Covariant derivative of a spinorin a metric-affine space

Lodovico Scarpa 1 and Hasan Sayginel 2

Under the supervision ofDr. Christian G. Bohmer

1lodovico.scarpa@wolfson.ox.ac.uk2hasan.sayginel@exeter.ox.ac.uk

Contents

1 Introduction 21.1 Why study spinors? . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 An intuitive understanding of spinors . . . . . . . . . . . . . . . . 21.3 Aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Index conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Tensor transformations and tensor densities 42.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Tensor densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Mixing and alternation . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Parallel transport and covariant derivative 6

4 Affine connection, torsion and non-metricity 84.1 The connection in general relativity . . . . . . . . . . . . . . . . 84.2 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Non-metricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 General form of the connection . . . . . . . . . . . . . . . . . . . 9

5 Lorentz group and transformations 105.1 Definition of the Lorentz group . . . . . . . . . . . . . . . . . . . 105.2 Infinitesimal Lorentz transformations . . . . . . . . . . . . . . . . 10

6 Spinors and spin space 116.1 Definition of 2-spinors . . . . . . . . . . . . . . . . . . . . . . . . 116.2 Spin-tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.3 Correspondence between R3 and spin space . . . . . . . . . . . . 13

6.3.1 Rotation about the z axis . . . . . . . . . . . . . . . . . . 136.3.2 Rotation about the x axis . . . . . . . . . . . . . . . . . . 146.3.3 Rotation about the y axis . . . . . . . . . . . . . . . . . . 15

6.4 Remark on spinor rotations . . . . . . . . . . . . . . . . . . . . . 176.5 The g-spin tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 176.6 Infinitesimal transformations of 2-spinors . . . . . . . . . . . . . 176.7 Definition of 4-spinors . . . . . . . . . . . . . . . . . . . . . . . . 186.8 Gamma Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.9 Infinitesimal transformation of 4-spinors . . . . . . . . . . . . . . 19

7 Tetrads 21

8 The covariant derivative of a spinor and the Dirac equation 238.1 Covariant derivative of a spinor . . . . . . . . . . . . . . . . . . . 238.2 The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1

1 Introduction

1.1 Why study spinors?

The concepts of scalars and vectors are introduced very early on in the studyof physics and maths; most secondary school students are familiar with them.If they move on to undertake a degree in physical sciences, they will encountertensors too. These objects are fundamental tools in understanding our universeand expressing it mathematically. Spinors, on the other hand, are reserved foradvanced level study, despite the fact that they are used to describe the mostfundamental objects of the universe.

A person with a physics background first encounters spinors when they startstudying relativistic quantum mechanics and in particular deal with the Diracrelativistic electron equation, given by:

i~ γµ ∂µ Ψ(x, t) = mcΨ(x, t) (1)

This elegant equation, claimed by many to be among the most beautiful equa-tions of physics, describes the behaviour of fermions, i.e. spin-1/2 particles. Ex-amples of fermionic particles are electrons, protons and neutrons; these particlesmake up the atoms and hence all matter in the universe. These fundamentalparticles cannot be described using typical objects like scalars, vectors or tensors;instead, spinors are required. This makes the study of spinors essential and afundamental aspect of particle physics. These objects are of interest to math-ematicians as well because they are associated with the Lorentz group andtransformations, thus are useful in differential geometry (further explored insection 6.3).

1.2 An intuitive understanding of spinors

Roughly speaking spinors can be thought of as the square root of a vector. Theyare either two component or four component vector-like objects that transformin a particular way under rotations. In fact, a spinor needs to be rotated by 720◦

to return to its original position, unlike a vector which obviously requires ‘only’360◦. To visualise how a spinor rotates, a helpful analogy is the movement of thedancers’ arms in Pandanggo sa ilaw, a Filipino candle dance; see [1, min. 1:43].When the dancer’s arm is turned once by 360◦ it is twisted, but another rotationof the same angle returns the arm to the initial state.

1.3 Aims and objectives

The aim of this text is to consider spinors on a metric-affine space, i.e. a spacewith torsion and non-metricity, and formulate an expression for the covariantderivative of a spinor on such a space.To do so, we first introduce the basic notions of tensors and parallel transport.

2

We then derive an expression for the connection in a metric-affine space. Sub-sequently, we explore the transformation properties of spinors by reviewing theproperties of the Lorentz group and exploring the correspondence between R4

and the spin space P1 in which spinors live. Finally, we utilise the concept ofparallel transport to define the covariant derivative of a spinor in Minkowskispace and then generalise this expression to curved space using the tetrad field.

1.4 Index conventions

In this text we will use lower-case Greek letters α, β, γ, . . . to label anholonomiccoordinates, i.e. coordinates in flat space. Instead, we will use lower-case Latinletters i, j, k, . . . for general labelling as well as to label holonomic coordinates,that is, coordinates in curved space.

Upper-case Latin letters from the beginning of the alphabet, A,B,C, . . ., willbe reserved for 2-spinors, and thus will range from 1 to 2. Similarly, lower-caseLatin letters from the beginning of the alphabet, a, b, c, . . ., will be used for 4-spinors, hence ranging from 1 to 4.

The Einstein summation convention is assumed throughout the text unless oth-erwise stated.

Finally, we will be working in a natural system of units so that c = G = ~ = 1.

3

2 Tensor transformations and tensor densities

2.1 Tensors

The general transformation law for a tensor of arbitrary rank (p, q) is:

T ′ij...kl... =∂x′i

∂xm∂x′j

∂xn. . .

∂xp

∂x′k∂xq

∂x′l. . . Tmn...pq... (2)

Note that a tensor of rank (p, q) has p contravariant and q covariant indices. Thepartial derivatives that appear in the transformation law give the componentsof the Jacobian matrix that is defined as follows:

Aij =∂x′i

∂xj(3)

The Jacobian matrix plays a role in determining how the volume element in a4-dimensional spacetime transforms. This transformation is determined by thedeterminant of the matrix as follows:

d4x′ =

∣∣∣∣∣∂x′i∂xj

∣∣∣∣∣ d4x (4)

2.2 Tensor densities

From the transformation of an infinitesimal volume element, we may definea scalar density such that its product with the volume element is invariantunder coordinate transformations, s′ d4x′ = s d4x 3. Therefore, a scalar densityis given by:

s′ =

∣∣∣∣∣ ∂xi∂x′j

∣∣∣∣∣ s (5)

The definition can be extended to define a tensor density which is a productof a tensor and a scalar density and transforms as:

T′ij...kl... =

∣∣∣∣∣ ∂xi∂x′k

∣∣∣∣∣ ∂x′i∂xm∂x′j

∂xn. . .

∂xp

∂x′k∂xq

∂x′lTmn...pq... (6)

A further extension can be made to define a weight. Densities could be of weightw.

T′ij...kl... =

∣∣∣∣∣ ∂xi∂x′k

∣∣∣∣∣w∂x′i

∂xm∂x′j

∂xn. . .

∂xp

∂x′k∂xq

∂x′lTmn...pq... (7)

Equation (7) is the definition of a relative tensor. An absolute tensor or just‘tensor’ given by equation (2) is a tensor density/relative tensor of weight 0.

3To denote densities we use a Gothic kernel with a tilde on top; Gothic kernels are writtenin LATEX using the command mathfrak, part of the amsfonts package. Note that T is a GothicT .

4

2.3 Mixing and alternation

Finally, we introduce two processes that will be used throughout this report.The mixing over n upper or n lower indices consists in constructing all n!isomers resulting from the permutation of these indices, summing these isomers,and dividing by n!. We denote this process with round brackets around theindices. To single out indices, we use the sign ||. Note that the effect of theround brackets or of the sign || is not stopped by any kind of ordinary brackets.An example of mixing is the following:

Q m(ij|k|l) =

1

3!(Q m

ijkl +Q mjlki +Q m

likj +Q milkj +Q m

ljki +Q mjikl )

The mixing of an object gives the symmetric part of that object.

The alternation over n upper or n lower indices is found in the same way asfor the mixing with the difference that odd permutation have a negative sign.We denote this process with square brackets around the indices. An example ofalternation is the following:

Q m[ij|k|l] =

1

3!(Q m

ijkl +Q mjlki +Q m

likj −Q milkj −Q m

ljki −Q mjikl )

The alternation of an object gives the alternating part of that object.

5

3 Parallel transport and covariant derivative

Let us consider two points separated infinitesimally in the space-time. We callthe first point P (xi) and the second point Q(xi + dxi). Consider a vector fieldΦ; this field takes the value Φk at P and Φk + dΦk at Q.

Now if we consider the difference in the vector field value dΦk, we notice that itis not a vector as the vectors at point P and point Q obey different transform-ation laws. In order to find the difference between the vectors in a meaningfulway, they must be brought to the same point. This is achieved by paralleltransport.

Let us parallel-transport the vector Φk at P to Q. It is then given by the vectorΦk + δΦk. This vector can now be subtracted from the original vector at Q:

(Φk + dΦk)− (Φk + δΦk) = dΦk − δΦk (8)

The resulting difference is the covariant differential of the vector between thetwo points:

DΦk = dΦk − δΦk (9)

Figure 1: Parallel transport of vector Φ at point P to point Q. The dashedline at Q represents the original vector at Q. The covariant differential is thedifference between these two vectors.

In order to define the parallel transport δΦ, we introduce the following condi-tions [8, p. 124]

(i) The covariant differential of a quantity Φ transforms in the same way asΦ.

(ii) The covariant differential of a sum of quantities is the sum of the covariantdifferentials of the terms.

(iii) The covariant differential obeys the Leibniz rule.

6

(iv) The covariant differential of a quantity is linear homogeneous in the dxi.

Therefore, δΦ for contravariant vector can be written as:

δΦk = −ΓkjiΦjdxi (10)

where Γkji is an object called connection, which will be discussed in section 4.The covariant differential and covariant derivative of a contravariant vector canthus be expressed as:

DV k = dV k + Γkji Vj dxi (11)

∇iV k = ∂iVk + Γkji V

j (12)

The same expressions for a covariant vector take the following form:

DWk = dWk − ΓjkiWj dxi (13)

∇iWk = ∂iWk − ΓjkiWj (14)

The covariant derivative can also act on higher-rank tensors. The general for-mula is given by [3]

∇i T j...`... = ∂i Tj...`... + Γjsi T

s...`... + · · · − Γsi` T

j...s... − . . . (15)

7

4 Affine connection, torsion and non-metricity

4.1 The connection in general relativity

We will now consider the connection Γkij . This is an object that connects nearbytangent spaces allowing us to define a covariant derivative on the manifold.

In general relativity, the connection respects the following properties [3]:

i It is metric-compatible, i.e. the covariant derivative commutes with theoperation of lowering and raising the indices

∇i gjk = 0 (16)

ii It is torsion-free, i.e.(∇i∇j −∇j∇i) f = 0 (17)

which implies that the connection is symmetric on the two lower indices

Γkij = Γkji (18)

Under these conditions, the connection is uniquely defined and coincides withthe Christoffel symbol {kij} 4, i.e. [3]

Γkij = {kij} =1

2gks(∂i gjs + ∂j gsi − ∂s gij). (19)

However, this is not the most general case possible on a pseudo-Riemannianmanifold.

4.2 Torsion

We define a tensor of asymmetry or torsion as the alternating part of the con-nection, i.e.

S kij = Γk[ij] ≡

1

2(Γkij − Γkji) (20)

The connection is said to be

i symmetric if S kij = 0

ii semi-symmetric if S kij = S[iA

kj]

iii asymmetric otherwise.

The torsion tensor can be interpreted in the following way. Consider two linearelements dξi

1and dξi

2, and parallelly transport them along each other. In general,

the figure so obtained is not a parallelogram, but a pentagon with a closingvector given by 2S k

ij dξi

1dξj2

[8].

4Note that the Christoffel symbol is not a tensor, but instead transforms as

{kij}′ =∂X′k

∂X`

∂Xm

∂X′i∂Xn

∂X′j {`mn}+

∂2Xs

∂X′i∂X′j∂X′k

∂Xs,

since it depends on the partial derivatives of a rank-2 tensor.

8

4.3 Non-metricity

Consider the metric tensor gij , which is symmetric and whose inverse is gij . Ingeneral, the covariant derivative of the inverse metric is given by

∇i gjk = Q jki (21)

which, together with the property gisgsj = δij , implies

∇i gjk = −Qijk (22)

where Qijk is the object of non-metricity [8].The connection is said to be

i metric if Qijk = 0

ii semi-metric if Qijk = Qi gij

iii non-metric otherwise.

4.4 General form of the connection

By writing explicitly the covariant derivative of the metric, i.e.

∇i gjk = ∂i gjk − Γsij gsk − Γsik gjs = −Qijk, (23)

we find the general form of the connection including torsion and non-metricity[8]:

Γkij = {kij}+ S kij − S k

j i + Skij +1

2(Q k

ij +Q kj i −Qkij), (24)

Equation (24) can be written in a more compact form by using the abbreviation

ψ{ijk} ≡ ψijk − ψjki + ψkij . (25)

Then, we can define χijk ≡ 12∂i gjk, so that we can finally write

Γkij = gks(χ{isj} − S{isj} +1

2Q{isj}). (26)

9

5 Lorentz group and transformations

5.1 Definition of the Lorentz group

The Lorentz group is the group of the transformations of the Minkowski space-time, which is the space-time of special relativity expressed as

ηµν xµ xν = −(x1)2 − (x2)2 − (x3)2 + (x4)2 (27)

Note that we are using the convention (−,−,−,+) and that x4 is the time-likecoordinate.

The homogeneous Lorentz group is the set of all real linear transformations

xµ′ = Λµν xν (28)

such that the Minkowski metric is left invariant, and past and future are notinterchanged, i.e.

ηµν Λµα Λµβ = ηαβ and L44 > 0 (29)

where Λµν is the Lorentz matrix representing a Lorentz transformation [5].Now taking the determinant of equation (29) gives

|Λµν | = ±1 (30)

• |Λµν | = +1 corresponds to proper Lorentz transformations.

• |Λµν | = −1 corresponds to improper Lorentz transformations.

In our investigation, we only focus on the proper Lorentz transformations, whichform an invariant subgroup.

5.2 Infinitesimal Lorentz transformations

Let’s consider infinitesimal Lorentz transformations which differ from the iden-tity transformation by infinitesimals ωµν , so the Lorentz matrix is expressedas:

Λµν = δµν + ωµν (31)

The orders higher than the linear order of ω are negligible.

In any representation of the Lorentz group the infinitesimal transformation equa-tion (31) is represented by:

1 +1

2ωµν Gµν (32)

where Gµν are the generators of the Lorentz group, 3 of which are given intable 1 in section 6.

10

6 Spinors and spin space

We can now finally introduce the objects of study, i.e. spinors. We will start bydefining them and stating their particular properties. Then, we will discuss auseful correspondence between R4 and the spin space, which is the space wherespinors live. Once this is done, we will be able to write down the infinitesimaltransformation of a 2-spinor. Finally, we will extend the theory to 4-spinors andexpress their infinitesimal transformation as well. These transformations willbe needed to define the covariant derivative of a spinor, which is our objective.

Note that we will be working in Minkowski space with metric ηµν . This isbecause it is much simpler to work in flat space and then generalise our resultsto curved space using the tetrads, which will be discussed in section 7.

6.1 Definition of 2-spinors

A spinor is a geometrical object ψA that is defined over a 2D complex spaceP1, called the spin space, and that transforms in the following way:

ψ′A = tBA ψB (A,B = 1, 2) (33)

where tBA is in general complex and non-singular.

For our purposes, i.e. physical applications of spinor theory in special relativity,we can restrict ourselves to unimodular matrices sAB , so that [5]

ψ′A = sBA ψB . (34)

A quantity that transforms in this way is called a covariant regular spinor offirst rank.

Analogously, a geometric object ψA that transforms according to

ψA′

= SAB ψB (35)

whereSAB s

BC = δAC (36)

is called a contravariant regular spinor of first rank. [5]

Covariant and contravariant components are related using the rules

ψA = εAB ψB (37)

ψA = εBA ψB (38)

where the metric spinors εAB and εAB are given by

εAB =

(0 1−1 0

)= εAB . (39)

11

Note that εBAεBC = δAC . This is the only way we have to raise and lower indicesin this space. [5]

If a quantity is written with one or more dotted indices 5, e.g. χBA

, then thecorresponding transformation matrix is complex conjugated. For instance,

χ′BA

= sCA SBD χD

C.

To raise or lower dotted indices, the metric spinors εAB and εAB have to beused in the rules

ψA = εAB ψB (40)

ψA = εBA ψB (41)

where

εAB =

(0 1−1 0

)= εAB . (42)

6.2 Spin-tensor

A quantity with both tensor and spinor indices is called a tensor-spinor or aspin-tensor. Consider, for instance, the object φµAB . It transforms as

φ′µAB

= aνµ φνAB

under coordinate transformations in R4, and as

φ′µAB

= sCA sDB φµCD

under constant spin transformations in P1.

The corresponding contravariant transformations are

φ′µAB = Aµν φνAB

φ′µAB = SAC SBD φ

µCD

Under combined (constant) coordinate transformations in R4 and P1, the spin-tensor φµAB transforms according to

φ′′µAB

= aνµ φ′νAB

= aνµ sCA s

DB φνCD

φ′′µAB = Aµν φ′νAB = Aµν S

AC S

BD φ

νCD

5We will call spinors with dotted indices conjugated or dotted.

12

6.3 Correspondence between R3 and spin space

We now approach the relation between 2-spinors and 4-tensors. In fact, thereexists a (1−1) correspondence between Hermitian matrices of second order andpoints of R4, thus every second-rank Hermitian matrix can be written in termsof four real parameters p, q, r, s, as, for instance, [5]

XAB =

(p+ q r + isr − is p− q

). (43)

Given a 4-vector xi, we use a formalism in which the identification chosen is

p = x4, q = x3, r = x1, s = x2 (44)

so that equation (43) takes the form

XAB =

(x4 + x3 x1 + ix2

x1 − ix2 x4 − x3). (45)

We will now show how this correspondence can be used to find the representationof spatial rotations in spin space.

6.3.1 Rotation about the z axis

For simplicity, consider a general 3D column vector v = (v1, v2, v3)T rotatedcounterclockwise about the z axis by an angle θ. Then, the new rotated vectoris given by v′ = R v, i.e.v′1v′2

v′3

=

cos θ − sin θ 0sin θ cos θ 0

0 0 1

v1v2v3

= (46)

=

cos θ v1 − sin θ v2sin θ v1 + cos θ v2

v3

(47)

Using equation (45), we can write the 3D vector v (setting p = x4 = 0) as

VAB =

(v3 v1 + iv2

v1 − iv2 −v3

)(48)

and similarly the 3D v′ as

V ′AB

(z) =

(v3 cos θ v1 − sin θ v2 + i(sin θ v1 + cos θ v2)

cos θ v1 − sin θ v2 − i(sin θ v1 + cos θ v2) −v3

)(49)

=

(v3 (cos θ + i sin θ) v1 + (− sin θ + i cos θ) v2

(cos θ − i sin θ) v1 − (sin θ + i cos θ) v2 −v3

)(50)

=

(v3 eiθ (v1 + iv2)

e−iθ (v1 − iv2) −v3

)(51)

13

where in the last passage we have used the following relations:

cos θ + i sin θ = eiθ (52)

cos θ − i sin θ = e−iθ (53)

sin θ + i cos θ = i(cos θ − i sin θ) = ie−iθ (54)

− sin θ + i cos θ = i(cos θ + i sin θ) = ieiθ (55)

The second-rank Hermitian matrix corresponding to the vector v′, i.e. V ′AB

,can be written as [5]

V′(z) = S(z) V S−1 (z). (56)

The change of basis matrix S can be found to be

S(z) =

(eiθ/2 0

0 e−iθ/2

)(57)

with S−1 = S†.

This is a 2D rotation by the angle θ/2, which is half of the angle by which wehad rotated the vector v in R3.

Finally, Taylor expanding to first order, we obtain

S(z) ≈(

1 + iθ/2 00 1− iθ/2

)= I + i

θ

2σz (58)

where I is the 2x2 identity matrix and σz is the Pauli matrix

σz =

(1 00 −1

). (59)

Now consider the 3D rotation matrix in equation (46). Taylor expanding to firstorder, we find

R ≈

1 −θ 0θ 1 00 0 1

= I− iθ

0 −i 0i 0 00 0 0

= I− iθ(σy 00 0

)(60)

where I is here the 3x3 identity matrix and σy is the Pauli matrix

σy =

(0 −ii 0

). (61)

6.3.2 Rotation about the x axis

Now, let’s consider the rotation of the general vector v = (v1, v2, v3)T by a smallangle θ about the x axis. This gives

14

v′1v′2v′3

=

1 0 00 cos θ − sin θ0 sin θ cos θ

v1v2v3

= (62)

=

v1cos θ v2 − sin θ v3sin θ v2 + cos θ v3

(63)

The corresponding Hermitian matrix representation of equation (62) is

V ′AB

(x) =

(sin θ v2 + cos θ v3 v1 + i(cos θ v2 − sin θ v3)

v1 − i(cos θ v2 − sin θ v3) − sin θ v2 − cos θ v3

)(64)

It can be shown that the change of basis matrix for the similarity transform,V′(x) = S(x) V S−1 (x), is:

S(x) =

(cos θ/2 i sin θ/2i sin θ/2 cos θ/2

)(65)

where the following double-angle identities were used:

sin θ = 2 sin θ/2 cos θ/2 (66)

cos θ = cos2 θ/2− sin2 θ/2 (67)

Taylor expanding S(x) to first order we obtain

S(x) ≈(

1 iθ/2iθ/2 1

)= I + i

θ

2σx (68)

where I is here the 3x3 identity matrix and σx is the Pauli matrix

σx =

(0 11 0

)(69)

6.3.3 Rotation about the y axis

Rotating the general vector v = (v1, v2, v3)T about the y-axis givesv′1v′2v′3

=

cos θ 0 sin θ0 1 0

− sin θ 0 cos θ

v1v2v3

= (70)

=

cos θ v1 + sin θ v3v2

− sin θ v1 + cos θ v3

(71)

The corresponding Hermitian matrix representation of equation (70) is

15

V ′AB

(y) =

(− sin θ v1 + cos θ v3 cos θ v1 + sin θ v3 + iv2

cos θ v1 + sin θ v3 − iv2 sin θ v1 − cos θ v3

)(72)

The corresponding similarity transformation for V′(y) is given by

S(y) =

(cos θ/2 − sin θ/2sin θ/2 cos θ/2

)(73)

Taylor expanding S to first order we obtain

S(y) ≈(

1 −θ/2θ/2 1

)= I + i

θ

2σy (74)

where I is here the 3x3 identity matrix and σy is the Pauli matrix

σy =

(0 i−i 0

)(75)

As a result, the Pauli matrices

σx =

(0 11 0

), σy =

(0 i−i 0

), σz =

(1 00 −1

)(76)

are shown to be the generators of infinitesimal rotations in spin space.

The extension to R4 is trivial when only rotations are considered. [2]

R4(x, y, z, t) Spin-space Generators

Rx

1 0 0 0

0 cos θ − sin θ 0

0 sin θ cos θ 0

0 0 0 1

(

cos θ/2 i sin θ/2

i sin θ/2 cos θ/2

)σx =

(0 1

1 0

)

Ry

cos θ 0 sin θ 0

0 1 0 0

− sin θ 0 cos θ 0

0 0 0 1

(

cos θ/2 − sin θ/2

sin θ/2 cos θ/2

)σy =

(0 i

−i 0

)

Rz

cos θ − sin θ 0 0

sin θ cos θ 0 0

0 0 1 0

0 0 0 1

(

exp(iθ/2

)0

0 exp(−iθ/2

)) σz =

(1 0

0 −1

)

Table 1: Generators of the Lorentz group and the correspondence between spinspace and R4

16

6.4 Remark on spinor rotations

Spinors appear to be similar to vectors, but differ from them by their behaviourunder rotations. Indeed, a vector is left unchanged when rotated by 360◦.However, a spinor is not; instead, we obtain its opposite. A rotation of 720◦ isrequired to return to the original position. This is because, as we have seen, aspatial rotation by an angle θ is mapped to a rotation in spin space by half thatangle.

6.5 The g-spin tensors

Consider again equation (45) and note it can be written as

XAB = gµAB xµ (77)

where gµ = (σx,σy,σz, I) and the bar represents complex conjugation.

The inverse equations can be written as

xµ =1

2gµAB XAB (78)

where gµ = (σx,σy,σz, I)T .

Note that we have used gµ =∥∥∥gµAB∥∥∥ and gµ =

∥∥∥gµAB∥∥∥.

Note also that tensor and spinor indices are raised or lowered using the metrictensor and the (skew) metric spinors, respectively.For a more complete analysis of g-spin tensors, see [5].

6.6 Infinitesimal transformations of 2-spinors

Let us consider Lorentz and spinor transformations; to every proper Lorentztransformation there exists a spin transformation such that the application oftwo together leaves the spin-tensors gµ

ABinvariant. [5]

Lorentz Spinorx′µ = xµ + ω ν

µ xν φ′A = φA + ηBAφBφ′A

= φA + ηBAφB

Table 2: Infinitesimal Lorentz and spinor transformations.

By the above statement, it should be noted that for every ωµν there exists ηABsuch that

ωµνgνCA + ηBAgµ

CB+ ηD

CgµDA

= 0 (79)

The solution is

ηAB =1

4gµCAg

Cν B (80)

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The transformation matrices are thus

δBA +1

4ωµνgµCAg

CBν

δBA

+1

4ωµνgµACg

BCν

(81)

By analogy with the representation of the Lorentz group,

||1 +1

2ωµνGµν || = δBA +

1

4ωµνgµCAg

CBν (82)

we find that,

||Gµν || =1

4(gµCAg

CBν − gνCAg

CBµ ) (83)

Now let us consider the infinitesimal transformation of a 2-spinor φA.

δφA = ηBAφB (84)

δφA = −1

4ωµν

{1

2(g Cµ Ag

BνC

− g Cν Ag

BµC

)

}φB (85)

We could express this equation in compact form as

δφ =1

2ωµνGµν op φ (86)

6.7 Definition of 4-spinors

4-spinors are a combination of two simple 2-spinors and can be expressed interms of one covariant regular spinor and one covariant conjugated spinor. [5]

Ψa =

∥∥∥∥∥∥∥∥Ψ1

Ψ2

Ψ3

Ψ4

∥∥∥∥∥∥∥∥ =

∥∥∥∥∥∥∥∥∥ψ1

ψ2

φ1

φ2

∥∥∥∥∥∥∥∥∥ (87)

6.8 Gamma Matrices

The gamma matrices can be expressed in terms of g-spin tensor components(which are 2× 2 matrices) as: [5]

γµab =

∥∥∥∥∥∥∥0 −i

∥∥∥gµAB

∥∥∥i∥∥∥gµAB∥∥∥ 0

∥∥∥∥∥∥∥ µ = {1, 2, 3, 4} (88)

We could also introduce a fifth gamma matrix that satisfies the following rela-tion:

γ5γµ + γµγ5 = 0 (89)

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This gamma matrix can be expressed as:

γ5 =

∥∥∥∥I 00 −I

∥∥∥∥ (90)

All gamma matrices are explicitly stated below:

γ1ab =

∥∥∥∥∥∥∥∥0 0 0 i0 0 i 00 i 0 0i 0 0 0

∥∥∥∥∥∥∥∥ γ2ab =

∥∥∥∥∥∥∥∥0 0 0 10 0 −1 00 1 0 0−1 0 0 0

∥∥∥∥∥∥∥∥γ3ab =

∥∥∥∥∥∥∥∥0 0 i 00 0 0 −ii 0 0 00 −i 0 0

∥∥∥∥∥∥∥∥ γ4ab =

∥∥∥∥∥∥∥∥0 0 −i 00 0 0 −ii 0 0 00 i 0 0

∥∥∥∥∥∥∥∥γ5ab =

∥∥∥∥∥∥∥∥1 0 0 00 1 0 00 0 −1 00 0 0 −1

∥∥∥∥∥∥∥∥In flat space, the gamma matrices obey the following anti-commutation relation:

{γµ, γν} = γµγν + γνγµ = 2ηµν (91)

In curved space, they obey:

{γi, γj} = γiγj + γjγi = 2gij (92)

which are interchangeable via the application of tetrads (see section ??):

gij = e µi eνj ηµν . (93)

6.9 Infinitesimal transformation of 4-spinors

In a similar manner to 2-spinors, we consider the infinitesimal transformationof 4-spinors 6 [5]

δΨ• =

∥∥∥∥∥δψBδφB

∥∥∥∥∥ =

∥∥∥∥∥ηABψA−ηBAφA

∥∥∥∥∥ =1

4ωµν

∥∥∥∥∥∥ gA

µCg Cν BψA

−gµCA

g CBν φA

∥∥∥∥∥∥ (94)

This can be written in matrix form as:

1

4ωµν

∥∥∥∥∥∥gA

µCg Cν B 0

0 −gµCA

g CBν

∥∥∥∥∥∥ ·∥∥∥∥∥ψAφA

∥∥∥∥∥ (95)

By the anti-symmetry of two indices of ωµν this is written as:

6The bullets correspond to any lower-case Latin (spinor) index.

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1

4ωµν

∥∥∥∥∥∥12 (g A

µCg Cν B − g A

νCg Cµ B ) 0

0 12 (g

νCAg CBµ − g

µCAg CBν )

∥∥∥∥∥∥Ψ• (96)

In compact form we have that the infinitesimal transformation of a 4-spinor isgiven by:

δΨ• =1

2ωµν G •

µν •Ψ• (97)

where ωµν is the infinitesimal rotation and G •µν • is the generator of the Lorentz

group, i.e.

Gµν =1

2γµν =

1

4[γµ, γν ]. (98)

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7 Tetrads

In this section we will show how the results obtained in Minkowski space in theprevious sections can be extended to a general curved space using the tetradframe.

This formalism corresponds to picturing spacetime as populated with observerswho can measure spatial and temporal distances, and relative orientations.

In fact, an orthonormal frame of four vectors is introduced at every point inspacetime [6]

eα(xk) = eiα ∂i with eα · eβ = ηαβ = diag(−1,−1,−1,+1). (99)

This tetrad field eα(xk) represents the local observers.

The dual frame (co-frame) is given by 7

eα = e αi dxi (100)

so that we have e αi eiβ = δαβ . Similarly, e αi ejα = δji .

Any vector can be expressed either using its components vi with respect to theholonomic coordinate system, or by its coordinate-invariant projections vα ontothe tetrad field e αi [7]

vα = e αi vi (101)

vi = eiα vα (102)

The metric tensor of the manifold can be related to the local Minkowski metricby using the tetrad

gij = e αi e βj ηαβ . (103)

Analogously, the Minkowski metric can be expressed as

ηαβ = eiα ejβ gij , (104)

thus the eiα matrices are similarity transformations that diagonalise the metricgij locally to the Minkowski metric.

The tetrad has 16 independent components which determine the 10 compon-ents of the metric tensor. The remaining 6 degrees of freedom do not affect themetric, they correspond to the 6 Lorentz transformations of the tangent space.

7Recall that a vector can be written using its basis ei or its dual basis ei as v = vi ei = vi ei

where vi and vi are respectively the contravariant and covariant components of the vector.The dual basis is such that ei · ej = δji , and we can identify ei ↔ ∂i and ei ↔ dxi.

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Note that holonomic indices are lowered and raised using the metric gij andits inverse gij , while anholonomic (coordinate-independent) indices are loweredand raised using the Minkowski metric ηαβ and its inverse ηαβ .

To change from a tetrad to another, we express the vectors of the new tetradas linear combinations of the vectors of the old one, i.e.

eiα = Λβα eiβ (105)

Applying equation (103) on the new tetrad eiα gives the orthogonality condition

Λγα Λδβ ηγδ = ηαβ (106)

which has to be obeyed by the matrix Λ. This corresponds to the conditiongiven in equation (29), thus Λ is a Lorentz matrix.

Hence, the Lorentz group can be thought as the group of tetrad rotations ingeneral relativity [7].

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8 The covariant derivative of a spinor and theDirac equation

8.1 Covariant derivative of a spinor

We will now use the results obtained in the previous sections to express thecovariant derivative of a spinor in a metric-affine space. We will first considerflat space, then extend the validity of the equation to a general curved spaceusing the tetrad field.Finally, we use our result to express the Dirac equation in a general metric-affinespace.

We have seen that the infinitesimal transformation of a 4-spinor is given by

δΨ =1

2ωµν G

µν Ψ (107)

where Gµν = 12γ

µν = 14 [γµ, γν ] and ωµν is an infinitesimal rotation.

Writing ωµν = ωαµν dxα, we obtain

δΨ =1

2ωαµν G

µν Ψ dxα = δΨα dxα. (108)

Then, the covariant differential is given by

DΨ = dΨ + δΨα dxα (109)

and hence the covariant derivative is 8

∇αΨ = ∂αΨ + δΨα = ∂αΨ +1

2ωαµν G

µν Ψ. (110)

Using the tetrad field, we can now express the covariant derivative in curvedspace:

∇iΨ = e αi ∇αΨ = e αi (∂αΨ +1

2ωαµν G

µν Ψ) (111)

∇iΨ = ∂iΨ +1

2ωiµν G

µν Ψ. (112)

We identify the object ωiµν as the affine spin connection. This is defined as [4]

ω αi β = e α

k Γkij ejβ + e α

j ∂i ejβ , (113)

where Γkij is the metric-affine connection given by equation (26).

We can lower anholonomic indices using the Minkowski metric, thus

ωiµν = ηµγ ωγi ν = ηµγ (e γ

k Γkis esν + e γ

s ∂i esν). (114)

8Cfr. section 3.

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8.2 The Dirac equation

Finally, substituting the partial derivative with the covariant derivative, we findthat the Dirac equation in a metric-affine space takes the form

i γi∇i Ψ = mΨ (115)

where ∇i Ψ is given by equation (112).

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References

[1] The Filipino Cultural Dance Troupe Wagga Wagga NSW Australia. Pan-danggo sa ilaw - Candle Dance. url: https://www.youtube.com/watch?v=4Gf8Xa0N2yk.

[2] S. Baskal, Y. S. Kim and M. E. Noz. Physics of the Lorentz Group. IOP,2015.

[3] C. G. Bohmer. Introduction To General Relativity And Cosmology. Essen-tial Textbooks In Physics. World Scientific, 2016.

[4] C. G. Bohmer and Y. Lee. Compatibility conditions of continua using Riemann-Cartan geometry. 2020. arXiv: 2006.06800 [math-ph].

[5] E. M. Corson. Introduction to tensors, spinors, and relativistic wave-equations.Blackie and son limited, 1955.

[6] F. W. Hehl. “Four Lectures on Poincare Gauge Field Theory”. In: Cos-mology and Gravitation. Spin, Torsion, Rotation, and Supergravity. Ed. byP. G. Bergmann and V. De Sabbata. Plenum Press, New York, 1980.

[7] N. J. Pop lawski. Covariant differentiation of spinors for a general affineconnection. 2007. arXiv: 0710.3982 [gr-qc].

[8] J. A. Schouten. Ricci-Calculus. 2nd ed. Grundlehren der mathematischenWissenschaften. Springer-Verlag Berlin Heidelberg, 1954. isbn: 978-3-642-05692-5. doi: 10.1007/978-3-662-12927-2.

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