# Quantum Field Theory. A Nonstandard Approach Based on Nonstandard Pointwise-Defined Quantum Fields. Part II. The field operators and the approximate vacuum

jaykov foukzon

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jaykov foukzon. Quantum Field Theory. A Nonstandard Approach Based on Nonstandard Pointwise-Defined Quantum Fields. Part II. The field operators and the approximate vacuum. 2024. hal-04681742v2

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# Model $P(\varphi)_4$ Quantum Field Theory. A Nonstandard Approach Based on Nonstandard Pointwise-Defined Quantum Fields. Part II. The field operators and the approximate vacuum

J. Foukzon

Center for Mathematical Sciences, Technion Israel Institute of Technology City, Haifa 3200003 Israel

jaykovfoukzon@list.ru

**Abstract.** A new non-Archimedean approach to interacted quantum fields is presented. In proposed approach, a field operator $\varphi(x, t)$ no longer a standard tempered operator-valued distribution, but a non-classical operator-valued function. We prove using this novel approach that the quantum field theory with Hamiltonian $P(\varphi)_4$ exists and that the corresponding $C^*$-algebra of bounded observables satisfies all the Haag-Kastler axioms except Lorentz covariance. We prove that the $\lambda(\varphi^4)_4$ quantum field theory model is Lorentz covariant.

## 1.Introduction

Remind that extending the classical real numbers $\mathbb{R}$ to include infinite and infinitesimal quantities originally enabled D. Laugwitz [1] to view the delta distribution $\delta(x)$ as a nonstandard point function. Independently A. Robinson [2] demonstrated that distributions could be viewed as generalized polynomials. Luxemburg [3] and Sloan [4] presented an alternate representative of distributions as internal functions within the context of canonical Robinson's theory of nonstandard analysis. For further information on classical nonstandard real analysis, we refer to [5]-[7]. The technique of nonstandard analysis (NSA) in constructive quantum field theory (QFT) originally were considered in P. J. Kelemen and A. Robinson papers [8],[9]. The methods of nonstandard analysis are demonstrated for the construction of the nonstandard $\lambda: \varphi_2^4$: model. J. Glimm and A. Jaffe's results [10],[11] were analysed from the nonstandard point of view. For further information on methods of classical nonstandard analysis in QFT, we refer to [12],[13]. However methods of classical nonstandard analysis cannot resolve this problem in physical dimension $d = 4$, in particular for the case of simplest scalar QFT model with interaction : $\varphi_2^4$, see concise explanation in ref. [15 Introduction, Remark 1.4] and ref.[17 sect.1, Remark 1.4]. Cardinally novel approach has been developed in author papers [14]-[19]. This approach based on non-conservative extension of the model theoretical NSA. In this paper we consider a some-what different hyperfinite cut-off theory, namely the $\lambda\varphi_4^4$ theory in a periodic box. This gives a cut-off interaction which is translation invariant, and therefore it is useful for the study of the vacuum state. In a hyperfinite interval we prove that the total Hamiltonian is self #-adjoint and has a complete set of normalizable eigenstates.

**Abbreviation 1.1** In this paper we adopt the following canonical notations. For a standard set $E$ we often write $E_{st}$. For a set $E_{st}$ let $\sigma E_{st}$ be a set $\sigma E_{st} = \{x|x \in E_{st}\}$. We identify $z$ with $\sigma z$ i.e., $z \equiv \sigma z$ for all $z \in \mathbb{C}$. Hence, $\sigma E_{st} = E_{st}$ if $E \subseteq \mathbb{C}$, e.g., $\sigma \mathbb{C} = \mathbb{C}$, $\sigma \mathbb{R} = \mathbb{R}$, $\sigma P = P$, $\sigma L_+^\uparrow = L_+^\uparrow$, etc.

Let $*\mathbb{R}_{\approx}$, $*\mathbb{R}_{\approx+}$, $*\mathbb{R}_{\text{fin}}$, $*\mathbb{R}_{\infty}$, and $*\mathbb{N}_{\infty}$ denote the sets of infinitesimal hyper-real numbers, positive infinitesimal hyper-real numbers, finite hyper-real numbers, infinite hyper-real numbers and infinite hyper natural numbers, respectively. Note that $*\mathbb{R}_{\text{fin}} = *\mathbb{R} \setminus *\mathbb{R}_{\infty}$, $*\mathbb{C} = *\mathbb{R} + i*\mathbb{R}$, $*\mathbb{C}_{\text{fin}} = *\mathbb{R}_{\text{fin}} + i*\mathbb{R}_{\text{fin}}$. Note that there is a natural imbedding $\mathbb{R} \hookrightarrow *\mathbb{R}$, see ref. [5].

**Notation 1.1** We denote #- completion of the non- Archimedean field $*\mathbb{R}$ by $*\mathbb{R}_c^\#$, see ref. [18],[19].

**Abbreviation 1.2** Let $*\mathbb{R}_{c\approx}^\#$, $*\mathbb{R}_{c+}^\#$, $*\mathbb{R}_{c\infty}^\#$, $*\mathbb{R}_{\infty}^\#$ denote the sets of infinitesimal hyper-real numbers, positive infinitesimal hyper-real numbers, finite hyper-real numbers, infinite hyper-real numbers in a non- Archimedean field $*\mathbb{R}_c^\#$. Note that there is a natural imbedding $*\mathbb{R} \hookrightarrow *\mathbb{R}_c^\#$, see ref. [18].

**Notation 1.2** We denote by $*\tilde{\mathbb{R}}_c^\#$ special extension of a non-Archimedean field $*\mathbb{R}_c^\#$, see ref. [19] and section 22 in this paper.

**Abbreviation 1.3** Let $*\mathbb{R}_{c\approx}^\#$, $*\mathbb{R}_{c+}^\#$, $*\mathbb{R}_{c\infty}^\#$, $*\mathbb{R}_{\infty}^\#$ denote the sets of infinitesimal hyperreal numbers, positive
infinitesimal hyper-real numbers, finite hyper-real numbers, infinite hyper-real numbers in non- Archimedean field *R_c^# respectively. Note that *R_c^#fin = *R_c^# \ *R_c^# , *C_c^# = *R_c^# + i *R_c^# , *C_c^#fin = *R_c^#fin + i *R_c^#fin.

**Abbreviation 1.3** Let $^{*\mathbb{R}}_{c\approx}$, $^{*\mathbb{R}}_{c+}$, $^{*\mathbb{R}}_{cfin}$, $^{*\mathbb{R}}_{c\infty}$ denote the sets of infinitesimal hyperreal numbers, positive infinitesimal hyper-real numbers, finite hyper-real numbers, infinite hyper-real numbers in non- Archimedean field respectively. Note that $^{*\mathbb{R}}_{cfin} = *\mathbb{R}^\# \setminus *\mathbb{R}^\#_{c\infty}$, $^{*\mathbb{C}}_{c} = *\mathbb{R}^\# + i *\mathbb{R}^\#_{c}$, $^{*\mathbb{C}}_{cfin} = *\mathbb{R}^\#_{cfin} + i *\mathbb{R}^\#_{cfin}$, see ref. [19]. Note that there is a natural imbedding $^{*\mathbb{R}}_{c} \hookrightarrow *\mathbb{R}^\#_{c}$, see ref. [19].

**Definition 1.1** The Schwartz space of essentially rapidly decreasing *C_c^#- valued test functions on *R_c^#n, n ∈ *N is the function space defined by

$$
\tilde{S}_{\text{fin}}^{\#} (*\tilde{\mathbb{R}}_{c}^{\#n}) = \tilde{S}_{\text{fin}}^{\#} (*\tilde{\mathbb{R}}_{c}^{\#n}, *\tilde{\mathbb{C}}_{c}^{\#}) = \\
\{ f \in C^{*\infty} (*\tilde{\mathbb{R}}_{c}^{\#n}, *\tilde{\mathbb{C}}_{c}^{\#}) | \forall (\alpha, \beta)(\alpha \in *\mathbb{N}^{n}, \beta \in *\mathbb{N}^{n}) \exists c_{\alpha\beta} (c_{\alpha\beta} \in *\tilde{\mathbb{R}}_{c,fin}^{\#}) \forall x (x \in *\tilde{\mathbb{R}}_{c}^{\#n}) \|x^{\alpha} (D^{\#\beta} f(x))\| < \\
c_{\alpha\beta} \}.
$$

**Definition 1.2** Let B be a non-Archimedean Banach space endowed with *R_c^# - valued #-norm ||°||_# . Let A be a linear operator A: B → B . We say that operator A is bounded in *R_c^# if there is positive constant δ ∈ *R_c^#n such that for any x ∈ B the inequality ||Ax||_# ≤ δ||x||_# holds.

**Definition 1.3** The Fock space F^# is the non-Archimedean Hilbert space #- completion of the symmetric tensor algebra over L_2^#(*R_c^#3)

$$
\mathcal{F}^{\#} = \mathfrak{C} \left( L_2^{\#} ( *\tilde{\mathbb{R}}_c^{\#3} ) \right) = \mathrm{Ext} - \bigoplus_{n=0}^{\infty} \mathcal{F}_n^{\#}, \qquad (1.1)
$$

where $\mathcal{F}_n^\#$ is the space of $n$ non-interacting particles,

$$
\mathcal{F}_n^\# = L_2^\# (*\tilde{\mathbb{R}}_c^{\#3}) \otimes_s L_2^\# (*\tilde{\mathbb{R}}_c^{\#3}) \otimes_s \cdots \otimes_s L_2^\# (*\tilde{\mathbb{R}}_c^{\#3}). \quad (1.2)
$$

The variable **k** = (k₁, k₂, k₃) ∈ *R_c^#3 denotes momentum vector. For ψ = {ψ₀, ψ₁, ...} ∈ F^# = F₀^# ⊕ F₁^# ⊕ ...

We define on Fock space F^# the *R_c^#3- valued #-norm ||·||_# by ||ψ||_#^2 = Ext- Σ_{n=0}^{∞} ||ψ_2||_#^2, where ||·||_# is a
#-norm in L_2^#(*R_c^#3) The no particle space F_0^# = *C^# is the complex numbers, and

$$
\Omega_0 = \{1,0,0, \dots\} \in \mathcal{F}^{\#} \qquad (1.3)
$$

is the (bare) vacuum or (bare) no-particle state vector. We define operators $\mathcal{N}$ and $H_{0,\kappa}$ by

$$
(N\psi)_n = n(\text{Ext} - \prod_{i=1}^{n} \theta(\|\mathbf{k}_i\|, \kappa) \psi_n), \qquad (1.4)
$$

$$
(H_{0,\kappa}\psi)_n(\mathbf{k}_1, \dots, \mathbf{k}_n) = \text{Ext} - \sum_{j=1}^{n} \theta(\|\mathbf{k}_j\|, \kappa) \mu(\mathbf{k}_j) \psi_n(\mathbf{k}_1, \dots, \mathbf{k}_n), \quad (1.5)
$$

where $\kappa \in *\mathbb{R}_{c+}^\# \setminus *\mathbb{R}_{c+}^\#$ and

$$
\theta(\|\mathbf{k}_j\|, \kappa) = 1 \text{ if } \|\mathbf{k}_j\| \le \kappa \text{ and } (\|\mathbf{k}_j\|, \kappa) = 0 \text{ if } \|\mathbf{k}_j\| > \kappa, \mu(\mathbf{k}_j) = \sqrt{(\langle \mathbf{k}_j, \mathbf{k}_j \rangle + m_0^2} \quad (1.6)
$$

Here N is the number of particles operator, and H₀,κ is the free energy operator (the free Hamiltonian). The rest mass of the non-interacting particles is m₀, and µ(k) is the energy of a free particle with momentum vector k . We use the standard annihilation and creation operators a(k) and a*(k),

$$
(a(\mathbf{k})\psi)_{n-1}(\mathbf{k}_1, \dots, \mathbf{k}_{n-1}) = \sqrt{n}\psi_n(\mathbf{k}, \mathbf{k}_1, \dots, \mathbf{k}_{n-1}).
$$
As a convenient minimal domain for $a(\mathbf{k})$, we use the set $\mathcal{E}^{\#}$ of vectors $\psi \in \mathcal{F}^{\#}$ with $\psi_n = 0$ for large $n \in \mathbb{N}$ and $\psi_n \in \tilde{S}_{\text{fin}}^{\#}(\mathbb{R}_c^{\#3})$ for all $n \in \mathbb{N}$.

$$ (a^*(\mathbf{k})\psi)_{n+1}(\mathbf{k}_1, \dots, \mathbf{k}_n, \mathbf{k}_{n+1}) = -^{1/2}\sqrt{n+1} \text{Ext} - \sum_{j=1}^{n+1} \delta^\#(\mathbf{k} - \mathbf{k}_j) \psi_n(\mathbf{k}_1, \dots, \hat{\mathbf{k}}_j, \dots \mathbf{k}_n). \quad (1.7) $$

Here the variable $\mathbf{k}_j$ is omitted. While $a^*((k)$ is not an operator, it is a densely defined bilinear form on $\mathcal{E}^{\#} \times \mathcal{E}^{\#}$. **Remark 1.1** Note for a $\mathbb{C}_c^{\#}$-valued function or $\mathbb{C}_c^{\#}$-valued distribution $b$ we can define $\mathbb{C}_c^{\#}$-valued bilinear form $B =$

$$ \text{Ext} - \int_{\mathbb{R}_c^{\#3}} \cdot \cdot \mathbb{R}_c^{\#3} b(\mathbf{k}_1, \dots, \mathbf{k}_m; \mathbf{k}'_1, \dots, \mathbf{k}'_n) a^*(\mathbf{k}_1) \cdots a^*(\mathbf{k}_m) a(-\mathbf{k}'_1) \cdots a(-\mathbf{k}'_n) d^{\#3} \mathbf{k}'_1 \cdots d^{\#3} \mathbf{k}'_n. \quad (1.8) $$

The integration helps in (1.8) and $B$ is not only a bilinear form, but often an operator. This is the case if, for example, $b$ is the kernel of a bounded operator $B_0$ from $\mathcal{F}_n^{\#}$ to $\mathcal{F}_m^{\#}$. In this case

$$ \|(N_\kappa + I)^{-\alpha/2} B (N_\kappa + I)^{-\beta/2}\|_{\#} \le \text{const} \cdot \|B_0\|_{\#}, \quad (1.9) $$

provided that $m+n \le \alpha + \beta$. The constant depends only on $\alpha, \beta, m$ and $n$. Intuitively we think of $B$ as being dominated by $N_\kappa^{(m+n)/2}$; in particular $B$ is an operator on $D(N_\kappa^{(m+n)/2})$ the domain of $N_\kappa^{(m+n)/2}$. The inequality (1.9) is one of our basic estimates and in using it we will often dominate $\|B_0\|_{\#}$ by the Hilbert Schmidt $\#$-norm $\|B_0\|_{\#HS} \le \|b\|_{\#2}, \|B_0\|_{\#HS} = \sqrt{\text{Ext} - \sum_{i \in \infty} \|Ae_i\|_{\#}}$, and where $\{e_i | i \in \infty\}$ is an orthonormal basis in $\mathcal{F}^{\#}$. By definition the field with hyperfinite momentum cut-off $\varphi_\kappa^{\#}(\mathbf{x}), \mathbf{x} = (x_1, x_2, x_3) \in \mathbb{R}_c^{\#3}, \kappa \in \mathbb{R}_c^{\#} \setminus \mathbb{R}_{\text{fin}}^{\#}$ is

$$ \begin{aligned} \varphi_\kappa^{\#}(\mathbf{x}) &= \text{Ext} - \int_{|\mathbf{k}| \le \kappa} (\text{Ext}\text{-exp}(-i\langle\mathbf{k}, \mathbf{x}\rangle)) \{a^*(\mathbf{k}) + a(\mathbf{k})\} [\mu(\mathbf{k})]^{-1/2} d^{\#3}k = \\ &= \text{Ext} - \int_{\mathbb{R}_c^{\#3}} \theta(\|\mathbf{k}\|, \kappa) (\text{Ext}\text{-exp}(-i\langle\mathbf{k}, \mathbf{x}\rangle)) \{a^*(\mathbf{k}) + a(\mathbf{k})\} [\mu(\mathbf{k})]^{-1/2} d^{\#3}k. \end{aligned} \quad (1.10) $$

We also define the bilinear form

$$ \begin{aligned} \pi_\kappa^\#(\mathbf{x}) &= \text{Ext} - \int_{|\mathbf{k}| \le \kappa} i(\text{Ext}\text{-exp}(-i\langle\mathbf{k}, \mathbf{x}\rangle)) \{a^*(\mathbf{k}) + a(\mathbf{k})\} [\mu(\mathbf{k})]^{\frac{1}{2}} d^{\#3}k = \\ &= \text{Ext} - \int_{\mathbb{R}_c^{\#3}} i\theta(\|\mathbf{k}\|, \kappa) (\text{Ext}\text{-exp}(-i\langle\mathbf{k}, \mathbf{x}\rangle)) \{a^*(\mathbf{k}) + a(\mathbf{k})\} [\mu(\mathbf{k})]^{\frac{1}{2}} d^{\#3}k, \end{aligned} \quad (1.11) $$

the conjugate momentum to $\varphi_\kappa^\#(\mathbf{x})$. Since the kernels $b(\mathbf{k}) = \theta(\|\mathbf{k}\|, \kappa)(\text{Ext}\text{-exp}(-i\langle\mathbf{k}, \mathbf{x}\rangle)) [\mu(\mathbf{k})]^{-\frac{1}{2}}$ in $L_2^\#$ the bilinear forms (1.10)-(111) define operator-valued functions $\varphi_\kappa^\#(\mathbf{x}): \mathbb{R}_c^{\#3} \to L(\mathcal{F}^\#)$ and $\pi_\kappa^\#(\mathbf{x}): \mathbb{R}_c^{\#3} \to L(\mathcal{F}^\#)$. For real $f(\mathbf{x}), g(\mathbf{x})$ such that $\theta(\|\mathbf{k}\|, \kappa)[\mu(\mathbf{k})]^{-\frac{1}{2}}\hat{f}(\mathbf{x}) \in L_2^\#$ and $\theta(\|\mathbf{k}\|, \kappa)[\mu(\mathbf{k})]^{\frac{1}{2}}\hat{g}(\mathbf{x}) \in L_2^\#$, the bilinear forms $\varphi_\kappa^\#(f)$ and $\pi_\kappa^\#(g)$ define operators whose $\#$-closures on $D(N_\kappa^{\frac{1}{2}})$ are self-$\#$-adjoint. They satisfy the canonical commutation relations

$$ \text{Ext}\text{-exp}(i\pi_\kappa^\#(g)) \text{Ext}\text{-exp}(i\varphi_\kappa^\#(g)) = \text{Ext}\text{-exp}(i\langle f, g \rangle_\#) \{ \text{Ext}\text{-exp}(i\varphi_\kappa^\#(g)) \text{Ext}\text{-exp}(i\pi_\kappa^\#(g)) \}. \quad (1.12) $$

It is furthermore possible to define polynomial functions of the field $\varphi_\kappa^\#(\mathbf{x})$, the Wick polynomials: $\varphi_\kappa^\#(\mathbf{x})$: (see chapter I for a definition of the Wick dots : :). Explicitly, as a bilinear form on $D(N_\kappa^{n/2}) \times D(N_\kappa^{n/2})$,

$$ : \varphi_\kappa^{\#n}(\mathbf{x}) := \sum_{j=0}^{n} \binom{n}{j} b_\kappa(\mathbf{k}_1, \dots, \mathbf{k}_n) a^*(\mathbf{k}_1) \cdots a^*(\mathbf{k}_j) a(-\mathbf{k}_{j+1}) \cdots a(-\mathbf{k}_n), \quad (1.13) $$
where

$$b_{\kappa}(\mathbf{k}_1, \dots, \mathbf{k}_n) = \prod_{j=1}^{n} \theta(\|\mathbf{k}_j\|, \kappa)[\mu(\mathbf{k}_j)]^{-1/2} \mathrm{Ext}\text{-exp}(-i\langle\sum_{j=1}^{n} \mathbf{k}_j, \mathbf{x}\rangle).$$

Thus for real $f(x) \in S^{\#}(\mathbb{R}_c^{\#3})$, the bilinear form

$$: \varphi_{\kappa}^{\#n}(f) := \mathrm{Ext} - \int_{\mathbb{R}_c^{\#3}} : \varphi_{\kappa}^{\#n}(\mathbf{x}) f(\mathbf{x}): d^{\#3}\mathbf{x}$$

has a kernel proportional to $\prod_{j=1}^{n} \theta(\|\mathbf{k}_j\|, \kappa)[\mu(\mathbf{k}_j)]^{-1/2} \hat{f}(\sum_{j=1}^{n} \mathbf{k}_j)$. Thus from (1.9) we conclude that : $\varphi_{\kappa}^{\#n}(f)$: defines a symmetric operator on the domain $D(N_{\kappa}^{n/2})$. It was shown in chapter I sect. 15 that : $\varphi_{\kappa}^{\#n}(f)$: is essentially self-#-adjoint on this domain.

## 2. The periodic hyperfinite approximation in configuration space. The cut-off Hamiltonian $H_{\kappa}(g)$.

The cut-off Hamiltonian $H_{\kappa}(g)$ acts on $\mathcal{F}_{V}^{\#}$ and can be written in terms of the field operator $\varphi_{\kappa}^{\#}(\mathbf{x})$, $\mathbf{x} = (x_1, x_2, x_3)$ as

$$H_{\kappa}(g) = H_{0,\kappa} + \mathrm{Ext} - \int_{\mathbb{R}_c^{\#3}} \varphi_{\kappa}^{\#4}(x) d^{\#3}x = H_{0,\kappa} + H_{I,\kappa,g} \quad (2.1)$$

where $H_{0,\kappa} = H_{\kappa}(0)$ is the free hamiltonian, and $0 \le g$. Let

$$C^{\infty}(H_{0,\kappa}) = \bigcap_{n=0}^{\infty} D(H_{0,\kappa}^{n})$$

be the set of $C^{\infty}$ vectors for $H_{0,\kappa}$. It was shown in [16]-[17] that $H_{\kappa}(g)$ and $H_{I,\kappa,g}$ are essentially self-#-adjoint on $C^{\infty}(H_{0,\kappa})$, that

$$D(H_{\kappa}(g)) = D(H_{0,\kappa}) \cap D(H_{I,\kappa,g}) \quad (2.2)$$

and that there are finite or hyperfinite $a, b = b(g)$ such that

$$\| H_{0,\kappa} \psi \|_{\#} + \| H_{I,\kappa,g} \psi \|_{\#} \le \|( H_{\kappa}(g) + b ) \psi \|_{\#} \quad (2.3)$$

$$\text{for all } \psi \in D(H_{\kappa}(g)).$$

Note that it is convenient to introduce a periodic hyperfinite approximation in configuration space. Under this approximation, the momentum space variable $\mathbf{k} = (k_1, k_2, k_3) \in *\mathbb{R}_c^{\#3}$ is replaced by a discrete variable $\mathbf{k} \in \Gamma_V^3$

$$\Gamma_V^3 = \{ \mathbf{k} = (k_1, k_2, k_3) | k_i = \frac{2\pi n_i}{V}, n_i \in \mathbb{Z}; i = 1, 2, 3 \}$$

with $V \in *\mathbb{R}_{c+}^{\#} \setminus *\mathbb{R}_{\text{fin}+}^{\#}$. Thus we define $\mathcal{F}_V^{\#}$, the Fock space for volume $V^3$ as

$$\mathcal{F}_V^{\#} = \mathfrak{C}(l_2^{\#}(\Gamma_V^3)) = *\mathfrak{C}^{\#} \oplus l_2^{\#}(\Gamma_V^3) \oplus \{l_2^{\#}(\Gamma_V^3) \otimes_s l_2^{\#}(\Gamma_V^3)\} \dots$$

We identify $\mathcal{F}_V^{\#}$ with the subspace of $\mathcal{F}^{\#}$ consisting of piecewise constant functions which are constant on each cube of volume $(2\pi/V)^{3j}$ cantered about a lattice point

$$\{\mathbf{k}_1, \dots, \mathbf{k}_j\} \in \Gamma_V^3 \times \Gamma_V^3 \times \dots \times \Gamma_V^3 = \Gamma_V^{3j}.$$

The periodic annihilation and creation operators $a(\mathbf{k})$ and $a^*(\mathbf{k})$ can be extended from $\mathcal{F}_V^{\#}$ to $\mathcal{F}^{\#}$ by the formulas
$$a_V(\mathbf{k}) = \left(\frac{V}{2\pi}\right)^{3/2} \left[ \mathrm{Ext} - \int_{-\pi/V}^{\pi/V} d^\# l_1 \mathrm{Ext} - \int_{-\pi/V}^{\pi/V} d^\# l_2 \mathrm{Ext} - \int_{-\pi/V}^{\pi/V} d^\# l_3 a(\mathbf{k} + \mathbf{l}) \right], \quad (2.4)$$

$$a_V^*(\mathbf{k}) = \left(\frac{V}{2\pi}\right)^{3/2} \left[ \mathrm{Ext} - \int_{-\pi/V}^{\pi/V} d^\# l_1 \mathrm{Ext} - \int_{-\pi/V}^{\pi/V} d^\# l_2 \mathrm{Ext} - \int_{-\pi/V}^{\pi/V} d^\# l_3 a^*(\mathbf{k} + \mathbf{l}) \right]. \quad (2.5)$$

Therefore the periodic field $\varphi_{\kappa, \nu}^{\#}(\mathbf{x})$ and the periodic Hamiltonian $H_{\kappa, \nu}(g)$ can be extended to act on $\mathcal{F}^{\#}$ by the formulas

$$\varphi_{\kappa, \nu}^{\#}(\mathbf{x}) = (2V)^{-3/2} \mathrm{Ext} - \sum_{\mathbf{k} \in \Gamma_V^3, |\mathbf{k}| \le \kappa} \mathrm{Ext} - \exp(-i\langle \mathbf{k}, \mathbf{x} \rangle) [a^*(\mathbf{k}) + a(-\mathbf{k})] (\mu(\mathbf{k}))^{-1/2}, \quad (2.6)$$

$$H_{\kappa, \nu} = H_{0, \kappa, \nu} + H_{I, \kappa, \nu}, \quad (2.7)$$

$$H_{I, \kappa, \nu} = \mathrm{Ext} - \int_{-V/2}^{V/2} \mathrm{Ext} - \int_{-V/2}^{V/2} \mathrm{Ext} - \int_{-V/2}^{V/2} \varphi_{\kappa, \nu}^{\#4}(\mathbf{x}) d^{\#3}x, \quad (2.8)$$

$$H_{0, \kappa, \nu} = \mathrm{Ext} - \int_{|\mathbf{k}| \le \kappa} a^*(\mathbf{k}) a(\mathbf{k}) \mu(\mathbf{k}_V) d^{\#3}k \quad (2.9)$$

with $\mathbf{k}_V$, a lattice point infinite close to $\mathbf{k}$,

$$\mathbf{k}_V \in \Gamma_V^3, \quad \|\mathbf{k} - \mathbf{k}_V\| \le \frac{\pi}{V} \approx 0. \quad (2.10)$$

**Remark 2.1** Note the absence of a $V$ in the $a(\mathbf{k})$ and $a^*(\mathbf{k})$ in (2.1.9). On $\mathcal{F}_V^{\#}$, this definition of $H_{0, \kappa, \nu}$ agrees with the standard definition

$$\mathrm{Ext} - \sum_{\mathbf{k} \in \Gamma_V^3, |\mathbf{k}| \le \kappa} a_V^*(\mathbf{k}) a_V(\mathbf{k}) \mu(\mathbf{k}).$$

The operators $H_{I, \kappa, \nu}$ and $H_{\kappa, \nu}$ are essentially self adjoint on $C^{*\infty}(H_{0, \kappa, \nu})$, and

$$D(H_{\kappa, \nu}) = D(H_{0, \kappa, \nu}) \cap D(H_{I, \kappa, \nu}). \quad (2.11)$$

For all $\psi \in D(H_{\kappa, \nu})$,

$$\| H_{0, \kappa, \nu} \psi \|_{\#} + \| H_{I, \kappa, \nu} \psi \|_{\#} \le a \|( H_{\kappa, \nu} + b ) \psi \|_{\#} \quad (2.12)$$

where $b$ depend on $V$. On $\mathcal{F}_V^{\#}$, the operator $H_{I, \kappa, \nu}$ has a \#-compact resolvent. We want to approximate $H_{I, \kappa}(g)$ by operators with \#-compact resolvents on $\mathcal{F}_V^{\#}$, so we define

$$\begin{align}
H_{\kappa}(g, V) &= H_{0, \kappa, \nu} + \mathrm{Ext} - \int_{\mathbb{R}_c^{\#3}} \varphi_{\kappa, \nu}^{\#4}(\mathbf{x}) g(\mathbf{x}) d^{\#3}x = \tag{2.13} \\
&= H_{0, \kappa, \nu} + H_{I, \kappa}(g, V). \nonumber
\end{align}$$

As in chapter I sect. we can show that $H_{\kappa}(g, V)$, and $H_{I, \kappa}(g, V)$ are essentially self-\#-adjoint on $C^{*\infty}(H_{0, \kappa, \nu})$, and that

$$D(H_{\kappa}(g, V)) = D(H_{0, \kappa, \nu}) \cap D(H_{I, \kappa}(g, V)). \quad (2.14)$$

Furthermore, for all $\psi \in D(H_{\kappa}(g, V))$,

$$\| H_{0, \kappa, \nu} \psi \|_{\#} + \| H_{I, \kappa}(g, V) \psi \|_{\#} \le a \|( H_{\kappa}(g, V) + b ) \psi \|_{\#}. \quad (2.15)$$
In this case both g and V serve as volume cutoffs, and the constant b = b(g, V) can be chosen independently of V for fixed g. On the space $\mathcal{F}_V^\#$, the operator $H_\chi(g, V)$ has a #-compact resolvent. Our hamiltonians are semi-bounded and for each $\varepsilon > 0$, there is a constant b such that

$$0 \leq \varepsilon H_{0,\chi} + H_{I,\chi}(g) + b, \qquad (2.16)$$

$$0 \leq \varepsilon H_{0,\chi,V} + H_{I,\chi,V} + b, \qquad (2.17)$$

$$0 \leq \varepsilon H_{0,\chi,V} + H_{I,\chi}(g, V) + b, \qquad (2.18)$$

see chapter I sect.18. In (2.18), the b can be chosen to be independent of V. Taking $\varepsilon = 1/2$, we have

$$\frac{1}{2} H_{0,\chi} \leq H_{I,\chi}(g) + b,$$

which implies that for all $\psi \in D((H_\chi(g))^2)$,

$$\| H_{0,\chi}^{1/2} \psi \|_{\#} \leq \sqrt{2} \|(H_\chi(g) + b(\chi))^{1/2} \psi\|_{\#}. \qquad (2.19)$$

Here we must choose $b(\chi)$ at least $|E_\chi(2g)|$, where $E_\chi(2g)$ is the vacuum energy for the cut-off $2g$.

### 3. The existence of a vacuum vector $\Omega_{\chi,g}$ for $H_\chi(g)$

In this section we prove the existence of a vacuum vector $\Omega_{\chi,g}$ for $H_\chi(g)$. Since the Hamiltonian $H_\chi(g)$ is bounded from below, we can define the vacuum energy $E_{\chi,g} \triangleq E(\chi, g)$ to be the infimum of the spectrum of $H_\chi(g)$ and we also refer to $E_{\chi,g}$ as the lower bound of $H_\chi(g)$. We show that $E_{\chi,g}$ is an isolated point in the spectrum. In a relativistic theory, the gap between the ground state and the first excited state is the mass of the interacting particle. For this reason we say that $H_\chi(g)$ has a mass gap. A vacuum vector $\Omega_{\chi,g}$ is defined as a normalized eigenvector of $H_\chi(g)$ corresponding to the eigenvalue $E_{\chi,g}$.

$$H_{\chi}(g)\Omega_{\chi,g} = E_{\chi,g}\Omega_{\chi,g}, \quad \|\Omega_{\chi,g}\|_{\#} = 1. \qquad (3.1)$$

**Theorem 3.1** There exists a vacuum vector $\Omega_{\chi,g}$ for Hamiltonian $H_\chi(g)$. For any $\varepsilon > 0, \varepsilon \approx 0$ the operator $H_\chi(g)$, restricted to the spectral interval $[E_{\chi,g}, E_{\chi,g} + m_0 - \varepsilon]$ is #-compact.

**Theorem 3.2** The approximate Hamiltonian $H_{\chi,V}(g)$, has a vacuum vector $\Omega_{\chi,g,V}$. Any hyperfinite sequence of volumes $V_j$ tending to hyperinfinity $^{*\infty}$ has a hyper infinite subsequence $V_l, l \in *\mathbb{N}$ such that #-limit

$$\Omega_{\chi,g} = \#\text{-lim}_{l \to *\infty} \Omega_{\chi,g, V_l} \qquad (3.2)$$

exists and satisfies (3.1).

**Remark 3.1** Let $E_{\chi,g,V}$ be the lower bound of $H_{\chi,V}(g)$ on $\mathcal{F}_V^\#$. Since $H_{\chi,V}(g)$ has a #-compact resolvent on $\mathcal{F}_V^\#$, there is a vacuum vector $\Omega_{\chi,g,V}$ for $H_{\chi,V}(g) \uparrow \mathcal{F}_V^\#$. We now see that $E_{\chi,g,V}$ is the lower bound for $H_{\chi,V}(g)$ on $\mathcal{F}_V^\#$, so that $\Omega_{\chi,g,V}$ is a vacuum vector for $H_{\chi,V}(g)$.

**Remark 3.2** Let $\mathcal{F}_V^{\#\perp}$ be the orthogonal complement of $\mathcal{F}_V^\#$. Since $H_{\chi,V}(g)$ leaves $\mathcal{F}_V^\#$ invariant and is self-adjoint, $H_{\chi,V}(g)$ also leaves $\mathcal{F}_V^{\#\perp}$ invariant.

**Theorem 3.3** The lower bound of $H_{\chi,V}(g)$ on $\mathcal{F}_V^\#$ is $E_{\chi,g,V} + m_0$, where $m_0$, is the rest mass of the Fock space bosons.

**Remark 3.3** Theorem 3.3 shows that $\Omega_{\chi,g,V}$ is a vacuum for $H_{\chi,V}(g)$.

**Proof** We have an orthogonal decomposition in the single particle space

$$\mathcal{F}_1^\# = L_2^\# ( * \mathbb{R}_c^{\#3} ) = \mathcal{F}_{1V}^\# \oplus \mathcal{F}_{1V}^{\#\perp}. \qquad (3.3)$$
Here $\mathcal{F}_{1V}^{\#} = \mathcal{F}_{1}^{\#} \cap \mathcal{F}_{V}^{\#}$ consists of functions piecewise constant on intervals cantered at lattice points. Thus we may write

$$ \mathcal{F}^{\#} = \mathrm{Ext} - \bigoplus_{j=0}^{\infty} \mathcal{F}^{\#(j)}, \quad \mathcal{F}_{V}^{\#\perp} = \mathrm{Ext} - \bigoplus_{j=1}^{\infty} \mathcal{F}^{\#(j)}, \qquad (3.4) $$

where $\mathcal{F}^{\#(j)}$ consists of vectors with exactly $j$ particles from $\mathcal{F}_{1V}^{\#\perp}$ and

$$ \mathcal{F}^{\#(j)} = (\mathrm{Ext} - \mathcal{F}_{1V}^{\#\perp} \otimes_s \cdots \otimes_s \mathcal{F}_{1V}^{\#\perp}) \otimes_s \mathcal{F}_{V}^{\#} \qquad (3.5) $$

In this tensor product decomposition there are $j$ factors $\mathcal{F}_{1V}^{\#\perp}$. The Hamiltonian $H_{\kappa,V}(g)$ leaves each subspace $\mathcal{F}^{\#(j)}$ invariant, and on $\mathcal{F}^{\#(j)}$ we have $H_{\kappa,V}(g) = I \otimes A + B \otimes I$, where $A = H_{\kappa,V}(g) \uparrow \mathcal{F}_{V}^{\#}$ and $B$ is a sum of $j$ copies of $H_{0,\kappa,V}$ each acting on a single factor $\mathcal{F}_{1V}^{\#\perp}$. Since

$$ jm_0 \le B, \qquad (3.6) $$

the Theorem follows from this decomposition.

**Theorem 3.4** For $V \le \mathcal{N}$, and for $b$ sufficiently large we have

$$ D(H_{0,\kappa}) \subset D\left(H_{0,\kappa}^{\frac{1}{2}}\right) \cap D(N_{\kappa}) \subset D(H_{\kappa,V}(g) + b), \qquad (3.7) $$

$$ D(H_{0,\kappa}) \subset D\left(\left[(N_{\kappa} + I)^{-1}(H_{\kappa,V}(g) + b)\right]^{\#-}\right). \qquad (3.8) $$

Here we denote by $A^{\#-}$ #-closure of the operator $A$.

**Proof** We take $b$ large enough so that $H_{\kappa,V}(g) + b$ is positive, see (2.1.18). By (1.9) and (2.14) we get

$$ D(H_{0,\kappa}) \cap D(N_{\kappa}^2) \subset D(H_{0,\kappa}) \cap D\left(H_{I,\kappa,V}(g)\right) = D\left(H_{\kappa,V}(g)\right) \subset D\left((H_{\kappa,V}(g) + b)^{\frac{1}{2}}\right). $$

Thus for all $\psi \in D(H_{0,\kappa}) \cap D(N_{\kappa}^2)$,

$$ \left\| (H_{\kappa,V}(g) + b)^{\frac{1}{2}} \psi \right\|_{\#}^{2} = \langle \psi, (H_{\kappa,V}(g) + b) \psi \rangle_{\#} \le \langle \psi, (H_{\kappa,V} + b) \psi \rangle_{\#} + \\ + \left\| (N_{\kappa} + I)^{-1} H_{I,\kappa,V} (N_{\kappa} + I)^{-1} \right\|_{\#} \| (N_{\kappa} + I) \psi \|_{\#}^{2}. $$

Since $(H_{\kappa,V}(g) + b)^{\frac{1}{2}}$ is a #-closed operator, we can extend this inequality by #-continuity. As $N_{\kappa}$ and $H_{0,\kappa,V}$ commute, the inequality extends by #-continuity to all $\psi \in D(H_{0,\kappa}^{1/2}) \cap D(N_{\kappa}) \supset D(H_{0,\kappa})$. The proof of (3.8) is similar.

**Theorem 3.5** Let $z$ be non-real or real and sufficiently negative. Then as $V$ tends to hyper infinity $\mathcal{N}$,

$$ \left\| (H_{\kappa,V}(g) - zI)^{-1} - (H_{\kappa}(g) - zI)^{-1} \right\|_{\#} = O(V^{-1}). \qquad (3.9) $$

**Proof** Let us fix $g$ and $z$ and suppress $g$ when possible. In chapter I sect 16 we have shown that $H_{\kappa}(g)$ is essentially self-#-adjoint on $C^{\infty}(H_{0,\kappa})$. Thus vectors of the form $\chi = (H_{\kappa} - zI)\psi, \psi \in C^{\infty}(H_{0,\kappa})$, are #-dense in $\mathcal{F}^{\#}$. On these vectors

$$ \{ (H_{\kappa,V} - zI)^{-1} - (H_{\kappa} - zI)^{-1} \} \chi = (H_{\kappa,V} - zI)^{-1} \{ (H_{\kappa} - zI)\psi - (H_{\kappa,V} - zI)\psi \} = \\ = (H_{\kappa,V} - zI)^{-1} (H_{\kappa} - H_{\kappa,V}) (H_{\kappa} - zI)^{-1} \chi = $$
$$ = (H_{\kappa,V} - zI)^{-1} (N_{\kappa} + I) (N_{\kappa} + I)^{-1} (H_{\kappa} - H_{\kappa,V}) (N_{\kappa} + I)^{-1} (N_{\kappa} + I) (H_{\kappa} - zI)^{-1} \chi. $$

For $\theta \in \mathcal{F}^{\#}$,

$$ \begin{aligned} & \left| \langle \theta, \left\{ (H_{\kappa,V} - zI)^{-1} - (H_{\kappa} - zI)^{-1} \right\} \chi \rangle_{\#} \right| \le \\ & \le \left\| (N_{\kappa} + I) (H_{\kappa,V} - \bar{z}I)^{-1} \right\|_{\#} \cdot \|\theta\|_{\#} \cdot \left\| (N_{\kappa} + I)^{-1} (H_{\kappa} - H_{\kappa,V}) (N_{\kappa} + I)^{-1} \right\|_{\#} \times \\ & \quad \times \left\| (N_{\kappa} + I) (H_{\kappa} - zI)^{-1} \right\|_{\#} \|\chi\|_{\#}. \end{aligned} \quad (3.10) $$

Using (2.15), we find that $\|(N_\kappa + I)(H_{\kappa,V} - \bar{z}I)^{-1}\|_\#$ is bounded uniformly in $V$, since

$$ \begin{aligned} \|(N_{\kappa} + I)(H_{\kappa,V} - \bar{z}I)^{-1} \psi\|_{\#} & \le \text{const} \cdot \|(H_{0,\kappa,V} + I)(H_{\kappa,V} - \bar{z}I)^{-1} \psi\|_{\#} \le \\ & \le \text{const} \cdot \| H_{\kappa,V} (H_{\kappa,V} - \bar{z}I)^{-1} \psi \|_{\#} + \text{const} \cdot \|(H_{\kappa,V} - \bar{z}I)^{-1} \psi \|_{\#}, \end{aligned} $$

where the constants can be chosen independently of $V$. By a similar consideration, the orthogonal decomposition (3.3) shows that $(N_\kappa + I)(H_{\kappa,V} - zI)^{-1}$ is a bounded operator. Thus from (3.10), and the fact that the $\chi$ are \#-dense, we infer

$$ \|(H_{\kappa,V} - zI)^{-1} - (H_{\kappa} - zI)^{-1}\|_{\#} \le \text{const} \cdot \|(N_{\kappa} + I)^{-1} (H_{\kappa} - H_{\kappa,V}) (N_{\kappa} + I)^{-1}\|_{\#} \quad (3.11) $$

with a constant independent of $V$. The difference $H_\kappa - H_{\kappa,V} = (H_{0,\kappa} - H_{0,\kappa,V}) + (H_{I,\kappa}(g) - H_{I,\kappa,V}(g))$ and for infinite large $V$,

$$ \|(N_{\kappa} + I)^{-1/2} (H_{0,\kappa} - H_{0,\kappa,V}) (N_{\kappa} + I)^{-1}\|_{\#} = O(V^{-1}). \quad (3.12) $$

This is a simple direct computation, using $|\mu(\mathbf{k}_V) - \mu(\mathbf{k})| = O(V^{-1})$. For the interaction terms, we use (1.10) to estimate

$$ \|(N_{\kappa} + I)^{-1/2} (H_{I,\kappa}(g) - H_{I,\kappa,V}(g)) (N_{\kappa} + I)^{-1}\|_{\#} = O(V^{-1}). \quad (3.13) $$

The kernel $b(\mathbf{k}_1, \dots, \mathbf{k}_4)$ corresponding to a monomial in $H_{I,\kappa}(g)$ is

$$ b(\mathbf{k}_1, \dots, \mathbf{k}_4) = \binom{4}{j} \prod_{j=1}^{4} \theta(\|\mathbf{k}_j\|, \kappa) [\mu(\mathbf{k}_j)]^{-1/2} \hat{g} \left( k_1^{(1)} + k_2^{(1)} + k_3^{(1)} + k_4^{(1)}, \dots, k_1^{(3)} + k_2^{(3)} + k_3^{(3)} + k_4^{(3)} \right), $$

$0 \le j \le 4$. The kernel $b_V(\mathbf{k}_1, \dots, \mathbf{k}_4)$ for the corresponding monomial in $H_{I,\kappa,V}(g)$ is obtained by replacing the factor $\prod_{j=1}^4 \theta(\|\mathbf{k}_j\|, \kappa) [\mu(\mathbf{k}_j)]^{-1/2}$ by the factor $\prod_{j=1}^4 \theta(\|\mathbf{k}_{jV}\|, \kappa) [\mu(\mathbf{k}_{jV})]^{-1/2}$. Inspection of the difference $b(\mathbf{k}_1, \dots, \mathbf{k}_4) - b_V(\mathbf{k}_1, \dots, \mathbf{k}_4)$ shows that $\|b(\mathbf{k}_1, \dots, \mathbf{k}_4) - b_V(\mathbf{k}_1, \dots, \mathbf{k}_4)\|_{L_2^\#} = O(V^{-1})$. as $V \to *\infty$, from which we conclude that (3.13) is $O(V^{-1})$. The \#-convergence of the resolvents follows from (3.11)-(3.13). The \#-limit

$$ E_{\kappa,g,V} \to_{\#} E_{\kappa,g} $$

follows from the \#-convergence of the resolvents, since for large positive $b$,

$$ (E_{\kappa,g,V} + b)^{-1} = \|(H_{\kappa,V}(g) + b)^{-1}\|_{\#}. $$
**Proof of the theorems 3.1 and 3.2** Let $f(x)$ be a \#-smooth positive function with support in the interval $[-\epsilon, m_0 - \epsilon]$. Then $f(H_{\kappa,V}(g) - E_{\kappa,g,V}) \uparrow \mathcal{F}_V^{\#}$ is \#-compact, since the resolvent of $H_{\kappa,V}(g) \uparrow \mathcal{F}_V^{\#}$ is \#-compact on $\mathcal{F}_V^{\#}$. By Theorem 3.3, $f(H_{\kappa,V}(g) - E_{\kappa,g,V}) \uparrow \mathcal{F}_V^{\#\perp} = 0$ and therefore \#-compact on the full Fock space $\mathcal{F}_V^{\#}$. By Theorem 3.5, the resolvent $(H_{\kappa,V}(g) - E_{\kappa,g,V} - z)^{-1}$ \#-converge in \#-norm as $V \to ^{*\infty}$, and therefore

$$ \|f(H_{\kappa,V}(g) - E_{\kappa,g,V}) - f(H_{\kappa}(g) - E_{\kappa,g})\|_{\#} \to_{\#} 0, $$

since $f(H_{\kappa}(g) - E_{\kappa,g})$ is a bounded function of $(H_{\kappa}(g) - E_{\kappa,g} - z)^{-1}$ which vanishes at hyper infinity. Since the uniform \#-limit of \#-compact operators is \#-compact, $H_{\kappa}(g)$ restricted to the spectral interval $[-\epsilon, m_0 - \epsilon]$ is \#-compact. This means furthermore that only a finite or hyperfinite number of eigenvalues of $H_{\kappa,V}(g)$ \#-converge to $E_{\kappa,g}$. Theorem 3.6 shows that the projection onto the corresponding set of eigenvectors of $H_{\kappa,V}(g)$ \#-converge as $V \to ^{*\infty}$. Since $\Omega_{\kappa,g,V}$ is an eigenvector of $f(H_{\kappa,V}(g) - E_{\kappa,g,V})$ a hyperinfinite subsequence of the $\Omega_{\kappa,g,V}$ \#-converge to a \#-limit as $V \to ^{*\infty}$. For this \#-limit

$$ \begin{aligned} (E_{\kappa,g} + b)^{-1} \Omega_{\kappa,g} &= \#-\lim_{l \to ^{*\infty}} (E_{\kappa,g,V_l} + b)^{-1} \Omega_{\kappa,g,V_l} = \\ &= \#-\lim_{l \to ^{*\infty}} (H_{\kappa,V_l}(g) + b)^{-1} \Omega_{\kappa,g,V_l} = (H_{\kappa}(g) + b)^{-1} \Omega_{\kappa,g} \end{aligned} $$

by Theorem 3.5. Hence $\Omega_{\kappa,g} \in D(H_{\kappa}(g))$, $H_{\kappa}(g)\Omega_{\kappa,g} = E_{\kappa,g}\Omega_{\kappa,g}$ and $\Omega_{\kappa,g}$ is a vacuum vector for $H_{\kappa}(g)$. In the following section we will see that $\Omega_{\kappa,g}$ and $\Omega_{\kappa,g,V}$ are unique except for an arbitrary phase multiple $Ext-\exp(i\theta)$, and that there is a natural choice for this arbitrary phase. With this choice, we then will prove that the $\Omega_{\kappa,g,V}$ \#-converge to $\Omega_{\kappa,g}$ as $V \to ^{*\infty}$.

**4. Uniqueness of the vacuum.**

In this subsection we prove the uniqueness of a vacuum vector $\Omega_{\kappa,g}$ for $H_{\kappa}(g)$.

**Theorem 4.1** The vacuum vector $\Omega_{\kappa,g,V}$ for $H_{\kappa}(g)$ is unique.

**Remark 4.1** In other words $E_{\kappa,g}$, the lower bound of $H_{\kappa}(g)$ is a simple eigenvalue.

**Definition 4.1** Let $\mathcal{H}^{\#} = L_2^{\#}(Q, d^{\#}\mu)$ be a non-Archimedean Hilbert space. We say that a bounded operator $A: \mathcal{H}^{\#} \to \mathcal{H}^{\#}$ has a strictly positive kernel provided that

$$ \langle \psi, A\chi \rangle_{\#} > 0 \qquad (4.1) $$

whenever $\psi$ and $\chi$ are non-negative $L_2^{\#}$ functions with non-zero \#-norms. Such an operator transforms a function $\chi \ge 0$, $\|\chi\|_{\#} \ne 0$ into a function $A\chi$ which is strictly positive \#-almost everywhere.

**Definition 4.2** Let $\mathcal{H}^{\#} = L_2^{\#}(Q, d^{\#}\mu)$ be a non-Archimedean Hilbert space. We say that a bounded operator $A: \mathcal{H}^{\#} \to \mathcal{H}^{\#}$ has a positive, ergodic kernel if for each $\psi, \chi$ as above $\langle \psi, A\chi \rangle \ge 0$ and

$$ \langle \psi, A^j \chi \rangle_{\#} > 0 \qquad (4.2) $$

for some $j$, depending on $\psi$ and $\chi$. Clearly every $A$ with a strictly positive kernel has a positive, ergodic kernel. **Theorem 4.2** Let $A$ have a positive ergodic kernel, and suppose that $\|A\|_{\#}$ is an eigenvalue of $A$. Then $\|A\|_{\#}$ is a simple eigenvalue and the corresponding eigenvector can be chosen to be a strictly positive function.

**Proof** Since $A$ maps positive functions into positive functions it also maps real functions into real functions. If $\psi \in \mathcal{H}^{\#}$ satisfies $A\psi = \|A\|_{\#} \cdot \psi$, then so do Re$\psi$ and Im$\psi$. Therefore without loss of generality we may assume that $\psi$ is real. Since $\|A^j\|_{\#} = \|A\|_{\#}^j$, and $A^j\psi = \|A\|_{\#}^j \cdot \psi$, we infer that

$$ \|A^j\|_{\#} \cdot \|\psi\|_{\#}^2 = \langle \psi, A^j \psi \rangle_{\#} \le \langle |\psi|, A^j |\psi| \rangle_{\#} \le \|A\|_{\#}^j \cdot \|\psi\|_{\#}^2, $$

$$ \langle \psi, A^j \psi \rangle_{\#} = \langle |\psi|, A^j |\psi| \rangle_{\#}. $$
Writing now $\psi = \psi^+ - \psi^-$, where $\psi^+$ and $\psi^-$ are the positive and negative parts of $\psi$,

$$ \begin{aligned} & \langle \psi^+, A^j \psi^+ \rangle_{\#} - \langle \psi^+, A^j \psi^- \rangle_{\#} - \langle \psi^-, A^j \psi^+ \rangle_{\#} + \langle \psi^-, A^j \psi^- \rangle_{\#} = \\ & = \langle \psi^+, A^j \psi^+ \rangle_{\#} + \langle \psi^+, A^j \psi^- \rangle_{\#} + \langle \psi^-, A^j \psi^+ \rangle_{\#} + \langle \psi^-, A^j \psi^- \rangle_{\#} \end{aligned} $$

or

$$ \langle \psi^+, A^j \psi^- \rangle_{\#} + \langle \psi^-, A^j \psi^+ \rangle_{\#} = 0. \quad (4.3) $$

Unless $\psi^+ = 0$ or $\psi^- = 0$, each term of (4.3) could be made strictly positive by choosing an appropriate $j$. Thus either $\psi^+$ or $\psi^-$ must vanish, and we may choose the eigenvector $\psi$ to be non-negative. If $\chi \ge 0$, $\|\chi\|_{\#} \ne 0$, then for some integer $j$, $0 < \langle \chi, A^j \psi \rangle_{\#} = \|A\|_{\#}^j \cdot \langle \chi, \psi \rangle_{\#}$. This proves that $\chi\psi$ is not zero almost everywhere, and that $\psi$ is strictly positive #-almost everywhere. Finally, if $\psi$ and $\chi$ were linearly independent eigenvectors of $A$ with the eigenvalue $\|A\|_{\#}$, then we could repeat the above argument with the component of $\chi$ orthogonal to $\psi$. This would yield two positive, orthogonal eigenvectors, which would be impossible, and the proof is complete.

**Remark 4.2** Let $\varphi_{\chi}^{\#}(h) = \text{Ext} - \int_{\mathbb{R}_c^{\#4}} \varphi_{\chi}^{\#}(x)h(x) d^{\#3}x$ denote the smeared, time zero free field operators. The spectral projections of the $\varphi_{\chi}^{\#}(h)$, or the functions $\text{Ext}-\exp(i\varphi_{\chi}^{\#}(h))$ generate a maximal abelian algebra $M^{\#}$ of bounded operators on $\mathcal{F}^{\#}$. Let $Q$ be the spectrum of the algebra $M^{\#}$. The no particle vector $\Omega_0 \in \mathcal{F}^{\#}$ is a cyclic vector for 'DR, namely $\mathcal{F}^{\#} = \#-(M^{\#}\Omega_0)$. Therefore we may introduce a #-measure $d^{\#}\mu$ on $Q$ so that $\mathcal{F}^{\#}$ is unitarily equivalent to $L_2^{\#}(Q, d^{\#}\mu)$ and so that the equivalence carries $M^{\#}$ into $L_{\infty}^{\#}$ and takes $\Omega_0$ into the function 1.

**Theorem 4.3** With $\mathcal{F}^{\#}$ represented as $L_2^{\#}(Q, d^{\#}\mu)$, $\text{Ext}-\exp(-H_{0,\chi})$ has a positive, ergodic kernel.

**Proof** Let $\psi$ and $\chi$ be non-negative. Write $\psi = \psi_1 + \psi_2$, where $\psi_1$ is the component of $\psi$ along $\Omega_0$. Thus the $L_1^{\#}$ #-norm of $\psi$ is given by $\|\psi\|_{\#1} = \langle \psi, \Omega_0 \rangle_{\#} = \langle \psi_1, \Omega_0 \rangle_{\#}$. Note $\|\psi\|_{\#1} \ne 0$ whenever $\psi$ is non-zero, and $\|\text{Ext}-\exp(-tH_{0,\chi})\psi_2\|_{\#1} \le (\text{Ext}-\exp(-tm_0))\|\psi_2\|_{\#1}$, where $m_0$ is the boson mass. Thus

$$ \langle \psi, \text{Ext}-\exp(-tH_{0,\chi})\chi \rangle_{\#} \ge \|\psi\|_{\#1} \cdot \|\chi\|_{\#1} - \|\psi_2\|_{\#1} \cdot \|\chi_2\|_{\#1} (\text{Ext}-\exp(-tm_0)). \quad (4.4) $$

By choosing $t$ sufficiently large, (4.4) is positive, which proves (4.2). If the following inequality holds

$$ \text{Ext}-\exp(-tm_0) < \frac{\frac{1}{2} \|\psi\|_{\#1} \cdot \|\chi\|_{\#1}}{\|\psi_2\|_{\#1} \cdot \|\chi_2\|_{\#1}} = \frac{\frac{\|\psi\|_{\#1} \cdot \|\chi\|_{\#1}}{2}}{\left(\frac{\|\psi\|_{\#}^2 - \|\psi\|_{\#1}^2}{\|\chi\|_{\#}^2 - \|\chi\|_{\#1}^2}\right)^{1/2}} \quad (4.5) $$

then

$$ \langle \psi, \text{Ext}-\exp(-tH_{0,\chi})\chi \rangle_{\#} \ge \frac{1}{2} \|\psi\|_{\#1} \cdot \|\chi\|_{\#1}. \quad (4.6) $$

We need to show that $\langle \psi, \text{Ext}-\exp(-tH_{0,\chi})\chi \rangle_{\#} > 0$ for all finite $t$. In fact, it is sufficient to prove this for a #-dense set of non-negative $\psi$ and $\chi$. Let us consider an approximate free energy operator

$$ H_{0,\chi,V} = \text{Ext} - \int_{|k| \le \chi} a^*(\mathbf{k}) a(\mathbf{k}) \mu(\mathbf{k}_V) d^{\#3}k. \quad (4.7) $$

For vectors $\psi \in C^{*\infty}(H_{0,\chi})$, as $V \to *\infty$, $\| H_{0,\chi,V}\psi - H_{0,\chi}\psi \|_{\#} \to 0$. Since $H_{0,\chi}$ is essentially self-#-adjoint on $C^{*\infty}(H_{0,\chi})$, the resolvents of $H_{0,\chi,V}$ converge strongly [18, p. 429]. Thus the generalized semigroup #-convergence theorem [18, p. 502] ensures that for all $\psi \in \mathcal{F}^{\#}$

$$ \| \text{Ext}-\exp(-tH_{0,\chi,V})\psi - \text{Ext}-\exp(-tH_{0,\chi})\psi \|_{\#} \to 0 $$

as $V \to *\infty$, and the #-convergence is uniform on #-compact sets of $t$. Therefore we need only show that for a #-dense set of non-negative $\psi$ and $\chi$ $\langle \psi, \text{Ext}-\exp(-tH_{0,\chi,V})\chi \rangle_{\#} \ge 0$. Let $F(x_1, ..., x_n)$ be a non-negative, hyper infinitely #-differentiable function with #-compact support, and let
$$ \psi = F(\varphi_{\kappa}^{\#}(f_1), \dots, \varphi_{\kappa}^{\#}(f_n))\Omega_0, \quad (4.8) $$

where $f_1, \dots, f_n$ are real. The set of all such vectors are #-dense in $\mathcal{F}^{#+}$, the non-negative vectors in $\mathcal{F}^{#}$. Furthermore, we define

$$ \psi_{\kappa, \nu} = F(\varphi_{\kappa, \nu}^{\#}(f_1), \dots, \varphi_{\kappa, \nu}^{\#}(f_n)) \Omega_0, \quad (4.9) $$

where $\varphi_{\kappa, \nu}^{\#}(f_1)$ is defined by restricting the sum in (2.6) to those

$$ \mathbf{k} \in \Gamma_{\kappa, \nu}^{3} = \Gamma_{\nu}^{3} \cap \{\mathbf{k} \mid \|\mathbf{k}\| \le \kappa\}. $$

Then $\psi_{\kappa, \nu} \in \mathcal{F}_{\kappa, \nu}^{\#+} \subset \mathcal{F}^{\#+}$ where $\mathcal{F}_{\kappa, \nu}^{\#+}$ is the Fock space corresponding to the modes $\mathbf{k} \in \Gamma_{\kappa, \nu}^{3}$. For any vector $\chi \in C^{*\infty}(H_{0,\kappa})$

$$ \|\varphi_{\kappa, \nu}^{\#}(f)\chi - \varphi_{\kappa}^{\#}(f)\chi\|_{\#} \to_{\#} 0, \text{ as } V \to^{*\infty}, $$

and as $C^{*\infty}(H_{0,\kappa})$ is a #-core for $\varphi_{\kappa}^{\#}(f)$, the resolvents of $\varphi_{\kappa, \nu}^{\#}(f)$ #-converge strongly to the resolvent of $\varphi_{\kappa}^{\#}(f)$. [18, p. 429]. Thus the generalized semigroup #-convergence theorem [19] ensures that for each $\chi \in \mathcal{F}^{#}$, $s$ real

$$ \| \mathrm{Ext}\text{-exp}(is\varphi_{\kappa,\nu}^{\#}(f))\psi - \mathrm{Ext}\text{-exp}(is\varphi_{\kappa}^{\#}(f))\psi \|_{\#} \to_{\#} 0, \text{ as } V \to^{*\infty}, $$

and the #-convergence is uniform for #-compact sets of $s$. By (4.9)

$$ \psi_{\kappa, \nu} = \mathrm{Ext} - \int \hat{F}(s_1, \dots, s_n) \left[ i \sum_{j=1}^{n} \mathrm{Ext}\text{-exp} \left( is \varphi_{\kappa, \nu}^{\#} (f_j) \right) \right] d^{\#} s_1 \cdots d^{\#} s_n, $$

and $\hat{F}(s_1, \dots, s_n)$ vanishes rapidly at hyper infinity, so we conclude that

$$ \|\psi_{\kappa, \nu} - \psi\|_{\#} \to_{\#} 0, \text{ as } V \to^{*\infty}. $$

Thus for such vectors $\psi, \chi$,

$$ \langle \psi, \mathrm{Ext}\text{-exp}(-tH_{0,\kappa})\chi \rangle_{\#} = \#\text{-lim}_{V \to^{*\infty}} \langle \psi_{\kappa,\nu}, \mathrm{Ext}\text{-exp}(-tH_{0,\kappa,\nu})\chi_{\kappa,\nu} \rangle_{\#} $$

and we need only show that

$$ \langle \psi_{\kappa, \nu}, \mathrm{Ext}\text{-exp}(-tH_{0, \kappa, \nu}) \chi_{\kappa, \nu} \rangle_{\#} \geq 0. \quad (4.10) $$

However on $\mathcal{F}_{\kappa, \nu}^{\#}$

$$ H_{0,\kappa,\nu} = \mathrm{Ext} - \sum_{\mathbf{k} \in \Gamma_{\kappa,\nu}^{3}} a_{\nu}^{*}(\mathbf{k}) a_{\nu}(\mathbf{k}) \mu(\mathbf{k}) = \mathrm{Ext} - \sum_{\mathbf{k} \in \Gamma_{\kappa,\nu}^{3}} H_{0,\kappa,\nu}, $$

so $\mathrm{Ext}\text{-exp}(-tH_{0,\kappa,\nu}) = \mathrm{Ext} - \prod_{\mathbf{k} \in \Gamma_{\kappa,\nu}^{3}} \exp(-tH_{0,\kappa,\nu})$. It easily verify by explicit computation that each operator $\exp(-tH_{0,\kappa,\nu})$ have a strictly positive kernel, so (4.10) holds and the proof is complete.

**Theorem 4.4** With $\mathcal{F}^{\#}$ represented as $L_2^{\#}(Q, d^{\#}\mu)$, the operator $\mathrm{Ext}\text{-exp}(-H_{\kappa}(g))$ has a positive, ergodic kernel.

**Remark 4.3** We expect that $\mathrm{Ext}\text{-exp}(-H_{0,\kappa})$ and $\mathrm{Ext}\text{-exp}(-H_{\kappa}(g))$ have strictly positive kernels.

**Proof** As in Theorem 4.3, formula (4.7), we consider $H_{\kappa,\nu}(g) = H_{0,\kappa} + H_{I,\kappa,\nu}(g)$. The approximate interaction $H_{I,\kappa,\nu}(g)$ is constructed with $\varphi_{\kappa,\nu}^{\#}$ in place of $\varphi_{\kappa}^{\#}$. Since $C^{*\infty}(H_{0,\kappa})$ is a #-core for $H_{\kappa}(g)$, we can argue as in the previous theorem that for all $\psi \in \mathcal{F}^{\#}$
$$ \mathrm{Ext}\text{-exp}(-tH_{\kappa,V}(g))\psi \to_{\#} \mathrm{Ext}\text{-exp}(-tH_{\kappa}(g))\psi, \text{ as } V \to {}^{*}\infty. $$

Thus we need only prove that for $\psi, \chi$ as in Theorem 4.3

$$ 0 < \varepsilon < \langle \psi_{\kappa,V}, \mathrm{Ext}\text{-exp}(-tH_{\kappa,V}(g)) \chi_{\kappa,V} \rangle_{\#}. \qquad (4.11) $$

and that for sufficiently large $t$, the constant $\varepsilon = \varepsilon(\psi, \chi, \kappa, V)$ can be chosen independently of $\kappa$ and $V$. On $\mathcal{F}_{\kappa,V}^{\#}$ we have an explicit representation of $\mathrm{Ext}\text{-exp}(-tH_{\kappa,V}(g))$ given by generalized Feynman-Kac integral formula

$$ \langle \psi_{\kappa,V}, \mathrm{Ext}\text{-exp}(-tH_{\kappa,V}(g)) \chi_{\kappa,V} \rangle_{\#} = \qquad (4.12) $$

$$ \mathrm{Ext}\text{-}\int_{C_{\kappa,V}} \mathrm{Ext}\text{-exp} \left( - \left[ \mathrm{Ext}\text{-}\int_{0}^{t} H_{I,g,\kappa,V}(q(s)) d^{\#}t \right] \right) \psi_{\kappa,V}(q(0)) \chi_{\kappa,V}(q(t)) D^{\#}q(\cdot). $$

Here $q(s)$ denotes a points in the spectrum of the modes

$$ q_{V}(\mathbf{k}) = a_{V}(\mathbf{k}) + a_{V}^{*}(\mathbf{k}) + a_{V}^{*}(\mathbf{k}) + a_{V}^{*}(-\mathbf{k}) $$

$$ q'_{V}(\mathbf{k}) = a_{V}(\mathbf{k}) - a_{V}(-\mathbf{k}) + a_{V}^{*}(\mathbf{k}) - a_{V}^{*}(-\mathbf{k}) $$

for $\mathbf{k} \in \Gamma_{\kappa,V}^{3} = \{\mathbf{k} | \mathbf{k} \in \Gamma_{V}^{3} \wedge |\mathbf{k}| \le \kappa\}$, and $C_{\kappa,V}$ is the path space for these modes. Since $\mathrm{Ext}\text{-exp}(-tH_{0,\kappa})$ has a strictly positive kernel, (4.12) exhibits $\mathrm{Ext}\text{-exp}(-tH_{\kappa,V}(g))$ explicitly as an operator with a strictly positive kernel. Thus (4.11) is valid, and taking the $\#$-limit as $V \to {}^{*}\infty$ shows that

$$ \langle \psi, \mathrm{Ext}\text{-exp}(-tH_{\kappa}(g))\chi \rangle_{\#} \ge 0. \qquad (4.13) $$

We now establish a uniform lower bound on $\varepsilon$ in (4.11) to prove that for $t$ sufficiently large (4.13) is strictly positive. Given any positive $M$ we can split the integral (4.13) into two parts. Let $C_{\kappa,V}^{(1)}$ be those paths such that the exponent in the Feynman-Kac formula satisfies $-[\mathrm{Ext}\text{-}\int_{0}^{t} H_{I,g,\kappa,V}(q(s)) d^{\#}t] \ge -M$, and let $C_{\kappa,V}^{(2)}$ be the complementary set of paths. Thus

$$ \langle \psi_{\kappa,V}, \mathrm{Ext}\text{-exp}(-tH_{\kappa,V}(g)) \chi_{\kappa,V} \rangle_{\#} \ge (\mathrm{Ext}\text{-exp}(-M)) \mathrm{Ext}\text{-}\int_{C_{\kappa,V}^{(1)}} \psi_{\kappa,V}(q(0)) \chi_{\kappa,V}(q(t)) D^{\#}q(\cdot) = \\ = (\mathrm{Ext}\text{-exp}(-M)) \left\{ \langle \psi_{\kappa,V}, \mathrm{Ext}\text{-exp}(-tH_{0,\kappa,V}) \chi_{\kappa,V} \rangle_{\#} - \mathrm{Ext}\text{-}\int_{C_{\kappa,V}^{(2)}} \psi_{\kappa,V}(q(0)) \chi_{\kappa,V}(q(t)) D^{\#}q(\cdot) \right\}. \qquad (4.14) $$

First we choose $t$ by (4.5) so that (4.6) holds. Then for sufficiently infinitely large $V$ (depending on $t$),

$$ \langle \psi_{\kappa,V}, \mathrm{Ext}\text{-exp}(-tH_{0,\kappa,V}) \chi_{\kappa,V} \rangle_{\#} \ge \frac{1}{2} \langle \psi, \mathrm{Ext}\text{-exp}(-tH_{0,\kappa}) \chi \rangle_{\#} \ge \frac{1}{4} \|\psi\|_{\#1} \cdot \|\chi\|_{\#1}. $$

Thus (4.14) becomes

$$ \langle \psi_{\kappa,V}, \mathrm{Ext}\text{-exp}(-tH_{0,\kappa,V}) \chi_{\kappa,V} \rangle_{\#} \ge \\ \ge \mathrm{Ext}\text{-exp}(-M) \left\{ \frac{1}{4} \|\psi\|_{\#1} \cdot \|\chi\|_{\#1} - \mathrm{Ext}\text{-}\int_{C_{\kappa,V}^{(2)}} \psi_{\kappa,V}(q(0)) \chi_{\kappa,V}(q(t)) D^{\#}q(\cdot) \right\}. $$

Let $\mathrm{Pr}\{\cdot\}$ denote the $\#$-measure on path space, so that by the generalized Holder inequality

$$ \left| \mathrm{Ext}\text{-}\int_{C_{\kappa,V}^{(2)}} \psi_{\kappa,V}(q(0)) \chi_{\kappa,V}(q(t)) D^{\#}q(\cdot) \right| \le \left( \mathrm{Pr}\{C_{\kappa,V}^{(2)}\} \right)^{\frac{(r-1)}{r}} \| \psi_{\kappa,V}(q(0)) \chi_{\kappa,V}(q(t)) \|_{\#r}, $$
where $1 < r < 2$. By the smoothing property of $Ext-\exp(-tH_{0,\chi,V})$ for sufficiently large $t$

$$ \|\psi_{\chi,V}(q(0))\chi_{\chi,V}(q(t))\|_{\#r} \le \|\psi_{\chi,V}\|_{\#2} \times \|\chi_{\chi,V}\|_{\#2} $$

and for $V$ sufficiently infinitely large, this is dominated by $2\|\psi\|_{\#2} \cdot \|\chi\|_{\#2}$. Thus with the choices so far made for $V, t, M$,

$$ \begin{aligned} \langle \psi_{\chi,V}, \mathrm{Ext-}\exp(-tH_{\chi,V}(g)) \chi_{\chi,V} \rangle_{\#} &\ge \mathrm{Ext-}\exp(-M) \left\{ \frac{1}{4} \|\psi\|_{\#1} \cdot \|\chi\|_{\#1} - 2\|\psi\|_{\#2} \cdot \|\chi\|_{\#2} \left( \mathrm{Pr}\{C_{\chi,V}^{(2)}\} \right)^{\frac{(r-1)}{r}} \right\} \\ &\ge \frac{1}{8} (\mathrm{Ext-}\exp(-M)) \|\psi\|_{\#1} \cdot \|\chi\|_{\#1} > \varepsilon(\psi, \chi, t) \end{aligned} $$

provided in addition that

$$ \mathrm{Pr}\{C_{\chi,V}^{(2)}\} \le \left( \frac{\|\psi\|_{\#1} \cdot \|\chi\|_{\#1}}{16 \|\psi\|_{\#2} \cdot \|\chi\|_{\#2}} \right)^{\frac{r}{(r-1)}}. \quad (4.15) $$

We now show that for $M$ sufficiently large, (4.15) is satisfied and therefore theorem is proved.

Note that

$$ \begin{aligned} \mathrm{Pr}\{C_{\chi,V}^{(2)}\} &= \mathrm{Pr}\{M \le \mathrm{Ext}-\int_{0}^{t} H_{I,\chi,g,V} q(s) d^{\#} s\} = \mathrm{Pr}\{1 \le M^{-1} (\mathrm{Ext}-\int_{0}^{t} H_{I,\chi,g,V} q(s) d^{\#} s)\} \le \\ &\le M^{-2} \left[ \mathrm{Ext}-\int_{C_{\chi,V}} \left| \mathrm{Ext}-\int_{0}^{t} H_{I,\chi,g,V} q(s) d^{\#} s \right|^{2} d^{\#} q(\cdot) \right]. \end{aligned} $$

Replacing the integral over $s$ by a $\#$-limit of hyperfinite Riemann sums, we obtain a bound in terms of generalized Wiener integrals depending on a hyperfinite number of times.

$$ \mathrm{Pr}\{C_{\chi,V}^{(2)}\} \le \#\lim_{n \to \infty} \left(\frac{t}{nM}\right)^2 \left[ \mathrm{Ext}-\sum_{i,j=1}^{n} \mathrm{Ext}-\int_{C_{\chi,V}} H_{I,\chi,g,V} q\left(\frac{ti}{n}\right) H_{I,\chi,g,V} q\left(\frac{tj}{n}\right) d^\# q(\cdot) \right]. $$

By the definition of the generalized Wiener integral, this expression can be evaluated in terms of no-particle expectation values, and it equals

$$ \#\lim_{n \to \infty} \left(\frac{t}{nM}\right)^2 \mathrm{Ext}-\sum_{i,j=1}^{n} \langle \Omega_0, \{\mathrm{Ext-}\exp(-|i-j|tH_{0,\chi}/n)\} \cdot H_{I,\chi,g,V}\Omega_0 \rangle_{\#}. $$

By the generalized Schwarz inequality

$$ \mathrm{Pr}\{C_{\chi,V}^{(2)}\} \le \left(\frac{t}{M}\right)^2 \|H_{I,\chi,g,V}\Omega_0\|_{\#}^2 \le \left(\frac{tD_{\chi}}{M}\right)^2 $$

for some constant $D_\chi$ independent of $V$. Thus we choose

$$ M \ge Dt \left( \frac{16 \|\psi\|_{\#1} \cdot \|\chi\|_{\#1}}{16 \|\psi\|_{\#2} \cdot \|\chi\|_{\#2}} \right)^{\frac{r}{(r-1)}}. $$

Combining Theorem 4.4 with Theorem 4.2 yields a proof of Theorem 4.1. Clearly the same proof applies to $H_{\chi,V}(g)$, to show that its vacuum is unique.

**Corollary 4.5.** Let $\Omega_{\chi,g,V}$ be the vacuum for $H_{\chi,g,V}$, with its phase determined by the requirement

$$ \langle \Omega_0, \Omega_{\chi,g,V} \rangle_{\#} > 0. \quad (4.16) $$

Then $\#\lim_{V \to \infty} \Omega_{\chi,g,V} = \Omega_{\chi,g}$ exists, $\Omega_{\chi,g}$ is the vacuum for $H_{\chi,g}$, and
$$ \langle \Omega_0, \Omega_{\kappa,g} \rangle_{\#} > 0. \qquad (4.17) $$

**Proof** A hyper infinite sequence $\Omega_{\kappa,g,V_j}$ with $V_j \to ^{*}\infty$ has a #-convergent hyper infinite subsequence by Theorem 4.2, #-converging to a vacuum for $H_{\kappa,g}$. The phase (4.16) fixes the phase (4.17) so every #-convergent hyper infinite subsequence has the same #-limit $\Omega_{\kappa,g}$. Thus the $\Omega_{\kappa,g,V}$ #-converge to $\Omega_{\kappa,g}$, as required.

**Corollary 4.6** The vacuum $\Omega_{\kappa,g}$ is a cyclic vector for $\mathcal{M}$.

**Proof** The function $\Omega_{\kappa,g}$ is positive for #-almost all $q \in Q^{\#}$, and $\mathcal{M} = L_{*\infty}^{\#}(Q^{\#})$ in the $L_2^{\#}(Q^{\#}, d^{\#}f)$ representation of $\mathcal{F}^{\#}$.

## 5. The Heisenberg picture field operators

In the Heisenberg picture operators have the time dependence

$$ A(t) = \mathrm{Ext-}\exp(it H_{\kappa}(g))A(0)\mathrm{Ext-}\exp(-it H_{\kappa}(g)) \qquad (5.1) $$

This definition of the dynamics contains the cut-off function $g(\mathbf{x})$ explicitly. For an important class of operators $A(0)$, however, $A(t)$ is independent of $g(\mathbf{x})$ provided that $g(\mathbf{x}) = \lambda$, the coupling constant, on a suitably large set. For example, we take $A(0)$ to be an observable representing a measurement performed in some 3-dimensional region $B \subset *\mathbb{R}_c^{\#3}$ of space (at time $t=0$). Then $A(t)$ represents the same measurement performed at time $t$. A Hamiltonian with a hyperfinite ultraviolet cut-off $\kappa \in *\mathbb{R}_{c+}^{\#} \setminus *\mathbb{R}_{\mathrm{fin}+}^{\#}$, such as $H_{\kappa}(g)$, propagates information with at most the speed of light. Therefore if $g(\mathbf{x}) = \lambda$ on a region containing $B$, and $t$ is sufficiently small, the fact that $g(\mathbf{x})$ does not equal $\lambda$ everywhere will never be recorded by a measurement $A(t)$. For each localized observable $A(0)$ and each $t$, we make an appropriate choice for $g(\mathbf{x})$. Therefore (5.1) provides the correct dynamics for the $(\varphi^4)_4$ quantum field theory with the cut quantum field theory with the cut-off removed. In this section we consider the quantum field operators $\varphi_{\kappa}^{\#}(\mathbf{x}, t)$ or $*\mathbb{R}_c^{\#4}$

$$ \varphi_{\kappa}^{\#}(f) = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#4}} \varphi_{\kappa}^{\#}(\mathbf{x}, t) d^{\#3}x d^{\#}t. \qquad (5.2) $$

We see that integration helps in (4.2) because $\varphi_{\kappa}^{\#}(f)$ is an operator while $\varphi_{\kappa}^{\#}(\mathbf{x}, t)$ is a bilinear form. Actually the time integration is not required and for real $f$,

$$ A(t) = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3}} \varphi_{\kappa}^{\#}(\mathbf{x}, t) d^{\#3}x \qquad (5.3) $$

is also a self-#-adjoint operator depending #-continuously on $t$. We expect that this is a special feature of the two dimensional model we are considering and that sharp time fields will not be operators in four dimensions. For this reason, basic physical concepts have been formulated in terms of the time averaged fields (5.2) rather than the sharp time fields (5.3). For example, Wightman's axioms for a quantum field theory are expressed in terms of the opera tors (5.2), and we will show that many of his axioms are satisfied for our model. The time integration in (5.2) presents some new difficulties (for example in the proof of self-#-adjointness or of locality) which would not occur if we considered only the sharp time operators (5.3). An advantage of the time averaged field is that products $\varphi(f_1) \cdots \varphi(f_n)$ can be defined on vectors with finite energy (Corollary 6.5). In fact we will construct a dense domain which the localized field operator $\varphi(f)$ leaves invariant, and on which $p(f)$ is essentially self-#-adjoint for real $f$.

## 6. An invariant domain for localized quantum fields.

In this section we study the Heisenberg picture field localized in a 4-dimensional region of space time $\mathcal{B}$. We find that $\varphi_{\kappa}^{\#}(\mathbf{x}, t)$ is a bilinear form and that for real $f$, $\varphi_{\kappa}^{\#}(f)$ is a #-densely defined symmetric operator. We start with the region $B$, a bounded open subset of space time. We require that $H_{\kappa}(g)$ be a Hamiltonian for $\mathcal{B}$. This means that the spatial cut-off $g(\mathbf{x})$ equals the coupling constant $\lambda$ on a sufficiently large interval to contain the domain of dependence of $\mathcal{B}$. In other words, assuming that the velocity of light is one, for every point $(\mathbf{y}, t) \in \mathcal{B}$,

$$ g(\mathbf{x}) = \lambda, \text{ if } \|\mathbf{x} - \mathbf{y}\| < t. \qquad (6.1) $$
It is convenient to deal with the field

$$ \varphi_{\kappa,g}^{\#}(\mathbf{x}, t) = \mathrm{Ext}\text{-exp}(it H_{\kappa}(g)) \varphi_{\kappa}^{\#}(\mathbf{x}) \mathrm{Ext}\text{-exp}(-i t H_{\kappa}(g)) $$

and its time #-derivative

$$ \pi_{\kappa,g}^{\#}(\mathbf{x}, t) = \mathrm{Ext}\text{-exp}(it H_{\kappa}(g)) \pi_{\kappa}^{\#}(\mathbf{x}) \mathrm{Ext}\text{-exp}(-i t H_{\kappa}(g)) = \partial^{\#} \varphi_{\kappa,g}^{\#}(\mathbf{x}, t) / \partial^{\#} t. $$

The time zero fields $\varphi_{\kappa}^{\#}(\mathbf{x})$ and its conjugate momentum $\pi_{\kappa}^{\#}(\mathbf{x})$ were defined in chapter I. We shall see that for $(\mathbf{x}, t) \in \mathcal{B}$, $\varphi_{\kappa,g}^{\#}(\mathbf{x}, t)$ is independent of $g$, and equals the field $\varphi_{\kappa}^{\#}(\mathbf{x}, t)$. Thus all the cut-offs have been removed in the definition of $\varphi_{\kappa}^{\#}(\mathbf{x}, t)$. For each $C^{*\infty}$-function $f(\mathbf{x}, t)$ with support in $\mathcal{B}$, we show that

$$ \varphi_{\kappa}^{\#}(f) = \mathrm{Ext}\text{-}\int_{\mathbb{R}_c^{\#4}} \varphi_{\kappa}^{\#}(\mathbf{x}, t) f(\mathbf{x}, t) d^{\#3}xd^{\#}t \quad (6.2) $$

is an operator whose domain contains

$$ D_{\kappa,g}^{\#} = C^{*\infty}(H_{\kappa}(g)) = \cap_{n=0}^{*\infty} D(H_{\kappa}^{n}(g)), \quad (6.3) $$

In fact $D_{\kappa,g}^{\#}$ is an invariant domain, i.e.

$$ \varphi_{\kappa}^{\#}(f) D_{\kappa,g}^{\#} \subset D_{\kappa,g}^{\#}, \quad (6.4) $$

so that $D_{\kappa,g}^{\#} \subset C^{*\infty}(\varphi_{\kappa}^{\#}(f))$. We note that this invariant domain may depend on the region $\mathcal{B}$ in which the field $\varphi_{\kappa}^{\#}(f)$ is localized. For $\psi \in D_{\kappa,g}^{\#}$ the expectation values

$$ \langle \psi, \varphi_{\kappa}^{\#}(\mathbf{x}_1, t_1) \cdots \varphi_{\kappa}^{\#}(\mathbf{x}_n, t_n) \psi \rangle_{\#} \quad (6.5) $$

is $C_c^{\#4}$-valued Schwartz distribution in $D^{\#'}(\mathcal{B} \times \cdots \times \mathcal{B})$. If $f(\mathbf{x}, t)$ is a function in $S^{\#}(\mathbb{R}_c^{\#4})$, then $\varphi_{\kappa,g}^{\#}(f)$ still is defined on $D_{\kappa,g}^{\#}$ and leaves it invariant. The expectation values (6.5) of $\varphi_{\kappa,g}^{\#}(\mathbf{x}, t)$ are tempered distributions in $S^{\#'}(\mathbb{R}_c^{\#4})$. However, the fields $\varphi_{\kappa,g}^{\#}(f)$ may depend on $g$.

**Lemma 6.1.** The field $\varphi_{\kappa,g}^{\#}(\mathbf{x}, t)$ is a bilinear form on $D((H_{\kappa}(g) + b)^{1/2}) \times D((H_{\kappa}(g) + b)^{1/2})$ #-continuous in $\mathbf{x}$ and $t$. Namely for $\psi \in D((H_{\kappa}(g) + b)^{1/2})$, $\langle \psi, \varphi_{\kappa}^{\#}(\mathbf{x}, t) \psi \rangle_{\#}$ is a #-continuous function. Furthermore

$$ \left| \mathrm{Ext}\text{-}\int_{\mathbb{R}_c^{\#3}} \langle \psi, \varphi_{\kappa}^{\#}, g(\mathbf{x}, t) \psi \rangle_{\#} \frac{\partial^{\#}}{\partial^{\#}x_i} f(\mathbf{x}) \right| \le \mathrm{const} \cdot \|f\|_{\#2} \langle \psi, (H_{\kappa}(g) + b)\psi \rangle_{\#}, \quad i = 1,2,3. \quad (6.6) $$

**Proof** The free field $\varphi_{\kappa}^{\#}(\mathbf{x}, 0)$ is the sum of two expressions of the form (1.8). The kernels $\theta(\mathbf{k}, \kappa)b(\mathbf{k})$ are in $L_2^1$. Furthermore we have $\theta(\mathbf{k}, \kappa)b(\mathbf{k})[\mu(\mathbf{k})]^{-1/2} \in L_2^1$. The estimate (1.9) has been generalized to cover such kernels, giving us

$$ \left\| (H_{0,\kappa} + I)^{-1/2} \varphi_{\kappa,g}^{\#}(\mathbf{x}, 0) (H_{0,\kappa} + I)^{-1/2} \right\|_{\#} \le \mathrm{const} \cdot \left\| \theta(\mathbf{k}, \kappa) b(\mathbf{k}) [\mu(\mathbf{k})]^{-1/2} \right\|_{\#} < *\infty. \quad (6.7) $$

Thus for $\psi \in D((H_{\kappa}(g) + b)^{1/2})$, $\mathrm{Ext}\text{-exp}(-it H_{\kappa}(g))\psi \in D((H_{\kappa}(g) + b)^{1/2}) \subset D(H_{0,\kappa}^{1/2})$, by (2.19) and therefore $\langle \psi, \varphi_{\kappa,g}^{\#}(\mathbf{x}, t) \psi \rangle_{\#} = \langle \mathrm{Ext}\text{-exp}(-it H_{\kappa}(g))\psi, \varphi_{\kappa,g}^{\#}(\mathbf{x}, 0) \mathrm{Ext}\text{-exp}(-it H_{\kappa}(g))\psi \rangle_{\#}$ is defined and

$$ |\langle \psi, \varphi_{\kappa,g}^{\#}(\mathbf{x}, t) \psi \rangle_{\#}| \le \mathrm{const} \cdot \left\| \theta(\mathbf{k}, \kappa) b(\mathbf{k}) [\mu(\mathbf{k})]^{-1/2} \right\|_{\#} \cdot \langle \psi, (H_{\kappa}(g) + b) \psi \rangle_{\#} $$

Since $\|\theta(\mathbf{k}, \kappa)b(\mathbf{k})[\mu(\mathbf{k})]^{-1/2}|\mathbf{k}|^2\hat{f}(\mathbf{k})\|_{\#} \le \|\theta(\mathbf{k}, \kappa)b(\mathbf{k})[\mu(\mathbf{k})]^{-1/2}|\mathbf{k}|^2\|_{*\infty} \cdot \|f\|_{\#2} \le \mathrm{const} \cdot \|f\|_{\#2}$ the inequality (4.2.6) holds. Let us write $b_{x_i}$, $i = 1,2,3$ for $b$ to denote the dependence of $b$ on $x_i$. Then
$\|(b_{x_i} - b_{y_i})\theta(\mathbf{k}, \mathcal{K})[\mu(\mathbf{k})]^{-1/2}\|_{\#2}$ is a function of $(x_i - y_i)$ only and it $\#$-tends to zero as $|\mathbf{x} - \mathbf{y}| \to_{\#} 0$.

Since

$$ \begin{aligned} \left|\left\langle\psi, \left(\varphi_{\mathbf{k},g}^{\#}(\mathbf{x}, t)-\varphi_{\mathbf{k},g}^{\#}(\mathbf{y}, t)\right) \psi\right\rangle_{\#}\right| & \le \operatorname{const} \cdot \left\|\left(I+H_{0, \mathbf{k}}\right)^{1 / 2} \psi\right\|_{\#}^{2} \cdot \left\|\left(b_{x_i}-b_{y_i}\right) \theta(\mathbf{k}, \mathcal{K})[\mu(\mathbf{k})]^{-1 / 2}\right\|_{\# 2} \\ & \le \operatorname{const} \cdot \left\|(H_{\mathbf{k}}(g)+b)^{1 / 2} \psi\right\|_{\#}^{2} \cdot \left\|\left(b_{x_i}-b_{y_i}\right) \theta(\mathbf{k}, \mathcal{K})[\mu(\mathbf{k})]^{-1 / 2}\right\|_{\# 2} \end{aligned} \quad (6.8) $$

we have continuity with respect to $\mathbf{x}$. Also

$$ \left\|\left(H_{\mathbf{k}}(g)+b\right)^{\frac{1}{2}}\left(\operatorname{Ext}\operatorname{exp}(-i t H_{\mathbf{k}}(g)) - \operatorname{Ext}\operatorname{exp}(-i s H_{\mathbf{k}}(g))\right)\right\|_{\#} \to_{\#} 0 $$

as $|t-s| \to_{\#} 0$. Thus

$$ \begin{aligned} \left|\left\langle\psi, \left(\varphi_{\mathbf{k},g}^{\#}(\mathbf{x}, t)-\varphi_{\mathbf{k},g}^{\#}(\mathbf{x}, s)\right) \psi\right\rangle_{\#}\right| & \le \\ & \le \operatorname{const} \cdot \left\|\left(I+H_{0, \mathbf{k}}\right)^{1 / 2}\left(\operatorname{Ext}\operatorname{exp}(-i t H_{\mathbf{k}}(g)) - \operatorname{Ext}\operatorname{exp}(-i s H_{\mathbf{k}}(g))\right) \psi\right\|_{\#} \cdot \left\|\left(b_{x_i}\right) \theta(\mathbf{k}, \mathcal{K})[\mu(\mathbf{k})]^{-1 / 2}\right\|_{\# 2} \times \\ & \times \left\{\left\|\left(I+H_{0, \mathbf{k}}\right)^{1 / 2}\left(\operatorname{Ext}\operatorname{exp}(-i t H_{\mathbf{k}}(g))\right) \psi\right\|_{\#}+\left\|\left(I+H_{0, \mathbf{k}}\right)^{1 / 2}\left(\operatorname{Ext}\operatorname{exp}(-i s H_{\mathbf{k}}(g))\right) \psi\right\|_{\#}\right\} \\ & \le \\ & \operatorname{const} \cdot \left\|\left(H_{\mathbf{k}}(g)+b\right)^{1 / 2}\left(\operatorname{Ext}\operatorname{exp}(-i t H_{\mathbf{k}}(g)) - \operatorname{Ext}\operatorname{exp}(-i s H_{\mathbf{k}}(g))\right) \psi\right\|_{\#} \cdot \left\|\left(b_{x_i}\right) \theta(\mathbf{k}, \mathcal{K})[\mu(\mathbf{k})]^{-1 / 2}\right\|_{\# 2} \times \\ & \times \left\|(H_{\mathbf{k}}(g)+b)^{1 / 2} \psi\right\|_{\#} \to_{\#} 0 \end{aligned} \quad (6.9) $$

as $|t-s| \to_{\#} 0$.

From (6.8)-(6.9) we see that $\varphi_{\mathbf{k},g}^{\#}(\mathbf{x}, t)$ is jointly $\#$-continuous in $\mathbf{x}$ and $t$. Probably $\pi_{\mathbf{k},g}^{\#}(\mathbf{x}, t)$ is a bilinear form on $D((H_{\mathbf{k}}(g)+b)^{3/2}) \times D((H_{\mathbf{k}}(g)+b)^{3/2})$ $\#$-continuous in $\mathbf{x}$ and $t$, but our estimates are not strong enough to prove this. The functions $f(\mathbf{x}, t)$ in $S_{\text{fin}}^{\#}(*\mathbb{R}^{\#4})$ determine bounded

$\#$-measures $d^\#v = f(\mathbf{x}, t)d^\#x d^\#t$, so $\varphi_{\mathbf{k},g}^\#(f) = \operatorname{Ext}- \int \varphi_{\mathbf{k},g}^\#(\mathbf{x}, t) f(\mathbf{x}, t)d^\#x d^\#t$ is a bilinear form. If $d^\#v_n \to_{\#} d^\#v$ in the weak topology for $\#$-measures, then $\operatorname{Ext}- \int \varphi_{\mathbf{k},g}^\#(\mathbf{x}, t) d^\#v_n \to_{\#} \operatorname{Ext}- \int \varphi_{\mathbf{k},g}^\#(\mathbf{x}, t) d^\#v$ in the weak sense that for $\psi \in D((H_{\mathbf{k}}(g)+b)^{1/2})$

$$ \langle \psi, \operatorname{Ext}- \int \varphi_{\mathbf{k},g}^{\#}(\mathbf{x}, t) d^{\#}v_n \psi \rangle_{\#} \to_{\#} \langle \psi, \operatorname{Ext}- \int \varphi_{\mathbf{k},g}^{\#}(\mathbf{x}, t) d^{\#}v \psi \rangle_{\#}. \quad (6.10) $$

We define also the sharp time field

$$ A_{\mathbf{k},g}^{\#}(t) = \operatorname{Ext}- \int \varphi_{\mathbf{k},g}^{\#}(\mathbf{x}, t) f(\mathbf{x}, t) d^{\#3}x \quad (6.11) $$

and

$$ B_{\mathbf{k},g}^{\#}(t) = \operatorname{Ext}- \int \pi_{\mathbf{k},g}^{\#}(\mathbf{x}, t) f(\mathbf{x}, t) d^{\#3}x. \quad (6.12) $$

**Lemma 6.2** Let function $f(\mathbf{x}, t)$ in $S_{\text{fin}}^{\#}(*\mathbb{R}^{\#4})$ be real. Then $A_{\mathbf{k},g}^{\#}(t)$ and $B_{\mathbf{k},g}^{\#}(t)$ define self- $\#$-adjoint operators, and their domains include $D((H_{\mathbf{k}}(g)+b)^{1/2})$. With a constant $c$ independent of $g$,

$$ \|A_{\mathbf{k},g}^{\#}(t)\psi\|_{\#} + \|B_{\mathbf{k},g}^{\#}(t)\psi\|_{\#} \le c\{\|f(\cdot, t)\|_{\#2} + \|D_{\mathbf{x}}^{\#}f(\cdot, t)\|_{\#2}\} \|(H_{\mathbf{k}}(g)+b)^{\frac{1}{2}}\psi\|_{\#}, \quad (6.13) $$

for all $\psi \in D((H_{\mathbf{k}}(g)+b)^{1/2})$.

**Proof** It is sufficient to consider $\varphi_{\mathbf{k}}^{\#}(f_t) = \operatorname{Ext}- \int \varphi_{\mathbf{k}}^{\#}(\mathbf{x}) f(\mathbf{x}, t) d^{\#3}x$ in place of $A_{\mathbf{k},g}^{\#}(t)$ and $\pi_{\mathbf{k}}^{\#}(f_t) =$
$Ext- \int \pi_{\kappa}^{\#}(\mathbf{x}) f(\mathbf{x}, t) d^{\#3}\mathbf{x}$ in place of $B_{\kappa,g}^{\#}(t)$, as they are unitarily equivalent by the unitary operator $Ext-\exp(-it H_{\kappa}(g))$, and this unitary leaves $D((H_{\kappa}(g) + b)^{1/2})$ invariant. The same is true for $\pi_{\kappa}^{\#}(f_t)$ By (2.19) we have

$$ \|(I + N_{\kappa})^{1/2} \psi\|_{\#}^{2} \leq m_{0}^{-1} \|(I + H_{0, \kappa})^{1/2} \psi\|_{\#}^{2} \leq 2m_{0}^{-1} \|(H_{\kappa}(g) + b)^{1/2} \psi\|_{\#}^{2} $$

so we need only prove that

$$ \|\varphi_{\kappa}^{\#}(f_t)\psi\|_{\#} + \|\pi_{\kappa}^{\#}(f_t)\psi\|_{\#} \leq c\{\|f(\cdot, t)\|_{\#2} + \|D_{\kappa}^{\#}f(\cdot, t)\|_{\#2}\} \|(I + N_{\kappa})^{\frac{1}{2}}\psi\|_{\#}. \quad (6.14) $$

The lemma now follows from (1.9). For example, $\pi_{\kappa}^{\#}(f_t)$ is the sum of two operators of the form (1.8) with kernels $b(\mathbf{k}) = \theta(\mathbf{k}, \kappa)b(\mathbf{k})[\mu(\mathbf{k})]^{\frac{1}{2}}(Ext-\int(Ext-\exp(-i\langle\mathbf{k}, \mathbf{x}\rangle))f(\mathbf{x}, t)d^{\#3}\mathbf{x})$, satisfying the inequality

$$ \|b\|_{\#2} = \|(D_{\kappa}^{\#2} + m_0^2)^{1/4} f(\cdot, t)\|_{\#2} \leq \text{const} \cdot (\|f(\cdot, t)\|_{\#2} + \|D_{\kappa}^{\#} f(\cdot, t)\|_{\#2}). $$

The kernel for $\varphi_{\kappa}^{\#}(f_t)$ can be bounded by the $\|f(\cdot, t)\|_{\#2}$ #-norm alone. The estimate (6.13) now follows from (6.14). The self-#-adjointness of $A_{\kappa,g}^{\#}(t)$ and $B_{\kappa,g}^{\#}(t)$ can be proven by showing that $\varphi_{\kappa}^{\#}(f_t)$ and $\pi_{\kappa}^{\#}(f_t)$ are self-#-adjoint. But (4.14) ensures that every vector in $\mathcal{F}^{\#}$ with a finite or hyperfinite number of particles is #-analytic for $\varphi_{\kappa}^{\#}(f_t)$ and for $\pi_{\kappa}^{\#}(f_t)$, so these operators are essentially self-#-adjoint on the domain of vectors with a finite or hyperfinite number of particles. Hence they are uniquely determined by their definition on that domain.

We now explain the sense in which the integral (6.12) #-converges, since we did not show that $\pi_{\kappa,g}^{\#}(f_t)$ was a bilinear form. If $\psi \in D((H_{\kappa}(g) + b)^{1/2})$, then $\psi \in D((N_{\kappa} + I)^{1/2})$ and

$$ \langle \psi, \mathrm{Ext}-\hat{\pi}_{\kappa}^{\#}(\mathbf{k})\psi \rangle_{\#} = \theta(\mathbf{k}, \kappa)[\mu(\mathbf{k})]^{\frac{1}{2}}\{(a(\mathbf{k})\psi, \psi)_{\#} + \langle \psi, a(-\mathbf{k})\psi \rangle_{\#}\} \quad (6.15) $$

is a slowly increasing, locally summable function, and hence a tempered distribution in $S_{\text{fin}}^{\#'}(*\mathbb{R}^{\#4})$. Thus $\langle \psi, \pi_{\kappa}^{\#}(\mathbf{x})\psi \rangle_{\#}$ is by definition the distribution Fourier transform of (6.15), and hence a tempered distribution in $S_{\text{fin}}^{\#'}(*\mathbb{R}^{\#4})$. Finally (6.12) is the weak integral

$$ \langle \psi, B_{\kappa,g}^{\#}(t)\psi \rangle_{\#} = \mathrm{Ext}-\int d^{\#3}xf(\mathbf{x}, t) \langle (\mathrm{Ext}-\exp(-it H_{\kappa}(g)))\psi, \pi_{\kappa}^{\#}(\mathbf{x}) (\mathrm{Ext}-\exp(-it H_{\kappa}(g)))\psi \rangle_{\#}. $$

**Theorem 6.3** Let $\psi \in D((H_{\kappa}(g) + b)^{1/2})$, and let $f(\mathbf{x}, t)$ be a real function in $S_{\text{fin}}^{\#}(*\mathbb{R}^{\#4})$. Then the vectors $A_{\kappa,g}^{\#}(t)\psi$ and $B_{\kappa,g}^{\#}(t)\psi$ are strongly #-continuous and are rapidly decreasing functions of $t$. The integrals $\mathrm{Ext}-\int A_{\kappa,g}^{\#}(t)\psi d^{\#}t = \varphi_{\kappa,g}^{\#}(f)\psi$ and $\mathrm{Ext}-\int B_{\kappa,g}^{\#}(t)\psi d^{\#}t = \pi_{\kappa,g}^{\#}(f)\psi$ exist and define $\varphi_{\kappa,g}^{\#}(f)$ and $\pi_{\kappa,g}^{\#}(f)$ as #-closed symmetric operators with domains containing $D((H_{\kappa}(g) + b)^{1/2})$. We have the estimate

$$ \|\varphi_{\kappa,g}^{\#}(f)\psi\|_{\#} + \|\pi_{\kappa,g}^{\#}(f)\psi\|_{\#} \leq c(\mathrm{Ext}-\int\{\|f(\cdot, t)\|_{\#2} + \|D_{\kappa}^{\#}f(\cdot, t)\|_{\#2}\}d^{\#}t)(H_{\kappa}(g) + b)^{1/2}\psi \quad (6.16) $$

with a constant $c$ independent of $f$ and $g$.

**Proof** We write

$$ |f(\cdot, t)|_1 = c\|f(\cdot, t)\|_{\#2} + c\|D_{\kappa}^{\#}f(\cdot, t)\|_{\#2} \quad (6.17) $$

and

$$ A_{\kappa,g}^{\#}(s)\psi - A_{\kappa,g}^{\#}(t)\psi = (I - \mathrm{Ext}-\exp(-i(t-s)H_{\kappa}(g))) A_{\kappa,g}^{\#}(s)\psi + \\ + [\mathrm{Ext}-\exp(itH_{\kappa}(g))]\{\mathrm{Ext}-\int \varphi_{\kappa}^{\#}(\mathbf{x}) (f(\mathbf{x},s) - f(\mathbf{x},t))d^{\#3}\mathbf{x}\} [\mathrm{Ext}-\exp(-isH_{\kappa}(g))]\psi + $$
$$A_{\mathcal{H},g}^{\#}(s)[\mathrm{Ext}\text{-exp}(-i(t-s)H_{\mathcal{H}}(g))]\psi$$

Thus by (6.13),

$$ \begin{aligned} \| A_{\mathcal{H},g}^{\#}(s)\psi - A_{\mathcal{H},g}^{\#}(t)\psi \|_{\#} \le & \|(I - \mathrm{Ext}\text{-exp}(-i(t-s)H_{\mathcal{H}}(g))) A_{\mathcal{H},g}^{\#}(s)\psi \|_{\#} + \\ & + |f(\cdot, s) - f(\cdot, t)|_1 \|(H_{\mathcal{H}}(g) + b)^{\frac{1}{2}}\psi \|_{\#} + \\ & + |f(\cdot, t)|_1 \|(\mathrm{Ext}\text{-exp}(-i(t-s)H_{\mathcal{H}}(g)) - I)(H_{\mathcal{H}}(g) + b)^{\frac{1}{2}}\psi \|_{\#} \to_{\#} 0 \end{aligned} $$

as $t \to_{\#} s$. This proves the #-continuity. The rapid decrease is ensured by (6.13) and the fact that $f \in S_{\mathrm{fin}}^{\#}(\mathbb{R}^{\#4})$. A similar argument works for $B_{\mathcal{H},g}^{\#}(t)\psi$. The integrals defining $\varphi_{\mathcal{H},g}^{\#}(f)$ and $\pi_{\mathcal{H},g}^{\#}(f)$ now exist; (6.16) follows from integrating (6.13). Since $A_{\mathcal{H},g}^{\#}(t)$ and $B_{\mathcal{H},g}^{\#}(t)$ are self-#-adjoint, for $\psi \in D((H_{\mathcal{H}}(g) + b)^{1/2}) \subset$

$$(\psi, \varphi_{\mathcal{H},g}^{\#}(f)\psi)_{\#} = \mathrm{Ext}\text{-}\int (\psi, A_{\mathcal{H},g}^{\#}(t)\psi)_{\#} d^{\#}t = \mathrm{Ext}\text{-}\int (A_{\mathcal{H},g}^{\#}(t)\psi, \psi)_{\#} d^{\#}t$$

is a real, and similarly for $\pi_{\mathcal{H},g}^{\#}(f)$. Symmetric operators are #-closable and we now define $\varphi_{\mathcal{H},g}^{\#}(f)$ and $\pi_{\mathcal{H},g}^{\#}(f)$ as the #-closure of the above operators on the domain

$$D((H_{\mathcal{H}}(g) + b)^{1/2}). \qquad (6.18)$$

**Remark 6.1** (a) The integrals defining $\varphi_{\mathcal{H},g}^{\#}(f)\psi$ and $\pi_{\mathcal{H},g}^{\#}(f)\psi$ are strong Riemann integrals, $\varphi_{\mathcal{H},g}^{\#}(f)$ is a strong #-limit of operators of the form

$$\mathrm{Ext}\text{-}\sum_{i=1}^{n} A_{\mathcal{H},g}^{\#}(t_i), n \in \mathbb{N}_{\infty}. \qquad (6.19)$$

Conversely using the #-continuity of $A_{\mathcal{H},g}^{\#}(t)\psi$, we see that an operator of the form (6.19) is a strong #-limit of a hyper infinite sequence $\varphi_{\mathcal{H},g}^{\#}(f_j)$, $j \in \mathbb{N}_{\infty}$ and the $f_j$ can be chosen with the #-norm

$$|f|_1 = c \left( \mathrm{Ext}\text{-}\int_{\mathbb{R}_c^{\#3}} \{ \|f(\cdot, t)\|_{\#2} + \sum_{i=1}^{3} \|\partial_{x_i}^\# f(\cdot, t)\|_{\#2} \} d^\#t \right) \qquad (6.20)$$

uniformly bounded. For both #-limits the #-convergence occurs on the domain (6.18) and similar considerations apply to $\pi_{\mathcal{H},g}^{\#}(f)$. Furthermore $\varphi_{\mathcal{H},g}^{\#}(f)$ and $\pi_{\mathcal{H},g}^{\#}(f)$ can be defined whenever $|f|_{\#1} < *\infty$.

(b) Using (2.19) in order to estimate $H_{0,\mathcal{H}}$, we have from (6.16),

$$\|\varphi_{\mathcal{H},g}^{\#}(f)\Omega_{\mathcal{H},g}\|_{\#} + \|\pi_{\mathcal{H},g}^{\#}(f)\Omega_{\mathcal{H},g}\|_{\#} \le |f|_1 (|E_{\mathcal{H},2g} - E_{\mathcal{H},g}| + 1)^{\frac{1}{2}}, \qquad (6.21)$$

but the bound on the right grows in the diameter of the support of $g$.

**Theorem 6.4** [18] Let $|f|_{\#1}$ be the #-norm $|f|_{\#1} = c (\mathrm{Ext}\text{-}\int_{\mathbb{R}_c^{\#3}} \{ \|f(\cdot, t)\|_{\#2} + \sum_{i=1}^{3} \|\partial_{x_i}^\# f(\cdot, t)\|_{\#2} \} d^\#t)$.

Let $|f|_{\#1}$ is finite. Then on the domain $D((H_{\mathcal{H},g} + b)^{\frac{3}{2}})$, the field $\varphi_{\mathcal{H}}^{\#}(f)$ satisfies the following equation

$$(\partial_t^\# \varphi_{\mathcal{H}}^{\#})(f) = -\varphi_{\mathcal{H}}^{\#}(\partial_t^\# f) = \pi_{\mathcal{H}}^{\#}(f) = [iH_{\mathcal{H},g}, \varphi_{\mathcal{H}}^{\#}(f)]. \qquad (6.22)$$

**Proof** Note that the first equality in (6.22) is the definition of a distribution #-derivative. The out the difference quotient $\Delta_{\varepsilon}f(x, t)$ to #-derivative $\partial_t^\# f$ reads $\Delta_{\varepsilon}f(x, t) = \frac{|f(x+\varepsilon, t)-f(x,t)|}{\varepsilon}$, $\varepsilon \approx 0$, note that
\#-lim_{\epsilon \to \#0} \Delta_{\epsilon} f(x, t) = \partial_t^{\#} f(x, t). \text{ Note that for any vector } \psi \text{ such that } \psi \in D\left((H_{\kappa,g} + b)\right) \text{ by canonical consideration we get}

$$ \#\text{-lim}_{\epsilon \to 0} \| \varphi_{\kappa}^{\#}(\partial_{t}^{\#} f) \psi - \varphi_{\kappa}^{\#}(\Delta_{\epsilon} f(x, t)) \psi \|_{\#} = 0. $$

We have for $\psi \in D\left((H_{\kappa,g} + b)\right)^{\frac{3}{2}}$ that

$$ \varphi_{\kappa}^{\#}(\Delta_{\epsilon} f(x, t))\psi = \epsilon^{-1}(I - \text{Ext-exp}[i\epsilon H]) \left\{ \text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#3}} \varphi_{\kappa}^{\#}(x, t - \epsilon)f(x, t)d^{\#3}x\psi d^{\#}t \right\} + \\ + \epsilon^{-1} \left\{ \text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#3}} A_{\kappa}(f, t)(\text{Ext-exp}[i\epsilon H_{\kappa,g}] - I)\psi d^{\#}t \right\}. \quad (6.23) $$

Here the last term \#-converges as $\epsilon \to \#0$ and it \#-limit is: $i(\text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#3}} A_{\kappa}(f, t)H_{\kappa,g}\psi d^{\#}t)$.

Since $\varphi_{\kappa}^{\#}(\Delta_{\epsilon} f(x, t))\psi$ \#-converges as $\epsilon \to \#0$, the remaining term in expression for $\varphi_{\kappa}^{\#}(\Delta_{\epsilon} f(x, t))\psi$ \#-converges also to a \#-limit $\psi_1$. For $\chi \in D(H_{\kappa,g})$ we obtain that

$$ \langle \chi, \psi_1 \rangle = \#\text{-lim}_{\epsilon \to 0} \langle \chi, \epsilon^{-1}(I - \text{Ext-exp}[i\epsilon H_{\kappa,g}]) \{\text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#3}} \varphi_{\kappa}^{\#}(x, t - \epsilon)f(x, t)d^{\#3}x\psi d^{\#}t \} \rangle = \\ \langle iH_{\kappa,g}\chi, \varphi_{\kappa}^{\#}(f)\psi \rangle. $$

Since $H_{\kappa,g} = H_{\kappa,g}^*$, it follows that $\varphi_{\kappa,g}^{\#}(f)\psi \in D(H_{\kappa,g})$ and $\psi_1 = iH_{\kappa,g}\varphi_{\kappa,g}^{\#}(f)\psi$ and therefore:

$$ -\varphi_{\kappa}^{\#}(\partial_{t}^{\#} f)\psi = [iH_{\kappa,g}, \varphi_{\kappa}^{\#}(f)]\psi. $$

From the above equation we obtain

$$ i\langle\psi, \varphi_{\kappa}^{\#}(\partial_{t}^{\#} f)\psi\rangle = \text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#}} \langle H_{\kappa}\psi(t), \text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#3}} \varphi_{\kappa}^{\#}(x, 0)f(x, t)d^{\#3}x\psi(t) \rangle d^{\#}t - \\ \text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#}} \langle \text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#3}} \varphi_{\kappa}^{\#}(x, 0)f(x, t)d^{\#3}x\psi(t), H_{\kappa,g}\psi(t) \rangle d^{\#}t. \quad (6.24) $$

Here $\psi(t) = \text{Ext-exp}[itH_{\kappa,g}]\psi$. Note that $\psi(t) \in D(H_{0,\kappa}) \cap D(H_{I,\kappa,g})$, and

$$ \|H_{I,\kappa,g}(\psi(t) - \psi(s))\|_{\#} \le a \|(H_{\kappa,g} + b)(\psi(t) - \psi(s))\|_{\#} \to_{\#} 0, $$

as $|t-s| \to \#0$. Therefore we may substitute $H_{0,\kappa} + H_{I,\kappa}$ for $H_{\kappa,g}$ and consider each term separately. Note that the operators $H_{I,\kappa}$ and $\text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#3}} \varphi_{\kappa}^{\#}(x, 0)f(x, t)d^{\#3}x$ commute and therefore $H_{I,\kappa}$ contribute zero to equality above. The following identity by canonical computation holds for any $\psi \in D(H_{0,\kappa})$, in particular for $\psi(t) = \text{Ext-exp}[itH]\psi \in D(H_{0,\kappa})$

$$ \langle H_{0,\kappa}\psi, \text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#3}} \varphi_{\kappa}^{\#}(x, 0)f(x, t)d^{\#3}x\psi \rangle - \langle \text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#3}} \varphi_{\kappa}^{\#}(x, 0)f(x, t)d^{\#3}x \rangle \psi, H_{0,\kappa}\psi \rangle = \\ \langle \psi, -i \int_{\mathbb{R}_{\epsilon}^{\#3}} \pi_{\kappa}^{\#}(x, 0)f(x, t)d^{\#3}x \rangle \psi. $$

Therefore finally we get

$$ i\langle\psi, \varphi_{\kappa}^{\#}(\partial_{t}^{\#} f)\psi\rangle = \text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#}} \langle \psi(t), -i\text{Ext-} \int_{\mathbb{R}_{\epsilon}^{\#3}} \pi_{\kappa}^{\#}(x, 0)f(x, t)d^{\#3}x\psi \rangle d^{\#}t = \langle\psi, -i\pi_{\kappa}^{\#}(f)\psi\rangle. $$

This equality finalized the proof.

**Remark 6.2** (a) in exactly the same fashion one proves that

$$ \partial_{tt}^{\#2} \varphi_{\kappa,g}^{\#}(f) = \partial_{t}^{\#} \pi_{\kappa,g}^{\#}(f) = -\pi_{\kappa,g}^{\#}(\partial_{t}^{\#} f) = [iH_{\kappa,g}, \pi_{\kappa,g}^{\#}(f)] $$
if $|\partial_{tt}^{\#2} f|_1$ is also finite or hyperfinite. The commutator is a bilinear form on $D_{\kappa,g}^{\#} \times D_{\kappa,g}^{\#}$, $D_{\kappa,g}^{\#} = C^{*\infty}(H_{\kappa,g})$, namely

$$ \partial_{tt}^{\#2} \varphi_{\kappa,g}^{\#}(f) = \sum_{i=1}^{i=3} \partial_{x_i}^{\#2} \varphi_{\kappa,g}^{\#}(f) - m_0^2 \varphi_{\kappa,g}^{\#}(f) - 4 \mathrm{Ext} - \int_{\mathbb{R}_c^{\#4}} : \varphi_{\kappa,g}^{\#3}(\mathbf{x}, t) : f(\mathbf{x}, t) g(\mathbf{x}) d^{\#3}x \, d^{\#}t. \quad (6.25) $$

Here we define the $: \varphi_{\kappa,g}^{\#3}(\mathbf{x}, t) : \text{product by}$

$$ : \varphi_{\kappa,g}^{\#3}(\mathbf{x}, t) := (\mathrm{Ext}-\exp[itH_{\kappa,g}]): \varphi_{\kappa,g}^{\#3}(\mathbf{x}): (\mathrm{Ext}-\exp[-itH_{\kappa,g}]), \quad (6.26) $$

which we now prove is an operator valued non - Archimedean distribution. First we note that

$$ \mathrm{Ext} - \int_{\mathbb{R}_c^{\#3}} : \varphi_{\kappa,g}^{\#3}(\mathbf{x}, t) : f(\mathbf{x}, t) d^{\#3}x $$

is a sum of monomials in creation and annihilation operators with kernels in $L_2^\#$, and their $L_2^\#$ #-norms are #-continuous in $t$. Thus by (2.9)

$$ \mathrm{Ext} - \int_{\mathbb{R}_c^{\#3}} : \varphi_{\kappa}^{\#3}(\mathbf{x}, t) : f(\mathbf{x}, t) g(\mathbf{x}) d^{\#3}x $$

is a bilinear form on $D_{\kappa,g}^{\#} \times D_{\kappa,g}^{\#}$. By (2.3),

$$ (H + b)^{-1} \{ \mathrm{Ext} - \int_{\mathbb{R}_c^{\#3}} : \varphi_{\kappa}^{\#3}(\mathbf{x}, t) : f(\mathbf{x}, t) g(\mathbf{x}) d^{\#3}x \} (H + b)^{-1} $$

is a bounded operator, #-norm #-continuous in $t$. Thus on $D(H_{\kappa,g}) \times D(H_{\kappa,g})$,

$$ : \varphi_{\kappa,g}^{\#3}(f) := \mathrm{Ext} - \int_{\mathbb{R}_c^{\#4}} : \varphi_{\kappa,g}^{\#3}(\mathbf{x}, t) : f(\mathbf{x}, t) g(\mathbf{x}) d^{\#3}x \, d^{\#}t $$

is defined as a bilinear form. Hence (6.25) holds as an equation for bilinear forms on $D(H_{\kappa,g}) \times D(H_{\kappa,g})$. But each term except the last is an operator defined on $D((H+b)^{\frac{1}{2}})$. Thus $: \varphi_{\kappa,g}^{\#3}(f) :$ is actually an operator on $D((H+b)^{\frac{1}{2}})$, and in fact for real $f(x, t)$ it is essentially self #-adjoint. Furthermore, on $D((H+b)^{\frac{1}{2}}) \times D((H+b)^{\frac{1}{2}})$ each term in (6.25) except the last is a bilinear form which is a distribution of order two. Thus the same is true for $\langle \psi, : \varphi_{\kappa}^{\#3}(\mathbf{x}, t) : \psi \rangle_{\#}$. We have used (6.26) to define the cube of the interacting field. It would be interesting to determine whether this definition agrees with conventional notions involving the separation of points. We shall see in this section and the following one that the $g(\mathbf{x})$ in equation (6.25) can be removed if $f(\mathbf{x}, t)$ has #-compact support and $g(\mathbf{x}) = \lambda$ on a sufficiently large set. Then (6.25) becomes

$$ \left( \frac{\partial^{\#2}}{\partial t^{\#2}} + \sum_{i=1}^{i=3} \frac{\partial^{\#2}}{\partial x_i^{\#2}} + m_0^2 \right) \varphi_{\kappa}^{\#3}(f) = -4\lambda : \varphi_{\kappa}^{\#3}(f) :; $$

which is a non-linear equation for a self-#-adjoint operator valued distribution.

(b) The identity $\pi_{\kappa}^{\#}(f) = [iH_{\kappa,g}, \varphi_{\kappa}^{\#}(f)]$ implies

$$ B_{\kappa,g}^{\#}(t) = [iH_{\kappa,g}, A_{\kappa,g}^{\#}(t)] \quad (6.27) $$

provided that the right and left sides of (6.27) make sense and are #-continuous in $t$. They are certainly defined and are #-continuous as bilinear forms on $D(H_{\kappa,g}) \times D(H_{\kappa,g})$. To see that (6.27) makes sense as operators on $D((H_{\kappa,g} + b)^{\frac{3}{2}})$ we need only show that $A_{\kappa,g}^{\#}(t)$ maps $D((H_{\kappa,g} + b)^{\frac{3}{2}})$ into $D(H_{\kappa,g})$. We choose a hyper infinite sequence $f_j(\mathbf{x}, t) = f(\mathbf{x}, t_0)\delta_j(t - t_0)$, $j \in \mathbb{N}$ where $\delta_j(t - t_0)$ is a hyper infinite sequence of #-smooth functions #-converging to $\delta^{\#}(t - t_0)$ in the $w^*$-#-topology on #-measures and with the #-norms $|f_j|_{\#1}$, uniformly bounded. Then the bilinear forms #-converge, which means that the inner products
$$ \langle \theta, i H_{x,g}^m \varphi_{x,g}^{\#}(f) \psi \rangle_{\#} = \langle \theta, i \varphi_{x,g}^{\#}(f) H_{x,g}^m \psi \rangle_{\#} + \langle \theta, \pi_{x,g}^{\#}(f) \psi \rangle_{\#} $$

\#-converge for $\theta \in D(H_{x,g})$. However the \#-norms

$$ \| H_{x,g}^m \varphi_{x,g}^{\#}(f) \psi \|_{\#} \le \| i \varphi_{x,g}^{\#}(f) H_{x,g}^m \psi \|_{\#} + \| \pi_{x,g}^{\#}(f) \psi \|_{\#} + |f|_1 \|(H_{x,g} + b)^{\frac{3}{2}} \psi\|_{\#} $$

are uniformly bounded, and so the inner products \#-converge for all $\theta \in \mathbb{F}^n$. Thus the \#-limit $A_{\kappa}^{\#}(t) = \text{weak} \#-\lim \varphi_{\kappa}^{\#}(f_j)\psi$ is in $D(H_{x,g}^*) = D(H_{x,g})$ which proves (6.27) on the domain $D((H_{x,g} + b)^{\frac{3}{2}})$.

**Corollary 6.5** Let $f \in S_{\text{fin}}^{\#}(\mathbb{R}^{\#4})$. Then $D_{x,g}^{\#} = C^{\#\infty}(H_{x,g}) \subset C^{\#\infty}(\varphi_{x,g}^{\#}(f))$, and $\varphi_{x,g}^{\#}(f) D_{x,g}^{\#} \subset D_{x,g}^{\#}$.

**Proof** Using Theorem 6.4, we prove by hyper infinite induction on $m \in \mathbb{N}$ that $\varphi_{x,g}^{\#}(f) D_{x,g}^{\#} \subset D(H_{x,g}^m)$ and that for $\psi \in D_{x,g}^{\#}$,

$$ H_{x,g}^m \varphi_{x,g}^{\#}(f) \psi = \varphi_{x,g}^{\#}(f) H_{x,g}^m \psi + \text{Ext-} \sum_{j=1}^m \binom{m}{j} i^j \varphi_{x,g}^{\#}(\partial_t^{\#j} f) H_{x,g}^{m-j} \psi. $$

This formula is a special case of the identity $A^m B = \text{Ext-} \sum_{j=1}^m \binom{m}{j} [(\text{ad}A)^j B] A^{m-j}$, $m \in \mathbb{N}$. Thus we obtain

$$ \| H_{x,g}^m \varphi_{x,g}^{\#}(f) \psi \|_{\#} \le \text{Ext-} \sum_{j=1}^m \binom{m}{j} |\partial_t^{\#j} f|_{\#1} \|(H_{x,g} + b)^{m-j+\frac{1}{2}} \psi\|_{\#}. \quad (6.28) $$

**Theorem 6.5.** Let $f \in C_0^{\#\infty}(B_4^\#)$, that is $f$ is $C_0^{\#\infty}$ with support in the \#-open region of space time $B_4^\#$. Let $H_{x,g}$ be a Hamiltonian for $B_4^\#$, so that $g(x) = \lambda$ on a large set. Then

$$ \varphi_{x,g}^{\#}(f) = \varphi_{x,g}^{\#}(f) \quad \text{and} \quad \pi_{x,g}^{\#}(f) = \pi_{x,g}^{\#}(f) \text{ are independent of } g/\lambda. $$

**Proof** The spectral projections $E_{\#}^t(\lambda, \kappa)$ of the sharp time field

$$ A_{0,\kappa}(t) = \text{Ext-} \int_{\mathbb{R}_c^{\#3}} \varphi_{x,g}^{\#}(\mathbf{x}, t) f(\mathbf{x}, t_0) d^{\#3}\mathbf{x} = \text{Ext-} \int_{\mathbb{R}_c^{\#}} \lambda d^{\#} E_{\#}^t(\lambda, \kappa) $$

are given by the formula

$$ E_{\#}^{t}(\lambda, \kappa) = (\text{Ext-exp}[itH_{x,g}]) E_{\#}^{0}(\lambda, \kappa) (\text{Ext-exp}[-itH_{x,g}]) $$

and are independent of $g$. Thus $A_{0,\kappa}(t)$ is independent of $g$ and so is $A_{\kappa}(t)$. By (3.1.7), for all $g$,

$$ D_{0,\kappa} = D(H_{0,\kappa}^{1/2}) \cap D(N_{\kappa}) \subset D((H_{x,g} + b)^{\frac{1}{2}}) \subset D(\varphi_{x,g}^{\#}(f)) $$

so that $\varphi_{x,g}^{\#}(f) \uparrow D_{0,\kappa}$ is independent of $g$. Thus to complete the proof, we only need to show that the domain of $\varphi_{x,g}^{\#}(f) = \#-\overline{(\varphi_{x,g}^{\#}(f) \uparrow D((H_{x,g} + b)^{\frac{1}{2}}))}$ is independent of $g$. Since $H_{x,g}$ is essentially self-\#-adjoint on the domain $C^{\#\infty}(H_{0,\kappa}) \subset D_{0,\kappa}$, so is $(H_{x,g} + b)^{\frac{1}{2}}$. Thus by (6.16) $\varphi_{x,g}^{\#}(f) \uparrow D((H_{x,g} + b)^{\frac{1}{2}}) \subset \#-\overline{(\varphi_{x}^{\#}(f) \uparrow D_{0,\kappa})}$. Therefore $\#-\overline{(\varphi_{x}^{\#}(f) \uparrow D_{0,\kappa})} = \varphi_{x,g}^{\#}(f)$, so $\varphi_{x,g}^{\#}(f) = \varphi_{x}^{\#}(f)$ is independent of $g$. Similarly $\pi_{x,g}^{\#}(f) = \pi_{x}^{\#}(f)$ is independent of $g$.

**Theorem 6.7** Let $\psi \in D_{x,g}^{\#}$, with $H_{x,g}$ a Hamiltonian for $B_4^\#$. Then

$$ \langle \psi, \varphi_{x}^{\#}(\mathbf{x}_1, t_1) \cdots \varphi_{x}^{\#}(\mathbf{x}_n, t_n) \psi \rangle_{\#} $$

is a distribution in $D^{\#}(\mathcal{B} \times \cdots \times \mathcal{B})$

**Proof** This follows directly from our previous estimates (6.16) and (6.28).
### 7. Essential self-#-adjointness.

The main result of this section is the proof that for real test functions $f = f(x, t)$ with #-compact support, the field $\varphi_{\kappa}^{\#}(f)$ is self-#-adjoint, and essentially self-#-adjoint on $D_{0,\kappa} = D(H_{0,\kappa}^{1/2}) \cap D(N_\kappa)$, or on any $D_{\kappa,g}$ where $H_\kappa(g)$ is a Hamiltonian for the support of $f$. We see furthermore that if $f$ is real and $|f|_{\#1}$ defined by (6.20) is finite or hyperfinite, then, $\varphi_{\kappa,g}^{\#}(f)$ is self-#-adjoint and essentially self-#-adjoint on $D_{\kappa,g}$. The proof has three main steps. First, we assume that $f$ is a regular function of $t$; in that case we use an analytic vector argument to show that $\varphi_{\kappa,g}^{\#}(f)$ is essentially self-#-adjoint on $D_{\kappa,g}$. As a second step, we take #-limits in the resolvents $(\varphi_{\kappa,g}^{\#}(f) - z)^{-1}$ as $f$ tends to a more general function. In this way, we obtain a self-#-adjoint operator $\varphi_{\kappa,g}^{\#}(f)$. As a third step, we show that $\varphi_{\kappa,g}^{\#}(f)$ is essentially self-adjoint on $D(H_\kappa(g))$. The regularity we impose on $f$ is the requirement that its Fourier transform be a #-smooth function with #-compact support, or more generally that for the #-norm $|f|_{\#1}$, of (6.20) there exist constants $\alpha = \alpha(f)$ and $\beta = \beta(f)$

$$|D_t^r f| \le \alpha \beta^r, r \in \mathbb{N}. \qquad (7.1)$$

For a vector $\psi$, we consider the conditions

$$\|(H_\kappa(g) + b)^r \psi\|_{\#} \le a d^r, r \in \mathbb{N}. \qquad (7.2)$$

**Lemma 7.1** Assume (7.1) and (7.2). Then

$$\|(H_\kappa(g) + b)^r \varphi_{\kappa,g}^{\#}(f)\psi\|_{\#} \le \alpha a d c(b + \beta)^r \qquad (7.3)$$

for some constant $c$ independent of $r, \psi, f$.

**Proof.** By (6.28) and (7.1)-(7.2)

$$\begin{aligned} \|(H_\kappa(g) + b)^r \varphi_{\kappa,g}^{\#}(f)\psi\|_{\#} &\le \mathrm{Ext} - \sum_{j=0}^r \binom{r}{j} c \alpha \beta^j \|(H_\kappa(g) + b)^{r-j+1} \varphi_{\kappa,g}^{\#}(f)\psi\|_{\#} \\ &\le \mathrm{Ext} - \sum_{j=0}^r \binom{r}{j} c \alpha \beta^j a d^{r-j+1} \le \alpha a d c(b + \beta)^r. \end{aligned}$$

**Lemma 7.2** Assume (7.1) and (7.2). Then $\psi$ is an #-analytic vector for $\varphi_{\kappa,g}^{\#}(f)$. In particular for real $f(x, t)$, $\varphi_{\kappa,g}^{\#}(f)$ is essentially self-#-adjoint on $D_{\kappa,g}$.

**Proof** We applying the preceding lemma successively. We see that multiplication by $\varphi_{\kappa,g}^{\#}(f)$ changes the constants $a$ and $d$ of (7.2) as follows: $\to \alpha a d c \to d + \beta$. Thus

$$\|(H_\kappa(g) + b)^r [\varphi_{\kappa,g}^{\#}(f)]^k \psi\|_{\#} \le a(\alpha c)^k [\mathrm{Ext} - \prod_{j=0}^{k-1} (d + j\beta)] (d + k\beta)^r,$$

and

$$\|[\varphi_{\kappa,g}^{\#}(f)]^k \psi\|_{\#} \le a B^k k!$$

for some constant $B$, which proves that $\psi$ is #-analytic for $\varphi_{\kappa,g}^{\#}(f)$. The essential self-#-adjointness of $\varphi_{\kappa,g}^{\#}(f)$ follows from generalized Nelson's analytic vector theorem, see ref.[17]-[18]. We can draw more information from (7.3). If we write $H_\kappa(g) + b = \mathrm{Ext} - \int_{\mathbb{R}^d \setminus C} \lambda d^\# E_\#(\lambda, \kappa)$, then (7.2) is equivalent to $\psi \in \mathrm{Range}\{E_\#(d, \kappa)\}$ and (7.3) gives that

$$\varphi_{\kappa,g}^{\#}(f) \mathrm{Range}\{E_\#(d, \kappa)\} \subset \mathrm{Range}\{E_\#(d + b, \kappa)\}. \qquad (7.4)$$

Because $\varphi_{\kappa,g}^{\#}(f)$ is self-#-adjoint we have

$$\varphi_{\kappa,g}^{\#}(f) \mathrm{Range}\{(I - E_{\#}(d, \kappa))\} \subset \mathrm{Range}\{(I - E_{\#}(d - b, \kappa))\}. \qquad (7.5)$$
These two inclusions have simple physical interpretations. We imagine that $\phi_{\kappa,g}^{\#}(f)$ is written as a sum of two operators, one creating physical wave packets associated with $H_{\kappa}(g)$, and the other annihilating them. Because of (7.1) the wave packets have energy at most $\beta$, and so $\phi_{\kappa,g}^{\#}(f)$ can increase or decrease the total energy $H_{\kappa}(g)$, by at most $\beta$.

We note that $\phi_{\kappa,g}^{\#}(f)$ is essentially self-#-adjoint on the domain $U_l \text{Range}\{E_{\#}(l, \kappa)\}$, by the proof of Lemma 7.2 and the remarks above. Our next step is to take #-limits with respect to $f$ in the resolvents $R = R(f, z) = (\phi_{\kappa,g}^{\#}(f) - z)^{-1}$. As preparation, we now prove that $R$ preserves #-regularity, which means

$$ \|(H_{\kappa}(g) + b)^{1/2} R(f, z) \psi\|_{\#} \le M \|(H_{\kappa}(g) + b)^{1/2} \psi\|_{\#}. \quad (7.6) $$

**Lemma 7.3.** Let $f$ be real and satisfy (7.1). Then the estimate (7.6) holds for $\text{Im } z \ne 0$. The constants $M$ and $b$ depend only on $z, g$, and $|f|_1$.

**Proof** To prove this lemma, we obtain uniform estimates on approximating operators $R_n$. If (7.6) holds for $R_n$, with $M$ independent of $n \in *\mathbb{N}$, and

$$ R = \text{strong-#-lim } R_n, \quad (7.7) $$

then (7.6) also holds for $R$. In fact $\|(H_{\kappa}(g) + b)^{1/2} \psi\|_{\#}$ defines a #-norm on the domain $D((H_{\kappa}(g) + b)^{1/2}) = \mathcal{H}_1$, which makes it into a non-Archimedean Hilbert space. The inequality (7.6) in equivalent to $R_n$, being a bounded operator on $\mathcal{H}_1$, and the #-norm $\|R_n\|_{\#1,1}$, of $R_n$, as an operator from $\mathcal{H}_1$ to itself is defined by

$$ \|R_n\|_{\#1,1} = \|(H_{\kappa}(g) + b)^{1/2} R_n (H_{\kappa}(g) + b)^{-1/2}\|_{\#} \le M. \quad (7.8) $$

From the strong #-convergence (7.7) on $\mathcal{F}^{\#}$, we conclude that on a #-dense set of vectors in $\mathcal{H}_1$, $R_n$ #-converges weakly to $R$. Since the operators $R_n, n \in *\mathbb{N}$ are uniformly bounded on $\mathcal{H}_1$, $R_n \to_{\#} R$ in weak operator #-convergence on $\mathcal{H}_1$. Thus the #-norm $\|R_n\|_{\#1,1}$ is bounded by the #-lim sup of the $\|R_n\|_{\#1,1}$ and (7.6) holds for $R$. Let

$$ H_{\kappa}(g) + b = \text{Ext-} \int_{\mathbb{R}_c^{\#}} \lambda d^{\#} E_{\#}(\lambda, \kappa). \quad (7.9) $$

We approximate $\phi_{\kappa,g}^{\#}(f)$ by the bounded self-#-adjoint operator $C_n = E_{\#}(n, \kappa)\phi_{\kappa,g}^{\#}(f)E_{\#}(n, \kappa)$, $n \in *\mathbb{N}$. From (7.4) it is clear that $C_n \to_{\#} \phi_{\kappa,g}^{\#}(f)$ on vectors with #-compact support in the energy. Since $\phi_{\kappa,g}^{\#}(f)$ is essentially self-#-adjoint on this domain, the resolvents also #-converge strongly [18]

$$ \#\text{-lim}_{n \to *\infty} R_n(z) = \#\text{-lim}_{n \to *\infty} (C_n - z)^{-1} = R(z), $$

proving (7.7). We now show that (7.8) holds and it is sufficient to prove

$$ \|(H_{\kappa}(g) + b)^{1/2} R_n (H_{\kappa}(g) + b)^{-1/2} \psi\|_{\#} \le M \|\psi\|_{\#} \quad (7.10) $$

for $\psi$ in the #-dense set $D_{\kappa,g}$. Since $(H_{\kappa}(g) + b)^{-1/2}$ and $R_n$ both map $D_{\kappa,g}$ onto $D_{\kappa,g}$ we need only prove that on the domain $D_{\kappa,g} \times D_{\kappa,g}$,

$$ \begin{aligned} H_{\kappa}(g) + b &\le M^2(C_n - \bar{z})(H_{\kappa}(g) + b)(C_n - z) = \\ &= M^2(C_n - x)(H_{\kappa}(g) + b)(C_n - x) + (My)^2(H_{\kappa}(g) + b) + iM^2y[H_{\kappa}(g), C_n], \end{aligned} \quad (7.11) $$

where $z = x + iy$. As the first term is positive, it is sufficient to show that

$$ 0 \le [(My)^2 - 1](H_{\kappa}(g) + b) + M^2y[iH_{\kappa}(g), C_n]. \quad (7.12) $$
But

$$[iH_{\kappa}(g), C_n] = E_{\#}(n, \kappa)[iH_{\kappa}(g), \varphi_{\kappa,g}^{\#}(f)]E_{\#}(n, \kappa) = E_{\#}(n, \kappa)\pi_{\kappa,g}^{\#}(f)E_{\#}(n, \kappa).$$

By Theorem 6.3,

$$ \begin{aligned} |\langle\psi, [iH_{\kappa}(g), C_n]\psi\rangle_{\#}| &= |\langle E_{\#}(n, \kappa)\psi, \pi_{\kappa,g}^{\#}(f)E_{\#}(n, \kappa)\psi\rangle_{\#}| \le \|\psi\|_{\#} \cdot \|\pi_{\kappa,g}^{\#}(f)E_{\#}(n, \kappa)\psi\|_{\#} \le \\ &\le \|\psi\|_{\#} \cdot |f|_{\#1} \cdot \|(H_{\kappa}(g) + b)^{\frac{1}{2}} E_{\#}(n, \kappa)\psi\|_{\#} \le \frac{1}{2} |f|_{\#1} \cdot \{\varepsilon \langle\psi, (H_{\kappa}(g) + b)^{\frac{1}{2}}\psi\rangle_{\#} + \varepsilon^{-1} \langle\psi, \psi\rangle_{\#}\} \end{aligned} $$

for any $\varepsilon > 0$. Furthermore, the #-norm $|f|_{\#1}$ of (6.20) can be chosen independent of $b$ for large $b$, since for $b_1 \le b_2$ we have that $H_{\kappa}(g) + b_1 \le H_{\kappa}(g) + b_2$. Therefore (7.12) is valid as long as

$$0 \le \{(My)^2 - 1 - \frac{1}{2}M^2y\varepsilon |f|_{\#1}\} (H_{\kappa}(g) + b) - \frac{M^2y}{2\varepsilon} |f|_{\#1}.$$

For each $|f|_{\#1}, y \ne 0$, we can pic $M$ large so that $(My)^2 > 3$, $\varepsilon$ small enough so that $\frac{1}{2}M^2y\varepsilon |f|_{\#1} < 1$, and $b$ large enough so that the inequality is valid. This completes the proof.

We now show that the resolvents of approximate field operators #-converge. We use the spectral projections $E_{\#}(n, \kappa)$, $n \in \mathbb{N}$ defined by (7.9) to cut-off the field. If $\varphi_{\kappa,g}^{\#}(f)$ is a #-closed symmetric field operator, then $E_{\#}(n, \kappa)\varphi_{\kappa,g}^{\#}(f)E_{\#}(n, \kappa)$ is a bounded, self-#-adjoint approximation to $\varphi_{\kappa,g}^{\#}(f)$.

**Lemma 7.4.** Let $f_n, n \in \mathbb{N}$ be a hyper infinite sequence of real functions satisfying (7.1) with $\beta$ depending on $n$. If the graphs $G(\varphi_{\kappa,g}^{\#}(f_n))$ #-converge to the graph of a #-densely defined operator, if

$$\|(H_{\kappa}(g) + b)^{-\frac{1}{2}}\{\varphi_{\kappa,g}^{\#}(f_n) - \varphi_{\kappa,g}^{\#}(f_m)\}(H_{\kappa}(g) + b)^{-\frac{1}{2}}\|_{\#} \to_{\#} 0 \quad (7.13)$$

and if the #-norms $|f_n|_{\#1}$, are uniformly bounded, then the resolvents

$$R_n(z) = (C_n - z)^{-1} \quad (7.14)$$

of

$$C_n = E_{\#}(n, \kappa)\varphi_{\kappa,g}^{\#}(f_n)E_{\#}(n, \kappa). \quad (7.15)$$

#-converge strongly to the resolvent of a self-#-adjoint operator $C$.

**Proof** This result is a special case of [ref. [19], Th. 5 and Cor. 6]. See that paper for notation. Recall that #-measure $E_{\#}(n, \kappa)$ is defined by (4.3.9). Note that

$$\|(H_{\kappa}(g) + b)^{-\frac{1}{2}}(C_n - \varphi_{\kappa,g}^{\#}(f_n))(H_{\kappa}(g) + b)^{-\frac{1}{2}}\|_{\#} \le \quad (7.16)$$

$$ \begin{aligned} & \|(E_{\#}(n, \kappa) - I)(H_{\kappa}(g) + b)^{-\frac{1}{2}}\|_{\#} \cdot \{\|\varphi_{\kappa,g}^{\#}(f_n)(H_{\kappa}(g) + b)^{-\frac{1}{2}}\|_{\#} + \|(H_{\kappa}(g) + b)^{-\frac{1}{2}}\varphi_{\kappa,g}^{\#}(f_n)\|_{\#}\} \le \\ & \le n^{-1/2} \{\|\varphi_{\kappa,g}^{\#}(f_n)(H_{\kappa}(g) + b)^{-\frac{1}{2}}\|_{\#} + \|(H_{\kappa}(g) + b)^{-\frac{1}{2}}\varphi_{\kappa,g}^{\#}(f_n)\|_{\#}\}. \end{aligned} $$

By Theorem 6.3 the operator $\varphi_{\kappa,g}^{\#}(f_n)(H_{\kappa}(g) + b)^{-\frac{1}{2}}$ and its #-adjoint are bounded with

$$\|\varphi_{\kappa,g}^{\#}(f_n)(H_{\kappa}(g) + b)^{-\frac{1}{2}}\|_{\#} \le c |f_n|_{\#1},$$

which is bounded uniformly in $n, \in \mathbb{N}$. Then
$$ \|(H_\kappa(g) + b)^{-\frac{1}{2}}(C_n - \varphi_{\kappa,g}^{\#}(f_n))(H_\kappa(g) + b)^{-\frac{1}{2}}\|_{\#} = O(n^{-1/2}) \quad (7.17) $$

and so by (7.13),

$$ \|(H_\kappa(g) + b)^{-\frac{1}{2}}(C_n - C_m)(H_\kappa(g) + b)^{-\frac{1}{2}}\|_{\#} \to_{\#} 0 $$

as $n, m \to *\infty$. The required uniform boundedness of the resolvents $\|(C_n - z)^{-1}\|_{\#1,1} < \text{const}$ follows from Lemma 7.3.

We now discuss when the hypotheses of Lemma 7.4 are satisfied. If the $\varphi_{\kappa,g}^{\#}(f_n)$ #-converge strongly on a #-dense domain, then the graphs #-converge. The $\varphi_{\kappa,g}^{\#}(f_n)$ will #-converge on $D_{\kappa,g}$ if $f_n \to_{\#} f$ as $n \to *\infty$ in the #-norm $|\cdot|_{\#1}$; they will also #-converge for some hyper infinite sequence $f_n, n \in *\mathbb{N}$

$$ f_n \to_{\#} \text{Ext-} \sum_{i=1}^{i=n} f(\cdot, t_i) \delta^{\#} (t - t_i) \quad (7.18) $$

with $f(\cdot, t_i) \in S_{\text{fin}}^{\#} (*\mathbb{R}^{\#4})$. We can choose $f(\cdot, t)$ to have the form

$$ f(\mathbf{x}, t) = \text{Ext-} \sum_{i=1}^{i=n} f(\cdot, t_i) \delta_n (t - t_i), $$

where $\delta_n(t) \ge 0$ has support in $|t| < n^{-1}$, and $\text{Ext-} \int \delta_n(t) d^\# t$. For such a sequence $|f_n|_{\#1}$ is uniformly bounded in $n, n \in *\mathbb{N}$. From (6.9) we see that $w^*$ #-convergence of the $\delta_n(t)$ as bounded #-measures implies (7.13). Thus the hypotheses are satisfied for the sequence (7.18). They are also satisfied if the $f_n$ #-converge in the #-norm $|\cdot|_{\#1}$, and every $f_n$ with finite $|f_n|_{\#1}$ is the #-limit of such a hyper infinite sequence.

**Theorem 7.5.** Let $f$ be real and $|f|_{\#1}$ finite. Then the operator $\varphi_{\kappa,g}^{\#}(f)$ is self-#-adjoint and essentially self-#-adjoint on $D_{\kappa,g}$. A real linear combination of sharp time fields with real test functions in $S_{\text{fin}}^{\#} (*\mathbb{R}^{\#3})$, $\text{Ext-} \sum_{i=1}^{i=r} A_{\kappa,g}(t_i)$, is also essentially self-#-adjoint on $D_{\kappa,g}$.

**Proof** The two cases are similar and we only consider $\varphi_{\kappa,g}^{\#}(f)$. We first prove that the operator $C$ of Lemma 7.4 extends $\varphi_{\kappa,g}^{\#}(f)$

$$ \varphi_{\kappa,g}^{\#}(f) \subset C. \quad (7.19) $$

As in the proof of Theorem 6.6, we have

$$ \varphi_{\kappa,g}^{\#}(f) = (\varphi_{\kappa,g}^{\#}(f) \upharpoonright D_{\kappa})^{\#-} = (\varphi_{\kappa,g}^{\#}(f) \upharpoonright D_0)^{\#-} \quad (7.20) $$

where $(A)^{\#-}$ is #-closure of the operator $A$ and $D_0 = D(H_\kappa^{1/2}) \cap D(N_\kappa)$. Let $R_n(z)$ be defined by (7.14), where $f_n$ approximates $f$ and satisfies the hypotheses of Lemma 7.4. Thus $R(z) = \# - \lim_{n \to *\infty} R_n(z)$ exists and is the resolvent of a self-#-adjoint operator $C$. For $\psi \in D_{\kappa,g}$, $C_n\psi = E(n) \varphi_{\kappa,g}^{\#}(f)E(n)\psi \to_{\#} \varphi_{\kappa,g}^{\#}(f)\psi$, and #-convergence can be shown on $D((H(g) + b)^{3/2})$. For $\chi = (\varphi_{\kappa,g}^{\#}(f) - z)\psi$,

$$ R(z)\chi = \# - \lim_{n \to *\infty} R_n(z)\chi = \# - \lim_{n \to *\infty} R_n(z)(C_n - z)\psi = \psi. $$

Thus we obtain that $(C - z)^{-1}(\varphi_{\kappa,g}^{\#}(f) \upharpoonright D_g) = I \upharpoonright D_g$. And therefore by (7.20), (7.19) is valid.

We now show that $\varphi_{\kappa,g}^{\#}(f)$ is equal to $C$, which completes the proof. We need only show that if $\psi \in D(C)$, then $\psi \in D(\varphi_{\kappa,g}^{\#}(f))$. We first notice that

$$ R(z)D((H_\kappa(g) + b)^{\frac{1}{2}}) \subset D((H_\kappa(g) + b)^{\frac{1}{2}}) \quad (7.21) $$
and that (7.8) is valid for $R(z)$. The argument for this is the same as the proof of Lemma 7.3, but the approximate operator $C_n = E_\#(n, \kappa)\varphi_{\kappa,g}^\#(f_n)E_\#(n, \kappa)$ replaces the $C_n$. of the former proof. The remaining calculation is the same since the $|f_n|_{\#1}, n \in \mathbb{N}$, are assumed uniformly bounded. We now introduce the smoothing operator

$$P_j = \left(1 + \frac{1}{j}(H_\kappa(g) + b)^2\right)^{-1} \qquad (7.22)$$

with the properties $\|P_j\|_{\#} \le 1$

$$\text{strong } \#\text{-lim}_{n \to \infty} P_j = I, \qquad (7.23)$$

and for $r < 2$,

$$\|P_j(H_\kappa(g) + b)^r\|_{\#} = \|(H_\kappa(g) + b)^r P_j\|_{\#} \le j^{\frac{r}{2}}. \qquad (7.24)$$

Let $\psi \in D(C)$ and $\psi = R(z)\chi$. Then $P_j\psi = \psi_j \to_{\#} \psi$, as $j \to \infty$ and $\psi_j \in D(H_\kappa(g)) \subset D(\varphi_{\kappa,g}^\#(f))$. If $\varphi_{\kappa,g}^\#(f)\psi_j$ #-converges and then $\psi$ is in the domain of the #-closed operator $\varphi_{\kappa,g}^\#(f)$, so we prove this

$$(\varphi_{\kappa,g}^{\#}(f) - z)\psi_j = (\varphi_{\kappa,g}^{\#}(f) - z) P_j \psi = (\varphi_{\kappa,g}^{\#}(f) - z) P_j R(z) \chi = \qquad (7.25) \\ = (\varphi_{\kappa,g}^{\#}(f) - z) R(z) P_j \chi + (\varphi_{\kappa,g}^{\#}(f) - z) [P_j, R(z)] \chi.$$

The last equality is valid since $P_j\chi \in D(H_\kappa(g)^2) \subset D((H_\kappa(g) + b)^{1/2})$ and by (7.21)

$$R(z) P_j \chi \in D \left( (H_\kappa(g) + b)^{\frac{1}{2}} \right) \subset D \left( \varphi_{\kappa,g}^{\#}(f) \right).$$

Since $C$ extends $\varphi_{\kappa,g}^\#(f)$, $(\varphi_{\kappa,g}^\#(f) - z)\psi_j = P_j\chi + (\varphi_{\kappa,g}^\#(f) - z)[P_j, R(z)]\chi$. As $P_j\chi \to_{\#} \chi = (C-z)\psi$ to conclude that $\psi \in D(\varphi_{\kappa,g}^\#(f))$, we need to show that

$$\Lambda_j = (\varphi_{\kappa,g}^{\#}(f) - z) [P_j, R(z)] \chi \to_{\#} 0. \qquad (7.26)$$

We now claim that

$$\Lambda_j = \#\text{-lim}_{n \to \infty} (C_n - z) [P_j, R_n(z)] \chi, \qquad (7.27)$$

where $C_n$ and $R_n$ are defined in (7.14)-(7.15). Since $(C_n - z)R_n(z) = (C - z)R(z) = I$, we need only prove the existence of the limit (7.27) with the commutator removed. As observed in the first part of the proof, for

$\psi \in D((H_\kappa(g) + b)^{\frac{3}{2}})$, $\|C_n\psi - C\psi\|_{\#} = \|C_n\psi - \varphi_{\kappa,g}^\#(f)\psi\|_{\#} \to_{\#} 0$. Since $P_jR(z)\chi \in D(H_\kappa(g)^2) \subset D((H_\kappa(g) + b)^{3/2})$, as $n \to \infty$, $(C_n - z)P_jR(z)\chi \to_{\#} (\varphi_{\kappa,g}^\#(f) - z)P_jR(z)\chi$. Also $R_n(z)\chi \to_{\#} R(z)\chi$, and by Theorem 6.3 and (7.24), $\|(C_n - z)P_j\|_{\#} \le \text{const} \cdot |f_n|_{\#1} \cdot j^{\frac{1}{4}}$, which is bounded uniformly in $n \in \mathbb{N}$. Therefore

$$(C_n - z) P_j R_n(z) \chi \to_{\#} (\varphi_{\kappa,g}^{\#}(f) - z) P_j R(z) \chi,$$

and (7.27) is established. Thus $\Lambda_j = \#\text{-lim}_{n \to \infty} \Lambda_{j,n}$, where

$$\Lambda_{j,n} = (C_n - z) [P_j, R_n(z)] \chi = (C_n - z) R_n(z) P_j [P_j^{-1}, (C_n - z)] P_j R(z) \chi = \\ = j^{-1} P_j [(H_\kappa(g) + b)^2, (C_n - z)] P_j R(z) \chi =$$
$$
\begin{align*}
&= j^{-1} P_j\{(H_\kappa(g) + b)[H_\kappa(g), C_n] + [H_\kappa(g), C_n](H_\kappa(g) + b)\} P_j R(z)\chi = \\
&= -ij^{-1} P_j\{(H_\kappa(g) + b)E_\#(n, \kappa)\pi_{\kappa,g}^\#(f_n)E_\#(n, \kappa) + E_\#(n, \kappa)\pi_{\kappa,g}^\#(f_n)E_\#(n, \kappa)(H_\kappa(g) + \\
&\qquad b)\}P_j R(z)\chi.
\end{align*}
$$

Now by (7.24) we obtain

$$
\| P_j (H_\kappa(g) + b) \|_{\#} = \|(H_\kappa(g) + b)P_j \|_{\#} \leq j^{-1/2}
$$

and

$$
\begin{align*}
& \| E_{\#}(n, \kappa) \pi_{\kappa,g}^{\#}(f_n) E_{\#}(n, \kappa) P_j \|_{\#} + \| P_j E_{\#}(n, \kappa) \pi_{\kappa,g}^{\#}(f_n) E_{\#}(n, \kappa) \|_{\#} \le \text{const} \cdot |f_n|_{\#1} \cdot j^{1/4} \\
& \le \text{const} \cdot j^{1/4}
\end{align*}
$$

as the $|f_n|_{\#1}$ are assumed uniformly bounded. The constant is independent of $j$ and $n$. Therefore
$$ \| \Lambda_{j,n} \|_{\#} \le \text{const} \cdot j^{-1/4}, \text{ and} $$

$$
\#\text{-lim}_{j \to \infty} \| \Lambda_j \|_{\#} \le \#\text{-lim}_{n \to \infty} \left( \#\text{-lim}_{n \to \infty} \| \Lambda_{j,n} \|_{\#} \right) = 0.
$$

Thus (4.3.26) is established and the proof is complete.

8. The field as a tempered distribution in S#( *R#⁴).

In the previous sections we studied the quantum field $\varphi_{\kappa,g}^{\#}(f)$ corresponding to the Hamiltonian $H_{\kappa}(g)$. We found that if $\varphi_{\kappa,g}^{\#}(f)$ is localized, namely if $f$ has finitely bounded #-compact support in $B$ and $H_{\kappa}(g)$ is a Hamiltonian for $B$, then $\varphi_{\kappa,g}^{\#}(f) = \varphi_{\kappa}^{\#}(f)$ is independent of the spatial cut-off $g$. In this section we show that there is a cut-off independent field $\varphi_{\kappa}^{\#}(f)$ defined for all $f \in S_{\text{fin}}^{\#}(\mathbb{R}^{\#4})$, and $\varphi_{\kappa}^{\#}(f)$ agrees with the previous one when $f$ has #-compact support. The domain of $\varphi_{\kappa}^{\#}(f)$ includes $D_{0,\kappa} = D(H_{0,\kappa}^{1/2}) \cap D(N_{0,\kappa})$, and on this domain $\varphi_{\kappa}^{\#}(f)$ is a tempered distribution in $S_{\text{fin}}^{\#}(\mathbb{R}^{\#4})$.

**Lemma 8.1.** Let $D_{0,\kappa} = D(H_{0,\kappa}^{1/2}) \cap D(N_{0,\kappa})$. For $\psi \in D_{0,\kappa}$, $\langle \psi, \varphi_{\kappa}^{\#}(\mathbf{x}, t)\psi \rangle_{\#}$ is a #-continuous, polynomially bounded function and

$$
| \mathrm{Ext} - \int_{\mathbb{R}^{\#3}} \langle \psi, \varphi_{\kappa}^{\#}(\mathbf{x}, t) \psi \rangle_{\#} D_{\mathbf{x}}^{\#'} f(\mathbf{x}, t) d^{\#3} \mathbf{x} | \le O(t) \|f(\cdot, t)\|_{\#2} \langle \psi, (H_{0,\kappa} + N_{0,\kappa}^2 + I)\psi \rangle_{\#}. \quad (8.1)
$$

**Proof** We divide space time into a number of similar regions with a partition of unity. Let $\xi(\mathbf{x}, t)$, $\mathbf{x} =$
$(x_1, x_2, x_3)$, be a $C^{*\infty}$ function satisfying

$$
0 \le \xi(\mathbf{x}, t) \le 1, \tag{8.2}
$$

$$
\operatorname{supp}(\xi) \subset \{(\mathbf{x}, t) \mid |\mathbf{x}| \le 1, |t| \le \xi\}, \quad (8.3)
$$

and such that

$$
\mathrm{Ext} - \sum_{ij} \xi_{ij}(\mathbf{x}, t) = \mathrm{Ext} - \sum_{ij} \xi(x_1 - i, x_2 - i, x_3 - i, t - j) = 1. \quad (8.4)
$$

Thus if $f(\mathbf{x}, t) \in S_{\text{fin}}^{\#}(\mathbb{R}^{\#4})$,

$$
f = \mathrm{Ext} - \sum_{ij} f_{ij}(\mathbf{x}, t) = \mathrm{Ext} - \sum_{ij} f(\mathbf{x}, t) \xi_{ij}(\mathbf{x}, t) \quad (8.5)
$$

with $f_{ij}(\mathbf{x}, t)$ a $C^{*\infty}$ function with support in the cube

$$
B_{ij} = \{(\mathbf{x}, t) \mid |x_1 - i| \le 1, |x_2 - i| \le 1, |x_3 - i| \le 1, |t - j| \le 1\}. \quad (8.6)
$$
We also pick a $C^*\infty$ function $g_0(\mathbf{x})$ such that

$$g_0(\mathbf{x}) = \lambda, \text{ if } |\mathbf{x}| \le 2, \qquad (8.7)$$

and

$$g_0(\mathbf{x}) = 0, \text{ if } |\mathbf{x}| \ge 3. \qquad (8.8)$$

Thus $H_\kappa(g_{ij})$ is a Hamiltonian for $B_{ij}$ when

$$g_{ij}(\mathbf{x}) = g_0\left(\frac{x_1-i}{1+|j|}, \frac{x_2-i}{1+|j|}, \frac{x_3-i}{1+|j|}\right). \qquad (8.9)$$

Furthermore

$$\|(I + N_{0,\kappa})^{-1} H_{l,\kappa,g_{ij}} (I + N_{0,\kappa})^{-1}\|_{\#} = O(j) \qquad (8.10)$$

as the kernels of operators contributing to $H_{l,\kappa,g_{ij}}$ have $L_2^\#$-norms with are $O(j)$. For $(\mathbf{x}, t) \in B_{ij}$ and $\psi \in D_{0,\kappa}$, we have by Lemma 6.1, that $\langle \psi, \varphi_\kappa^\#(\mathbf{x}, t)\psi\rangle_\#$ is $\#$-continuous and

$$|\langle \psi, \varphi_\kappa^\#(\mathbf{x}, t) \psi \rangle_\#| \le \text{const} \cdot \langle \psi, (H_\kappa(g_{ij}) + \tilde{b}(\kappa)) \psi \rangle_\# \qquad (8.11)$$

where the constant is independent of $\mathbf{x}, t, i$, and $j$. Here $\tilde{b}(\kappa)$ is hyperfinite constant proportional to the lower bound $b(\kappa)$ of $H_\kappa(g_{ij})$, see (2.19). Note that that the lower bound of $H_\kappa(g_{ij})$, is proportional to the diameter of the support of $g_{ij}$, namely $O(j)$). Thus (8.11) gives the bound for $(\mathbf{x}, t) \in B_{ij}, \psi \in D_{0,\kappa}$

$$|\langle \psi, \varphi_\kappa^\#(\mathbf{x}, t) \psi \rangle_\#| \le \text{const} \cdot \|\| H_{0,\kappa}^{1/2} \psi \|_\#^2 + \|(I + N_{0,\kappa})\|_\#^2 \cdot \|(I + N_{0,\kappa})^{-1} H_{l,\kappa,g_{ij}} (I + N_{0,\kappa})^{-1} \|_\# + \tilde{b}(\kappa) \|\psi\|_\#^2 \le \text{const} \cdot \|f(\cdot, t)\|_{\#2} \langle \psi, (H_{0,\kappa} + N_{0,\kappa}^2 + I)\psi \rangle_\# \cdot b(\kappa) \cdot O(j), \qquad (8.12)$$

$$\text{const} \cdot \left\{ \| H_{0,\kappa}^{1/2} \psi \|_{\#}^2 + \|(I + N_{0,\kappa})\|_{\#}^2 \cdot \|(I + N_{0,\kappa})^{-1} H_{l,\kappa,g_{ij}} (I + N_{0,\kappa})^{-1} \|_{\#} + \tilde{b}(\kappa) \|\psi\|_{\#}^2 \right\} \le \text{const} \cdot \|f(\cdot, t)\|_{\#2} \langle \psi, (H_{0,\kappa} + N_{0,\kappa}^2 + I)\psi \rangle_{\#} \cdot b(\kappa) \cdot O(j),$$

by (8.10) and the above discussion of $\tilde{b}(\kappa)$. Since $O(j) = O(|t|)$, we have proved polynomial boundedness. Thus, as in Lemma 6.1,

$$|\text{Ext} - \int_{\mathbb{R}^{\#3}} \langle \psi, \varphi_{\kappa}^{\#}(\mathbf{x}, t) \psi \rangle_{\#} D_{\mathbf{x}}^{\#} f(\mathbf{x}, t) d^{\#3}x| \le O(t) \|f(\cdot, t)\|_{\#2} \langle \psi, (H_{0,\kappa} + N_{0,\kappa}^2 + I)\psi \rangle_{\#},$$

which yields (8.1). We now define the sharp time fields

$$A_\kappa(f, t) = \text{Ext} - \int_{\mathbb{R}^{\#3}} \varphi_\kappa^\#(\mathbf{x}, t) f(\mathbf{x}, t) d^{\#3}x \qquad (8.13)$$

and

$$B_\kappa(f, t) = \text{Ext} - \int_{\mathbb{R}^{\#3}} \pi_\kappa^\#(\mathbf{x}, t) f(\mathbf{x}, t) d^{\#3}x. \qquad (8.14)$$

**Lemma 8.2** Let $f(\mathbf{x}, t) \in S_{\text{fin}}^{\#}(*\mathbb{R}^{\#4})$ be real. Then $A_\kappa(f, t)$ and $B_\kappa(f, t)$ define self-$\#$-adjoint operators, and their domain includes $D_{0,\kappa}$. For $\psi \in D_{0,\kappa}$, $\|A_\kappa(f, t)\psi\|_{\#} + \|B_\kappa(f, t)\psi\|_{\#} \le |f(\cdot, t)|_{\#2} \cdot \|F_\kappa\psi\|_{\#}$, where

$$F_\kappa = (H_{0,\kappa} + N_{0,\kappa}^2 + I)^{1/2},$$

and $|f(\cdot, t)|_{\#2} = c(1 + |t|) \{\|f(\cdot, t)\|_{\#2} + \|D_{\mathbf{x}}^{\#'} f(\cdot, t)\|_{\#2}\}$.

**Proof** The proof is similar to that of Lemmas 6.2 and 8.1.

**Theorem 8.3** Let $f(\mathbf{x}, t) \in S_{\text{fin}}^{\#}(*\mathbb{R}^{\#4})$ be a real function in $S_{\text{fin}}^{\#}$. The vectors $A_\kappa(t)\psi$ and $B_\kappa(t)\psi$, where
$\psi \in D_{0,\kappa}$, are #-continuous and rapidly decreasing in $t$. Their integrals over $t$ exist and define #-closed symmetric operators $\varphi_{\kappa}^{\#}(f)$ and $\pi_{\kappa}^{\#}(f)$ with domains containing $D_{0,\kappa}$. The fields $\varphi_{\kappa}^{\#}(f)$, $\pi_{\kappa}^{\#}(f)$, $A_{\kappa}(f, t)$ and $B_{\kappa}(f, t)$ are all independent of $g(\mathbf{x})$. For any vector $\psi \in D_{0,\kappa}$ we have

$$ \| \varphi_{\kappa}^{\#}(f) \psi \|_{\#} + \| \pi_{\kappa}^{\#}(f) \psi \|_{\#} \le |f|_{\#2} \cdot \| F_{\kappa} \psi \|_{\#}, $$

where $|f|_{\#2} = \text{Ext-} \int |f(\cdot, t)|_{\#2} d^{\#}t$ and $F_{\kappa} = (H_{0,\kappa} + N_{0,\kappa}^2 + I)^{1/2}$.

**Proof** This proof is based on the proofs of Lemma 8.1, Theorem 6.3, and Theorem 6.6. The fields $\varphi_{\kappa}^{\#}(f)$ and $\pi_{\kappa}^{\#}(f)$ are defined as their #-closures on $D_{0,\kappa}$.

## 9. Locality

In this section we derive locality of the field operators. Locality means that two field operators $\varphi_{\kappa}^{\#}(f)$ and $\varphi_{\kappa}^{\#}(h)$ commute provided the supports of $f$ and $h$ are spacelike separated. In other words, whenever

$$ (\mathbf{x}, t) \in \text{supp}(f) \text{ and } (\mathbf{y}, s) \in \text{supp}(h), $$

we have that

$$ |\mathbf{x} - \mathbf{y}| > |t - s|. $$

Under this hypothesis a signal originating in $\text{supp}(f)$ (caused, for example, by the process of performing the measurement of $\varphi_{\kappa}^{\#}(f)$ cannot be recorded by the measurement of $\varphi_{\kappa}^{\#}(h)$. Thus one expects that the measurement of $\varphi_{\kappa}^{\#}(f)$ does not interfere with the measurement of $\varphi_{\kappa}^{\#}(h)$, and that the joint measurement of $\varphi_{\kappa}^{\#}(f)$ and $\varphi_{\kappa}^{\#}(h)$ can be performed in either order. The rigorous mathematical statement that the measurements can be performed in either order is that $\varphi_{\kappa}^{\#}(f)$ and $\varphi_{\kappa}^{\#}(h)$ commute. For any #-closed operator $A$, a #-core $D^{\#}(A)$ of $A$ is defined to be a #-dense domain contained in $D(A)$ such that $A = \#-(A \uparrow D)$.

Self-#-adjoint operators $A$ and $B$ commute if and only if for any spectral projection $E$ of $B$, and #-core $D$ of $A$, $ED \subset D(A)$ and for $\psi \in D$, $EA\psi = AE\psi$.

**Definition 9.1** (i) Let $\psi \in D^{\#}(A)$, we say that vector $\psi$ is a near standard vector if $\|\psi\|_{\#} \in \mathbb{R}_{\text{fin}}^{\#}$.

(ii) A near standard #-core $D_{\text{fin}}^{\#}(A)$ of $A$ is defined to be a subdomain $D_{\text{fin}}^{\#}(A) \subset D^{\#}(A)$ which contains all near standard vectors $\psi$ such that: (a) $\psi \in D^{\#}(A)$ and (b) vector $A\psi$ is a near standard vector.

(iii) A near standard domain $D_{\text{fin}}(A)$ of $A$ is defined to be a subdomain $D_{\text{fin}}(A) \subset D(A)$ which contains all near standard vectors $\psi$ such that: (a) $\psi \in D(A)$ and (b) vector $A\psi$ is a near standard vector.

**Definition 9.2** Self-#-adjoint operators $A$ and $B$ -commute on domain $D_{\text{fin}}^{\#}(A) \cap D_{\text{fin}}^{\#}(B)$ if for any near standard vector $\psi \in D_{\text{fin}}^{\#}(A) \cap D_{\text{fin}}^{\#}(B)$ the following condition holds $AB\psi \approx BA\psi$.

**Lemma 9.1** Self-#-adjoint operators $A$ and $B$ -commute on domain $D_{\text{fin}}^{\#}(A) \cap D_{\text{fin}}^{\#}(B)$ if and only if for any spectral projection $E_B$ of $B$, and near standard #-core $D_{\text{fin}}^{\#}(A)$ of $A$, $E_B D_{\text{fin}}^{\#}(A) \subset D_{\text{fin}}(A)$ and for all $\psi \in D_{\text{fin}}^{\#}(A)$, $E_B A \psi \approx A E_B \psi$.

**Theorem 9.1** If $\text{supp}(f)$ and $\text{supp}(h)$ are spacelike separated, $\varphi_{\kappa}^{\#}(f)$ and $\varphi_{\kappa}^{\#}(h) \approx$ -commute.

**Proof** Let

$$ A_{\kappa,g}(f,s) = \text{Ext-} \int_{\mathbb{R}_{\text{fin}}^{\#3}} \varphi_{\kappa}^{\#}(\mathbf{x}, t) f(\mathbf{x}, t) d^{\#3}x $$

and

$$ B_{\kappa,g}(h,t) = \text{Ext-} \int_{\mathbb{R}_{\text{fin}}^{\#3}} \pi_{\kappa}^{\#}(\mathbf{x}, t) f(\mathbf{x}, t) d^{\#3}x $$

be the sharp time fields obtained from the test functions $f$ and $h$ correspondingly. First we prove that $A_{\kappa,g}(f, t)$ and $B_{\kappa,g}(h, t)$ commute. For any #-open set $\mathfrak{I}$ in space, we define the algebra $\mathfrak{C}^{\#}(\mathfrak{I})$ as the weak #-closure of the finitely bounded functions of the $t=0$ fields

$$ \text{Ext-} \int_{\mathbb{R}_{\text{fin}}^{\#3}} \varphi_{\kappa}^{\#}(\mathbf{x}, 0) f_0(\mathbf{x}) d^{\#3}x \text{ and } \text{Ext-} \int_{\mathbb{R}_{\text{fin}}^{\#3}} \pi_{\kappa}^{\#}(\mathbf{x}, 0) f_0(\mathbf{x}) d^{\#3}x $$
as $f_0$ runs over the $C^{+\infty}$-functions with support in $\mathfrak{I}$. If $\mathfrak{I}_1$ and $\mathfrak{I}_2$ are disjoint #-open sets, then elements of $\mathfrak{C}^\#(\mathfrak{I}_1)$ and $\mathfrak{C}^\#(\mathfrak{I}_2)$ commute, and it was shown in [17] that

$$ [\mathrm{Ext}\-\exp(-i\sigma H_\kappa(g))]\mathfrak{C}^\#(\mathfrak{I})[\mathrm{Ext}\-\exp(-i\sigma H_\kappa(g))] \subset \mathfrak{C}^\#(\mathfrak{I}_\sigma), \quad (9.1) $$

where $\mathfrak{I}_\sigma$ is the set of all points in space with distance less than $|\sigma|$ from $\mathfrak{I}$. The proof in [17] is valid whether or not $g(x) = \text{const}$ on the set $\mathfrak{I}$. If $\mathfrak{I}_1$ is a small neighbourhood of supp $(f) \cap \{\text{time} = s\}$ and $\mathfrak{I}_2$ is similarly defined with respect to $h$ at time $t$, then $\mathfrak{I}_1$ and $(\mathfrak{I}_2)_{t-s}$ are disjoint. Since the finitely bounded functions of $A_{\kappa,g}(f,s)$ belong to $[\mathrm{Ext}\-\exp(-is H_\kappa(g))]\mathfrak{C}^\#(\mathfrak{I}_1)[\mathrm{Ext}\-\exp(-is H_\kappa(g))]$

and the bounded functions of $B_{\kappa,g}(h,t)$ belong to

$$ [\mathrm{Ext}\-\exp(-it H_\kappa(g))]\mathfrak{C}^\#(\mathfrak{I}_2)[\mathrm{Ext}\-\exp(-it H_\kappa(g))] \subset \\ \subset [\mathrm{Ext}\-\exp(-it H_\kappa(g))]\mathfrak{C}^\#((\mathfrak{I}_2)_{s-t})[\mathrm{Ext}\-\exp(-it H_\kappa(g))], $$

$A_{\kappa,g}(f,s)$ and $B_{\kappa,g}(h,t) \approx$ -commute. Let $E$ be a spectral projection of $A_{\kappa,g}(f,s)$ and let $\psi \in D_{\kappa,g,\text{fin}}$ a near standard #-core for $\varphi_{\kappa,g}^{\#}(h)$. Then $E\psi \in D_{\text{fin}}(B_{\kappa,g}(h,t))$ for all $t$ and

$$ \langle \varphi_{\kappa,g}^{\#}(h)\theta, E\psi \rangle_{\#} \approx \mathrm{Ext}\-\int \langle\theta, B_{\kappa,g}(h,t)E\psi\rangle_{\#} d^{\#}t \approx \langle\theta, E[\mathrm{Ext}\-\int B_{\kappa,g}(h,t)\psi d^{\#}t]\rangle_{\#} \approx \langle\theta, E\varphi_{\kappa,g}^{\#}(h)\rangle_{\#} \quad (9.2) $$

for all $\theta \in D_{\kappa,g,\text{fin}}$. Thus

$$ E\psi \in D_{\text{fin}}(\varphi_{\kappa,g}^{\#}(h)^*) = D_{\text{fin}}(\varphi_{\kappa,g}^{\#}(h)), $$

$$ \varphi_{\kappa,g}^{\#}(h)E\psi \approx E\varphi_{\kappa,g}^{\#}(h)\psi, $$

and $A_{\kappa,g}(f,s) \approx$ -commutes with $\varphi_{\kappa,g}^{\#}(h)$. Now let $F$ be a spectral projection for $\varphi_{\kappa,g}^{\#}(h)$.

Then $F\psi \in D_{\text{fin}}(A_{\kappa,g}(f,s))$ for all $s$ and

$$ \langle \varphi_{\kappa,g}^{\#}(f)\theta, E\psi \rangle_{\#} \approx \mathrm{Ext}\-\int \langle\theta, A_{\kappa,g}(f,t)E\psi\rangle_{\#} d^{\#}t \approx \langle\theta, F[\mathrm{Ext}\-\int A_{\kappa,g}(h,t)\psi d^{\#}t]\rangle_{\#} \quad (9.3) $$

as before in (4.5.2), so that $E\psi \in D_{\text{fin}}(\varphi_{\kappa,g}^{\#}(f)^*) = D_{\text{fin}}(\varphi_{\kappa,g}^{\#}(f))$ and $\varphi_{\kappa,g}^{\#}(f)E\psi \approx E\varphi_{\kappa,g}^{\#}(f)\psi$. Therefore, $\varphi_{\kappa}^{\#}(f)$ and $\varphi_{\kappa}^{\#}(h) \approx$ -commute.

## 10. Space time covariance

Space time covariance means that the field transforms in the expected fashion under the space time translation $x' = (x'_1, x'_2, x'_3), t'$,

$$ \varphi_{\kappa}^{\#}(x, t) \rightarrow \varphi_{\kappa}^{\#}(x + x', t + t') \quad (10.1) $$

By its canonical definition the field transforms correctly under time translation. Let $U(x')$ be the unitary operator on $\mathcal{F}^\#$ which implements the free field space translation $x \rightarrow x + x'$. By definition, $U(x')$ acts on each vector $\theta_j(k_1, \dots, k_j)$ in the $j$ particle subspace $\mathcal{F}_j^\#$ by

$$ U(x')\theta(k_1, \dots, k_j) = (\mathrm{Ext}\-\exp[i\langle x', \sum_{i=1}^j k_i \rangle])\theta(k_1, \dots, k_j). \quad (10.2) $$

We use the convention that $U(x')$ is the Schrödinger picture operator. On a suitable domain,

$$ U(-x')a^*(k)U(x') = (\mathrm{Ext}\-\exp[i\langle k, x' \rangle])a^*(k), \quad (10.3) $$

$$ U(-x')a(k)U(x') = (\mathrm{Ext}\-\exp[i\langle k, x' \rangle])a(k), \quad (10.4) $$
and from the definition (2.10) we have

$$U(-\mathbf{x}')\varphi_{\mathbf{x}}^{\#}(\mathbf{x})U(\mathbf{x}') = \varphi_{\mathbf{x}}^{\#}(\mathbf{x} + \mathbf{x}'). \qquad (10.5)$$

Now $U(\mathbf{x}')$ does not commute with $H_\kappa(g)$, but in fact

$$U(-\mathbf{x}') H_\kappa(g) U(\mathbf{x}') = H_\kappa(g'),$$

where $g'(\mathbf{x}) = g(\mathbf{x} - \mathbf{x}')$. But $\varphi_{\mathbf{x}}^{\#}(\mathbf{x}, t)$ is independent of the space cutoff function $g(\mathbf{y})$ provided that $g(\mathbf{y}) = \lambda$ for $|\mathbf{x} - \mathbf{y}| > |t|$. Thus if $g(\mathbf{y}) = \lambda$ for $|\mathbf{x} + \mathbf{x}' - \mathbf{y}| > |t|$,

$$\begin{aligned} U(-\mathbf{x}')\varphi_{\mathbf{x}}^{\#}(\mathbf{x}, t)U(\mathbf{x}') &= U(-\mathbf{x}')[\mathrm{Ext}\text{-exp}(it H_{\kappa}(g))]\varphi_{\mathbf{x}}^{\#}(\mathbf{x})[\mathrm{Ext}\text{-exp}(-it H_{\kappa}(g))]U(\mathbf{x}') = \\ &= [\mathrm{Ext}\text{-exp}(it H_{\kappa}(g'))]\varphi_{\mathbf{x}}^{\#}(\mathbf{x} + \mathbf{x}')[\mathrm{Ext}\text{-exp}(-it H_{\kappa}(g'))] = \varphi_{\mathbf{x}}^{\#}(\mathbf{x} + \mathbf{x}', t) \end{aligned} \qquad (10.6)$$

on a suitable domain, for example on $D_{0,\kappa} \times D_{0,\kappa}$. Thus $U(\mathbf{x}')$ implements the space translation for fields $\varphi_{\mathbf{x}}^{\#}(f)$.

## 11. The algebra of local observables

To each #-open region $B \subset *\mathbb{R}_c^{\#4}$ of space time, we associate a non-Archimedean $C_{\#}^*$ algebra $\mathfrak{C}^{\#}(B)$ in such a way that the self-#-adjoint elements of $\mathfrak{C}^{\#}(B)$ are exactly the operators corresponding to experiments which may be performed in $B$.

**Definition 11.1** Let $A \in \mathfrak{C}^{\#}(B)$, we say that operator is near-standard if $\|A\|_{\#} \in *\mathbb{R}_{c,\mathrm{fin}}^{\#}$ and $\mathrm{st}(\|A\|_{\#}) \neq 0$. The sub algebra of the all near-standard operators in $\mathfrak{C}^{\#}(B)$ will be denoted by $\mathfrak{C}_{\approx}^{\#}(B)$.

**Definition 11.2** The $C^*$ algebra of standard local observables $\mathrm{st}(\mathfrak{C}_{\approx}^{\#}(B))$ is defined by

$$\mathrm{st}(\mathfrak{C}_{\approx}^{\#}(B)) = \{\mathrm{st}(A)|A \in \mathfrak{C}_{\approx}^{\#}(B)\}.$$

Remind that the requirements for a local quantum theory are: to each bounded open region $B$ of space time, there is an associated non-Archimedean $C_{\#}^*$ algebra $\mathfrak{C}^{\#}(B)$ containing the identity.

(a) Isotony: if $B_1 \supset B_2$, then $\mathfrak{C}^{\#}(B_1) \supset \mathfrak{C}^{\#}(B_2)$.

(b) Locality: $B_1$ and $B_2$ are space like separated, then $\mathrm{st}(\mathfrak{C}_{\approx}^{\#}(B_1))$ commutes with $\mathrm{st}(\mathfrak{C}_{\approx}^{\#}(B_2))$.

(c) The algebra of local observables $\mathfrak{C}^{\#}$ is defined as the #-norm #-closure of the union of the $\mathfrak{C}^{\#}(B)$.

(d) The algebra is primitive; in other words, it has a faithful, irreducible representation.

(e) Lorentz covariance: Let $\{a, \Lambda\}$ be an element of the inhomogeneous Lorentz group $L_+^\uparrow$. Then there is a representation $\sigma_{\{a, \Lambda\}}$ of $L_+^\uparrow$ by a group of $*$-automorphisms of $\mathfrak{C}^{\#}$, such that for a bounded region $B$

$$\sigma_{\{a, \Lambda\}} \mathfrak{C}_{\approx}^{\#}(B) \approx \mathfrak{C}_{\approx}^{\#}(\{a, \Lambda\}B). \qquad (11.1)$$

In this section we consider several possible definitions for the non-Archimedean algebra $\mathfrak{C}^{\#}(B)$. The different definitions undoubtedly lead to different $C_{\#}^*$ algebras. In order to arrive at a natural and aesthetic definition, we prove that all reasonable candidates for $\mathfrak{C}^{\#}(B)$ have the same weak #-closure; we take this weakly #-closed algebra as the definition of $\mathfrak{C}^{\#}(B)$.

**Definition 11.3** $\mathfrak{C}^{\#}(B)$ is the weakly #-closed operator algebra generated by the operators

$$\{F[\varphi_{\mathbf{x}}^{\#}(f)]|F \in L_{\infty}^{\#}, \mathrm{supp}(f) \subset B, |f|_{\#1} \in *\mathbb{R}_{c,\mathrm{fin}}^{\#}\}.$$

The definition is unchanged if we replace $L_{\infty}^{\#}$, by some non-Archimedean $*$-algebra which is #-dense in the weak operator topology. It is also unchanged if we replace the class of test functions by another (for example $D_{\mathrm{fin}}^{\#}(B)$) having the same #-closure in the $|\cdot|_{\#1}$ #-norm. In fact, if $|f_n - f|_{\#1} \to_\# 0$, then $(\varphi_{\mathbf{x}}^{\#}(f_n) -
$z)^{-1} \to_{\#} (\varphi_{\kappa}^{\#}(f) - z)^{-1}$ in the strong operator topology by Lemma 7.4 and by the generalized semigroup convergence theorem $Ext-\exp(i\varphi_{\kappa}^{\#}(f_n)) \to_{\#} Ext-\exp(i\varphi_{\kappa}^{\#}(f))$. Thus $Ext-\exp(i\varphi_{\kappa}^{\#}(f))$ and $F(\varphi_{\kappa}^{\#}(f))$ belong to the weak $\#$-closure if each $f_n, n \in \mathbb{N}$ is admitted as a test function in definition of non-Archimedean $C_{\#}^*$ algebra $\mathfrak{C}^{\#}(B)$. The same algebra $\mathfrak{C}^{\#}(B)$ is generated by the finitely bounded functions of sharp time fields $A_{\kappa}(t) = Ext-\int_{\mathbb{R}_{c}^{\#3}} \varphi_{\kappa}^{\#}(\mathbf{x}, t)f(\mathbf{x}, t)d^{\#3}\mathbf{x}, f \in D_{\text{fin}}^{\#}(B)$. In fact, using a hyper infinite sequence $f_n, n \in \mathbb{N}$ such as (4.3.18), we have the resolvents $\#\text{-converging } (\varphi_{\kappa}^{\#}(f_n) - z)^{-1} \to_{\#} (A_{\kappa}(t) - z)^{-1}$, and so $F(A_{\kappa}(t)) \in \mathfrak{C}^{\#}(B)$. Thus the sharp time fields generate a smaller algebra. However, if $f \in D_{\text{fin}}^{\#}(B)$, we can approximate $i\varphi_{\kappa}^{\#}(f_n)$ by the following hyperfinite sum $Ext-\sum_{i=1}^{i=n} A_{\kappa}(t_i)\Delta t_i$, with strong $\#\text{-convergence on } D_{g}^{\#}$. By Lemma 4.3.4 the resolvents $\#\text{-converge, so } F(\varphi_{\kappa}^{\#}(f))$ belongs to the weakly $\#\text{-closed algebra generated by the finitely bounded functions of hyperfinite linear combinations of the sharp time fields. We now see that all such } F(\varphi_{\kappa}^{\#}(f))$ belong to the algebra generated by the finitely bounded functions of the sharp time fields themselves. Let $F$ be a finitely bounded operator commuting with $A_{\kappa}(t_1), A_{\kappa}(t_2), \dots$, and $A_{\kappa}(t_n), n \in \mathbb{N}$. Then by the generalized spectral theorem, $F$ commutes with $Ext-\sum_{i=1}^{i=n} A_{\kappa}(t_i)\Delta t_i$ on the domain $D_{\kappa,g}^{\#}$, which by Theorem 4.3.5 is a $\#\text{-core for } Ext-\sum_{i=1}^{i=n} A_{\kappa}(t_i)\Delta t_i$. Thus $F$ commutes with $Ext-\sum_{i=1}^{i=n} A_{\kappa}(t_i)\Delta t_i$. Thus the commutant of $Ext-\sum_{i=1}^{i=n} A_{\kappa}(t_i)\Delta t_i$ is larger than that of $Ext-\sum_{i=1}^{i=n} A_{\kappa}(t_i)\Delta t_i$, and the double commutant smaller. Therefore, the sharp time fields generate $\mathfrak{C}^{\#}(B)$ as asserted.

**Theorem 11.1** With mentioned above definition of $\mathfrak{C}^{\#}(B)$, the axioms (a)-(f) are satisfied

## 12. Estimates on the interaction Hamiltonian

Let $\mathcal{F}^{\#}$ be the Pock space for a massive, neutral scalar Geld in two-dimensional space-time. The elements of $\mathcal{F}^{\#}$ are sequences of functions on momentum space. Let the annihilation and creation operators be normalized by the relation

$$ [a(\mathbf{k}), a^*(\mathbf{k}')] = \delta^{\#}(\mathbf{k} - \mathbf{k}'). \qquad (12.1) $$

Thus the free-field Hamiltonian is

$$ H_{0,\kappa} = Ext - \int_{|\mathbf{k}| \le \kappa} a^*(\mathbf{k}) a(\mathbf{k}) \omega(\mathbf{k}) d^{\#3} \mathbf{k}. \qquad (12.2) $$

The $t=0$ field with hyperfinite ultraviolet cut-off $\kappa$ is

$$ \varphi_{\kappa}^{\#}(\mathbf{x}) = Ext - \int_{|\mathbf{k}| \le \kappa} Ext-\exp(-i\langle\mathbf{k}, \mathbf{x}\rangle)[a^*(\mathbf{k}) + a(\mathbf{k})] d^{\#3}\mathbf{k} \qquad (12.3) $$

The spatially cut-off interaction Hamiltonian reads

$$ H_{I,\kappa}(g) = Ext - \int_{\mathbb{R}_{c}^{\#3}} \varphi_{\kappa}^{\#4}(\mathbf{x}) : g(\mathbf{x}) d^{\#3}x = \qquad (12.4) $$

$$ \sum_{j=0}^{4} \binom{4}{j} \{ Ext - \int_{|k_1| \le \kappa} d^{\#3}k_1 \cdots Ext - \int_{|k_m| \le \kappa} d^{\#3}k_m a^*(k_1) \cdots a^*(k_j) a(-k_{j+1}) \cdots \\ \times a(-k_4) \hat{g} \left( \sum_{i=1}^{4} k_i^{(1)}, \sum_{i=1}^{4} k_i^{(2)}, \sum_{i=1}^{4} k_i^{(3)} \right) \prod_{i=1}^{4} [\omega(k_i)]^{-1/2} d^{\#3}k_1 \cdots d^{\#3}k_4 \}, $$

where we let $\mathbf{k}_i = (k_i^{(1)}, k_i^{(2)}, k_i^{(3)})$, $i = 1,2,3$.

The total Hamiltonian reads

$$ H_{\kappa}(g) = H_{0,\kappa} + H_{I,\kappa}(g) \qquad (12.5) $$

We let

$$ N_{\kappa} = Ext - \int_{|\mathbf{k}| \le \kappa} a^*(\mathbf{k}) a(\mathbf{k}) d^{\#3} \mathbf{k}, \qquad (12.6) $$

and
$$D_{0,\kappa}^{\#} = \cap_{n=0}^{\infty} D(H_{0,\kappa}^{n}). \qquad (12.7)$$

**Theorem 12.1** For any $\epsilon \in \mathbb{R}_{\text{fin}+}^{\#}$ and for fixed $g(x) \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#3})$ there is a constant $b$ such that as bilinear forms on $D_{0,\kappa}^{\#} \times D_{0,\kappa}^{\#}$

$$- \left[ H_{0,\kappa'}^{\frac{1}{2}}, [H_{0,\kappa'}^{\frac{1}{2}}, H_{I,\kappa}(g)] \right] \leq \epsilon H_{0,\kappa}^{2} + b, \qquad (12.8)$$

$$- [N_{\kappa}, [N_{\kappa}, H_{I,\kappa}(g)]] \leq \epsilon N_{\kappa}^2 + b. \qquad (12.9)$$

**Theorem 12.2** Let $W: \mathcal{F}^{\#} \to \mathcal{F}^{\#}$ be an operator of the form

$$W = \text{Ext-} \int_{|k_1| \le \kappa} d^{\#3} k_1 \cdots \text{Ext-} \int_{|k_m| \le \kappa} d^{\#3} k_m w(k_1, \dots, k_m) a^*(k_1) \cdots a(-k_m), \qquad (12.10)$$

where $w(k_1, \dots, k_m) \in L_2^{\#}((\mathbb{R}_c^{\#3m}))$. Then

$$\| (N_{\kappa} + I)^{-j/2} W (N_{\kappa} + I)^{-(m-j)/2} \|_{\#} \le \text{const} \| w(k_1, \dots, k_m) \|_{L_2^{\#}}, \qquad (12.11)$$

$$\left\| (H_{0,\kappa} + I)^{-1} \left[ H_{0,\kappa'}^{\frac{1}{2}}, [H_{0,\kappa'}^{\frac{1}{2}}, W] \right] (H_{0,\kappa} + I)^{-1} (N_{\kappa} + I)^{-\frac{(m-4)}{2}} \right\|_{\#} \le \\ \le \text{const} \left\| \omega^{\frac{1}{2}} (\sum_{i=1}^{4} k_i^{(1)}, \sum_{i=1}^{4} k_i^{(2)}, \sum_{i=1}^{4} k_i^{(3)}) w(k_1, \dots, k_m) \right\|_{L_2^{\#}}, \qquad (12.12)$$

$$\left\| \left[ H_{0,\kappa'}^{\frac{1}{2}}, [H_{0,\kappa'}^{\frac{1}{2}}, W] \right] (N_{\kappa} + I)^{-m/2} \right\|_{\#} \le \text{const} \times \kappa^4 \left\| \sum_{i=1}^{m} \omega(k_i) w(k_1, \dots, k_m) \right\|_{L_2^{\#}}. \qquad (12.13)$$

**Theorem 12.3** Let the operator $W$ be as above. Then

$$\left\| \left[ H_{0,\kappa'}^{\frac{1}{2}}, [H_{0,\kappa'}^{\frac{1}{2}}, W] \right] (N_{\kappa} + I)^{-m/2} \right\|_{\#} \le \text{const} \|w(\mathbf{k}_1, \dots, \mathbf{k}_m)\|_{L_2^{\#}}. \qquad (12.14)$$

**Proof** of Theorem 12.1. Introduce the $t=0$ field $\varphi_{\mu}^{\#}(x)$ with an hyperfinite ultraviolet cut-off $\mu < \kappa$:

$$\varphi_{\mu}^{\#}(x) = \text{Ext-} \int_{|k| \le \mu} \text{Ext-}\exp(-i\langle \mathbf{k}, \mathbf{x} \rangle) [a^*(\mathbf{k}) + a(\mathbf{k})] d^{\#3}k$$

The spatially cut-off interaction Hamiltonian $H_{I,\mu}(g)$ corresponding to the $t=0$ field $\varphi_{\mu}^{\#}(x)$ reads

$$H_{I,\kappa}(g) = \text{Ext-} \int_{\mathbb{R}_c^{\#3}} \varphi_{\kappa}^{\#4}(\mathbf{x}): g(\mathbf{x}) d^{\#3}\mathbf{x}. \qquad (12.15)$$

Note that

$$H_{I,\kappa}(g) = \text{strong} \# - \lim_{\mu \to \#\kappa} H_{I,\mu}(g). \qquad (12.16)$$

If we write $H_{I,\kappa}(g)$ as a sum of five operators of the form $W$ in (12.10), then by Theorem 12.2 taken for the case $m=4$ we get

$$\left\| (H_{0,\kappa} + I)^{-1} \left[ H_{0,\kappa'}^{\frac{1}{2}}, [H_{0,\kappa'}^{\frac{1}{2}}, W] \right] (H_{0,\kappa} + I)^{-1} \right\|_{\#} \le$$
$$ \leq \text{const} \left\| \omega^{\frac{1}{2}} \left( \sum_{i=1}^{4} k_i^{(1)}, \sum_{i=1}^{4} k_i^{(2)}, \sum_{i=1}^{4} k_i^{(3)} \right) w(\mathbf{k}_1, \dots, \mathbf{k}_4) \right\|_{L_2^{\#}}. \quad (12.17) $$

Since the kernel $w(\mathbf{k}_1, \dots, \mathbf{k}_4)$ has an over-all factor $\hat{g}(\sum_{i=1}^4 k_i^{(1)}, \sum_{i=1}^4 k_i^{(2)}, \sum_{i=1}^4 k_i^{(3)})$, where $\hat{g}(\mathbf{k})$ is the Fourier transform of the spatial cut-off $g(\mathbf{x})$, the fast decrease of $\hat{g}(\mathbf{k})$ ensures that

$$ \omega^{\frac{1}{2}} \left( \sum_{i=1}^{4} k_i^{(1)}, \sum_{i=1}^{4} k_i^{(2)}, \sum_{i=1}^{4} k_i^{(3)} \right) w(\mathbf{k}_1, \dots, \mathbf{k}_4) \in L_2^{\#} $$

Thus the kernel for the corresponding cut-off interaction term $w_\mu$ approximates $w_\kappa$ in the sense that

$$ \left\| \omega^{\frac{1}{2}} \left( \sum_{i=1}^{4} k_i^{(1)}, \sum_{i=1}^{4} k_i^{(2)}, \sum_{i=1}^{4} k_i^{(3)} \right) (w_{\kappa}(\mathbf{k}_1, \dots, \mathbf{k}_4) - w_{\mu}(\mathbf{k}_1, \dots, \mathbf{k}_4)) \right\|_{L_2^{\#}} \to_{\#} 0 \quad (12.18) $$

as $\mu \to \# \kappa$. This is holds for each $W$ making up $H_{I,\kappa}(g)$, so we infer that there exists a $\mu_0$ such that for any $\mu$ such that: $\mu_0 < \mu < \kappa$

$$ \left\| (H_{0,\kappa} + I)^{-1} \left[ H_{0,\kappa'}^{\frac{1}{2}}, H_{0,\kappa'}^{\frac{1}{2}} (H_{I,\kappa}(g) - H_{I,\mu}(g)) \right] (H_{0,\kappa} + I)^{-1} \right\|_{\#} \leq \frac{1}{2} \varepsilon. \quad (12.19) $$

## 13. Self #-adjointness of the interaction Hamiltonian

For a real spatial cut-off $g(\mathbf{x})$ in the Schwartz space $S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#3})$, the interaction part of the Hamiltonian $H_{I,\kappa}(g)$ is self #-adjoint.

**Theorem 13.1** If $g \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#3})$ is real, then

$$ H_{I,\kappa}(g) = \text{Ext} - \int_{\mathbb{R}_c^{\#3}} \varphi_{\kappa}^{\#4}(\mathbf{x}) : g(\mathbf{x}) \, d^{\#3}\mathbf{x} \quad (13.1) $$

is essentially self #-adjoint on $D_{0,\kappa}^{\#} = \bigcap_{n=0}^{\infty} D(H_{0,\kappa}^n)$.

Let us introduce a domain $D_{1,\kappa}^{\#}$ obtained by applying any polynomial of the $t=0$ fields $\varphi_{\kappa}^{\#}(f_i)$, for real $f_i \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#3})$ the no particle state $\Omega_0$. Clearly $D_{1,\kappa}^{\#} \subset D_{0,\kappa}^{\#}$, and any vector $\Omega$ in $D_{1,\kappa}^{\#}$ is an entire vector for $\varphi_{\kappa}^{\#}(f)$, which means that the hyperinfinite power series

$$ \text{Ext} - \sum_{n=0}^{\infty} \frac{\| \varphi_{\kappa}^{\#n}(f) \Omega \|_{\#}}{n!} Z^n \quad (13.2) $$

defines an entire function of s. Since $D_{1,\kappa}^{\#}$ is #-dense in Fock space, Generalized Nelson's analytic vector theorem (see [19] Chapt.vi, sect.5) shows that for real $f$, $\varphi_{\kappa}^{\#}(f)$ is essentially self #-adjoint on $D_{1,\kappa}^{\#}$. A similar argument can be made for the canonically conjugate $t=0$ fields $\pi_{\kappa}^{\#}(f)$. Let $\mathcal{M}_{\kappa}^{\#}$ denote the von Neumann algebra of operators generated by the spectral projections of all the $t=0$ field $\varphi_{\kappa}^{\#}(f)$, $f \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#3})$. The algebra $\mathcal{M}_{\kappa}^{\#}$ is maximal Abelian. In other words, a bounded operator which commutes with all operators in $\mathcal{M}_{\kappa}^{\#}$ is itself in $\mathcal{M}_{\kappa}^{\#}$. Let us consider $\varphi_{\kappa}^{\#}(f)$ for $\text{supp}(f) \subset \mathbf{O} \subset \mathbb{R}_c^{\#3}$, where $\mathbf{O}$ is an #-open region of space. (The support of a function is the smallest #-closed set outside of which the function vanishes identically.) Define $\mathcal{C}_{\kappa}^{\#}(\mathbf{O})$ as the von Neumann algebra of operators generated by the spectral projections of all the fields $\varphi_{\kappa}^{\#}(f)$ and $\pi_{\kappa}^{\#}(f)$ with $\text{supp}(f) \subset \mathbf{O}$. Since

$$ \begin{aligned} \varphi_{\kappa}^{\#}(\mathbf{x}, t) &= \text{Ext}-\text{exp}(itH_{0,\kappa}) \varphi_{\kappa}^{\#}(\mathbf{x}) \text{Ext}-\text{exp}(-itH_{0,\kappa}) = \\ &= \text{Ext} - \int_{\mathbb{R}_c^{\#3}} d^{\#3}\mathbf{y} \{ \Delta_{\#}(\mathbf{x} - \mathbf{y}, t; m) \pi_{\kappa}^{\#}(\mathbf{y}) - \left[ \frac{\partial^{\#}}{\partial^{\#}t} \Delta_{\#}(\mathbf{x} - \mathbf{y}, t; m) \right] \varphi_{\kappa}^{\#}(\mathbf{y}) \}, \end{aligned} \quad (13.3) $$

where $\Delta_{\#}(\mathbf{x}, t; m)$ is the solution of the generalized Klein-Gordon equation with Cauchy data $\Delta_{\#}(\mathbf{x}, 0; m) = 0$, $\frac{\partial^{\#}}{\partial^{\#}t} \Delta_{\#}(\mathbf{x}, 0; m) = \delta^{\#}(\mathbf{x})$. (see [15],Eq.111) and $\Delta_{\#}(\mathbf{x}, t; m)$ vanishes outside the light cone, we infer that
$$ \mathrm{Ext}\text{-exp}(itH_{0,\kappa})\mathcal{C}_{\kappa}^{\#}(\mathbf{O})\mathrm{Ext}\text{-exp}(-itH_{0,\kappa}) \subset \mathcal{C}_{\kappa}^{\#}(\mathbf{O}_t), \qquad (13.4) $$

where $\mathbf{O}_t$ is the region $\mathbf{O}$ expanded by $t$.

**Theorem 13.2** If $g(x) \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#3})$ is real and has its support in an #-open rectangular parallelepiped $\mathbf{O} \subset \mathbb{R}_c^{\#3}$, then for the $H_{I,\kappa}(g)$ of (13.1)

$$ \mathrm{Ext}\text{-exp}(itH_{I,\kappa}(g)) \in \mathcal{C}_{\kappa}^{\#} \cap \mathcal{M}_{\kappa}^{\#}. $$

**Theorem 13.3** Let $T$ be any operator with domain $D_{1,\kappa}^{\#}$ such that

$$ T D_{1,\kappa}^{\#} \subset D(\varphi_{\kappa}^{\#n}(f)), \qquad (13.5) $$

$$ T D_{1,\kappa}^{\#} \subset D((T \uparrow D_{1,\kappa}^{\#})^{*}), \qquad (13.6) $$

$$ [T, \varphi_{\kappa}^{\#n}(f)] D_{1,\kappa}^{\#} = 0. \qquad (13.7) $$

Then

$$ \mathcal{M}_{\kappa}^{\#} D_{1,\kappa}^{\#} \subset D(T \uparrow D_{1,\kappa}^{\#}), \qquad (13.8) $$

$$ [\# - \bar{T}, \mathcal{M}_{\kappa}^{\#}] D_{1,\kappa}^{\#} = 0. \qquad (13.9) $$

**Proof** For $\Omega \in D_{1,\kappa}^{\#}$, from (13.5) and (13.7) we get

$$ T \varphi_{\kappa}^{\#n}(f)\Omega = \varphi_{\kappa}^{\#n}(f)T\Omega. $$

But by (13.6), for real $f$

$$ \|T \varphi_{\kappa}^{\#n}(f)\Omega\|_{\#}^{2} = \langle T\Omega, \varphi_{\kappa}^{\#2n}(f)T\Omega \rangle_{\#} = \langle T^{*}T\Omega, \varphi_{\kappa}^{\#2n}(f)\Omega \rangle_{\#} \le \|T^{*}T\Omega\|_{\#} \|\varphi_{\kappa}^{\#2n}(f)\Omega\|_{\#}. $$

Thus the #-convergent power series (3.2) shows that for $\Omega \in D_{1,\kappa}^{\#}$,

$$ \# - \bar{T} (\mathrm{Ext}\text{-exp}(i\varphi_{\kappa}^{\#}(f)))\Omega = \mathrm{Ext}\text{-exp}(i\varphi_{\kappa}^{\#}(f))T\Omega. \qquad (13.10) $$

It is clear that (13.10) is still valid with $\mathrm{Ext}\text{-exp}(i\varphi_{\kappa}^{\#}(f))$ replaced by strong #-limits of sums of such exponentials, and hence (13.8) and (13.9).

**Theorem 13.4** Let $\mathcal{M}$ is a maximal Abelian algebra of bounded operators on a non-Archimedean Hilbert space $\mathcal{H}$ with a cyclic vector $\Omega_0$. Let $T$ be a symmetric operator with domain $\mathcal{M}\Omega_0$, and let $T$ commute with $\mathcal{M}$. Then $T$ is essentially self #-adjoint.

**Proof** Without loss of generality, $\mathcal{M} = L_{\infty}^{\#}(X)$ and $\mathcal{H} = L_2^{\#}(X)$ for some #-measure space $(X, \Sigma, \mu)$, and $\Omega_0$ is the function 1. Let $f \in L_2^{\#}(X)$. Then $t \in L_2^{\#}(X)$ and $T$ is multiplication by $t$, with domain $L_{\infty}^{\#}(X)$. Let $f \in L_2^{\#}(X)$ and suppose $tf \in L_2^{\#}(X)$ also and let $f_n(x) = f(x)$ if $|f(x)| \le n, n \in \mathbb{N}$ and $f_n(x) \equiv 0$ otherwise. Then $f_n \in L_{\infty}^{\#} = D(T)$ and $f_n \to_{\#} f$, $tf_n \to_{\#} tf$ in $L_2^{\#}$ norm by the bounded #-convergence theorem. Thus $\{f, tf\}$ is in the graph of the #-closure of $T$. Thus the #-closure of $T$ is self #-adjoint, and $T$ is essentially self #-adjoint.

**Remark 13.1** Let $T_n, n \in \mathbb{N}$ be a hyperinfinite sequence of operators with the property of $T$ in the Theorem 13.4. Then $T_n \to_{\#} T$ strongly on the domain $\mathcal{M}\Omega_0$ if and only if $T_n\Omega_0 \to_{\#} T\Omega_0$.

**Proof** of the Theorems 13.1 and 13.2. We apply now the Theorems 13.3 and 13.4 with the case $T = H_{I,\kappa}(g)$, $\mathcal{M}$ in Theorem 13.4 as in Theorem 13.3, the non-Archimedean Hilbert space Fock space $\mathcal{F}^{\#}$, and $\Omega_0$ the Fock no-particle state. The hypotheses (13.5) and (13.6) can be verified by a direct computation. Thus $H_{I,\kappa}(g)$ is essentially self #-adjoint on $D_{1,\kappa}^{\#} \subset D_{0,\kappa}^{\#}$, and hence $H_{I,\kappa}(g)$ is essentially self #-adjoint on $D_{0,\kappa}^{\#}$.

If we assume that $\sup(g) \subset \mathbf{O}$, then as $\mathbf{O}$ is an #-open region, $\sup(g) \subset \mathbf{O}_1$ where $\mathbf{O}_1$ is $\mathbf{O}$ contracted by some small amount $\varepsilon > 0, \varepsilon \approx 0$. Since $H_{I,\kappa}(g)$ commutes with $\mathcal{M}$, and $\mathcal{M}$ is maximal Abelian, $-\exp(itH_{I,\kappa}(g))$
∈ M. Furthermore the argument in the proof of Theorem 13.3, can be repeated to show that $H_{1,\kappa}(g)$ commutes with $\mathcal{C}_{\kappa}^{\#}(\mathbf{o}'_{1})$, where $\mathbf{o}'_{1}$ is the complement of the #-closure of $\mathbf{o}_{1}$. Since $\mathcal{C}_{\kappa}^{\#}(\mathbb{R}_{c}^{\#3})$ is irreducible and $H_{1,\kappa}(g)$ commutes with $\mathcal{C}_{\kappa}^{\#}(\mathbf{o}'_{1})$, $\mathrm{Ext}\text{-exp}(itH_{1,\kappa}(g)) \in \mathcal{C}_{\kappa}^{\#}(\mathbf{o}_{2})$ where $\mathbf{o}_{2}$ is $\mathbf{o}_{1}$ expanded by any amount $\varepsilon' > 0$. Taking $\varepsilon' < \varepsilon$, we have $\mathrm{Ext}\text{-exp}(itH_{1,\kappa}(g)) \in \mathcal{C}_{\kappa}^{\#}(\mathbf{o})$, which completes the proof.

## 14. #-Self adjointness of the total Hamiltonian

**Theorem 14.1** (a) For real $g(x) \in S_{\mathrm{fin}}^{\#}(\mathbb{R}_{c}^{\#3})$, the total Hamiltonian $H_{\kappa}(g) = H_{0,\kappa}(g) + H_{1,\kappa}(g)$ is self #-adjoint with the domain $D(H_{\kappa}(g)) = D(H_{0,\kappa}(g)) \cap D(H_{1,\kappa}(g))$.

(b) The total Hamiltonian $H_{\kappa}(g)$ is essentially self #-adjoint on the domain

$$D_{0,\kappa}^{\#} = \bigcap_{n=0}^{\infty} D(H_{0,\kappa}^{n}).$$

**Remark 14.1** In order to prove the self #-adjointness of $H_{\kappa}$, we combine the estimates of section 14, the #-self adjointness of $H_{1,\kappa}(g)$ proved in section 15, and a singular perturbation theory developed in [19], see also section 21 in this paper. We need the following result which is a special case of Theorem 21.6 from section 21 in this paper.

**Theorem 14.2** Under the hypotheses (i)-(iii) below, the operator $H_{\kappa} = H_{0,\kappa} + H_{1,\kappa}$ is self #-adjoint.

(i) Both $H_{0,\kappa}$ and $H_{1,\kappa}$ are self #-adjoint. The domain $D_{0,\kappa}^{\#}$ is contained in the domain of $H_{1,\kappa}$, and $H_{1,\kappa}$ is essentially self #-adjoint on $D_{0,\kappa}^{\#}$.

(ii) Let $N_{\kappa}$ be a positive self #-adjoint operator, commuting with $H_{0,\kappa}$, and such that $N_{\kappa} \le \mathrm{const} H_{0,\kappa}$. Suppose that the operators $(N_{\kappa} + I)^{-1}H_{1,\kappa}(N_{\kappa} + I)^{-1}$ and $(N_{\kappa} + I)^{-1}H_{1,\kappa}(N_{\kappa} + I)^{-3}$ are bounded.

(iii) Suppose that for any $\varepsilon > 0$, there exists a number $b \in \mathbb{R}_{c}^{\#}$ such that as bilinear forms on $D_{0,\kappa}^{\#} \times D_{0,\kappa}^{\#}$,

$$-H_{1,\kappa} \le \varepsilon N_{\kappa} + bI, \qquad (14.1)$$

$$-\left[ H_{0,\kappa}^{\frac{1}{2}} \left[ H_{0,\kappa}^{\frac{1}{2}} H_{1,\kappa} \right] \right] \le \varepsilon H_{0,\kappa}^{2} + bI, \qquad (14.2)$$

$$-\left[ N_{\kappa}, [N_{\kappa}, H_{1,\kappa}] \right] \le \varepsilon N_{\kappa}^{3} + bI. \qquad (14.3)$$

**Proof of Theorem 14.1** In order to prove that $H_{\kappa}(g)$ is self #-adjoint, we apply Theorem 16.2 in the case that $H_{0,\kappa}$ is the free Hamiltonian, $N_{\kappa}$ is the number operator, and $H_{1,\kappa}$ is the interaction Hamiltonian $H_{1,\kappa}(g)$. Thus we need to verify (i)-(iii). Condition (i) was dealt with in Theorem 13.1, while condition (ii) is a consequence of (14.11). In refs. [15] and [17] it is shown that for any $\varepsilon > 0$, there is a number $b \in \mathbb{R}_{c}^{\#}$ such that

$$-H_{1,\kappa}(g) \le \varepsilon H_{0,\kappa} + bI.$$

By following that proof, but using the smoothing operator $\mathrm{Ext}\text{-exp}(-tN_{\kappa})$, in place of $\mathrm{Ext}\text{-exp}(-tH_{0,\kappa})$, one arrives at the estimate (16.1) required in (iii). The remaining estimates (14.2) and (14.3) were established in Theorem 14.1. Thus we conclude from Theorem 14.2 that $H_{\kappa}(g)$ is self #-adjoint on the domain $D(H_{0,\kappa}) \cap D(H_{1,\kappa}(g))$. We now show that $H_{\kappa}(g)$ is essentially self #-adjoint on $D(H_{0,\kappa})$. We first show that $H_{\kappa}(g)$ is essentially self #-adjoint on $D_{2} = D(H_{0,\kappa}) \cap D(N_{\kappa}^{2})$. By (14.11) it is clear that the domain of $H_{\kappa}(g)$ contains $D_{2}$. For $\psi \in D(H_{\kappa}(g)) = D(H_{0,\kappa}) \cap D(H_{1,\kappa}(g))$, consider hyperinfinite sequence $\psi_{n} \in D_{2}$, $n \in \mathbb{N}$ defined by

$$\psi_n = n(nI + N_\kappa)^{-1}\psi. \qquad (14.4)$$

Thus $\|\psi_n - \psi\|_{\#} + \|H_{0,\kappa}\psi_n - H_{0,\kappa}\psi\|_{\#} \to_{\#} 0$ as $n \to \infty$.
We need to study the following differences

$$H_{I,\kappa}\psi_n - H_{I,\kappa}\psi = -N_{\kappa}(nI + N_{\kappa})^{-1} H_{I,\kappa}\psi + n[H_{I,\kappa}, (nI + N_{\kappa})^{-1}]\psi, n \in \mathbb{N}. \quad (14.5)$$

Since $N_{\kappa}(nI + N_{\kappa})^{-1}, n \in \mathbb{N}$ is a uniformly bounded hyperinfinite sequence #-converging to zero on the #-dense set $D(N_{\kappa})$, it #-converges to zero and $\|N_{\kappa}(nI + N_{\kappa})^{-1} H_{I,\kappa}\psi\|_{\#}$ as $n \to \mathbb{N}$. But for the second term in (14.5) we get

$$\begin{aligned} n[H_{I,\kappa}, (nI + N_{\kappa})^{-1}]\psi &= [H_{I,\kappa}, (nI + N_{\kappa})^{-1}](nI + N_{\kappa})n(nI + N_{\kappa})^{-1}\psi = \\ &= (nI + N_{\kappa})^{-1}[N_{\kappa}, H_{I,\kappa}]n(nI + N_{\kappa})^{-1}\psi = \\ &= (nI + N_{\kappa})^{-1}(I + N_{\kappa})(I + N_{\kappa})^{-1}[N_{\kappa}, H_{I,\kappa}] \times \\ &\quad (I + N_{\kappa})^{-1}n(nI + N_{\kappa})^{-1}(I + N_{\kappa})\psi. \end{aligned} \quad (14.6)$$

Note that as $n \to \mathbb{N}$, hyperinfinite sequence $\delta_n = n(nI + N_{\kappa})^{-1}(I + N_{\kappa})\psi, n \in \mathbb{N}$ #-converges strongly to $(I + N_{\kappa})\psi$, that by (14.11), $(I + N_{\kappa})^{-1}[N_{\kappa}, H_{I,\kappa}](I + N_{\kappa})^{-1}$ is bounded, and hyperinfinite sequence $\gamma_n = (nI + N_{\kappa})^{-1}(I + N_{\kappa})\psi, n \in \mathbb{N}$ #-converges strongly to zero. Thus we get $\|[H_{I,\kappa}, (nI + N_{\kappa})^{-1}\psi]\|_{\#} \to \mathbb{N}$ as $n \to \mathbb{N}$, and so $\|H_{I,\kappa}\psi_n - H_{I,\kappa}\psi\|_{\#} \to \mathbb{N}$ as $n \to \mathbb{N}$. Thus we can conclude that $H_{\kappa}(g)$ is the #-closure of $H_{\kappa}(g)$ restricted to $D_2$, so $H_{\kappa}(g)$ is essentially self #-adjoint on $D_2$. Let $D_2$ be a Hilbert space endowed with the #-norm $\|\cdot\|_{\#}$ such that

$$(\|\psi\|_{\#})^2 = \|\psi\|_{\#}^2 + \|H_{0,\kappa}\psi\|_{\#}^2 + \|N_{\kappa}\psi\|_{\#}^2. \quad (14.7)$$

From (14.11) we infer that

$$\|H_{\kappa}(g) \psi\|_{\#} \le \text{const}\|\psi\|_{\#}^{\prime},$$

so that $H_{\kappa}(g)$ is essentially self #-adjoint on any subset of $D_2$ which is #-dense in the Hilbert space $D_2$. For any $\psi \in D_2$, $\psi_{\lambda} = \text{Ext}\text{-exp}(-\lambda H_{0,\kappa})\psi \in D_{0,\kappa}^{\#} = \bigcap_{n=0}^{\infty} D(H_{0,\kappa}^{n})$, and $\|\psi - \psi_{\lambda}\|_{D_{1,\kappa}^{\#}} \to \mathbb{N}$ as $\lambda \to \mathbb{N}$. Thus $H_{\kappa}(g)$ is essentially self #-adjoint on $D_0$.

## 15. Removing the spatial cut-off and locality

For the reader's convenience, we sketch a proof of generalized Segal's theorem that the self #-adjointness of $H_{\kappa}(g)$ allows the removal of the spatial cut-off. In fact, if $A$ is a bounded function of the free fields localized in a bounded region of space at $t = 0$, then

$$\sigma_t(A) = \text{Ext}\text{-exp}(itH_{\kappa}(g))A\text{Ext}\text{-exp}(-itH_{\kappa}(g))$$

is independent of $g(x)$ provided that $g(x) = \lambda$, the desired coupling constant, on a sufficiently large region, depending on $t$. Furthermore, if $A$ is localized in the region of space $O$, then $\sigma_t(A)$ is localized in the region $O_t$, where $O_t$ is the region $O$ expanded by $t$. (We have taken the velocity of light to be one.) In other words, the time translation $\sigma_t$ gives rise to a local theory. If one chooses for the operator $A$ a spectral projection of the $t = 0$ field $\phi_{\kappa}^{\#}(f)$, one can piece together the time translation operator for the fields themselves. In section 16, we showed that $H_{\kappa} = H_{0,\kappa} + H_{I,\kappa}$, which is sum of two self #-adjoint operators, is itself self #-adjoint. As a consequence of this fact, the generalized Trotter product formula (see [19] Chapt.6, section 5.10) says that for all $\psi \in \mathcal{F}^{\#}$

$$\text{Ext}\text{-exp}(itH_{\kappa}(g))\psi = \#\text{-lim}_{n \to \infty} \left( \left[ \text{Ext}\text{-exp}\left(\frac{itH_{0,\kappa}(g)}{n}\right) \right] \left[ \text{Ext}\text{-exp}\left(\frac{itH_{I,\kappa}(g)}{n}\right) \right] \right) \psi.$$

And therefore we obtain
$$
\sigma_t(A)\psi = \#\lim_{n \to *\infty} \left( \left[ \mathrm{Ext}\-\exp\left(\frac{itH_{0,\mathcal{X}}(g)}{n}\right) \right] \left[ \mathrm{Ext}\-\exp\left(\frac{itH_{I,\mathcal{X}}(g)}{n}\right) \right] \right)^n A \left( \left[ \mathrm{Ext}\-\exp\left(\frac{-itH_{0,\mathcal{X}}(g)}{n}\right) \right] \left[ \mathrm{Ext}\-\exp\left(\frac{-itH_{I,\mathcal{X}}(g)}{n}\right) \right] \right)^n \psi.
$$

Let $\mathbf{O}$ be the region of space defined by $|\mathbf{x}| < M, t = 0$, and let $A \in \mathbf{C}_{\mathcal{X}}^{\#}(\mathbf{O})$, where $\mathbf{C}_{\mathcal{X}}^{\#}(\mathbf{O})$ is defined above in section 13. Given an arbitrary, positive $\varepsilon$, split $g(\mathbf{x})$ into two infinitely $\#$-differentiable parts $g_1(\mathbf{x})$, $g_2(\mathbf{x})$ such that

$$
g(\mathbf{x}) = g_1(\mathbf{x}) + g_2(\mathbf{x}),
$$

where $\mathrm{supp}(g_1(\mathbf{x})) \subset \mathbf{O}_{\varepsilon}$ and $\mathrm{supp}(g_2(\mathbf{x})) \cap \mathbf{O}_{\frac{\varepsilon}{2}} = \emptyset$ is empty. Write now

$$
H_{I,\chi}(g) = H_{I,\chi}(g_1) + H_{I,\chi}(g_2),
$$

so that as a consequence of theorems 15.1 and 15.2, $H_{I,\chi}(g_1)$ and $H_{I,\chi}(g_2)$ commute, and

$$
\mathrm{Ext}\-\exp\left(\frac{itH_{I,\chi}(g)}{n}\right) = \left[\mathrm{Ext}\-\exp\left(\frac{itH_{I,\chi}(g_1)}{n}\right)\right] \left[\mathrm{Ext}\-\exp\left(\frac{itH_{I,\chi}(g_2)}{n}\right)\right].
$$

Furthermore,

$$
\mathrm{Ext}\-\exp\left(\frac{itH_{I,\chi}(g_1)}{n}\right) \in \mathbf{C}_{\mathcal{X}}^{\#}(\mathbf{O}_{\varepsilon}),
$$

and $\mathrm{Ext}\-\exp\left(\frac{itH_{I,\chi}(g_2)}{n}\right)$ commutes with $\mathbf{C}_{\mathcal{X}}^{\#}(\mathbf{O}_{\varepsilon/4})$. Therefore,

$$
A_1(t) = \left[ \mathrm{Ext}\-\exp\left(\frac{itH_{0,\chi}(g_1)}{n}\right) \right] \left[ \mathrm{Ext}\-\exp\left(\frac{itH_{I,\chi}(g_2)}{n}\right) \right] A \left[ \mathrm{Ext}\-\exp\left(-\frac{itH_{I,\chi}(g_1)}{n}\right) \right] \left[ \mathrm{Ext}\-\exp\left(-\frac{itH_{0,\chi}(g_2)}{n}\right) \right]
$$

depends on $g(\mathbf{x})$ only in the region $\mathbf{O}_{\varepsilon}$, and by the free propagation property (15.4),

$$
A_1 \in \mathcal{C}_{\chi}^{\#}(\mathbf{O}_{(t/n)+\epsilon})
$$

We continue step by step, and after $n \in *\mathbb{N} \setminus \mathbb{N}$ steps by using hyperinfinite induction principle, see ref. [10], we
conclude that

$$
A_n(t) = \left( \left[ \mathrm{Ext}\-\exp\left(\frac{itH_{0,\chi}(g_1)}{n}\right) \right] \left[ \mathrm{Ext}\-\exp\left(\frac{itH_{I,\chi}(g_2)}{n}\right) \right] \right)^n A \times \\
\times \left( \left[ \mathrm{Ext}\-\exp\left(-\frac{itH_{I,\chi}(g_1)}{n}\right) \right] \left[ \mathrm{Ext}\-\exp\left(-\frac{itH_{0,\chi}(g_2)}{n}\right) \right] \right)^n
$$

depends on $g(\mathbf{x})$ only in the region $\mathbf{O}_{t+n\epsilon}$ and

$$
A_1(t) \in \mathbf{C}_{\mathcal{X}}^{\#}(\mathbf{O}_{t+n\epsilon})
$$

Since $\epsilon$ can be chosen arbitrarily, $A_n(t)$ depends on $g(\mathbf{x})$ only in the region $\#-\bar{\mathbf{O}}_t$, the $\#$-closure of $\mathbf{O}_t$, and

$$
A_n(t) \in \bigcap_{\epsilon>0} \mathcal{C}_{\chi}^{\#}(\mathbf{O}_{t+\epsilon}).
$$

Thus $A_n(t)$ commutes with any local observable $B$ localized in $\#$-open region of space $\mathbf{O}'$ such that $\mathbf{O}'$ and $\mathbf{O}_t$
are disjoint. As this is true for each $n \in *\mathbb{N}\setminus\mathbb{N}$, it is true for

$$
\sigma_t(A) = \text{strong } \#\text{-lim}_{n \to *\infty} A_n(t).
$$

Hence $\sigma_t(A)$ is local and it depends on $g(\mathbf{x})$ only in the region $\#-\bar{\mathbf{O}}_t$, where we choose $g(\mathbf{x}) = \lambda$. Thus we conclude that the spatial cut-off has been removed and the resulting theory is local.
16. Semi-boundedness of the total Hamiltonian

**16. Reduction to a Problem with Discrete Momentum** We use the non-Archimedean Fock space representation for our field $\varphi_{\kappa}^{\#}(\mathbf{x})$, $\mathbf{x} \in *\mathbb{R}_c^{\#3}$. The Fock non-Archimedean Hubert space $\mathcal{F}^{\#}$ is a direct sum

$$ \mathcal{F}^{\#} = \mathrm{Ext} - \bigoplus_{n=0}^{\infty} \mathcal{F}_n^{\#}, $$

where $\mathcal{F}_n^{\#}$ is the space of $n$ non-interacting particles, i.e. $\mathcal{F}_n^{\#}$ is the space of symmetric square #-integrable functions, i.e. $L_2^{\#}(*\mathbb{R}_c^{\#3})$ functions of $n$ variables. Let $\mathbf{k} = (k_1, k_2, k_3) \in *\mathbb{R}_c^{\#3}$

$$ \mu(\mathbf{k}) = (\mathbf{k}^2 + \mu_0^2)^{1/2} = (k_1^2 + k_2^2 + k_3^2 + m_0^2)^{1/2}, $$

$$ \varphi_{\kappa}^{\# -}(\mathbf{x}) = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3}} \mathrm{Ext}\text{-exp}(i(\mathbf{k}, \mathbf{x})) a(\mathbf{k})\theta(|\mathbf{k}|, \kappa)[\mu(\mathbf{k})]^{-1/2} d^{\#3}\mathbf{k}, \quad (16.1) $$

$$ \varphi_{\kappa}^{\# +}(\mathbf{x}) = \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3}} \mathrm{Ext}\text{-exp}(i(\mathbf{k}, \mathbf{x})) a^*(-\mathbf{k})\theta(|\mathbf{k}|, \kappa)[\mu(\mathbf{k})]^{-1/2} d^{\#3}\mathbf{k}, \quad (16.2) $$

$$ \theta(|\mathbf{k}|, \kappa) = 1 \text{ if } |\mathbf{k}| \le \kappa \text{ and } (|\mathbf{k}|, \kappa) = 0 \text{ if } |\mathbf{k}| > \kappa, $$

and $\varphi_{\kappa}^{\#}(\mathbf{x}) = \varphi_{\kappa}^{\# -}(\mathbf{x}) + \varphi_{\kappa}^{\# +}(\mathbf{x})$, where $a(\mathbf{k})$ and $a^*(\mathbf{k})$ are the annihilation and creation operators,

$$ [a(\mathbf{k}), a^*(\mathbf{k}')] = \delta^{\#}(\mathbf{k} - \mathbf{k}'). \quad (16.3) $$

By definition,

$$ : \varphi_{\kappa}^{\#p}(\mathbf{x}) := \sum_j \binom{p}{j} \varphi_{\kappa}^{\# +}(\mathbf{x})^j \varphi_{\kappa}^{\# -}(\mathbf{x})^{p-j}. \quad (16.4) $$

**Remark 16.1** Remind that Wick product differs from the ordinary product in that all the annihilators are placed to the right and the creators are placed to the left. : $\varphi_{\kappa}^{\#p}(\mathbf{x})$: is not an operator, but it is a densely defined bilinear form. We take Fourier transforms to compute

$$ \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3}} : \varphi_{\kappa}^{\#p}(\mathbf{x}) : h(\mathbf{x}) d^{\#3}\mathbf{x} = \sum_j \binom{p}{j} \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3p}} a^*(-\mathbf{k}_1) \cdots a^*(-\mathbf{k}_j) a(\mathbf{k}_j) \cdots a(\mathbf{k}_p) \times \\ \times \mathrm{Ext}\text{-}\hat{h}(\mathbf{k}_1 + \cdots + \mathbf{k}_p) \prod_{i=1}^p \theta(\|\mathbf{k}_i\|, \kappa) [\mu(\mathbf{k}_i)]^{-1/2} d^{\#3}\mathbf{k}_i, \quad (16.5) $$

where $\mathrm{Ext}\text{-}\hat{h}$ is the Fourier transform of $h$. We assume $h$ is in $L_2^{\#}$ and so $\mathrm{Ext}\text{-}\hat{h}$ is in $L_2^{\#}$ also. Since $\mu(\mathbf{k}) \sim |\mathbf{k}|$ for large $|\mathbf{k}|$, one can show that

$$ \mathrm{Ext}\text{-}\hat{h}(\mathbf{k}_1 + \cdots + \mathbf{k}_p) \prod_{i=1}^p \theta(\|\mathbf{k}_i\|, \kappa) [\mu(\mathbf{k}_i)]^{-1/2} \in L_2^{\#}. \quad (16.6) $$

It is well known that (16.6) implies that each integral on the right side of (16.5) is an operator defined on the domain $D(N^{p/2})$ of $N^{p/2}$. This domain is the set of $\psi = \psi_0, \psi_1, \dots, \psi_j \in \mathcal{F}_j^{\#}$ with

$$ \mathrm{Ext} - \sum_n n^{p/2} \| \mathrm{Ext} - \prod_{i=1}^n \theta(\|\mathbf{k}_i\|, \kappa) \psi_n \|_{\#2}^2 < \infty. \quad (16.7) $$

Thus (16.5) is an operator defined on $D(N^{p/2})$. Similarly $H_{0,\kappa} + \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#3}} P(\varphi_{\kappa}^{\#}(\mathbf{x})) : d^{\#3}\mathbf{x}$ is an operator defined on the #-dense domain, $D(H_{0,\kappa}) \cap D(N^{d/2})$, where $d$ is the degree of the polynomial $P$. We approximate now (16.5) by a hyperfinite sum. Choose numbers $\delta \approx 0$ and $\kappa \in *\mathbb{R}_c^{\#} \setminus *\mathbb{R}_{c,\mathrm{fin}}^{\#}$. We define now an hyperfinite approximation in configuration space. Under this approximation, the momentum space variable $\mathbf{k} = (k_1, k_2, k_3) \in *\mathbb{R}_c^{\#3}$ is replaced by a discrete variable $\mathbf{k} \in \Gamma_{\delta}^3$

$$ \Gamma_{\delta}^{3} = \{\mathbf{k} = (k_1, k_2, k_3) | k_i = \delta n_i, n_i \in \mathbb{Z}; i = 1, 2, 3\} \quad (16.8) $$

Thus we define $\mathcal{F}_V^{\#}$, the Fock space for hyperfinite volume $V^3 = \delta^{-3}$ as
$$
\mathcal{F}_V^\# = \mathbb{C}(l_2^\#(\Gamma_V^3)) = {}^*\mathbb{C}^\# \oplus l_2^\#(\Gamma_V^3) \oplus \{l_2^\#(\Gamma_V^3) \otimes_s l_2^\#(\Gamma_V^3)\} \cdots \quad (16.9)
$$

We choose now one to one correspondence *Z $\leftrightarrow$ *Z$\delta \times$ *Z$\delta \times$ *Z$\delta = \Gamma_{\delta}^3$ given by vector-function $\wp(m)$

$$
\wp(m) = \{k_1(m), k_2(m), k_3(m)\} = \mathbf{k}(m) \tag{16.10}
$$

and such that

$$
\wp(-m) = -\wp(m). \tag{16.11}
$$

And we define now

$$
\Gamma_{\kappa, \delta}^{3} = \{ \mathbf{k} \in \Gamma_{\delta}^{3} \mid |\mathbf{k}| \le \kappa \}. \qquad (16.12)
$$

We set now

$$
a_{\delta}(\mathbf{k}(m)) = (\delta)^{-3/2} \left[ \mathrm{Ext} - \int_{0}^{\delta} d^{\#}l_{1} \mathrm{Ext} - \int_{0}^{\delta} d^{\#}l_{2} \mathrm{Ext} - \int_{0}^{\delta} d^{\#}l_{3} a(\mathbf{k}(m) + \mathbf{l}) \right], \quad (16.13)
$$

$$
a_{\delta}^{*}(\mathbf{k}(m)) = (\delta)^{-3/2} \left[ \mathrm{Ext} - \int_{0}^{\delta} d^{\#}l_{1} \mathrm{Ext} - \int_{0}^{\delta} d^{\#}l_{2} \mathrm{Ext} - \int_{0}^{\delta} d^{\#}l_{3} a^{*}(\mathbf{k}(m) + \mathbf{l}) \right]. \quad (16.14)
$$

Then one obtains

$$
[a_{\delta}^{*}(\mathbf{k}(m_1)), a_{\delta}(\mathbf{k}(m_2))] = \delta_{m_1 m_2} = \begin{cases} 1 & \text{if } m_1 = m_2 \\ 0 & \text{if } m_1 \neq m_2 \end{cases} \quad (16.15)
$$

Let

$$
H_{0,\kappa,\delta} = \mathrm{Ext} - \sum_{\mathbf{k} \in \Gamma_{\kappa,\delta}^3} \mu(\mathbf{k}) a_{\delta}^{*}(\mathbf{k}) a_{\delta}(\mathbf{k}). \quad (16.16)
$$

One can check that each $\psi$ in $D(H_{0,\kappa})$ is in $D(H_{0,\kappa,\delta})$ also and that

$$
\#\lim_{\delta \to 0} H_{0, \kappa, \delta} \psi = H_{0, \kappa} \psi. \qquad (16.17)
$$

Next we approximate (16.5) by

$$
\begin{equation}
\begin{split}
\phi_{\kappa, \delta}^{\#p}(\mathbf{x}) := \delta^{3p/2} \sum_j \binom{p}{j} \operatorname{Ext} - \sum_{\mathbf{k} \in \Gamma_{\kappa, \delta}^3} & a_{\delta}^{*}(-\mathbf{k}_1) \cdots a_{\delta}^{*}(-\mathbf{k}_j) a_{\delta}(\mathbf{k}_j) \cdots a_{\delta}(\mathbf{k}_p) \times \\
& \times \operatorname{Ext} \hat{h}([\mathbf{k}_1] + \cdots + [\mathbf{k}_p]) \prod_i [\mu([\mathbf{k}_i])]^{-1/2},
\end{split}
\tag{16.18}
\end{equation}
$$

where

$$
\hat{h}_{\delta}(\mathbf{k}) = \mathrm{Ext} - \int_{-\pi_{\#}/\delta}^{\pi_{\#}/\delta} \int_{-\pi_{\#}/\delta}^{\pi_{\#}/\delta} \int_{-\pi_{\#}/\delta}^{\pi_{\#}/\delta} (\mathrm{Ext} - \exp(i\langle\mathbf{k}, \mathbf{x}\rangle)) h(\mathbf{x}) d^{\#3}\mathbf{x}
$$

and $[\mathbf{k}] = ([k_1], [k_2], [k_3])$, where

$$
[k_1] = \sup\{l_1 | (l_1, l_2, l_3) \in \Gamma_{\delta}^3, l_1 \le k_1\}, \quad [k_2] = \sup\{l_2 | (l_1, l_2, l_3) \in \Gamma_{\delta}^3, l_2 \le k_2\},
$$

$$
[k_3] = \sup\{l_3 | (l_1, l_2, l_3) \in \Gamma_{\delta}^3, l_3 \le k_3\}
$$

is the integral part of **k** relative to the lattice $\Gamma_{\delta}^{3}$. Since $h \in L_{1}^{\#}$, $\hat{h}_{\delta}$ is #-continuous and

$$
\mathrm{Ext} - \hat{h}([\mathbf{k}_1] + \cdots + [\mathbf{k}_p]) \prod_i [\mu([\mathbf{k}_i})]^{-1/2} \xrightarrow{\#} \mathrm{Ext} - \hat{h}(\mathbf{k}_1 + \cdots + \mathbf{k}_p) \prod_{i=1}^p \theta(\|\mathbf{k}_i\|, \varkappa) [\mu(\mathbf{k}_i)]^{-1/2}
$$

uniformly. Let $D_0^\#$ be the set of states $\psi = \{\psi_0, \psi_1, \dots\}$ with $\psi_n(\mathbf{k}_1, \dots, \mathbf{k}_n) = 0$ for $n < *\infty$ or $\mathrm{Ext}-\sum_i |k_i| < *\infty$. If $\phi, \psi \in D_0^\#$ then
$$ \#\lim_{\delta \to 0} \langle \phi, \phi^{\#p}(\mathbf{x}) : \psi \rangle_{\#} = \langle \phi, \mathrm{Ext} - \int_{\mathbb{R}_c^{\#3}} \phi^{\#p}(\mathbf{x}) : d^{\#3}x \psi \rangle. \quad (16.18) $$

Thus the bilinear form of

$$ H_{\mathcal{H},\delta} = H_{0,\mathcal{H},\delta} + \sum_p b_p : \phi_{\mathcal{H},\delta}^{\#p}(h): \quad (16.19) $$

\#-converges to $H_{\mathcal{H}}$ on $D_0^{\#} \times D_0^{\#}$ where $b_0, ..., b_r$ are the coefficients of $y_0, y_1, ..., y_r$ in the polynomial $P(y)$. Hence if the $H_{\mathcal{H},\delta}$ are semibounded with a lower bound independent of $\delta$ then $H_{\mathcal{H}}$ is semibounded also. Let $\mathcal{F}_{\delta}^{\#}$ be the subspace of $\mathcal{F}^{\#}$ consisting of functions which are piece wise constant between lattice points. In other words,

$$ \psi = \{\psi_0, \psi_1, ..., \psi_n, ...\} \in \mathcal{F}_{\delta}^{\#} \text{ if} \\ \psi_n(\mathbf{k}_1, ..., \mathbf{k}_n) = \psi_n([\mathbf{k}_1], ..., [\mathbf{k}_n]). $$

Let $\mathcal{F}_{\mathcal{H},\delta}^{\#}$ be the subspace of $\mathcal{F}_{\delta}^{\#}$ defined by the restriction

$$ \psi_n(\mathbf{k}_1, ..., \mathbf{k}_n) = 0 \text{ if } [\mathbf{k}_i] \in \Gamma_{\mathcal{H},\delta}^3 $$

for some $i, 1 \le i \le n$.

The operators $a_{\delta}^{*}(\mathbf{k})$ and $a_{\delta}(\mathbf{k})$, $\mathbf{k} \in \Gamma_{\mathcal{H},\delta}^{3}$, leave $\mathcal{F}_{\mathcal{H},\delta}^{\#}$ invariant and act irreducibly on $\mathcal{F}_{\mathcal{H},\delta}^{\#}$. We set now $\delta = 2^{-v}$, $\mathcal{H} = 2^v$ and observe that $\mathcal{F}_{2^v, 2^{-v}}^{\#}$ increases monotonically with $v$ and that

$$ D_0^{\#'} = D_0^{\#} \cap \bigcup_v \mathcal{F}_{2^v, 2^{-v}}^{\#} $$

is \#-dense in $\mathcal{F}^{\#}$ and $H_{\mathcal{H}} \subset -(H_{\mathcal{H}} \upharpoonright D_0^{\#'})$. Thus it is sufficient to prove the semiboundedness of

$$ H_{\mathcal{H},\delta} \upharpoonright (D(H_{0,\mathcal{H}}) \cap D(N_{\mathcal{H}}^{d/2}) \cap \mathcal{F}_{\mathcal{H},\delta}^{\#}) $$

with a lower bound independent of $\delta$.

**17. Diagonalizing the potential.** In this subsection we give a new representation of $\mathcal{F}_{\mathcal{H},\delta}^{\#}$ in which the interaction term of $: \phi_{\mathcal{H}}^{\#p}(h):$ is a multiplication operator. Let

$$ \begin{align*} q(\mathbf{k}(|m|)) &= (2^{-2}\mu(\mathbf{k}(m)))^{1/2} [a_{\delta}(\mathbf{k}(m)) + a_{\delta}^{*}(\mathbf{k}(m)) + a_{\delta}(-\mathbf{k}(m)) + a_{\delta}^{*}(-\mathbf{k}(m))], \\ q(\mathbf{k}(-|m|)) &= i(2^{-2}\mu(\mathbf{k}(m)))^{1/2} [-a_{\delta}(\mathbf{k}(|m|)) + a_{\delta}^{*}(\mathbf{k}(|m|)) + a_{\delta}(-\mathbf{k}(|m|)) - a_{\delta}^{*}(-\mathbf{k}(|m|))], \\ p(\mathbf{k}(|m|)) &= i(2^{-2}\mu(\mathbf{k}(m)))^{1/2} [a_{\delta}(\mathbf{k}(m)) - a_{\delta}^{*}(\mathbf{k}(m)) + a_{\delta}(-\mathbf{k}(m)) - a_{\delta}^{*}(-\mathbf{k}(m))], \\ p(\mathbf{k}(-|m|)) &= (2^{-2}\mu(\mathbf{k}(m)))^{1/2} [a_{\delta}(\mathbf{k}(|m|)) + a_{\delta}^{*}(\mathbf{k}(|m|)) - a_{\delta}(-\mathbf{k}(|m|)) - a_{\delta}^{*}(-\mathbf{k}(|m|))], \\ p_m &= p(\mathbf{k}(m)), \quad q_m = q(\mathbf{k}(m)), \\ q(\mathbf{k}(m)) &= \begin{cases} q(\mathbf{k}(|m|)) & \text{if } m > 0, \\ q(-\mathbf{k}(|m|)) & \text{if } m < 0, \end{cases} \end{align*} $$

for $0 \neq \mathbf{k} \in \Gamma_{\delta}^{3}$ and let

$$ \begin{align*} q_0 &= (\mu_0/2)^{1/2} [a_{\delta}(\mathbf{0}) + \mathbf{a}_{\delta}^{*}(\mathbf{0})], \\ p_0 &= i(\mu_0/2)^{1/2} [a_{\delta}(\mathbf{0}) - \mathbf{a}_{\delta}^{*}(\mathbf{0})]. \end{align*} $$
Using the equations mentioned above one can compute that

$$H_{0,\kappa,\delta} = \mathrm{Ext} - \sum_{m \in \mathbb{Z}, |k(m)| \le \kappa} 2^{-1} [p_m^2 + \mu^2(\mathbf{k}(m))q_m^2 - \mu(\mathbf{k}(m))]. \quad (17.1)$$

We replace now $p_m$ and $q_m$ by unitarily equivalent operators. Let

$$\mathcal{H}_{\kappa, \delta}^{\#} = \mathrm{Ext} - \bigotimes_{\mathbf{k} \in \Gamma_{\kappa, \delta}^{3}} \mathcal{H}_{\mathbf{k}}^{\#},$$

where $\mathcal{H}_{\mathbf{k}}^{\#}$ is $L_2^*(\mathbb{R}_c^\#)$ with respect to the Gaussian #-measure

$$\phi_{\mathbf{k}}^{2}(q) d^{\#} q = (\mu(\mathbf{k})/\pi_{\#})^{1/2} (\mathrm{Ext}\text{-exp}(-\mu(\mathbf{k})q^2)) d^{\#} q. \quad (17.2)$$

There is a unitary equivalence between $\mathcal{H}_{\kappa, \delta}^{\#}$ and $\mathcal{F}_{\kappa, \delta}^{\#}$ which sends $q_m$ into multiplication by $q$ in the factor $\mathcal{H}_{\mathbf{k}(m)}^{\#}$ and $p_m$ into the operator

$$\phi_{\mathbf{k}}^{-1}(q) i \left( \frac{d^{\#}}{d^{\#}q} \right) \phi_{\mathbf{k}}(q)$$

again acting in the factor $\mathcal{H}_{\mathbf{k}}^{\#}$. The proof of this statement is essentially generalized von Neumann's uniqueness theorem for irreducible representations of the commutation relations. We identify $\mathcal{H}_{\kappa, \delta}^{\#}$ and $\mathcal{F}_{\kappa, \delta}^{\#}$ and we identify $q_m$, etc. with its image, multiplication by $q$, etc. Let

$$\begin{align}
H_{\mu(\mathbf{k})} = 2^{-1} \phi_{\mathbf{k}}^{-1}(q) \left[ -\left( \frac{d^{\#}}{d^{\#}q} \right)^2 + \mu(\mathbf{k}) q^2 \right] \phi_{\mathbf{k}}(q) &= \tag{17.3} \\
&= -2^{-1} \left( \frac{d^{\#}}{d^{\#}q} \right)^2 + \mu(\mathbf{k}) q \left( \frac{d^{\#}}{d^{\#}q} \right) \nonumber
\end{align}$$

acting on $\mathcal{H}_{\mathbf{k}}^{\#}$. Now $-H_{\mu(\mathbf{k})}$ is the #-infinitesimal generator of a known Markoff process and furthermore the operator $\mathrm{Ext}\text{-exp}(-H_{\mu(\mathbf{k})})$ is an integral operator and the kernel can be computed explicitly. In particular

$$(\mathrm{Ext}\text{-exp}(-H_{\mu(\mathbf{k})})\psi)(q) = \mathrm{Ext}\text{-} \int_{\mathbb{R}_c^\#} p^t(q, q')\psi(q')\phi_{\mathbf{k}}^2(q')d^\#q' \quad (17.4)$$

for $\psi \in \mathcal{H}_{\mathbf{k}}^{\#}$, where

$$p^t(q, q') = [1 - \mathrm{Ext}\text{-exp}(-\mu t)] \left\{ \mathrm{Ext}\text{-exp} \left[ - \frac{\mu(q' - (\mathrm{Ext}\text{-exp}(-\mu t))q)^2}{1 - \mathrm{Ext}\text{-exp}(-2\mu t)} \right] + \mu q'^2 \right\}. \quad (17.5)$$

Let $q$ now denote a variable in a Euclidean space $E_\kappa$ and let $q$ have coordinates $q_m = q(\mathbf{k}(m))$. Then

$$\phi_{\kappa}^{2}(q) d^{\#3} q = \mathrm{Ext} - \prod_{\mathbf{k} \in \Gamma_{\kappa, \delta}^{3}} \phi_{\kappa}^{2}(q(\mathbf{k})) d^{\#} q(\mathbf{k}) \quad (17.6)$$

is the product of the #-measures (17.2) and

$$\mathcal{H}_{\kappa, \delta} = L_2^{\#}(\phi_{\kappa}^2(q) d^{\#} q).$$

In addition to the function space $L_2^\#$, we will have to consider $L_r^\#(\phi_\kappa^2(q) d^\# q)$. Since $\mathrm{Ext}\text{-} \int \phi_\kappa^2(q) d^\# q = 1$, we have $L_{r_2}^\# \subset L_{r_1}^\# \text{ if } r_1 \le r_2$.

**Lemma 17.1.** $\mathrm{Ext}\text{-exp}(-H_{0,\kappa,\delta})$ is a contraction operator on $L_r^\#$, $1 \le r \le *\infty$. If $T \le t, 1 < p$ and $r < *\infty$ it is a contraction from $L_p^\#$ to $L_r^\#$, for some $T$ not depending on $\delta$. If $p$ is bounded away from one and $r$ is bounded then $T$ does not depend on $p$ or $r$.

Now we show that the interaction term : $\phi_{\kappa, \delta}^{\#p}(\mathbf{x})$: is a polynomial in the $q'$'s. Let

$$\phi_{\kappa, \delta}^{\#}(\mathbf{x}) = \delta^{3/2} \mathrm{Ext} - \sum_{\mathbf{k}(m) \in \Gamma_{\kappa, \delta}^{3}} \mathrm{Ext}\text{-exp}(i(\mathbf{k}(m), \mathbf{x})) [\mu([\mathbf{k}(m)])]^{-1/2} (\mathbf{a}_{\delta}(\mathbf{k}(m)) + a_{\delta}^{*}(-\mathbf{k}(m))) \quad (17.7)$$
Since

$$[\mu([\mathbf{k}(m)])]^{-1/2} (\mathbf{a}_{\delta}(\mathbf{k}(m)) + a_{\delta}^{*}(-\mathbf{k}(m))) = \begin{cases} q(\mathbf{k}(|m|)) + iq(-\mathbf{k}(|m|)) & \text{if } m > 0 \\ \sqrt{2}q_0 & \text{if } m = 0 \\ q(\mathbf{k}(|m|)) - iq(-\mathbf{k}(|m|)) & \text{if } m < 0 \end{cases}$$

$\varphi_{\kappa, \delta}^{\#p}(\mathbf{x})$ and $\varphi_{\kappa, \delta}^{\#p}(\mathbf{x})$ are polynomials in the $q$'s. We use the canonical formula

$$\varphi_{\kappa, \delta}^{\#p}(\mathbf{x}) = \sum_{j=0}^{[p/2]} \frac{p!(2^{-j})}{(p-2j)!j!} c_{\kappa}^{j} : \varphi_{\kappa, \delta}^{\#p-2j}(\mathbf{x}): \quad (17.8)$$

to conclude by induction on $p$ that $: \varphi_{\kappa, \delta}^{\#p}(\mathbf{x}):$ is also a polynomial in the $q$'s. In (18.2.8) the coefficient

$$\frac{p!(2^{-j})}{(p-2j)!j!}$$

is just the number of ways of selecting $j$ unordered pairs from $p$ objects and $c_{\kappa}$ is defined by the formula

$$c_{\kappa} = \delta^3 \sum_{k \in \Gamma_{\kappa, \delta}^3} [\mu([\mathbf{k}])]^{-1/2}$$

we have the bound

$$c_{\kappa} \le K_1 \kappa^2 \quad (17.9)$$

where $K_1$ is independent of $\kappa$ and $\delta$. Thus

$$: \varphi_{\kappa, \delta}^{\#p}(h) : = \mathrm{Ext} - \int_{-\pi_{\#}/\delta}^{\pi_{\#}/\delta} \int_{-\pi_{\#}/\delta}^{\pi_{\#}/\delta} \int_{-\pi_{\#}/\delta}^{\pi_{\#}/\delta} : \varphi_{\kappa, \delta}^{\#p}(\mathbf{x}) : h(\mathbf{x}) d^{\#3}\mathbf{x}$$

is a polynomial in the $q$'s, as desired.

Let

$$P(y) = b_0 + b_1 y + \cdots + b_d y^d$$

be the polynomial in $y$ and let

$$V_{\kappa, \delta} = \sum_{0 \le p \le d} b_p : \varphi_{\kappa, \delta}^{\#p}(h) : \quad (17.10)$$

denote our approximate interaction term, as in (18.1.19).

**Lemma 17.2.** For some constant $K_2$, independent of $\delta$ and $\kappa$, we have

$$-K_2 \kappa^d \le V_{\kappa, \delta}. \quad (17.11)$$

**Proof** We use (17.8) to remove the Wick ordering in (17.10) and obtain

$$V_{\kappa, \delta} = \sum_p a_p(c_{\kappa}) \left\{ \mathrm{Ext} - \int_{-\pi_{\#}/\delta}^{\pi_{\#}/\delta} \int_{-\pi_{\#}/\delta}^{\pi_{\#}/\delta} \int_{-\pi_{\#}/\delta}^{\pi_{\#}/\delta} : \varphi_{\kappa, \delta}^{\#p}(\mathbf{x}) : h(\mathbf{x}) d^{\#3}\mathbf{x} \right\}$$

where $a_p(c_{\kappa})$ is a polynomial in $c_{\kappa}$ of degree at most $[(d-p)/2]$. The coefficients of $a_p$ depend only on the coefficients of $P$, and so we have an estimate

$$|a_p(c_{\kappa})| \le K' \times c_{\kappa}^{(d-p)/2}.$$

Since $a_d = b_d > 0$ and since $d$ is even by hypothesis, it follows that

$$0 < \sum_p a_p(c_{\kappa}) y^p \text{ for } K''(1 + c_{\kappa}) < |y|^2$$
and

$$-c_{\kappa}^{(d/2)} K''' \leq \sum_{p} a_p(c_{\kappa}) y^p$$

for all y. We bound $c_\kappa$ by (17.9) and the proof is complete.

**Lemma 17.3** Function $V_{\kappa, \delta} \in$ for all $r < *\infty$ and if $\lambda \le \kappa$, then

$$\|V_{\kappa, \delta} - V_{\lambda, \delta}\|_{\#2j}^{2j} \le (dj)! \times K_3^j \times (\kappa^{2d} - \lambda^{2d})^j, \quad (17.12)$$

where $K_3$ is a constant which is independent of $\delta, \lambda$ and $\kappa$.

**Proof** We use the particle representation, $\mathcal{F}_{\kappa, \delta}^{\#}$, in place of the representation $\mathcal{H}_{\kappa, \delta} = L_2^{\#}(\phi_{\kappa}^2(q)d^{\#}q)$. Now $1 \in \mathcal{H}_{\kappa, \delta}$ corresponds to the vacuum state $\Omega_0 = \{1,0,0, \dots \} \in \mathcal{F}^{\#}$

so

$$\begin{aligned} \|V_{\kappa, \delta} - V_{\lambda, \delta}\|_{\#2j}^{2j} &= \mathrm{Ext-} \int (V_{\kappa, \delta} - V_{\lambda, \delta})^{2j} \phi_{\kappa}^2(q) d^{\#}q = \\ &= \langle (V_{\kappa, \delta} - V_{\lambda, \delta})^j \Omega_0, (V_{\kappa, \delta} - V_{\lambda, \delta})^j \Omega_0 \rangle_{\#} = \|(V_{\kappa, \delta} - V_{\lambda, \delta})^j \Omega_0\|_{\#}^2. \end{aligned} \quad (17.13)$$

We set $\lambda = 0$ above and get

$$\|V_{\kappa, \delta}\|_{\#2j}^{2j} = \|V_{\kappa, \delta}^{2j} \Omega_0\|_{\#}^2$$

and so $V_{\kappa, \delta} \in L_r^{\#}$ for all $r < *\infty$. We return to (17.13) and note that $V_{\kappa, \delta} - V_{\lambda, \delta}$ is a sum of $d2^d$ terms of the form

$$A = b_p \delta^{3p/2} \mathrm{Ext-} \sum_{\lambda \le |k_i| \le \kappa} \left[ \mathrm{Ext-} \hat{h} \left( \sum_{i=1}^p k_i \right) a^{\#} \left( \pm k_i \right) \prod_{i=1}^p \left[ \mu(k_i) \right]^{-1/2} \right] \quad (17.14)$$

where in the summation over $k_i$ we have $k_i \in \Gamma_{\kappa, \delta}^3$ for $1 \le p \le d, p \le d$, and $k_i \notin \Gamma_{\lambda, \delta}^3$ for at least one $i$. Summing again over the same range of $k_i$ we get

$$\delta^{3p} \mathrm{Ext-} \sum_{\lambda \le |k_i| \le \kappa} \left[ \mathrm{Ext-} \hat{h} \left( \sum_{i=1}^p k_i \right) \prod_{i=1}^p \left[ \mu(k_i) \right]^{-1/2} \right] \le K_4 \times (\kappa^{2d} - \lambda^{2d}) \quad (17.15)$$

and $K_4$ is independent of $\lambda, \kappa$ and $\delta$.

Let $\psi$ be a state with at most $l$ particles. It follows from (17.15) and the form of $A$ that

$$\|A\Omega_0\|_{\#}^2 \le \left( \frac{(l+p)!}{l!} \right) \times_4 \left( \kappa^{2d} - \lambda^{2d} \right) \|\psi\|_{\#}^2$$

and furthermore $A\psi$ is a state with at most $l + p$ particles. Thus if we have operators $A_1, \dots, A_j$ of the form (17.14),

$$\|A_1 \cdots A_j \Omega_0\|_{\#}^2 \le (dj)! K_4^j \lambda^{2dj}.$$

Hence

$$\|(V_{\kappa, \delta} - V_{\lambda, \delta})^j \Omega_0\|_{\#}^2 \le (dj)! \times K_3^j \times (\kappa^{2d} - \lambda^{2d})^j,$$

and the proof is complete.

## 18. Path space and corresponding #-measure

Let $q$ now denote a variable in a Euclidean space $E_{\kappa}^3 = *\mathbb{R}_c^{\#3}$ and let $C^\#$ be the space of #-continuous paths
$q = q(s) \in E_{\mathcal{H}}^3, 0 \le s < *\infty$. There is a #-measure on $C^\#$ intrinsically associated with the semigroup $Ext\text{-exp}(-tH_{0,\mathcal{H},\delta})$. To define this #-measure we set $q_k = q(\mathbf{k})$ and

$$p_k^t(q_k, q'_k) = \phi_k^2(q'_k) d^\# q'_k = \mathrm{Pr}\{q_k(t) = q'_k | q_k(0) = q_k\} \quad (18.1)$$

the probability that $q_k(t) = q'_k$ if it is known that $q_k(0) = q_k \cdot p_k^t$ is defined by (17.5) we have added a subscript $\mathbf{k}$ to indicate the dependence on $\mu = \mu(\mathbf{k})$. Let

$$p_{\mathcal{H}}^{t}(q, q') = \mathrm{Ext} - \prod_{k \in \Gamma_{\mathcal{H}, \delta}^{3}} p_{k}^{t}(q_{k}, q'_{k}). \quad (18.2)$$

The $\sigma^\#$- field [19] of #-measurable subsets of $C^\#$ is generated by the sets

$$q(s_i) \in B_i, 1 \le i \le j, \quad (18.3)$$

where $B_i$ is a #-Borel subset of $E_{\mathcal{H}}^3$. The #-measure of (18.3) is

$$\mathrm{Ext} - \int_{B_j \times \dots \times B_j} \mathrm{Ext} - \prod_{i=2}^{j} p_{\mathcal{H}}^{s_i - s_{i-1}}(q(s_i), q(s_{i-1})) \phi_{\mathcal{H}}^2(q(s_i)) d^\# q(s_i) \phi_{\mathcal{H}}^2(q(0)) d^\# q(0) \quad (18.4)$$

if $s_1 = 0 < s_2 < \dots < s_j$. The definition (18.4) is forced by the definition (18.1) together with the Markov character of the process, the stipulation that each coordinate $q_k$ of $q$ defines an independent process and the specification of $\phi_k^2(q) d^\# q$ as the probability distribution of the initial point $q(0)$ of the path $q$. If $V_1, \dots, V_j \in L_j^\#(E_{\mathcal{H}}^3, \phi_k^2(q) d^\# Q)$ then we compute

$$\mathrm{Ext} - \int \mathrm{Ext} - \prod_i V_i(q(s_i)) d^\# Q = \mathrm{Ext} - \int V_1(q(0)) \phi_{\mathcal{H}}^2(q(0)) d^\# q(0) \times \quad (18.5)$$

$$\times [\mathrm{Ext}\text{-exp}(-(s_1 - s_0)H_{0,\mathcal{H},\delta})V_2\mathrm{Ext}\text{-exp}(-(s_2 - s_1)H_{0,\mathcal{H},\delta})(\dots(V_{j-1}\mathrm{Ext}\text{-exp}(-(s_i - s_{i-1})H_{0,\mathcal{H},\delta})V_j)\dots)](q(0))$$

and

$$|\mathrm{Ext} - \int \mathrm{Ext} - \prod_i V_i(q(s_i)) d^\# Q| \le \mathrm{Ext} - \prod_i \|V_i\|_{\#j} \quad (18.6)$$

using (18.4) and the fact that $\mathrm{Ext}\text{-exp}(-H_{0,\mathcal{H},\delta})$ is a contraction on $L_r^\#$. Furthermore (18.5) and (18.6) remain valid when some of the times $s_i$ coincide.

**Lemma 18.1** Let $V$ be a polynomial function on $E_{\mathcal{H}}^3$. Then $\mathrm{Ext} - \int_0^t V(q(s)) d^\# s \in L_p^\#(C^\#, d^\# Q)$ for all $p < *\infty$, and

$$\| \mathrm{Ext} - \int_0^t V(q(s)) d^\# s \|_{\#j} \le t \| V_i \|_{\#j}$$

for $j \in *\mathbb{N}$ an even positive integer.

**Lemma 18.2** Let $r \in [1,2)$. There is a finite $T$ independent of $\delta$ such that if $t \ge T$ and if $\phi$ and $\psi$ in $L_2^\#(L_2^\#(\phi_{\mathcal{H}}^2(q) d^\# q))$ then $\phi(q(0))\psi(q(t)) \in L_r^\#(C^\#, d^\# Q)$ and

$$\|\phi(q(0))\psi(q(t))\|_{\#r} \le \|\phi\|_{\#2} \times \|\psi\|_{\#2}.$$

The $T$ can be chosen independently of $r$ provided $r$ is bounded away from 2.

**19. The generalized Feynman Kac formula.** The generalized Feynman Kac formula states that

$$\langle \phi, \mathrm{Ext}\text{-exp}(-tH_{0,\mathcal{H},\delta})\psi \rangle_{\#} = \mathrm{Ext} - \int \phi(q(0)) \{ \mathrm{Ext}\text{-exp}(-[\mathrm{Ext} - \int_0^t V_{0,\mathcal{H},\delta}(q(s)) d^\# s]) \} \psi(q(s)) d^\# Q. \quad (19.1)$$

The RHS of (19.1) is bounded by
$$
\begin{align*}
\|\phi(q(0))\psi(q(s))\|_{\#p'} &\times \|\mathrm{Ext}\text{-exp}(-[\mathrm{Ext}\text{-}\int_0^t V_{\kappa,\delta}(q(s)) d^\#s]\|\|_{\#p} \\
&\le \|\phi\|_{\#2} \times \|\psi\|_{\#2} \times \|\mathrm{Ext}\text{-exp}(-[\mathrm{Ext}\text{-}\int_0^t V_{\kappa,\delta}(q(s)) d^\#s]\|\|_{\#p}
\end{align*}
$$

for $p > 2$ and for $t$ large, by Lemma 18.2. Thus

$$
\| \mathrm{Ext}\text{-exp}(-tH_{\kappa,\delta}) \|_{\#} \le \| \mathrm{Ext}\text{-exp}(-[\mathrm{Ext}\text{-}\int_{0}^{t} V_{\kappa,\delta}(q(s)) d^{\#} s] \|_{\#p}
$$

and therefore

$$
t^{-1} \left\{ \mathrm{Ext}\text{-ln} \left[ \left\| \mathrm{Ext}\text{-exp} \left( - \left[ \mathrm{Ext}\text{-}\int_{0}^{t} V_{\kappa,\delta}(q(s)) d^{\#} s \right] \right) \right\|_{\#p} \right] \right\} \le H_{\kappa,\delta}. \quad (19.2)
$$

Let

$$
I_{\lambda} = \mathrm{Ext} - \int_{0}^{t} V_{\kappa, \delta}(q(s)) d^{\#} s.
$$

Then by Lemma 17.2 we have

$-tK_2\lambda^d \le I_\lambda.$

Let $K_5, \dots$ denote positive constants depending only on $t$ and the polynomial $P$ and let $\mathrm{Pr}\{q|\cdot\}$ denote the
\#-measure defined by $d^\#Q$.

Then by Lemma 18.1 we get

$$
\mathbf{Pr}\{q|I_{\kappa} \le -tK_2\lambda^d - 1\} \le \mathbf{Pr}\{q| |I_{\kappa} - I_{\lambda}| \ge 1\} \le \mathrm{Ext}\text{-}\int |I_{\kappa} - I_{\lambda}|^{2j} d^{\#}Q \le t^{2j} \|I_{\kappa,\delta} - I_{\lambda,\delta}\|_{\#2j}^{2j} \quad (19.3)
$$

From (19.3) by Lemma 17.3,see (17.12) we get

$$
\mathbf{Pr}\{q|I_{\kappa} \le -tK_2\lambda^d - 1\} \le [(dj)!] \times t^{2j} \times K_3^j \times (\kappa^{2d} - \lambda^{2d})^j. \quad (19.3)
$$

**Lemma 19.1** Let $f$ be $\mathbb{R}_c^{\#}$-valued function on a probability \#-measure space $(M, \Sigma^{\#}, \mu^{\#})$, see ref.[19], and let $m_f(x) = \mu^{\#}\{q|f(q) \ge x\}$. Let $F: \mathbb{R}_c^{\#} \to \mathbb{R}_c^{\#}$ be a bounded positive, monotone non-decreasing $C^{\#1}$ function.
Then

$$
\mathrm{Ext} - \int_M F(f(q)) d^\# \mu^\# = \mathrm{Ext} - \int_{-\infty}^{\infty} F(x) d^\# m_f(x) = -F(-\infty) + \mathrm{Ext} - \int_{-\infty}^{\infty} m_f(x) F^{\#'}(x) d^\# x. \quad (19.4)
$$

In particular, by the generalized monotone convergence theorem:

$$
\mathrm{Ext} - \int_M [\mathrm{Ext}\text{-exp}(f(q))] d^\# \mu^\# = \mathrm{Ext} - \int_{-\infty}^{\infty} [\mathrm{Ext}\text{-exp}(x)] m_f(x) d^\# x. \quad (19.5)
$$

From (18.4.3) we get

$$
\mathbf{Pr}\{q|-I_{\kappa} \geq K_2\lambda^d + 1\} \leq [(dj)!] \times t^{2j} \times K_3^j \times (\kappa^{2d} - \lambda^{2d})^j. \quad (19.6)
$$

From (19.6) and (19.5) finally we get

$$
\mathrm{Ext} - \int [\mathrm{Ext}\text{-exp}(-pI_{\kappa}(q))] d^{\#}Q \le [(dj)!] \times t^{2j} \times K_3^j \mathrm{Ext} - \int_0^{\kappa} [\mathrm{Ext}\text{-exp}(x)] (\kappa^{2d} - x^{2d})^j d^{\#}x < *\infty. \quad (19.7)
$$

Thus $\mathrm{Ext} - \int [\mathrm{Ext}\text{-exp}(-pI_{\kappa}(q))] d^{\#}Q$ is bounded independently of $\delta$ and combining this with (7.2) we have $H_{\kappa,\delta}$ bounded below by a constant which is independent of $\delta$; according to section 18.1this proves Theorem 19.1.
**Theorem 19.1** Let $h$ be a nonnegative function in $L_1^\# \cap L_2^\#$. Suppose that the polynomial $P$ in ((18.2.10)) has even degree and that the leading coefficient is positive. Then Hamiltonian $H_\kappa$ is bounded from below.

**20. Alternate derivation without the use of functional integration**

We will give an alternate derivation of the results mentioned in sections above without the use of functional integration, central in section 19. We consider a Hamiltonian of the form

$$H_{\kappa} = H_{0,\kappa} + V_{\kappa}, \qquad (20.1)$$

where $H_{0,\kappa}$ is the free Hamiltonian of a particle of mass $\mu_0$ expressed in terms of the neutral scalar field $\varphi_\kappa^\#(\mathbf{x})$ and its momentum conjugate $\pi_\kappa^\#(\mathbf{x})$:

$$H_{0,\kappa} = \mathrm{Ext} - \int_0^1 d^\# x_1 (\mathrm{Ext} - \int_0^1 d^\# x_2 (\mathrm{Ext} - \int_0^1 d^\# x_3 : [\nabla \varphi_\kappa^{\#2}(\mathbf{x}) + \mu_0^2 \varphi_\kappa^{\#2}(\mathbf{x}) + \pi_\kappa^{\#2}(\mathbf{x})]: )) \quad (20.2)$$

As is evident from (17.2) we are working in a periodic box $B = [0,1]^3$. $V_\kappa$ is a polynomial function of the $\varphi_\kappa^\#(\mathbf{x})$. We denote by $^N H_{0,\kappa}$ and $^N V_\kappa$, $N \in *\mathbb{N}\setminus\mathbb{N}$ the parts of $H_{0,\kappa}$ and $V_\kappa$ depending only on the creation and annihilation operators of the $N$ lowest-energy modes of the free Hamiltonian and such that $|\mathbf{k}| \le \kappa$. We always imagine we are working with $^N H_{0,\kappa}$ and $^N V_\kappa$, but derive inequalities independent of $N$.

**Theorem 20.1** Assume for each finite $\alpha > 0$ that there is an $M_\alpha$ such that

$$\langle 0 | \mathrm{Ext} - \mathrm{exp}(-\alpha(^N V_\kappa)) | 0 \rangle \le M_\alpha,$$

where $|0\rangle$ denotes the vacuum of the free field. Then there is a $B$ such that

$$^N H_{0,\kappa} + ^N V_\kappa \ge B, \text{ for all } N.$$

Actually as will be seen it is not necessary to satisfy the condition above for all $\alpha$, but only for some sufficiently large $\alpha$ that one can calculate. We refer to section 18.3 for the result that the conditions of the theorem are satisfied for a large class of self-interactions.

We apply the notation

$$\varphi_{\kappa, \delta}^{\#}(\mathbf{x}) = \mathrm{Ext} - \sum_{k \in \Gamma_{\kappa, \delta}^3} \mathrm{Ext} - \mathrm{exp}(i\langle \mathbf{k}, \mathbf{x} \rangle) (a_{\delta}(\mathbf{k}) + a_{\delta}^{*}(-\mathbf{k})) \quad (20.3)$$

and define for $\mathbf{k} \in \Gamma_{\kappa, \delta}^3$

$$q_0 = (\mu_0/2)^{1/2} [a_{\delta}(\mathbf{0}) + a_{\delta}^{*}(\mathbf{0})], \quad p_0 = i(\mu_0/2)^{1/2} [a_{\delta}(\mathbf{0}) - a_{\delta}^{*}(\mathbf{0})], \quad (20.4)$$

$$q(\mathbf{k}(|m|)) = (2^{-2}\mu(\mathbf{k}(m)))^{1/2} [a_{\delta}(\mathbf{k}(m)) + a_{\delta}^{*}(\mathbf{k}(m)) + a_{\delta}(-\mathbf{k}(m)) + a_{\delta}^{*}(-\mathbf{k}(m))],$$

$$q(\mathbf{k}(-|m|)) = i(2^{-2}\mu(\mathbf{k}(m)))^{1/2} [-a_{\delta}(\mathbf{k}(|m|)) + a_{\delta}^{*}(\mathbf{k}(|m|)) + a_{\delta}(-\mathbf{k}(|m|)) - a_{\delta}^{*}(-\mathbf{k}(|m|))],$$

$$p(\mathbf{k}(|m|)) = i(2^{-2}\mu(\mathbf{k}(m)))^{1/2} [a_{\delta}(\mathbf{k}(m)) - a_{\delta}^{*}(\mathbf{k}(m)) + a_{\delta}(-\mathbf{k}(m)) - a_{\delta}^{*}(-\mathbf{k}(m))],$$

$$p(\mathbf{k}(-|m|)) = (2^{-2}\mu(\mathbf{k}(m)))^{1/2} [a_{\delta}(\mathbf{k}(|m|)) + a_{\delta}^{*}(\mathbf{k}(|m|)) - a_{\delta}(-\mathbf{k}(|m|)) - a_{\delta}^{*}(-\mathbf{k}(|m|))],$$

$$p_m = p(\mathbf{k}(m)), \quad q_m = q(\mathbf{k}(m)). \quad q(\mathbf{k}(m)) = \begin{cases} \mathbf{q}(\mathbf{k}(|m|)) & \text{if } m > 0, \\ \mathbf{q}(-\mathbf{k}(|m|)) & \text{if } m < 0. \end{cases}$$

In terms of these variables,
$$H_{0,\kappa,\delta} = \mathrm{Ext} - \sum_{m \in \mathbb{Z}, |k(m)| \le \kappa} 2^{-1} [p_m^2 + \mu^2(k(m))q_m^2 - \mu(k(m))] = \mathrm{Ext} - \sum_{m \in \mathbb{Z}, |k(m)| \le \kappa} H_m. \quad (20.5)$$

We represent these operators on the $L_2^\#$ space of $\mathbb{R}_c^\#N$ with #-measure $\mu$ the product of the #-measures $\mu_m$

$$d^\# \mu_m = (\omega_m / \pi_\#)^{1/2} (\mathrm{Ext} - \exp(-\omega_m q_m^2)) d^\# q_m \quad (20.6)$$

with $q_m$ a multiplicative operator and

$$p_m = i(\partial^\# / \partial^\# q_m) - \omega_m q_m. \quad (20.7)$$

Where

$$\omega_m = (\mathbf{k}^2(m) + \mu_0^2)^{1/2} = (k_1^2(m) + k_2^2(m) + k_3^2(m) + \mu_0^2)^{1/2}.$$

A complete set of eigenfunctions for $H_m$ is given by

$$\phi_{m,n}(q_m) = (2^n n!)^{-1/2} A_n(q_m(\omega_m)^{1/2}), n \in \mathbb{N}, \quad (20.8)$$

$$n! = \mathrm{Ext} - \prod_{0<p\le n} p, \quad 2^n = \mathrm{Ext} - \prod_{0<p\le n} 2,$$

$$A_n(z) = (-1)^n (\mathrm{Ext} - \exp(z^2)) \frac{d^{\#n}}{d^{\#z}n} (\mathrm{Ext} - \exp(-z^2)).$$

The chief inequality we will exploit is the following numerical inequality for $x, y \in \mathbb{R}_c^\#$, $y \ge 0$:

$$xy \le \mathrm{Ext} - \exp(x) + \mathrm{Ext} - \ln(y). \quad (20.9)$$

The expectation value of the interaction $V_\kappa$ in a state with $\mathbb{C}_c^\#$-function $F$ is given by

$$\langle F | V_\kappa | F \rangle = \mathrm{Ext} - \int (|F|^2 V_\kappa) d^\# \mu. \quad (20.10)$$

We apply (18.3.10) with $x = rV_\kappa$ and $y = r^{-1}F^2$ to derive the inequality

$$-\langle F | V_\kappa | F \rangle \le \mathrm{Ext} - \int (\mathrm{Ext} - \exp(-rV_\kappa)) d^\# \mu + \frac{1}{r} \left[ \mathrm{Ext} - \int |F|^2 (\mathrm{Ext} - \ln(|F|^2)) d^\# \mu \right] - \frac{1}{r} (\mathrm{Ext} - \ln(r)). \quad (20.11)$$

Here $r$ is a numerical factor to be fixed later. Note that

$$\mathrm{Ext} - \int (\mathrm{Ext} - \exp(-rV_\kappa)) d^\# \mu = \langle 0 | \mathrm{Ext} - \exp(-rV_\kappa) | 0 \rangle. \quad (20.12)$$

We intend now to bound the second term on the right side of (20.11) by the expectation value of $H_{0,\kappa}$ in the state $F$. We consider the following equation:

$$[\mathrm{Ext} - \int |F|^2 (\mathrm{Ext} - \ln(|F|^2)) d^\# \mu] = \quad (20.13)$$

$$= \frac{2}{\lambda} (\mathrm{Ext} - \int F^* H_{0,\kappa} F d^\# \mu) + \frac{1}{\lambda r} \frac{d^\#}{d^\# t} \left( \mathrm{Ext} - \int \left[ (\mathrm{Ext} - \exp(-tH_{0,\kappa}))^* (\mathrm{Ext} - \exp(-tH_{0,\kappa})) \right]^{1+\lambda t} d^\# \mu \right) \Big|_{t=0},$$

which easily follows for functions $F$ nice enough so that all the integrals exist and the differentiation may be moved inside the integral, a dense subspace in $L_2^\#$. We do not discuss domain questions. We rewrite (20.11) using (20.13):

$$-\langle F | V_\kappa | F \rangle \le \mathrm{Ext} - \int (\mathrm{Ext} - \exp(-rV_\kappa)) d^\# \mu + \frac{2}{\lambda r} \langle F | H_{0,\kappa} | F \rangle - \frac{1}{r} (\mathrm{Ext} - \ln(r)) + \quad (20.14)$$

$$+ \frac{1}{\lambda r} \frac{d^\#}{d^\# t} \left( \mathrm{Ext} - \int \left[ (\mathrm{Ext} - \exp(-tH_{0,\kappa}))^* (\mathrm{Ext} - \exp(-tH_{0,\kappa})) \right]^{1+\lambda t} d^\# \mu \right) \Big|_{t=0}.$$
The theorem we are after is established provided $\lambda r \ge 2$ and we can bound the last term in (20.14). The remainder of the paper is devoted to a study of

$$ \mathrm{Ext}-\int [(\mathrm{Ext}-\exp(-tH_{0,\kappa}))^*(\mathrm{Ext}-\exp(-tH_{0,\kappa}))]^{1+\lambda t} d^\#\mu = \mathrm{Ext}-\int |\mathrm{Ext}-\exp(-tH_{0,\kappa})|^{2+2\lambda t} d^\#\mu. \quad (20.15) $$

We consider, corresponding to any $g$ in $L_2^\#(\mu)$, its expression as a sum of products of the functions in (20.8):

$$ g(q) = \mathrm{Ext}-\sum_{i_{1,\cdot,\cdot},i_N} C_{i_{1,\cdot,\cdot},i_N} \{ \mathrm{Ext}-\prod_s (2^{i_s} i_s!)^{-1} (\mathrm{Ext}-\exp(i_s) A_{i_s} (q_s(\omega_s)^{1/2})) \}. \quad (20.16) $$

The $q_s$ are merely the $q_k$ in some order. The coefficient's $C_{i_{1,\cdot,\cdot},i_N}$ are now considered as functions on the discrete space whose points are the indices of the $C$'s. To the point $(i_{1,\cdot,\cdot},i_N)$ is associated the point mass $\mathrm{Ext}-\prod_s (\mathrm{Ext}-\exp(2i_s))$. With this measure, the transformation $T$ that carries a set of $C$'s into the corresponding function $g$ as in (20.16) is norm preserving as a map from $l_2^\#$ to $L_2^\#$. We will later show that $T$ is norm decreasing as a map from $l_1^\#$ to $L_4^\#$. Assuming this for a moment, we complete the proof of the theorem. We apply the generalized Riesz-Thorin convexity theorem to the transformation $T$ obtaining

$$ \begin{aligned} \mathrm{Ext}-\int |\mathrm{Ext}-\exp(-tH_{0,\kappa})|^{2+2\lambda t} d^\#\mu &\le \left[ \mathrm{Ext}-\left( \sum_{i_{1,\cdot,\cdot},i_N} \mathrm{Ext}-\prod_s (\mathrm{Ext}-\exp(2i_s)) \times \left| (\mathrm{Ext}-\exp(-\omega_{i_{1,\cdot,\cdot},i_N} t)) \times C_{i_{1,\cdot,\cdot},i_N} \right|^{\frac{2(1+\lambda t)}{1+3\lambda t}} \right) \right]^{\frac{1+3\lambda t}{2(1+\lambda t)}} \end{aligned} \quad (20.17) $$

with

$$ \omega_{i_{1,\cdot,\cdot},i_N} = \mathrm{Ext}-\sum_s i_s \omega_s. \quad (20.18) $$

In the right-hand side of (20.17) we apply the generalized Holder inequality to obtain an expression involving the weighted sum of the squares of the absolute values of the $C$'s which is equal to one:

$$ \begin{aligned} \mathrm{Ext}-\int |\mathrm{Ext}-\exp(-tH_{0,\kappa})|^{2+2\lambda t} d^\#\mu &\le \left[ \mathrm{Ext}-\left( \sum_{i_{1,\cdot,\cdot},i_N} \mathrm{Ext}-\prod_s (\mathrm{Ext}-\exp(2i_s)) \left( \mathrm{Ext}-\exp\left(-\omega_{i_{1,\cdot,\cdot},i_N} \times \frac{2(1+\lambda t)}{2\lambda} \right) \right) \right) \right]^{2\lambda t}. \end{aligned} \quad (20.19) $$

It follows that

$$ \frac{d^\#}{d^\# t} \left( \mathrm{Ext}-\int |\mathrm{Ext}-\exp(-tH_{0,\kappa})|^{2+2\lambda t} d^\#\mu \right) \Big|_{t=0} \le 2\lambda \times \left\{ \mathrm{Ext}-\ln \left[ \mathrm{Ext}-\left( \sum_{i_{1,\cdot,\cdot},i_N} \mathrm{Ext}-\prod_s (\mathrm{Ext}-\exp(2i_s)) \left( \mathrm{Ext}-\exp\left(-\omega_{i_{1,\cdot,\cdot},i_N} \times \frac{2(1+\lambda t)}{2\lambda} \right) \right) \right) \right] \right\}. \quad (20.20) $$

If $\mu_0/\lambda > 2$, this gives an inequality with finite right hand side in the #-limit $N \to *\infty$. It is clear that the theorem is now reduced to establishing that $T$ is #-norm decreasing from $l_1^\#$ to $L_4^\#$.

**Lemma 20.1.** Let $S$ be the space of sequences $\{C_\gamma\}$, $\gamma = 0, 1, \dots, N$ with #-measure at $\gamma$, $\mathrm{Ext}-\exp(2\gamma)$; and $Y$ the space of functions on $*\mathbb{R}_c^\#$ with #-measure

$$ (1/\pi_\#)^{1/2} (\mathrm{Ext}-\exp(-x^2)) d^\# x, \quad (20.21) $$

and $T$ the operator from $S$ to $Y$ given by

$$ T\{C_\gamma\} = \mathrm{Ext}-\sum_\gamma C_\gamma \frac{(\mathrm{Ext}-\exp(\gamma))A_\gamma(x)}{[2^\gamma \gamma!^\#]} \quad (20.22) $$
with $A_γ(x)$ the $γ$-th Hermite polynomial $A_γ(x) = (-1)^n(\mathrm{Ext}-\exp(x^2)) \frac{d^{\#n}}{d^{\#}x^n}(\mathrm{Ext}-\exp(-x^2))$; then, $T$ is
\#-norm decreasing from $l_1^\#$ to $l_4^\#$.

It is easy to see that this lemma would follow from establishing the inequality

$$
\left| \left(\frac{1}{\pi\#}\right)^{\frac{1}{2}} \left(\mathrm{Ext}-\exp(-a-b-c-d)\right) \times \left(\mathrm{Ext}-\int_{\mathbb{R}_c^\#} \left[2^{a+b+c+d}(a!)^2(b!)^2(c!)^2(d!)^2\right]^{-1/2} A_a(x)A_b(x)A_c(x)A_d(x)\left(\mathrm{Ext}-\exp(-x^2)\right)d^\#x\right) \right| \le 1_{\mathbb{R}_c^\#} \quad (20.23)
$$

for all integers $a, b, c \in *\mathbb{N}$ and $d \ge 0$; actually, it is sufficient to let $a = b = c = d$. We use the generating function

$$
\mathrm{Ext}-\exp(-t^2 + 2tZ) = \mathrm{Ext}-\sum_{N \in *\mathbb{N}} \frac{t^N}{N!} A_N(Z) \quad (20.24)
$$

to obtain

$$
\begin{align}
& \left(\frac{1}{\pi\#}\right)^{\frac{1}{2}} \left(\mathrm{Ext} - \int_{\mathbb{R}_c^\#} \left(\mathrm{Ext}-\exp(-x^2)\right) A_a(x) A_b(x) A_c(x) A_d(x) d^\# x\right) = \tag{20.25} \\
&= \frac{(a!)^2 (b!)^2 (c!)^2 (d!)^2}{\frac{1}{2} \mathbb{R}_c^\# (a+b+c+d)!^2} \times 2 \frac{1}{\mathbb{R}_c^\#} \frac{1}{2} (a+b+c+d) \times (rs + rt + ru + st + su + tu) \frac{1}{\text{pick-a-power}} \nonumber
\end{align}
$$

where pick-a-power means to find the coefficient of the monomial $r^a s^b t^c u^q$ in the expansion of the expression.
Note that $a + b + c + d$ is even or the integral vanishes. We make the crude estimate

$$
\begin{align}
& (rs + rt + ru + st + su + tu) \frac{1}{\text{pick-a-power}} \le \tag{20.26} \\
& \le 2 \frac{1}{\mathbb{R}_c^\#} \frac{1}{2} (a+b+c+d) \times (r+s+t+u) \frac{1}{\text{pick-a-power}} \nonumber
\end{align}
$$

Now we get

$$
(r + s + t + u)_{\text{pick-a-power}}^{\frac{1}{2}(a+b+c+d)} = \frac{(a+b+c+d)!^{\#}}{(a!)^{\#}(b!)^{\#}(c!)^{\#}(d!)^{\#}}. \quad (20.27)
$$

Denoting the left-hand side of (20.23) by $\mathfrak{I}$ and using (20.26) we obtain

$$
\mathfrak{I} \le (\mathrm{Ext}-\exp(-a-b-c-d)) \times \frac{[(a+b+c+d)!^\#] \times 2^{-\frac{1}{2}(a+b+c+d)}}{[(a!)^2(b!)^2(c!)^2(d!)^2]^{1/2} [1/(a+b+c+d)]^\#} \quad (20.28)
$$

It is easily verify that

$$
\mathfrak{I} \le 1_{\mathbb{R}_c^\#} \qquad (20.29)
$$

The inequality (20.29) finalized the proof of theorem.

**21. Strong \#-convergence of operators**

In this section we study the sum $A + B$ of two \#-selfadjoint operators on a non-Archimedean Banach spaces
over field $\bar{\mathbb{C}}_c^\#$, and we find sufficient conditions for $C = A + B$ to be \#-selfadjoint. Our technique is to
approximate $B$ by a hyper infinite sequence of bounded in $\mathbb{R}_c^\#$ \#-selfadjoint operators $\{B_n\}$, $n \in *\mathbb{N}$ and so to
approximate $C$ by \#-selfadjoint operators $C_n = A + B_n$. We answer these three questions separately:1.When do
the operators $C_n$ have a \#-lim $C$? 2.When is $C$ a \#-selfadjoint operator? 3.When is $C = A + B$? In Theorem 21.8
we give a set of estimates on the relative size of A and B which ensure a positive answer to all three questions. Hence these estimates show that $A + B = C$ is #-selfadjoint. In another paper [17], we used Theorem 21.8 to prove the existence of a self-interacting, causal quantum field in 4-dimensional space-time. Formally this field theory is Lorentz covariant and has non-trivial scattering; this application was the motivation for the present results. In order to investigate the meaning of #-lim$_{n \to *\infty}$ $C_n$, we give a new definition for the strong #-convergence of a hyper infinite sequence of operators. Consequences of this definition are worked out in this section below and we give also estimates on operators $C_n$ which are sufficient to ensure that the #-lim$_{n \to *\infty}$ $C_n = C$ exists and that operator C is maximal symmetric or #-selfadjoint. This result is given in Theorem 21.5 and Corollary 21.6. We investigate also whether #-lim$_{n \to *\infty}$ $C_n = C$ is equal to $A + B$. We combine this work in Theorem 21.8, our second main theorem, where B is a singular, but nearly positive #-selfadjoint perturbation of a positive #-selfadjoint operator A. To illustrate this theorem, let $A \ge I$ and let B be essentially #-selfadjoint on domain $D^# = \cap_{n \in *\mathbb{N}} D(A^n)$. Assume now that, for some $\beta > 0$ and some $\alpha$,

$$A^{-(1-\beta)}BA^{-(1-\beta)} \text{ and } A^{\beta}BA^{\alpha} \qquad (21.1)$$

are #-densely defined, bounded in $\tilde{\mathbb{R}}_c^\#$ operators. Also, for some positive $a, \varepsilon \in \tilde{\mathbb{R}}_{c+}^\#$ satisfying: $2a + \varepsilon < 1$, suppose that there is a constant $b \in \tilde{\mathbb{R}}_c^\#$ such that, as bilinear forms on $D^\# \times D^\#$,

$$0 \le aA + B + b, \qquad (21.2)$$

$$0 \le \varepsilon A^2 + [A^{1/2}, [A^{1/2}, B]] + b. \qquad (21.3)$$

Then $A + B$ is # -selfadjoint. We see from this example that neither the operator B nor the bilinear form B need be bounded relative to A. While it may not appear evident, the conditions (21.1)-(21.3) are closely related to a more easily understandable estimate on $D^\# \times D^\#$,

$$A^2 + B^2 c(A + B)^2 + c. \qquad (21.4)$$

In fact, estimates (21.1)-(21.3) are chosen because they allow us not only to prove (1.4), but also the similar inequality where B is replaced by $B_n$. Let us now see that if $A + B$ is #-selfadjoint, then (1.4) must hold for every vector in $D(A + B) = D(A) \cap D(B)$.

**Proposition 21.1** Let A and B be #-closed operators. Then A + B is #-closed if and only if there is a constant $c \in \tilde{\mathbb{R}}_c^\#$ such that for all $\psi \in D(A + B)$

$$\|A\psi\|_{\#} + \|B\psi\|_{\#} \le \|(A+B)\psi\|_{\#} + c\|\psi\|_{\#} \qquad (21.5)$$

and (21.5) is equivalent to (21.4) on $D(A + B) \times D(A + B)$.

*Proof:* Certainly (21.5) implies that $A + B$ is #-closed. Conversely, assume that $A + B$ is #-closed and introduce the #-norms on $D(A + B) = D(A) \cap D(B)$,

$$\|\psi\|_{\#1} \triangleq \|\psi\|_{\#} + \|A\psi\|_{\#} + \|B\psi\|_{\#}, \qquad (21.6)$$

$$\|\psi\|_{\#2} \triangleq \|\psi\|_{\#} + \|(A + B)\psi\|_{\#} \qquad (21.7)$$

Then $D(A + B)$, $\|\psi\|_{\#2}$ is a non-Archimedean Banach space because $A + B$ is #-closed. The identity map from $D(A + B)$, $\|\psi\|_{\#2}$ to $D(A + B)$, $\|\psi\|_{\#1}$ has a #-closed graph because A, B, and $A + B$ are #-closed. By the #-closed graph theorem, the identity map is #-continuous; hence

$$\|\psi\|_{\#1} \le c \|\psi\|_{\#2}. \qquad (21.8)$$

**Proposition 21.2** Let $A \ge I, B$ be #-selfadjoint operators with $D^\# \subset D(B)$ and suppose that (21.2) and (21.3) hold. Then (21.4) is valid on domain $D^\# \times D^\#$.

*Proof:* The operators $A^2, B^2, AB, BA$, and $A^{1/2}BA^{1/2}$ define bilinear forms on $D^\# \times D^\#$. Using (21.2) and (21.3), we have the inequality: