I was reading on some QFT and I came across the following paragraph:

In the same way that a generic state $|\psi\rangle$ of a particle can be described by giving its overlap with all the possible states where a particle has a completely defined position, $\langle \vec{x}|\psi\rangle$, a generic state $|\Psi\rangle$ of a quantum field can be described by its overlap with all the possible states where the field $\phi$ has a completely defined value en each point of space. This object is called wave field functional.

This is the first time I've heard about this term called "wave field functional", and I didn't understand what is the physical meaning and if it has a general expression. My guess is that it could be written like $\langle\phi|\Psi\rangle$ but I'm not sure if it can be expanded. I also couldn't find any information about it, which makes me think it is probably best known with another name (this is a translation from a non-English document).

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    $\begingroup$ The more common name is the (Schrodinger) wavefunctional. $\endgroup$ – knzhou Sep 19 '18 at 8:27
  • $\begingroup$ Possibly related. $\endgroup$ – Cosmas Zachos Sep 21 '18 at 1:24
  • $\begingroup$ Related. $\endgroup$ – Cosmas Zachos Sep 21 '18 at 1:30

In quantum field theories (especially in interacting theories), the Hilbert space of wavefunctions is naturally identified with a subspace of the space of function(al)s acting on the space of distributions $\mathscr{S}'(\Omega)$, corresponding to a suitable space $\Omega$.

In particular, the vacuum identifies a probability measure $\mu$ on $\mathscr{S}'(\Omega)$, and the Hilbert space of the theory is the space $L^2(\mathscr{S}'(\Omega),\mathrm{d}\mu)$ (the vacuum vector here corresponds to the constant function $1$).

In this representation, the wavefunctions are square-integrable functions $\Psi(\phi)$, that map (almost all) fields $\phi$ (distributions of $\mathscr{S}'(\Omega)$) to complex numbers. This is in analogy with the common square-integrable wavefunctions of quantum mechanics $\psi(x)$, that map (almost all) points of space to complex numbers.

I am not aware of the name "wave field functional", but I have seen the terminology wavefunctional for $\Psi(\phi)$ (probably in Steven Winberg's books on QFT).

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  • $\begingroup$ Indeed. Just one small comment: $\Omega$ here is just space and not spacetime. $\endgroup$ – Abdelmalek Abdesselam Sep 20 '18 at 14:33
  • $\begingroup$ @AbdelmalekAbdesselam Thanks, you are right. I edited it accordingly. $\endgroup$ – yuggib Sep 20 '18 at 20:30

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