Gamma matrices
In mathematical physics, the gamma matrices, {γ0, γ1, γ2, γ3}, also known as the Dirac matrices, form a matrixvalued representation of a set of orthogonal basis vectors for contravariant vectors in space time, from which can be constructed a Clifford algebra.
This in turn makes possible the introduction of spinors to represent spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin½ particles.
In Dirac representation, the four contravariant gamma matrices are
Contents
 1 Mathematical structure
 2 Physical structure
 3 Expressing the Dirac equation
 4 Identities
 5 The Fifth Gamma Matrix, γ5
 6 Other representations
 7 Euclidean Dirac matrices
 8 See also
 9 References
Mathematical structure
The defining property for the gamma matrices to form a Clifford algebra is the anticommutation relation
where is the Minkowski metric with signature (+ − − −) and is the unit matrix.
This defining property is considered to be more fundamental than the numerical values used in the gamma matrices, so other sign conventions for the metric necessitate a change in the definitions of the gamma matrices.
Covariant gamma matrices are defined by
 ,
and Einstein notation is assumed.
Physical structure
The 4tuple is often loosely described as a 4vector (where e0 to e3 are the basis vectors of the 4vector space). But this is misleading. Instead is more appropriately seen as a mapping operator, taking in a 4vector and mapping it to the corresponding vector in the Clifford algebra representation.
This is symbolised by the useful Feynman slash notation,
Slashed quantities like "live" in the multilinear Clifford algebra, with its own set of basis directions — they are immune to changes in the 4vector basis.
On the other hand, one can define a transformation identity for the mapping operator . If is the spinor representation of an arbitrary Lorentz transformation , then we have the identity
This says essentially that an operator mapping from the old 4vector basis to the old Clifford algebra basis is equivalent to a mapping from the new 4vector basis to a correspondingly transformed new Clifford algebra basis . Alternatively, in pure index terms, it shows that γμ transforms appropriately for an object with one contravariant 4vector index and one covariant and one contravariant Dirac spinor index.
Given the above transformation properties of γμ, if ψ is a Dirac spinor then the product γμψ transforms as if it were the product of a contravariant 4vector with a Dirac spinor. In expressions involving spinors, then, it is often appropriate to treat γμ as if it were simply a vector.
There remains a final key difference between γμ and any nonzero 4vector: γμ does not point in any direction. More precisely, the only way to make a true vector from γμ is to contract its spinor indices, leaving a vector of traces
This property of the gamma matrices is essential for them to serve as coefficients in the Dirac equation.
Expressing the Dirac equation
In natural units, the Dirac equation may be written as
where ψ is a Dirac spinor. Here, if γμ were an ordinary 4vector, 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
Applying to both sides yields
which is the KleinGordon equation. Thus, as the notation suggests, the Dirac particle has mass m.
Identities
The following identities follow from the fundamental anticommutation relation, so they hold in any basis.
Normalisation
Because of the above anticommutation relation, we can show:

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

 , and
These results can be generalized by the relation
Miscellaneous identities
 Num Identity 1 2 3 4
To show
one begins with the standard anticommutation relation
One can make this situation look similar by using the metric η:
To show
We again will use the standard commutation relation. So start:
Proof of 3To show
Use the anticommutator to shift γμ to the right
Using the relation γμγμ = 4I we can contract the last two gammas, and get
Finally using the anticommutator identity, we get
Proof of 4 (anticommutator identity) (using identity 3) (raising an index) (anticommutator identity) (2 terms cancel)Trace identities
 Num Identity 1 trace of any product of an odd number of γμ is zero 2 3 4 5
Proving the above involves use of four main properties of the Trace operator:
 tr(A + B) = tr(A) + tr(B)
 tr(rA) = r tr(A)
 tr(ABC) = tr(CAB) = tr(BCA)
To show
First note that
We'll also use two facts about the fifth gamma matrix (shown later in this article) that says:
So lets use these two facts to prove this identity for the first nontrivial case: the trace of three gamma matrices. Step one is to put in one pair of 's in front of the three original 's, and step two is to swap the matrix back to the original position, after making use of the cyclicity of the trace.
This can only be fulfilled if
Proof of 2To show
Begin with,
Proof of 3For the term on the right, we'll continue the pattern of swapping with its neighbor to the left,
Again, for the term on the right swap with its neighbor to the left,
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 2 to simplify terms like so:
So finally Eq (1), when you plug all this in information in gives
The terms inside the trace can be cycled, so
So really (4) is
or
Proof of 4To show

 ,
begin with

 (because ) (anticommute the with ) (rotate terms within trace) (remove 's)
Add to both sides of the above to see

 .
Now, this pattern can also be used to show

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

 .
For a proof of identity 5, the same trick still works unless 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 must be proportional to . The proportionality constant is , as can be checked by plugging in , writing out , and remembering that the trace of the identity is 4.
Feynman slash notation
The contraction of the mapping operator γμ with a vector aμ maps the vector out of the 4vector representation. So, it is common to write identities using the Feynman slash notation, defined by
Here are some similar identities to the ones above, but involving slash notation:
 where
 is the LeviCivita symbol and .
The Fifth Gamma Matrix, γ5
It is useful to define the product of the four gamma matrices as follows:
 (in the Dirac basis).
Although γ5 uses the letter gamma, it is not one of the gamma matrices. The number 5 is a relic of old notation in which γ0 was called "γ4".
γ5 has also an alternative form
due to the anticommutation relations of the (other) gamma matrices.
This matrix is useful in discussions of quantum mechanical chirality. For example, a Dirac field can be projected onto its lefthanded and righthanded components by:
 .
Some properties are:
 It is hermitian:

 ,
 Its eigenvalues are ±1, because:
 It anticommutes with the four gamma matrices:

 ,
Other representations
The matrices are also sometimes written using the 2x2 identity matrix, I, and
where i runs from 1 to 3 and the σi 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:
Weyl basis
Another common choice is the Weyl or chiral basis, in which γi remains the same but γ0 is different, and so γ5 is also different:
The Weyl basis has the advantage that its chiral projections take a simple form:
By a slight abuse of notation we can then identify
where now ψL and ψR are lefthanded and righthanded twocomponent Weyl spinors.
Majorana basis
There's also a Majorana basis, in which all of the Dirac matrices are imaginary. In terms of the Pauli matrices, it can be written as
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
Different from Minkowski space, in Euclidean space,
 γ5 = γ1γ2γ3γ4 = γ5 +
So in Chiral basis,
Nonrelativistic representation
See also
References
 Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0471887412.
 A. Zee, Quantum Field Theory in a Nutshell (2003), Princeton University Press: Princeton, New Jersey. ISBN 0691010196. See chapter II.1.
 M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995) See chapter 3.2.
 W. Pauli (1936). "Contributions mathématiques à la théorie des matrices de Dirac". Ann. Inst. Henri Poincaré 6: 109.
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