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On Fewster's notes on QFT on curved spacetimes he says:

Our goal is to find operators $\Phi(f)$ such that

$$\Phi(Pf) = 0$$

for all $f\in C^\infty_0(M)$ and so that the time zero fields

$$\varphi({\bf{x}})=\Phi(0,{\bf{x}}),\quad \pi({\bf{x}})=\dot{\Phi}(0,{\bf{x}})$$

obey the CCRs

$$[\varphi(f),\pi(g)]=i\langle \bar{f},g\rangle \mathbf{1},\quad [\varphi(f),\varphi(g)]=[\pi(f),\pi(g)]=0.$$

Now I confess I don't understand this. On the first equation the author is clearly using quantum fields as operator value distributions, i.e., as mappings

$$\Phi:C^\infty_0(M)\to \mathcal{L}(\mathcal{H})$$

then he talks about time zero fields. He then computes $\Phi(0,\mathbf{x})$ and $\dot{\Phi}(0,\mathbf{x})$. But wait a moment, now $\Phi$ seems to be a function on spacetime, it was an operator value distribution one line above.

In this same line, $\varphi,\pi$ seem to appear as fields defined on spacetime, that is $\varphi,\pi : M\to \mathbb{C}$.

But then on the line below they become operator value distributions, and this is quite confusing.

So what is really going on here? What is all of this about? How the fixed time fields appear in this operator value distributions formalism? Why and how the author is going back and forth between usual fields on spacetime as function defined on $M$ and operator value distributions, i.e., mappings on $C^\infty_0(M)$?

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  • $\begingroup$ I don't remember the way Chris followed in his lecture notes, but I suspect the he first introduced the fields $\phi$ and $\pi$ with fixed time CCR and next, using them, he defined the fied $\Phi$ in terms of the former fields through a procedure will be explained. So what you are reading presumably is just a declaration on author's goals. $\endgroup$ Commented Nov 5, 2017 at 8:16

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There are two scenarios that are usually considered, the second one being more general:

First, operators are distributions with respect to position, but regular functions with respect to time. In general, this approach is good enough for most purposes, and it seems this is the situation the author has in mind.

Second, operators are distributions with respect to all spacetime coordinates. In this situation, $\Phi(0,\boldsymbol x)$ is a meaningless expression. In principle, you can mimic the $t\to0$ limit by mollifying your operator with respect to an approximate Dirac delta with support $\{0\}$. It is also possible that this is what the author has in mind.

See for example 1602.00662, §§7,9. See also Spratte's thesis, §2.3.

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