Gordon decomposition

In mathematical physics, the Gordon decomposition[1] (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.

## Contents

* 1 Original statement
* 1.1 Proof
* 1.2 Utility
* 2 Massless generalization
* 3 Angular momentum density
* 4 Spin in Maxwell's equations
* 5 References

## Original statement

For any solution [math]\displaystyle{ \psi }[/math] of the massive Dirac equation,

[math]\displaystyle{ (i\gamma^\mu \nabla_\mu-m)\psi=0, }[/math]

the Lorentz covariant number-current [math]\displaystyle{ j^\mu=\bar\psi \gamma^\mu\psi }[/math] may be expressed as

[math]\displaystyle{ \bar\psi \gamma^\mu\psi =\frac{i}{2m} (\bar \psi \nabla^\mu\psi -(\nabla^\mu\bar \psi) \psi)+\frac{1}{m} \partial_\nu(\bar\psi \Sigma^{\mu\nu}\psi), }[/math]

where

[math]\displaystyle{ \Sigma^{\mu\nu} = \frac {i}{4} [\gamma^\mu,\gamma^\nu] }[/math]

is the spinor generator of Lorentz transformations, and

[math]\displaystyle{ \bar\psi = \psi^\dagger \gamma^0 }[/math]

is the Dirac adjoint.

The corresponding momentum-space version for plane wave solutions [math]\displaystyle{ u(p) }[/math] and [math]\displaystyle{ \bar u(p') }[/math] obeying

[math]\displaystyle{ (\gamma^\mu p_\mu -m)u(p)=0 }[/math]
[math]\displaystyle{ \bar u(p') (\gamma^\mu p'_\mu -m)=0, }[/math]

[math]\displaystyle{ \bar u(p') \gamma^\mu u(p)=\bar u(p') \left[\frac{(p+p')^\mu}{2m} +i \sigma^{\mu\nu}\frac{(p'-p)_\nu}{2m}\right]u(p) ~, }[/math]

where

[math]\displaystyle{ \sigma^{\mu\nu} = 2\Sigma^{\mu\nu}. }[/math]

### Proof

One sees that from Dirac's equation that

[math]\displaystyle{ \bar\psi \gamma^\mu (m \psi) = \bar\psi \gamma^\mu (i\gamma^\nu \nabla_\nu\psi) }[/math]

and, from the adjoint of Dirac's equation,

[math]\displaystyle{ (\bar\psi m )\gamma^\mu \psi = ((\nabla_\nu \bar\psi)(-i\gamma^\nu))\gamma^\mu\psi . }[/math]

Adding these two equations yields

[math]\displaystyle{ \bar\psi \gamma^\mu \psi = \frac{i}{2m}(\bar\psi \gamma^\mu \gamma^\nu \nabla_\nu\psi -(\nabla_\nu\bar\psi) \gamma^\nu \gamma^\mu\psi) . }[/math]

From Dirac algebra, one may show that Dirac matrices satisfy

[math]\displaystyle{ \gamma^\mu \gamma^\nu = \eta^{\mu\nu} - i\sigma^{\mu\nu}= \eta^{\nu\mu} + i\sigma^{\nu\mu}. }[/math]

Using this relation,

[math]\displaystyle{ \bar\psi \gamma^\mu \psi = \frac{i}{2m}(\bar\psi (\eta^{\mu\nu} - i\sigma^{\mu\nu})\nabla_\nu\psi -(\nabla_\nu\bar\psi)(\eta^{\mu\nu} + i\sigma^{\mu\nu})\psi), }[/math]

which amounts to just the Gordon decomposition, after some algebra.

### Utility

The second, spin-dependent, part of the current coupled to the photon field, [math]\displaystyle{ -A_\mu j^\mu }[/math] yields, up to an ignorable total divergence,

[math]\displaystyle{ -\frac{e\hbar}{2mc}\partial_\nu A_\mu \bar{\psi} \sigma^{\nu\mu}\psi= -\frac{e\hbar}{2mc}\tfrac{1}{2}F_{\mu\nu} \bar{\psi} \sigma^{\mu\nu}\psi , }[/math]

that is, an effective Pauli moment term, [math]\displaystyle{ -(e\hbar/2mc) \vec{B} \cdot \psi^\dagger \vec \sigma \psi }[/math].

## Massless generalization

This decomposition of the current into a particle number-flux (first term) and bound spin contribution (second term) requires [math]\displaystyle{ m\ne 0 }[/math].

If one assumed that the given solution has energy [math]\displaystyle{ E= \sqrt{|{\mathbf k}|^2+m^2} }[/math] so that [math]\displaystyle{ \psi(\mathbf r,t) = \psi({\mathbf r}) e^{-iEt} }[/math], one might obtain a decomposition that is valid for both massive and massless cases.[2]

Using the Dirac equation again, one finds that

[math]\displaystyle{ {\mathbf j}\equiv e\bar \psi {\boldsymbol \gamma} \psi = \frac{e}{2iE} \left(\psi^\dagger \nabla \psi - (\nabla \psi^\dagger)\psi\right) +\frac{e}{E} (\nabla \times{\mathbf S}). }[/math]

Here [math]\displaystyle{ {\boldsymbol \gamma}= (\gamma^1,\gamma^2,\gamma^3) }[/math], and [math]\displaystyle{ {\mathbf S} =\psi^\dagger \hat {\mathbf S}\psi }[/math] with [math]\displaystyle{ (\hat S_x,\hat S_y,\hat S_z)= (\Sigma^{23},\Sigma^{31},\Sigma^{12}), }[/math] so that

[math]\displaystyle{ \hat {\mathbf S}=\frac 12 \left[\begin{matrix}{\boldsymbol \sigma}&0 \\\ 0 &{\boldsymbol \sigma}\end{matrix}\right], }[/math]

where [math]\displaystyle{ {\boldsymbol \sigma}=(\sigma_x,\sigma_y,\sigma_z) }[/math] is the vector of Pauli matrices.

With the particle-number density identified with [math]\displaystyle{ \rho= \psi^\dagger\psi }[/math], and for a near plane-wave solution of finite extent, one may interpret the first term in the decomposition as the current [math]\displaystyle{ {\mathbf j}_{\rm free}= e\rho {\mathbf k}/E= e\rho {\mathbf v} }[/math], due to particles moving at speed [math]\displaystyle{ {\mathbf v}={\mathbf k}/E }[/math].

The second term, [math]\displaystyle{ {\mathbf j}_{\rm bound}= (e/E)\nabla\times {\mathbf S} }[/math] 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

[math]\displaystyle{ {\boldsymbol \mu}\stackrel{\rm }{=} \frac{1}{2}\int {\mathbf r}\times {\mathbf j}_{\rm bound}\,d^3x =\frac{1}{2}\int {\mathbf r}\times \left(\frac e E\nabla \times {\mathbf S}\right)\,d^3 x = \frac{e}{E}\int {\mathbf S}\,d^3 x ~. }[/math]

For a single massive particle in its rest frame, where [math]\displaystyle{ E=m }[/math], the magnetic moment reduces to

[math]\displaystyle{ {\boldsymbol \mu}_{\rm Dirac}=\left( \frac{e}{m}\right){\mathbf S}= \left(\frac{e g}{2m}\right) {\mathbf S}. }[/math]

where [math]\displaystyle{ |{\mathbf S}|=\hbar/2 }[/math] and [math]\displaystyle{ g=2 }[/math] 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 [math]\displaystyle{ \hat {\mathbf k} }[/math] of its kinetic momentum and the magnetic moment becomes[3]

[math]\displaystyle{ {\boldsymbol \mu}_{\rm Weyl}=\left( \frac{e}{E}\right) \frac{\hbar \hat {\mathbf k}}{2}. }[/math]

## 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

[math]\displaystyle{ T^{\mu\nu}_{\rm BR}= \frac{i}{4}(\bar \psi \gamma^\mu \nabla^\nu \psi - (\nabla^\nu \bar\psi) \gamma^\mu\psi +\bar \psi \gamma^\nu \nabla^\mu \psi-(\nabla^\mu \bar\psi) \gamma^\nu\psi). }[/math]

Using the Dirac equation one may evaluate [math]\displaystyle{ T^{0\mu}_{\rm BR}=({\mathcal E},{\mathbf P}) }[/math] to find the energy density to be [math]\displaystyle{ {\mathcal E}=E\psi^\dagger \psi }[/math], and the momentum density,

[math]\displaystyle{ {\mathbf P}= \frac 1{2i}\left (\psi^\dagger (\nabla \psi)- (\nabla \psi^\dagger)\psi\right) +\frac 12 \nabla\times {\mathbf S}. }[/math]

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

[math]\displaystyle{ T^{\mu\nu}_{\rm canonical}= \frac{i}{2}(\bar \psi \gamma^\mu \nabla^\nu \psi - (\nabla^\nu \bar\psi) \gamma^\mu\psi), }[/math]

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

[math]\displaystyle{ \int {\mathbf r}\times\left(\frac 12 \nabla\times {\mathbf S}\right)\,d^3x = \int {\mathbf S}\, d^3x. }[/math]

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 [math]\displaystyle{ g=2 }[/math] 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[4] 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 [math]\displaystyle{ \mathbf E = \mathbf E(\mathbf r) e^{-i\omega t} }[/math], [math]\displaystyle{ \mathbf H = \mathbf H(\mathbf r) e^{-i\omega t} }[/math]. The time average of the Poynting vector momentum density is then given by [math]\displaystyle{ \begin{align} \langle \mathbf P \rangle &=\frac 1{4c^2} [{\mathbf E}^*\times {\mathbf H}+ {\mathbf E}\times {\mathbf H}^*] \\\ &= \frac{\epsilon_0}{4i\omega }[{\mathbf E}^*\cdot(\nabla {\mathbf E})- (\nabla {\mathbf E}^*)\cdot{\mathbf E} +\nabla\times({\mathbf E}^*\times {\mathbf E})] \\\ &= \frac{\mu_0}{4i \omega }[{\mathbf H}^*\cdot(\nabla {\mathbf H})- (\nabla {\mathbf H}^*)\cdot{\mathbf H} + \nabla\times({\mathbf H}^*\times {\mathbf H})]. \end{align} }[/math] We have used Maxwell's equations in passing from the first to the second and third lines, and in expression such as [math]\displaystyle{ {\mathbf H}^*\cdot(\nabla {\mathbf H}) }[/math] the scalar product is between the fields so that the vector character is determined by the [math]\displaystyle{ \nabla }[/math].

As [math]\displaystyle{ \mathbf P_\text{tot} = \mathbf P_\text{free} + {\mathbf P}_\text{bound}, }[/math] and for a fluid with intrinsic angular momentum density [math]\displaystyle{ \mathbf S }[/math] we have [math]\displaystyle{ \mathbf P_\text{bound} = \frac 1 2 \nabla\times \mathbf S, }[/math] these identities suggest that the spin density can be identified as either [math]\displaystyle{ \mathbf S = \frac{\mu_0}{2i \omega } {\mathbf H}^*\times {\mathbf H} }[/math] or [math]\displaystyle{ \mathbf S = \frac{\epsilon_0}{2i \omega }{\mathbf E}^*\times {\mathbf E}. }[/math] The two decompositions coincide when the field is paraxial. They also coincide when the field is a pure helicity state – i.e. when [math]\displaystyle{ {\mathbf E}=i\sigma c {\mathbf B} }[/math] where the helicity [math]\displaystyle{ \sigma }[/math] takes the values [math]\displaystyle{ \pm 1 }[/math] 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 (9–10): 630–632. doi:10.1007/BF01327881. Bibcode: 1928ZPhy...50..630G.
2. ↑ M.Stone (2015). "Berry phase and anomalous velocity of Weyl fermions and Maxwell photons". International Journal of Modern Physics B 30 (2): 1550249. doi:10.1142/S0217979215502495.
3. ↑ D.T.Son, N.Yamamoto (2013). "Kinetic theory with Berry curvature from quantum field theories". Physical Review D 87 (8): 085016. doi:10.1103/PhysRevD.87.085016. Bibcode: 2013PhRvD..87h5016S.
4. ↑ M.V.Berry (2009). "Optical currents". J. Opt. A 11 (9): 094001 (12 pages). doi:10.1088/1464-4258/11/9/094001. Bibcode: 2009JOptA..11i4001B.

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