Gamma matrices

Short description: Generators of the Clifford algebra for relativistic quantum mechanics

In mathematical physics, the gamma matrices,  {γ0,γ1,γ2,γ3} , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(ℝ). It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin  1 2 particles. Gamma matrices were introduced by Paul Dirac in 1928.[1][2]

In the Dirac representation, the four contravariant gamma matrices are

γ0 =(1000010000−10000−1),γ1=(000100100−100−1000),γ2=i (000−100100100−1000),γ3=(0010000−1−10000100).

γ0 is the time-like, Hermitian matrix. The other three are space-like, anti-Hermitian matrices. More compactly,  γ0=σ3⊗I2 , and  γj=iσ2⊗σj , where  ⊗ denotes the Kronecker product and the  σj (for j = 1, 2, 3) denote the Pauli matrices.

In addition, for discussions of group theory the identity matrix (I) is sometimes included with the four gamma matricies, and there is an auxiliary, "fifth" traceless matrix used in conjunction with the regular gamma matrices

I4=(1000010000100001) ,γ5≡iγ0γ1γ2γ3=(0010000110000100).

The "fifth matrix"  γ5 is not a proper member of the main set of four; it is used for separating nominal left and right chiral representations.

The gamma matrices have a group structure, the gamma group, that is shared by all matrix representations of the group, in any dimension, for any signature of the metric. For example, the 2×2 Pauli matrices are a set of "gamma" matrices in three dimensional space with metric of Euclidean signature (3, 0). In five spacetime dimensions, the four gammas, above, together with the fifth gamma-matrix to be presented below generate the Clifford algebra.

## Contents

* 1 Mathematical structure
* 2 Physical structure
* 3 Expressing the Dirac equation
* 4 The fifth "gamma" matrix, γ5
* 4.1 Five dimensions
* 5 Identities
* 5.1 Miscellaneous identities
* 5.2 Trace identities
* 5.3 Normalization
* 5.4 Charge conjugation
* 5.5 Feynman slash notation
* 6 Other representations
* 6.1 Dirac basis
* 6.2 Weyl (chiral) basis
* 6.3 Weyl (chiral) basis (alternate form)
* 6.4 Majorana basis
* 6.5 Cl1,3(C) and Cl1,3(R)
* 7 Other representation-free properties
* 8 Euclidean Dirac matrices
* 8.1 Chiral representation
* 8.2 Non-relativistic representation
* 9 Footnotes
* 10 See also
* 11 Citations
* 12 References
* 13 External links

## Mathematical structure

The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation

{γμ,γν}=γμγν+γνγμ=2ημνI4 ,

where the curly brackets  {,} represent the anticommutator,  ημν is the Minkowski metric with signature (+ − − −), and I4 is the 4 × 4 identity matrix.

This defining property is more fundamental than the numerical values used in the specific representation of the gamma matrices. Covariant gamma matrices are defined by

γμ=ημνγν={γ0,−γ1,−γ2,−γ3} ,

and Einstein notation is assumed.

Note that the other sign convention for the metric, (− + + +) necessitates either a change in the defining equation:

{γμ,γν}=−2ημνI4

or a multiplication of all gamma matrices by i, which of course changes their hermiticity properties detailed below. Under the alternative sign convention for the metric the covariant gamma matrices are then defined by

γμ=ημνγν={−γ0,γ1,γ2,γ3}.

## Physical structure

The Clifford algebra  Cl1,3(ℝ) over spacetime V can be regarded as the set of real linear operators from V to itself, End(V), or more generally, when complexified to  Cl1,3(ℝ)ℂ , as the set of linear operators from any four-dimensional complex vector space to itself. More simply, given a basis for V,  Cl1,3(ℝ)ℂ is just the set of all 4×4 complex matrices, but endowed with a Clifford algebra structure. Spacetime is assumed to be endowed with the Minkowski metric ημν. A space of bispinors, Ux , is also assumed at every point in spacetime, endowed with the bispinor representation of the Lorentz group. The bispinor fields Ψ of the Dirac equations, evaluated at any point x in spacetime, are elements of Ux (see below). The Clifford algebra is assumed to act on Ux as well (by matrix multiplication with column vectors Ψ(x) in Ux for all x). This will be the primary view of elements of  Cl1,3(ℝ)ℂ in this section.

For each linear transformation S of Ux, there is a transformation of End(Ux) given by S E S−1 for E in  Cl1,3(ℝ)ℂ≈End⁡(Ux). If S belongs to a representation of the Lorentz group, then the induced action E ↦ S E S−1 will also belong to a representation of the Lorentz group, see Representation theory of the Lorentz group.

If S(Λ) is the bispinor representation acting on Ux of an arbitrary Lorentz transformation Λ in the standard (4 vector) representation acting on V, then there is a corresponding operator on  End⁡(Ux)=Cl1,3(ℝ)ℂ given by the equation:

γμ ↦ S(Λ) γμ S(Λ)−1=(Λ−1)μν γν=Λνμ γν ,

showing that the quantity of γμ can be viewed as a basis of a representation space of the 4 vector representation of the Lorentz group sitting inside the Clifford algebra. The last identity can be recognized as the defining relationship for matrices belonging to an indefinite orthogonal group, which is  ηΛTη=Λ−1 , written in indexed notation. This means that quantities of the form

a/≡aμγμ

should be treated as 4 vectors in manipulations. It also means that indices can be raised and lowered on the γ using the metric ημν as with any 4 vector. The notation is called the Feynman slash notation. The slash operation maps the basis eμ of V, or any 4 dimensional vector space, to basis vectors γμ. The transformation rule for slashed quantities is simply

a/μ↦Λμνa/ν.

This is different from the transformation rule for the γμ, which are now treated as (fixed) basis vectors. The designation of the 4 tuple (γμ)μ=03=(γ0,γ1,γ2,γ3) as a 4 vector sometimes found in the literature is thus a slight misnomer. The latter transformation corresponds to an active transformation of the components of a slashed quantity in terms of the basis γμ, and the former to a passive transformation of the basis γμ itself.

The elements  σμν=γμγν−γνγμ form a representation of the Lie algebra of the Lorentz group. This is a spin representation. When these matrices, and linear combinations of them, are exponentiated, they are bispinor representations of the Lorentz group, e.g., the S(Λ) of above are of this form. The 6 dimensional space the σμν span is the representation space of a tensor representation of the Lorentz group. For the higher order elements of the Clifford algebra in general and their transformation rules, see the article Dirac algebra. The spin representation of the Lorentz group is encoded in the spin group Spin(1, 3) (for real, uncharged spinors) and in the complexified spin group Spin(1, 3) for charged (Dirac) spinors.

## Expressing the Dirac equation

Main article: Dirac equation

In natural units, the Dirac equation may be written as

(iγμ∂μ−m)ψ=0

where  ψ is a Dirac spinor.

Switching to Feynman notation, the Dirac equation is

(i∂/−m)ψ=0.

## The fifth "gamma" matrix, γ5

It is useful to define a product of the four gamma matrices as γ5=σ1⊗I, so that

γ5≡iγ0γ1γ2γ3=(0010000110000100) (in the Dirac basis).

Although  γ5 uses the letter gamma, it is not one of the gamma matrices of  Cl1,3(ℝ). The index number 5 is a relic of old notation:  γ0 used to be called "γ4".

γ5 has also an alternative form:

γ5=i4!εμναβγμγνγαγβ

using the convention ε0123=1 , or

γ5=−i4!εμναβγμγνγαγβ

using the convention ε0123=1. Proof:

This can be seen by exploiting the fact that all the four gamma matrices anticommute, so

γ0γ1γ2γ3=γ[0γ1γ2γ3]=14!δμνϱσ0123γμγνγϱγσ ,

where δμνϱσαβγδ is the type (4,4) generalized Kronecker delta in 4 dimensions, in full antisymmetrization. If  εα…β denotes the Levi-Civita symbol in n dimensions, we can use the identity δμνϱσαβγδ=−εαβγδεμνϱσ. Then we get, using the convention  ε0123=1 ,

γ5=iγ0γ1γ2γ3=−i4!ε0123εμνϱσγμγνγϱγσ=−i4!εμνϱσγμγνγϱγσ=−i4!εμνϱσγμγνγϱγσ

This matrix is useful in discussions of quantum mechanical chirality. For example, a Dirac field can be projected onto its left-handed and right-handed components by:

ψL= I−γ5 2 ψ,ψR= I+γ5 2 ψ.

Some properties are:

* It is Hermitian:

(γ5)†=γ5.
* Its eigenvalues are ±1, because:

(γ5)2=I4.
* It anticommutes with the four gamma matrices:

{γ5,γμ}=γ5γμ+γμγ5=0.

In fact,  ψL and  ψR are eigenvectors of  γ5 since

γ5ψL= γ5−(γ5)2 2ψ=−ψL , and γ5ψR= γ5+(γ5)2 2ψ=ψR.

### Five dimensions

The Clifford algebra in odd dimensions behaves like two copies of the Clifford algebra of one less dimension, a left copy and a right copy.[3]: 68 Thus, one can employ a bit of a trick to repurpose i γ 5 as one of the generators of the Clifford algebra in five dimensions. In this case, the set {γ 0, γ 1, γ 2, γ 3, i γ 5} therefore, by the last two properties (keeping in mind that i 2 ≡ −1) and those of the ‘old’ gammas, forms the basis of the Clifford algebra in 5 spacetime dimensions for the metric signature (1,4).[lower-alpha 1] .[4]: 97 In metric signature (4,1), the set {γ 0, γ 1, γ 2, γ 3, γ 5} is used, where the γ μ are the appropriate ones for the (3,1) signature.[5] This pattern is repeated for spacetime dimension 2n even and the next odd dimension 2n \+ 1 for all n ≥ 1.[6]: 457 For more detail, see higher-dimensional gamma matrices.

## Identities

The following identities follow from the fundamental anticommutation relation, so they hold in any basis (although the last one depends on the sign choice for γ5).

### Miscellaneous identities

1\. γμγμ=4I4

Proof

Take the standard anticommutation relation:

{γμ,γν}=γμγν+γνγμ=2ημνI4.

One can make this situation look similar by using the metric η:

γμγμ
| =γμημνγν=ημνγμγν |

|
|

| =12(ημν+ηνμ)γμγν | (η symmetric)
| =12(ημνγμγν+ηνμγμγν) | (expanding)
| =12(ημνγμγν+ημνγνγμ) | (relabeling term on right)
| =12ημν{γμ,γν} |
| =12ημν(2ημνI4)=ημνημνI4=4I4. |

2\. γμγνγμ=−2γν

Proof

Similarly to the proof of 1, again beginning with the standard anticommutation relation:

γμγνγμ=γμ(2ημνI4−γμγν)=2γμημν−γμγμγν=2γν−4γν=−2γν.

3\. γμγνγργμ=4ηνρI4

Proof

To show

γμγνγργμ=4ηνρI4.

Use the anticommutator to shift γμ to the right

γμγνγργμ
| ={γμ,γν}γργμ−γνγμγργμ

|

| =2 ημνγργμ−γν{γμ,γρ}γμ+γνγργμγμ.

Using the relation γμγμ=4I we can contract the last two gammas, and get

γμγνγργμ
| =2γργν−γν2ημργμ+4γνγρ

|

| =2γργν−2γνγρ+4γνγρ
| =2(γργν+γνγρ)
| =2{γν,γρ}.

Finally using the anticommutator identity, we get

γμγνγργμ=4ηνρI4.

4\. γμγνγργσγμ=−2γσγργν

Proof

γμγνγργσγμ
| =(2ημν−γνγμ)γργσγμ (anticommutator identity)

|

| =2ημνγργσγμ−4γνηρσ (using identity 3)
| =2γργσγν−4γνηρσ (raising an index)
| =2(2ηρσ−γσγρ)γν−4γνηρσ (anticommutator identity)
| =−2γσγργν (2 terms cancel)

5\. γμγνγρ=ημνγρ+ηνργμ−ημργν−iϵσμνργσγ5

Proof

If μ=ν=ρ then ϵσμνρ=0 and it is easy to verify the identity. That is the case also when μ=ν≠ρ, μ=ρ≠ν or ν=ρ≠μ.

On the other hand, if all three indices are different, ημν=0, ημρ=0 and ηνρ=0 and both sides are completely antisymmetric; the left hand side because of the anticommutativity of the γ matrices, and on the right hand side because of the antisymmetry of ϵσμνρ. It thus suffices to verify the identities for the cases of γ0γ1γ2, γ0γ1γ3, γ0γ2γ3 and γ1γ2γ3.

−iϵσ012γσγ5=−iϵ3012(−γ3)(iγ0γ1γ2γ3)=−ϵ3012γ0γ1γ2=ϵ0123γ0γ1γ2−iϵσ013γσγ5=−iϵ2013(−γ2)(iγ0γ1γ2γ3)=ϵ2013γ0γ1γ3=ϵ0123γ0γ1γ3−iϵσ023γσγ5=−iϵ1023(−γ1)(iγ0γ1γ2γ3)=−ϵ1023γ0γ2γ3=ϵ0123γ0γ2γ3−iϵσ123γσγ5=−iϵ0123(γ0)(iγ0γ1γ2γ3)=ϵ0123γ1γ2γ3

6\. γ5σνρ=i2ϵσμνρσσμ , where  σμν=i2[γμ,γν]=i2(γμγν−γνγμ)

Proof

For  ν=ρ , ϵσμνρ=0 and both sides vanish. Otherwise, multiplying identity 5 by  γμ from the right gives that

γμγνγργμ
| =ημνγργμ+ηνργμγμ−ημργνγμ−iϵσμνργσγ5γμ

|

| =γργν+4ηνρI4−γνγρ−iϵσμνργσγ5γμ (raising indices and using identity 1)

where 4ηνρI4=0 since ν≠ρ. The left hand side of this equation also vanishes since γμγνγργμ=4ηνρI4 by property 3. Rearranging gives that

γργν−γνγρ
| =iϵσμνργσγ5γμ

|

| =−iϵσμνργ5γσγμ (since γ5 anticommutes with the gamma matrices)

Note that 2γσγμ=γσγμ−γμγσ=[γσ,γμ] for σ≠μ (for σ=μ, ϵσμνρ vanishes) by the standard anticommutation relation. It follows that

−[γν,γρ]
| =−i2ϵσμνργ5[γσ,γμ]

|

Multiplying from the left times  −i2γ5 and using that  (γ5)2=I4 yields the desired result.

### Trace identities

The gamma matrices obey the following trace identities:

1. tr⁡(γμ)=0
2. Trace of any product of an odd number of γμ is zero
3. Trace of γ5 times a product of an odd number of γμ is still zero
4. tr⁡(γμγν)=4ημν
5. tr⁡(γμγνγργσ)=4(ημνηρσ−ημρηνσ+ημσηνρ)
6. tr⁡(γ5)=tr⁡(γμγνγ5)=0
7. tr⁡(γμγνγργσγ5)=−4iϵμνρσ
8. tr⁡(γμ1…γμn)=tr⁡(γμn…γμ1)

Proving the above involves the use of three main properties of the trace operator:

* tr⁡(A+B)=tr⁡(A)+tr⁡(B)
* tr⁡(rA)=r⋅tr⁡(A)
* tr⁡(ABC)=tr⁡(BCA)=tr⁡(CAB)

Proof of 1

From the definition of the gamma matrices,

γμγν+γνγμ=2ημνI

We get

γμγμ=ημμI

or equivalently,

γμγμ ημμ =I

where  ημμ is a number, and  γμγμ is a matrix.

| tr⁡(γν)=1ημμtr⁡(γνγμγμ) (inserting the identity and using  tr(rA) = r tr(A) .)

=−1ημμtr⁡(γμγνγμ) (from anti-commutation relation, and given that we are free to select μ≠ν)
=−1ημμtr⁡(γνγμγμ) (using tr(ABC) = tr(BCA))
=−tr⁡(γν) (removing the identity)

This implies tr⁡(γν)=0

Proof of 2

To show

tr⁡(odd num of γ)=0

First note that

tr⁡(γμ)=0.

We'll also use two facts about the fifth gamma matrix γ5 that say:

(γ5)2=I4,andγμγ5=−γ5γμ

So let's use these two facts to prove this identity for the first non-trivial case: the trace of three gamma matrices. Step one is to put in one pair of γ5's in front of the three original γ's, and step two is to swap the γ5 matrix back to the original position, after making use of the cyclicity of the trace.

| tr⁡(γμγνγρ) | =tr⁡(γ5γ5γμγνγρ)

|

| =tr⁡(γ5γμγνγργ5) (using tr(ABC) = tr(BCA))
| =−tr⁡(γ5γ5γμγνγρ) (anticommuting γ5 three times)
| =−tr⁡(γμγνγρ)

This can only be fulfilled if

tr⁡(γμγνγρ)=0

The extension to 2n + 1 (n integer) gamma matrices is found by placing two gamma-5s after (say) the 2n-th gamma-matrix in the trace, commuting one out to the right (giving a minus sign) and commuting the other gamma-5 2n steps out to the left [with sign change (-1)^2n = 1]. Then we use cyclic identity to get the two gamma-5s together, and hence they square to identity, leaving us with the trace equalling minus itself, i.e., 0.

Proof of 3

If an odd number of gamma matrices appear in a trace followed by γ5, our goal is to move γ5 from the right side to the left. This will leave the trace invariant by the cyclic property. In order to do this move, we must anticommute it with all of the other gamma matrices. This means that we anticommute it an odd number of times and pick up a minus sign. A trace equal to the negative of itself must be zero.

Proof of 4

To show

tr⁡(γμγν)=4ημν

Begin with,

| tr⁡(γμγν) | =12(tr⁡(γμγν)+tr⁡(γνγμ))

|

| =12tr⁡(γμγν+γνγμ)=12tr⁡({γμ,γν})
| =122ημνtr⁡(I4)=4ημν
Proof of 5

| tr⁡(γμγνγργσ) | =tr⁡(γμγν(2ηρσ−γσγρ))

|

| =2ηρσtr⁡(γμγν)−tr⁡(γμγνγσγρ)(1)

For the term on the right, we'll continue the pattern of swapping γσ with its neighbor to the left,

tr⁡(γμγνγσγρ) | =tr⁡(γμ(2ηνσ−γσγν)γρ)

|

| =2ηνσtr⁡(γμγρ)−tr⁡(γμγσγνγρ)(2)

Again, for the term on the right swap γσ with its neighbor to the left,

tr⁡(γμγσγνγρ) | =tr⁡((2ημσ−γσγμ)γνγρ)

|

| =2ημσtr⁡(γνγρ)−tr⁡(γσγμγνγρ)(3)

Eq (3) is the term on the right of eq (2), and eq (2) is the term on the right of eq (1). We'll also use identity number 3 to simplify terms like so:

2ηρσtr⁡(γμγν)=2ηρσ(4ημν)=8ηρσημν.

So finally Eq (1), when you plug all this information in gives

tr⁡(γμγνγργσ)=8ηρσημν−8ηνσημρ+8ημσηνρ

− tr⁡(γσγμγνγρ)(4)

The terms inside the trace can be cycled, so

tr⁡(γσγμγνγρ)=tr⁡(γμγνγργσ).

So really (4) is

2 tr⁡(γμγνγργσ)=8ηρσημν−8ηνσημρ+8ημσηνρ

tr⁡(γμγνγργσ)=4(ηρσημν−ηνσημρ+ημσηνρ)
Proof of 6

To show

tr⁡(γ5)=0,

begin with

| tr⁡(γ5) | =tr⁡(γ0γ0γ5) | (because γ0γ0=I4)

|
|

| =−tr⁡(γ0γ5γ0) | (anti-commute the γ5 with γ0)
| =−tr⁡(γ0γ0γ5) | (rotate terms within trace)
| =−tr⁡(γ5) | (remove γ0's)

Add tr⁡(γ5) to both sides of the above to see

2tr⁡(γ5)=0.

Now, this pattern can also be used to show

tr⁡(γμγνγ5)=0.

Simply add two factors of γα, with α different from μ and ν. Anticommute three times instead of once, picking up three minus signs, and cycle using the cyclic property of the trace.

So,

tr⁡(γμγνγ5)=0.
Proof of 7

For a proof of identity 7, the same trick still works unless (μνρσ) is some permutation of (0123), so that all 4 gammas appear. The anticommutation rules imply that interchanging two of the indices changes the sign of the trace, so tr⁡(γμγνγργσγ5) must be proportional to ϵμνρσ (ϵ0123=η0μη1νη2ρη3σϵμνρσ=η00η11η22η33ϵ0123=−1). The proportionality constant is 4i, as can be checked by plugging in (μνρσ)=(0123), writing out γ5, and remembering that the trace of the identity is 4.

Proof of 8

Denote the product of n gamma matrices by Γ=γμ1γμ2…γμn. Consider the Hermitian conjugate of Γ:

| Γ† | =γμn†…γμ2†γμ1†

|

| =γ0γμnγ0…γ0γμ2γ0γ0γμ1γ0 | (since conjugating a gamma matrix with γ0 produces its Hermitian conjugate as described below)
| =γ0γμn…γμ2γμ1γ0 | (all γ0s except the first and the last drop out)

Conjugating with γ0 one more time to get rid of the two γ0s that are there, we see that γ0Γ†γ0 is the reverse of Γ. Now,

tr⁡(γ0Γ†γ0) | =tr⁡(Γ†) | (since trace is invariant under similarity transformations)

|
|

| =tr⁡(Γ*) | (since trace is invariant under transposition)
| =tr⁡(Γ) | (since the trace of a product of gamma matrices is real)

### Normalization

The gamma matrices can be chosen with extra hermiticity conditions which are restricted by the above anticommutation relations however. We can impose

(γ0)†=γ0, compatible with (γ0)2=I4

and for the other gamma matrices (for k = 1, 2, 3)

(γk)†=−γk, compatible with (γk)2=−I4.

One checks immediately that these hermiticity relations hold for the Dirac representation.

The above conditions can be combined in the relation

(γμ)†=γ0γμγ0.

### Charge conjugation

The charge conjugation operator, in any basis, may be defined as

CγμC−1=−(γμ)T

where (⋅)T denotes the matrix transpose. The explicit form that C takes is dependent on the specific representation chosen for the gamma matrices, up to an arbitrary phase factor. This is because although charge conjugation is an automorphism of the gamma group, it is not an inner automorphism (of the group). Conjugating matrices can be found, but they are representation-dependent.

Representation-independent identities include:

Cγ5C−1=+(γ5)TCσμνC−1=−(σμν)TCγ5γμC−1=+(γ5γμ)T

The charge conjugation operator is also unitary C−1=C†, while for Cl1,3(ℝ) it also holds that CT=−C for any representation. Given a representation of gamma matrices, the arbitrary phase factor for the charge conjugation operator can not always be chosen such that C†=CT, as is the case for the common four representations given below, known as Dirac, chiral and Majorana representation.

### Feynman slash notation

The Feynman slash notation is defined by

a/:=γμaμ

for any 4-vector a.

Here are some similar identities to the ones above, but involving slash notation:

* a/b/=[a⋅b−iaμσμνbν]I4
* a/a/=[aμaνγμγν]I4=[12aμaν(γμγν+γνγμ)]I4=[ημνaμaν]I4=a2I4
* tr⁡(a/b/)=4(a⋅b)
* tr⁡(a/b/c/d/)=4[(a⋅b)(c⋅d)−(a⋅c)(b⋅d)+(a⋅d)(b⋅c)]
* tr⁡(γ5a/b/)=0
* tr⁡(γ5a/b/c/d/)=−4iϵμνρσaμbνcρdσ
* γμa/γμ=−2a/[7]
* γμa/b/γμ=4(a⋅b)I4[7]
* γμa/b/c/γμ=−2c/b/a/[7]

where ϵμνρσ is the Levi-Civita symbol and σμν=i2[γμ,γν]. Actually traces of products of odd number of  γ is zero and thus
* tr⁡(a1/a2/⋯an/)=0 for n odd.[8]

Many follow directly from expanding out the slash notation and contracting expressions of the form  aμbνcρ … with the appropriate identity in terms of gamma matrices.

## Other representations

The matrices are also sometimes written using the 2×2 identity matrix, I2, and

γk=(0σk−σk0)

where k runs from 1 to 3 and the σk are Pauli matrices.

### Dirac basis

The gamma matrices we have written so far are appropriate for acting on Dirac spinors written in the Dirac basis; in fact, the Dirac basis is defined by these matrices. To summarize, in the Dirac basis:

γ0=(I200−I2),γk=(0σk−σk0),γ5=(0I2I20).

or using the Kronecker product:

γ0=(σ3⊗I2),γk=(iσ2⊗σk),γ5=(σ1⊗I2).

In the Dirac basis, the charge conjugation operator is real antisymmetric,[9]: 691–700

CD=iγ2γ0=(0−iσ2−iσ20)=(000−100100−1001000).

### Weyl (chiral) basis

Another common choice is the Weyl or chiral basis, in which γk remains the same but γ0 is different, and so γ5 is also different, and diagonal,

γ0=(0I2I20),γk=(0σk−σk0),γ5=(−I200I2),

or in more compact notation:

γμ=(0σμσ‾μ0),σμ≡(1,σi),σ‾μ≡(1,−σi).

The Weyl basis has the advantage that its chiral projections take a simple form,

ψL=12(1−γ5)ψ=(I2000)ψ,ψR=12(1+γ5)ψ=(000I2)ψ.

The idempotence of the chiral projections is manifest.

By slightly abusing the notation and reusing the symbols ψL/R we can then identify

ψ=(ψLψR),

where now ψL and ψR are left-handed and right-handed two-component Weyl spinors.

The charge conjugation operator in this basis is real antisymmetric,

CW=UCDUT=iγ2γ0=(iσ200−iσ2)

The Weyl basis can be obtained from the Dirac basis as

γWμ=UγDμU†,ψW=UψD

via the unitary transform

U=12 (1+γ5γ0)=12 (I2−I2I2I2).

### Weyl (chiral) basis (alternate form)

Another possible choice[10] of the Weyl basis has

γ0=(0−I2−I20),γk=(0σk−σk0),γ5=(I200−I2).

The chiral projections take a slightly different form from the other Weyl choice,

ψR=(I2000)ψ,ψL=(000I2)ψ.

In other words,

ψ=(ψRψL),

where ψL and ψR are the left-handed and right-handed two-component Weyl spinors, as before.

The charge conjugation operator in this basis is

C=iγ2γ0=(−iσ200iσ2)=(0−1001000000100−10)=−iσ3⊗σ2.

This basis can be obtained from the Dirac basis above as γWμ=UγDμU†,ψW=UψD via the unitary transform

U=12 (1−γ5γ0)=12 (I2I2−I2I2).

### Majorana basis

There is also the Majorana basis, in which all of the Dirac matrices are imaginary, and the spinors and Dirac equation are real. Using the Pauli matrices, the basis can be written as

γ0=(0σ2σ20) ,γ1=(iσ300iσ3) ,γ2=(0−σ2σ20),γ3=(−iσ100−iσ1) ,γ5=(σ200−σ2) ,C=(0−iσ2−iσ20) ,

where C is the charge conjugation matrix, which matches the Dirac version defined above.

The reason for making all gamma matrices imaginary is solely to obtain the particle physics metric (+, −, −, −), in which squared masses are positive. The Majorana representation, however, is real. One can factor out the  i to obtain a different representation with four component real spinors and real gamma matrices. The consequence of removing the  i is that the only possible metric with real gamma matrices is (−, +, +, +).

The Majorana basis can be obtained from the Dirac basis above as γMμ=UγDμU†,ψM=UψD via the unitary transform

U=U†=12 (I2σ2σ2−I2).

### Cl1,3(C) and Cl1,3(R)

Template:Summary style The Dirac algebra can be regarded as a complexification of the real algebra Cl1,3(ℝ), called the space time algebra:

Cl1,3(ℂ)=Cl1,3(ℝ)⊗ℂ

Cl1,3(ℝ) differs from Cl1,3(ℂ): in Cl1,3(ℝ) only real linear combinations of the gamma matrices and their products are allowed.

Two things deserve to be pointed out. As Clifford algebras, Cl1,3(ℂ) and Cl4(ℂ) are isomorphic, see classification of Clifford algebras. The reason is that the underlying signature of the spacetime metric loses its signature (1,3) upon passing to the complexification. However, the transformation required to bring the bilinear form to the complex canonical form is not a Lorentz transformation and hence not "permissible" (at the very least impractical) since all physics is tightly knit to the Lorentz symmetry and it is preferable to keep it manifest.

Proponents of geometric algebra strive to work with real algebras wherever that is possible. They argue that it is generally possible (and usually enlightening) to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in a real Clifford algebra that square to −1, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces. Some of these proponents also question whether it is necessary or even useful to introduce an additional imaginary unit in the context of the Dirac equation.[11]: x–xi

In the mathematics of Riemannian geometry, it is conventional to define the Clifford algebra Clp,q(ℝ) for arbitrary dimensions p,q. The Weyl spinors transform under the action of the spin group Spin(n). The complexification of the spin group, called the spinc group Spinℂ(n), is a product Spin(n)×ℤ2S1 of the spin group with the circle S1≅U(1). The product ×ℤ2 just a notational device to identify (a,u)∈Spin(n)×S1 with (−a,−u). The geometric point of this is that it disentangles the real spinor, which is covariant under Lorentz transformations, from the U(1) component, which can be identified with the U(1) fiber of the electromagnetic interaction. The ×ℤ2 is entangling parity and charge conjugation in a manner suitable for relating the Dirac particle/anti-particle states (equivalently, the chiral states in the Weyl basis). The bispinor, insofar as it has linearly independent left and right components, can interact with the electromagnetic field. This is in contrast to the Majorana spinor and the ELKO spinor (Eigenspinoren des Ladungskonjugationsoperators), which cannot (i.e. they are electrically neutral), as they explicitly constrain the spinor so as to not interact with the S1 part coming from the complexification. The ELKO spinor is a Lounesto class 5 spinor.[12]: 84

However, in contemporary practice in physics, the Dirac algebra rather than the space-time algebra continues to be the standard environment the spinors of the Dirac equation "live" in.

## Other representation-free properties

The gamma matrices are diagonalizable with eigenvalues ±1 for γ0, and eigenvalues ±i for γk.

Proof

This can be demonstrated for γ0 and follows similarly for γi. We can rewrite

(γ0)2=+1

(γ0+1)(γ0−1)=0.

By a well-known result in linear algebra, this means there is a basis in which γ0 is diagonal with eigenvalues {±1}.

In particular, this implies that γ0 is simultaneously Hermitian and unitary, while the γi are simultaneously anti–Hermitian and unitary.

Further, the multiplicity of each eigenvalue is two.

Proof

If v is an eigenvector of  γ0 , then  γ1v is an eigenvector with the opposite eigenvalue. Then eigenvectors can be paired off if they are related by multiplication by  γ1. Result follows similarly for  γi.

More generally, if  γμXμ is not null, a similar result holds. For concreteness, we restrict to the positive norm case  γμpμ=p/ with  p⋅p=m2>0. The negative case follows similarly.

Proof

It can be shown

p/2=m2 ,

so by the same argument as the first result,  p/ is diagonalizable with eigenvalues  ±m. We can adapt the argument for the second result slightly. We pick a non-null vector  qμ which is orthogonal to pμ. Then eigenvectors can be paired off similarly if they are related by multiplication by  q/.

It follows that the solution space to  p/−m=0 (that is, the kernel of the left-hand side) has dimension 2\. This means the solution space for plane wave solutions to Dirac's equation has dimension 2\.

This result still holds for the massless Dirac equation. In other words, if pμ null, then p/ has nullity 2\.

Proof

If pμ null, then p/p/=0. By generalized eigenvalue decomposition, this can be written in some basis as diagonal in 2×2 Jordan blocks with eigenvalue 0, with either 0, 1, or 2 blocks, and other diagonal entries zero. It turns out to be the 2 block case. The zero case is not possible as if  γμpμ=0 , by linear independence of the  γμ we must have  pμ=0. But null vectors are by definition non-zero. Consider (qμ)=(|𝐩|,−𝐩) and a zero-eigenvector v of p/. Note qμ is also null and satisfies

p/q/+q/p/=4|𝐩|.−(*)

If p/v=0, then it cannot simultaneously be a zero eigenvector of q/ by (*). Considering q/v, if we apply p/ then we get p/q/v=4|𝐩|v. Therefore, after a rescaling, v and q/v give a 2×2 Jordan block. This gives a pairing. There must be another zero eigenvector of

p/, which can be used to make the second Jordan block.

There is also a pleasant structure to these pairs. If left arrows correspond to application of p/, and right arrows to application of q/, and v is a zero eigenvector of p/, up to scalar factors we have

0<-v↔q/v->0.

## Euclidean Dirac matrices

In quantum field theory one can Wick rotate the time axis to transit from Minkowski space to Euclidean space. This is particularly useful in some renormalization procedures as well as lattice gauge theory. In Euclidean space, there are two commonly used representations of Dirac matrices:

### Chiral representation

γ1,2,3=(0iσ1,2,3−iσ1,2,30),γ4=(0I2I20)

Notice that the factors of i have been inserted in the spatial gamma matrices so that the Euclidean Clifford algebra

{γμ,γν}=2δμνI4

will emerge. It is also worth noting that there are variants of this which insert instead −i on one of the matrices, such as in lattice QCD codes which use the chiral basis.

In Euclidean space,

γM5=i(γ0γ1γ2γ3)M=1i2(γ4γ1γ2γ3)E=(γ1γ2γ3γ4)E=γE5.

Using the anti-commutator and noting that in Euclidean space (γμ)†=γμ, one shows that

(γ5)†=γ5

In chiral basis in Euclidean space,

γ5=(−I200I2)

which is unchanged from its Minkowski version.

### Non-relativistic representation

γ1,2,3=(0−iσ1,2,3iσ1,2,30) ,γ4=(I200−I2),γ5=(0−I2−I20)

## Footnotes

1. ↑ The set of matrices (Γa) = (γ μ, i γ 5 ) with a = (0, 1, 2, 3, 4)  satisfy the five-dimensional Clifford algebra {Γa, Γb} = 2 ηab

## See also

* Pauli matrices
* Gell-Mann matrices
* Higher-dimensional gamma matrices
* Fierz identity
* Spacetime algebra

## Citations

1. ↑ Kukin 2016.
2. ↑ Lonigro 2023.
3. ↑ Jost 2002.
4. ↑ Tong 2007, These introductory quantum field theory notes are for Part III (masters level) students..
5. ↑ Weinberg 2002, § 5.5.
6. ↑ de Wit & Smith 2012.
7. ↑ 7.0 7.1 7.2 Feynman, Richard P. (1949). "Space-time approach to quantum electrodynamics". Physical Review 76 (6): 769–789. doi:10.1103/PhysRev.76.769. Bibcode: 1949PhRv...76..769F. https://journals.aps.org/pr/abstract/10.1103/PhysRev.76.769.
8. ↑ Kaplunovsky 2008.
9. ↑ Itzykson & Zuber 2012.
10. ↑ Kaku 1993.
11. ↑ Hestenes 2015.
12. ↑ Rodrigues & Oliveira 2007.

## References

* de Wit, B.; Smith, J. (2 December 2012) (in en). Field Theory in Particle Physics, Volume 1. Elsevier. ISBN 978-0-444-59622-2. https://books.google.com/books?id=cCAGmj6XEfkC. [1]
* Halzen, Francis; Martin, Alan D. (17 May 2008) (in en). Quark & Leptons: An Introductory Course in Modern Particle Physics. Wiley India Pvt. Limited. ISBN 978-81-265-1656-8. https://archive.org/details/quarksleptonsint0000halz.
* Hestenes, David (2015) (in en). Space-Time Algebra. Springer International Publishing. ISBN 978-3-319-18412-8. https://books.google.com/books?id=SfRJrgEACAAJ.
* Itzykson, Claude; Zuber, Jean-Bernard (20 September 2012) (in en). Quantum Field Theory. Courier Corporation. Appendix A. ISBN 978-0-486-13469-7. https://books.google.com/books?id=CxYCMNrUnTEC.
* Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Springer. p. 68, Corollary 1.8.1. ISBN 978-3-540-42627-1. https://books.google.com/books?id=iDGzbwAACAAJ.
* Kaku, Michio (1993) (in en). Quantum Field Theory: A Modern Introduction. Oxford University Press. ISBN 978-0-19-509158-8. https://books.google.com/books?id=4mOMcRFYnzQC.
* Kaplunovsky, Vadim (2008). "Traceology". University of Texas at Austin. http://bolvan.ph.utexas.edu/~vadim/classes/2008f.homeworks/traceology.pdf.

* Kukin, V.D. (2016). "Dirac matrices - Encyclopedia of Mathematics". https://encyclopediaofmath.org/wiki/Dirac_matrices.

* Lonigro, Davide (2023). "Dimensional reduction of the Dirac equation in arbitrary spatial dimensions". The European Physical Journal Plus 138 (4): 324. doi:10.1140/epjp/s13360-023-03919-0. Bibcode: 2023EPJP..138..324L.
* Pauli, W. (1936). "Contributions mathématiques à la théorie des matrices de Dirac". Annales de l'Institut Henri Poincaré 6: 109. http://www.numdam.org/item?id=AIHP_1936__6_2_109_0.

* Peskin, M.; Schroeder, D. (1995). An Introduction to Quantum Field Theory. Westview Press. chapter 3.2. ISBN 0-201-50397-2. https://archive.org/details/introductiontoqu0000pesk.

* Rodrigues, Waldyr A.; Oliveira, Edmundo C. de (2007) (in en). The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach. Springer Science & Business Media. ISBN 978-3-540-71292-3. https://books.google.com/books?id=fW-39oNxz8wC.
* Tong, David (2007). Lectures on Quantum Field Theory (course lecture notes). David Tong at University of Cambridge. p. 93. http://www.damtp.cam.ac.uk/user/tong/qft.html. Retrieved 2015-03-07.

* Weinberg, S. (2002). The Quantum Theory of Fields. 1. Cambridge University Press. ISBN 0-521-55001-7. https://archive.org/details/quantumtheoryoff00stev.

* Zee, A. (2003). Quantum Field Theory in a Nutshell. Princeton, NJ: Princeton University Press. chapter II.1. ISBN 0-691-01019-6. https://archive.org/details/isbn_9780691140346.

## External links

* Dirac matrices on mathworld including their group properties
* Dirac matrices as an abstract group on GroupNames
* Hazewinkel, Michiel, ed. (2001), "Dirac matrices", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/d032690

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