$$[iH, M^{0k}] = P\left(\frac{a^{\#}(x_k g_0^{(k)})}{d^{\#}x_k}\right) + 4\lambda \mathrm{Ext} - \int_{\mathbb{R}_c^{\#3}} \{x_k g_1(x) - \alpha g_1(x) - x_k g_1(x) g_0^{(k)}(x)\} : \varphi_{\chi}^{\#3}(x) \pi_{\chi}^{\#}(x) : d^{\#3}x = \\ P\left(\frac{a^{\#}(x_k g_0^{(k)})}{d^{\#}x_k}\right),$$

This equality holds by the conditions (154). Hence the equality (143) holds on $D(H^2) \times D(H^2)$ and on the domain $D((M^{0k})^2) \times D((M^{0k})^2)$.

**Theorem 11.31** If $n \ge 2$, $D(H^n)$ is a \#-core for $M$ and $D((M^{0k})^n)$ is a \#-core for $H$.

**Theorem 11.32** Let $f \in S_{\mathrm{fin}}^{\#}(\mathbb{R}_c^{\#3})$ and $\mathrm{supp} f \subset \mathbb{R}_c^4_{[a,b]}$, then the operator $\varphi^{\#}(f)$ is defined on $D((M^{0k})^2)$, $\varphi^{\#}(f): ((M^{0k})^2) \to D(M^{0k})$, $k = 1,2,3$ and, as the operator equalities on $D(M^{0k})$, $k = 1,2,3$

$$[iM^{0k}, \varphi_{\chi}^{\#}(f)] = -\varphi_{\chi}^{\#} \left( t \frac{\partial^{\#} f}{\partial^{\#} x_k} + x_k \frac{\partial^{\#} f}{\partial^{\#} t} \right). \qquad (158)$$

**Remark 11.11** Note that for $f$ real, the operator $\varphi_{\chi}^{\#}(f)$ is essentially self \#-adjoint on $D(H^n)$ for any $n \ge 1/2$ and

$$\varphi_{\chi}^{\#}(f): D((H+b)^n) \to D((H+b)^{n-\frac{1}{2}}). \qquad (159)$$

**Proof** The terms in (158) are operators on $D(H^3)$ since $\varphi_{\chi}^{\#}(f)D(H^3) \subset D(H^2) \subset D(M^{0k})$, $k = 1,2,3$ and $M^{0k}D(H^3) \subset D(H) \subset D(\varphi_{\chi}^{\#}(f))$ by (157) and Theorem 11.26. Note that by Theorem 11.40 (158) holds on the domain $D(H^5)$. Assuming this, we now can to prove the theorem. Let $\psi \in D((M^{0k})^2)$, $k = 1,2,3$. By Theorem 11.29, $D((M^{0k})^2) \subset D(H)$ and by (159) we get $\psi \in D(\varphi_{\chi}^{\#}(f))$. Let us prove now that

$$\varphi_{\chi}^{\#}(f)\psi \in D(M^{0k}), k = 1,2,3. \qquad (160)$$

Note that $M^{0k}\psi \in D(M^{0k}) \subset D((H+b)^{\frac{1}{2}}) \subset D(\varphi_{\chi}^{\#}(f))$ by Theorem 11.29 and (159), also for $k = 1,2,3$

$$\psi \in D \left( \varphi_{\chi}^{\#} \left( t \frac{\partial^{\#} f}{\partial^{\#} x_k} + x_k \frac{\partial^{\#} f}{\partial^{\#} t} \right) \right).$$

Therefore by the assumption mentioned above that (158) holds on domain $D(H^5)$, we get for all $k = 1,2,3$ and for all $\chi \in D(H^5)$ that

$$\langle M^{0k}\chi, \varphi_{\chi}^{\#}(f)M^{0k}\psi \rangle = \langle \chi, \varphi_{\chi}^{\#}(f)M^{0k}\psi \rangle + i \langle \chi, \varphi_{\chi}^{\#} \left( t \frac{\partial^{\#} f}{\partial^{\#} x_k} + x_k \frac{\partial^{\#} f}{\partial^{\#} t} \right) \psi \rangle. \qquad (161)$$

So $\varphi_{\chi}^{\#}(f)\psi \in D((M^{0k} \upharpoonright D(H^5))^*)$ for $k = 1,2,3$. By Theorem 11.31, $D(H^5)$ is a \#-core for the $M^{0k}$, $k = 1,2,3$ and therefore we get inclusion (160). By using (160) we can rewrite (161) in the following equivalent form

$$\langle \chi, [M^{0k}, \varphi_{\chi}^{\#}(f)]\psi \rangle = \langle \chi, i\varphi_{\chi}^{\#} \left( t \frac{\partial^{\#} f}{\partial^{\#} x_k} + x_k \frac{\partial^{\#} f}{\partial^{\#} t} \right) \psi \rangle. \qquad (162)$$

Since $D(H^5)$ is \#-dense, we get $[M^{0k}, \varphi_{\chi}^{\#}(f)]\psi = i\varphi_{\chi}^{\#} \left( t \frac{\partial^{\#} f}{\partial^{\#} x_k} + x_k \frac{\partial^{\#} f}{\partial^{\#} t} \right) \psi$, proving (158) on the stated domain $D(M^{0k})$, $k = 1,2,3$.

**Remark 11.12** Let us consider the self \#-adjoint operators $M^{0k}(t) = \mathrm{Ext}-\exp(-itH)M^{0k}\mathrm{Ext}-\exp(itH)$, $k = 1,2,3$. Since the operator $\mathrm{Ext}-\exp(itH)$ leaves $D(H^n)$ invariant, we have by Theorem 11.29 and Theorem 11.26 that $D(H^2) \subset D(M^{0k}(t))$, $k = 1,2,3$. And for $l = 2,3,4$ we have that

$$M^{0k}(t): D(H^l) \to D(H^{l-2}), k = 1,2,3. \qquad (163)$$

Let $f \in S_{\mathrm{fin}}^{\#}(\mathbb{R}_c^{\#4})$ with $\mathrm{supp} f \subset \mathbb{R}_c^4$ for $l = [a,b]$. By (159) and (160) we can to conclude that $\varphi^{\#}(f)D(H^3) \subset D(H^2) \subset D(M^{0k}(t))$, $k = 1,2,3$ and $M^{0k}(t)D(H^3) \subset D(H) \subset D(\varphi_{\chi}^{\#}(f))$ or more generally, we can replace the operator $\varphi_{\chi}^{\#}(f)$ by $\mathrm{Ext}-\exp(itH)\varphi_{\chi}^{\#}(f)\mathrm{Ext}-\exp(-itH)$. Thus for $\psi \in D(H^3)$ and $f \in S_{\mathrm{fin}}^{\#}(\mathbb{R}_c^{\#4})$ with $\mathrm{supp} f \subset \mathbb{R}_c^4$, we can to define the functions

$$F_k(t) = \langle \psi, [iM^{0k}(t), \varphi_{\chi}^{\#}(f)]\psi \rangle = \langle \psi(t), [iM^{0k}, \mathrm{Ext}-\exp(itH)\varphi_{\chi}^{\#}(f)\mathrm{Ext}-\exp(-itH)]\psi(t) \rangle, \qquad (164)$$

$$\psi(t) = \mathrm{Ext}-\exp(itH)\psi. \qquad (165)$$

Let $I = [a, b]$, $I_{\delta} = [a - \delta, b + \delta]$ and let $\mathbb{R}_{\Delta_{\delta}}$ be the causal shadow of $\Delta_{\delta} = I_{\delta} \times I_{\delta} \times I_{\delta}$. Let $\mathbb{R}_s^4$ be a set

$$\mathbb{R}_s^4 = \mathbb{R}_{\Delta_{-|s|}} \cap \{(x, t) \mid |t| < \frac{1}{2} \varepsilon, a + |s| + |t| < b - |s| - |t|\}. \qquad (166)$$

Note that the points of $\mathbb{R}_s^4$ have small times, and $\mathbb{R}_s^4$ translated by times less than $|s|$ lies in $\mathbb{R}_s^4$.

**Theorem 11.33** Let $\psi \in D(H^5)$, then $F_k(t)$, $k = 1,2,3$ in (161) is twice \#-continuously differentiable. If function $f$ has \#-compact support in $\mathbb{R}_s$, then for $|t| \le |s|$, $\frac{a^{\#2}F_k(t)}{d^{\#}t^2} \equiv 0$.
**Proof** First we prove the differentiability of $F_k(t)$, $k=1,2,3$. Let $\Delta_n$ be the difference quotient for the $n$-derivative of $Ext-\exp(itH)$ at $t=0$. For instance, $\Delta_1(\epsilon) = \epsilon^{-1}(Ext-\exp(i\epsilon H) - I)$. Note that for a given vector $\psi \in D(H^n)$, and $m+j \le n$, as $\epsilon \to_\# 0$, we get $\|H^m\{\Delta_j(\epsilon) - (iH)^j\}\psi\|_\# = \|\{\Delta_j(\epsilon) - (iH)^j\}H^m\psi\|_\# \to_\# 0$. Hence, for $\psi \in D(H^n)$, the operator valued functions $M^{0k}(Ext-\exp(itH))$ is $n-2$ times #-differentiable, since for $j \le n-2$ we get $\|M^{0k}(Ext-\exp(itH))\{\Delta_j(\epsilon) - (iH)^j\}\psi\|_\# \le \|\{\Delta_j(\epsilon) - (iH)^j\}(H+b)^2\psi\|_\# \to_\# 0$. All these functions $F_k(t)$ has the following form

$$F_k(t) = i\langle M^{0k}(Ext-\exp(itH))\psi, Ext-\exp(itH)\phi_{\chi}^{\#}(f)\psi \rangle - i\langle Ext-\exp(itH)\phi_{\chi}^{\#}(f)\psi, M^{0k}(Ext-\exp(itH))\psi \rangle.$$

For a given vector $\psi \in D(H^5)$, $\phi_{\chi}^{\#}(f)\psi \in D(H^4)$ and $F_k(t)$ is three times #-continuously #-differentiable. Note that

$$\frac{d^{\#}F_k(t)}{d^{\#}t} = \langle M^{0k}H\psi(t), Ext-\exp(itH)\phi_{\chi}^{\#}(f)\psi \rangle - \langle M^{0k}\psi(t), H(Ext-\exp(itH))\psi \rangle - \\ \langle Ext-\exp(itH)\phi_{\chi}^{\#}(f)\psi, HM^{0k}\psi(t) \rangle + \langle Ext-\exp(itH)\phi_{\chi}^{\#}(f)\psi, M^{0k}H\psi(t) \rangle. \quad (167)$$

By rearranging the terms in (167) and using the domain relations of Theorem 11.26.1 we obtain by (143) that

$$\frac{d^{\#}F_k(t)}{d^{\#}t} = \langle \psi, [H, M^{0k}(t)]\phi_{\chi}^{\#}(f)\psi \rangle - \langle \phi_{\chi}^{\#}(f)\psi, [H, M^{0k}(t)]\psi \rangle = \\ -i \langle \psi, (Ext-\exp(-itH))P\left(\frac{d^{\#}(x_k g_0^{(k)})}{d^{\#}x_k}\right) (Ext-\exp(itH))\phi_{\chi}^{\#}(f)\psi \rangle + \\ i \langle \phi_{\chi}^{\#}(f)\psi, (Ext-\exp(-itH))P\left(\frac{d^{\#}(x_k g_0^{(k)})}{d^{\#}x_k}\right) (Ext-\exp(itH))\psi \rangle.$$

By #-differentiating (168) and writing $P_k$ for the operator $P\left(\frac{d^{\#}(x_k g_0^{(k)})}{d^{\#}x_k}\right)$ we obtain

$$\frac{d^{\#2}F_k(t)}{d^{\#}t^2} = -\langle \psi, (Ext-\exp(-itH))[H, P_k](Ext-\exp(itH))\psi \rangle + \\ \langle \phi_{\chi}^{\#}(f)\psi, (Ext-\exp(-itH))[H, P_k](Ext-\exp(itH))\psi \rangle = \\ i \langle \psi(t), \left[ \tilde{P} \left( \frac{d^{\#2}(x_k g_0^{(k)})}{d^{\#}x_k^2} \right) - T_I \left( \frac{d^{\#}(g_1)}{d^{\#}x_k} \right), (Ext-\exp(itH))\phi_{\chi}^{\#}(f)(Ext-\exp(-itH))\psi \right] \rangle. \quad (169)$$

Note that the all terms in (169) are well defined. For instance, $HP_k(Ext-\exp(itH))\phi_{\chi}^{\#}(f)\psi$ is well defined since, for a given vector $\psi \in D(H^5)$, $(Ext-\exp(itH))\phi_{\chi}^{\#}(f)\psi \in D(H^4)$, and by Theorem 11.26 for all $k=1,2,3$ we obtain

$$P_k(Ext-\exp(itH))\phi_{\chi}^{\#}(f)\psi = [iH, M^{0k}](Ext-\exp(itH))\phi_{\chi}^{\#}(f)\psi.$$

Note that $HM^{0k}(D(H^4)) \subset D(H)$ and $M^{0k}H(D(H^4)) \subset D(H)$, so $HP_k(Ext-\exp(itH))\phi_{\chi}^{\#}(f)\psi$ is well defined. Now, assuming that $\supp f \subset \mathbb{R}_s^4$, $|t| \le |s|$ we can to show that $\frac{d^{\#2}F_k(t)}{d^{\#}t^2} \equiv 0, k=1,2,3$, this proof is based on the locality of the operators $S_k, k=1,2,3$

$$S_k = \tilde{P}_\chi \left( \sum_{i=1}^{i=3} \frac{d^{\#2}(x_k g_0^{(k)})}{d^{\#}x_k^2} \right) - T_I \left( \sum_{i=1}^{i=3} \frac{d^{\#}g_1}{d^{\#}x_k} \right). \quad (170)$$

The operators $S_k$ are symmetric on $D(H_0N)$ and by (153) for $k=1,2,3$ and $i=1,2,3$ $\frac{d^{\#2}(x_k g_0^{(k)})}{d^{\#}x_k^2} = 0 = \frac{d^{\#}g_1}{d^{\#}x_k}$ in a neighbourhood of $\Delta=[a,b]^3$. We prove that $S_k, k=1,2,3$ commutes with the von Neumann algebra $W(I) = \{Ext-\exp(i\phi_{\chi}^{\#}(h_1) + i\pi_{\chi}^{\#}(h_2))|h_i = \bar{h}_i \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#3}), \text{supp}h_i \subset \mathbb{R}_l\}''$ generated by the spectral projections of the time zero fields $Ext-\int_{\mathbb{R}_c^{\#3}} \phi_{\chi}^{\#}(x)h_1(x)d^{\#3}x$ and $Ext-\int_{\mathbb{R}_c^{\#3}} \pi_{\chi}^{\#}(x)h_2(x)d^{\#3}x, h_i = \bar{h}_i \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#3})$, $\text{supp}h_i \subset \mathbb{R}_l$.

**Theorem 11.34** On the domain $D(H^2)$ for $k=1,2,3$ the equalities hold

$$[S_k, W(I)]D(H^2) = 0. \quad (171)$$

**Proof** Let $D^\#$ be the domain of well-behaved vectors.

$$D^\# = \{ \psi \in \mathcal{F}^\# | \psi^{(n)} \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#3n}), \psi^{(m)} = 0 \text{ for all sufficiently large m} \}. \quad (172)$$

For $\chi_1, \chi_2 \in D^\#$, direct momentum space computation gives for all $n \in \mathbb{N}$

$$\langle S_k \chi_1, (\phi_{\chi}^{\#}(h_1) + \pi_{\chi}^{\#}(h_2))^{n} \chi_2 \rangle = \langle (\phi_{\chi}^{\#}(h_1) + \pi_{\chi}^{\#}(h_2))^{n} \chi_1, S_k \chi_2 \rangle \quad (173)$$

By easy computation we get the inequality $\|(\phi_{\chi}^{\#}(h_1) + \pi_{\chi}^{\#}(h_2))^{n} \chi\| \le c_1 c_2^n (n!)^{\frac{1}{2}}$ for constants $c_1$ and $c_2$ depending on vector $\chi \in D^\#$. Therefore $\chi \in D^\#$ are entire vectors for the operator $(\phi_{\chi}^{\#}(h_1) + \pi_{\chi}^{\#}(h_2))$, and the sum
$$ U\chi = \mathrm{Ext} - \sum_{n=0}^{\infty} \frac{(i\phi_{\chi}^{\#}(h_1) + in\pi_{\chi}^{\#}(h_2))^{n}}{n!} \chi = \mathrm{Ext} - \exp[i(\phi_{\chi}^{\#}(h_1) + \pi_{\chi}^{\#}(h_2))]\chi \quad (174) $$

\#-converges strongly. Now, we multiply (173) by $i^n(n!)^{-1}$ and by summation over $n$ using the \#-convergence of the hyper infinite series (174) we get for all $k = 1,2,3$ that $\langle S_k\chi_1, U\chi_2 \rangle = \langle U^*\chi_1, S_k\chi_2 \rangle = \langle \chi_1, US_k\chi_2 \rangle$ for $\chi_i \in D^{\#}$, $i = 1,2$. Note that this equality extends to $\chi_i \in D(H_{0\chi}N)$, $i = 1,2$ since $D^{\#}$ is a core for operators $H_{0\chi}N$ and $S_k$ and $\|S_k\chi\|_{\#} \le \mu\|(H_{0\chi}N + I)\chi\|_{\#}$ where $\mu$ is finite constant. Therefore for $\chi \in D(H_{0\chi}N)$, we have proved that $U\chi \in D(S_k^*)$ and $S_k^*U\chi = US_k\chi$, $k = 1,2,3$. For the next step we now prove that $\chi \in D(H_{0\chi}N) \Rightarrow U\chi \in D(H_{0\chi}N)$, so that $S_kU\chi = US_k\chi$, $k = 1,2,3$, since the operators $S_k$ are symmetric on $D(H_{0\chi}N)$. We define on $D(H_{0\chi}N)$ a \#-norm by $\|\chi\|_{\#} = \|(H_{0\chi}N + I)\chi\|_{\#}$; Note that the corresponding scalar product makes $D(H_{0\chi}N)$ a non-Archimedean Hubert space, say $H_{\#1}$. For the next step we now prove that the operator $\mathcal{B} = \phi_{\chi}^{\#}(h_1) + \pi_{\chi}^{\#}(h_2)$ generates a one parameter group $U(\alpha) = \mathrm{Ext} - \exp(i\alpha\mathcal{B}) = \mathrm{Ext} - \exp[i\alpha(\mathcal{B} = \phi_{\chi}^{\#}(h_1) + \pi_{\chi}^{\#}(h_2))]$ on $H_{\#1}$ and therefore we need to prove that the operator

$$ \hat{\mathcal{B}} = (H_{0\chi}N + I)\mathcal{B}(H_{0\chi}N + I)^{-1} \quad (175) $$

is a generator to one parameter group on a corresponding Fock space. Since $\hat{\mathcal{B}}$ is essentially self \#-adjoint on $D^{\#}$, and on this domain we have that

$$ \hat{\mathcal{B}} = \mathcal{B} + [H_{0\chi}N, \mathcal{B}](H_{0\chi}N + I)^{-1} = \mathcal{B} + [N, \mathcal{B}]H_{0\chi}(H_{0\chi}N + I)^{-1} + N[H_{0\chi}, \mathcal{B}](H_{0\chi}N + I)^{-1} = \mathcal{B} + A. $$

Hear $A$ is bounded operator. Note that $\hat{\mathcal{B}} \uparrow D^{\#}$ is a bounded perturbation of an essentially self \#-adjoint operator. Hence it \#- closure \#- $(\hat{\mathcal{B}} \uparrow D^{\#})$ generates a one parameter group on Fock space $\mathcal{F}^{\#}$, and operator $\mathcal{B} \uparrow (H_{0\chi}N + I)D^{\#}$ has a \#- closure in $H_{\#1}$ that generates a one parameter group on $H_{\#1}$. Since the topology of $H_{\#1}$ is stronger than that of $\mathcal{F}^{\#}$, the \#-closure of $\mathcal{B} \uparrow (H_{0\chi}N + I)D^{\#}$ in $H_{\#1}$ is a restriction of \#- $\hat{\mathcal{B}}$ in $\mathcal{F}^{\#}$ and the one parameter group in $H_{\#1}$ is a restriction of the one parameter group generated by \#- $\hat{\mathcal{B}}$ in $\mathcal{F}^{\#}$. This proves that

$$ U: D(H_{0\chi}N) \to D(H_{0\chi}N) $$

Therefore we have proved that $S_kU\chi = US_k\chi$, $k = 1,2,3$. Now by passing to strong limits of linear combinations of such operators $U$ we obtain (165) on restricting to the domain $D(H^2) \subset D(H_{0\chi}N)$. This makes precise the statement that operators $S_k$, $k = 1,2,3$ are localized outside $\Delta = [a, b]^3$.

**Remark 11.13** Note that for each $t_1$, $|t_1| \le |s_1|$, the spectral projections of $\mathrm{Ext} - \int_{\mathbb{R}^{\#3}_c} \phi_{\chi}^{\#}(x)f(x, t_1)d^{\#3}x$ belong to $W(\#\mathrm{int}(\Delta_{-|s|}))$, where $\#\mathrm{int}(\Delta_{-|s|})$ is the \#-interior of $\Delta_{-|s|} = \{x | (x, t_1) \in \mathbb{R}^4_s\} = \{(x_1, x_2, x_3) | a + |s| < x_k < b - |s|\}$. Note that $\mathrm{supp}f \subset \mathbb{R}^4_s$, hence the spectral projections of

$$ \mathrm{Ext} - \exp[iH(t + t_1)] (\mathrm{Ext} - \int_{\mathbb{R}^{\#3}_c} \phi_{\chi}^{\#}(x)f(x, t_1)d^{\#3}x) \mathrm{Ext} - \exp[-iH(t + t_1)] \quad (176) $$

belong to $W(\#\mathrm{int}(\Delta_{|t|-|s|}))$. For $|t| \le |s|$, $\#\mathrm{int}(\Delta_{|t|-|s|}) \subset \Delta$; so the spectral projections of (170) belong to $W(\Delta)$. Now we use the locality property of the operators $S_k$, $k = 1,2,3$. Note that for vector $\chi \in D(H^2), \psi \in D(H^3)$ we have that $\psi \in D(\mathrm{Ext} - \int_{\mathbb{R}^{\#3}_c} \phi_{\chi}^{\#}(x, 0)f(x, t_1)d^{\#3}x)$, and for $\phi_{\chi}^{\#}(f) = \mathrm{Ext} - \int_{\mathbb{R}^{\#3}_c} \phi_{\chi}^{\#}(x, t)f(x, t)d^{\#3}x d^{\#}t$, by (159) it follows

$$ \mathrm{Ext} - \exp[iH] \phi_{\chi}^{\#}(f) \mathrm{Ext} - \exp[iH] \psi \in D(H^2). \quad (177) $$

Therefore by (171) and the localization of (176) for all $k = 1,2,3$ we get

$$ \langle S_k\chi, \mathrm{Ext} - \exp[iH(t + t_1)] \rangle (\mathrm{Ext} - \int_{\mathbb{R}^{\#3}_c} \phi_{\chi}^{\#}(x)f(x, t_1)d^{\#3}x) \mathrm{Ext} - \exp[-iH(t + t_1)]\psi = \quad (178) $$

$$ \langle \mathrm{Ext} - \exp[iH(t + t_1)] \rangle (\mathrm{Ext} - \int_{\mathbb{R}^{\#3}_c} \phi_{\chi}^{\#}(x)f(x, t_1)d^{\#3}x) \mathrm{Ext} - \exp[-iH(t + t_1)]\chi, S_k\psi \rangle. $$

Note that for $|t| \le |s|$ and $f \in S_{\mathrm{fin}}^{\#}(\mathbb{R}^{\#4}_c)$ with $\mathrm{supp}f \subset \mathbb{R}^4_s$ we can integrate the equality (178) over $t_1$ to obtain

$$ \langle S_k\chi, \mathrm{Ext} - \exp[iH(t)]\phi_{\chi}^{\#}(f)\mathrm{Ext} - \exp[-iH(t)]\psi \rangle = \langle \mathrm{Ext} - \exp[iH(t)]\phi_{\chi}^{\#}(f)\mathrm{Ext} - \exp[-iH(t)]\chi, S_k\psi \rangle = \quad (179) $$

$$ \langle \chi, S_k \mathrm{Ext} - \exp[iH(t)] \phi_{\chi}^{\#}(f) \mathrm{Ext} - \exp[-iH(t)] \psi \rangle. $$

Here the last equality in (179) follows by (177) and the fact that $S_k$ is a symmetric operator on $D(H_{0\chi}N) \supset D(H^2)$. From (179) we obtain that $S_k\psi \in D((\mathrm{Ext} - \exp[iH(t)]\phi_{\chi}^{\#}(f)\mathrm{Ext} - \exp[-iH(t)]) \uparrow D(H^2))^*$ and therefore that $S_k\psi \in D(\mathrm{Ext} - \exp[iH(t)]\phi_{\chi}^{\#}(f)\mathrm{Ext} - \exp[-iH(t)])$, since $D(H^2)$ is a \#-core for $\phi_{\chi}^{\#}(f)$. Finally from (179) we get for $|t| \le |s|$ and $f \in S_{\mathrm{fin}}^{\#}(\mathbb{R}^{\#4}_c)$ with $\mathrm{supp}f \subset \mathbb{R}^4_s$ for all $k = 1,2,3$ that

$$ S_k \mathrm{Ext} - \exp[iH(t)] \phi_{\chi}^{\#}(f) \mathrm{Ext} - \exp[-iH(t)] \psi = \mathrm{Ext} - \exp[iH(t)] \phi_{\chi}^{\#}(f) \mathrm{Ext} - \exp[-iH(t)] S_k \psi. \quad (180) $$

We apply the relation (180) to (169). In that case $\psi(t) \in D(H^5) \subset D(H^3)$, so $\frac{d^{\#2}F_k(t)}{d^{\#}t^2} \equiv 0$, for $|t| \le |s|$.
**Theorem 11.35** [15] Let $f \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#4})$ and $\text{supp} f \subset \mathbb{R}_s^4$, then on domain $D(H^5)$ the operator equalities hold for all $k = 1,2,3$

$$[iM^{0k}(s), \varphi_{\chi}^{\#}(f)] = [iM^{0k}, \varphi_{\chi}^{\#}(f)] - s \left[ P \left( \frac{d^{\#}(x_k g_0^{(k)})}{d^{\#}x_k} \right), \varphi_{\chi}^{\#}(f) \right]. \quad (181)$$

The next step in the proof of Theorem 11.32 is to pass to the sharp time #-limit of Theorem 11.35, thus we need to choose a hyper infinite sequence of functions $f_n \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#4})$, $n \in \mathbb{N}$ which pick out a time zero contribution in the #-limit. Let us define now

$$A_{\chi}(f, t) = \text{Ext} - \int_{\mathbb{R}_c^{\#3}} \varphi_{\chi}^{\#}(x)f(x, t)d^{\#3}x, \quad (182)$$

$$B_{\chi}(f, t) = \text{Ext} - \int_{\mathbb{R}_c^{\#3}} \pi_{\chi}^{\#}(x)f(x, t)d^{\#3}x. \quad (183)$$

Where $\varphi_{\chi}^{\#}(x)$ and $\pi_{\chi}^{\#}(x)$ the canonical time-zero fields. For real $f \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#4})$, with #-compact support, $A_{\chi}(f, t)$ and $B_{\chi}(f, t)$ are essentially self-#-adjoint on $D((H+b)^{\frac{1}{2}})$. Let $f \in C_0^{*\infty}(\mathbb{R}_l^4)$ and let $f_n(x, t) \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#4})$, $n \in \mathbb{N}$ be a hyper infinite sequence of functions of the following form $f_n(x, t) = f_n(x, s)\delta_n(t)$ with support in $\mathbb{R}_s^4$ and #-converging in the weak sense to $f_n(x, s)\delta(t)$ as $n \to \infty$. For the vector $\psi \in D(H^5)$, the vectors $M^{0k}(s)\psi, k = 1,2,3$, and the vectors $M^{0k}\psi, P(\frac{d^{\#}(x_k g_0^{(k)})}{d^{\#}x_k})\psi$ the same as in the proof of Theorem 11.35. Note that the bilinear form $\varphi_{\chi}^{\#}(x, t)$ for $(x, t) \in \mathbb{R}_l^4$ determines a bounded operator

$$G(x, t) = (H + b)^{\frac{1}{2}} \varphi_{\chi}^{\#}(x, t)(H + b)^{-\frac{1}{2}}. \quad (184)$$

Note that the operator valued function $G(x, t)$ is #-continuous in variable $(x, t)$.

**Theorem 11.36** Let $f \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#4})$ and $\text{supp} f \subset \mathbb{R}_l^4$. Then, in the sense of bilinear forms on $D(H^5)$, for all $k = 1,2,3$

$$[iM^{0k}(s), A_{\chi}(f, s)] = [iM^{0k}, A_{\chi}(f, s)] - s[iP_{\chi}, A_{\chi}(f, s)] \quad (185)$$

$$\text{Here } P_k = P \left( \frac{d^{\#}(x_k g_0^{(k)})}{d^{\#}x_k} \right).$$

**Theorem 11.37** [15] Let $f \in C_0^{*\infty}(\mathbb{R}_l^4)$. As an equality of bilinear forms on $D(H) \times D(H)$

$$[i P_k, A_{\chi}(f, s)] = A \left( \frac{d^{\#}f}{d^{\#}x_k}, s \right). \quad (186)$$

And where $P_k$ is defined in Theorem 11.36.

**Theorem 11.38** As the equalities of bilinear forms on $D(H^2) \times D(H^2)$ for all $k = 1,2,3$

$$[iM^{0k}, A_{\chi}(f, s)] = [iH, A_{\chi}(x_k f, s)] = B_{\chi}(x_k f, s). \quad (187)$$

**Theorem 11.39** [15] Let $|f|_{\#1}$ be the #-norm $|f|_{\#1} = c (\text{Ext} - \int_{\mathbb{R}_c^{\#3}} \{\|f(\cdot, t)\|_{\#2} + \sum_{i=1}^3 \|\partial_{x_i}^\# f(\cdot, t)\|_{\#2}\} d^\#t)$.

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

$$(\partial_t^\# \varphi_{\chi}^{\#})(f) = -\varphi_{\chi}^{\#}(\partial_t^\# f) = \pi_{\chi}^{\#}(f) = [iH, \varphi_{\chi}^{\#}(f)]. \quad (188)$$

**Proof** Note that the first equality in (188) is the definition of a distribution #-derivative. The out the difference quotient $\Delta_{\varepsilon}f(x, t)$ to #-derivative $\partial_t^\# f$ reads $\Delta_{\varepsilon}f(x, t) = \frac{|f(x+\varepsilon, t)-f(x,t)|}{\varepsilon}$, note that $\#\text{-lim}_{\varepsilon\to\#0} \Delta_{\varepsilon}f(x, t) = \partial_t^\# f(x, t)$.

Note that for any vector $\psi$ such that $\psi \in D((H+b)^{\frac{1}{2}})$ by canonical consideration we get

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

We have for $\psi \in D((H+b)^{\frac{3}{2}})$ that

$$\varphi_{\chi}^{\#}(\Delta_{\varepsilon}f(x, t))\psi = \varepsilon^{-1}(I - \text{Ext}-\text{exp}[i\varepsilon H]) \{\text{Ext} - \int_{\mathbb{R}_c^{\#3}} \varphi_{\chi}^{\#}(x, t - \varepsilon)f(x, t)d^{\#3}x\psi d^{\#}t\} + \\ \varepsilon^{-1} \{\text{Ext} - \int_{\mathbb{R}_c^{\#3}} A_{\chi}(f, t)(\text{Ext}-\text{exp}[i\varepsilon H] - I)\psi d^{\#}t\}.$$

Here the last term #-converges as $\varepsilon \to \#0$ and it #-limit is: $i (\text{Ext} - \int_{\mathbb{R}_c^{\#3}} A_{\chi}(f, t)H\psi d^{\#}t)$. Since $\varphi_{\chi}^{\#}(\Delta_{\varepsilon}f(x, t))\psi$ #-converges as $\varepsilon \to \#0$, the remaining term in expression for $\varphi_{\chi}^{\#}(\Delta_{\varepsilon}f(x, t))\psi$ #-converges also to a #-limit $\psi_1$. For $\chi \in D(H)$ we obtain that

$$\langle \chi, \psi_1 \rangle = \#\text{-lim}_{\varepsilon \to \#0} \langle \chi, \varepsilon^{-1}(I - \text{Ext}-\text{exp}[i\varepsilon H]) \{\text{Ext} - \int_{\mathbb{R}_c^{\#3}} \varphi_{\chi}^{\#}(x, t - \varepsilon)f(x, t)d^{\#3}x\psi d^{\#}t\} \rangle = \langle iH\chi, \varphi_{\chi}^{\#}(f)\psi \rangle.$$
Since $H = H^*$, it follows that $\varphi_{\chi}^{\#}(f)\psi \in D(H)$ and $\psi_1 = iH\varphi_{\chi}^{\#}(f)\psi$ and therefore: $-\varphi_{\chi}^{\#}(\partial_t^{\#}f)\psi = [iH, \varphi_{\chi}^{\#}(f)]\psi$. From the above equation we obtain

$$ \langle \psi, \varphi_{\chi}^{\#}(\partial_t^{\#} f)\psi \rangle = \operatorname{Ext}- \int_{\mathbb{R}_c^{\#}} \langle H\psi(t), \operatorname{Ext}- \int_{\mathbb{R}_c^{\#3}} \varphi_{\chi}^{\#}(x, 0)f(x, t)d^{\#3}x\psi(t) \rangle d^{\#}t - \\ \operatorname{Ext}- \int_{\mathbb{R}_c^{\#}} \langle \operatorname{Ext}- \int_{\mathbb{R}_c^{\#3}} \varphi_{\chi}^{\#}(x, 0)f(x, t)d^{\#3}x\psi(t), H\psi(t) \rangle d^{\#}t. $$

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

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

Therefore finally we get

$$ i\langle \psi, \varphi_{\chi}^{\#}(\partial_t^{\#} f)\psi \rangle = \operatorname{Ext}- \int_{\mathbb{R}_c^{\#}} \langle \psi(t), -i\operatorname{Ext}- \int_{\mathbb{R}_c^{\#3}} \pi_{\chi}^{\#}(x, 0)f(x, t)d^{\#3}x\psi \rangle d^{\#}t = \langle \psi, -i\pi_{\chi}^{\#}(f)\psi \rangle. $$

This equality finalized the proof.

**Theorem 11.40** As the operator equalities on $D(H^5)$ for all $k = 1,2,3$

$$ [iM^{0k}, \varphi_{\chi}^{\#}(f)] = -\varphi_{\chi}^{\#} \left( t \frac{\partial^{\#}f}{\partial^{\#}x_k} + x_k \frac{\partial^{\#}f}{\partial^{\#}t} \right). \qquad (189) $$

**Proof** We first prove (189) as equalities of bilinear forms on $D(H^5) \times D(H^5)$. Let $\psi$ is a near standard vector and $\psi \in D(H^5)$. By Theorems 11.37-11.39, for all $k = 1,2,3$ we get

$$ \langle \psi, iM^{0k}(s), A_{\chi}(f, s)\psi \rangle = \langle \psi, B_{\chi}(x_k f, s)\psi, \rangle - \langle \psi, A \left( \frac{d^{\#}f}{d^{\#}x_k}, s \right) \psi \rangle. $$

Substituting $\operatorname{Ext}-\operatorname{exp}(iHs)$ for $\psi$, we obtain that

$$ \langle \psi, [iM^{0k}, \operatorname{Ext}-\operatorname{exp}(iHs)A_{\chi}(f, s)\operatorname{Ext}-\operatorname{exp}(-iHs)]\psi \rangle = \\ \langle \psi, \operatorname{Ext}-\operatorname{exp}(iHs) \left\{ B_{\chi}(x_k f, s) - A \left( s \frac{d^{\#}f}{d^{\#}x_k}, s \right) \right\} \operatorname{Ext}-\operatorname{exp}(-iHs)\psi \rangle. \qquad (190) $$

From (188) we get

$$ \operatorname{Ext}- \int_{\mathbb{R}_c^{\#4}} \operatorname{Ext}-\operatorname{exp}(iHt) \pi_{\chi}^{\#}(x) \operatorname{Ext}-\operatorname{exp}(iHt) f(x, t) d^{\#3}x d^{\#}t = -\varphi_{\chi}^{\#} \left( \frac{\partial^{\#}f}{\partial^{\#}t} \right). \qquad (191) $$

Using (191) we integrate (190) over $s$ to obtain for all $k = 1,2,3$ the equalities of bilinear forms

$$ \langle \psi, iM^{0k}, \varphi_{\chi}^{\#}(f)\psi \rangle = - \langle \psi, \varphi_{\chi}^{\#} \left( t \frac{\partial^{\#}f}{\partial^{\#}x_k} + x_k \frac{\partial^{\#}f}{\partial^{\#}t} \right) \psi \rangle. \qquad (192) $$

Since $M^{0k}\varphi_{\chi}^{\#}(f)$, $\varphi_{\chi}^{\#}(f)M^{0k}$, and $\varphi_{\chi}^{\#} \left( t \frac{\partial^{\#}f}{\partial^{\#}x_k} + x_k \frac{\partial^{\#}f}{\partial^{\#}t} \right)$ are operators on $D(H^5)$ for all $k = 1,2,3$, the operator equalities (189) follows by polarization and the #-density of $D(H^5)$. This final remark completes the proof of the theorem and hence it completes the proof of Theorem 11.32.

**Theorem 11.41** [15] Let $\mathcal{R} \subset *\mathbb{R}_{c,\text{fin}}^{\#4}$ be an bounded region in $*\mathbb{R}_{c,\text{fin}}^{\#4}$ and let $F_k(\beta, x, t)$, $k = 1,2,3$ be a functions such that $F_k(\beta, x, t)$, $\beta \in *\mathbb{R}_{c,\text{fin}}^{\#4}$ and $\frac{\partial^{\#}F_k(\beta, x, t)}{\partial^{\#}\beta}$ are #-continuous in $(\beta, x, t)$, where the partial #-derivative exists for each point $(x, t) \in *\mathbb{R}_{c,\text{fin}}^{\#4}$. Assume that for all $f(x, t) \in C_{0,\text{fin}}^{*\infty}(\mathcal{R})$ the following equalities hold for all $k = 1,2,3$,

$$ \operatorname{Ext}- \int_{\mathbb{R}_c^{\#3}} \frac{\partial^{\#}F_k(\beta, x, t)}{\partial^{\#}\beta} f(x, t) d^{\#3}x d^{\#}t = - \operatorname{Ext}- \int_{\mathbb{R}_c^{\#3}} F_k(\beta, x, t) \left[ x_k \frac{\partial^{\#}f}{\partial^{\#}t} + t \frac{\partial^{\#}f}{\partial^{\#}x_k} \right] d^{\#3}x d^{\#}t. \qquad (193) $$

Then for all $(\beta, x, t)$ such that $\Lambda_{\gamma\beta}(x, t) \in \mathcal{R}$ for $0 \le \gamma \le 1$, $k = 1,2,3$

$$ F_k(\beta, x, t) = F_k(0, \Lambda_{\gamma\beta}(x, t)) + \delta(\beta, x, t) = \\ F_k(0, x_k \cosh \beta + t \sinh \beta, x_k \sinh \beta + t \cosh \beta) + \delta(\beta, x, t). \qquad (194) $$

Here $\delta(\beta, x, t)$ is a nonzero function such that $\delta(\beta, x, t) \ne 0$ and $\delta(\beta, x, t)$ is #-differentiable with zero partial #-derivatives $\delta_{\beta}^{\prime\#}(\beta, x, t) \equiv 0$, $\delta_{x_k}^{\prime\#}(\beta, x, t) \equiv 0$, $\delta_{t}^{\prime\#}(\beta, x, t) \equiv 0$.
**Proof** Obviously (194) is a solution to the equations (193). Thus we need prove uniqueness (194) for a given function $\delta(\beta, x, t)$ and for all $k = 1,2,3$ and it is sufficient to prove uniqueness for the case $F_k(0, x, t) = \delta(0, x, t)$. Let $A_k$ be the operator $A_k = x_k \frac{\partial}{\partial x_k} + t \frac{\partial}{\partial x_k}$. Note that by (177), provided $\text{supp} f(\Lambda_{\gamma\beta'}(x, t)) \subset \mathbb{R}$ we get

$$ \text{Ext-} \int_{\mathbb{R}_c^{\#3}} \left\{ \frac{\partial^\# F_k(\beta', x, t)}{\partial^\# \beta'} f(\Lambda_{\gamma\beta'}(x, t)) d^{\#3}x d^\#t \right\} = \text{Ext-} \int_{\mathbb{R}_c^{\#3}} F_k(\beta', x, t) f(\Lambda_{\gamma\beta'}(x, t)) d^{\#3}x d^\#t = 0. \quad (195) $$

Let $\tilde{\mathbb{R}} = \cap_{0 \le \gamma \le 1} \Lambda_{\gamma\beta} \mathbb{R}$ and $f(x, t) \in C_{0,\text{fin}}^{*\infty}(\tilde{\mathbb{R}})$, then (195) holds for all $\beta'$ such that $0 \le \beta' \le \beta$. Note that for all functions $f(x, t) \in C_{0,\text{fin}}^{*\infty}(\mathbb{R})$ the following equalities (196) hold for all $k = 1,2,3$,

$$ \text{Ext-} \int_{\mathbb{R}_c^{\#3}} F_k(\beta, x, t) f(\Lambda_{\gamma\beta'}(x, t)) d^{\#3}x d^\#t = 0. \quad (196) $$

Thus, in the sense of distributions we obtain that

$$ F_k(\beta, x, t) = 0, (x, t) \in \tilde{\mathbb{R}}. \quad (197) $$

Since $F_k(\beta, x, t)$ is #-continuous, (197) holds in usual sense everywhere in $\tilde{\mathbb{R}}$. This establishes required uniqueness, and completes the proof of the theorem.

**Definition 11.18** (1) Let $(H_\#)$, $\|\cdot\|_\#$ be a linear normed space over field $\mathbb{C}_c^\#$. An element $x \in H_\#$ is called finite or norm finite if $\|x\|_\# \in \mathbb{R}_{c,\text{fin}}^\#$ and we let $\text{Fin}(H_\#)$ denote the set of the all finite elements of $H_\#$; the element $x \in H_\#$ is called infinitesimal if $\|x\|_\# \approx 0$ and we write $x \approx y$ for $\|x - y\|_\# \approx 0$. (2) Let $(H_\#), \langle \cdot, \cdot \rangle_\#$ be a non-Archimedean Hilbert space over field $\mathbb{C}_c^\#$ endowed with a canonical #-norm $\|x\|_\# = \sqrt{\langle x, x \rangle_\#}$, then we apply the same definition as in (1).

**Definition 11.19** Let $A$ be a linear operator $A: H_\# \to H_\#$ with domain $D(A)$. Let $D_{\text{fin}}(A) \subset D(A)$ be a subdomain such that for all $\psi \in D(A)$: $\psi \in D_{\text{fin}}(A) \Leftrightarrow \|x\|_\# \in \mathbb{R}_{c,\text{fin}}^\#$ and let $D_{\text{fin}}^\#(A)$ be a subdomain $D_{\text{fin}}^\#(A) \subset D_{\text{fin}}(A)$ such that for all $\psi \in D_{\text{fin}}(A)$: $\psi \in D_{\text{fin}}^\#(A) \Leftrightarrow \|Ax\|_\# \in \mathbb{R}_{c,\text{fin}}^\#$.

**Definition 11.20** Let $q(\cdot, \cdot)$ be a bilinear form with domain $D(q) \times D(q)$ on $H_\#$ such that $D(q) \times D(q) \subset H_\# \times H_\#$ and $D(q) \times D(q) \to \mathbb{C}_c^\#$. Let $D_{\text{fin}}(q) \times D_{\text{fin}}(q) \subset D(q) \times D(q)$ be a subdomain such that for all $\{\psi_1, \psi_2\} \in D_{\text{fin}}(q) \times D_{\text{fin}}(q) \Leftrightarrow |\langle \psi_1, \psi_2 \rangle_\#| \in \mathbb{R}_{c,\text{fin}}^\#$. Let $D_{\text{fin}}^\#(q) \times D_{\text{fin}}^\#(q) \subset D_{\text{fin}}(q) \times D_{\text{fin}}(q)$ be a subdomain such that for all $\{\psi_1, \psi_2\} \in D_{\text{fin}}(q) \times D_{\text{fin}}(q): \{\psi_1, \psi_2\} \in D_{\text{fin}}^\#(q) \times D_{\text{fin}}^\#(q) \Leftrightarrow q(\psi_1, \psi_2) \in \mathbb{C}_{c,\text{fin}}^\#$.

**Theorem 11.42** [15] Assume that the operators $M^{0k} = M_{\lambda}^{0k} = M_{0,\lambda}^{0k} + M_{I,\lambda}^{0k}$, $k = 1,2,3$ satisfy conditions (152)-(154) and where the operators $M_{0,\lambda}^{0k}$ are defined by (125). We set now $\delta(\beta, x, t) \approx 0$.

(1) If $f \in S_{\text{fin}}^\#(*\mathbb{R}_c^\#)$, $\text{supp} f \subset \#\text{-int}(\mathbb{R}_\Delta^4)$, $\Delta = [a, b]^3$ and $\text{supp} f_{\Lambda(\beta)} \subseteq \#\text{-int}(\mathbb{R}_\Delta^4) = \wp_\Delta^4$, then for all $k = 1,2,3$ on domains $D_{\text{fin}}((M^{0k})^2)$

$$ \text{Ext-} \exp(iM^{0k}\beta)\varphi_{\lambda}^{\#}(f)\text{Ext-} \exp(-iM^{0k}\beta) \approx \varphi_{\lambda}^{\#}(f_{\Lambda(\beta)}). \quad (198) $$

Here the $\approx$ - equalities (198) hold as $\approx$ - equalites for self #-adjoint operators.

(2) If $(x, t) \in \mathbb{R}_\Delta^4$ and $\Lambda_\beta(x, t) \in \mathbb{R}_\Delta^4$, then for all $k = 1,2,3$

$$ \text{Ext-} \exp(iM^{0k}\beta)\varphi_{\lambda}^{\#}(x, t)\text{Ext-} \exp(-iM^{0k}\beta) \approx \varphi_{\lambda}^{\#}(\Lambda_{\beta}(x, t)) \quad (199) $$

Here the $\approx$ - equalities (199) hold in the sense of $*\mathbb{R}_{c,\text{fin}}^\#$- valued bilinear forms on domains $D_{\text{fin}}^\#(M^{0k}) \times D_{\text{fin}}^\#(M^{0k})$ and on domains $D_{\text{fin}}^\#(M^{0k}) \times D_{\text{fin}}^\#(M^{0k})$.

**Remark 11.15** Note that (1) for real-valued $f \in S_{\text{fin}}^\#(*\mathbb{R}_c^\#)$ is a self-#-adjoint operator $\varphi_\lambda^\#(f)$, essentially self-#-adjoint operator on a variety of appropriate domains. It is for this self #-adjoint operator that (198) is valid; (2) on the subdomains $D_{\text{fin}}^\#((M^{0k})^2) \approx$ -equalites (198) entail for all $k = 1,2,3$ the equalities

$$ \text{st}(\text{Ext-} \exp(iM^{0k}\beta)\varphi_{\lambda}^{\#}(x, t)\text{Ext-} \exp(-iM^{0k}\beta)) = \text{st}(\varphi_{\lambda}^{\#}(\Lambda_{\beta}(x, t))); $$

(3) on the subdomains $D_{\text{fin}}^\#((M^{0k})^2)$ the $\approx$ -equalites (198) entail for all $k = 1,2,3$ the equalities

$$ \text{st}(\text{Ext-} \exp(iM^{0k}\beta)\varphi_{\lambda}^{\#}(f)\text{Ext-} \exp(-iM^{0k}\beta)) = \text{st}(\varphi_{\lambda}^{\#}(f_{\Lambda(\beta)})). $$

**Proof** Let $\psi \in D(M^{0k})$ and let $F_k(\beta, x, t)$ be the function is defined by

$$ F_k(\beta, x, t) = \langle \text{Ext-} \exp(-iM^{0k}\beta)\psi, \varphi_{\lambda}^{\#}(x, t)(\text{Ext-} \exp(-iM^{0k}\beta)\psi) \rangle. \quad (200) $$

For all $(\beta, x, t) \in *\mathbb{R}_{c,\text{fin}}^\# \times *\mathbb{R}_{c,\text{fin}}^\#$ and for $f \in S_{\text{fin}}^\#(*\mathbb{R}_c^\#)$, let $F_k(\beta, f)$ be the function is defined by

$$ F_k(\beta, f) = \langle \text{Ext-} \exp(-iM^{0k}\beta)\psi, \varphi_{\lambda}^{\#}(f)(\text{Ext-} \exp(-iM^{0k}\beta)\psi) \rangle = $$
$$ \mathrm{Ext} - \int_{\wp_{\Delta}^{4}} F_{k}(\beta, x, t)f(x, t)d^{\#3}xd^{\#}t. \qquad (201) $$

Note that $\varphi_{x}^{\#}(x, t)$ is a bilinear form defined on $D((H+b)^{\frac{3}{2}}) \times D((H+b)^{\frac{3}{2}})$, \#-continuous in $(x, t) \in \mathbb{R}_{c, \text{fin}}^{\#4}$.
By Theorem 11.29 $D(M^{0k}) \subset D((H+b)^{\frac{1}{2}})$ and therefore $F_k(\beta, x, t)$ is well defined and \#-continuous in $(x, t)$.
Note that a function $F_k(\beta, x, t)$ is \#-continuously \#-differentiable in $\beta \in \mathbb{R}_{c, \text{fin}}^{\#4}$ and for all $k = 1,2,3$

$$ \begin{aligned} \frac{\partial^{\#} F_k(\beta, x, t)}{\partial^{\#} \beta} = & -\langle \mathrm{Ext} - \exp(-iM^{0k}\beta)iM^{0k}\psi, \varphi_{x}^{\#}(f)(\mathrm{Ext} - \exp(-iM^{0k}\beta)\psi) \rangle \\ & -\langle \mathrm{Ext} - \exp(-iM^{0k}\beta)\psi, \varphi_{x}^{\#}(f)(\mathrm{Ext} - \exp(-iM^{0k}\beta)iM^{0k}\psi) \rangle. \end{aligned} \qquad (202) $$

By the canonical argument, we have for all $k = 1,2,3$ that

$$ \frac{\partial^{\#} F_k(\beta, f)}{\partial^{\#} \beta} = \langle \mathrm{Ext} - \exp(-iM^{0k}\beta)\psi, [iM^{0k}, \varphi_{x}^{\#}(f)](\mathrm{Ext} - \exp(-iM^{0k}\beta)\psi) \rangle = \qquad (203) \\ \mathrm{Ext} - \int_{\wp_{\Delta}^{4}} F_k(\beta, x, t)f(x, t)d^{\#3}xd^{\#}t. $$

By Theorem 11.40 under the condition $\mathrm{supp} f \subset \#\text{-int}(\mathbb{R}_{\Delta}^{4})$ we have for all $k = 1,2,3$ that

$$ \begin{aligned} \frac{\partial^{\#} F_k(\beta, f)}{\partial^{\#} \beta} = & -\langle \mathrm{Ext} - \exp(-iM^{0k}\beta)\psi, \varphi_{x}^{\#} \left( x_k \frac{\partial^{\#} f}{\partial^{\#} t} + t \frac{\partial^{\#} f}{\partial^{\#} x_k} \right) \mathrm{Ext} - \exp(-iM^{0k}\beta)\psi \rangle = \\ & -\mathrm{Ext} - \int_{\mathbb{R}_{c}^{\#3}} F_k(\beta, x, t) \left( x_k \frac{\partial^{\#} f}{\partial^{\#} t} + t \frac{\partial^{\#} f}{\partial^{\#} x_k} \right) f(x, t) d^{\#3}xd^{\#}t. \end{aligned} \qquad (204) $$

Therefore by Theorem 11.40 under the condition

$$ U_{0 \le \gamma \le 1} \Lambda_{\gamma\beta}(x, t) \in \mathbb{R}_{\Delta}^{4} \qquad (205) $$

we have for all $k = 1,2,3$ that

$$ F_k(\beta, x, t) = F_k(0, \Lambda_{\gamma\beta}(x, t)) + \delta(\beta, x, t) \qquad (206) $$

That is, if (205) holds, then (206) also holds for all $k = 1,2,3$ and finally we get

$$ \mathrm{Ext} - \exp(iM^{0k}\beta)\varphi_{x}^{\#}(x, t)\mathrm{Ext} - \exp(-iM^{0k}\beta) = \varphi_{x}^{\#}(\Lambda_{\beta}(x, t)) + \delta(\beta, x, t). \qquad (207) $$

Here the equations (207) hold in the sense of bilinear forms on $D((M^{0k})^2) \times D((M^{0k})^2)$, i.e.

$$ \langle \psi_1, \mathrm{Ext} - \exp(iM^{0k}\beta)\varphi_{x}^{\#}(x, t)\mathrm{Ext} - \exp(-iM^{0k}\beta)\psi_2 \rangle = \langle \psi_1, \varphi_{x}^{\#}(\Lambda_{\beta}(x, t))\psi_2 \rangle + \delta(\beta, x, t)\langle \psi_1, \psi_2 \rangle. \qquad (208) $$

From (208) on the domain $D_{\text{fin}}^{\#}((M^{0k})^2) \times D_{\text{fin}}^{\#}((M^{0k})^2) \subset D_{\text{fin}}((M^{0k})^2) \times D_{\text{fin}}((M^{0k})^2) \subset D((M^{0k})^2) \times D((M^{0k})^2)$ we get the $\approx$ -equality

$$ \langle \psi_1, \mathrm{Ext} - \exp(iM^{0k}\beta)\varphi_{x}^{\#}(x, t)\mathrm{Ext} - \exp(-iM^{0k}\beta)\psi_2 \rangle \approx \langle \psi_1, \varphi_{x}^{\#}(\Lambda_{\beta}(x, t))\psi_2 \rangle, \qquad (209) $$

since $\langle \psi_1, \psi_2 \rangle$ is finite and therefore $\delta(\beta, x, t)\langle \psi_1, \psi_2 \rangle \approx 0$.

Note that in the \#-limit $\lambda \to 0$ by (125) we get

$$ \#\text{-lim}_{\lambda \to 0} M^{0k} = M_{x}^{0k}. \qquad (210) $$

Therefore in the \#-limit $\lambda \to 0$ from (208) and (210) we obtain that

$$ \lim_{\lambda \to 0} \langle \psi_1, \mathrm{Ext} - \exp(iM^{0k}\beta)\varphi_{x}^{\#}(x, t)\mathrm{Ext} - \exp(-iM^{0k}\beta)\psi_2 \rangle = \qquad (211) $$

$$ \langle \psi_1, \mathrm{Ext} - \exp(iM_{x}^{0k}\beta)\varphi_{0,x}^{\#}(x, t)\mathrm{Ext} - \exp(-iM_{x}^{0k}\beta)\psi_2 \rangle = $$

$$ \lim_{\lambda \to 0} \langle \psi_1, \varphi_{x}^{\#}(\Lambda_{\beta}(x, t))\psi_2 \rangle + \delta(\beta, x, t)\langle \psi_1, \psi_2 \rangle = \langle \psi_1, \varphi_{0,x}^{\#}(\Lambda_{\beta}(x, t))\psi \rangle + \delta(\beta, x, t)\langle \psi_1, \psi_2 \rangle. $$

From (211) on the domain $D_{\text{fin}}^{\#}((M^{0k})^2) \times D_{\text{fin}}^{\#}((M^{0k})^2) \subset D_{\text{fin}}((M^{0k})^2) \times D_{\text{fin}}((M^{0k})^2) \subset D((M^{0k})^2) \times D((M^{0k})^2)$ we get the $\approx$ -equality for free quantum field $\varphi_{0,x}^{\#}(x, t)$

$$ \langle \psi_1, \mathrm{Ext} - \exp(iM_{x}^{0k}\beta)\varphi_{0,x}^{\#}(x, t)\mathrm{Ext} - \exp(-iM_{x}^{0k}\beta)\psi_2 \rangle \approx \langle \psi_1, \varphi_{0,x}^{\#}(\Lambda_{\beta}(x, t))\psi_2 \rangle. \qquad (212) $$

**Remark 11.16** Note that the $\approx$ -equality required by (212) is necessary, see Remark 9.2.

The $\approx$ -equality (209) extends by \#-closure to $D_{\text{fin}}^{\#}(M) \times D_{\text{fin}}^{\#}(M)$, since $D_{\text{fin}}^{\#}(M) \subset D_{\text{fin}}^{\#}((H+b)^{1/2})$ by Theorem 11.29, and the estimate

$$ |\langle \psi, \mathrm{Ext} - \exp(iM^{0k}\beta)\varphi_{x}^{\#}(x, t)\mathrm{Ext} - \exp(-iM^{0k}\beta)\psi \rangle| \approx \qquad (213) $$

$$ |\langle \psi, \varphi_{x}^{\#}(\Lambda_{\beta}(x, t))\psi \rangle| \le c \|(H+b)^{1/2}\psi\|^2. $$

Here c is finite constant. Furthermore $D((M^{0k})^2)$ for any $k = 1,2,3$ is a \#-core for $H$, by Theorem 11.31, and therefore a \#-core for $(H+b)^{\frac{1}{2}}$. Thus (208) extends to $D((M^{0k})^2) \times D((M^{0k})^2)$ and on this domain we also have \#-continuity of the form in $(x, t) \in \mathbb{R}_{c, \text{fin}}^{\#4}$. Note that it is necessary to assume that $U_{0 \le \gamma \le 1} \Lambda_{\gamma\beta}(x, t) \in \mathbb{R}_{\Delta}^{4}$.
However for the regions $\mathbb{R}_\Delta^4$ this statement follows from the condition $(x, t) \in \mathbb{R}_\Delta^4 \Rightarrow \Lambda_\beta(x, t) \in \mathbb{R}_\Delta^4$. This final remark completes the proof of this theorem part (2). Now we go to prove the operator $\approx$ -equality (198) for the case $f \in S_{\text{fin}}^{\#}(\mathbb{R}_c^{\#4})$, $\text{supp}f \cup \text{supp}f_{\Lambda_\beta}$. By Theorem 11.29, the operators $\varphi_\kappa^\#(f)$ and $\varphi_\kappa^\#(f_{\Lambda_\beta})$ are defined on domain $D((M^{0k})^2)$. Integrating (207) against $f(x, t)$, we get the equalities

$$ \operatorname{Ext-\exp}(iM^{0k}\beta)\varphi_{\kappa}^{\#}(f)\operatorname{Ext-\exp}(-iM^{0k}\beta) = \varphi_{\kappa}^{\#}(f_{\Lambda_{\beta}}) + \operatorname{Ext-}\int_{\mathbb{R}_{\Delta}^{4}} \delta(\beta, x, t)f(x, t)d^{\#3}xd^{\#}t. \quad (214) $$

Obviously the equalities (213) hold on the domains $D((M^{0k})^2)$ with $k = 1,2,3$ correspondingly. For any vector $\psi$ such that $\psi \in D((M^{0k})^2)$ from (207) we obtain the equalities

$$ \varphi_{\kappa}^{\#}(f)\operatorname{Ext-\exp}(-iM^{0k}\beta)\psi = \operatorname{Ext-\exp}(-iM^{0k}\beta)\varphi_{\kappa}^{\#}(f_{\Lambda_{\beta}})\psi + (\operatorname{Ext-}\int_{\mathbb{R}_{\Delta}^{4}} \delta(\beta, x, t)f(x, t)d^{\#3}xd^{\#}t)\psi. \quad (215) $$

Since $\|\varphi_\kappa^\#(f_{\Lambda_\beta})\psi\| \le c_1 \|(H+b)^{\frac{1}{2}}\psi\|$ and $D((M^{0k})^2)$ for any $k = 1,2,3$ is a \#-core for $H$, by Theorem 11.31, the equalities (215) extends by \#-closure to $D(H)$ and (215) holds for $\psi \in D(H)$. Since the domain $D(H)$ is a \#-core for the operator $\varphi_\kappa^\#(f_{\Lambda_\beta})$, we conclude that (214) extends by \#-closure to $D(\varphi_\kappa^\#(f_{\Lambda_\beta}))$ and therefore the equalities (215) hold for all $k = 1,2,3$ and for any $\psi$ such that $\psi \in D(\varphi_\kappa^\#(f_{\Lambda_\beta}))$. Thus we have proved that

$$ \operatorname{Ext-\exp}(-iM^{0k}\beta)D(\varphi_{\kappa}^{\#}(f_{\Lambda_{\beta}})) \subset D(\varphi_{\kappa}^{\#}(f)). $$

By similar consideration one obtains that

$$ \operatorname{Ext-\exp}(-iM^{0k}\beta)D(\varphi_{\kappa}^{\#}(f_{\Lambda_{\beta}})) \subset D(\varphi_{\kappa}^{\#}(f)). $$

This proves (214) as an equality between self- \#-adjoint operators, completing the proof of the theorem.

## CONCLUSION

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 canonical $C^*$-algebra of bounded observables corresponding to this model satisfies all the Haag-Kastler axioms except Lorentz covariance. We prove that the $\lambda(\varphi^4)_4$ quantum field theory model is Lorentz covariant. For each Poincare transformation $a, \Lambda$ and each bounded region $O$ of Minkowski space we obtain a unitary operator $U$ which correctly transforms the field bilinear forms $\varphi(x, t)$ for $(x, t) \in O$. The von Neumann algebra $C(O)$ of local observables is obtained as standard part of external nonstandard algebra $B_\#(O)$.

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