Gamma matrices
Encyclopedia
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
Cℓ(1,3). 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 space time
acts. This in turn makes it possible to represent infinitesimal spatial rotation
s and Lorentz boosts. Spinors facilitate space-time 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
Analogue 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).
is the anticommutation relation
where
is the Minkowski metric with signature (+ − − −) and 
is the unit matrix.
This defining property is considered to be more fundamental than the numerical values used in 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
.
is often loosely described as a 4-vector (where e0 to e3 are the basis vectors of the 4-vector space). But this is misleading. Instead 
is more appropriately seen as a mapping operator, taking in a 4-vector 
and mapping it to the corresponding matrix in the Clifford algebra representation.
This is symbolised by the useful Feynman slash notation
,
Slashed quantities like
"live" in the multilinear Clifford algebra, with its own set of basis directions — they are immune to changes in the 4-vector basis.
On the other hand, one can define a transformation identity for the mapping operator
. If 
is the spinor
representation of an arbitrary Lorentz transformation
, then we have the identity
This says essentially that an operator mapping from the old 4-vector basis
to the old Clifford algebra basis 
is equivalent to a mapping from the new 4-vector basis 
to a correspondingly transformed new Clifford algebra basis 
. Alternatively, in pure index terms, it shows that 
transforms appropriately for an object with one contravariant 4-vector index and one covariant and one contravariant Dirac spinor index.
Given the above transformation properties of
, if 
is a Dirac spinor then the product 
transforms as if it were the product of a contravariant 4-vector with a Dirac spinor. In expressions involving spinors, then, it is often appropriate to treat 
as if it were simply a vector.
There remains a final key difference between
and any nonzero 4-vector: 
does not point in any direction. More precisely, the only way to make a true vector from 
is to contract its spinor indices, leaving a vector of traces
This property of the gamma matrices is essential for them to serve as coefficients in the Dirac equation.
, the Dirac equation may be written as
where ψ is a Dirac spinor. Here, if
were an ordinary 4-vector, then it would pick out a preferred direction in spacetime, and the Dirac equation would not be Lorentz invariant.
Switching to Feynman notation, the Dirac equation is
Applying
to both sides yields
which is the Klein-Gordon equation
. Thus, as the notation suggests, the Dirac particle has mass m.
The Fifth Gamma Matrix,
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. 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:
).
Proving the above involves the use of three main properties of the Trace
operator:
by the above anticommutation relations however. We can impose
and for the other gamma matrices (for k=1,2,3)
One checks immediately that these hermicity relations hold for the Dirac representation.
The above conditions can be combined in the relation
The hermicity conditions are not invariant under the action
of a Lorentz transformation because 
is not a unitary transformation. This is intuitively clear because time and space are treated on unequal footing.
with a vector 
maps the vector out of the 4-vector representation.
So, it is common to write identities using the Feynman slash notation
, defined by
Here are some similar identities to the ones above, but involving slash notation:
, I, and
where k runs from 1 to 3 and the σk are Pauli matrices
.
remains the same but 
is different, and so 
is also different:
The Weyl
basis has the advantage that its chiral projections
take a simple form:
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.
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 (-,+,+,+).
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 spinor
s of the Dirac equation "live" in.
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:
Different from Minkowski space, in Euclidean space,
So in Chiral basis,
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, the gamma matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
,
, also known as the DiracPaul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate
Generating set
In mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings:...
a matrix representation of the Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
Cℓ(1,3). It is also possible to define higher-dimensional Gamma matrices. When interpreted as the matrices of the action of a set of orthogonal
Orthogonality
Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...
basis vectors for contravariant vectors
Vector (mathematics and physics)
In mathematics and physics, a vector is an element of a vector space. If n is a non negative integer and K is either the field of the real numbers or the field of the complex number, then K^n is naturally endowed with a structure of vector space, where K^n is the set of the ordered sequences of n...
in Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
of space time
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
acts. This in turn makes it possible to represent infinitesimal spatial rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s and Lorentz boosts. Spinors facilitate space-time computations in general, and in particular are fundamental to the Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
for relativistic spin-½
Spin-½
In quantum mechanics, spin is an intrinsic property of all elementary particles. Fermions, the particles that constitute ordinary matter, have half-integer spin. Spin-½ particles constitute an important subset of such fermions. All known elementary fermions have a spin of ½.- Overview :Particles...
particles.
In Dirac representation, the four contravariant
Covariance and contravariance
In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. When one coordinate system is just a rotation of the other, this...
gamma matrices are
Analogue sets of gamma matrices can be defined in any dimension and signature of the metric. For example the Pauli matrices
Pauli matrices
The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...
are a set of "gamma" matrices in dimension 3 with metric of Euclidean signature (3,0).
Mathematical structure
The defining property for the gamma matrices to generate a Clifford algebraClifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
is the anticommutation relation
where
is the Minkowski metric with signature (+ − − −) and is the unit matrix.This defining property is considered to be more fundamental than the numerical values used in the gamma matrices.
Covariant gamma matrices are defined by
and Einstein notation
Einstein notation
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...
is assumed.
Note that the other sign convention
Sign convention
In physics, a sign convention is a choice of the physical significance of signs for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of...
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 4-tuple is often loosely described as a 4-vector (where e0 to e3 are the basis vectors of the 4-vector space). But this is misleading. Instead is more appropriately seen as a mapping operator, taking in a 4-vector and mapping it to the corresponding matrix in the Clifford algebra representation.This is symbolised by the useful Feynman slash notation
Feynman slash notation
In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation...
,
Slashed quantities like
"live" in the multilinear Clifford algebra, with its own set of basis directions — they are immune to changes in the 4-vector basis.On the other hand, one can define a transformation identity for the mapping operator
. If is the spinorSpinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
representation of an arbitrary Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
, then we have the identityThis says essentially that an operator mapping from the old 4-vector basis
to the old Clifford algebra basis is equivalent to a mapping from the new 4-vector basis to a correspondingly transformed new Clifford algebra basis . Alternatively, in pure index terms, it shows that transforms appropriately for an object with one contravariant 4-vector index and one covariant and one contravariant Dirac spinor index.Given the above transformation properties of
, if is a Dirac spinor then the product transforms as if it were the product of a contravariant 4-vector with a Dirac spinor. In expressions involving spinors, then, it is often appropriate to treat as if it were simply a vector.There remains a final key difference between
and any nonzero 4-vector: does not point in any direction. More precisely, the only way to make a true vector from is to contract its spinor indices, leaving a vector of tracesTrace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
This property of the gamma matrices is essential for them to serve as coefficients in the Dirac equation.
Expressing the Dirac equation
In natural unitsNatural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...
, the Dirac equation may be written as
where ψ is a Dirac spinor. Here, if
were an ordinary 4-vector, then it would pick out a preferred direction in spacetime, and the Dirac equation would not be Lorentz invariant.Switching to Feynman notation, the Dirac equation is
Applying
to both sides yieldswhich is the Klein-Gordon equation
Klein-Gordon equation
The Klein–Gordon equation is a relativistic version of the Schrödinger equation....
. Thus, as the notation suggests, the Dirac particle has mass m.
The Fifth Gamma Matrix, 
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. 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
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...
. 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:
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
| Num | | Identity |
|---|---|
| 1 |  |
| 2 |  |
| 3 |  |
| 4 |  |
| 5 |  |
Trace identities
| Num | | Identity |
|---|---|
| 0 |  |
| 1 | trace of any product of an odd number of  is zero |
| 2 |  |
| 3 |  |
| 4 |  |
| 5 |  |
Proving the above involves the use of three main properties of the Trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
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 hermicity conditions which are restrictedby 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 hermicity relations hold for the Dirac representation.
The above conditions can be combined in the relation
The hermicity conditions are not invariant under the action
of a Lorentz transformation because is not a unitary transformation. This is intuitively clear because time and space are treated on unequal footing.Feynman slash notation
The contraction of the mapping operator with a vector maps the vector out of the 4-vector representation.So, it is common to write identities using the Feynman slash notation
Feynman slash notation
In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation...
, defined by
Here are some similar identities to the ones above, but involving slash notation:
-
- where
is the Levi-Civita symbolLevi-Civita symbolThe Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...
and
Other representations
The matrices are also sometimes written using the 2x2 identity matrixIdentity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
, I, and
where k runs from 1 to 3 and the σk are Pauli matrices
Pauli matrices
The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...
.
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:Weyl basis
Another common choice is the Weyl or chiral basis, in which remains the same but is different, and so is also different:The Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
basis has the advantage that its chiral projections
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...
take a simple form:
By slightly abusing the notation
Abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition . Abuse of notation should be contrasted with misuse of notation, which should be avoided...
and reusing the symbols
we can then identifywhere now
and are left-handed and right-handedtwo-component Weyl spinors.
Another possible choice of the Weyl basis has:
The chiral projections
Chirality (physics)
A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...
take a slightly different form from the other Weyl choice:
In other words:
where
and are the left-handed and right-handedtwo-component Weyl spinors as before.
Majorana basis
There's also a MajoranaMajorana
Majorana may refer to:* Majorana equation, a relativistic wave equation* Majorana fermion, a concept in particle physics* Majorana spinor, a concept in quantum field theory* Origanum majorana, a somewhat cold-sensitive perennial herb...
basis, in which all of the Dirac matrices are imaginary and spinors are real. In terms of the Pauli matrices
Pauli matrices
The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...
, 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 complexificationComplexification
In mathematics, the complexification of a real vector space V is a vector space VC over the complex number field obtained by formally extending scalar multiplication to include multiplication by complex numbers. Any basis for V over the real numbers serves as a basis for VC over the complex...
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
Geometric algebra
Geometric algebra , together with the associated Geometric calculus, provides a comprehensive alternative approach to the algebraic representation of classical, computational and relativistic geometry. GA now finds application in all of physics, in graphics and in robotics...
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 spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s of the Dirac equation "live" in.
Euclidean Dirac matrices
In quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
one can Wick rotate the time axis to transit from Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
to Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, this is particularly useful in some renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
procedures as well as lattice gauge theory
Lattice gauge theory
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice. Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum electrodynamics, quantum chromodynamics and the Standard...
. In Euclidean space, there are two commonly used representations of Dirac Matrices:
Chiral representation
Different from Minkowski space, in Euclidean space,
So in Chiral basis,
Non-relativistic representation
External links
- Dirac matrices on mathworld including their group properties