Covariant Differentiation of Spinors for a General Affine Connection

Home , Affine connection, Covariant derivative, Metric tensor, Nonmetricity tensor, Ricci calculus

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a general affine connection. An example of a physical theory with such a connection is the generalized Einstein–Maxwell theory with the electromagnetic field tensor represented by the second Ricci tensor (the homothetic curvature tensor) [12]. In this paper we present a derivation of the covariant derivative of a spinor for a general connection. We show how the projective invariance of the spinor connection allows to introduce gauge fields interacting with spinors in curved spacetime. We also derive the formula for the curvature spinor in the presence of a general connection.

2 Tetrads

In order to construct a generally covariant Dirac equation, we must regard the components of a spinor as invariant under coordinate transformations [13]. In addition to the coordinate systems, at each spacetime point we set up four orthogonal vectors (a tetrad) and the spinor gives a representation for the Lorentz transformations that rotate the tetrad [14,15]. Any vector V can be specified by its components V µ with respect to the coordinate system or by the coordinate-invariant projections V a a of the vector onto the tetrad field eµ: a a µ µ µ a V = eµV , V = ea V , (1) µ a where the tetrad field ea is inverse of eµ:

µ a µ µ b b ea eν = δν , ea eµ = δa. (2)

The metric tensor gµν of general relativity is related to the coordinate-invariant metric tensor of special relativity ηab = diag(1, 1, 1, 1) through the tetrad: − − − a b

gµν = eµeν ηab. (3) e Accordingly, the determinant g of the metric tensor gµν is related to the determinant of the tetrad a

eµ by e √ g = . (4) − µν We can use gµν and its inverse g to lower and raise coordinate-based indices, and ηab and its inverse ηab to lower and raise coordinate-invariant (Lorentz) indices. Eq. (3) imposes 10 constraints on the 16 components of the tetrad, leaving 6 components arbitrary. If µ µ we change from one tetrad ea to another,e ˜b , then the vectors of the new tetrad are linear combinations of the vectors of the old tetrad: µ b µ e˜a = Λ aeb . (5) µ Eq. (3) applied to the tetrad ﬁelde ˜b imposes on the matrix Λ the orthogonality condition:

c d Λ aΛ bηcd = ηab, (6) so Λ is a Lorentz matrix. Consequently, the Lorentz group can be regarded as the group of tetrad rotations in general relativity [4,14].

3 Spinors

Let γa be the coordinate-invariant Dirac matrices: γaγb + γbγa =2ηab. (7)

µ µ a Accordingly, the spacetime-dependent Dirac matrices, γ = ea γ , satisfy γµγν + γν γµ =2gµν . (8)

Let L be the spinor representation of a tetrad rotation (5):

a a b −1 γ˜ = Λ bLγ L . (9) 3

Since the Dirac matrices γa are constant in some chosen representation, the conditionγ ˜a = γa gives a the matrix L as a function of Λ b [14,15]. For inﬁnitesimal Lorentz transformations: a a a Λ b = δb + ǫ b, (10)

where the antisymmetry of the 6 infinitesimal Lorentz coefficients, ǫab = ǫba, follows from Eq. (6), the solution for L is: − 1 ab −1 1 ab L =1+ ǫabG ,L =1 ǫabG , (11) 2 − 2 where Gab are the generators of the spinor representation of the Lorentz group: 1 Gab = (γaγb γbγa). (12) 4 − A spinor ψ is defined to be a quantity that, under tetrad rotations, transforms according to [15]

ψ˜ = Lψ. (13)

An adjoint spinor ψ¯ is deﬁned to be a quantity that transforms according to

ψ˜¯ = ψL¯ −1. (14)

Consequently, the Dirac matrices γa can be regarded as quantities that have, in addition to the invariant index a, one spinor index and one adjoint-spinor index. The derivative of a spinor does not transform like a spinor since ψ˜,µ = Lψ,µ + L,µψ. (15)

If we introduce the spinor connection Γµ that transforms according to

−1 −1 Γ˜µ = LΓµL + L,µL , (16) then the covariant derivative of a spinor [14]:

ψ µ = ψ,µ Γµψ, (17) : − is a spinor: ψ˜:µ = Lψ:µ. (18) Similarly, one can show that the spinor-covariant derivative of the Dirac matrices γa is

a a γ = [Γµ,γ ] (19) :µ − µ µ −1 a sinceγ ˜ = Lγ L due to Eq. (9) and γ ,µ = 0.

4 Lorentz connection

µ Covariant differentiation of a contravariant vector V and a covariant vector Wµ in a relativistic theory ρ of gravitation introduces the affine connection Γµ ν : µ µ µ ρ ρ V = V + Γ V , Wµ ν = Wµ,ν Γ Wρ, (20) ;ν ,ν ρ ν ; − µ ν where the semicolon denotes the covariant derivative with respect to coordinate indices.1 The affine ρ ρ connection in general relativity is constrained to be symmetric, Γµ ν = Γν µ, and metric compatible, gµν;ρ = 0. For a general spacetime we do not impose these constraints. As a result, raising and lowering ρ of coordinate indices does not commute with covariant differentiation with respect to Γµ ν . Let us define: µ µ µ µ ρ ω aν = ea;ν = ea,ν + Γρ ν ea. (21)

1 µ µ The deﬁnitions (20) are related to one another via the condition (V Wµ);ν =(V Wµ),ν . 4

ab a bc ρ The quantities ω µ = eρη ω cµ transform like tensors under coordinate transformations. We can extend the notion of covariant diﬀerentiation to quantities with Lorentz coordinate-invariant indices ab by regarding ω µ as a connection [4,14]:

a a a b V |µ = V ,µ + ω bµV . (22)

a The covariant derivative of a scalar V Wa coincides with its ordinary derivative:

a a (V Wa)|µ = (V Wa),µ, (23) which gives b W = Wa,µ ω Wb. (24) a|µ − aµ We can also assume that the covariant derivative recognizes coordinate and spinor indices, acting on them like ; and :, respectively. Accordingly, the covariant| derivative of the Dirac matrices γa is [16]

a a b a γ = ω γ [Γµ,γ ]. (25) |µ bµ − The deﬁnition (21) can be written as [14]

eµ = eµ + Γ µ eρ ωb eµ =0. (26) a|ν a,ν ρ ν a − aν b Eq. (26) implies that the total covariant differentiation commutes with converting between coordi- ab nate and Lorentz indices. This equation also determines the Lorentz connection ω µ, also called the spin connection, in terms of the affine connection, tetrad and its derivatives. Conversely, the affine connection is determined by the Lorentz connection, tetrad and its derivatives [17]:

ρ ρ a ρ Γµ ν = ω µν + eµ,ν ea. (27)

ρ ρ The Cartan torsion tensor S µν = Γ[µ ν] is then

ρ ρ a ρ S µν = ω [µν] + e[µ,ν]ea, (28)

ν from which we obtain the torsion vector Sµ = S µν : ν a ν Sµ = ω [µν] + e[µ,ν]ea. (29)

5 Spinor connection

We now derive the spinor connection Γµ for a general aﬃne connection. Deviations of such a connection from the Levi-Civita metric-compatible connection are characterized by the nonmetricity tensor:

Nµνρ = gµν;ρ. (30) Eq. (30) yields: ab ab η = Nabρ, η = N , (31) ab|ρ |ρ − ρ from which it follows that 1 ω = Nabµ, (32) (ab)µ −2 i.e. the Lorentz connection is antisymmetric in first two indices only for a metric-compatible affine connection [4].2 In the presence of nonmetricity (lack of metric compatibility) the covariant derivative of the Dirac matrices deviates from zero. From Eqs. (7) and (31) it follows that [18] 1 γa = N a γb. (33) |µ −2 bµ 2 As a result, raising and lowering of Lorentz indices does not commute with covariant differentiation; it commutes only with ordinary differentiation. 5

Multiplying both sides of this equation by γa (from the left) yields

a b a 1 c ωabµγ γ γaΓµγ +4Γµ = N . (34) − −2 cµ We seek the solution of Eq. (34) in the form:

1 a b Γµ = ω γ γ Aµ, (35) −4 [ab]µ −

where Aµ is a spinor quantity with one vector index. Substituting Eq. (35) to (34) and using the a b c ab identity γcγ γ γ =4η give

a 1 c c γaAµγ +4Aµ = N + ω . (36) − 2 cµ cµ

The right-hand side of Eq. (36) vanishes because of Eq. (32) so Aµ is simply an arbitrary vector multiple of the unit matrix [14,19]. We can write Eq. (35) as

1 (A) a b Γµ = ω γ γ , (37) −4 abµ

(A) 3 where ωabµ is related to ω[ab]µ by a projective transformation:

(A) ωabµ = ω[ab]µ + ηabAµ. (38)

If we assume that Aµ represents some non-gravitational field then we can, for spinors in a purely 4 gravitational field, set Aµ = 0. Therefore the spinor connection Γµ for a general affine connection has the form of the Fock–Ivanenko coefficients of general relativity [9,11,14,18,21,22]:

1 a b Γµ = ω γ γ , (39) −4 [ab]µ with the antisymmetric part of the Lorentz connection.5 Using the deﬁnition (21), we can also write Eq. (39) as 1 ν c 1 ν Γµ = e [γν ,γ ]= [γ ,γν ]. (40) −8 c;µ 8 ;µ

6 Curvature spinor

The commutator of the covariant derivatives of a vector with respect to the aﬃne connection deﬁnes the curvature tensor Rρ = Γ ρ Γ ρ + Γ κ Γ ρ Γ κ Γ ρ :6 σµν σ ν,µ − σ µ,ν σ ν κ µ − σ µ κ ν ρ ρ ρ σ σ ρ σ σ V V = R V +2S V , Vρ νµ Vρ µν = R Vσ +2S Vρ σ . (41) ;νµ − ;µν σµν µν ;σ ; − ; − ρµν µν ; In analogous fashion, the commutator of the total covariant derivatives of a spinor:

ρ ψ ψ = Kµν ψ +2S ψ , (42) |νµ − |µν µν |ρ 3 The transformation (38) is related, because of Eq. (27), to a projective transformation of the affine ρ ρ ρ connection: Γµν → Γµν + δµAν [20]. 4 The invariance of Eq. (34) under the addition of a vector multiple Aµ of the unit matrix to the spinor connection allows to introduce gauge fields interacting with spinors [19]. Eq. (17) can be written as ψ:µ = 1 a b (∂µ + Aµ)ψ + 4 ω[ab]µγ γ ψ, from which it follows that if the vector Aµ is imaginary then we can treat it as a gauge field. 5 The Fock–Ivanenko coefficients (39) can also be written in terms of the generators of the spinor represen- − 1 ab tation of the Lorentz group (12): Γµ = 2 ωabµG . 6 µ The curvature tensor can also be defined through a parallel displacement δV = (V,µ − V;µ)dx of a µν ρ − 1 ρ σ µν vector V along the boundary of an infinitesimal surface element ∆f [23]: H δV = 2 R σµν V ∆f and 1 σ µν H δVρ = 2 R ρµν Vσ∆f . These equations are related to one another since the parallel displacement of a scalar along a closed curve vanishes. 6

deﬁnes the antisymmetric curvature spinor Kµν [14]:

Kµν = Γµ,ν Γν,µ + [Γµ, Γν ]. (43) − a From Eq. (16) it follows that Kµν transforms under tetrad rotations like the Dirac matrices γ (with one spinor index and one adjoint-spinor index):

−1 K˜µν = LKµν L . (44)

Eq. (33) is equivalent to 1 γρ = N ρ γσ. (45) |µ −2 σµ The commutator of the covariant derivatives of the spacetime-dependent Dirac matrices with respect to the aﬃne connection is then: 2γρ = (N ρ γσ) . (46) |[νµ] − σ[ν |µ] Multiplying both sides of this equation by γρ (from the left) and using

ρ ρ σ σ ρ ρ 2γ |[νµ] = R σµν γ +2S µν γ |σ + [Kµν ,γ ] (47) and Eq. (8) yield

ρ σ σ ρ ρ ρ σ Rρσµν γ γ S N + γρKµν γ 4Kµν = γρ(N γ ) − µν ρσ − − σ[ν |µ] ρ 1 λρ σ ρ σ ρ 1 ρλ σ = N + N N γλγ = N S N + N N γλγ . (48) − ρ[ν;µ] 2 ρσ[ν µ] − ρ[ν,µ] − µν ρσ 2 ρσ[ν µ] We seek the solution of Eq. (48) in the form:

1 ρ σ 1 ρλ σ Kµν = R γ γ N N γλγ + Bµν , (49) 4 [ρσ]µν − 8 ρσ[ν µ] where Bµν is a spinor quantity with two vector indices. Substituting Eq. (49) to (48) gives

ρ ρ γρBµν γ 4Bµν = Qµν N , (50) − − − ρ[ν,µ] where ρσ ρ ρ Qµν = Rρσµν g = Γ Γ (51) ρ ν,µ − ρ µ,ν is the second Ricci tensor, also called the tensor of homothetic curvature [12] or the segmental curvature tensor [18]. The right-hand side of Eq. (50) vanishes because of Eq. (30) so Bµν is simply an antisymmetric-tensor multiple of the unit matrix. The tensor Bµν is related to the vector Aµ in Eq. (35) by7 Bµν = Aν,µ Aµ,ν + [Aµ, Aν ]. (52) − 8 Setting Aµ = 0, which corresponds to the absence of non-gravitational ﬁelds, yields Bµν = 0. Therefore the curvature spinor for a general aﬃne connection is:9

1 ρ σ 1 ρσ λ Kµν = R γ γ N N γ γσ. (53) 4 [ρσ]µν − 8 ρλ[µ ν]

7 If ψ is a pure spinor, i.e. has no non-spinor indices, then Aµ is a pure vector, like the electromagnetic potential [19], and [Aµ,Aν ] = 0. If ψ has non-spinor indices corresponding to some symmetries, e.g., the electron–neutrino symmetry, then Aµ is also assigned these indices and [Aµ,Aν ] can be different from zero. 8 Eq. (52) resembles the definition of a field strength in terms of a field potential for a non-commutative gauge field. The invariance of Eq. (48) under the addition of an antisymmetric-tensor multiple Bµν of the unit matrix to the curvature spinor is related to the invariance of Eq. (34) under the addition of a vector multiple Aµ of the unit matrix to the spinor connection. 9 The curvature spinor (53) can also be written in terms of the generators of the spinor representation of 1 ρσ − 1 ρσ λ the Lorentz group (12): Kµν = 2 Rρσµν G 4 NρλµN ν G σ. 7

7 Curvature and Ricci tensors

Finally, we examine the curvature tensor for a general aﬃne connection. The commutator of the covariant derivatives of a tetrad with respect to the aﬃne connection is

ρ ρ σ σ ρ 2ea;[νµ] = R σµν ea +2S µν ea;σ. (54)

This commutator can also be expressed in terms of the Lorentz connection:

ρ ρ ρ b ρb b ρ ρb b ρ σ ρ ea;[νµ] = ω a[ν;µ] = (eb ω a[ν );µ] = ωba[ν ω µ] + ω a[ν;µ]eb = ωba[ν ω µ] + ω a[ν,µ]eb + S µν ω aσ. (55)

Consequently, the curvature tensor with two Lorentz and two coordinate indices depends only on the Lorentz connection and its ﬁrst derivatives [10,14,21]:

Rab = ωab ωab + ωa ωcb ωa ωcb . (56) µν ν,µ − µ,ν cµ ν − cν µ The contraction of the tensor (56) with a tetrad leads to two Lorentz-connection Ricci tensors:

a [ab] ν Rµ = R µν eb , (57) a (ab) ν Pµ = R µν eb , (58)

ρ that are related to the sum and diﬀerence, respectively, of the Ricci tensor, Rµν = R µρν , and the ρσ a tensor Cµν = Rµρνσ g . The contraction of the tensor Rµ with a tetrad gives the Ricci scalar,

a µ ab µ ν R = Rµea = R µν ea eb , (59)

a while the contraction of Pµ gives zero. The contraction of the curvature tensor (56) with the Lorentz 10 metric tensor ηab gives the second Ricci tensor (51),

c c Qµν = ω ω . (60) cν,µ − cµ,ν Under the projective transformation (38) the tensor (56) changes according to

ab ab ab R R + η Bµν , (61) µν → µν

(ab) a where Bµν is given by Eq. (52), and so does R µν . Consequently, the tensor Pµ changes according to a a aν P P + e Bµν . (62) µ → µ [ab] a 11 The tensor R µν is projectively invariant and so are Rµ and R. The second Ricci tensor changes according to

Qµν Qµν +8A , (63) → [ν,µ] so it is invariant only under special projective transformations [20] with Aµ = λ,µ, where λ is a scalar. If [Aµ, Aν ] = 0 then Eqs. (62) and (63) yield the projective invariance of the tensor:

a a aν I =4P Qµν e . (64) µ µ − 10 If the Lorentz connection is antisymmetric, the curvature tensor (56) is antisymmetric in the Lorentz a indices and the tensors Pµ and Qµν vanish.

11 The Einstein–Cartan–Kibble–Sciama formulation of gravitation [4,10], where the tetrad and Lorentz con- e nection are dynamical variables, is based on the Lagrangian density Ä = R. 8

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