Dirac equation in the algebra of physical space

The Dirac equation, as the relativistic equation that describes spin 1/2 particles in quantum mechanics, can be written in terms of the Algebra of physical space (APS), which is a case of a Clifford algebra or geometric algebra that is based on the use of paravectors.

The Dirac equation in APS, including the electromagnetic interaction, reads

i∂¯Ψ𝐞3+eA¯Ψ=mΨ¯†

Another form of the Dirac equation in terms of the Space time algebra was given earlier by David Hestenes.

In general, the Dirac equation in the formalism of geometric algebra has the advantage of providing a direct geometric interpretation.

## Contents

* 1 Relation with the standard form
* 2 Electromagnetic gauge
* 3 Current
* 4 Second order Dirac equation
* 5 Free particle solutions
* 5.1 Positive energy solutions
* 5.2 Negative energy solutions
* 6 Dirac Lagrangian
* 7 See also
* 8 References
* 8.1 Textbooks
* 8.2 Articles

## Relation with the standard form

The spinor can be written in a null basis as

Ψ=ψ11P3−ψ12P3𝐞1+ψ21𝐞1P3+ψ22P¯3,

such that the representation of the spinor in terms of the Pauli matrices is

Ψ->(ψ11ψ12ψ21ψ22)
Ψ¯†->(ψ22*−ψ21*−ψ12*ψ11*)

The standard form of the Dirac equation can be recovered by decomposing the spinor in its right and left-handed spinor components, which are extracted with the help of the projector

P3=12(1+𝐞3),

such that

ΨL=Ψ¯†P3
ΨR=ΨP3

with the following matrix representation

ΨL->(ψ22*0−ψ12*0)
ΨR->(ψ110ψ210)

The Dirac equation can be also written as

i∂Ψ¯†𝐞3+eAΨ¯†=mΨ

Without electromagnetic interaction, the following equation is obtained from the two equivalent forms of the Dirac equation

(0i∂¯i∂0)(Ψ¯†P3ΨP3)=m(Ψ¯†P3ΨP3)

so that

(0i∂0+i∇i∂0−i∇0)(ΨLΨR)=m(ΨLΨR)

or in matrix representation

i((0110)∂0+(0σ−σ0)⋅∇)(ψLψR)=m(ψLψR),

where the second column of the right and left spinors can be dropped by defining the single column chiral spinors as

ψL->(ψ22*−ψ12*)
ψR->(ψ11ψ21)

The standard relativistic covariant form of the Dirac equation in the Weyl representation can be easily identified iγμ∂μψ=mψ, such that

ψ=(ψ22*−ψ12*ψ11ψ21)

Given two spinors Ψ and Φ in APS and their respective spinors in the standard form as ψ and ϕ, one can verify the following identity

ϕ†γ0ψ=⟨Φ¯Ψ+(Ψ¯Φ)†⟩S,

such that

ψ†γ0ψ=2⟨Ψ¯Ψ⟩SR

## Electromagnetic gauge

The Dirac equation is invariant under a global right rotation applied on the spinor of the type

Ψ->Ψ′=ΨR0

so that the kinetic term of the Dirac equation transforms as

i∂¯Ψ𝐞3->i∂¯ΨR0𝐞3R0†R0=(i∂¯Ψ𝐞3′)R0,

where we identify the following rotation

𝐞3->𝐞3′=R0𝐞3R0†

The mass term transforms as

mΨ†‾->m(ΨR0)†‾=mΨ†‾R0,

so that we can verify the invariance of the form of the Dirac equation. A more demanding requirement is that the Dirac equation should be invariant under a local gauge transformation of the type R=exp⁡(−ieχ𝐞3)

In this case, the kinetic term transforms as

i∂¯Ψ𝐞3->(i∂¯Ψ)R𝐞3+(e∂¯χ)ΨR,

so that the left side of the Dirac equation transforms covariantly as

i∂¯Ψ𝐞3−eA¯Ψ->(i∂¯ΨR𝐞3R†−e(A+∂χ)‾Ψ)R,

where we identify the need to perform an electromagnetic gauge transformation. The mass term transforms as in the case with global rotation, so, the form of the Dirac equation remains invariant.

## Current

The current is defined as

J=ΨΨ†,

which satisfies the continuity equation

⟨∂¯J⟩S=0

## Second order Dirac equation

An application of the Dirac equation on itself leads to the second order Dirac equation

(−∂∂¯+AA¯)Ψ−i(2e⟨A∂¯⟩S+eF)Ψ𝐞3=m2Ψ

## Free particle solutions

### Positive energy solutions

A solution for the free particle with momentum p=p0+𝐩 and positive energy p0>0 is

Ψ=pmR(0)exp⁡(−i⟨px¯⟩S𝐞3).

This solution is unimodular

ΨΨ¯=1

and the current resembles the classical proper velocity

u=pm
J=ΨΨ†=pm

### Negative energy solutions

A solution for the free particle with negative energy and momentum p=−|p0|−𝐩=−p′ is

Ψ=ip′mR(0)exp⁡(i⟨p′x¯⟩S𝐞3),

This solution is anti-unimodular

ΨΨ¯=−1

and the current resembles the classical proper velocity u=pm

J=ΨΨ†=−pm,

but with a remarkable feature: "the time runs backwards"

dtdτ=⟨pm⟩S<0

## Dirac Lagrangian

The Dirac Lagrangian is

L=⟨i∂Ψ¯†𝐞3Ψ¯−eAΨ¯†Ψ¯−mΨΨ¯⟩S

## See also

* Paravector
* Algebra of physical space
* Geometric algebra
* Two-body Dirac equations

## References

### Textbooks

* Baylis, William (2002). Electrodynamics: A Modern Geometric Approach (2nd ed.). Birkhäuser. ISBN 0-8176-4025-8
* W. E. Baylis, editor, Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering, Birkhäuser, Boston 1996. ISBN 0-8176-3868-7

### Articles

* Baylis, W. E. (1 March 1992). "Classical eigenspinors and the Dirac equation". Physical Review A (American Physical Society (APS)) 45 (7): 4293–4302. doi:10.1103/physreva.45.4293. ISSN 1050-2947.
* Hestenes, David (1975). "Observables, operators, and complex numbers in the Dirac theory". Journal of Mathematical Physics (AIP Publishing) 16 (3): 556–572. doi:10.1063/1.522554. ISSN 0022-2488.

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