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]

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 the mathematics of Riemannian geometry, it is conventional to define the Clifford algebra Cℓ_p,q(ℝ) for arbitrary dimensions p,q; the anti-commutation of the Weyl spinors emerges naturally from the Clifford algebra.[4] The Weyl spinors transform under the action of the spin group ${\displaystyle \mathrm {Spin} (n)}$. The complexification of the spin group, called the spinc group ${\displaystyle \mathrm {Spin} ^{\mathbb {C} }(n)}$, is a product ${\displaystyle \mathrm {Spin} (n)\times _{\mathbb {Z} _{2}}S^{1}}$ of the spin group with the circle ${\displaystyle S^{1}\cong U(1)}$ with the product ${\displaystyle \times _{\mathbb {Z} _{2}}}$ just a notational device to identify ${\displaystyle (a,u)\in \mathrm {Spin} (n)\times S^{1}}$ with ${\displaystyle (-a,-u).}$ The geometric point of this is that it disentangles the real spinor, which is covariant under Lorentz transformations, from the ${\displaystyle U(1)}$ component, which can be identified with the ${\displaystyle U(1)}$ fiber of the electromagnetic interaction. The ${\displaystyle \times _{\mathbb {Z} _{2}}}$ is entangling parity and charge conjugation in a manner suitable for relating the Dirac particle/anti-particle states (equivalently, the chiral states in the Weyl basis). The bispinor, insofar as it has linearly independent left and right components, can interact with the electromagnetic field. This is in contrast to the Majorana spinor and the ELKO spinor, which cannot (i.e. they are electrically neutral), as they explicitly constrain the spinor so as to not interact with the ${\displaystyle S^{1}}$ part coming from the complexification.

Insofar as the presentation of charge and parity can be a confusing topic in conventional quantum field theory textbooks, the more careful dissection of these topics in a general geometric setting can be elucidating. Standard expositions of the Clifford algebra construct the Weyl spinors from first principles; that they "automatically" anti-commute is an elegant geometric by-product of the construction, completely by-passing any arguments that appeal to the Pauli exclusion principle (or the sometimes common sensation that Grassmann variables have been introduced via ad hoc argumentation.)

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

See also

• Higher-dimensional gamma matrices
• Fierz identity

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
4. Jurgen Jost (2002) "Riemannian Geometry and Geometric Analysis (3rd Edition)", Springer Universitext. See section 1.8