In mathematical physics, the Gordon decomposition (named after Walter Gordon) 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.
Original statement.
For any solution formula_1 of the massive Dirac equation, 
formula_2
the Lorentz covariant number-current formula_3 may be expressed as
formula_4
where 
formula_5
is the spinor generator of Lorentz transformations, 
and
formula_6
is the Dirac adjoint.
The corresponding momentum-space version for plane wave solutions formula_7 and formula_8 obeying 
formula_9
formula_10
is 
formula_11
where
formula_12
Proof.
One sees that from Dirac's equation that
formula_13
and, from the adjoint of Dirac's equation,
formula_14
Adding these two equations yields
formula_15
From Dirac algebra, one may show that Dirac matrices satisfy
formula_16
Using this relation,
formula_17
which amounts to just the Gordon decomposition, after some algebra.
Utility.
The second, spin-dependent, part of the current coupled to the photon field, formula_18 yields, up to an ignorable total divergence,
formula_19
that is, an effective Pauli moment term, formula_20.
Massless generalization.
This decomposition of the current into a particle number-flux (first term) and bound spin contribution (second term) requires formula_21. 
If one assumed that the given solution has energy formula_22 so that formula_23, one might obtain a decomposition that is valid for both massive and massless cases.
Using the Dirac equation again, one finds that 
formula_24
Here formula_25, and 
formula_26
with 
formula_27
so that 
formula_28
where formula_29 is the vector of Pauli matrices.
With the particle-number density identified with formula_30, and for a near plane-wave
solution of finite extent, one may interpret the first term in the decomposition as the current formula_31, due to particles moving at speed formula_32. 
The second term, formula_33 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
formula_34
For a single massive particle in its rest frame, where formula_35, the magnetic moment reduces to
formula_36
where formula_37 and formula_38 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 formula_39 of its kinetic momentum and the magnetic moment becomes
formula_40
Angular momentum density.
For both the massive and massless cases, one also has an expression for the momentum density as part of the symmetric Belinfante–Rosenfeld stress–energy tensor 
formula_41
Using the Dirac equation one may evaluate formula_42 to find the energy density to be formula_43, and the momentum density, 
formula_44
If one used the non-symmetric canonical energy-momentum tensor 
formula_45
one would not find the bound spin-momentum contribution.
By an integration by parts one finds that the spin contribution to the total angular momentum is 
formula_46
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 formula_38 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 formula_48, formula_49. The time average of the Poynting vector momentum density is then given by 
formula_50
We have used Maxwell's equations in passing from the first to the second and third lines, and in expression such as formula_51 the scalar product is between the fields so that the vector character is determined by the formula_52.
As 
formula_53
and for a fluid with intrinsic angular momentum density formula_54 we have 
formula_55
these identities suggest that the spin density can be identified as either
formula_56
or 
formula_57 
The two decompositions coincide when the field is paraxial. They also coincide when the field is a pure helicity state – i.e. when formula_58 where the helicity formula_59 takes the values formula_60 for light that is right or left circularly polarized respectively. In other cases they may differ.
References.
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