Quantization of Gauge theories ?

[mhill]

Quantization of Gauge theories ??

Hi , i am trying to learn the math formalism of Gauge Theories

as far as i know they begin with the 1-form

$$A= \sum_{i} T^{i}A_{\mu}^{i}$$

where 'T_i ' are the generators of the Lie Group

then we define the 2-form $$F= dA + (1/2)[A,A]$$

and the equation of motion are $$dF =0$$ (exterior derivative of F ) and $$*d *F = J$$

with J being an external source and $$*F_{ij}=e_{ijkl}F^{kl}$$ Hodge Star operator

QUESTION:
=========

How can you define a Hamiltonian of Your Gauge theory if the Lagrangian is equal to $$tr[F^{ab}F_{ab}]$$

how can you apply the Quantization to these theories ??

 

[Angryphysicist]

How can you define a Hamiltonian of Your Gauge theory if the Lagrangian is equal to $$tr[F^{ab}F_{ab}]$$

how can you apply the Quantization to these theories ??

Two ways: one is the sum over histories approach which requires knowledge only of the Lagrangian.

The other is to treat the potential $$A$$ as the "position" coordinate, so we can find the canonically conjugate momenta

$$\pi = \frac{\partial L}{\partial \dot{A}}$$

and then use the Legendre transform to get the Hamiltonian

$$H = \pi\partial_{0}A - L$$.

NOTE NOTE NOTE these are Lagrangian, Hamiltonian, and momentum DENSITIES and if you want to get the Lagrangian, Hamiltonian, or momentum you merely integrate the corresponding density over the spatial volume.

Usually you end up with constraints in the canonical approach and you either quantize then constrain (Dirac's approach) or constrain then quantize (reduced phase space approach). It's a whole problem...

A good book or two to refer to would be Quantization of Fields with Constraints by D. M. Gitman and I.V. Tyutin (the latter was the "T" in the BRST technique of gauge quantization), or Quantization of Gauge Systems by Henneaux and Teitelboim. The latter is a more difficult read (my notes consist of elucidating what they say, and providing more detailed proofs; it's very elegant but also very short and kinda choppy in the beginning in my opinion...).

 

[Jim Kata]

Look at chapter 15 in Weinberg or Dirac's little primer "Lectures in quantum mechanics".