# functional representations of free quantum fields

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?

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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? – Pavel Safronov Jan 18 '12 at 23:30
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. – Peter Morgan Jan 19 '12 at 1:24