Dirac algebra について

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Dirac algebra ：ウィキペディア英語版

Dirac algebra

In mathematical physics, the Dirac algebra is the Clifford algebra ''C''ℓ_1,3(C). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation with the Dirac gamma matrices, which represent the generators of the algebra.
The gamma elements have the defining relation
:

\displaystyle\ = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^ \bold

where

\eta^ \,

are the components of the Minkowski metric with signature (+ − − −) and

\bold

is the identity element of the algebra (the identity matrix in the case of a matrix representation). This allows the definition of a scalar product
:

\displaystyle \langle a , b \rangle  = \sum_ \eta^ a_ b^\dagger_\nu

where
:

\, a = \sum_ a_ \gamma^

and

\, b = \sum_ b_ \gamma^

.

==Derivation starting from the Dirac and Klein–Gordon equation==

The defining form of the gamma elements can be derived if one assumes the covariant form of the Dirac equation:
:

-i \hbar \gamma^\mu \partial_\mu \psi + m c \psi = 0 \,.

and the Klein–Gordon equation:
:

- \partial_t^2 \psi + \nabla^2 \psi = m^2 \psi

to be given, and requires that these equations lead to consistent results.

Derivation from consistency requirement (proof)

Multiplying the Dirac equation by its conjugate equation yields:
:

\psi^  ( i \hbar \gamma^\mu \partial_\mu \psi + m c ) ( -i \hbar \gamma^\nu \partial_\nu \psi + m c ) \psi = 0 \,.

The demand of consistency with the Klein–Gordon equation leads immediately to:
:

\displaystyle\ = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^ I_4

where

\

is the anticommutator,

\eta^ \,

is the Minkowski metric with signature (+ − − −) and

\ I_4 \,

is the 4x4 unit matrix.〔see also: Victoria Martin, (Lecture Notes SH Particle Physics 2012 ), Lecture Notes 5–7, (Section 5.5 The gamma matrices )〕

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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