• 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.


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


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,064 questions , 2,215 unanswered
5,347 answers , 22,734 comments
1,470 users with positive rep
818 active unimported users
More ...

  functional representations of free quantum fields

+ 3 like - 0 dislike

The free real quantum field, satisfying $[\hat\phi(x),\hat\phi(y)]=\mathrm{i}\!\Delta(x-y)$, $\hat\phi(x)^\dagger=\hat\phi(x)$, with the conventional vacuum state, which has a moment generating function $\omega(\mathrm{e}^{\mathrm{i}\hat\phi(f)})=\mathrm{e}^{-(f^*,f)/2}$ , where $(f,g)$ is the inner product $(f,g)=\int f^*(x)\mathsf{C}(x-y)g(y)\mathrm{d}^4x\mathrm{d}^4y$, $\mathsf{C}(x-y)-\mathsf{C}(y-x)=\mathrm{i}\!\Delta(x-y)$, has a representation as $$\hat\phi_r(x)^\dagger=\hat\phi_r(x)=\frac{\delta}{\delta\alpha(x)}+\pi\alpha^*(x) +\int \mathsf{C}(z-x)\alpha(z)\mathrm{d}^4z,$$ in terms of multiplication by $\alpha(x)$ and functional differentiation $\frac{\delta}{\delta\alpha(x)}$. Because $$\left[\frac{\delta}{\delta\alpha(x)}+\pi\alpha^*(x), \int \mathsf{C}(z-y)\alpha(z)\mathrm{d}^4z\right]=\mathsf{C}(x-y),$$ it is straightforward to show that $\hat\phi_r(x)$ verifies the commutation relation of the free real quantum field. For the Gaussian functional integral $$\omega(\hat A_r)=\int\hat A_r\mathrm{e}^{-\pi\int\alpha^*(x)\alpha(x)\mathrm{d}^4x} \prod\limits_z\mathrm{d}^4z\mathrm{d}\alpha^*(z)\mathrm{d}\alpha(z),$$ we find, as required, $$\begin{eqnarray} \omega(\mathrm{e}^{\mathrm{i}\hat\phi_r(f)})&=&\int\mathrm{e}^{\mathrm{i}\hat\phi_r(f)} \mathrm{e}^{-\pi\int\alpha^*(x)\alpha(x)\mathrm{d}^4x} \prod\limits_z\mathrm{d}^4z\mathrm{d}\alpha^*(z)\mathrm{d}\alpha(z)\cr &=&\int\exp\left[\mathrm{i}\!\!\int\!\! \mathsf{C}(z-x)\alpha(z)\mathrm{d}^4z\right] \exp\left[\mathrm{i}\left(\!\!\frac{\delta}{\delta\alpha(x)}+\pi\alpha^*(x) \right)\right]\cr &&\qquad\qquad\times\mathrm{e}^{-(f^*,f)/2} \mathrm{e}^{-\pi\int\alpha^*(x)\alpha(x)\mathrm{d}^4x} \prod\limits_z\mathrm{d}^4z\mathrm{d}\alpha^*(z)\mathrm{d}\alpha(z) =\mathrm{e}^{-(f^*,f)/2}. \end{eqnarray}$$ The last equality is a result of $\frac{\delta}{\delta\alpha(x)}+\pi\alpha^*(x)$ annihilating $\mathrm{e}^{-\pi\int\alpha^*(x)\alpha(x)\mathrm{d}^4x}$, and the Gaussian integral annihilates powers of $\int\!\! \mathsf{C}(z-x)\alpha(z)\mathrm{d}^4z$.

This is a largely elementary transposition of the usual representation of the SHO in terms of differential operators, with a not very sophisticated organization of the relationships between points in space-time and operators, so I imagine something quite like this might be found in the literature. References please, if anyone knows of any?

This post has been migrated from (A51.SE)
asked Jan 18, 2012 in Theoretical Physics by Peter Morgan (1,230 points) [ no revision ]
retagged Mar 18, 2014 by dimension10
Isn't that a version of the usual functional representation of the Hilbert space as the space of functionals of fixed-time configurations? The only difference seems to be that you are not requiring that $\phi$ satisfies the equations of motion. Is Schweber, chapter 7e similar to what you are saying?

This post has been migrated from (A51.SE)
Thanks @Pavel, perhaps there doesn't need to be any mention of equations of motion because this construction is in 4-space. I don't know the Schweber, but should! Library. Thanks.

This post has been migrated from (A51.SE)
Was this an answer? If so, it should be posted as an answer.

This post has been migrated from (A51.SE)
I suspect that Pavel felt uncertain what the Question was asking, but I think, András, now I've looked at the section in Schweber that Pavel cited, that it's a Useful Answer, at least to me, so thanks again. The Schweber is of course hugely different in its details.

This post has been migrated from (A51.SE)

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:
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