A space-time calculus based on symmetric 2-spinors

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.


Introduction
When working with tensorial expressions, one usually encounters difficulties handling index manipulations due to complicated symmetries.Techniques including group theoretical calculations 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, g ab ) be a 4-dimensional manifold with metric g ab of Lorentzian signature and admitting a spin structure with spin metric ǫ AB .It is well known that any tensor field on M can 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 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 K i operators 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 for symmetric spinors K, L, M, ϕ.For arbitrary ϕ a systematic computation, using the techniques of this paper, shows that the conditions on K, L, M are of the form 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 2 we 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 in Section 2.5.The computer algebra implementation is discussed in Section 3 and Section 4 contains some conclusions.

Symmetric spinor algebra
Let S k,l be the space of symmetric valence (k, l) spinors.In abstract index notation, elements are of the form φ A1...A k A ′ 1 ...A ′ l ∈ S k,l .Sometimes it is convenient to suppress the valence and/or indices and we write e.g.φ ∈ S or φ ∈ S k,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 For φ ∈ S k,l , ψ ∈ S n,m , it is given by For many commutator relations we will need the following coefficients.
Definition 2. Define the associativity coefficients 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 ω Theorem 3. Let φ ∈ S i,j , ω ∈ S r,s , ϕ ∈ S k,l .The symmetric product ⊙ of Definition 1 has the following properties: (1) It is graded anti-commutative: (3) It is Hermitian: Combining the first two points, we get the following useful relation.

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.
Lemma 6.For φ ∈ S i,j , ϕ ∈ S k,l with p unprimed and q primed contractions, we have the irreducible decomposition Proof.Let φ and ϕ be 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 Taking a trace of the summand, we find by partial expansions of the symmetrizations that Recursively for p ≤ min(i, k) traces we get By complex conjugation we get the corresponding decomposition for the primed indices.

Proof of Theorem 3.
To proof the main theorem and in particular (7b), we need the following intermediate identities.We restrict to unprimed indices, as the effect of primed indices can be superimposed.We begin with a partial expansion of symmetrization of B indices.
Proposition 7. Let ω ∈ S r,0 , ϕ ∈ S k,0 .We have the partial expansion The sum can be limited to the range max(0, t + m − r) ≤ p ≤ min(t, k − m).

Proof. Partial expansion of the symmetry for the indices
which can be simplified to (17).
We aso need to make an irreducible decomposition of a product of two spinors with some contractions and symmetrizations.
Proof.Let ≏ mean equal after lowering the A indices, raising the B indices and symmetrizing over the A and B index sets separately.Using Lemma 6, performing a partial expansion of the symmetries and noticing that ǫ Repeatedly expanding, we find After rearranging the indices, and simplifying, we get (19).
Proof of Theorem 3. Part 1 follows from the zee-zaw rule on the m + n contracted indices and part 3 follows from complex conjugation of (5).Part 2 follows from the following argument.Proposition 7, a renaming of the contracted indices and using the zee-zaw rule gives Using Proposition 8, contracting the spin metrics, and using the zee-zaw rule, we get Hence (φ 2.4.Derivatives.In [4], the irreducible decomposition of the covariant derivative of a symmetric spinor was done in terms of fundamental spinor operators.By extending the symmetric product to the space of linear, symmetric differential operators of valence (k, l), O k,l , we can express the fundamental spinor operators in a compact way.
Remark.For ∇ ∈ O 1,1 we have the fundamental spinor operators [4, Definition 13] On ϕ ∈ S k,l we have the irreducible decomposition of the covariant derivative into fundamental operators [4, Lemma 15], Next, we write the commutators in the new notation.Define the operator = −(∇ and its complex conjugate ∈ O 0,2 . In index notation, it reads AB = ∇ (A|A ′ | ∇ B) A ′ .Acting on ϕ ∈ S k,l it can be expressed in terms of curvature via 0,0 2,0 Lemma 10.For symmetric spinors φ ∈ S i,j , ϕ ∈ S k,l we have the following Leibniz rules.

GHP expansion.
In this section we collect equations to efficiently expand symmetric spinorial equations into GHP components.Let us first briefly review the formalism, see [6] for details.Introducing a normalized spinor dyad (o A , ι A ), o A ι A = 1, a two dimensional subgroup of the Lorentz group is given by with non-vanishing, complex scalar field λ.A field φ is said to be of GHP weight {p, q} if it transforms via φ → λ p λq φ (34) under (33) and its complex conjugate.The Levi-Civita connection has a natural lift of the form and is of weight zero in the sense that it maps {p, q} weighted fields to {p, q} fields.The GHP operators are given by the dyad expansion of (35), The connection coefficients are defined as follows, To express the dyad expansion of a general symmetric spinor, it is convenient to define a symmetric spinor basis B n,k m,l of weight {2n − k, 2m − l} by In particular this allows us to mostly avoid spinor indices for the rest of this section.For a full contraction of two basis elements we find where δ a b = 1 if a = b and zero otherwise.Now any φ ∈ S k,l can be expanded into where the scalar components of weight {k − 2i, l − 2j} are defined by The following two lemmas yield component expressions for general symmetric products and derivatives of symmetric spinors.This allows to expand general symmetric spinor differential equations into dyad components, without expanding the symmetrizations.
Lemma 11.For φ ∈ S i,j , ϕ ∈ S k,l the symmetric product has components with coefficients given by Proof.For ease of notation we assume φ ∈ S i,0 , ϕ ∈ S k,0 .Using the observation that B p,k 0,0 = B 0,k−p 0,0 0,0 ⊙ B p,p 0,0 , where B 0,k−p 0,0 is a symmetric product of ι A and B p,p 0,0 is a symmetric product of o A , we can use (17) to obtain Using this in the expansion (40), we find Contracting the B indices, symmetrizing and using (39) yield The relation (39) then gives The primed indices gives an analogous expansion and the combination yields (42).
Lemma 12.The GHP components of fundamental spinor operators (26) on φ ∈ S k,l take the form Proof.To prove (48a), we start by expanding the argument of Dφ using (40) and contract with a symmetric basis as in (41), Next, we use the Leibniz rule (31d), but switch to the GHP connection Θ AA ′ (so the fundamental spinor operators are with respect to Θ AA ′ instead of ∇ AA ′ ) as the GHP components and the basis elements are GHP weighted, From ( 36) and (37) we have and Inserting ( 51), (52) back into (50) and expanding Γ, Γ ′ into the basis we can use the contraction rules which are easily verified by expanding out the symmetries.The result can now be substituted into (49).Each term has a full contraction of the form (39) which cancels the double sum due to the δ factors.After some elementary algebra, the end result is given by (48a).The other expansions can be verified along the same lines, the only minor computation that needs to be done is the analog of ( 52) and (53).

SymSpin: A computer algebra implementation in xAct
The xAct [8] suite for Mathematica is an open source project mainly devoted to symbolic computation in differential geometry and tensor algebra.In this section we introduce our contributed package SymSpin [5] which contains the formalism of Section 2. For syntax and more examples, see SymSpinDoc.nb on that page.(56) 3.2.Example: Coefficients.Assume that K, L and M are symmetric spinor fields, and we want to find under which conditions of K, L and M the equation holds for all symmetric spinor fields ϕ.The following calculation leads to the conditions We first define the symmetric spinor fields.For clarity we have added the valence numbers to the names of the spinors, but not the display form.
To convert this to the new formalism, we need the irreducible decomposition of the product of the L and φ spinor.
It is convenient to work with the expanded and canonicalized version To work efficiently we turn the original equation into an index-free version.One could also use the index-free version as a starting point.
In := IndexFreeEq=ToIndexFree[ToCanonical@ContractMetric[OriginalEq /.EqToRule@L20ϕ20IrrDecEq]//.SymHToSymMultRule] /.MultScalToSymMultRule[Spin]/.SortSymMult[Not@FreeQ[#,ϕ20]&] We can turn the spinor valued equation into a scalar equation by contracting it with a dummy spinor T to turn the free indices into contracted dummy indices.This dummy spinor is defined by In := DefSymmetricSpinor[T20,2,0,Spin,"T"] (64) As the field ϕ and the dummy spinor T both should be arbitrary, we see that the irreducible components of their product can be treated as independent arbitrary fields.For convenience we make a list of them with We can now contract our index-free equation with T .
In := SymMult[T20,2,0]/@IndexFreeEq Commute T inside, so that T is directly contracted with the field ϕ, so we obtain the independent spinors in the list IrrDecComps.
From this one can conclude that the coefficients of (T 1,0 ⊙ ϕ) and (T 0,0 ⊙ ϕ) both have to be zero.As a convenience, we have implemented all of the steps from the index-free equation to the final list of equations in one function.
In := ExtractCoeffsIndexFree[IndexFreeEq,ϕ20] This can be translated back to the indexed form with In := ToIndexed/@% Out = {0 == Sym Performing this kind of calculation in the indexed form would require expansions of symmetries and several steps of irreducible decompositions of different products.This new method was heavily used in [7].