Dirac algebra

In mathematical physics, the Dirac algebra is the Clifford algebra Cℓ₄(C), which may be thought of as Cℓ_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

${\displaystyle \displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\eta ^{\mu \nu }\mathbf {1} }$

where ${\displaystyle \eta ^{\mu \nu }\,}$ are the components of the Minkowski metric with signature (+ − − −) and ${\displaystyle \mathbf {1} }$ 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

${\displaystyle \displaystyle \langle a,b\rangle =\sum _{\mu \nu }\eta ^{\mu \nu }a_{\mu }b_{\nu }^{\dagger }}$

where

${\displaystyle \,a=\sum _{\mu }a_{\mu }\gamma ^{\mu }}$ and ${\displaystyle \,b=\sum _{\nu }b_{\nu }\gamma ^{\nu }}$.

Higher powers

The sigmas[1]

${\displaystyle \sigma ^{\mu \nu }=-{\frac {i}{4}}\left[\gamma ^{\mu },\gamma ^{\nu }\right],}$

(I4)

only 6 of which are non-zero due to antisymmetry of the bracket, span the six-dimensional representation space of the tensor (1, 0) ⊕ (0, 1)-representation of the Lorentz algebra inside ${\displaystyle {\mathcal {Cl}}_{1,3}(\mathbb {R} )}$. Moreover, they have the commutation relations of the Lie algebra,[2]

${\displaystyle i\left[\sigma ^{\mu \nu },\sigma ^{\rho \tau }\right]=\eta ^{\nu \rho }\sigma ^{\mu \tau }-\eta ^{\mu \rho }\sigma ^{\nu \tau }-\eta ^{\tau \mu }\sigma ^{\rho \nu }+\eta ^{\tau \nu }\sigma ^{\rho \mu },}$

(I5)

and hence constitute a representation of the Lorentz algebra (in addition to spanning a representation space) sitting inside ${\displaystyle {\mathcal {Cl}}_{1,3}(\mathbb {R} ),}$ the (1/2, 0) ⊕ (0, 1/2) spin representation.

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:

${\displaystyle -i\hbar \gamma ^{\mu }\partial _{\mu }\psi +mc\psi =0\,.}$

and the Klein–Gordon equation:

${\displaystyle -\partial _{t}^{2}\psi +\nabla ^{2}\psi =m^{2}\psi }$

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:

${\displaystyle \psi ^{\dagger }(i\hbar \gamma ^{\mu }\partial _{\mu }+mc)(-i\hbar \gamma ^{\nu }\partial _{\nu }+mc)\psi =0\,.}$

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

${\displaystyle \displaystyle \{\gamma ^{\mu },\gamma ^{\nu }\}=\gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2\eta ^{\mu \nu }I_{4}}$

where ${\displaystyle \{,\}}$ is the anticommutator, ${\displaystyle \eta ^{\mu \nu }\,}$ is the Minkowski metric with signature (+ − − −) and ${\displaystyle \ I_{4}\,}$ is the 4x4 unit matrix.[3]

See also: Dirac equation § Dirac's coup

Cℓ_1,3(ℂ) and Cℓ_1,3(ℝ)

The Dirac algebra can be regarded as a complexification of the real spacetime algebra Cℓ_1,3(ℝ):

${\displaystyle \mathrm {C\ell } _{1,3}(\mathbb {C} )=\mathrm {C\ell } _{1,3}(\mathbb {R} )\otimes \mathbb {C} .}$

Cℓ_1,3(ℝ) differs from Cℓ_1,3(ℂ): in Cℓ_1,3(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.

In contemporary practice, the Dirac algebra continues to be the standard environment the spinors of the Dirac equation "live" in, rather than the spacetime algebra.

References

1. Weinberg 2002, Equation 5.4.6 harvnb error: no target: CITEREFWeinberg2002 (help)
2. Weinberg 2002, Equation 5.4.4 harvnb error: no target: CITEREFWeinberg2002 (help) Section 5.4.
3. see also: Victoria Martin, Lecture Notes SH Particle Physics 2012, Lecture Notes 5–7, Section 5.5 The gamma matrices