Second order symmetry operators for the massive Dirac equation

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Second order symmetry operators for the massive Dirac

equation

Simon Jacobsson

E-mail: simon.jacobsson@kuleuven.be

Chalmers University of Technology, SE-412 96 Gothenburg, Sweden

KU Leuven, BE-3000 Leuven, Belgium

Thomas B¨ackdahl

E-mail: thomas.backdahl@chalmers.se

Mathematical Sciences, Chalmers University of Technology and University of Gothenburg,

SE-412 96 Gothenburg, Sweden

Abstract

Employing the covariant language of two-spinors, we ﬁnd what conditions a curved Lorentzian

spacetime must satisfy for existence of a second order symmetry operator for the massive Dirac

equation. The conditions are formulated as existence of a set of Killing spinors satisfying a set

of covariant linear diﬀerential equations. Using these Killing spinors, we then state the most

general form of such an operator. Partial results for the zeroth and ﬁrst order are presented and

interpreted as well. Computer algebra tools from the Mathematica package suite xAct were used

for the calculations.

1. Introduction

Asymmetry operator is a linear diﬀerential operator mapping solutions to solutions of a

diﬀerential equation. Such operators can be very useful for detailed studies of the solutions.

However, the existence of such operators is not trivial and is linked to the existence of diﬀerent

kinds of symmetries of the curved spacetime geometry that the diﬀerential equation is deﬁned

on. This paper aims to elucidate this for the massive Dirac equation.

Many partial diﬀerential equations from physics, such as the Schr¨odinger and Helmholtz

equations, lend themselves naturally to separation of variables, but also the Dirac equation has

been separated in some cases. This is closely related to the existence of symmetry operators. For

instance, Kalnins et al. [16, section 3], explain the separation of the Dirac equation on the Kerr

spacetime in terms of the existence of symmetry operators associated with a Killing tensor by

identifying a set of separation constants as eigenvalues of said symmetry operators.

Symmetry operators also have other uses, for instance, given a conserved energy, or an

energy estimate, one can easily construct higher order versions by inserting a symmetry operator.

More advanced uses have also been found. Andersson and Blue [7] used higher order symmetry

operators for the scalar wave equation on the Kerr spacetime to handle the complicated trapping

phenomena when proving decay estimates.

For many diﬀerential equations, a Lie derivative along a Killing vector gives a symmetry

operator, i.e. a symmetry of the spacetime gives a symmetry operator. However, in many cases

there are also other less obvious symmetries sometimes called hidden symmetries that can give

rise to symmetry operators. In general these are described in terms of Killing spinors. An

important example is a second order symmetry operator related to the Carter constant [13] used

by Andersson and Blue in [7]. This symmetry operator can not be built from Killing vectors.

To know that all symmetry operators have been found, a systematic study is required. If

the set of symmetry operators is not large enough, the methods described above will not give

arXiv:2210.05216v1 [gr-qc] 11 Oct 2022

Second order symmetry operators for the massive Dirac equation 2

satisfactory results.

The conditions for existence of symmetry operators we present here are described as existence

of a set of Killing spinors satisfying a set of covariant diﬀerential equations. This can be interpreted

as conditions on the spacetime geometry. Assuming the spacetime is a suﬃciently smooth four-

dimensional Lorentzian manifold that allows for a spin structure, these conditions are both

necessary and suﬃcient.

The spin structure allow us to decompose tensorial objects into irreducible components.

Using the covariant two-spinor formalism described by Penrose and Rindler [19,20], these

decompositions are used to decompose equations into independent subequations that must be

satisﬁed simultaneously.

It is in general a time-consuming and nontrivial task to ﬁnd these irreducible decompositions.

Thus, for this task, computer algebra systems such as the Mathematica packages Sym-

Manipulator [8] and SymSpin [2] have been developed. While there is considerable power in basic

Mathematica,SymManipulator lets the user handle abstract symmetrized tensor expressions, and

automatically decompose spinors into irreducible symmetric spinors. SymSpin allows the user to

handle complicated expressions with such spinors in an eﬃcient way.

The massive Dirac equation is, in spinor form,

∇AA0φA=mχA0,(1a)

∇AA0

χA0=−mφA,(1b)

where φAand χA0are spinor ﬁelds. The mass mis assumed to be nonzero. The ﬁrst result in this

article is that there are no nontrivial zeroth order symmetry operators. The second result is that

there exists a ﬁrst order symmetry operator if and only if there exist Killing spinors satisfying

auxiliary condition A.

Deﬁnition 1. Let SAA0,TA0B0,UAB , and RAA0be Killing spinors on a Lorentzian manifold.

They satisfy auxiliary condition A if

∇(AA0

SB)A0= 0,(2a)

∇A(A0S|A|B0)= 0,(2b)

∇AA0RAA0= 0,(2c)

∇AB0TA0B0+∇BA0UAB= 0.(2d)

The third result is that there exists a second order symmetry operator if and only if there

exist Killing spinors satisfying auxiliary condition B, which we will state later in deﬁnition 21

after some notation has been introduced.

For this article, we have used Mathematica version 13.1.0, xTensor version 1.2.0, Spinors

version 1.0.6, SymManipulator version 0.9.5, SymSpin version 0.1.1, and TexAct version 0.4.3. A

notebook used for creating all of the results presented in the following sections are available in a

GitHub repository [15].

1.1. Previous work

Michel, Radoux and ˇ

Silhan [18] analysed the symmetry operators for the conformal wave equation.

In [5] a method was developed to ﬁnd all second order symmetry operators for the conformal

wave equation, the Dirac–Weyl equation, and the Maxwell equation. Their results are also

formulated as existence of a set of Killing spinors satisfying a set of covariant diﬀerential equations.

We use the same method here. As we are dealing with a more complicated system of equations,

we will however take advantage of the recent development of the SymSpin package.

The conditions (2a) to (2d) for the existence of a ﬁrst order symmetry operator and the

form of that operator, presented in theorem 20, is a reformulation of a result by Kamran and

McLenaghan [17, theorem II] into covariant spinor language. Benn and Kress [10] have showed

Second order symmetry operators for the massive Dirac equation 3

that this result is the most general one of the ﬁrst order in the sense that it extends to arbitrary

spin manifolds.

A special case of the second order symmetry operator presented in this article has been

derived by Fels and Kamran [14, theorem 4.1].

Auxiliary condition A can be interpreted very geometrically. In section 3.2.1, we show that

(2d) implies the existence of a Killing–Yano tensor ﬁeld. If an operator commutes with the Dirac

operator, then it is a symmetry operator, and so the set of operators commutating with the

Dirac equation is a subset of the symmetry operators. Previous work has been able to relate

such operators to Killing–Yano tensors [11,12]. But also general symmetry operators have been

studied in terms of Killing–Yano tensors [9,1].

2. Preliminaries

In this section, the notation and concepts used in this article are presented. Abstract index

notation [19, chapter 2] is used throughout and conventions are consistent with Penrose and

Rindler [19,20]. Lowercase latin letters are used for Lorentzian tensor indices while uppercase

latin letters are used for spinorial tensor indices, with a prime to indicate indices in the conjugate

space.

2.1. Killing tensors

A Killing vector is a vector ﬁeld Kcsuch that taking the Lie derivative of the metric with respect

to it is zero, which can be written as ∇(aKb)= 0. The following deﬁnitions are then natural

generalizations,

Deﬁnition 2. A vector Kcis a conformal Killing vector if

∇(aKb)=λgab (3)

for some scalar ﬁeld λ.

Deﬁnition 3. A totally symmetric tensor Kb...q is a Killing tensor if

∇(aKb...q)= 0.(4)

Deﬁnition 4. A totally symmetric spinor SB0...Q0

B...Q is a Killing spinor if

∇(A0

(ASB0...Q0)

B...Q)= 0.(5)

Another type of geometrical quantitity of interest is Killing–Yano tensors. They are used to

construct valence 2 Killing tensors and sometimes they are easier to ﬁnd than the Killing tensors

they correspond to.

Deﬁnition 5. A totally antisymmetric tensor fb0...bnis a Killing–Yano tensor if

∇(afb0)b1...bn= 0.(6)

Lastly for this subsection, we will deﬁne the conformally weighted Lie derivative [4, (15)], [5,

(2.5)]. It will be used to interpret some of the terms in the symmetry operators.

Deﬁnition 6. If ξA0

Ais a Killing vector, and ϕA1...Akis a totally symmetric valence (k, 0) spinor,

then

LξϕA1...Ak=ξBB0∇BB0ϕA1...Ak+k

2ϕB(A2...Ak∇A1)B0ξBB0+2−k

8ϕA1...Ak∇BB0

ξBB0.(7)

If ϕis instead of valence (0, k), then ˆ

Lξϕis deﬁned as ˆ

Lξϕ.

Second order symmetry operators for the massive Dirac equation 4

2.2. Decomposing spinors

We formulated the Dirac equation in (1a) and (1b) using two-spinors. Two-spinors transform

under the universal covering group, SL(2,C), of the proper Lorentz group. Something that

greatly simpliﬁes discussions about two-spinors is that, when working over SL(2,C), the only

spinorial tensor that is antisymmetric in more than two indices is 0 and the only spinorial tensor

antisymmetric in two indices is the spin-metric AB and its multiples. From this follows a very

useful result, proved in Penrose and Rindler [19, proposition 3.3.54].

Theorem 7. Any spinor TA1...Ap

1...A0

qis the sum of T(A1...Ap)(A0

1...A0

q)and linear combinations

of outer products of symmetric spinors of lower valence with spin-metrics.

As an example of this theorem, the spinorial Riemann tensor, RAA0BB0CC0DD0, can be

decomposed as [21, (13.2.25)]

RAA0BB0CC0DD0= ΨABCDA0B0C0D0+ Λ (AC BD +BC AD )A0B0C0D0

+ ΦABC0D0A0B0CD + complex conjugate.(8)

ΨABCD =1

4RX0Y0

(ABCD)X0Y0is the Weyl spinor, Λ = 1

24 RY X0X Y 0

X X0Y Y 0is the Ricci scalar,

and ΦABC0D0=1

4RX0X

(AB)X0X(C0D0)is the Ricci spinor.

2.3. Index-free notation

Theorem 7allows us to decompose spinors into sums of outer products of symmetric spinors and

:s, but if an expression is symmetric in all of its free indices, then, after applying theorem 7,

every will have at least one index contracted. So the expression may be written only in terms of

partially contracted outer products of symmetric spinors. If two symmetric spinors are multiplied

and partially contracted, it does not matter which indices are contracted, only how many.

Hence the following deﬁnition.

Deﬁnition 8 ([3] deﬁnition 1).Let TA0

1...A0

A1...Akand SA0

1...A0

A1...Anbe totally symmetric

spinors. Then the symmetric pq-multiplication of TA0

1...A0

A1...Akwith SA0

1...A0

A1...Anis the totally

symmetrized outer product of Tand Swhere punprimed indices are contracted and qprimed

indices are contracted:

(Tp,q

S)A0

1...A0

l+n−2q

A1...Ak+m−2p

=TB1...Bp(A0

1...A0

l−q|B0

1...B0

(A1...Ak−pSA0

l−q+1...A0

l+n−2q)

Ak−p+1...Ak+m−2p)B1...BpB0

1...B0

q.(9)

With this operator, we don’t need to write out the indices in partially contracted outer

products of symmetric spinors. We will call this index-free notation.

2.4. Fundamental derivatives

Another application of theorem 7is to the covariant spinor derivative of a totally symmetric

spinor. Such an expression has four irreducible parts and we will name them as follows.

Deﬁnition 9 ([5] deﬁnition 13).Let Sk,l denote the space of smooth symmetric spinor ﬁelds

of valence (k, l) and let ψA1...Ak

1...A0

l∈ Sk,l. Then there are four fundamental derivatives: the

divergence D:Sk,l → Sk−1,l−1which acts by

(Dψ)A0

1...A0

l−1

A1...Ak−1=∇BB0

ψA0

1...A0

l−1

A1...Ak−1B B0for k≥1, l ≥1,(10a)

the curl C:Sk,l → Sk−1,l+1 which acts by

(Cψ)A0

1...A0

l−1

A1...Ak+1 =∇B0

(A1ψA0

1...A0

l−1

A2...Ak+1)B0for k≥0, l ≥1,(10b)

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SecondordersymmetryoperatorsforthemassiveDiracequationSimonJacobssonE-mail:simon.jacobsson@kuleuven.beChalmersUniversityofTechnology,SE-41296Gothenburg,SwedenKULeuven,BE-3000Leuven,BelgiumThomasBackdahlE-mail:thomas.backdahl@chalmers.seMathematicalSciences,ChalmersUniversityofTechnologyandUniversit...

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Second order symmetry operators for the massive Dirac equation

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