
IJI′J ′ also to take complex values.
Thus, in light of (4.16), the first part of the Theorem now follows from the observation that
if W[π] is the spinorial symmetry corresponding to W[F ; ph], then W[iπ] corresponds to the
symmetry Ŵ[F ; ph] = −∗W[F ; ph].
Finally, the dimension count follows from the dimension count in Theorem 1. This concludes
the proof of the Theorem. �
Remark 1. The new chiral symmetries W[F ; ph] are physically interesting since, as evidenced
by equation (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 sec-
ond 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.
5 Applications to the Dirac operator on Minkowski space
Recall that a Dirac spinor consists of a pair
Ψ =
(
ψA
′
ϕA
)
of spinor fields of type (0, 1) and (1, 0) (see, e.g., [20]), with the conjugate spinor Ψ∗ given by
Ψ∗ =
(
ϕA
′
ψA
)
.
A vector vi ∈ M determines a linear transformation /v on Dirac spinors given by
/vΨ =
(
viσAA
′
i ϕA
viσiAA′ψ
A′
)
.
Then the Dirac γ-matrices are defined by the condition
viγiΨ = /vΨ,
so that
γi =
(
0 σA
′B
i
σiAB′ 0
)
, 0 ≤ i ≤ 3.
Moreover, as is customary, we will write
γ5 = −iγ0γ1γ2γ3 =
(
I 0
0 −I
)
.
Generalized Symmetries of Massless Free Fields on Minkowski Space 15
The Weyl system, or the massless Dirac equation, is given by
/∂Ψ = 0, (5.1)
where
/∂ = γi
∂
∂xi
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
ϕ AA′
A, = 0, ψA
′
,AA′ = 0
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
X = QA
′
∂ψA′ +Q
A
∂ψA +RA∂ϕA +RA′∂ϕA′ ,
where the characteristic
Q =
(
QA
′
RA
)
can be assumed to depend on the spacetime variables xi and the exact sets of fields, that is, the
symmetrized derivatives
ψ
A′A′
p
Ap
= ψ
(A′A′
p)
,Ap
, ϕ
A′
p
AAp
= ϕ
A′
p
(A,Ap)
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
S[Ψ] = Ψ, S[iΨ] = iΨ.
Let ξCC
′
be a real conformal Killing vector, πB
′C′
a type (0, 2) Killing spinor. We write
Z[Ψ; ξ] =
ξCC
′
ψA
′
CC′ +
1
2
(∂A
′
C ξ
CC′
)ψC′ +
1
8
(∂CC′ξCC
′
)ψA
′
ξCC
′
ϕACC′ +
1
2
(∂AC′ξCC
′
)ϕC +
1
8
(∂CC′ξCC
′
)ϕA
,
W[Ψ;π] =
πBCϕA
′
BC +
2
3
∂A
′
C π
BCϕB
πB
′C′
ψAB′C′ +
2
3
∂AC′πB
′C′
ψB′
for the characteristics of the basic conformal and chiral symmetries of the Weyl system (5.1).
Moreover, we define Z[iΨ; ξ] in the obvious fashion, and denote the componentwise Lie deriva-
tives of Z[Ψ; ξ], Z[iΨ; ξ], W[Ψ;π] with respect to the conformal symmetries corresponding to
ζ1,. . . ,ζp by
Z[Ψ; ξ, ζ1, . . . , ζp], Z[iΨ; ξ, ζ1, . . . , ζp], W[Ψ;π; ζ1, . . . , ζp].
16 J. Pohjanpelto and S.C. Anco
Theorem 3. Let Q be a generalized symmetry of order r of the Weyl system (5.1) in evolutionary
form. Then Q is equivalent to a symmetry Q̂ of order at most r in evolutionary form which can
be written as
Q̂ = V + E [Ψ],
where E [Ψ] is an elementary symmetry corresponding to a solution Ψ = Ψ(xi) of the Weyl
system, and where V is equivalent to a symmetry that is a linear combination of the symmetries
S[Ψ], S[iΨ], γ5S[Ψ], γ5S[iΨ],
S[Ψ∗], S[iΨ∗], γ5S[Ψ∗], γ5S[iΨ∗],
Z[Ψ; ξ, ζ1, . . . , ζp], γ5Z[Ψ; ξ, ζ1, . . . , ζp],
Z[iΨ; ξ, ζ1, . . . , ζp], γ5Z[iΨ; ξ, ζ1, . . . , ζp], (5.2)
Z[Ψ∗; ξ, ζ1, . . . , ζp], γ5Z[Ψ∗; ξ, ζ1, . . . , ζp],
Z[iΨ∗; ξ, ζ1, . . . , ζp], γ5Z[iΨ∗; ξ, ζ1, . . . , ζp],
W[Ψ;π; ζ1, . . . , ζp], γ5W[Ψ;π; ζ1, . . . , ζp],
W[Ψ∗;π; ζ1, . . . , ζp], γ5W[Ψ∗;π; ζ1, . . . , ζp],
where p ≤ r − 1. In particular, the dimension dr of the real vector space of equivalence classes
of spinorial symmetries of order at most r spanned by the symmetries (5.2) is
dr =
2
9
[
(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. �
Acknowledgements
The research of the first author is supported in part by the NSF Grants DMS 04–53304 and
OCE 06–21134, and that of the second author by an NSERC grant.
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Generalized Symmetries of Massless Free Fields on Minkowski Space 17
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1 Introduction
2 Preliminaries
3 Main results
4 Symmetries of Maxwell's equations
5 Applications to the Dirac operator on Minkowski space
References
|
| id | nasplib_isofts_kiev_ua-123456789-149003 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:08:58Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Pohjanpelto, J. Anco, S.C. 2019-02-19T12:53:38Z 2019-02-19T12:53:38Z 2008 Generalized Symmetries of Massless Free Fields on Minkowski Space / J. Pohjanpelto, S.C. Anco // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 20 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 58J70; 70S10 https://nasplib.isofts.kiev.ua/handle/123456789/149003 A complete and explicit classification of generalized, or local, symmetries of massless free fields of spin s ≥ 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. This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). The research of the first author is supported in part by the NSF Grants DMS 04–53304 and OCE 06–21134, and that of the second author by an NSERC grant. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Generalized Symmetries of Massless Free Fields on Minkowski Space Article published earlier |
| spellingShingle | Generalized Symmetries of Massless Free Fields on Minkowski Space Pohjanpelto, J. Anco, S.C. |
| title | Generalized Symmetries of Massless Free Fields on Minkowski Space |
| title_full | Generalized Symmetries of Massless Free Fields on Minkowski Space |
| title_fullStr | Generalized Symmetries of Massless Free Fields on Minkowski Space |
| title_full_unstemmed | Generalized Symmetries of Massless Free Fields on Minkowski Space |
| title_short | Generalized Symmetries of Massless Free Fields on Minkowski Space |
| title_sort | generalized symmetries of massless free fields on minkowski space |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149003 |
| work_keys_str_mv | AT pohjanpeltoj generalizedsymmetriesofmasslessfreefieldsonminkowskispace AT ancosc generalizedsymmetriesofmasslessfreefieldsonminkowskispace |