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.

Original Statement

For any solution $\psi$ of the massive Dirac equation

(i\gamma ^{\mu }\nabla _{\mu }-m)\psi =0,

the Lorentz covariant number-current $j^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi$ can be expressed as

\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),

where

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

is the spinor generator of Lorentz transformations.

The corresponding momentum-space version for plane wave solutions $u(p)$ and ${\bar {u}}(p')$ obeying

(\gamma ^{\mu }p_{\mu }-m)u(p)=0

{\bar {u}}(p')(\gamma ^{\mu }p'_{\mu }-m)=0,

is

{\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)

where

\sigma ^{\mu \nu }=2\Sigma ^{\mu \nu }.

Proof

You can see that from Dirac equation,

{\bar {\psi }}\gamma ^{\mu }(m\psi )={\bar {\psi }}\gamma ^{\mu }(i\gamma ^{\nu }\nabla _{\nu }\psi )

and from conjugation of Dirac equation

({\bar {\psi }}m)\gamma ^{\mu }\psi =((\nabla _{\nu }{\bar {\psi }})(-i\gamma ^{\nu }))\gamma ^{\mu }\psi

Adding two equations yields

{\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 )

From Dirac algebra, you can show that Dirac matrices satisfy

\gamma ^{\mu }\gamma ^{\nu }=\eta ^{\mu \nu }-i\sigma ^{\mu \nu }=\eta ^{\nu \mu }+i\sigma ^{\nu \mu }

Using this relation,

{\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 )

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 $m\ne 0$ . If we assume that the given solution has energy $E={\sqrt {|{\mathbf {k} }|^{2}+m^{2}}}$ so that $\psi({\mathbf r},t)=\psi({\mathbf r})\exp\{-iEt\}$ , we can obtain a decomposition that is valid for both massive and massless cases. Using the Dirac equation again we find that

{\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}).

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

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

where ${\boldsymbol \sigma}=(\sigma_x,\sigma_y,\sigma_z)$ is the vector of Pauli matrices.

With the particle-number density identified with $\rho= \psi^\dagger\psi$ , and for a near plane-wave solution of finite extent, we can interpret the first term in the decomposition as the current ${\mathbf j}_{\rm free}= e\rho {\mathbf k}/E= e\rho {\mathbf v}$ due to particles moving at speed ${\mathbf v}={\mathbf k}/E$ . The second term, ${\mathbf j}_{\rm bound}= (e/E)\nabla\times {\mathbf S}$ 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

{\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.

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

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

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

{\boldsymbol {\mu }}_{\rm {Weyl}}=\left({\frac {e}{E}}\right){\frac {\hbar {\hat {\mathbf {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^{\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).

Using the Dirac equation we can evaluate $T^{0\mu}_{\rm BR}=({\mathcal E},{\mathbf P})$ to find the energy density to be ${\mathcal E}=E\psi^\dagger \psi$ , and the momentum density to be given by

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

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

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

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

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

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 ${\mathbf E}={\mathbf E}({\mathbf r})e^{-i\omega t}$ , ${\mathbf H}={\mathbf H}({\mathbf r})e^{-i\omega t}$ . The time average of the Poynting vector momentum density is then given by

<\mathbf P>=\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})].

We have used Maxwell's equations in passing from the first to the second and third lines, and in expression such as ${\mathbf {H} }^{*}\cdot (\nabla {\mathbf {H} })$ the scalar product is between the fields so that the vector character is determined by the $\nabla$ .

As

{\mathbf P}_{\rm tot}= {\mathbf P}_{\rm free}+ {\mathbf P}_{\rm bound},

and for a fluid with instrinsic angular momentum density ${\mathbf S}$ we have

{\mathbf P}_{\rm bound}= \frac 12 \nabla\times {\mathbf S},

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

{\mathbf S}= \frac{\mu_0}{2i \omega }{\mathbf H}^*\times {\mathbf H}

or

{\mathbf S}= \frac{\epsilon_0}{2i \omega }{\mathbf E}^*\times {\mathbf E}.

The two decompositions coincide when the field is paraxial. They also coincide when the field is a pure helicity state --- i.e. when ${\mathbf E}=i\sigma c {\mathbf B}$ where the helicity $\sigma$ takes the values $\pm 1$ for light that is right or left circularly polarized respectively. In other cases they may differ.

References

↑ W. Gordon (1928). "Der Strom der Diracschen Elektronentheorie". Z. Phys. 50: 630–632. Bibcode:1928ZPhy...50..630G. doi:10.1007/BF01327881.
↑ D.T.Son, N.Yamamoto (2013). "Kinetic theory with Berry curvature from quantum field theories". Physical Review D. 87: 085016. arXiv:1210.8158. Bibcode:2013PhRvD..87h5016S. doi:10.1103/PhysRevD.87.085016.
↑ M.V.Berry (2009). "Optical currents". J. Opt. A. 11: 094001 (12 pages). Bibcode:2009JOptA..11i4001B. doi:10.1088/1464-4258/11/9/094001.

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[gordan-1] W. Gordon (1928). "Der Strom der Diracschen Elektronentheorie". Z. Phys. 50: 630–632. Bibcode:1928ZPhy...50..630G. doi:10.1007/BF01327881.

[son-2] D.T.Son, N.Yamamoto (2013). "Kinetic theory with Berry curvature from quantum field theories". Physical Review D. 87: 085016. arXiv:1210.8158. Bibcode:2013PhRvD..87h5016S. doi:10.1103/PhysRevD.87.085016.

[berry-3] M.V.Berry (2009). "Optical currents". J. Opt. A. 11: 094001 (12 pages). Bibcode:2009JOptA..11i4001B. doi:10.1088/1464-4258/11/9/094001.