Gamma matrices

In mathematical physics, the gamma matrices, 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 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 particles. Gamma matrices were introduced by Paul Dirac in 1928.[1][2]

In Dirac representation, the four contravariant gamma matrices are

is the time-like, Hermitian matrix. The other three are space-like, anti-Hermitian matrices. More compactly, and where denotes the Kronecker product and the (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

The "fifth matrix" 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.

Mathematical structure

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

where the curly brackets represent 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

Physical structure

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The Clifford algebra 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 as the set of linear operators from any four-dimensional complex vector space to itself. More simply, given a basis for V, 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 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 If S belongs to a representation of the Lorentz group, then the induced action ES 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 given by equation:

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 written in indexed notation. 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 basis 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 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 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

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In natural units, the Dirac equation may be written as

where is a Dirac spinor.

Switching to Feynman notation, the Dirac equation is

The fifth "gamma" matrix, γ5

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It is useful to define a product of the four gamma matrices as , so that

(in the Dirac basis).

Although uses the letter gamma, it is not one of the gamma matrices of The index number 5 is a relic of old notation: used to be called "".

has also an alternative form:

using the convention or

using the convention Proof:

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

where is the type (4,4) generalized Kronecker delta in 4 dimensions, in full antisymmetrization. If denotes the Levi-Civita symbol in n dimensions, we can use the identity . Then we get, using the convention

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:

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

In fact, and are eigenvectors of since

and

Five dimensions

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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).[a] .[4]: 97  In metric signature (4,1), the set {γ 0, γ 1, γ 2, γ 3, γ 5} is used, where the γμ 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

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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 ).

Miscellaneous identities

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1.

2.

3.

4.

5.

6. where

Trace identities

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The gamma matrices obey the following trace identities:

  1. Trace of any product of an odd number of is zero
  2. Trace of times a product of an odd number of is still zero

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)

Normalization

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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 non-compactness of the Lorentz group.[citation needed]

Charge conjugation

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

where denotes the matrix transpose. The explicit form that 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:

The charge conjugation operator is also unitary , while for it also holds that for any representation. Given a representation of gamma matrices, the arbitrary phase factor for the charge conjugation operator can also be chosen such that , as is the case for the four representations given below (Dirac, Majorana and both chiral variants).

Feynman slash notation

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The Feynman slash notation is defined by

for any 4-vector .

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

  • [7]
  • [7]
  • [7]
    where is the Levi-Civita symbol and Actually traces of products of odd number of is zero and thus
  • for n odd.[8]

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

Other representations

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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.

Dirac basis

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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:

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