In mathematical physics, the gamma matrices, , also known as 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-½ particles.
In Dirac representation, the four contravariant gamma matrices are
Analogous sets of gamma matrices can be defined in any dimension and signature of the metric. For example the Pauli matrices are a set of "gamma" matrices in dimension 3 with metric of Euclidean signature (3,0).
The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation
where is the anticommutator, is the Minkowski metric with signature (+ − − −) and 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
and Einstein notation is assumed.
Note that the other sign convention for the metric, (− + + +) necessitates either a change in the defining equation:
or a multiplication of all gamma matrices by , 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
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 4-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 SES−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 ↦ SES−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
showing that the 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. This means that quantities of the form
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 unit vectors eμ of V, or any 4-dimensional vector space, to basis vectors γμ. The transformation rule for slashed quantities is simply
One should note that 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) = (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. But it is noted here that the Clifford algebra has no subspace being the representation space of a spin representation of the Lorentz group in the context used here.
Expressing the Dirac equation
In natural units, the Dirac equation may be written as
where is a Dirac spinor.
Switching to Feynman notation, the Dirac equation is
It is useful to define the product of the four gamma matrices as follows:
- (in the Dirac basis).
Although uses the letter gamma, it is not one of the gamma matrices of Cℓ1,3(R). The number 5 is a relic of old notation in which was called "".
has also an alternative form:
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:
Some properties are:
- Its eigenvalues are ±1, because:
- It anticommutes with the four gamma matrices:
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 ).
The gamma matrices obey the following trace identities:
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(CAB) = tr(BCA)
Proof of 1
First note that
We'll also use two facts about the fifth gamma matrix that says:
So lets 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 '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
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 2
If an odd number of gamma matrices appear in a trace followed by , our goal is to move 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
For 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 3 to simplify terms like so:
So finally Eq (1), when you plug all this information in gives
The terms inside the trace can be cycled, so
So really (4) is
Proof of 5
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.
Proof of 6
For a proof of identity 6, 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.
The gamma matrices can be chosen with extra hermiticity conditions which are restricted
by the above anticommutation relations however. We can impose
- , compatible with
and for the other gamma matrices (for k = 1, 2, 3)
- , compatible with
One checks immediately that these hermiticity relations hold for the Dirac representation.
The above conditions can be combined in the relation
The hermiticity conditions are not invariant under the action of a Lorentz transformation because is not necessarily a unitary transformation due to the noncompactness of the Lorentz group.
Feynman slash notation
The Feynman slash notation is defined by
for any 4-vector a.
Here are some similar identities to the ones above, but involving slash notation:
- is the Levi-Civita symbol and
The matrices are also sometimes written using the 2×2 identity matrix, , and
where k runs from 1 to 3 and the σk are Pauli matrices.
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:
Another common choice is the Weyl or chiral basis, in which remains the same but is different, and so is also different, and diagonal,
or in more compact notation:
The Weyl basis has the advantage that its chiral projections take a simple form,
The idempotence of the chiral projections is manifest.
By slightly abusing the notation and reusing the symbols we can then identify
where now and are left-handed and right-handed
two-component Weyl spinors.
Another possible choice of the Weyl basis has
The chiral projections take a slightly different form from the other Weyl choice,
In other words,
where and are the left-handed and right-handed
two-component Weyl spinors, as before.
There is also the Majorana basis, in which all of the Dirac matrices are imaginary and spinors are real. In terms of the Pauli matrices, it can be written as
The reason for making the 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 to obtain a different representation with four component real spinors and real gamma matrices. The consequence of removing the is that the only possible metric with real gamma matrices is (−,+,+,+).
Cℓ1,3(C) and Cℓ1,3(R)
The Dirac algebra can be regarded as a complexification of the real algebra Cℓ1,3(R), called the space time algebra:
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.
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.
However, in contemporary practice, the Dirac algebra rather than the space time algebra continues to be the standard environment the spinors of the Dirac equation "live" in.
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:
Notice that the factors of have been inserted in the spatial gamma matrices so that the Euclidean Clifford algebra
will emerge. It is also worth noting that there are variants of this which insert instead on one of the matrices, such as in lattice QCD codes which use the chiral basis.
Different from Minkowski space, in Euclidean space,
So in Chiral basis,
- ↑ Michio Kaku, Quantum Field Theory, ISBN 0-19-509158-2, appendix A
- 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) [ISBN 0-201-50397-2] See chapter 3.2.
- Dirac matrices on mathworld including their group properties