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functional representations of free quantum fields

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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 (1,075 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)

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