Wikipedia / Gamma matrices
Wikipedia / 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 rotationss and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic \tfrac{\ 1\ }{2} particles. Gamma matrices were introduced by Paul Dirac in 1928.
In the Dirac basis of the Dirac representation, the four contravariant gamma matrices are
is the time-like, Hermitian matrix. The other three are space-like, anti-Hermitian matrices denotes the Kronecker product (for ) denote the Pauli matrices.
In addition, for discussions of group theory the identity matrix () is sometimes included with the four gamma matrices, and there is an auxiliary, "fifth" traceless matrix used in conjunction with the regular gamma matrices
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
The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation
represent the anticommutator is the Minkowski metric with signature is the 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:
, which subsequently changes their hermiticity properties detailed below. Under the alternative sign convention for the metric the covariant gamma matrices are then defined by
Physical structure
over spacetime can be regarded as the set of real linear operators from to itself, , or more generally, when complexified as the set of linear operators from any four-dimensional complex vector space to itself. More simply, given a basis for is just the set of all complex matrices, but endowed with a Clifford algebra structure. Spacetime is assumed to be endowed with the Minkowski metric . A space of Dirac spinors, , is also assumed at every point in spacetime, endowed with the Dirac spinor representation of the Lorentz group. The Dirac spinor fields of the Dirac equations, evaluated at any point in spacetime, are elements of (see below). The Clifford algebra is assumed to act on as well (by matrix multiplication with column vectors in for all in this section.
For each linear transformation of , there is a transformation of given by for If belongs to a representation of the Lorentz group, then the induced action will also belong to a representation of the Lorentz group, see Representation theory of the Lorentz group.
If is the Dirac spinor representation acting on of an arbitrary Lorentz transformation in the standard (4vector) representation acting on given by the equation:
showing that the quantity of can be viewed as a basis of a representation space of the 4vector 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 written in indexed notation. This means that quantities of the form
should be treated as 4vectors in manipulations. It also means that indices can be raised and lowered on the using the metric as with any 4vector. The notation is called the Feynman slash notation. The slash operation maps the basis of , or any 4dimensional vector space, to basis vectors . The transformation rule for slashed quantities is simply
This is different from the transformation rule for the , which are now treated as (fixed) basis vectors. The designation of the 4 as a 4vector 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 , and the former to a passive transformation of the basis itself.
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 Dirac spinor representations of the Lorentz group, e.g., the of above are of this form. The 6dimensional space the 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 (for real, uncharged spinors) and in the complexified spin group for charged (Dirac) spinors.
Expressing the Dirac equation
In natural units, the Dirac equation may be written as
is a Dirac spinor.
Switching to Feynman notation, the Dirac equation is
The fifth "gamma" matrix, γ5
, so that
(in the Dirac basis).
uses the letter gamma, it is not one of the The index number".
has also an alternative form:
or
Proof:
This can be seen by exploiting the fact that all the four gamma matrices anticommute, so
is the type (4,4) generalized Kronecker delta in 4 dimensions, in full antisymmetrization denotes the Levi-Civita symbol in
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:
since
Five dimensions
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. Thus, one can employ a bit of a trick to repurpose as one of the generators of the Clifford algebra in five dimensions. In this case, the set therefore, by the last two properties (keeping in mind that ) and those of the ‘old’ gammas, forms the basis of the Clifford algebra in spacetime dimensions for the metric signature .. In metric signature , the set is used, where the are the appropriate ones for the signature. This pattern is repeated for spacetime dimension even and the next odd dimension for all . For more detail, see higher-dimensional gamma matrices.
Identities
).
Miscellaneous identities
Trace identities
The gamma matrices obey the following trace identities:
Proving the above involves the use of three main properties of the trace operator:
|}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. |}|} with its neighbor to the left,
{|
|}
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
or
|} to both sides of the above to see
.
Now, this pattern can also be used to show
.
. Anticommute three times instead of once, picking up three minus signs, and cycle using the cyclic property of the trace.
So,
.
|}. Now,
{|
|(since the trace of a product of gamma matrices is real) |} |}
Normalization
The gamma matrices can be chosen with extra hermiticity conditions which are restricted by the above anticommutation relations however. We can impose
and for the other gamma matrices (for )
One checks immediately that these hermiticity relations hold for the Dirac representation.
The above conditions can be combined in the relation
is not necessarily a unitary transformation due to the non-compactness of the Lorentz group.
Charge conjugation
The charge conjugation operator, in any basis, may be defined as
denotes the matrix transpose 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:
, as is the case for the common four representations given below, known as Dirac, chiral and Majorana representation.
Feynman slash notation
The Feynman slash notation is defined by
.
Here are some similar identities to the ones above, but involving slash notation:
is the Levi-Civita symbol is zero and thus
for odd.
with the appropriate identity in terms of gamma matrices.
Other representations
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
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:
or using the Kronecker product:
In the Dirac basis, the charge conjugation operator is real antisymmetric,
Weyl (chiral) basis
Another common choice is the Weyl or chiral basis 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 we can then identify
are left-handed and right-handed two-component Weyl spinors.
The charge conjugation operator in this basis is real antisymmetric,
The Weyl basis can be obtained from the Dirac basis as
via the unitary transform
Weyl (chiral) basis (alternate form)
Another possible choice of the Weyl basis has
The chiral projections take a slightly different form from the other Weyl choice,
In other words,
are the left-handed and right-handed two-component Weyl spinors, as before.
The charge conjugation operator in this basis is
via the unitary transform
Majorana basis
There is also the Majorana basis, in which all of the Dirac matrices are imaginary, and the spinors and Dirac equation are real. Using the Pauli matrices, the basis can be written as
is the charge conjugation matrix, which matches the Dirac version defined above.
The reason for making all gamma matrices imaginary is solely to obtain the particle physics metric is that the only possible metric with real gamma matrices is .
via the unitary transform
Cl1,3(C) and Cl1,3(R)
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.
In the mathematics of Riemannian geometry, it is conventional to define the Clifford algebra Clp,q) for arbitrary dimensions . The Weyl spinors transform under the action of the spin group. The complexification of the spin group, called the spinc group 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 Dirac spinor, 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 (Eigenspinoren des Ladungskonjugationsoperators), which cannot (i.e. part coming from the complexification. The ELKO spinor is a Lounesto class 5 spinor.
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
. are simultaneously anti–Hermitian and unitary.
Further, the multiplicity of each eigenvalue is two. The negative case follows similarly. (that is, the kernel of the left-hand side) has dimension2. This means the solution space for plane wave solutions to Dirac's equation has dimension2.
has nullity2.
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:
Chiral representation
have been inserted in the spatial gamma matrices so that the Euclidean Clifford algebra
on one of the matrices, such as in lattice QCD codes which use the chiral basis.
In Euclidean space,
, one shows that
In chiral basis in Euclidean space,
which is unchanged from its Minkowski version.
Non-relativistic representation
Footnotes
See also
Citations
References
External links
Dirac matrices on mathworld including their group properties