Appendix A: Domain Questions
Lemma A.1
Let
\(n\in \mathbb {N}\). There exist constants 0≤a<1 and
b≥0 s.t. for any
\(\varPsi\in \mathcal{C}^{(n)}\)
there holds the bound
$$ \bigl\| H_{\mathrm{I}}^{(n)}\varPsi\bigr\| \leq a \bigl\| H_{\mathrm {fr}}^{(n)} \varPsi\bigr\| +b\|\varPsi\|, $$
(A.1)
where
\(H_{\mathrm{I}}^{(n)}=H_{\mathrm{I}}|_{\mathcal{C}^{(n)}}\), \(H_{\mathrm {fr}}^{(n)}=H_{\mathrm {fr}}|_{\mathcal{C}^{(n)}}\).
Proof
Let us use the form of the interaction Hamiltonian appearing in formula (1.33). We have \(H_{\mathrm{I}}^{(n)}=H_{\mathrm{I}}^{\mathrm{a}, (n)}+H_{\mathrm{I}}^{\mathrm{c}, (n)}\), where
$$ H_{\mathrm{I}}^{\mathrm{a}, (n)}:=\sum_{i=1}^n \int d^3k v_{\overline {\alpha }}(k) e^{ikx_i}a(k) $$
(A.2)
and \(H_{\mathrm{I}}^{\mathrm{c}, (n)}=(H_{\mathrm{I}}^{\mathrm{a}, (n)})^{*}\). Let us set \(C_{n}(k):= \sum_{l=1}^{n} e^{ikx_{l}}\) and compute for some \(\varPsi \in \mathcal{C}^{(n)}\)
$$\begin{aligned} \bigl\| H_{\mathrm{I}}^{\mathrm{a},(n)}\varPsi\bigr\| \leq& \int d^3k v_{\overline {\alpha }}(k) \bigl\| C_n(k) a(k)\varPsi \bigr\| \leq n \int d^3k v_{\overline {\alpha }}(k) \bigl\| a(k)\varPsi\bigr\| \\ \leq& n \bigl\| \omega ^{-1/2}v_{\overline {\alpha }}\bigr\| _2\langle \varPsi,H_{\mathrm {f}} \varPsi \rangle ^{ \frac {1}{2}}\leq \frac {1}{4}\|H_{\mathrm {fr}}\varPsi \|+n^2\bigl\| \omega ^{-1/2}v_{\overline {\alpha }}\bigr\| _2^2\|\varPsi \|, \end{aligned}$$
(A.3)
where in the last step we anticipate that (A.1) should hold with 0<a<1.
Let us now consider the creation part of \(H_{\mathrm{I}}^{(n)}\). Making use of the canonical commutation relations, we get
$$\begin{aligned} \bigl\| H_{\mathrm{I}}^{\mathrm{c},(n)}\varPsi\bigr\| ^2 =&\int d^3k_1d^3k_2 v_{\overline {\alpha }}(k_1)v_{\overline {\alpha }}(k_2) \bigl\langle C_n(k_1)^*a^*(k_1) \varPsi,C_n(k_2)^*a^*(k_2)\varPsi\bigr\rangle \\ \leq& \bigl\| H_{\mathrm{I}}^{\mathrm{a},(n)}\varPsi\bigr\| ^2+ n^2 \|v_{\overline {\alpha }}\|_2^2 \|\varPsi\|^2, \end{aligned}$$
(A.4)
which, together with (A.3), concludes the proof. □
Lemma A.2
The domain
\(\mathcal{D}\), defined in (2.19), is contained in the domains of
H, H
e, H
f, \(H_{\mathrm{I}}^{\mathrm{a}/\mathrm{c}}\), \(\check{H}_{\mathrm{I},\sigma }\)
and
\(\hat{\eta}^{*}(h)\), \(h\in C_{0}^{2}(S)\). Moreover, these operators leave
\(\mathcal{D}\)
invariant.
Proof
Let \(\varPsi_{l}^{r_{1},r_{2}}\) be a vector of the form (2.17). Then
$$\begin{aligned} & \hat{\eta}^*_{\sigma }(h)\varPsi_l^{r_1,r_2} \\ &\quad {}=\sum_{m,m'}\frac{1}{\sqrt{m'!}} \frac{1}{\sqrt{m!}} \int d^3p' d^{3l}p d^{3m'}k' d^{3m}k F'_{1,m'} \bigl(p';k'\bigr)F_{l,m}(p;k) \\ &\qquad {}\times \eta ^*\bigl(p'\bigr) \eta ^*(p)^l a^*\bigl(k'\bigr)^{m'} a^*(k)^m\varOmega \\ &\quad {}= \sum_{\tilde{m}=0}^{\infty}\frac{1}{\sqrt{\tilde{m}!}} \int d^{3(l+1)}\tilde{p} d^{3\tilde{m}}\tilde{k} \tilde{F}_{\tilde{m}}(\tilde{p};\tilde{k}) \eta ^*(\tilde{p})^{l+1} a^*(\tilde{k})^{\tilde{m}}\varOmega , \end{aligned}$$
(A.5)
where \(F'_{1,m'}(p',k'):= h(p'+ \underline{k}') f^{m'}_{p'+ \underline{k},\sigma }(k')\), \(\tilde{m}:=m+m'\), \(\tilde{k}:=(k,k')\), \(\tilde{p}:=(p,p')\) and
$$ \tilde{F}_{\tilde{m}}(\tilde{p};\tilde{k})=\sum_{m=0}^{\tilde{m}} \frac{ \sqrt{\tilde{m}!} }{ \sqrt{(\tilde{m}-m)!} \sqrt{m!} }\bigl(F'_{1,(\tilde{m}-m) }F_{l,m} \bigr)_{\mathrm {sym} } (\tilde{p};\tilde{k}), $$
(A.6)
where the symmetrization is performed in the \(\tilde{p}\) and \(\tilde{k}\) variables separately. By Theorem 1.2, \(F'_{1,m'}\) satisfies the bound (2.18), and therefore
$$ \|\tilde{F}_{\tilde{m}}\|_2\leq\frac{c^{\tilde{m}}}{\sqrt{\tilde{m}!}}, $$
(A.7)
for some constant c. Hence \(\hat{\eta}^{*}_{\sigma }(h)\varPsi_{l}^{r_{1},r_{2}}\) is well defined and belongs to \(\mathcal{D}\).
Next, we note that
$$\begin{aligned} &H_{\mathrm{e}}\varPsi_l^{r_1,r_2}=\sum _{m=0}^{\infty}\frac{1}{\sqrt{m!}}\int d^{3l}p d^{3m}k\bigl(\varOmega (p_1)+\cdots+\varOmega (p_l) \bigr) F_{l,m}(p;k)\eta ^*(p)^la^*(k)^m\varOmega , \\ \end{aligned}$$
(A.8)
$$\begin{aligned} &H_{\mathrm {f}}\varPsi_l^{r_1,r_2}=\sum _{m=0}^{\infty}\frac{1}{\sqrt{m!}}\int d^{3l}p d^{3m}k \bigl(\omega (k_1)+\cdots+\omega (k_m) \bigr) F_{l,m}(p;k)\eta ^*(p)^la^*(k)^m\varOmega . \end{aligned}$$
(A.9)
Due to the support properties of F
l,m
(p;k) these vectors are well defined and belong to \(\mathcal{D}\).
Finally, we consider the operators \(H_{\mathrm{I}}^{\mathrm{a}/\mathrm{c}}\). We recall that the interaction Hamiltonian restricted to \(\mathcal {H}^{(l)}\) has the form
$$ H_{\mathrm{I}}^{(l)}:=\sum_{i=1}^l \int d^3k v_{\overline {\alpha }}(k) \bigl( e^{ikx_i}a(k)+e^{-ikx_i}a^*(k) \bigr) $$
(A.10)
and we express \(\varPsi_{l}^{r_{1},r_{2}}\) in terms of its m-photon components, i.e.,
$$ \bigl\{ \varPsi_l^{r_1,r_2}\bigr\} ^{(l,m)}(p_1, \ldots, p_l;k_1,\ldots ,k_m)=F_{l,m}(p_1, \ldots, p_l;k_1,\ldots, k_m). $$
(A.11)
Now we can write
$$\begin{aligned} &\bigl(H_{\mathrm{I}}^{\mathrm{a}}\varPsi_l^{r_1,r_2} \bigr)^{(l,m)}(p_1,\ldots, p_l; k_1,\ldots, k_m) \\ &\quad {}=\sqrt{m+1}\int d^3k v_{\overline {\alpha }}(k)\sum_{i=1}^l \bigl(\varPsi _l^{r_1,r_2}\bigr)^{(l,m+1)} \\ &\qquad {}\times (p_1, \ldots, p_i-k,\ldots, p_l; k, k_1,\ldots, k_m). \end{aligned}$$
(A.12)
It is easy to see that for some constant c, independent of m,
$$ \bigl\| \bigl(H_{\mathrm{I}}^{\mathrm{a}}\varPsi_l^{r_1,r_2} \bigr)^{(l,m)}\bigr\| _2\leq\frac{c^m}{\sqrt{m!}}. $$
(A.13)
Similarly, we obtain that
$$\begin{aligned} &\bigl(H_{\mathrm{I}}^{\mathrm{c}}\varPsi_l^{r_1,r_2} \bigr)^{(l,m)}(p_1,\ldots, p_l; k_1,\ldots, k_m) \\ &\quad {}=\frac{1}{\sqrt{m}}\sum_{i=1}^m\sum _{j=1}^lv_{\overline {\alpha }}(k_i) \bigl(\varPsi _l^{r_1,r_2}\bigr)^{(l,m-1)} \\ &\qquad {}\times (p_1, \ldots,p_j+k_i,\ldots, p_l; k_1,\ldots ,k_{i_*},\ldots, k_m), \end{aligned}$$
(A.14)
where \(k_{i_{*}}\) means omission of the i-th variable. This gives, again, a bound of the form (A.13). Since the case of \(\check{H}_{\mathrm{I},\sigma }\) is analogous, this concludes the proof. □
Appendix B: Fock Space Combinatorics
In Lemmas B.1, B.3 and B.5 below we deal first with \(G_{i,m}, G_{i,m}', F_{n,m}, F'_{n,m}\) of Schwartz class and then extend the results to square integrable functions using Theorem X.44 of [26].
Lemma B.1
Let
\(G_{m}, G_{m}'\in L^{2}(\mathbb {R}^{3}\times \mathbb {R}^{3m})\)
be symmetric in their photon variables, see (1.68). Let us define as operators on
\(\mathcal{C}\)
$$ B_m^*(G_m):=\int d^3p d^{3m}k G_m(p;k) \eta ^*(p- \underline{k})a^*(k)^m $$
(B.1)
and
\(B_{m}(G_{m}):=(B_{m}^{*}(G_{m}))^{*}\). Then there holds the identity
$$ \bigl\langle \varOmega , B_m\bigl(G_m'\bigr) B_m^*(G_m) \varOmega \bigr\rangle =m!\int d^3p d^{3m}k \overline{G}_{m}'(p; k) G_{m}(p; k ) . $$
(B.2)
Proof
We compute
$$\begin{aligned} &\bigl\langle \varOmega , B_m\bigl(G_m'\bigr) B_m^*(G_m) \varOmega \bigr\rangle \\ &\quad {}=\int d^3p d^{3m}k\int d^3p' d^{3m}k' G_m(p;k) \overline{G}_{m}' \bigl(p'; k'\bigr) \\ &\qquad {}\times \bigl\langle \varOmega , a\bigl(k' \bigr)^m\eta \bigl(p'- \underline{k}'\bigr) \eta ^*(p- \underline{k})a^*(k)^m\varOmega \bigr\rangle \\ &\quad {}=\int d^3p d^{3m}k\int d^3p' d^{3m}k' G_m(p;k) \overline{G}_{m}' \bigl(p'; k'\bigr) \delta \bigl(p- \underline{k}-p'+ \underline{k}'\bigr) \\ &\qquad {}\times \bigl\langle \varOmega , a\bigl(k'\bigr)^m a^*(k)^m\varOmega \bigr\rangle \\ &\quad {}=\int d^3p d^{3m}k\int d^{3m}k' G_m(p;k) \overline{G}_{m}'\bigl(p- \underline{k}+\underline{k}'; k'\bigr) \bigl\langle \varOmega , a\bigl(k' \bigr)^m a^*(k)^m\varOmega \bigr\rangle \\ &\quad {}= \int d^3p d^{3m}k\int d^{3m}k' G_m(p;k) \overline{G}_{m}'\bigl(p- \underline{k}+\underline{k}'; k'\bigr) \sum_{\rho\in S_m} \prod_{i=1}^m\delta \bigl(k_{\rho(i)}-k'_i \bigr) \\ &\quad {}= m!\int d^3p d^{3m}k G_m(p;k) \overline{G}_{m}'(p; k), \end{aligned}$$
(B.3)
where S
m
is the set of all permutations of an m-element set and in the last step we exploited the fact that \(G_{m}'\) is symmetric in its photon variables. □
Lemma B.2
Let
\(n,m,\tilde{n},\tilde{m}\in \mathbb {N}_{0}\)
be s.t. \(n+m=\tilde{n}+\tilde{m}\). Let us choose
$$\begin{aligned} &r=(r_1,\ldots,r_n)\in \mathbb {R}^{3n}, \qquad k=(k_1,\ldots, k_m)\in \mathbb {R}^{3m}, \end{aligned}$$
(B.4)
$$\begin{aligned} &\tilde{r}=(\tilde{r}_1,\ldots, \tilde{r}_{\tilde{n}})\in \mathbb {R}^{3\tilde{n}}, \qquad \tilde{k}=(\tilde{k}_1,\ldots,\tilde{k}_{\tilde{m}} )\in \mathbb {R}^{3\tilde{m}} \end{aligned}$$
(B.5)
and define the sets
$$\begin{aligned} &C_n:=\{1,\ldots, n\}, \qquad C_n':= \{n+1,\ldots,n+m\}, \end{aligned}$$
(B.6)
$$\begin{aligned} &C_{\tilde{n}}:=\{1,\ldots,\tilde{n}\}, \qquad C_{\tilde{n}}':= \{\tilde{n}+1,\ldots ,\tilde{n}+ \tilde{m}\}. \end{aligned}$$
(B.7)
(Note that
\(C_{n}'\)
is the complement of
C
n
in {1,…,n+m}. Similarly for
\(C_{\tilde{n}}'\)). Let
S
m+n
be the set of permutations of an
m+n
element set. For any
ρ∈S
m+n
we introduce the following notation:
$$\begin{aligned} &\hat{r} :=(r_i)_{(i,\rho (i))\in C_n\times C_{\tilde{n}}}, \qquad \check{r} :=(r_i)_{(i,\rho (i))\in C_n\times C_{\tilde{n}}' }, \end{aligned}$$
(B.8)
$$\begin{aligned} &\hat{k} :=(k_{i-n})_{(i,\rho (i))\in C_n'\times C_{\tilde{n}}' }, \qquad \check{k} :=(k_{i-n})_{(i,\rho (i))\in C_n'\times C_{\tilde{n}} }, \end{aligned}$$
(B.9)
so that
\(r=(\hat{r} ,\check{r} )\), \(k=(\hat{k} ,\check{k} )\). Similarly,
$$\begin{aligned} &\hat{\tilde{r}}:=(\tilde {r}_{\rho (i)})_{(i,\rho (i))\in C_n\times C_{\tilde{n}}},\qquad\check{\tilde{r}}:=(\tilde{r}_{\rho (i)})_{(i,\rho (i))\in C_n'\times C_{\tilde{n}} }, \end{aligned}$$
(B.10)
$$\begin{aligned} &\hat{\tilde{k}}:=(\tilde{k}_{\rho (i)-\tilde{n}})_{(i,\rho (i))\in C_n'\times C_{\tilde{n}}' },\qquad\check{\tilde{k}}:=(\tilde{k}_{\rho (i)-\tilde{n} })_{(i,\rho (i))\in C_n\times C_{\tilde{n}}' }, \end{aligned}$$
(B.11)
so that
\(\tilde{r}=(\hat{\tilde{r}}, \check{\tilde{r}})\)
and
\(\tilde{k}=(\hat{\tilde{k}}, \check{\tilde{k}})\). (If
\(\{ i\,|\, (i,\rho (i))\in C_{n}\times C_{\tilde{n}}\} =\emptyset\)
then we say that
\(\hat{r} \)
is empty, and analogously for other collections of photon variables introduced above). Finally, we define
$$\begin{aligned} \delta (\hat{r} -\hat{\tilde{r}}) :=&\prod_{(i,\rho (i))\in C_n\times C_{\tilde{n}} } \delta (r_i-\tilde{r}_{\rho (i)}), \end{aligned}$$
(B.12)
$$\begin{aligned} \delta (\check{r} -\check{\tilde{k}}) :=&\prod_{(i,\rho (i))\in C_n\times C_{\tilde{n}}' } \delta (r_i-\tilde{k}_{\rho (i)-\tilde{n}} ), \end{aligned}$$
(B.13)
$$\begin{aligned} \delta (\check{k} -\check{\tilde{r}}) :=&\prod_{(i,\rho (i))\in C_n'\times C_{\tilde{n}} } \delta (k_{i-n}-\tilde{r}_{\rho (i)} ), \end{aligned}$$
(B.14)
$$\begin{aligned} \delta (\hat{k} -\hat{\tilde{k}} ) :=&\prod_{(i,\rho (i))\in C_n'\times C_{\tilde{n}}' } \delta (k_{i-n}-\tilde{k}_{\rho (i)-\tilde{n}} ). \end{aligned}$$
(B.15)
Then there holds
$$\begin{aligned} \bigl\langle \varOmega , a(\tilde{r})^{\tilde{n}} a(\tilde{k})^{\tilde{m}} a^*(r)^{n} a^*(k)^{m}\varOmega \bigr\rangle =\sum _{\rho \in S_{m+n}} \delta (\hat{r} -\hat{\tilde{r}})\delta (\check{r} -\check{\tilde{k}})\delta (\check{k} - \check{\tilde{r}})\delta (\hat{k} -\hat{\tilde{k}} ). \end{aligned}$$
(B.16)
Proof
Let (v
1,…,v
n+m
)=(r
1,…,r
n
,k
1,…,k
m
) and \((\tilde{v}_{1},\ldots,\tilde{v}_{n+m})=(\tilde{r}_{1},\ldots,\tilde{r}_{\tilde{n}}, \tilde{k}_{1},\ldots, \tilde{k}_{\tilde{m}})\). There holds
$$\begin{aligned} & \bigl\langle \varOmega , a(\tilde{r})^{\tilde{n}} a(\tilde{k})^{\tilde{m}} a^*(r)^{n} a^*(k)^{m}\varOmega \bigr\rangle \\ &\quad {} =\sum _{\rho \in S_{m+n}}\prod_{j=1}^{m+n} \delta (v_{j}-\tilde{v}_{\rho (j)}) \\ &\quad {}=\sum_{\rho \in S_{m+n}} \delta (r_1-\tilde{v}_{\rho (1)} )\cdots \delta (r_n-\tilde{v}_{\rho (n)}) \delta (k_1-\tilde{v}_{\rho (n+1)})\cdots \delta (k_m-\tilde{v}_{\rho (n+m)}) \end{aligned}$$
(B.17)
$$\begin{aligned} &\quad {}=\sum_{\rho \in S_{m+n}} \biggl(\prod _{(i, \rho (i))\in C_n\times C_{\tilde{n}}} \delta (r_i-\tilde{r}_{\rho (i)} ) \biggr) \biggl(\prod_{(i,\rho (i))\in C_n\times C_{\tilde{n}}' } \delta (r_i-\tilde{k}_{\rho (i)-\tilde{n}} ) \biggr) \\ &\qquad {}\times \biggl(\prod_{(i,\rho (i))\in C_n'\times C_{\tilde{n}} } \delta (k_{i-n}-\tilde{r}_{\rho (i)} ) \biggr) \biggl(\prod _{(i,\rho (i))\in C_n'\times C_{\tilde{n}}' } \delta (k_{i-n}-\tilde{k}_{\rho (i)-\tilde{n}} ) \biggr), \end{aligned}$$
(B.18)
which concludes the proof. □
Lemma B.3
Let
\(F_{n,m}, F'_{n,m} \in L^{2}((\mathbb {R}^{3}\times \mathbb {R}^{3n})\times(\mathbb {R}^{3}\times \mathbb {R}^{3m}))\)
be symmetric in the photon variables for any
\(n,m\in \mathbb {N}\). Let us introduce the following operators on
\(\mathcal{C}\)
$$ B^*_{n,m}(F_{n,m}):=\int d^3q d^3p \int d^{3n}r d^{3m}k F_{n,m}(q;r \,|\,p ; k) a^*(r)^{n}a^*(k)^m\eta ^*(p- \underline{k})\eta ^*(q- \underline{r}) $$
(B.19)
and set
\(B_{n,m}(F_{n,m}):=(B^{*}_{n,m}(F_{n,m}))^{*}\). There holds
$$\begin{aligned} &\bigl\langle B_{\tilde{n},\tilde{m}}^*\bigl(F'_{\tilde{n},\tilde{m}}\bigr)\varOmega , B_{n,m}^*(F_{n,m})\varOmega \big\rangle \\ &\quad {}=\sum _{\rho \in S_{m+n}}\int d^3 q d^3p \int d^{3n}r d^{3m}k F_{n,m}(q;r \,|\,p; k) \\ &\qquad {}\times \bigl( \overline{F}_{\tilde{n},\tilde{m}}'(p- \underline {\hat{k}} + \underline {\hat{r}} ; \hat{r} , \check{k} \,|\,q+\underline {\hat{k}} - \underline {\hat{r}} ; \hat{k} , \check{r} ) \\ &\qquad {} + \overline{F}_{\tilde{n},\tilde{m}}'(q+ \underline {\check{k}} - \underline {\check{r}} ; \hat{r} , \check{k} \,|\,p- \underline {\check{k}} + \underline {\check{r}} ; \hat{k} , \check{r} ) \bigr) \end{aligned}$$
(B.20)
for any
\(\tilde{n},\tilde{m}\in \mathbb {N}\)
s.t. \(n+m=\tilde{n}+\tilde{m}\). Otherwise the expression on the l.h.s. is zero. Here
S
m+n
is the set of permutations of an
m+n
element set and the notation
\(\hat{k} \), \(\check{k} \), \(\hat{r} \), \(\check{r} \)
is explained in Lemma B.2.
Proof
We compute the expectation value
$$\begin{aligned} &\bigl\langle B_{\tilde{n},\tilde{m}}^*\bigl(F'_{\tilde{n},\tilde{m}}\bigr)\varOmega , B_{n,m}^*(F_{n,m})\varOmega \bigr\rangle \end{aligned}$$
(B.21)
$$\begin{aligned} &\quad {}=\int d^3\tilde{q} d^3\tilde{p} d^3 q d^3p \int d^{3\tilde{n}}\tilde{r} d^{3\tilde{m}}\tilde{k} d^{3n}r d^{3m}k \overline{F}'_{\tilde{n},\tilde{m}}(\tilde{q}; \tilde{r} \,|\,\tilde{p}; \tilde{k}) F_{n,m}(q;r \,|\,p; k) \\ &\qquad {}\times \bigl( \delta (\tilde{q}-p+\tilde { \underline{k}}- \underline{r})+\delta (\tilde{q}- q-\tilde { \underline{r}}+\underline{r}) \bigr)\delta (\tilde{p}+\tilde{q}-p-q) \\ &\qquad {}\times \bigl\langle \varOmega , a(\tilde{r})^{\tilde{n}} a(\tilde{k})^{\tilde{m}} a^*(r)^{n} a^*(k)^{m}\varOmega \bigr\rangle . \end{aligned}$$
(B.22)
The last factor is non-zero only if \(\tilde{n}+\tilde{m}=n+m\). Moreover,
$$ \bigl\langle \varOmega , a(\tilde{r})^{\tilde{n}} a(\tilde{k})^{\tilde{m}} a^*(r)^{n} a^*(k)^{m}\varOmega \bigr\rangle = \sum _{\rho \in S_{m+n}}\delta (\hat{r} -\hat{\tilde{r}})\delta (\check{r} -\check{\tilde{k}})\delta (\check{k} - \check{\tilde{r}})\delta (\hat{k} -\hat{\tilde{k}} ), $$
(B.23)
where we made use of Lemma B.2. Thus the r.h.s. of (B.22) is a sum over ρ∈S
m+n
of terms of the form:
$$\begin{aligned} &\int d^3\tilde{q} d^3\tilde{p} d^3 q d^3p \int d^{3n}r d^{3m}k \overline{F}'_{\tilde{n},\tilde{m}}( \tilde{q}; \hat{r} , \check{k} \,|\,\tilde{p}; \hat{k} , \check{r} ) F_{n,m}(q; r \,|\,p; k ) \\ &\qquad {}\times \delta (\tilde{p}+\tilde{q}-p-q) \bigl( \delta (\tilde{q}-p+\underline {\hat{k}} - \underline {\hat{r}} )+\delta (\tilde{q}- q- \underline {\check{k}} + \underline {\check{r}} ) \bigr) \\ &\quad {}=\int d^3 q d^3p \int d^{3n}r d^{3m}k F_{n,m}(q; r \,|\,p; k ) \bigl(\overline{F}_{\tilde{n},\tilde{m}}'(p- \underline {\hat{k}} + \underline {\hat{r}} ; \hat{r} , \check{k} \,|\,q+ \underline {\hat{k}} - \underline {\hat{r}} ; \hat{k} , \check{r} ) \\ &\qquad {}+ \overline{F}_{\tilde{n},\tilde{m}}'( q+ \underline {\check{k}} -\underline {\check{r}} ; \hat{r} , \check{k} \,|\,p- \underline {\check{k}} + \underline {\check{r}} ; \hat{k} , \check{r} ) \bigr) \end{aligned}$$
(B.24)
which concludes the proof. □
Lemma B.4
Let
\(G_{1,m},G_{1,m}',G_{2,m}, G_{2,m}'\in L^{2}(\mathbb {R}^{3}\times \mathbb {R}^{3m})\)
be symmetric in the photon variables for any
\(m\in \mathbb {N}\). We define, as an operator on
\(\mathcal{C}\),
$$ B_m^*(G_{i,m}):=\int d^3p d^{3m}k G_{i,m}(p;k) \eta ^*(p- \underline{k})a^*(k)^m $$
(B.25)
and set
\(B_{m}(G_{i,m})=(B_{m}^{*}(G_{i,m}))^{*}\). There holds the identity
$$\begin{aligned} &\bigl\langle \varOmega , B_{\tilde{n}}\bigl(G_{1,\tilde{n}}' \bigr)B_{\tilde{m}}\bigl(G_{2,\tilde{m}}' \bigr)B_{n}^*(G_{1,n}) B_{m}^*(G_{2,m}) \varOmega \bigr\rangle \\ &\quad {}=\sum_{\rho \in S_{m+n}}\int d^3 q d^3p \int d^{3n}r d^{3m}k G_{1,n}(q; r) G_{2,m}(p; k)\phantom {444} \\ &\qquad {}\times \bigl(\overline{G}_{1,\tilde{n}}'(p- \underline {\hat{k}} + \underline {\hat{r}} ; \hat{r} , \check{k} ) \overline{G}_{2,\tilde{m}}'(q+ \underline {\hat{k}} - \underline {\hat{r}} ; \hat{k} , \check{r} ) \\ &\qquad {} +\overline{G}_{1,\tilde{n}}'( q+ \underline {\check{k}} - \underline {\check{r}} ; \hat{r} , \check{k} ) \overline{G}_{2,\tilde{m}}'(p- \underline {\check{k}} + \underline {\check{r}} ; \hat{k} , \check{r} ) \bigr), \end{aligned}$$
(B.26)
for any
\(n,\tilde{n},\tilde{m}\in \mathbb {N}\)
s.t. \(n+m=\tilde{n}+\tilde{m}\). Otherwise the expression on the l.h.s. is zero. Here
S
m+n
is the set of permutations of an
m+n
element set and the notation
\(\hat{k} , \check{k} , \hat{r} , \check{r} \)
is explained in Lemma B.2.
Proof
Follows immediately from Lemma B.3. □
Lemma B.5
Let
\(G_{1,m},G_{2,m} \in L^{2}(\mathbb {R}^{3}\times \mathbb {R}^{3m})\)
be supported in
\(\mathbb {R}^{3}\times\{ k\in \mathbb {R}^{3}\,|\, |k|\geq \sigma \}^{\times m}\)
and symmetric in their photon variables. There holds the identity
$$\begin{aligned} &\bigl\langle \varOmega , B_{ \tilde{n}}(G_{2,\tilde{n}}) \bigl(\check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} \bigr)^* B_{ \tilde{m}}(G_{1,\tilde{m}})B_{n}^*(G_{1,n}) \check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} B_{m}^*(G_{2,m})\varOmega \bigr\rangle \\ &\quad {}=\sum_{\rho \in S_{m+n}}\int d^3 q d^3p \int d^{3n}r d^{3m}k G_{1,n}(q;r) G_{2,m}(p; k) \\ &\qquad {}\times \biggl( \int d^3\tilde{p} \check {v}_{\overline {\alpha }} ^{\sigma }(\tilde{p})^2 \overline{G}_{2,\tilde{n}} (p-\tilde{p}- \underline {\hat{k}} + \underline {\hat{r}} ;\hat{r} ,\check{k} ) \overline{G}_{1,\tilde{m}} (\tilde{p}+q+ \underline {\hat{k}} - \underline {\hat{r}} ; \hat{k} ,\check{r} ) \\ &\qquad {}+ \bigl\| \check {v}_{\overline {\alpha }} ^{\sigma }\bigr\| _2^2 \overline{G}_{2,\tilde{n}} (q+\underline {\check{k} }-\underline {\check{r} };\hat{r} ,\check{k} ) \overline{G}_{1,\tilde{m}} (p-\underline {\check{k} }+\underline {\check{r} }; \hat{k} ,\check{r} ) \biggr) \end{aligned}$$
(B.27)
for
\(m+n= \tilde{m}+ \tilde{n}\), otherwise the l.h.s. is zero. Here
S
m+n
is the set of permutations of an
m+n
element set and the notation
\(\hat{k} , \check{k} , \hat{r} , \check{r} \)
is explained in Lemma B.2.
Proof
We compute the expectation value
$$\begin{aligned} &\bigl\langle \varOmega , B_{ \tilde{n}}(G_{2,\tilde{n}}) \bigl(\check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} \bigr)^* B_{ \tilde{m}}(G_{1,\tilde{m}})B_{n}^*(G_{1,n}) \check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} B_{m}^*(G_{2,m})\varOmega \bigr\rangle \\ &\quad {}=\int d^3\tilde{u} d^3\tilde{w} d^3u d^3w\int d^3\tilde{q} d^3\tilde{p} d^3 q d^3p \int d^{3 \tilde{n}}\tilde{r} d^{3 \tilde{m}}\tilde{k} d^{3n}r d^{3m}k \\ &\qquad {}\times \check {v}_{\overline {\alpha }} ^{\sigma }(\tilde{w}) \check {v}_{\overline {\alpha }} ^{\sigma }(w) \overline{G}_{2,\tilde{n}} (\tilde{q};\tilde{r}) \overline{G}_{1,\tilde{m}} (\tilde{p}; \tilde{k}) G_{1,n}(q;r) G_{2,m}(p; k) \\ &\qquad {}\times \bigl\langle \varOmega , \eta (\tilde{p}-\tilde { \underline{k}}) \eta ^*(\tilde{u}) \eta (\tilde{u}-\tilde{w}) \eta ( \tilde{q}-\tilde { \underline{r}}) \eta ^*(q- \underline{r})\eta ^*(u-w)\eta (u)\eta ^*(p-\underline{k}) \varOmega \bigr\rangle \\ &\qquad {}\times \bigl\langle \varOmega , a(\tilde{r})^{ \tilde{n}}a(\tilde{w}) a(\tilde{k})^{ \tilde{m}} a^*(r)^{n} a^*(w) a^*(k)^{m}\varOmega \bigr\rangle . \end{aligned}$$
(B.28)
We note that
$$\begin{aligned} &\bigl\langle \varOmega , \eta (\tilde{p}-\tilde { \underline{k}}) \eta ^*(\tilde{u}) \eta (\tilde{u}-\tilde{w}) \eta (\tilde{q}- \tilde { \underline{r}}) \eta ^*(q- \underline{r}) \eta ^*(u-w)\eta (u) \eta ^*(p- \underline{k}) \varOmega \bigr\rangle \\ &\quad {}=\delta (\tilde{p}-\tilde { \underline{k}}-\tilde{u}) \delta (p- \underline{k}- u) \bigl\langle \varOmega ,\eta (\tilde{u}-\tilde{w}) \eta (\tilde{q}-\tilde { \underline{r}}) \eta ^*(q- \underline{r})\eta ^*(u-w) \varOmega \bigr\rangle \\ &\quad {}=\delta (\tilde{p}-\tilde { \underline{k}}-\tilde{u})\delta (p- \underline{k}- u) \bigl( \delta (\tilde{u}-\tilde{w}-q+\underline{r})\delta ( \tilde{q}-\tilde { \underline{r}}-u+w) \\ &\qquad {}+\delta (\tilde{u}-\tilde{w}-u+w)\delta (\tilde{q}-\tilde { \underline{r}}-q+ \underline{r}) \bigr). \end{aligned}$$
(B.29)
Let us consider the contribution to (B.28) of the first term in the bracket in (B.29):
$$\begin{aligned} &\bigl\langle \varOmega , B_{ \tilde{n}}(G_{2,\tilde{n}}) \bigl(\check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} \bigr)^* B_{ \tilde{m}}(G_{1,\tilde{m}})B_{n}^*(G_{1,n}) \check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} B_{m}^*(G_{2,m})\varOmega \bigr\rangle _1 \\ &\quad {}:=\int d^3\tilde{u} d^3\tilde{w} d^3u d^3w\int d^3\tilde{q} d^3\tilde{p} d^3 q d^3p \int d^{3 \tilde{n}}\tilde{r} d^{3 \tilde{m}}\tilde{k} d^{3n}r d^{3m}k \\ &\qquad {}\times \check {v}_{\overline {\alpha }} ^{\sigma }(\tilde{w}) \check {v}_{\overline {\alpha }} ^{\sigma }(w) \overline{G}_{2,\tilde{n}} (\tilde{q};\tilde{r}) \overline{G}_{1,\tilde{m}} (\tilde{p}; \tilde{k}) G_{1,n}(q;r) G_{2,m}(p; k) \\ &\qquad {}\times \delta (\tilde{p}-\tilde { \underline{k}}-\tilde{u})\delta (p- \underline{k}- u) \delta (\tilde{u}-\tilde{w}-q+\underline{r})\delta ( \tilde{q}-\tilde { \underline{r}}-u+w) \\ &\qquad {}\times \bigl\langle \varOmega , a(\tilde{r})^{ \tilde{n}}a(\tilde{w}) a(\tilde{k})^{ \tilde{m}} a^*(r)^{n} a^*(w) a^*(k)^{m}\varOmega \bigr\rangle \\ &\quad {}=\int d^3\tilde{q} d^3\tilde{p} d^3 q d^3p \int d^{3 \tilde{n}}\tilde{r} d^{3 \tilde{m}}\tilde{k} d^{3n}r d^{3m}k \\ &\qquad {}\times \check {v}_{\overline {\alpha }} ^{\sigma }(\tilde{w}_*) \check {v}_{\overline {\alpha }} ^{\sigma }(w_*) \overline{G}_{2,\tilde{n}} (\tilde{q};\tilde{r}) \overline{G}_{1,\tilde{m}} (\tilde{p}; \tilde{k}) G_{1,n}(q;r) G_{2,m}(p; k) \\ &\qquad {}\times \bigl\langle \varOmega , a(\tilde{r})^{ \tilde{n}} a(\tilde{k})^{ \tilde{m}} a(\tilde{w}_* )a^*(w_*) a^*(r)^{n} a^*(k)^{m}\varOmega \bigr\rangle , \end{aligned}$$
(B.30)
where in the last step we integrated over \(u, w, \tilde{u}, \tilde{w}\) and set \(w_{*}:=p- \underline{k}-\tilde{q}+\tilde { \underline{r}}\), \(\tilde{w}_{*}:=\tilde{p}-\tilde { \underline{k}}-q+ \underline{r}\). Now we consider the expectation value of the photon creation operators:
$$\begin{aligned} &\bigl\langle \varOmega , a(\tilde{r})^{\tilde{n}} a(\tilde{k})^{\tilde{m}} a(\tilde{w}_*) a^*(w_*) a^*(r)^{n} a^*(k)^{m}\varOmega \bigr\rangle \\ &\quad {} =\delta (w_*-\tilde{w}_*) \bigl\langle \varOmega , a(\tilde{r})^{\tilde{n}} a(\tilde{k})^{\tilde{m}} a^*(r)^{n} a^*(k)^{m}\varOmega \bigr\rangle , \end{aligned}$$
(B.31)
for \(r,k,w, \tilde{r},\tilde{k},\tilde{w}\) in the supports of the respective functions. (Here we made use of the fact that \(|\tilde{w}_{*}|\leq \sigma \), whereas |r
i
|≥σ, |k
j
|≥σ). Let us now substitute the r.h.s. of (B.31) to (B.30). Making use of Lemma B.2, we obtain
$$\begin{aligned} &\bigl\langle \varOmega , B_{ \tilde{n}}(G_{2,\tilde{n}}) \bigl( \check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} \bigr)^* B_{ \tilde{m}}(G_{1,\tilde{m}})B_{n}^*(G_{1,n}) \check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} B_{m}^*(G_{2,m})\varOmega \bigr\rangle _{1} \\ &\quad {}=\sum_{\rho \in S_{m+n}}\int d^3\tilde{q} d^3\tilde{p} d^3 q d^3p \int d^{3\tilde{n}}\tilde{r} d^{3 \tilde{m}}\tilde{k} d^{3n}r d^{3m}k \\ &\qquad {}\times \check {v}_{\overline {\alpha }} ^{\sigma }(\tilde{p}-\tilde { \underline{k}}-q+ \underline{r}) \check {v}_{\overline {\alpha }} ^{\sigma }(p- \underline{k}-\tilde{q}+ \tilde { \underline{r}}) \overline{G}_{2,\tilde{n}} (\tilde{q} ; \tilde{r}) \overline{G}_{1,\tilde{m}} (\tilde{p}; \tilde{k}) G_{1,n}(q ;r) G_{2,m}(p; k) \\ &\qquad {}\times \delta (p+q-\tilde{p}-\tilde{q}) \delta (\hat{\tilde {r}}-\hat{r} )\delta (\check{\tilde {r}}- \check{k})\delta (\hat{\tilde {k}}-\hat{k} ) \delta (\check{\tilde {k}}-\check{r}). \end{aligned}$$
(B.32)
By integrating over \(\tilde{q}\), \(\tilde{r}\), \(\tilde{k}\), we obtain
$$\begin{aligned} &\bigl\langle \varOmega , B_{ \tilde{n}}(G_{2,\tilde{n}}) \bigl( \check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} \bigr)^* B_{ \tilde{m}}(G_{1,\tilde{m}})B_{n}^*(G_{1,n}) \check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} B_{m}^*(G_{2,m})\varOmega \bigr\rangle _{1} \\ &\quad {}=\sum_{\rho \in S_{m+n}}\int d^3\tilde{p} d^3 q d^3p \int d^{3n}r d^{3m}k \check {v}_{\overline {\alpha }} ^{\sigma }(\tilde{p}- q - \underline {\hat{k}} + \underline {\hat{r}} )^2 \\ &\qquad {}\times \overline{G}_{2,\tilde{n}} (p+q-\tilde{p};\hat{r} ,\check{k} ) \overline{G}_{1,\tilde{m}} (\tilde{p}; \hat{k} ,\check{r} ) G_{1,n}(q;r) G_{2,m}(p; k) \\ &\quad {}=\sum_{\rho \in S_{m+n}}\int d^3\tilde{p} d^3 q d^3p \int d^{3n}r d^{3m}k \check {v}_{\overline {\alpha }} ^{\sigma }(\tilde{p})^2 \\ &\qquad {}\times \overline{G}_{2,\tilde{n}} (p-\tilde{p}- \underline {\hat{k}} + \underline {\hat{r}} ;\hat{r} ,\check{k} ) \overline{G}_{1,\tilde{m}} (\tilde{p}+q+ \underline {\hat{k}} - \underline {\hat{r}} ; \hat{k} ,\check{r} ) G_{1,n}(q;r) G_{2,m}(p; k), \end{aligned}$$
(B.33)
where in the last step we made a change of variables \(\tilde{p} \to \tilde{p}+q+ \underline {\hat{k}} - \underline {\hat{r}} \). This gives the first term on the r.h.s. of (B.27).
Let us now consider the contribution of the second term in the bracket on the r.h.s. of formula (B.29):
$$\begin{aligned} & \bigl\langle \varOmega , B_{ \tilde{n}}(G_{2,\tilde{n}}) \bigl(\check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} \bigr)^* B_{ \tilde{m}}(G_{1,\tilde{m}})B_{n}^*(G_{1,n}) \check{H}_{\mathrm{I},\sigma }^{\mathrm{c}} B_{m}^*(G_{2,m})\varOmega \bigr\rangle _2 \\ &\quad {}:=\int d^3w\int d^3\tilde{p} d^3 q d^3p \int d^{3 \tilde{n}}\tilde{r} d^{3 \tilde{m}} \tilde{k} d^{3n}r d^{3m}k \check {v}_{\overline {\alpha }} ^{\sigma }(\tilde{p}-\tilde { \underline{k}}-p+ \underline{k}+w) \check {v}_{\overline {\alpha }} ^{\sigma }(w) \\ &\qquad {}\times \overline{G}_{2,\tilde{n}} (q+\tilde { \underline{r}}- \underline{r};\tilde{r}) \overline{G}_{1,\tilde{m}} (\tilde{p}; \tilde{k}) G_{1,n}(q;r) G_{2,m}(p; k) \\ &\qquad {}\times\bigl\langle \varOmega , a(\tilde{r})^{ \tilde{n}}a(\tilde{p}-\tilde { \underline{k}}-p+ \underline{k}+w) a( \tilde{k})^{ \tilde{m}} a^*(r)^{n} a^*(w) a^*(k)^{m} \varOmega \bigr\rangle \\ &\quad {}= \int d^3w\int d^3\tilde{p} d^3 q d^3p \int d^{3 \tilde{n}}\tilde{r} d^{3 \tilde{m}}\tilde{k} d^{3n}r d^{3m}k \check {v}_{\overline {\alpha }} ^{\sigma }(\tilde{p}-\tilde { \underline{k}}-p+ \underline{k}+w) \check {v}_{\overline {\alpha }} ^{\sigma }(w) \\ &\qquad {}\times \overline{G}_{2,\tilde{n}} (q+\tilde { \underline{r}}- \underline{r};\tilde{r}) \overline{G}_{1,\tilde{m}} (\tilde{p}; \tilde{k}) G_{1,n}(q;r) G_{2,m}(p; k)\delta (\tilde{p}-\tilde { \underline{k}}-p+ \underline{k}) \\ &\qquad {}\times \bigl\langle \varOmega , a(\tilde{r})^{ \tilde{n}}a(\tilde{k})^{ \tilde{m}} a^*(r)^{n} a^*(k)^{m}\varOmega \bigr\rangle \\ &\quad {}= \bigl\| \check {v}_{\overline {\alpha }} ^{\sigma }\bigr\| _2^2 \sum _{\rho \in S_{m+n}} \int d^3 q d^3p \int d^{3n}r d^{3m}k G_{1,n}(q;r) G_{2,m}(p; k) \\ &\qquad {}\times \overline{G}_{2,\tilde{n}} (q+\underline {\check{k} }-\underline {\check{r} };\hat{r} , \check{k} ) \overline{G}_{1,\tilde{m}} (p-\underline {\check{k} }+\underline {\check{r} }; \hat{k} ,\check{r} ), \end{aligned}$$
(B.34)
where in the first step we integrated over \(\tilde{u}, \tilde{w}, u, \tilde{q}\) and in the last step we made use again of Lemma B.2. This gives the second term on the r.h.s. of (B.27) and concludes the proof. □
