Gamma matrices

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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 Cl1,3(). 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-1/2 particles.

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.

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 Pauli matrices are a set of "gamma" matrices in dimension 3 with metric of Euclidean signature (3, 0). In 5 spacetime dimensions, the 4 gammas above together with the fifth gamma-matrix to be presented below generate the Clifford algebra.

Mathematical structure[edit]

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

Physical structure[edit]

The Clifford algebra Cl1,3() 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(), as the set of linear operators from any 4 dimensional complex vector space to itself. More simply, given a basis for V, Cl1,3() 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() 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() ≈ End(Ux). If S belongs to a representation of the Lorentz group, then the induced action ESES−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() 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 (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 = γμγνγνγμ 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[edit]

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[edit]

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 Cl1,3(). The number 5 is a relic of old notation, in which was 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[edit]

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.[1] Thus, one can employ a bit of a trick to repurpose iγ5 as one of the generators of the Clifford algebra in five dimensions. In this case, the set {γ0, γ1, γ2, γ3, 5} therefore, by the last two properties (keeping in mind that i2 ≡ −1) and those of the old gammas, forms the basis of the Clifford algebra in 5 spacetime dimensions for the metric signature (1,4).[2] In metric signature (4,1), the set {γ0, γ1, γ2, γ3, γ5} is used, where the γμ are the appropriate ones for the (3,1) signature.[3] This pattern is repeated for spacetime dimension 2n even and the next odd dimension 2n + 1 for all n ≥ 1.[4] For more detail, see higher-dimensional gamma matrices.

Identities[edit]

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[edit]

1.

2.

3.

4.

5.

6. , where

Trace identities[edit]

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[edit]

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[edit]

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 (its form expressed as product of the gamma matrices is representation dependent, while it can be seen in the Dirac basis, this fails to hold in the Majorana basis, for example). 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:

In addition, for all four representations given below (Dirac, Majorana and both chiral variants), one has

Feynman slash notation[edit]

The Feynman slash notation is defined by

for any 4-vector .

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

  • where is the Levi-Civita symbol and Actually traces of products of odd number of is zero and thus
  • for odd.[5]

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[edit]

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[edit]

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,[6]

Weyl (chiral) basis[edit]

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.

The charge conjugation operator in this basis is real antisymmetric,

The Dirac basis can be obtained from the Weyl basis as

via the unitary transform

Weyl (chiral) basis (alternate form)[edit]

Another possible choice[6][7] 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.

The charge conjugation operator in this basis is

This basis can be obtained from the Dirac basis above as via the unitary transform

Majorana basis[edit]

There is also the Majorana basis, in which all of the Dirac matrices are imaginary, and the spinors and Dirac equation are real. Regarding the Pauli matrices, the basis can be written as[6]

where is the charge conjugation matrix, as defined above.

(The reason for making all 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 i 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 (−, +, +, +).)

The Majorana basis can be obtained from the Dirac basis above as via the unitary transform

Cl1,3(ℂ) and Cl1,3(ℝ)[edit]

The Dirac algebra can be regarded as a complexification of the real algebra Cl1,3(), called the space time algebra:

Cl1,3() differs from Cl1,3(): in Cl1,3() only real linear combinations of the gamma matrices and their products are allowed.

Two things deserve to be pointed out. As Clifford algebras, Cl1,3() and Cl4() are isomorphic, see classification of Clifford algebras. The reason is that the underlying signature of the spacetime metric loses its signature (1,3) upon passing to the complexification. However, the transformation required to bring the bilinear form to the complex canonical form is not a Lorentz transformation and hence not "permissible" (at the very least impractical) since all physics is tightly knit to the Lorentz symmetry and it is preferable to keep it manifest.

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.[8]

In the mathematics of Riemannian geometry, it is conventional to define the Clifford algebra Clp,q() for arbitrary dimensions p,q. The Weyl spinors transform under the action of the spin group . The complexification of the spin group, called the spinc group , is a product of the spin group with the circle The product just a notational device to identify with The geometric point of this is that it disentangles the real spinor, which is covariant under Lorentz transformations, from the component, which can be identified with the fiber of the electromagnetic interaction. The is entangling parity and charge conjugation in a manner suitable for relating the Dirac particle/anti-particle states (equivalently, the chiral states in the Weyl basis). The bispinor, insofar as it has linearly independent left and right components, can interact with the electromagnetic field. This is in contrast to the Majorana spinor and the ELKO spinor, which cannot (i.e. they are electrically neutral), as they explicitly constrain the spinor so as to not interact with the part coming from the complexification.

However, in contemporary practice in physics, the Dirac algebra rather than the space-time algebra continues to be the standard environment the spinors of the Dirac equation "live" in.

Other representation-free properties[edit]

The gamma matrices are diagonalizable with eigenvalues for , and eigenvalues for .

In particular, this implies that is simultaneously Hermitian and unitary, while the are simultaneously anti–Hermitian and unitary.

Further, the multiplicity of each eigenvalue is two.

More generally, if is not null, a similar result holds. For concreteness, we restrict to the positive norm case with . The negative case follows similarly.

It follows that the solution space to (that is, the kernel of the left-hand side) has dimension 2. This means the solution space for plane wave solutions to Dirac's equation has dimension 2.

This result still holds for the massless Dirac equation. In other words, if null, then has nullity 2.

Euclidean Dirac matrices[edit]

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:

Chiral representation[edit]

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.

In Euclidean space,

Using the anti-commutator and noting that in Euclidean space , one shows that

In chiral basis in Euclidean space,

which is unchanged from its Minkowski version.

Non-relativistic representation[edit]

See also[edit]

References[edit]

  1. ^ Jost, Jurgen (2002). Riemannian Geometry and Geometric Analysis (3rd ed.). Springer Universitext. p. 68, Corollary 1.8.1.
  2. ^ The set of matrices (ΓA) = (γμ, 5) with A = (0, 1, 2, 3, 4) satisfy the five-dimensional Clifford algebra {ΓA, ΓB} = 2ηAB. See Tong 2007, p. 93.
  3. ^ Weinberg 2002 Section 5.5.
  4. ^ de Wit & Smith 1986, p. 679.
  5. ^ vadim [U.T.A. user name] (Fall 2008). "Traceology" (PDF) (Lecture note). Homework. University of Texas at Austin. Archived from the original (PDF) on 2019-11-13. Retrieved 2021-11-04. {{cite web}}: |author= has generic name (help)
  6. ^ a b c Claude Itzykson and Jean-Bernard Zuber, (1980) "Quantum Field Theory", MacGraw-Hill (See Appendix A)
  7. ^ Michio Kaku, Quantum Field Theory, ISBN 0-19-509158-2, appendix A
  8. ^ See e.g. Hestenes (1996). "Real Dirac" (PDF). Tempe, AZ: Arizona State University.

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