In
mathematical physics
Mathematical physics refers to 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 t ...
, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional
Gamma matrices
In mathematical physics, the gamma matrices, \left\ , 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(\ma ...
of
Dirac
Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety o ...
, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically invariant wave equations for fermions (such as spinors) in arbitrary space-time dimensions, notably in string theory and supergravity. The
Weyl–Brauer matrices
In mathematics, particularly in the theory of spinors, the Weyl–Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra of matrices. They generalize the Pauli matrices to dimensions, and are a specific constructi ...
provide an explicit construction of higher-dimensional gamma matrices for
Weyl spinor
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
s. Gamma matrices also appear in generic settings in
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...
, particularly when a
spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematical ...
can be defined.
Introduction
Consider a space-time of dimension with the flat
Minkowski metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
,
:
with
positive entries,
negative entries,
and . Set . The standard
Dirac matrices
In mathematical physics, the gamma matrices, \left\ , 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(\ma ...
correspond to taking and or .
In higher (and lower) dimensions, one may define a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, the gamma group, behaving in the same fashion as the Dirac matrices.
[
] More precisely, if one selects a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
for the (complexified)
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. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
, then the gamma group
generated by is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the ''multiplicative'' subgroup generated by the basis elements
(ignoring the additive aspect of the Clifford algebra).
By convention, the gamma group is realized as a collection of matrices, the gamma matrices, although the group definition does not require this. In particular, many important properties, including the
C,
P and
T symmetries do not require a specific matrix representation, and one obtains a clearer definition of
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 from ...
in this way.
Several matrix representations are possible, some given below, and others in the article on the
Weyl–Brauer matrices
In mathematics, particularly in the theory of spinors, the Weyl–Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra of matrices. They generalize the Pauli matrices to dimensions, and are a specific constructi ...
. In the matrix representation, the spinors are
-dimensional, with the gamma matrices acting on the spinors. A detailed construction of spinors is given in the article on
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. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
. Jost provides a standard reference for spinors in the general setting of Riemmannian geometry.
[Jurgen Jost, (2002) "Riemannian Geometry and Geometric Analysis (3rd edition)", Springer. ''See Chapter 1, section 1.8.'']
Gamma group
Most of the properties of the gamma matrices can be captured by a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, the gamma group. This group can be defined without reference to the real numbers, the complex numbers, or even any direct appeal to 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. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
.
The matrix representations of this group then provide a concrete realization that can be used to specify the action of the
gamma matrices
In mathematical physics, the gamma matrices, \left\ , 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(\ma ...
on
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
s. For
dimensions, the matrix products behave just as the conventional
Dirac matrices
In mathematical physics, the gamma matrices, \left\ , 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(\ma ...
. The
Pauli group
In physics and mathematics, the Pauli group G_1 on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I and all of the Pauli matrices
:X = \sigma_1 =
\begin
0&1\\
1&0
\end,\quad
Y = \sigma_2 =
\begin ...
is a
representation of the gamma group for
although the Pauli group has more relationships (is
less free); see the note about the chiral element below for an example. The
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s provide a representation for
The
presentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
of the gamma group
is as follows.
* A
neutral element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
is denoted as
.
* The element
with
is a stand-in for the complex number
; it
commutes with all other elements,
* There is a collection of generators
indexed by
with
* The remaining generators
obey
* The anticommutator is defined as
for
These generators completely define the gamma group. It can be shown that, for all
that
and so
Every element
can be uniquely written as a product of a finite number of generators placed in canonical order as
:
with the indexes in ascending order
:
and
The gamma group is finite, and has at most
elements in it.
The gamma group is a
2-group but not a
regular p-group
In mathematical finite group theory, the concept of regular ''p''-group captures some of the more important properties of abelian ''p''-groups, but is general enough to include most "small" ''p''-groups. Regular ''p''-groups were introduced by . ...
. The
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
(derived subgroup) is
therefore it is not a
powerful p-group
In mathematics, in the field of group theory, especially in the study of ''p''-groups and pro-''p''-groups, the concept of powerful ''p''-groups plays an important role. They were introduced in , where a number of applications are given, includi ...
. In general, 2-groups have a large number of
involutions; the gamma group does likewise. Three particular ones are singled out below, as they have a specific interpretation in the context of
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. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
s, in the context of the representations of the gamma group (where transposition and Hermitian conjugation literally correspond to those actions on matrices), and in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, where the "main involution"
corresponds to a combined
P-symmetry
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
and
T-symmetry
T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal,
: T: t \mapsto -t.
Since the second law of thermodynamics states that entropy increases as time flows toward the future ...
.
Transposition
Given elements
of the generating set of the gamma group, the
transposition or ''reversal'' is given by
:
If there are
elements
all distinct, then
:
Hermitian conjugation
Another
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 automorphisms ...
of the gamma group is given by conjugation, defined on the generators as
:
supplemented with
and
For general elements in the group, one takes the transpose:
From the properties of transposition, it follows that, for all elements
that either
or that
that is, all elements are either Hermitian or unitary.
If one interprets the
dimensions as being "time-like", and the
dimensions as being "space-like", then this corresponds to
P-symmetry
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
in physics. That this is the "correct" identification follows from the conventional Dirac matrices, where
is associated with the time-like direction, and the
the spatial directions, with the "conventional" (+−−−) metric. Other metric and representational choices suggest other interpretations.
Main involution
The main involution is the map that "flips" the generators:
but leaves
alone:
This map corresponds to the combined
P-symmetry
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
and
T-symmetry
T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal,
: T: t \mapsto -t.
Since the second law of thermodynamics states that entropy increases as time flows toward the future ...
in physics; all directions are reversed.
Chiral element
Define the chiral element
as
:
where
. The chiral element commutes with the generators as
:
It squares to
:
For the Dirac matrices, the chiral element corresponds to
thus its name, as it plays an important role in distinguishing the chirality of spinors.
For the
Pauli group
In physics and mathematics, the Pauli group G_1 on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I and all of the Pauli matrices
:X = \sigma_1 =
\begin
0&1\\
1&0
\end,\quad
Y = \sigma_2 =
\begin ...
, the chiral element is
whereas for the gamma group
, one cannot deduce any such relationship for
other than that it squares to
This is an example of where a representation may have more identities the represented group. For the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s, which provide a representation of
the chiral element is
Charge conjugation
None of the above automorphisms (transpose, conjugation, main involution) are
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
s; that is they ''cannot'' be represented in the form
for some existing element
in the gamma group, as presented above. Charge conjugation requires extending the gamma group with two new elements; by convention, these are
:
and
:
The above relations are not sufficient to define a group;
and other products are undetermined.
Matrix representation
The gamma group has a matrix representation given by complex
matrices with
and
and
the
floor function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
, the largest integer less than
The group presentation for the matrices can be written compactly in terms of the
anticommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
relation from 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. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
:
where the matrix 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.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
in dimensions. Transposition and Hermitian conjugation correspond to their usual meaning on matrices.
Charge conjugation
For the remainder of this article,it is assumed that
and so
. That is, 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. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
is assumed. In this case, the gamma matrices have the following property under
Hermitian conjugation
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
where ...
,
:
Transposition will be denoted with a minor change of notation, by mapping
where the element on the left is the abstract group element, and the one on the right is the literal
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 ...
.
As before, the generators all generate the same group (the generated groups are all
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
; the operations are still
involutions). However, since the are now matrices, it becomes plausible to ask whether there is a matrix that can act as a
similarity transformation that embodies the automorphisms. In general, such a matrix can be found. By convention, there are two of interest; in the physics literature, both referred to as
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-symm ...
matrices. Explicitly, these are
:
They can be constructed as real matrices in various dimensions, as the following table shows. In even dimension both
exist, in odd dimension just one.
Note that
is a basis choice.
Symmetry properties
We denote a product of gamma matrices by
:
and note that the anti-commutation property allows us to simplify any such sequence to one in which the indices are distinct and increasing. Since distinct
anti-commute this motivates the introduction of an anti-symmetric "average". We introduce the anti-symmetrised products of distinct -tuples from 0, ..., − 1:
:
where runs over all the
permutations
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
of symbols, and is the
alternating character
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total o ...
. There are 2
''d'' such products, but only
2 are independent, spanning the space of × matrices.
Typically, provide the (bi)spinor representation of the generators of the
higher-dimensional Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
, , generalizing the
6 matrices ''σμν'' of the
spin representation
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equ ...
of the Lorentz group in four dimensions.
For even , one may further define the hermitian ''chiral matrix''
:
such that and . (In odd dimensions, such a matrix would commute with all
''a''s and would thus be proportional to the identity, so it is not considered.)
A matrix is called symmetric if
:
otherwise, for a − sign, it is called antisymmetric.
In the previous expression, can be either
or
. In odd dimension, there is no ambiguity, but in even dimension it is better to choose whichever one of
or
allows for Majorana spinors. In = 6, there is no such criterion and therefore we consider both.
Identities
The proof of the trace identities for gamma matrices hold for all even dimension. One therefore only needs to remember the
4D case and then change the overall factor of 4 to
. For
other identities (the ones that involve a contraction), explicit functions of
will appear.
Even when the number of physical dimensions is four, these more general identities are ubiquitous in loop calculations due to
dimensional regularization
__NOTOC__
In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini as well as – independently and more comprehensively – by 't Hooft and Veltman for regularizing integrals in the evaluation of Fe ...
.
Example of an explicit construction in the chiral basis
The matrices can be constructed recursively, first in all even dimensions, = 2, and thence in odd ones, 2 + 1.
''d'' = 2
Using the
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
, take
:
and one may easily check that the charge conjugation matrices are
:
One may finally define the hermitian chiral
chir to be
:
Generic even ''d'' = 2''k''
One may now construct the , matrices and the charge conjugations
(±) in + 2 dimensions, starting from the , (), and
(±) matrices in dimensions.
Explicitly,
:
One may then construct the charge conjugation matrices,
:
with the following properties,
:
Starting from the sign values for = 2,
(2,+) = +1 and
(2,−) = −1, one may fix all subsequent signs
(''d'',±) which have periodicity 8; explicitly, one finds
Again, one may define the hermitian chiral matrix in +2 dimensions as
:
which is diagonal by construction and transforms under charge conjugation as
:
It is thus evident that = 0.
Generic odd ''d'' = 2''k'' + 1
Consider the previous construction for − 1 (which is even) and simply take all matrices, to which append its . (The is required in order to yield an antihermitian matrix, and extend into the spacelike metric).
Finally, compute the charge conjugation matrix: choose between
and
, in such a way that transforms as all the other matrices. Explicitly, require
:
As the dimension ranges, patterns typically repeat themselves with period 8. (cf. the
Clifford algebra clock.)
See also
*
Weyl–Brauer matrices
In mathematics, particularly in the theory of spinors, the Weyl–Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra of matrices. They generalize the Pauli matrices to dimensions, and are a specific constructi ...
*
Bispinor
In physics, and specifically in quantum field theory, a bispinor, is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons. It is a specific embodiment of a spinor, speci ...
*
Clifford module In mathematics, a Clifford module is a representation of a Clifford algebra. In general a Clifford algebra ''C'' is a central simple algebra over some field extension ''L'' of the field ''K'' over which the quadratic form ''Q'' defining ''C'' is d ...
Notes
References
General reading
*
*
*
*
* de Wit, Bryce and Smith, J. (1986). ''Field Theory in Particle Physics'' (North-Holland Personal Library), Volume 1, Paperback
Appendix EArchived from original,
* Murayama, H. (2007).
"Notes on Clifford Algebra and Spin(N) Representations"* Pietro Giuseppe Frè (2012). "Gravity, a Geometrical Course: Volume 1: Development of the Theory and Basic Physical Applications." Springer-Verlag. {{ISBN, 9400753608. ''See pp 315ff.''
Spinors
Matrices
Clifford algebras