All the properties of gamma matrix and sigma matrix
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Gamma matrices
This appendix reviews the properties of γ-matrices. In 4 space-time dimensions we
have already given an explicit representation of these matrices in chapter 5. The set-
up of this appendix is kept more general; motivated by dimensional regularization
and by recent discussions of higher-dimensional theories in the context of Kaluza-
Klein supergravity and superstrings we summarize the properties of γ-matrices in
arbitrary space time dimension D. For this reason we adopt a notation which is
different from that used in the main text. In D-dimensional Minkowski space the
space components carry indices 1, 2, ...D −1, and the purely imaginary time compo-
nent carries index D. Readers who are emotionally attached to 4-dimensional space
time can simply insert D = 4, or, if they only need a certain 4-dimensional formula,
they are advised to consult section E.3 and parts of the later sections where explicit
results for D = 4 are listed.
E.1. The Clifford algebra
We consider a representation of the D-dimensional Clifford algebra
γaγb + γbγa = 2δabI, a, b = 1, ....D . (E.1)
Repeated multiplication of the γ-matrices leads to a set of 2D matrices Γ
A
Γ
A
: I, γa , γab , γabc . . . , (E.2)
where
γab = γaγb (a < b), γabc = γaγbγc (a < b < c) , etc . (E.3)
In (E.2) we only include ordered strings of different γ-matrices; products in which
the γ-matrices appear in different order or the same γ-matrix appears more than
once can be reduced to one of these by using the anticommutation relation (E.1). On
account of (E.1) the matrices γa1...an are antisymmetric in the indices a1, . . . , an, so
they can also be defined as an antisymmetrized product of γ-matrices
γa1...an =
1
n!
X
perm
[a1...an]
(−)
P
γa1 γa2
. . . γan . (E.4)
As there are `n
D
´
different ways of selecting n different indices between 1 and D,
there are `n
D
´
matrices γa1...an . Therefore the total number of matrices Γ
A is
XD
n=0
D
n
!
= 2D . (E.5)
Gamma matrices
This appendix reviews the properties of γ-matrices. In 4 space-time dimensions we
have already given an explicit representation of these matrices in chapter 5. The set-
up of this appendix is kept more general; motivated by dimensional regularization
and by recent discussions of higher-dimensional theories in the context of Kaluza-
Klein supergravity and superstrings we summarize the properties of γ-matrices in
arbitrary space time dimension D. For this reason we adopt a notation which is
different from that used in the main text. In D-dimensional Minkowski space the
space components carry indices 1, 2, ...D −1, and the purely imaginary time compo-
nent carries index D. Readers who are emotionally attached to 4-dimensional space
time can simply insert D = 4, or, if they only need a certain 4-dimensional formula,
they are advised to consult section E.3 and parts of the later sections where explicit
results for D = 4 are listed.
E.1. The Clifford algebra
We consider a representation of the D-dimensional Clifford algebra
γaγb + γbγa = 2δabI, a, b = 1, ....D . (E.1)
Repeated multiplication of the γ-matrices leads to a set of 2D matrices Γ
A
Γ
A
: I, γa , γab , γabc . . . , (E.2)
where
γab = γaγb (a < b), γabc = γaγbγc (a < b < c) , etc . (E.3)
In (E.2) we only include ordered strings of different γ-matrices; products in which
the γ-matrices appear in different order or the same γ-matrix appears more than
once can be reduced to one of these by using the anticommutation relation (E.1). On
account of (E.1) the matrices γa1...an are antisymmetric in the indices a1, . . . , an, so
they can also be defined as an antisymmetrized product of γ-matrices
γa1...an =
1
n!
X
perm
[a1...an]
(−)
P
γa1 γa2
. . . γan . (E.4)
As there are `n
D
´
different ways of selecting n different indices between 1 and D,
there are `n
D
´
matrices γa1...an . Therefore the total number of matrices Γ
A is
XD
n=0
D
n
!
= 2D . (E.5)
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