In
mathematical physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
, 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
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...
It is also possible to define
higher-dimensional gamma matrices
In mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Paul Dirac, Dirac, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically i ...
. When interpreted as the matrices of the action of a set of
orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
basis vectors for
contravariant vectors in
Minkowski space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
, the column vectors on which the matrices act become a space of
spinors
In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
, on which the Clifford algebra of
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
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
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
for relativistic
particles. Gamma matrices were introduced by
Paul Dirac
Paul Adrien Maurice Dirac ( ; 8 August 1902 – 20 October 1984) was an English mathematician and Theoretical physics, theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for bot ...
in 1928.
In
Dirac representation, the four
contravariant gamma matrices are
:
is the time-like,
Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the ...
. The other three are space-like,
anti-Hermitian matrices. More compactly,
and
where
denotes the
Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product (which is denoted by the same symbol) from vector ...
and the
(for ) denote the
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
.
In addition, for discussions of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
() 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
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
are a set of "gamma" matrices in three dimensional space with metric of Euclidean signature (3, 0). In five
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
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
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...
is the anticommutation relation
:
where the curly brackets
represent the
anticommutator,
is the
Minkowski metric
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model ...
with signature , and
is the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
.
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
In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies ...
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
The Clifford algebra
over spacetime can be regarded as the set of real linear operators from to itself, , 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 ,
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 bispinors, , is also assumed at every point in spacetime, endowed with the
bispinor representation of the
Lorentz group. The bispinor 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 ). This will be the primary view of elements of
in this section.
For each linear transformation of , there is a transformation of given by for in
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
bispinor representation acting on of an arbitrary
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
in the standard (4 vector) representation acting on , then there is a corresponding operator on
given by equation:
:
showing that the quantity of 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
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimension (vector space), dimensional real number, real vector space that leave invariant a nondegenerate form, nondegenerate, symmetric bi ...
, 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 using the metric as with any 4 vector. The notation is called the
Feynman slash notation. The slash operation maps the basis of , or any 4 dimensional vector space, to basis vectors . The transformation rule for slashed quantities is simply
:
One should note that this is different from the transformation rule for the , 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 , and the former to a passive transformation of the basis itself.
The elements
form a representation of the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
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 of above are of this form. The 6 dimensional 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
In physics, natural unit systems are measurement systems for which selected physical constants have been set to 1 through nondimensionalization of physical units. For example, the speed of light may be set to 1, and it may then be omitted, equa ...
, the Dirac equation may be written as
:
where
is a Dirac spinor.
Switching to
Feynman notation, the Dirac equation is
:
The fifth "gamma" matrix, 5
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 dimensions, we can use the identity
.
Then we get, using the convention
:
This matrix is useful in discussions of quantum mechanical
chirality
Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable fro ...
. 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
The
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...
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
In mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Paul Dirac, Dirac, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically i ...
.
Identities
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
1.
2.
3.
4.
5.
6.
where
Trace identities
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'')
Normalization
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 )
:
, 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.
Charge conjugation
The
charge conjugation
In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C- ...
operator, in any basis, may be defined as
:
where
denotes the
matrix transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
. 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
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
of the
gamma group, it is ''not'' an
inner automorphism
In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within thos ...
(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 not always be chosen such that
, 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
:
for any 4-vector
.
Here are some similar identities to the ones above, but involving slash notation:
*
*
*
*