Deriving 3 Momentum & Angular Momentum Operators of Maxwell Lagrangian

In summary: Springer Verlag, 2nd ed., 2007), by D. Bohm and J. Clauser, has a very nice derivation in ch.2, pp. 182-192.
  • #1
Petraa
21
0
Hello,

Where can I find a good explanation (book) of the derivation via Noether's theorem of the three momentum and angular momentum operators of the usual maxwell lagrangian ?

Thank you!
 
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  • #2
This is standard QFT (actually QED) material, any thorough book should have it. Check out a nice treatment in Chapter 2 of F. Gross' "Relativistic Quantum Mechanics and Field Theory", Wiley, 1999.

In purely classical context (no operators), advanced electrodynamics books should also have this.
 
  • #3
dextercioby said:
This is standard QFT (actually QED) material, any thorough book should have it. Check out a nice treatment in Chapter 2 of F. Gross' "Relativistic Quantum Mechanics and Field Theory", Wiley, 1999.

In purely classical context (no operators), advanced electrodynamics books should also have this.

I've been watching the book and yes, the book treats it but don't deduce them. He just announces and perform some calculations with them
 
  • #4
Can you calculate [itex] T^{\mu\nu} [/itex] and [itex] M^{\lambda}_{~~\mu\nu} [/itex] from the Lagrangian and the general Noether formula which for the energy-momentum 4 tensor reads:

[itex] T^{\mu}_{~~\nu} [/itex] = ([itex] \frac{\partial \mathcal{L}}{\partial(\partial_{\mu}A_{\rho})}[/itex] [itex] -\mathcal{L}\delta^{\mu}_{\lambda} [/itex]) X [itex] \frac{\partial x'^{\lambda}}{\partial\epsilon^{\nu}} [/itex],

where

[tex] x'^{\mu} = x^{\mu} + \epsilon^{\mu} [/tex]
 
Last edited:
  • #5
dextercioby said:
Can you calculate [itex] T^{\mu\nu} [/itex] and [itex] M^{\lambda}_{~~\mu\nu} [/itex] from the Lagrangian and the general Noether formula which for the energy-momentum 4 tensor reads:

[itex] T^{\mu}_{~~\nu} [/itex] = ([itex] \frac{\partial \mathcal{L}}{\partial(\partial_{\mu}A_{\rho})}[/itex] [itex] -\mathcal{L}\delta^{\mu}_{\lambda} [/itex]) X [itex] \frac{\partial x'^{\lambda}}{\partial\epsilon^{\nu}} [/itex],

where

[tex] x'^{\mu} = x^{\mu} + \epsilon^{\mu} [/tex]

I'll try it.
 
  • #6
dextercioby said:
Can you calculate [itex] T^{\mu\nu} [/itex] and [itex] M^{\lambda}_{~~\mu\nu} [/itex] from the Lagrangian and the general Noether formula which for the energy-momentum 4 tensor reads:

[itex] T^{\mu}_{~~\nu} [/itex] = ([itex] \frac{\partial \mathcal{L}}{\partial(\partial_{\mu}A_{\rho})}[/itex] [itex] -\mathcal{L}\delta^{\mu}_{\lambda} [/itex]) X [itex] \frac{\partial x'^{\lambda}}{\partial\epsilon^{\nu}} [/itex],

where

[tex] x'^{\mu} = x^{\mu} + \epsilon^{\mu} [/tex]

[tex] T^{\mu\nu}=-F^{\mu\nu}\partial^{\nu}A_{\rho}+\frac{1}{4}F^{2}g^{\mu\nu}[/tex]

And now? How I relate this to the momentum and total angular momentum operators ?
 
  • #7
The momentum should be [itex] T^{0i} [/itex], just like energy is [itex] T^{00} [/itex]. For angular momentum, you should derive the general formula using the linearized version of a general Lorentz transformation (i.e. a linearized space-time rotation):

x'μ=xμμ ν xν, where

ϵμν = - ϵνμ

A minor change

Tμν=−FμρνAρ+1/4 F2gμν
 
  • #8
Petraa said:
Hello,

Where can I find a good explanation (book) of the derivation via Noether's theorem of the three momentum and angular momentum operators of the usual maxwell lagrangian ?
Thank you!
I too would be interested in seeing this for EM angular momentum.
Every place, I have looked seems to use the result in some form without actually deriving it.
 
  • #9
Petraa said:
Hello,

Where can I find a good explanation (book) of the derivation via Noether's theorem of the three momentum and angular momentum operators of the usual maxwell lagrangian ?

Thank you!

Maggiore "Modern introduction in QFT"
 

1. What is the Maxwell Lagrangian?

The Maxwell Lagrangian is a mathematical expression that describes the dynamics of electromagnetic fields. It is derived from the principles of classical mechanics and is used to understand the behavior of electromagnetic fields in various situations.

2. What is the significance of deriving the momentum and angular momentum operators of the Maxwell Lagrangian?

The momentum and angular momentum operators of the Maxwell Lagrangian allow for the calculation of the momentum and angular momentum of electromagnetic fields. This is important in understanding the behavior of fields in different situations, such as in the presence of charges or in the presence of moving objects.

3. How are the momentum and angular momentum operators of the Maxwell Lagrangian derived?

The momentum and angular momentum operators are derived by applying the principles of classical mechanics to the Maxwell Lagrangian. This involves using the equations of motion and the Hamiltonian formalism to derive the operators.

4. What is the physical interpretation of the momentum and angular momentum operators in the Maxwell Lagrangian?

The momentum operator represents the flow of momentum in electromagnetic fields, while the angular momentum operator represents the rotational motion of the fields. Both operators have physical significance in understanding the behavior and interactions of electromagnetic fields.

5. How are the momentum and angular momentum operators used in practical applications?

The momentum and angular momentum operators are used in various practical applications, such as in the study of electromagnetic waves, the behavior of particles in magnetic fields, and the interactions between charged particles. They are also used in the development of technologies such as MRI machines and particle accelerators.

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