Higher-dimensional gamma matrices

In mathematical physics, **higher-dimensional gamma matrices** generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically invariant wave equations for fermions (such as spinors) in arbitrary space-time dimensions, notably in string theory and supergravity.

Consider a space-time of dimension d with the flat Minkowski metric,

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \eta = \parallel \eta\_{a b} \parallel = \text{diag}(+1,-1, \dots, -1) ~, }**

where  *a,b* = 0,1, ..., *d*−1. Set  *N*= 2⌊*d*/2⌋. The standard Dirac matrices correspond to taking *d* = *N* = 4.

The higher gamma matrices are a d-long sequence of complex *N*×*N* matrices **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \Gamma\_i,\ i=0,\ldots,d-1}**
 which satisfy the anticommutator relation from the Clifford algebra *Cℓ**1,d−1*(**R**) (generating a representation for it),

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \{ \Gamma\_a ~,~ \Gamma\_b \} = \Gamma\_a\Gamma\_b + \Gamma\_b\Gamma\_a = 2 \eta\_{a b} I\_N ~,}**

where *IN* is the identity matrix in N dimensions. (The spinors acted on by these matrices have N components in d dimensions.) Such a sequence exists for all values of d and can be constructed explicitly, as provided below.

The gamma matrices have the following property under hermitian conjugation,

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \Gamma\_0^\dagger= +\Gamma\_0 ~,~ \Gamma\_i^\dagger= -\Gamma\_i ~(i=1,\dots,d-1) ~. }**

## Charge conjugation

Since the groups generated by  *Γa, −ΓaT, ΓaT* are the same, we can look for a similarity transformation which connects them all. This transformation is generated by a respective charge conjugation matrix.

Explicitly, we can introduce the following matrices

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(+)} \Gamma\_a C\_{(+)}^{-1} = + \Gamma\_a^T }**

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(-)} \Gamma\_a C\_{(-)}^{-1} = - \Gamma\_a^T ~.}**

They can be constructed as real matrices in various dimensions, as the following table shows. In even dimension both **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_\pm}**
 exist, in odd dimension just one.

Note that **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C^\*\_{(\pm)}= C\_{(\pm)} }**
 is a basis choice.

## Symmetry properties

We denote a product of gamma matrices by

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \Gamma\_{abc \ldots} = \Gamma\_a \cdot \Gamma\_b \cdot \Gamma\_c \cdots \ }**

and note that the anti-commutation property allows us to simplify any such sequence to one in which the indices are distinct and increasing. Since distinct **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \Gamma\_a}**
 anti-commute this motivates the introduction of an anti-symmetric "average". We introduce the anti-symmetrised products of distinct n-tuples from 0,...,d−1:

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \Gamma\_{a\_1 \dots a\_n} = \frac{1}{n!} \sum\_{\pi \in S\_n} \epsilon(\pi) \Gamma\_{a\_{\pi(1)}} \cdots \Gamma\_{a\_{\pi(n)}} ~,}**

where π runs over all the permutations of n symbols, and  ϵ is the alternating character. There are 2*d* such products, but only N2 are independent, spanning the space of N×N matrices.

Typically, Γ*ab* provide the (bi)spinor representation of the *d(d−1)*/2 generators of the higher-dimensional Lorentz group, SO+(1,*d*−1), generalizing the 6 matrices *σμν* of the spin representation of the Lorentz group in four dimensions.

For even d, one may further define the hermitian *chiral matrix*

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \Gamma\_\text{chir}= i^{d/2-1} \Gamma\_0 \Gamma\_1 \dots \Gamma\_{d-1} ~,}**

such that {*Γ*chir , *Γa*} = 0 and *Γ*chir2=1. (In odd dimensions, such a matrix would commute with all Γ*a*s and would thus be proportional to the identity, so it is not considered.)

A Γ matrix is called symmetric if

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle ( C \Gamma\_{a\_1 \dots a\_n} )^T = + ( C \Gamma\_{a\_1 \dots a\_n} ) ~;}**

otherwise, for a − sign, it is called antisymmetric.
In the previous expression, C can be either **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(+)} }**

or **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(-)} }**
. In odd dimension, there is no ambiguity, but
in even dimension it is better to choose whichever one of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(+)} }**

or **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(-)} }**
 allows for Majorana spinors. In d=6, there is no such
criterion and therefore we consider both.

## Identities

The proof of the trace identities for gamma matrices is independent of dimension. One therefore only needs to remember the 4D case and then change the overall factor of 4 to **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \operatorname{tr}(I\_N)}**
. For other identities (the ones that involve a contraction), explicit functions of **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle d}**
 will appear.

1. **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \gamma^\mu\gamma\_\mu = d I\_N}**
2. **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \gamma^\mu\gamma^\nu\gamma\_\mu = (2 - d) \gamma^\nu}**
3. **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \gamma^\mu\gamma^\nu\gamma^\rho\gamma\_\mu = 2 \gamma^\rho\gamma^\nu + (d - 2) \gamma^\nu\gamma^\rho}**
4. **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\gamma\_\mu = 2 \gamma^\rho\gamma^\sigma\gamma^\nu - 2 \gamma^\nu\gamma^\sigma\gamma^\rho - (d - 2) \gamma^\nu\gamma^\rho\gamma^\sigma }**

Even when the number of physical dimensions is four, these more general identities are ubiquitous in loop calculations due to dimensional regularization.

## Example of an explicit construction in the chiral basis

The Γ matrices can be constructed recursively, first in all even dimensions, d= 2k, and thence in odd ones, 2k+1.

### *d* = 2

Using the Pauli matrices, take

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \gamma\_0= \sigma\_1 ~,~ \gamma\_1= -i \sigma\_2 }**

and one may easily check that the charge conjugation matrices are

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(+)}= \sigma\_1 = C\_{(+)}^\* = s\_{(2,+)} C\_{(+)}^T = s\_{(2,+)} C\_{(+)}^{-1} \qquad s\_{(2,+)}=+1 }**

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(-)}= i \sigma\_2 = C\_{(-)}^\* = s\_{(2,-)} C\_{(-)}^T = s\_{(2,-)} C\_{(-)}^{-1} \qquad s\_{(2,-)}=-1 ~.}**

One may finally define the hermitian chiral γchir to be

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \gamma\_\text{chir}= \gamma\_0 \gamma\_1 = \sigma\_3 = \gamma\_\text{chir}^\dagger ~.}**

### Generic even *d* = 2*k*

One may now construct the  *Γa* , (*a*=0, ... , *d*+1), matrices and the charge conjugations C(±) in d+2 dimensions, starting from the *γa'* , ( *a'* =0, ... , *d*−1), and c(±) matrices in d dimensions.

Explicitly,

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \Gamma\_{a'} = \gamma\_{a'} \otimes \sigma\_3 ~~~(a'=0, \dots, d-1)~,~~~ \Gamma\_{d} = I \otimes (i \sigma\_1)~,~~~ \Gamma\_{d+1}= I \otimes (i \sigma\_2) ~.}**

One may then construct the charge conjugation matrices,

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(+)} = c\_{(-)} \otimes \sigma\_1 ~, \qquad C\_{(-)} = c\_{(+)} \otimes (i \sigma\_2) ~,}**

with the following properties,

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(+)}= C\_{(+)}^\* = s\_{(d+2,+)} C\_{(+)}^T = s\_{(d+2,+)} C\_{(+)}^{-1} \qquad s\_{(d+2,+)}= s\_{(d,-)} }**

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(-)}= C\_{(-)}^\* = s\_{(d+2,-)} C\_{(-)}^T = s\_{(d+2,-)} C\_{(-)}^{-1} \qquad s\_{(d+2,-)}=-s\_{(d,+)} ~.}**

Starting from the sign values for d=2, s(2,+)=+1 and s(2,−)=−1, one may fix all subsequent signs s(*d*,±)  which have periodicity 8; explicitly, one finds

 **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle s\_{(d,-)} }** | +1

Again, one may define the hermitian chiral matrix in d+2 dimensions as

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle \Gamma\_\text{chir}= \alpha\_{d+2} \Gamma\_0 \Gamma\_1 \dots \Gamma\_{d+1} = \gamma\_\text{chir} \otimes \sigma\_3 ~,~~~~~~ \alpha\_d= i^{d/2-1} ~,}**

which is diagonal by construction and transforms under charge conjugation as

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(\pm)} \Gamma\_\text{chir} C\_{(\pm)}^{-1} = \beta\_{d+2} \Gamma\_\text{chir}^T ~~~~~~~~ \beta\_d= (-)^{d(d-1)/2} ~. }**

It is thus evident that {*Γ*chir , *Γa*}  = 0.

### Generic odd *d* = 2*k* + 1

Consider the previous construction for  d−1 (which is even) and simply take all  *Γa* (*a*=0, ..., *d*−2) matrices, to which append its  *iΓ*chir ≡ *Γd−1*. (The i is required in order to yield an antihermitian matrix, and extend into the spacelike metric).

Finally, compute the charge conjugation matrix: choose between **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(+)} }**
 and **Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(-)} }**
, in such a way that  *Γd−1* transforms as all the other  *Γ* matrices. Explicitly, require

**Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest\_v1/":): {\displaystyle C\_{(s)} \Gamma\_\text{chir} C\_{(s)}^{-1} = \beta\_{d} \Gamma\_\text{chir}^T = s \Gamma\_\text{chir}^T ~.}**

As the dimension d ranges, patterns typically repeat themselves with period 8. (cf. the Clifford algebra clock.)