- #1

- 2,116

- 19

I've had some trouble with the argument "spin angular mometum has no classical analogue", that everyone seems to be repeating. I used Noether's theorem to calculate angular momentum of a classical Dirac's field, and found that it has internal angular mometum density, that cannot be written as a cross produc of position and momentum. To be more presice, if I rotate Dirac's field around an angle [tex](\theta_1,\theta_2,\theta_3)=\theta(n_1,n_2,n_3)[/tex], I calculated [tex]d\psi/d\theta[/tex] to be (in Weyl representation)

[tex]

n_k\Big((\boldsymbol{x}\times\nabla)^k + \frac{i}{2}\left[\begin{array}{cc}\sigma^k & 0 \\ 0 & \sigma^k \\ \end{array}\right]\Big)\psi

[/tex]

The first term is kind of transformation that also scalar fields have, and the second term comes from property of Dirac's field. Ignoring the first term, I then solved a vector

[tex]

\frac{\hbar c}{2}\psi^\dagger \left[\begin{array}{cc}\boldsymbol{\sigma} & 0 \\ 0 & \boldsymbol{\sigma} \\ \end{array}\right]\psi

[/tex]

to be a density of a conserved quantity. Isn't it justified to call this the internal angular mometum of classical Dirac's field? I also noted, that I can get spin operators by substituing [tex]\psi[/tex] operators into this expression, so it seems very much that this internal angular mometum is quite analogous to the spin of electrons.

[tex]

n_k\Big((\boldsymbol{x}\times\nabla)^k + \frac{i}{2}\left[\begin{array}{cc}\sigma^k & 0 \\ 0 & \sigma^k \\ \end{array}\right]\Big)\psi

[/tex]

The first term is kind of transformation that also scalar fields have, and the second term comes from property of Dirac's field. Ignoring the first term, I then solved a vector

[tex]

\frac{\hbar c}{2}\psi^\dagger \left[\begin{array}{cc}\boldsymbol{\sigma} & 0 \\ 0 & \boldsymbol{\sigma} \\ \end{array}\right]\psi

[/tex]

to be a density of a conserved quantity. Isn't it justified to call this the internal angular mometum of classical Dirac's field? I also noted, that I can get spin operators by substituing [tex]\psi[/tex] operators into this expression, so it seems very much that this internal angular mometum is quite analogous to the spin of electrons.

Last edited: