Gordon Decomposition

In mathematical physics, the Gordon-decomposition[1] (named after Walter Gordon one of the discoverers of the Klein-Gordon equation) of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of the Dirac equation and so it applies only to "on-shell" solutions of the Dirac equation.

## Contents

* 1 Original Statement
* 1.1 Proof
* 2 Massless Generalization
* 3 Angular Momentum Density
* 4 Spin in Maxwell's equations
* 5 References

## Original Statement

For any solution ψ of the massive Dirac equation

(iγμ∇μ−m)ψ=0,

the Lorentz covariant number-current jμ=ψ¯γμψ can be expressed as

ψ¯γμψ=i2m(ψ¯∇μψ−(∇μψ¯)ψ)+1m∂ν(ψ¯Σμνψ),

where

Σμν=i4[γμ,γν]

is the spinor generator of Lorentz transformations.

The corresponding momentum-space version for plane wave solutions u(p) and u¯(p′) obeying

(γμpμ−m)u(p)=0
u¯(p′)(γμp'μ−m)=0,

u¯(p′)γμu(p)=u¯(p′)[(p+p′)μ2m+iσμν(p′−p)ν2m]u(p)

where

σμν=2Σμν.

### Proof

You can see that from Dirac equation,

ψ¯γμ(mψ)=ψ¯γμ(iγν∇νψ)

and from conjugation of Dirac equation

(ψ¯m)γμψ=((∇νψ¯)(−iγν))γμψ

Adding two equations yields

ψ¯γμψ=i2m(ψ¯γμγν∇νψ−(∇νψ¯)γνγμψ)

From Dirac algebra, you can show that Dirac matrices satisfy

γμγν=ημν−iσμν=ηνμ+iσνμ

Using this relation,

ψ¯γμψ=i2m(ψ¯(ημν−iσμν)∇νψ−(∇νψ¯)(ημν+iσμν)ψ)

which is just Gordon decomposition after some algebra.

## Massless Generalization

This decomposition of the current into a particle number-flux (first term) and bound spin contribution (second term) requires m≠0. If we assume that the given solution has energy E=|𝐤|2+m2 so that ψ(𝐫,t)=ψ(𝐫)exp⁡{−iEt}, we can obtain a decomposition that is valid for both massive and massless cases. Using the Dirac equation again we find that

𝐣≡eψ¯γψ=e2iE(ψ†∇ψ−(∇ψ†)ψ)+eE(∇×𝐒).

Here γ=(γ1,γ2,γ3), and 𝐒=ψ†𝐒^ψ with (S^x,S^y,S^z)=(Σ23,Σ31,Σ12) so that

𝐒^=12[σ00σ],

where σ=(σx,σy,σz) is the vector of Pauli matrices.

With the particle-number density identified with ρ=ψ†ψ, and for a near plane-wave solution of finite extent, we can interpret the first term in the decomposition as the current 𝐣free=eρ𝐤/E=eρ𝐯 due to particles moving at speed 𝐯=𝐤/E. The second term, 𝐣bound=(e/E)∇×𝐒 is the current due to the gradients in the intrinsic magnetic moment density. The magnetic moment itself is found by integrating by parts to show that

μ=12∫𝐫×𝐣boundd3x=12∫𝐫×(eE∇×𝐒)d3x=eE∫𝐒d3x.

For a single massive particle in its rest frame, where E=m, the magnetic moment becomes

μDirac=(em)𝐒=(eg2m)𝐒.

where |𝐒|=ℏ/2 and g=2 is the Dirac value of the gyromagnetic ratio.

For a single massless particle obeying the right-handed Weyl equation the spin-1/2 is locked to the direction 𝐤^ of its kinetic momentum and the magnetic moment becomes[2]

μWeyl=(eE)ℏ𝐤^2.

## Angular Momentum Density

For the both massive and massless case we also have an expression for the momentum density as part of the symmetric Belinfante-Rosenfeld stress-energy tensor

TBRμν=i4(ψ¯γμ∇νψ−(∇νψ¯)γμψ+ψ¯γν∇μψ−(∇μψ¯)γνψ).

Using the Dirac equation we can evaluate TBR0μ=(ℰ,𝐏) to find the energy density to be ℰ=Eψ†ψ, and the momentum density to be given by

𝐏=12i(ψ†(∇ψ)−(∇ψ†)ψ)+12∇×𝐒.

If we used the non-symmetric canonical energy-momentum tensor

Tcanonicalμν=i2(ψ¯γμ∇νψ−(∇νψ¯)γμψ),

we would not find the bound spin-momentum contribution.

By an integration by parts we find that the spin contribution to the total angular momentum is

∫𝐫×(12∇×𝐒)d3x=∫𝐒d3x.

This is what is expected, so the division by 2 in the spin contribution to the momentum density is necessary. The absence of a division by 2 in the formula for the current reflects the g=2 gyromagnetic ratio of the electron. In other words, a spin-density gradient is twice as effective at making an electric current as it is at contributing to the linear momentum.

## Spin in Maxwell's equations

Motivated by the Riemann-Silberstein vector form of Maxwell's equations, Michael Berry[3] uses the Gordon strategy to obtain gauge-invariant expressions for the intrinsic spin angular-momentum density for solutions to Maxwell's equations.

He assumes that the solutions are monochromatic and uses the phasor expressions 𝐄=𝐄(𝐫)e−iωt, 𝐇=𝐇(𝐫)e−iωt. The time average of the Poynting vector momentum density is then given by

<𝐏>=14c2[𝐄*×𝐇+𝐄×𝐇*]
=ϵ04iω[𝐄*⋅(∇𝐄)−(∇𝐄*)⋅𝐄+∇×(𝐄*×𝐄)]
=μ04iω[𝐇*⋅(∇𝐇)−(∇𝐇*)⋅𝐇+∇×(𝐇*×𝐇)].

We have used Maxwell's equations in passing from the first to the second and third lines, and in expression such as 𝐇*⋅(∇𝐇) the scalar product is between the fields so that the vector character is determined by the ∇.

𝐏tot=𝐏free+𝐏bound,

and for a fluid with instrinsic angular momentum density 𝐒 we have

𝐏bound=12∇×𝐒,

these identities suggest that the spin density can be identified as either

𝐒=μ02iω𝐇*×𝐇

𝐒=ϵ02iω𝐄*×𝐄.

The two decompositions coincide when the field is paraxial. They also coincide when the field is a pure helicity state 
 i.e. when 𝐄=iσc𝐁 where the helicity σ takes the values ±1 for light that is right or left circularly polarized respectively. In other cases they may differ.

## References

1. ↑ W. Gordon (1928). "Der Strom der Diracschen Elektronentheorie". Z. Phys. 50: 630–632. doi:10.1007/BF01327881. Bibcode: 1928ZPhy...50..630G.
2. ↑ D.T.Son, N.Yamamoto (2013). "Kinetic theory with Berry curvature from quantum field theories". Physical Review D 87: 085016. doi:10.1103/PhysRevD.87.085016. Bibcode: 2013PhRvD..87h5016S.
3. ↑ M.V.Berry (2009). "Optical currents". J. Opt. A 11: 094001 (12 pages). doi:10.1088/1464-4258/11/9/094001. Bibcode: 2009JOptA..11i4001B.

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