Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  From quantization under external classical gauge field to a fully quantized theory

+ 7 like - 0 dislike
1410 views

Let me take QED for example to clarify my question: The textbook-approach(at least for Peskin&Schroeder) to quantize ED is to first quantize EM field and Dirac field as free fields respectively, and then couple them together perturbatively to represent the interaction, and we will have a fully quantized theory. Here by "fully quantized" I mean both EM field and Dirac field are quantized.

On the other hand, we may quantize the Dirac field under an given external classical EM field: briefly speaking, one solves minimally coupled Dirac equation and take the solution space as 1-particle Hilbert space, and then second quantize the theory by building a Fock space based on this 1-particle space.(A more detailed description can be found in chap 10 of "the Dirac equation" by Thaller.B) Such a theory is defined non-perturbatively, and describes many-electron systems interacting with an external classical EM field but non-interacting among themselves, so in this sense it's a "better" theory than a quantized free Dirac field. However I cannot see a way to go from this theory to a fully quantized one, if there is, do we get exactly the same theory as given by the text-book approach, or possibly a better one? Any answer, comment or reference will be appreciated.

This post imported from StackExchange Physics at 2014-04-07 08:27 (UCT), posted by SE-user Jia Yiyang
asked Jul 7, 2012 in Theoretical Physics by Jia Yiyang (2,640 points) [ no revision ]
I noticed a possibly related comment by Xiao-Gang Wen from this post, at the bottom of the page:

This post imported from StackExchange Physics at 2014-04-07 08:27 (UCT), posted by SE-user Jia Yiyang
"There are three kinds of gauge theories: (1) Classical gauge theory where both gauge field and charged matter are treated classically. (2) fake quantum gauge theory where gauge field is treated classically and charged matter is treated quantum mechanically. (3) real quantum gauge theory where both gauge field and charged matter are treated quantum mechanically. Most papers and books deal with the fake quantum gauge theory..."

This post imported from StackExchange Physics at 2014-04-07 08:27 (UCT), posted by SE-user Jia Yiyang
Does the background field method go towards answering your question? en.wikipedia.org/wiki/Background_field_method

This post imported from StackExchange Physics at 2014-04-07 08:27 (UCT), posted by SE-user Michael Brown
I just want to say that do quantization like that, you have to be able to solve the Dirac equation for arbitrary gauge field configuration, and slightly even more than that. I guess the reason people usually go the other way is the computational feasibility. However, in some cases, it is possible to follow you approach. For example, one of the infinite number of solutions of the Schwinger model goes precisely that way. If it is of any interest, I can write up an answer in some time.

This post imported from StackExchange Physics at 2014-04-07 08:27 (UCT), posted by SE-user Peter Kravchuk
@PeterKravchuk: Yes please, I'm quite interested.

This post imported from StackExchange Physics at 2014-04-07 08:27 (UCT), posted by SE-user Jia Yiyang

Bump up for new discussions.

1 Answer

+ 1 like - 0 dislike

This may not be a satisfactory answer to your question but hopefully it will be of some help : In order to quantize a classical field theory general procedure (apart from path integral method ) is to first solve the classical equations of motion, and define a Fock space$^1$ out of these (Just as you mentioned). Now for a free theory, i.e. a classical theory for which solutions can be explicitly found this procedure certainly gives a nonperturbative definition of QFT; but if your classical equations are nonlinear then (in general) it in not possible to solve them exactly and you have to resort to perturbative methods. The same thing happens in case of Dirac equation when you couple it with electromagnetic field. In principle you can still apply the same procedure as you used for the case of free Dirac field; all you need to do is to find space of exact solutions and use some quantization method like geometric quantization. But that is at least very difficult, if not impossible. So one makes use of perturbative methods$^2$.


$1$ : When the classical configuration space is $Q$ (so that phase space is $T^*Q$) then space of states is space of functions on $Q$. In the case when $Q$ is a linear space (or can be made linear through choice of some linear structure) then space of (polynomial) functions on $Q$ can be written as a Fock space.

$2$. See e.g. Peskin-Schroeder

This post imported from StackExchange Physics at 2014-04-07 08:27 (UCT), posted by SE-user user10001
answered Jul 7, 2012 by user10001 (635 points) [ no revision ]
I agree with what you said, but it doesn't answer my question.

This post imported from StackExchange Physics at 2014-04-07 08:27 (UCT), posted by SE-user Jia Yiyang

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysi$\varnothing$sOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...