# Self-Adjointness criterion for operators in Fock spaces

### Math. Phys. Anal. Geom. 18, No.1 2015

The full published version is available (under subscription) at SpringerLink.

### arXiv 1405.6570

The preprint in pdf is available at arXiv.org.

## Abstract

In this paper we provide a criterion of essential self-adjointness for operators in the tensor product of a separable Hilbert space and a Fock space. The class of operators we consider may contain a self-adjoint part, a part that preserves the number of Fock space particles and a non-diagonal part that is at most quadratic with respect to the creation and annihilation operators. The hypotheses of the criterion are satisfied in several interesting applications.

## Introduction

[The introduction is copied here for rapid consultation]

Let $$\mathscr{H}_1$$, $$\mathscr{H}_2$$ be separable Hilbert spaces. We consider the following space:

$$\label{eq:2} \mathscr{H}=\mathscr{H}_1\otimes \Gamma_s(\mathscr{H}_2)\; ;$$

where $$\Gamma_s(\mathscr{K})$$ is the symmetric Fock space based on $$\mathscr{K}$$ [see Cook 1951, DerGer 2013, ReeSim 1975 for mathematical presentations of Fock spaces and second quantization]. The symmetric structure of the Fock space does not play a role in the argument: in principle it is possible to formulate the same criterion for anti-symmetric Fock spaces $$\mathscr{H}_1\otimes\Gamma_a(\mathscr{H}_2)$$. We focus on symmetric spaces, the corresponding antisymmetric results should be deduced without effort.

We are interested in proving a criterion of essential self-adjointness for densely defined operators of the form:

$$\label{eq:1} H= H_{01}\otimes 1 + 1\otimes H_{02}+H_I\; ;$$

with suitable assumptions on $$H_{01}$$, $$H_{02}$$ and $$H_I$$. Operators based on these spaces and with such structure are crucial in physics, to describe the quantum dynamics of interacting particles and fields.

Self-adjointness of operators in Fock spaces has been widely studied, in particular in the context of Constructive Quantum Field Theory [e.g. GinVel 1970, Glimm 1967, GliJaf 1985, Rosen 1970, Segal 1970] and Quantum ElectroDynamics [e.g. Ammari 2000, BacFroSig 1998-1999, HasHer 2008, Hiroshima 2002, Nelson 1964, Spohn 2004]. A variety of advanced tools has been utilized, for even "simple" systems present technical difficulties to overcome: many questions still remain unsolved.

In some favourable situations, however, it is possible to take advantage of the peculiar structure of the Fock space and prove essential self-adjointness with almost no effort. The idea first appeared in a paper by Ginibre and Velo [1970]; and the author utilized it in [AmmFal 2014, Falconi 2012] for the Nelson model with cut off: essential self-adjointness can be proved with less assumptions than using the Kato-Rellich Theorem (and that becomes particularly significative in dimension two), see Section 4.2. Another remarkable application is the Pauli-Fierz Hamiltonian describing particles coupled with a radiation field. For general coupling constants, essential self-adjointness has been first proved in a probabilistic setting, using stochastic integration [Hiroshima 2000, Hiroshima 2002]. In this paper we prove the same result directly in Section 4.3, applying the criterion formulated in Assumptions $$A_0$$, $$A_{I}$$ and Theorem 3.1.

In the literature, self-adjointness of operators in Fock spaces has been studied using various tools of functional analysis: the Kato-Rellich and functional integration arguments mentioned above are two examples, as well as the Nelson commutator theorem. For each particular system, a strategy is utilized ad hoc: the more complicated is the correlation between $$\mathscr{H}_1$$ and $$\Gamma_s(\mathscr{H}_2)$$, the more difficult is the strategy. We realized that, if we take suitable advantage of the fibered structure of the Fock space, the type of interaction between the spaces is not so relevant. This was a strong motivation to study the problem from a general perspective. Due to the variety of possible applications, an effort has been made to formulate the necessary assumptions in a general form. Roughly speaking, the essential requirement is that the part of $$H_I$$ that does not commute with the number operator of $$\Gamma_s(\mathscr{H}_2)$$ is at most quadratic with respect to the creation and annihilation operators. As anticipated, the space $$\mathscr{H}_1$$ does not play a particular role, as long as $$H_I$$ behaves sufficiently well with respect to $$H_{01}$$.