Gamma Matrices

\gamma^0 = \begin{pmatrix} I & 0 \ 0 & -I \end{pmatrix} = 
\begin{pmatrix} 
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \ 
0 & 0 & -1 & 0 \
0 & 0 & 0 & -1 \end{pmatrix}

where ''I'' is the 2x2 Identity Matrix and
:

\gamma^i = \begin{pmatrix} 0 & \sigma^i \ -\sigma^i & 0 \end{pmatrix}

where ''i'' runs from 1 to 3 and the σⁱ are Pauli Matrices . Explicitly,

\gamma^1 \!=\! \begin{pmatrix}
0 & 0 & 0 & 1 \
0 & 0 & 1 & 0 \
0 & -1 & 0 & 0 \
-1 & 0 & 0 & 0 \end{pmatrix},
 \gamma^2 \!=\! \begin{pmatrix}
0 & 0 & 0 & -i \
0 & 0 & i & 0 \
0 & i & 0 & 0 \
-i & 0 & 0 & 0 \end{pmatrix},
\gamma^3 \!=\! \begin{pmatrix}
0 & 0 & 1 & 0 \
0 & 0 & 0 & -1 \
-1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \end{pmatrix}.

MATHEMATICAL STRUCTURE

The gamma matrices form a Clifford Algebra whose defining property is
:

\displaystyle\{ \gamma^\mu, \gamma^
u \} = 2 \eta^{\mu 
u} I

where the curly brackets denote the Anticommutator , η is the Minkowski Metric with signature (+ − − −), and ''I'' is the 4x4 identity matrix. This defining property is considered to be more fundamental than the numerical values used in the gamma matrices, so other Sign Convention s for the metric necessitate a change in the definitions of the gamma matrices.

Covariant gamma matrices are defined by
:

\displaystyle \gamma_\mu := \eta_{\mu 
u} \gamma^
u

,
and Einstein Notation is assumed.

PHYSICAL STRUCTURE

The 4-tuple

\displaystyle\gamma^\mu=(\gamma^0,\gamma^1,\gamma^2,\gamma^3)

has the same numerical values regardless of reference frame, so it is not a vector but a Scalar quantity with the trivial transformation

\gamma^\mu	o\gamma^\mu

. On the other hand, if

\displaystyle\lambda

is the Spinor representation of an arbitrary Lorentz Transformation

\displaystyle\Lambda

, then we have the identity
:

\displaystyle\gamma^\mu=\Lambda^\mu{}_
u\lambda\gamma^
u\lambda^{-1}

and so

\gamma^\mu

also transforms appropriately as an object with one Contravariant 4-vector index and one covariant and one contravariant Dirac Spinor index.

Given the above transformation properties of

\gamma^\mu

, if

\psi

is a Dirac spinor then the product

\gamma^\mu\psi

transforms ''as if'' it were the product of a contravariant 4-vector with a Dirac spinor. In expressions involving spinors, then, it is often appropriate to treat

\gamma^\mu

as if it were simply a vector.

There remains a key difference between

\gamma^\mu

and any nonzero 4-vector:

\gamma^\mu

does not point in any direction. More precisely, the only way to make a true vector from

\gamma^\mu

is to contract its spinor indices, leaving a vector of Traces
:

\displaystyle tr(\gamma^\mu)= (0, 0, 0, 0)

This property of the gamma matrices is essential for them to serve as coefficients in the Dirac equation.

FEYNMAN NOTATION

Since the gamma matrices act like a vector, their contraction with a vector acts like a scalar.
Such a contraction can be denoted using a notation due to Feynman:
:

ot\!\,\, a := \gamma^\mu a_\mu

.
For more, see '' Feynman Slash Notation ''.

EXPRESSING THE DIRAC EQUATION

In Natural Units , the Dirac equation may be written as
:

(i \gamma^\mu \partial_\mu - m) \psi = 0

where ψ is a Dirac spinor. Here, if

\gamma^\mu

were an ordinary 4-vector, then it would pick out a preferred direction in spacetime, and the Dirac equation would not be Lorentz invariant.

Switching to Feynman notation, the Dirac equation is
:

(i 
ot\!\;\partial - m) \psi = 0.

Applying

-(i 
ot\!\;\partial + m)

to both sides yields
:

(
ot\!\;\partial^2 + m^2) \psi = (\partial^2 + m^2) \psi = 0,

which is the Klein-Gordon Equation . Thus, as the notation suggests, the Dirac particle has mass ''m''.

THE FIFTH GAMMA MATRIX, <MATH> \GAMMA^5 </MATH>

It is useful to define the product of the four gamma matrices as follows:
:

\gamma^5 := i\gamma^0\gamma^1\gamma^2\gamma^3 = \begin{pmatrix}
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \end{pmatrix}

.
Although

\gamma^5

uses the letter gamma, it is not one of ''the'' gamma matrices. The number 5 is a relic of old notation in which

\gamma^0

was called "

\gamma^4

".

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:
:

\psi_L= rac{1-\gamma^5}{2}\psi, \qquad\psi_R= rac{1+\gamma^5}{2}\psi

.

OTHER REPRESENTATIONS

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:

:

\gamma^0 = \begin{pmatrix} I & 0 \ 0 & -I \end{pmatrix},\quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \ -\sigma^i & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} 0 & I \ I & 0 \end{pmatrix}.

Weyl basis

Another common choice is the ''Weyl'' or ''chiral basis'', in which

\gamma^i

remains the same but

\gamma^0

is different, and so

\gamma^5

is also different:

:

\gamma^0 = \begin{pmatrix} 0 & I \ I & 0 \end{pmatrix},\quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \ -\sigma^i & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} -I & 0 \ 0 & I \end{pmatrix}.

The Weyl basis has the advantage that its Chiral Projections take a simple form:

:

\psi_L=\begin{pmatrix} I & 0 \0 & 0 \end{pmatrix}\psi,\quad \psi_R=\begin{pmatrix} 0 & 0 \0 & I \end{pmatrix}\psi.

By a slight Abuse Of Notation we can then identify

:

\psi=\begin{pmatrix} \psi_L \\psi_R \end{pmatrix},

where now

\psi_L

and

\psi_R

are left-handed and right-handed
two-component Weyl Spinors .

Majorana basis

There's also a Majorana basis, in which all of the Dirac matrices are imaginary.

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 representation of Dirac Matrices:

Chiral representation

:

\gamma^{1,2,3} = \begin{pmatrix} 0 & -i \sigma^{1,2,3} \ i \sigma^{1,2,3} & 0 \end{pmatrix}, \quad
\gamma^4=\begin{pmatrix} 0 & I \ I & 0 \end{pmatrix}

Different from Minkowski space, in Euclidean space,

:

\gamma^5 = \gamma^1 \gamma^2 \gamma^3 \gamma^4 = \gamma^{5+}

So in Chiral basis,

:

\gamma^5=\gamma^1 \gamma^2 \gamma^3 \gamma^4 = \begin{pmatrix} I & 0 \ 0 & -I \end{pmatrix}

Non-relativistic representation

:

\gamma^{1,2,3} = \begin{pmatrix} 0 & -i \sigma^{1,2,3} \ i \sigma^{1,2,3} & 0 \end{pmatrix}, \quad
\gamma^4=\begin{pmatrix} I & 0 \ 0 & -I \end{pmatrix}, \quad 
\gamma^5=\begin{pmatrix} 0 & -I \ -I & 0 \end{pmatrix}

IDENTITIES

The following identities are useful for calculations involving gamma matrices. They follow from the fundamental anticommutation relation, so they hold in any basis.

The trace of any product of an odd number of gamma matrices is zero.

$\operatorname{tr} (\gamma^\mu\gamma^$

$\operatorname{tr} (\gamma^\mu\gamma^$

u}\eta^{ ho\sigma} - \eta^{\mu ho}\eta^{ u\sigma} + \eta^{\mu\sigma}\eta^{ u ho})

$\operatorname{tr}(\gamma^5)=\operatorname{tr} (\gamma^\mu\gamma^$

$\operatorname{tr} (\gamma^\mu\gamma^$

u ho\sigma}

$\displaystyle\gamma^\mu\gamma_\mu=4$

$\displaystyle\gamma^\mu\gamma^$

$\displaystyle\gamma^\mu\gamma^$

u ho}

$\displaystyle\gamma^\mu\gamma^$

u

SEE ALSO

Pauli Matrices

Gell-Mann Matrices

REFERENCES

A. Zee, ''Quantum Field Theory in a Nutshell'' (2003), Princeton University Press: Princeton, New Jersey. ISBN 0-691-01019-6. ''See chapter II.1''.

M. Peskin, D. Schroeder, ''An Introduction to Quantum Field Theory'' (Westview Press, 1995) 0201503972 ''See chapter 3.2''.

	CATEGORIES ABOUT GAMMA MATRICES

	quantum field theory

	spinors

	matrices

	clifford algebras

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Information About

Gamma Matrices