Girish Mahajan (Editor)

Gordon Decomposition

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In mathematical physics, the Gordon-decomposition (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

Original Statement

For any solution ψ of the massive Dirac equation

( i γ μ ∇ μ − m ) ψ = 0 ,

the Lorentz covariant number-current j μ = ψ ¯ γ μ ψ can be expressed as

ψ ¯ γ μ ψ = i 2 m ( ψ ¯ ∇ μ ψ − ( ∇ μ ψ ¯ ) ψ ) + 1 m ∂ ν ( ψ ¯ Σ μ ν ψ ) ,

where

Σ μ ν = i 4 [ γ μ , γ ν ]

is the spinor generator of Lorentz transformations.

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 + m 2 so that ψ ( r , t ) = ψ ( r ) exp ⁡ { − i E t } , we can obtain a decomposition that is valid for both massive and massless cases. Using the Dirac equation again we find that

j ≡ e ψ ¯ γ ψ = e 2 i E ( ψ † ∇ ψ − ( ∇ ψ † ) ψ ) + e E ( ∇ × S ) .

Here γ = ( γ 1 , γ 2 , γ 3 ) , and S = ψ † S ^ ψ with ( S ^ x , S ^ y , S ^ z ) = ( Σ 23 , Σ 31 , Σ 12 ) so that

S ^ = 1 2 [ σ 0 0 σ ] ,

where σ = ( σ x , σ y , σ z ) 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 j f r e e = e ρ k / E = e ρ v due to particles moving at speed v = k / E . The second term, j b o u n d = ( 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 = 1 2 ∫ r × j b o u n d d 3 x = 1 2 ∫ r × ( e E ∇ × S ) d 3 x = e E ∫ S d 3 x .

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

μ D i r a c = ( e m ) S = ( e g 2 m ) 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

μ W e y l = ( e E ) ℏ k ^ 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

T B R μ ν = i 4 ( ψ ¯ γ μ ∇ ν ψ − ( ∇ ν ψ ¯ ) γ μ ψ + ψ ¯ γ ν ∇ μ ψ − ( ∇ μ ψ ¯ ) γ ν ψ ) .

Using the Dirac equation we can evaluate T B R 0 μ = ( E , P ) to find the energy density to be E = E ψ † ψ , and the momentum density to be given by

P = 1 2 i ( ψ † ( ∇ ψ ) − ( ∇ ψ † ) ψ ) + 1 2 ∇ × S .

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

T c a n o n i c a l μ ν = i 2 ( ψ ¯ γ μ ∇ ν ψ − ( ∇ ν ψ ¯ ) γ μ ψ ) ,

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 × ( 1 2 ∇ × S ) d 3 x = ∫ S d 3 x .

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 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 − i ω t , H = H ( r ) e − i ω t . The time average of the Poynting vector momentum density is then given by

< P >= 1 4 c 2 [ E ∗ × H + E × H ∗ ] = ϵ 0 4 i ω [ E ∗ ⋅ ( ∇ E ) − ( ∇ E ∗ ) ⋅ E + ∇ × ( E ∗ × E ) ] = μ 0 4 i ω [ 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 ∇ .

As

P t o t = P f r e e + P b o u n d ,

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

P b o u n d = 1 2 ∇ × S ,

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

S = μ 0 2 i ω H ∗ × H

or

S = ϵ 0 2 i ω E ∗ × 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 = i σ c B 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

Gordon Decomposition Wikipedia
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