I need some suggestions and/or corrections if I understand this correct? My questions are based on the book by Mandl and Shaw.(adsbygoogle = window.adsbygoogle || []).push({});

Conserved currents are based on Noethers theorem and directly connected to spacetime and field transformations (rotations, translations, phase, ...). One can therefore derive the general expression for the Hamiltonian and angular momentum as two conserved quantities.

The Hamiltonian density can be written as,

[tex] \mathcal{H} = \pi_r \times \frac{\mathrm{d}}{\mathrm{d}t}\phi_r - \mathcal{L}, r=1...n,[/tex]

where

[tex]n,[/tex]

is the number of fields.

In a charge KG theory you can have the fields as either

[tex]\phi, \phi^\dagger,[/tex]

or

[tex]\phi_1 = Re(\phi), \phi_2 = Im(\phi),[/tex]

but the easiest way is to use the

[tex]\phi, \phi^\dagger,[/tex]

for a particle interpretation in a canonical quantization and one starts with

[tex] \mathcal{H} = \pi \times \frac{\mathrm{d}}{\mathrm{d}t}\phi + \pi^\dagger \times \frac{\mathrm{d}}{\mathrm{d}t}\phi^\dagger - \mathcal{L}.[/tex]

QUESTION: How does one deal with the expression of the angular momentum? The same question for the Dirac case as follows.

The Dirac theory imposes the use of the same principle as it is a complex field but we now have

[tex]\Psi, \bar{\Psi},[/tex]

and thus

[tex] \mathcal{H} = \pi \times \frac{\mathrm{d}}{\mathrm{d}t}\Psi + \bar{\pi} \times \frac{\mathrm{d}}{\mathrm{d}t}\bar{\Psi}

- \mathcal{L}.[/tex]

QUESTION: However when the book speaks about the angular momentum it does not use the adjoint field at all, only the Dirac field. Is it correct to think that it means that it treats the Dirac field as the complex one directly? Without the need of the adjoint in this sense?

NOTE: The reason for my question is that the spinpart of the expression on the angular momentum is as well a kind of sum of present fields but also a transformationmatrix is involved. It yields,

[tex]\mathcal{M}^{0\alpha\beta} = x^\alpha\mathcal{F}^{0\beta} - x^\beta\mathcal{F}^{0\alpha} + \pi_rS^{\alpha\beta}_{rs}\phi_s.[/tex]

It is stated that one uses the sum as the elements of the Dirac spinor instead of the fields. But, this is not the case for the Hamiltonian nor the momentum where one uses the fields.

Any suggestions of other resources to read from. I have looked in a lot of books and people speak differently when dealing with these issues. It seems somehow that I miss an important part?

/C

EDIT: Did the formulas with LaTex as I discovered it was possible.

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# Angular momentum vs the Hamiltonian problem in the Dirac field theory (canonical)

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