# Fock space vs. wavefunctionals

There are at least two representations of the Hilbert spaces of quantum field theory. For a scalar field, we have

The Fock space representation, such that every state is represented as the Fock space of 'one particle' representations,

$$\mathcal H = \bigoplus_{n = 0}^\infty \text{Sym}(\mathcal H_1^{\otimes n})$$

with $\mathcal H_1^0 = \mathbb C$ and $\mathcal H_1 = \mathcal L^2(\Bbb R^3)$.

The wavefunctional representation, such that every state is represented by a functional on the space of configurations of the field on $\Bbb R^3$ which is square integrable with respect to some measure

$$\mathcal H = \mathcal L^2(\mathcal F[C(\Bbb R^3)], d\mu[\phi])$$

Since those two representations describe the same theory, I am guessing that there is some isomorphism between the two, but what would it be?

• IIRC, there are subtelties to define the "position representation" in the QFT case, because it's not so easy to come up with a measure on an infinite-dimensional configuration space. In particular, there are no Lebesgue measure (aka translation-invariant measure) on such a space. Also, bcs it's QFT, I expect the $L_2$ spaces associated to different measures to not be unitarily equivalent (in contrast to the finite-dimensional case). May 30, 2017 at 12:56
• That being said, using a suitable Gaussian measure (that would correspond to the Fock vacuum), maybe we can get the isomorphism you are looking for. I will look into it if nobody else does... (but I'm on my phone at the moment) May 30, 2017 at 13:00
• The vacuum is the function 1, and the rest of the correspondence is built on the field and momentum operators, as illustrated in the Reed Simon book, Vol. II, p.228-230. May 30, 2017 at 13:52
• And yes @Luzanne , for free fields the corresponding measure is the Gaussian on $\mathbb{R}^{\infty}$ (the topological space given by the countable product of $\mathbb{R}$ with the product topology). For interacting fields, the representation is usually non-Fock, and we usually look for a representation of the type $L^2(\mathscr{S}'(\mathbb{R}^d),\mu)$, for a suitable probability measure $\mu$ on the distributions. This representation is successfully built only in very few situations, see the Glimm-Jaffe book for more details. May 30, 2017 at 14:03
• @yuggib I see. Yes, the construction you just outlined is what I suspected: with a measure that matches our vacuum, clearly the latter will be represented as the constant $1$, and we can reconstruct all other states by aplying the right operators, à la GNS construction. I presume the field conjugate momentum has to be represented as derivation + an extra divergence term to compensate for the measure being non-translation-invariant? May 30, 2017 at 14:52