A space-time calculus based on symmetric 2-spinors_2

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arXiv:2210.02423v1 [gr-qc] 5 Oct 2022

A SPACE-TIME CALCULUS BASED ON SYMMETRIC 2-SPINORS

STEFFEN AKSTEINER AND THOMAS B ¨

ACKDAHL

Abstract. In this paper we present a space-time calculus for symmetric spinors, including a

product with a number of index contractions followed by symmetrization. As all operations

stay within the class of symmetric spinors, no involved index manipulations are needed. In fact

spinor indices are not needed in the formalism. It is also general because any covariant tensor

expression in a 4-dimensional Lorentzian spacetime can be translated to this formalism. The

computer algebra implementation SymSpin as part of xAct for Mathematica is also presented.

1. Introduction

When working with tensorial expressions, one usually encounters diﬃculties handling index

manipulations due to complicated symmetries. Techniques including group theoretical calcula-

tions and Young tableaux have been introduced to try to tackle these problems. However, their

complexity grows quickly with the size of the problem. The purpose of this paper is to present

a formalism based on 2-spinors that aims to simplify the situation by utilizing the symmetry

properties of irreducible spinors.

Let (M, gab) be a 4-dimensional manifold with metric gab of Lorentzian signature and admitting

a spin structure with spin metric ǫAB . It is well known that any tensor ﬁeld on Mcan be expressed

in terms of 2-spinors, which in turn can be decomposed into irreducible symmetric spinors [9,

Prop 3.3.54]. For instance a valence (3,0) spinor can be decomposed as

TABC =T(ABC)+1

3TDD(BǫC)A−1

3ǫA(BTDC)D−1

2TADDǫBC .(1)

Therefore, it is suﬃcient to work with with symmetric spinors. To fully establish this perspective,

a symmetric product for symmetric spinors with a number of contractions is needed. It is the

intention of this work to introduce the corresponding algebra and to derive its basic properties.

In particular, with these operations we stay within the algebra of symmetric spinors. This oﬀers

great simpliﬁcations, and speeds up the calculations. Furtheremore, no relevant information is

left in the indices, and we therefore get an index-free compact formalism.

We have previously described the decomposition of the covariant derivative [4], leading to four

fundamental spinor operators, which can be viewed as a special case. Also, the symmetric product

is a generalization of some special operators, like the Kioperators deﬁned in [1, Deﬁnition II.4].

Therefore, all properties of such operators can easily be derived from the corresponding properties

of the symmetric product described in this paper.

The formalism has many potential applications, see [2],[3]. As a simple example, consider a

condition of the form

0 = KAB F H LFCϕHC +M(ACϕB)C,(2)

for symmetric spinors K, L, M, ϕ. For arbitrary ϕa systematic computation, using the techniques

of this paper, shows that the conditions on K, L, M are of the form

KG(ABC L|G|F)= 0, MAB =1

2KCF AB LCF ,(3)

see Section 3.2 for details. The same techniques have been used in [7] to derive conditions on the

spacetime for the existence of second order symmetry operators for the massive Dirac equation.

The formalism is implemented in the SymSpin [5] package for xAct [8] for Mathematica.

In Section 2we introduce the symmetric product and state basic properties in Theorem 3.

The expansion of a product into symmetric products is discussed in Lemma 6. The irreducible

parts of the Levi-Civita connection, its commutators, curvature and Leibniz rules are discussed

in Section 2.4. A concise form the the dyad components of such symmetric spinors is given

2 S. AKSTEINER AND T. B ¨

ACKDAHL

in Section 2.5. The computer algebra implementation is discussed in Section 3and Section 4

contains some conclusions.

2. Symmetric spinor algebra

Let Sk,l be the space of symmetric valence (k, l) spinors. In abstract index notation, elements

are of the form φA1...AkA′

1...A′

l∈ Sk,l. Sometimes it is convenient to suppress the valence and/or

indices and we write e.g. φ∈ S or φ∈ Sk,l.

2.1. Symmetric product. Given two symmetric spinors, we introduce a product which involves

a given number of contractions and symmetrization afterwards.

Deﬁnition 1. Let k, l, n, m, i, j be integers with i≤min(k, n)and j≤min(l, m). The symmetric

product is a bilinear form

i,j

⊙:Sk,l × Sn,m → Sk+n−2i,l+m−2j.(4)

For φ∈ Sk,l, ψ ∈ Sn,m, it is given by

(φi,j

⊙ψ)A′

1...A′

l+m−2j

A1...Ak+n−2i=φ(A′

1...A′

l−j−1|B1...BiB′

1...B′

(A1...Ak−i−1ψA′

l−j...A′

l+m−2j)

Ak−i...Ak+n−2i)B1...BiB′

1...B′

j(5)

For many commutator relations we will need the following coeﬃcients.

Deﬁnition 2. Deﬁne the associativity coeﬃcients

Ft,m,M

i,r,k =

p=0

M−p

q=0

(−1)t−p+qk−m

p m

M−p−qk−m−p

qr−m

t−p i−t

M−p−qt−p

q

i+k−M−p+1

M−pM−p

qk−2m+r

t.(6)

Observe that the limits can be restricted to max(0, m −r+t)≤p≤min(k−m, M, t) and

max(0, M −m−p, M −i−p+t)≤q≤min(k−m−p, M −p, t −p) because the terms are zero

outside this range.

For multiple products we will use the convention ωm,n

⊙ϕt,u

⊙φ=ωm,n

⊙(ϕt,u

⊙φ).

Theorem 3. Let φ∈ Si,j , ω ∈ Sr,s, ϕ ∈ Sk,l. The symmetric product ⊙of Deﬁnition 1has the

following properties:

(1) It is graded anti-commutative:

φm,n

⊙ω= (−1)m+nωm,n

⊙φ(7a)

(2) It is non-associative:

(ωm,n

⊙ϕ)t,u

⊙φ=

min(i,k)

M=0

min(j,l)

N=0

(−1)t+u+M+NFt,m,M

i,r,k Fu,n,N

j,s,l ωt+m−M,u+n−N

⊙ϕM,N

⊙φ. (7b)

(3) It is Hermitian:

φm,n

⊙ω=φn,m

⊙ω(7c)

Combining the ﬁrst two points, we get the following useful relation.

Corollary 4.

φt,u

⊙ωm,n

⊙ϕ=

min(i,k)

M=0

min(j,l)

N=0

Ft,m,M

i,r,k Fu,n,N

j,s,l ωt+m−M,u+n−N

⊙φM,N

⊙ϕ(8)

A SPACE-TIME CALCULUS BASED ON SYMMETRIC 2-SPINORS 3

2.2. Irreducible decomposition. A key property of the symmetric product is that the product

of two symmetric spinors can always be decomposed in terms of symmetric products and spin

metrics.

Deﬁnition 5. We will use the following notation for for products of spin metrics.

ǫB1...Bp

A1...Ap=ǫA1

B1...ǫAp

Bp,(9a)

¯ǫB′

1...B′

A′

1...A′

q= ¯ǫA′

B′

1...¯ǫA′

B′

q.(9b)

Lemma 6. For φ∈ Si,j , ϕ ∈ Sk,l with punprimed and qprimed contractions, we have the

irreducible decomposition

φC1...CpC′

1...C′

A1...Ai−pA′

1...A′

j−qϕB1...Bk−pB′

1...B′

l−q

C1...CpC′

1...C′

= (−1)p+q

min(i,k)

m=p

min(j,l)

n=q(−1)m+ni−p

m−pk−p

m−pj−q

n−ql−q

n−q

i+k−m−p+1

m−pj+l−n−q+1

n−q

×ǫ(B1...Bm−p

(A1...Am−p(φm,n

⊙ϕ)Bm−p+1...Bk−p)(B′

n−q+1...B′

l−q

Am−p+1...Ai−p)(A′

n−p+1...A′

j−q¯ǫB′

1...B′

n−q)

A′

1...A′

n−q).(10)

Proof. Let φand ϕbe symmetric of valence (i, 0) and (k, 0) respectively. By [9, Prop 3.3.54] the

irreducible decomposition of the product must have the following form

φA1...AiϕB1...Bk=

min(i,k)

m=0

cmǫ(B1...Bm

(A1...Am(φm,0

⊙ϕ)Bm+1...Bk)

Am+1...Ai)(11)

Taking a trace of the summand, we ﬁnd by partial expansions of the symmetrizations that

ǫ(B1...Bm

(A1...Am(φm,0

⊙ϕ)Bm+1...Bk−1Ai)

Am+1...Ai)

iǫ(B1...Bm

Ai(A1...Am−1(φm,0

⊙ϕ)Bm+1...Bk−1Ai)

Am...Ai−1)+i−m

iǫ(B1...Bm

(A1...Am(φm,0

⊙ϕ)Bm+1...Bk−1Ai)

Am+1...Ai−1)Ai

ik ǫAi(B1...Bm−1

Ai(A1...Am−1(φm,0

⊙ϕ)Bm...Bk−1)

Am...Ai−1)+m(m−1)

ik ǫ(B1|Ai|...Bm−1

Ai(A1...Am−1(φm,0

⊙ϕ)Bm...Bk−1)

Am...Ai+1)

+m(k−m)

ik ǫ(B1...Bm

Ai(A1...Am−1(φm,0

⊙ϕ)Bm+1...Bk−1)Ai

Am...Ai−1)+(i−m)m

ik ǫAi(B1...Bm−1

(A1...Am(φm,0

⊙ϕ)Bm...Bk−1)

Am+1...Ai−1)Ai

=m(i+k−m+1)

ik ǫ(B1...Bm−1

(A1...Am−1(φm,0

⊙ϕ)Bm...Bk−1)

Am...Ai−1).(12)

Recursively for p≤min(i, k) traces we get

ǫ(B1...Bm

(A1...Am(φm,0

⊙ϕ)Bm+1...Bk−pAi−p+1...Ai)

Am+1...Ai)

=m(i+k−m+1)

ik ǫ(B1...Bm−1

(A1...Am−1(φm,0

⊙ϕ)Bm...Bk−pAi−p+1...Ai−1)

Am...Ai−1)

=m(i+k−m+1)

(m−1)(i+k−m)

(i−1)(k−1) ǫ(B1...Bm−2

(A1...Am−2(φm,0

⊙ϕ)Bm−1...Bk−pAi−p+1...Ai−2)

Am−1...Ai−2)

=ǫ(B1...Bm−p

(A1...Am−p(φm,0

⊙ϕ)Bm−p+1...Bk−p)

Am−p+1...Ai−p)

p−1

q=0

(m−q)(i+k−m+1−q)

(i−q)(k−q).

=ǫ(B1...Bm−p

(A1...Am−p(φm,0

⊙ϕ)Bm−p+1...Bk−p)

Am−p+1...Ai−p)1+i+k−m

pm

p

i

pk

p(13)

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摘要：

arXiv:2210.02423v1[gr-qc]5Oct2022ASPACE-TIMECALCULUSBASEDONSYMMETRIC2-SPINORSSTEFFENAKSTEINERANDTHOMASB¨ACKDAHLAbstract.Inthispaperwepresentaspace-timecalculusforsymmetricspinors,includingaproductwithanumberofindexcontractionsfollowedbysymmetrization.Asalloperationsstaywithintheclassofsymmetricspino...

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A space-time calculus based on symmetric 2-spinors_2

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