# What is the wave field functional?

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).

• The more common name is the (Schrodinger) wavefunctional. – knzhou Sep 19 '18 at 8:27
• Possibly related. – Cosmas Zachos Sep 21 '18 at 1:24
• – 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).
• Indeed. Just one small comment: $\Omega$ here is just space and not spacetime. – Abdelmalek Abdesselam Sep 20 '18 at 14:33