fa.functional analysis - Spectral theory for self-adjoint field operators on a symmetric Fock space

One convenient way to do analysis on the symmetric Fock space is to use its isomorphism to
the Bargmann (reproducuing Kernel Hilbert) space (sometimes called the Bargmann-Fock pace)
of analytic functions on $mathbb C^s$ (with respect to the Gaussian measure) defined in the classical paper:

Bargman V. On a Hilbert space of analytic functions and associated integral transform I,
Pure Appl. Math. 14(1961), 187-214.

An introduction to the Bargmann space may be found in chapter 4 of the
book by Uri Neretin

On the Bargmann space the creation and anihilation operators
are just the multiplication $a_j = z_j$ and the derivation $a^*_j = d/dZ_j$
and consequently, the theory of several complex variables can be used for the analysis on this space,
for example the trace of (a trace class) operator can be represented as an integral on its symbol.

Remark: The isomorphism between the symmetric Fock and Bargmann spaces is not proved in the Book. It can be found for example in the references of the following article:

Regarding the question about $a(f)+a^*(f)$, it is proportional to the position operator of quantum mechanics.
This is an unbounded operator, its spectrum is the whole real line, but it does not have
eigenfunvectors within the Fock space (Loosly speaking, they are Dirac delta functions), however one can find a series of vectors which approximate arbitrarily closely its eigenvectors. Using the corresponding projectors, one can approximate the spectral decomposition of this operator.
The case of the momentum operator $i(a(f)-a^*(f))$ is used more frequently, a possible choice of the approximate eigenvectors is by means of wave packets.