Appendix 1: Derivation of the path integral
For simplicity, in this section, we assume a quantum hamiltonian given by
$$\begin{aligned} \hat{H}=\sum \limits _{\alpha ,\beta }^{}h_{\alpha \beta }\hat{c}_{\alpha }^{\dagger }\hat{c}_{\beta }^{}+{\mathop {\mathop {\sum }\limits _{\alpha ,\beta }}\limits _{\alpha \ne \beta }}^{}U_{\alpha \beta }\hat{c}_{\alpha }^{\dagger }\hat{c}_{\beta }^{\dagger }\hat{c}_{\beta }^{}\hat{c}_{\alpha }^{}. \end{aligned}$$
(85)
The result for a non-diagonal interaction \(U_{\alpha \beta \gamma \nu }\), however, is given in Appendix 3 In order to get from Eq. (17) to the complex path integral Eq. (19), the following two integrals with \(j,j^\prime \in \mathbb {N}_0\), will be inserted:
$$\begin{aligned}&\int \limits _{0}^{2\pi }\!\mathrm{d}\theta \int \limits _{}^{}\!\mathrm{d}^2\phi \exp \left( -\left| \phi \right| ^2+\left. \phi \right. ^*\mathrm{e}^{\mathrm{i}\theta }-\mathrm{i}j\theta \right) \phi ^{j^\prime }=2\pi ^2\delta _{j,j^\prime } \end{aligned}$$
(86)
$$\begin{aligned}&\int \limits _{}^{}\!\mathrm{d}^2\phi \int \limits _{}^{}\!\mathrm{d}^2\mu \exp \left( -\left| \phi \right| ^2-\left| \mu \right| ^2+\left. \phi \right. ^*\mu \right) \phi ^j\left( \left. \mu \right. ^*\right) ^{j^\prime }=\pi ^2j!\delta _{j,j^\prime }, \end{aligned}$$
(87)
Thereby \(\mathrm{d}^2\mu =\mathrm{d}\mathfrak {R}{\mu }\mathrm{d}\mathfrak {I}{\mu }\), i.e., the integrations over \(\phi\) and, in the second case, over \(\mu\) run over the whole complex plane. One should notice, that the first of these two integrals is just the second one, but with the modulus of \(\mu\) already integrated out.
The first of these two integrals is used to decouple \({\varvec{\zeta }}^{(0)}\) from \({\varvec{\zeta }}^{(1)}\) by the following identity:
$$\begin{aligned}&\int \mathrm{d}^{2J}\zeta ^{(0)}_{}\exp \left( -\left. {\varvec{\zeta }}^{(0)}\right. ^*\cdot {\varvec{\zeta }}^{(0)}\right) \left[ \prod \limits _{j=1}^{J}\left( 1+\left. \chi ^{(0)}_{j}\right. ^*\zeta ^{(0)}_{j}\right) \right] \prod \limits _{j=1}^{J}\left( \left. \zeta ^{(0)}_{j}\right. ^*\right) ^{n^{(i)}_{j}}= \\&\int \frac{\mathrm{d}^{2N_i}\phi ^{(0)}_{}}{\pi ^{N_i}}\int \limits _{0}^{2\pi }\frac{\mathrm{d}^{N_i}\theta ^{(i)}_{}}{\left( 2\pi \right) ^{N_i}}\int \mathrm{d}^{2J}\zeta ^{(0)}_{}\exp \left( -\left. {\varvec{\zeta }}^{(0)}\right. ^*\cdot {\varvec{\zeta }}^{(0)}-\left| {{\phi }}^{(0)}\right| ^2+\left. {{\phi }}^{(0)}\right. ^*\cdot {{\mu }}^{(0)}\right) \\&\left[ \prod \limits _{j=1}^{J}\left( 1+\left. \chi ^{(0)}_{j}\right. ^*\phi ^{(0)}_{j}\right) \right] \left[ \prod \limits _{j=0}^{J-1}\left( 1+\zeta ^{(0)}_{J-j}\left. \mu ^{(0)}_{J-j}\right. ^*\right) \right] \prod \limits _{j=1}^{J}\left( \left. \zeta ^{(0)}_{j}\right. ^*\right) ^{n^{(i)}_{j}}, \end{aligned}$$
(88)
with \(\mu ^{(0)}_{j}=n^{(i)}_{j}\exp ( \mathrm{i}\theta ^{(i)}_{j})\) for all \(j\in \{1,\ldots ,J\}\), where \(J\) is the number of single-particle states taken into account. Note that here, for the initially unoccupied single-particle states, the phases \(\theta ^{(i)}_{j}\) are arbitrary but fixed, e.g. to zero, while the integration runs only over those initial phases \(\theta ^{(i)}_{j}\), for which \(n^{(i)}_{l}=1\). In this way, the integrals, that have to be performed exactly, in order to get a reasonable and correct semiclassical approximation for the propagator are already done, and do not have to be carried out later.
For the \(N_i=\sum _{j=1}^{J}n^{(i)}_{j}\) initially occupied single-particle states, the identity follows directly from Eq. (86), while for the unoccupied ones, it is important to notice, that the term \(\left. \chi ^{(0)}_{j}\right. ^*\zeta ^{(0)}_{j}\) does vanish when integrating over \({\varvec{\zeta }}^{(0)}\). This is because of the properties of the Grassmann integrals Eq. (10) and the fact, that there is no \(\left. \zeta ^{(0)}_{j}\right. ^*\) for those components, for which \(n^{(i)}_{j}=0\).
The thus obtained expression is the starting point for an iterative insertion of integrals of the form of Eq. (87). For \(1\le m<M\), an evaluation of the overlaps and matrix elements of Eq. (17) containing \({\varvec{\zeta }}^{(m)}\) yields the following expression:
$$\begin{aligned}&\left[ \prod \limits _{j=1}^{J}\left( 1+\left. \chi ^{(m)}_{j}\right. ^*\zeta ^{(m)}_{j}\right) \right]\\&\times \left[ 1-\frac{\mathrm{i}\tau }{\hbar }\sum \limits _{\alpha ,\beta =1}^{J}\left( h_{\alpha \beta }^{(m-1)}\left. \zeta ^{(m)}_{\alpha }\right. ^*\chi ^{(m-1)}_{\beta }+U_{\alpha \beta }^{(m-1)}\left. \zeta ^{(m)}_{\alpha }\right. ^*\left. \zeta ^{(m)}_{\beta }\right. ^*\chi ^{(m-1)}_{\beta }\chi ^{(m-1)}_{\alpha }\right) \right] \prod \limits _{j=1}^{J}\left( 1+\left. \zeta ^{(m)}_{j}\right. ^*\chi ^{(m-1)}_{j}\right)\\ &= \left[ a^{(m)}-\frac{\mathrm{i}\tau }{\hbar }\sum \limits _{\alpha }^{}h_{\alpha \alpha }^{(m-1)}b_\alpha ^{(m)}-\frac{\mathrm{i}\tau }{\hbar }{\mathop {\mathop {\sum }\limits _{\alpha ,\beta }}\limits _{\alpha \ne \beta }}^{}h_{\alpha \beta }^{(m-1)}c_{\alpha \beta }^{(m)}-\frac{\mathrm{i}\tau }{\hbar }{\mathop {\mathop {\sum }\limits _{\alpha ,\beta }}\limits _{\alpha \ne \beta }}^{}U_{\alpha \beta }^{(m-1)}d_{\alpha \beta }^{(m)}\right] \\&\prod \limits _{j=1}^{J}\left( 1+\left. \zeta ^{(m)}_{j}\right. ^*\chi ^{(m-1)}_{j}\right) . \end{aligned}$$
(89)
With the help of the integral Eq. (87), the coefficients \(a^{(m)},\)
\(b^{(m)}\), \(c^{(m)}\) and \(d^{(m)}\) can successively—starting from \(m=1\) – be written as
$$\begin{aligned} a^{(m)}&= \int \frac{\mathrm{d}^{2J}\mu ^{(m)}_{}}{\pi ^J}\int \frac{\mathrm{d}^{2J}\phi ^{(m)}_{}}{\pi ^J}\left[ \prod \limits _{j=1}^{J}\left( 1+\left. \chi ^{(m)}_{j}\right. ^*\phi ^{(m)}_{j}\right) \right] \\&\times \exp\left( -\left| {{\phi }}^{(m)}\right| ^2-\left| {{\mu }}^{(m)}\right| ^2+\left. {{\phi }}^{(m)}\right. ^*\cdot {{\mu }}^{(m)}\right) \\&\times \prod \limits _{j=0}^{J-1}\left[ 1+\zeta ^{(m)}_{J-j}\sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}\right] , \end{aligned}$$
(90)
$$\begin{aligned} b_{\alpha }^{(m)}&= \int \frac{\mathrm{d}^{2J}\mu ^{(m)}_{}}{\pi ^J}\int \frac{\mathrm{d}^{2J}\phi ^{(m)}_{}}{\pi ^J}\exp \left( -\left| {{\phi }}^{(m)}\right| ^2 \right) \\ &\times \exp\left(-\left| {{\mu }}^{(m)}\right| ^2+\left. {{\phi }}^{(m)}\right. ^*\cdot {{\mu }}^{(m)}\right) \left. \zeta ^{(m)}_{\alpha }\right. ^*\chi ^{(m-1)}_{\alpha } \\&\times \left\{ \prod \limits _{j=0}^{J-\alpha -1}\left[ 1+\zeta ^{(m)}_{J-j}\sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}\right] \right\} \\&\quad \times \left[ 1+\zeta ^{(m)}_{\alpha }\sum \limits _{k=1}^{\infty }c_k^{(1)}\left( \phi ^{(m-1)}_{\alpha }\right) \left( \left. \mu ^{(m)}_{\alpha }\right. ^*\right) ^{k}\right] \\&\times \left\{ \prod \limits _{j=J-\alpha +1}^{J-1}\left[ 1+\zeta ^{(m)}_{J-j}\sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}\right] \right\} \\&\times \left[ \prod \limits _{j=1}^{J}\left( 1+\left. \chi ^{(m)}_{j}\right. ^*\phi ^{(m)}_{j}\right) \right] , \end{aligned}$$
(91)
$$\begin{aligned} c_{\alpha \beta }^{(m)}&= \int \frac{\mathrm{d}^{2J}\mu ^{(m)}_{}}{\pi ^J}\int \frac{\mathrm{d}^{2J}\phi ^{(m)}_{}}{\pi ^J} \\&\quad \times \exp\left( -\left| {{\phi }}^{(m)}\right| ^2-\left| {{\mu }}^{(m)}\right| ^2+\left. {{\phi }}^{(m)}\right. ^*\cdot {{\mu }}^{(m)}\right) \left. \zeta ^{(m)}_{\alpha }\right. ^*\chi ^{(m-1)}_{\beta } \\&\quad \times \left\{ \prod \limits _{j=0}^{J-\max \left( \alpha ,\beta \right) -1}\left[ 1+\zeta ^{(m)}_{J-j}\sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}\right] \right\} \\&\quad \times \left[ 1+\zeta ^{(m)}_{\max \left( \alpha ,\beta \right) }\sum \limits _{k=1}^{\infty }c_k^{(2)}\left( \phi ^{(m-1)}_{\max \left( \alpha ,\beta \right) }\right) \left( \left. \mu ^{(m)}_{\max \left( \alpha ,\beta \right) }\right. ^*\right) ^k\right] \\&\quad \times \left\{ \prod \limits _{j=J-\max \left( \alpha ,\beta \right) +1}^{J-\min \left( \alpha ,\beta \right) -1}\left[ 1+\zeta ^{(m)}_{J-j}\sum \limits _{k=1}^{\infty }c_k^{(3)}\left( \phi ^{(m-1)}_{J-j}\right) \left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\right] \right\} \\&\quad \times \left[ 1+\zeta ^{(m)}_{\min \left( \alpha ,\beta \right) }\sum \limits _{k=1}^{\infty }c_k^{(2)}\left( \phi ^{(m-1)}_{\min \left( \alpha ,\beta \right) }\right) \left( \left. \mu ^{(m)}_{\min \left( \alpha ,\beta \right) }\right. ^*\right) ^k\right] \\&\quad \times \left\{ \prod \limits _{j=J-\min \left( \alpha ,\beta \right) +1}^{J-1}\left[ 1+\zeta ^{(m)}_{J-j}\sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}\right] \right\} \\&\quad \times \left[ \prod \limits _{j=1}^{J}\left( 1+\left. \chi ^{(m)}_{j}\right. ^*\phi ^{(m)}_{j}\right) \right] \end{aligned}$$
(92)
$$\begin{aligned} d_{\alpha \beta }^{(m)}&= \int \frac{\mathrm{d}^{2J}\mu ^{(m)}_{}}{\pi ^J}\int \frac{\mathrm{d}^{2J}\phi ^{(m)}_{}}{\pi ^J}\exp \left( -\left| {{\phi }}^{(m)}\right| ^2-\left| {{\mu }}^{(m)}\right| ^2+\left. {{\phi }}^{(m)}\right. ^*\cdot {{\mu }}^{(m)}\right) \\&\quad \times \left[ \prod \limits _{j=1}^{J}\left( 1+\left. \chi ^{(m)}_{j}\right. ^*\phi ^{(m)}_{j}\right) \right] \left. \zeta ^{(m)}_{\alpha }\right. ^*\left. \zeta ^{(m)}_{\beta }\right. ^*\chi ^{(m-1)}_{\beta }\chi ^{(m-1)}_{\alpha } \\&\quad \times \left\{ \prod \limits _{j=0}^{J-\max \left( \alpha ,\beta \right) -1}\left[ 1+\zeta ^{(m)}_{J-j}\sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}\right] \right\} \\&\quad \times \left[ 1+\zeta ^{(m)}_{\max \left( \alpha ,\beta \right) }\sum \limits _{k=1}^{\infty }c_k^{(4)}\left( \phi ^{(m-1)}_{\max \left( \alpha ,\beta \right) }\right) \left( \left. \mu ^{(m)}_{\max \left( \alpha ,\beta \right) }\right. ^*\right) ^k\right] \\&\quad \times \left\{ \prod \limits _{j=J-\max \left( \alpha ,\beta \right) +1}^{J-\min \left( \alpha ,\beta \right) -1}\left[ 1+\zeta ^{(m)}_{J-j}\sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}\right] \right\} \\&\quad \times \left[ 1+\zeta ^{(m)}_{\min \left( \alpha ,\beta \right) }\sum \limits _{k=1}^{\infty }c_k^{(4)}\left( \phi ^{(m-1)}_{\min \left( \alpha ,\beta \right) }\right) \left( \left. \mu ^{(m)}_{\min \left( \alpha ,\beta \right) }\right. ^*\right) ^k\right] \\&\quad \times \left\{ \prod \limits _{j=J-\min \left( \alpha ,\beta \right) +1}^{J-1}\left[ 1+\zeta ^{(m)}_{J-j}\sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}\right] \right\} , \end{aligned}$$
(93)
with \(c_1^{(1)}=c_1^{(2)}=c_1^{(3)}=c_1^{(4)}=1\).
It is important to notice, that the integral over \({{\phi }}^{(m)}\) and \({{\mu }}^{(m)}\) selects only the \(k=1\) terms of the occurring sums. Therefore, the terms with \(k\ge 2\) can be varied, in order to modify the final path integral in the desired way.
Finally, for \(m=M\), a similar argument as for \(m=0\) allows to restrict the integrals over \({{\phi }}^{(M)}\) again to those \(N_f=\sum _{j=1}^{J}n^{(f)}_{j}\) components with \(n^{(f)}_{j}=1\), while setting all the other components of \({{\phi }}^{(M)}\) to zero.
After this the \(m\)-th factor in the product over the timesteps only depends on \({\varvec{\zeta }}^{(m+1)}\) and \({\varvec{\chi }}^{(m)}\), such that one can easily integrate out the intermediate Grassmann variables \({\varvec{\zeta }}^{(1)},\ldots ,{\varvec{\zeta }}^{(M)}\) and \({\varvec{\chi }}^{(0)},\ldots ,{\varvec{\chi }}^{(M-1)}\) by using
$$\begin{aligned}&\int \mathrm{d}^{2J}\zeta \int \limits _{}^{}\!\mathrm{d}^{2J}\chi \exp \left( -\left. \varvec{\zeta }\right. ^*\cdot \varvec{\zeta }-\left. \varvec{\chi }\right. ^*\cdot \varvec{\chi }\right) \\&\quad \times \left[ \prod \limits _{j=0}^{J-1}\left( 1+\zeta _{J-j}f_{J-j}^{(m)}\right) \right] \\&\quad \times \left[ \prod \limits _{j=1}^{J}\left( 1+\left. \zeta _{j}\right. ^*\chi _{j}\right) \right] \left[ \prod \limits _{j=1}^{J}\left( 1+\left. \chi _{j}\right. ^*\phi ^{(m)}_{j}\right) \right] =\prod \limits _{j=1}^{J}\left( 1+f_j^{(m)}\phi ^{(m)}_{j}\right) , \end{aligned}$$
(94)
$$\begin{aligned}&\int \mathrm{d}^{2J}\zeta \int \limits _{}^{}\!\mathrm{d}^{2J}\chi \exp \left( -\left. \varvec{\zeta }\right. ^*\cdot \varvec{\zeta }-\left. \varvec{\chi }\right. ^*\cdot \varvec{\chi }\right) \\&\quad \times \left[ \prod \limits _{j=0}^{J-1}\left( 1+\zeta _{J-j}f_{J-j}^{(m)}\right) \right] \\&\quad \times \left[ \prod \limits _{j=1}^{J}\left( 1+\left. \zeta _{j}\right. ^*\chi _{j}\right) \right] \left[ \prod \limits _{j=1}^{J}\left( 1+\left. \chi _{j}\right. ^*\phi ^{(m)}_{j}\right) \right] \left. \zeta _{\alpha }\right. ^*\chi _{\beta } \\&\quad =f_\alpha ^{(m)}\phi ^{(m)}_{\beta } \\&\quad \times \left[ \prod \limits _{j=1}^{\min \left( \alpha ,\beta \right) -1}\left( 1+f_j^{(m)}\phi ^{(m)}_{j}\right) \right] \\&\quad \times \left[ \prod \limits _{j=\max \left( \alpha ,\beta \right) +1}^{J}\left( 1+f_j^{(m)}\phi ^{(m)}_{j}\right) \right] \prod \limits _{j=\min \left( \alpha ,\beta \right) +1}^{\max \left( \alpha ,\beta \right) -1}\left( 1-f_j^{(m)}\phi ^{(m)}_{j}\right) , \end{aligned}$$
(95)
$$\begin{aligned}&\int \mathrm{d}^{2J}\zeta \int \limits _{}^{}\!\mathrm{d}^{2J}\chi \exp \left( -\left. \varvec{\zeta }\right. ^*\cdot \varvec{\zeta }-\left. \varvec{\chi }\right. ^*\cdot \varvec{\chi }\right) \\&\quad \times \left[ \prod \limits _{j=0}^{J-1}\left( 1+\zeta _{J-j}f_{J-j}^{(m)}\right) \right] \\&\quad \times \left[ \prod \limits _{j=1}^{J}\left( 1+\left. \zeta _{j}\right. ^*\chi _{j}\right) \right] \left[ \prod \limits _{j=1}^{J}\left( 1+\left. \chi _{j}\right. ^*\phi ^{(m)}_{j}\right) \right] \left. \zeta _{\alpha }\right. ^*\left. \zeta _{\beta }\right. ^*\chi _{\beta }\chi _{\alpha } \\&\quad = f_\alpha ^{(m)}f_\beta ^{(m)}\phi ^{(m)}_{\beta }\phi ^{(m)}_{\alpha }{\mathop {\mathop {\prod }\limits _{ j=1 }}\limits _{j\ne \alpha ,\beta }^{J}}\left( 1+f_j^{(m)}\phi ^{(m)}_{j}\right) . \end{aligned}$$
(96)
Moreover, the integrals over \({\varvec{\zeta }}^{(0)}\) and \({\varvec{\chi }}^{(M)}\) yield
$$\begin{aligned}&\int \mathrm{d}^{2J}\zeta ^{(0)}_{}\exp \left( -\left. {\varvec{\zeta }}^{(0)}\right. ^*\cdot {\varvec{\zeta }}^{(0)}\right) \left[ \prod \limits _{j=0}^{J-1}\left( 1+\zeta ^{(0)}_{J-j}\left. \mu ^{(0)}_{J-j}\right. ^*\right) \right] \\ & \quad \times \prod \limits _{j=1}^{J}\left( \left. \zeta ^{(0)}_{j}\right. ^*\right) ^{n^{(i)}_{j}}=\prod \limits _{j:n^{(i)}_{j}=1}^{}\left. \mu ^{(0)}_{j}\right. ^* \end{aligned}$$
(97)
$$\begin{aligned}&\int \mathrm{d}^{2J}\chi ^{(M)}_{}\exp \left( -\left. {\varvec{\chi }}^{(M)}\right. ^*\cdot {\varvec{\chi }}^{(M)}\right) \left[ \prod \limits _{j=0}^{J-1}\left( \chi ^{(M)}_{J-j}\right) ^{n^{(f)}_{J-j}}\right]\\ &\quad \times \prod \limits _{j=1}^{J}\left( 1+\chi ^{(M)}_{j}\left. \phi ^{(M)}_{j}\right. ^*\right) =\prod \limits _{j:n^{(f)}_{j}=1}^{}\phi ^{(M)}_{j} \end{aligned}$$
(98)
After performing these integrals, one notices, that the inserted integrals have been chosen such, that the resulting sums can be performed and yield exponentials, such that the propagator is, after integrating out \({{\mu }}^{(1)},\ldots ,{{\mu }}^{(M)}\) as well as \({{\phi }}^{(0)}\) and undo the expansion in \(\tau\), given by the path integral Eq. (19), where the classical Hamiltonian is given by
$$\begin{aligned}& H_{cl}\left( \left. {\mu }\right. ^*,{\phi }\right)= \sum \limits _{\alpha }^{}h_{\alpha \alpha }\left. \mu _\alpha \right. ^*\phi _\alpha f_1\left( \left. \mu _\alpha \right. ^*,\phi _\alpha \right) \\&+{\mathop {\mathop {\sum }\limits _{\alpha ,\beta }}\limits _{\alpha \ne \beta }}^{}U_{\alpha \beta }\left. \mu _\alpha \right. ^*\left. \mu _\beta \right. ^*\phi _\alpha \phi _\beta f_3\left( \left. \mu _\alpha \right. ^*,\phi _\alpha \right) f_3\left( \left. \mu _\beta \right. ^*,\phi _\beta \right) \\&+{\mathop {\mathop {\sum }\limits _{\alpha ,\beta }}\limits _{\alpha \ne \beta }}^{}h_{\alpha \beta }\left. \mu _\alpha \right. ^*\phi _\beta f_2\left( \left. \mu _\alpha \right. ^*,\phi _\alpha \right) \exp \left( -\left. \mu _\beta \right. ^*\phi _\beta \right) {\prod \limits _{l}}^{\alpha ,\beta }g\left( \left. \mu _l\right. ^*,\phi _l\right) , \end{aligned}$$
(99)
where \(f_1,\, f_2,\, f_3\) and \(g\) are arbitrary analytic functions satisfying the following conditions:
$$\begin{aligned} f_1\left( 0,\phi \right) =f_2\left( 0,\phi \right) =f_3\left( 0,\phi \right)&= 1 \end{aligned}$$
(100)
$$\begin{aligned} g\left( 0,\phi \right)&= 1 \end{aligned}$$
(101)
$$\begin{aligned} \left. \frac{\partial }{\partial \mu ^*}g\left( \mu ^*,\phi \right) \right| _{\mu ^*=0}&= -2\phi . \end{aligned}$$
(102)
Moreover, as in Sect. 3, the product in the third line runs only over those values of \(j\), which are lying between \(\alpha\) and \(\beta\), excluding \(\alpha\) and \(\beta\) themselves,
$$\begin{aligned} {\prod \limits _{j}}^{\alpha ,\beta }\ldots =\prod \limits _{j=\min \left( \alpha ,\beta \right) +1}^{\max \left( \alpha ,\beta \right) -1}\ldots \end{aligned}$$
(103)
Appendix 2: The semiclassical amplitude
The semiclassical amplitude is given by the integral over the exponential of the second variation of the path integral around the classical path which can be written as,
$$\begin{aligned} \fancyscript{A}_\gamma&= \lim \limits _{M\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{2N-1+\left( M-1\right) J}}\int \limits _{}^{}\!\mathrm{d}^{N-1}\delta \theta ^{(0)}_{}\int \limits _{}^{}\!\mathrm{d}^N\delta J^{(M)}_{}\int \limits _{}^{}\!\mathrm{d}^N\delta \theta ^{(M)}_{}\int \limits _{}^{}\!\mathrm{d}^J\delta J^{(1)}_{}\int \limits _{}^{}\!\mathrm{d}^J\delta \theta ^{(1)}_{} \\&\cdots \int \limits _{}^{}\!\mathrm{d}^J\delta J^{(M-1)}_{}\int \limits _{}^{}\!\mathrm{d}^J\delta \theta ^{(M-1)}_{} \\&\quad \exp \Bigg \{-\frac{1}{2}\delta {\varvec{\theta }}^{(0)}\mathbf {P}_i^\prime \frac{\partial {{\phi }}^{(0)}}{\partial {\varvec{\theta }}^{(i)}}\left[ -\exp \left[ -2\mathrm{i}\mathrm{diag}\left( {\varvec{\theta }}^{(i)}\right) \right] +\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial {{{\phi }}^{(0)}}^2}\right] \frac{\partial {{\phi }}^{(0)}}{\partial {\varvec{\theta }}^{(i)}}{\mathbf {P}_i^\prime }^\mathrm{T}\delta {\varvec{\theta }}^{(0)} \\&\quad -\frac{1}{2}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(M)}\mathbf {P}_f \\ \delta \mathbf{J}^{(M)}\mathbf {P}_f \end{array}\right) {\mathbf {O}^{(M)}}^\mathrm{T}\left( \begin{array}{cc} \exp \left[ -2\mathrm{i}\mathrm{diag}\left( {\varvec{\theta }}^{(M)}\right) \right] & \mathbf {I}_{J} \\ \mathbf {I}_{J} & \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(M-1)}}{\partial {\left. {{\phi }}^{(M)}\right. ^*}^2} \end{array}\right) \mathbf {O}^{(M)}\left( \begin{array}{c} \mathbf {P}_f^\mathrm{T}\delta {\varvec{\theta }}^{(M)} \\ \mathbf {P}_f^\mathrm{T}\delta \mathbf{J}^{(M)} \end{array}\right) \\&\quad -\frac{1}{2}\sum \limits _{m=1}^{M-1}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m)} \\ \delta \mathbf{J}^{(m)} \end{array}\right) {\mathbf {O}^{(m)}}^\mathrm{T}\left( \begin{array}{cc} \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial {{{\phi }}^{(m)}}^2} & \mathbf {I}_{J} \\ \mathbf {I}_{J} & \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m-1)}}{\partial {\left. {{\phi }}^{(m)}\right. ^*}^2} \end{array}\right) \mathbf {O}^{(m)}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m)} \\ \delta \mathbf{J}^{(m)} \end{array}\right) \\&\quad +\left( \begin{array}{c} \delta {\varvec{\theta }}^{(1)} \\ \delta \mathbf{J}^{(1)} \end{array}\right) {\mathbf {O}^{(1)}}^\mathrm{T}\left( \begin{array}{c} 0 \\ \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial \left. {{\phi }}^{(1)}\right. ^*\partial {{\phi }}^{(0)}} \end{array}\right) \frac{\partial {{\phi }}^{(0)}}{\partial {\varvec{\theta }}^{(i)}}{\mathbf {P}_i^\prime }^\mathrm{T}\delta {\varvec{\theta }}^{(i)} \\&\quad +\left( \begin{array}{c} \delta {\varvec{\theta }}^{(M)}\mathbf {P}_f \\ \delta \mathbf{J}^{(M)}\mathbf {P}_f \end{array}\right) {\mathbf {O}^{(M)}}^\mathrm{T}\left( \begin{array}{cc} 0 &{} 0 \\ \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(M-1)}}{\partial \left. {{\phi }}^{(M)}\right. ^*\partial {{\phi }}^{(M-1)}} &{} 0 \end{array}\right) \mathbf {O}^{(M-1)}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(M-1)} \\ \delta \mathbf{J}^{(M-1)} \end{array}\right) \\&\quad +\sum \limits _{m=1}^{M-2}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m)} \\ \delta \mathbf{J}^{(m)} \end{array}\right) {\mathbf {O}^{(m)}}^\mathrm{T}\left( \begin{array}{cc} 0 &{} 0 \\ \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial \left. {{\phi }}^{(m+1)}\right. ^*\partial {{\phi }}^{(m)}} &{} 0 \end{array}\right) \mathbf {O}^{(m)}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m)} \\ \delta \mathbf{J}^{(m)} \end{array}\right) \Bigg \}, \end{aligned}$$
(104)
with
$$\begin{aligned} \mathbf {O}^{(m)}=\left( \begin{array}{cc} \frac{\partial {{\phi }}^{(m)}}{\partial {\varvec{\theta }}^{(m)}} &{} \frac{\partial {{\phi }}^{(m)}}{\partial \mathbf{J}^{(m)}} \\ \frac{\partial \left. {{\phi }}^{(m)}\right. ^*}{\partial {\varvec{\theta }}^{(m)}} &{} \frac{\partial \left. {{\phi }}^{(m)}\right. ^*}{\partial \mathbf{J}^{(m)}} \end{array}\right) . \end{aligned}$$
(105)
Moreover, \(\mathrm{diag}\left( \mathbf{v}\right)\) is the diagonal \(d\times d\)-matrix for which the \((j,j)\)-th entry is equal to \(v_j\), where \(d\) is the dimensionality of the vector \(\mathbf{v}\) and \(\mathbf {P}_{i/f}\) and \(\mathbf {P}_{i/f}^\prime\) are defined as the \(N\times J\) and \((N-1)\times J\)-matrices, respectively, which project onto the subspace of initially and finally occupied single-particle states, with the latter excluding the first occupied one,
$$\begin{aligned} \left( \mathbf {P}_{i/f}\right) _{lj}=&\delta _{j_{l}^{(\prime )},j} \end{aligned}$$
(106)
$$\begin{aligned} \left( \mathbf {P}_{i/f}^\prime \right) _{lj}=&\delta _{j_{l+1}^{(\prime )},j}, \end{aligned}$$
(107)
where \(j_1<\cdots <j_{N}\in \left\{ j\in \{1,\ldots ,J\}:n^{(i)}_{j}=1\right\}\) and \(j_1^\prime <\cdots <j_{N}^\prime \in \left\{ j\in \{1,\ldots ,J\}:n^{(f)}_{j}=1\right\}\) are the initially, respectively finally, occupied single-particle states.
For later reference, we also define \(\bar{\mathbf {P}}_{i/f}\) as the complement of \(\mathbf {P}_{i/f}\) as well as
$$\begin{aligned} \mathbf {Q}_{i/f}=\left( \begin{array}{c} \bar{\mathbf {P}}_{i/f} \\ \mathbf {P}_{i/f} \end{array}\right) , \end{aligned}$$
(108)
which are the (orthogonal) matrices, which put the components corresponding to initially and finally unoccupied single-particle states to the first \(J-N\) positions, and those corresponding to occupied single-particle states to the last \(N\) positions, i.e.,
$$\begin{aligned} \mathbf {Q}_{i/f}\mathbf{n}^{(i/f)}=(\underbrace{0,\ldots ,0}_{J-N},\underbrace{1,\ldots ,1}_{N})^\mathrm{T}. \end{aligned}$$
(109)
The integral over \(\delta {\varvec{\theta }}^{(0)}\) is given by
$$\begin{aligned}&\frac{1}{\left( 2\pi \right) ^{N-1}}\int \limits _{}^{}\!\mathrm{d}^{N-1}\delta \theta ^{(0)}_{}\exp \Bigg \{-\frac{1}{2}\delta {\varvec{\theta }}^{(0)}\mathbf {P}_i^\prime \frac{\partial {{\phi }}^{(0)}}{\partial {\varvec{\theta }}^{(i)}}\left( -\exp \left[ -2\mathrm{i}\mathrm{diag}\left( {\varvec{\theta }}^{(i)}\right) \right] +\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial {{{\phi }}^{(0)}}^2}\right) \frac{\partial {{\phi }}^{(0)}}{\partial {\varvec{\theta }}^{(i)}}{\mathbf {P}_i^\prime }^\mathrm{T}\delta {\varvec{\theta }}^{(0)} \\&\quad +\left( \begin{array}{c} \delta {\varvec{\theta }}^{(1)} \\ \delta \mathbf{J}^{(1)} \end{array}\right) {\mathbf {O}^{(1)}}^\mathrm{T}\left( \begin{array}{c} 0 \\ \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial \left. {{\phi }}^{(1)}\right. ^*\partial {{\phi }}^{(0)}} \end{array}\right) \frac{\partial {{\phi }}^{(0)}}{\partial {\varvec{\theta }}^{(i)}}{\mathbf {P}_i^\prime }^\mathrm{T}\delta {\varvec{\theta }}^{(i)} \\&\quad -\frac{1}{2}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(1)} \\ \delta \mathbf{J}^{(1)} \end{array}\right) {\mathbf {O}^{(1)}}^\mathrm{T}\left( \begin{array}{cc} \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(1)}}{\partial {{{\phi }}^{(1)}}^2} &{} \mathbf {I}_{J} \\ \mathbf {I}_{J} &{} \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial {\left. {{\phi }}^{(1)}\right. ^*}^2} \end{array}\right) \mathbf {O}^{(1)}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(1)} \\ \delta \mathbf{J}^{(1)} \end{array}\right) \Bigg \} \\&\quad =\frac{1}{\sqrt{2\pi }^{N-1}}\left\{ \det \left[ \mathbf {I}_{J}-\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial \left( \mathbf {P}_i{{\phi }}^{(0)}\right) ^2}\exp \left[ 2\mathrm{i}\mathrm{diag}\left( {\varvec{\theta }}^{(i)}\right) \right] \right] \right\} ^{-1} \\&\quad \times \exp \left\{ -\frac{1}{2}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(1)} \\ \delta \mathbf{J}^{(1)} \end{array}\right) {\mathbf {O}^{(1)}}^\mathrm{T}\left( \begin{array}{cc} \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(1)}}{\partial {{{\phi }}^{(1)}}^2} &{} \mathbf {I}_{J} \\ \mathbf {I}_{J} &{} \mathbf {X}^{(1)} \end{array}\right) \mathbf {O}^{(1)}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(1)} \\ \delta \mathbf{J}^{(1)} \end{array}\right) \right\} , \end{aligned}$$
(110)
where \(\mathbf {X}^{(1)}\) is defined as
$$\begin{aligned} \mathbf {X}^{(1)}&= \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial {\left. {{\phi }}^{(1)}\right. ^*}^2}+\left( \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial \left. {{\phi }}^{(1)}\right. ^*\partial {{\phi }}^{(0)}}\right) {\mathbf {P}_i^\prime }^\mathrm{{T}} \\&\quad \times \left\{ \exp \left[ -2\mathrm{i}\mathrm{diag}\left( \mathbf {P}_i^\prime {\varvec{\theta }}^{(i)}\right) \right] -\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial \left( \mathbf {P}_i^\prime {{\phi }}^{(0)}\right) ^2}\right\} ^{-1}\mathbf {P}_i^\prime \left( \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial {{\phi }}^{(0)}\partial \left. {{\phi }}^{(1)}\right. ^*}\right) \end{aligned}$$
(111)
It can be shown, that Eq. (111) can also be written as
$$\begin{aligned} \mathbf {X}^{(1)}&= \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial {\left. {{\phi }}^{(1)}\right. ^*}^2} \\& + \left( \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial \left. {{\phi }}^{(1)}\right. ^*\partial {{\phi }}^{(0)}}\right) \mathbf {X}^{(0)} \\&\times \left( \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial {{{\phi }}^{(0)}}^2}\mathbf {X}^{(0)}\right) ^{-1}\left( \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(0)}}{\partial {{\phi }}^{(0)}\partial \left. {{\phi }}^{(1)}\right. ^*}\right) , \end{aligned}$$
(112)
with
$$\begin{aligned} \mathbf {X}^{(0)}={\mathbf {Q}_i}^\mathrm{T}\left( \begin{array}{cc} 0 \\ &{} \exp \left[ 2\mathrm{i}\mathrm{diag}\left( \mathbf {P}_i^\prime {\varvec{\theta }}^{(i)}\right) \right] \end{array}\right) \mathbf {Q}_i. \end{aligned}$$
(113)
Now, consider the integral
$$\begin{aligned}&\frac{1}{\left( 2\pi \right) ^{J}}\int \limits _{}^{}\!\mathrm{d}^J\delta J^{(m)}_{}\int \limits _{}^{}\!\mathrm{d}^J\delta \theta ^{(m)}_{}\exp \Bigg \{-\frac{1}{2}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m+1)} \\ \delta \mathbf{J}^{(m+1)} \end{array}\right) \\&\quad \times {\mathbf {O}^{(m+1)}}^\mathrm{T}\left( \begin{array}{cc} \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m+1)}}{\partial {{{\phi }}^{(m+1)}}^2} &{} \mathbf {I}_{J} \\ \mathbf {I}_{J} &{} \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial {\left. {{\phi }}^{(m+1)}\right. ^*}^2} \end{array}\right) {\mathbf {O}^{(m+1)}}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m+1)} \\ \delta \mathbf{J}^{(m+1)} \end{array}\right) \\&\quad -\frac{1}{2}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m)} \\ \delta \mathbf{J}^{(m)} \end{array}\right) {\mathbf {O}^{(m)}}^\mathrm{T}\left( \begin{array}{cc} \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial {{{\phi }}^{(m)}}^2} &{} \mathbf {I}_{J} \\ \mathbf {I}_{J} &{} X^{(m)} \end{array}\right) {\mathbf {O}^{(m)}}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m)} \\ \delta \mathbf{J}^{(m)} \end{array}\right) \\&\quad +\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m+1)} \\ \delta \mathbf{J}^{(m+1)} \end{array}\right) {\mathbf {O}^{(m+1)}}^\mathrm{T}\left( \begin{array}{cc} 0 &{} 0 \\ \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial \left. {{\phi }}^{(m+1)}\right. ^*\partial {{\phi }}^{(m)}} &{} 0 \end{array}\right) {\mathbf {O}^{(m)}}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m)} \\ \delta \mathbf{J}^{(m)} \end{array}\right) \Bigg \} \\&\quad =\left\{ \det \left[ \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial {{{\phi }}^{(m)}}^2}\mathbf {X}^{(m)}\right] \right\} ^{-1} \\&\quad \exp \left\{ -\frac{1}{2}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m+1)} \\ \delta \mathbf{J}^{(m+1)} \end{array}\right) {\mathbf {O}^{(m+1)}}^\mathrm{T}\left( \begin{array}{cc} \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m+1)}}{\partial {{{\phi }}^{(m+1)}}^2} &{} \mathbf {I}_{J} \\ \mathbf {I}_{J} &{} \mathbf {X}^{(m+1)} \end{array}\right) {\mathbf {O}^{(m+1)}}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(m+1)} \\ \delta \mathbf{J}^{(m+1)} \end{array}\right) \right\} \end{aligned}$$
(114)
with
$$\begin{aligned} \mathbf {X}^{(m+1)}&= \frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial {\left. {{\phi }}^{(m+1)}\right. ^*}^2}+\left( \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial \left. {{\phi }}^{(m+1)}\right. ^*\partial {{\phi }}^{(m)}}\right) \mathbf {X}^{(m)} \\&\times \left( \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial {{{\phi }}^{(m)}}^2}\mathbf {X}^{(m)}\right) ^{-1}\left( \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial {{\phi }}^{(m)}\partial \left. {{\phi }}^{(m+1)}\right. ^*}\right) . \end{aligned}$$
(115)
For \(m=1\) this is exactly the integral in Eq. (104) after integrating out \(\delta {\varvec{\theta }}^{(0)}\) and thus defines \(\mathbf {X}^{(2)}\). One then recognizes, that after the \(m\)-th integration, the integral is again of the form of Eq. (114) up to the \((M-1)\)-th integration. With this observation, the semiclassical amplitude is given by
$$\begin{aligned} \fancyscript{A}_\gamma&= \lim \limits _{M\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{\frac{3N-1}{2}}}\int \limits _{}^{}\!\mathrm{d}^NJ^{(M)}_{}\int \limits _{}^{}\!\mathrm{d}^N\theta ^{(M)}_{}\prod \limits _{m=0}^{M-1}\sqrt{\det \left( \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial {{{\phi }}^{(m)}}^2}\mathbf {X}^{(m)}\right) }^{-1} \\&\qquad \times \exp \left\{ -\frac{1}{2}\left( \begin{array}{c} \delta {\varvec{\theta }}^{(M)}\mathbf {P}_f \\ \delta \mathbf{J}^{(M)}\mathbf {P}_f \end{array}\right) {\mathbf {O}^{(M)}}^\mathrm{T}\left( \begin{array}{cc} \exp \left[ -2\mathrm{i}\mathrm{diag}\left( {\varvec{\theta }}^{(M)}\right) \right] &{} \mathbf {I}_{J} \\ \mathbf {I}_{J} &{} X^{(M)} \end{array}\right) \mathbf {O}^{(M)}\left( \begin{array}{c} \mathbf {P}_f^\mathrm{T}\delta {\varvec{\theta }}^{(M)} \\ \mathbf {P}_f^\mathrm{T}\delta \mathbf{J}^{(M)} \end{array}\right) \right\} \\&= \lim \limits _{M\rightarrow \infty }\frac{1}{\sqrt{2\pi }^{N-1}}\left[ \prod \limits _{m=0}^{M-1}\sqrt{\det \left( \mathbf {I}_{J}-\frac{\mathrm{i}\tau }{\hbar }\frac{\partial ^2{H^{(cl)}}^{(m)}}{\partial {{{\phi }}^{(m)}}^2}\mathbf {X}^{(m)}\right) }^{-1}\right] \sqrt{\det \left( \mathbf {I}_{N}-\exp \left[ -2\mathrm{i}\mathrm{diag}\left( \mathbf {P}_{f}{\varvec{\theta }}^{(M)}\right) \right] \mathbf {P}_f\mathbf {X}^{(M)}\mathbf {P}_f^\mathrm{T}\right) }^{-1}. \end{aligned}$$
(116)
In the continuous limit, the discrete set of \(\mathbf {X}^{(m)}\) turns into a function of time \(\mathbf {X}(t)\), and (by expanding it up to first order in \(\tau\)) is given by Eq. (60), and the semiclassical amplitude can be written in the form given in Eq. (57).
Appendix 3: Possible classical Hamiltonians
In this part, we state different possibilities for the classical hamiltonian as can be derived out of similar calculations as in Appendix 1 without going further into detail.
1.1 Appendix 3.1: Classical Hamiltonians in the particle picture
First, we present two possibilities arising directly from the derivation presented in Appendix 1, but restrict ourselves to those, which contain \({\mu }\) and \({\phi }\) in a symmetric way and omitting the one already stated in Sect. 3. These examples shall just illustrate, which kinds of classical Hamiltonians are possible:
$$\begin{aligned} H_{cl}^{(1)}&\left( \left. {\mu }\right. ^*,{\phi }\right) \\&\quad = \sum \limits _{\alpha }^{}h_{\alpha \alpha }\left. \mu _\alpha \right. ^*\phi _\alpha \cos \left( \left. \mu _\alpha \right. ^*\phi _\alpha \right) +{\mathop {\mathop {\sum }\limits _{\alpha ,\beta }}\limits _{\alpha \ne \beta }}^{}U_{\alpha \beta }\left. \mu _\alpha \right. ^*\left. \mu _\beta \right. ^*\phi _\alpha \phi _\beta \\&\quad +{\mathop {\mathop {\sum }\limits _{\alpha ,\beta }}\limits _{\alpha \ne \beta }}^{}h_{\alpha \beta }\left. \mu _\alpha \right. ^*\phi _\beta \exp \left( -\sum \limits _{l=\min \left( \alpha ,\beta \right) }^{\max \left( \alpha ,\beta \right) }\left. \mu _l\right. ^*\phi _l\right) , \end{aligned}$$
(117)
$$\begin{aligned}&H_{cl}^{(2)}\left( \left. {\mu }\right. ^*,{\phi }\right) \\&\quad =\sum \limits _{\alpha }^{}h_{\alpha \alpha }\left. \mu _\alpha \right. ^*\phi _\alpha \exp \left( \left. \mu _\alpha \right. ^*\phi _\alpha \right) \\&\quad +{\mathop {\mathop {\sum }\limits _{\alpha ,\beta }}\limits _{\alpha \ne \beta }}^{}h_{\alpha \beta }\left. \mu _\alpha \right. ^*\phi _\beta \exp \left( -\left. \mu _\beta \right. ^*\phi _\beta -\left. \mu _\alpha \right. ^*\phi _\alpha \right) \\& \quad \times \prod \limits _{l=\min \left( \alpha ,\beta \right) +1}^{\max \left( \alpha ,\beta \right) -1}\left[ 1-\sinh \left( 2\left. \mu _l\right. ^*\phi _l\right) \right] \\& \quad +{\mathop {\mathop {\sum }\limits _{\alpha ,\beta }}\limits _{\alpha \ne \beta }}^{}U_{\alpha \beta }\left. \mu _\alpha \right. ^*\left. \mu _\beta \right. ^*\phi _\alpha \phi _\beta \cosh \left( \left. \mu _\alpha \right. ^*\phi _\alpha \right) \cosh \left( \left. \mu _\beta \right. ^*\phi _\beta \right) , \end{aligned}$$
(118)
Next, consider the more general case, that the quantum Hamiltonian is written in the form
$$\begin{aligned} \hat{H}=\sum \limits _{\alpha ,\beta }^{}h_{\alpha \beta }\hat{c}_\alpha ^\dagger \hat{c}_\beta ^{}+{\mathop {\mathop {\sum }\limits _{\alpha ,\beta ,\rho ,\nu }}\limits _{\alpha \ne \beta ,\rho \ne \nu }}^{}U_{\alpha \beta \rho \nu }\hat{c}_\alpha ^\dagger \hat{c}_\beta ^\dagger \hat{c}_\rho ^{}\hat{c}_\nu . \end{aligned}$$
(119)
By splitting the interaction term also into (pairwise) diagonal and non-diagonal terms, one can in a similar way as in Sect. 7 construct the following classical Hamiltonian
$$\begin{aligned} H_{cl}\left( \left. {\mu }\right. ^*,{\phi }\right)&= \sum \limits _{\alpha }^{}h_{\alpha \alpha }\left. \mu _\alpha \right. ^*\phi _\alpha f_1\left( \left. \mu _\alpha \right. ^*,\phi _\alpha \right) \\&\quad +{\mathop {\mathop {\sum }\limits _{\alpha ,\beta }}\limits _{\alpha \ne \beta }}^{}h_{\alpha \beta }\left. \mu _\alpha \right. ^*\phi _\beta f_2\left( \left. \mu _\alpha \right. ^*,\phi _\alpha \right) \exp \left( -\left. \mu _\beta \right. ^*\phi _\beta \right) \prod \limits _{l=\min \left( \alpha ,\beta \right) +1}^{\max \left( \alpha ,\beta \right) -1}g\left( \left. \mu _l\right. ^*,\phi _l\right) \\&\quad +{\mathop {\mathop {\sum }\limits _{\alpha ,\beta }}\limits _{\alpha \ne \beta }}^{}U_{\alpha \beta \beta \alpha }\left. \mu _\alpha \right. ^*\left. \mu _\beta \right. ^*\phi _\alpha \phi _\beta f_3\left( \left. \mu _\alpha \right. ^*,\phi _\alpha \right) f_3\left( \left. \mu _\beta \right. ^*,\phi _\beta \right) \\&\quad +{\mathop {\mathop {\sum }\limits _{\alpha ,\beta ,\rho }}\limits _{\alpha \ne \beta ,\rho \ne \alpha ,\rho \ne \beta }}^{}\left[ \Theta \left( \beta -\alpha \right) \Theta \left( \beta -\rho \right) +\Theta \left( \alpha -\beta \right) \Theta \left( \rho -\beta \right) \right. \\&\quad \left. -\Theta \left( \alpha -\beta \right) \Theta \left( \beta -\rho \right) -\Theta \left( \beta -\alpha \right) \Theta \left( \rho -\beta \right) \right] \\&\quad\times \left( U_{\alpha \beta \beta \rho }-U_{\alpha \beta \rho \beta }\right) \left. \mu _\alpha \right. ^*\left. \mu _\beta \right. ^*\phi _\beta \phi _\rho f_1\left( \left. \mu _\alpha \right. ^*,\phi _\alpha \right) f_2\left( \left. \mu _\alpha \right. ^*,\phi _\alpha \right) \\&\quad \times \exp \left( -\left. \mu _\rho \right. ^*\phi _\rho \right) \prod \limits _{j=\min \left( \alpha ,\rho \right) +1}^{\max \left( \alpha ,\rho \right) -1}g\left( \left. \mu _j\right. ^*,\phi _j\right) \\&\quad +{\mathop {\mathop {\sum }\limits _{\alpha ,\beta ,\rho }}\limits _{ \alpha \ne \beta ,\rho \ne \alpha ,\rho \ne \beta }}^{}\left[ \Theta \left( \beta -\alpha \right) \Theta \left( \rho -\alpha \right) \right. \\&\quad \left. +\Theta \left( \alpha -\beta \right) \Theta \left( \alpha -\rho \right) -\Theta \left( \alpha -\beta \right) \Theta \left( \rho -\alpha \right) -\Theta \left( \beta -\alpha \right) \Theta \left( \alpha -\rho \right) \right] \\&\quad \left( U_{\alpha \beta \rho \alpha }-U_{\alpha \beta \alpha \rho }\right) \left. \mu _\alpha \right. ^*\left. \mu _\beta \right. ^*\phi _\alpha \phi _\rho f_1\left( \left. \mu _\alpha \right. ^*,\phi _\alpha \right) f_2\left( \left. \mu _\beta \right. ^*,\phi _\beta \right) \\&\quad \times\exp \left( -\left. \mu _\rho \right. ^*\phi _\rho \right) \prod \limits _{j=\min \left( \beta ,\rho \right) +1}^{\max \left( \beta ,\rho \right) -1}g\left( \left. \mu _j\right. ^*,\phi _j\right) \\&\quad +{\mathop {\mathop {\sum }\limits _{\alpha ,\beta ,\rho ,\nu }}\limits _{\alpha \ne \beta ,\alpha \ne \rho ,\alpha \ne \nu ,\beta \ne \rho ,\beta \ne \nu ,\rho \ne \nu }}^{}\left[ \Theta \left( \beta -\alpha \right) \right. \\&\quad \left. -\Theta \left( \alpha -\beta \right) \right] \left[ \Theta \left( \rho -\nu \right) -\Theta \left( \nu -\rho \right) \right] U_{\alpha \beta \rho \nu }\left. \mu _\alpha \right. ^*\left. \mu _\beta \right. ^*\phi _\rho \phi _\nu f_2 \\&\left( \left. \mu _\alpha \right. ^*,\phi _\alpha \right) f_2\left( \left. \mu _\beta \right. ^*,\phi _\beta \right) \\&\quad \exp \left( -\left. \mu _\rho \right. ^*\phi _\rho -\left. \mu _\nu \right. ^*\phi _\nu \right) \\&\quad \times \left[ \prod \limits _{l=\min \left( \alpha ,\beta ,\rho ,\nu \right) +1}^{\min \big \{\left\{ \alpha ,\beta ,\rho ,\nu \right\} \setminus \left\{ \min \left( \alpha ,\beta ,\rho ,\nu \right) \right\} \big \}-1}g\left( \left. \mu _j\right. ^*,\phi _j\right) \right] \left[ \prod \limits _{l=\max \big \{\left\{ \alpha ,\beta ,\rho ,\nu \right\} \setminus \left\{ \max \left( \alpha ,\beta ,\rho ,\nu \right) \right\} \big \}+1}^{\max \left( \alpha ,\beta ,\rho ,\nu \right) -1}g\left( \left. \mu _j\right. ^*,\phi _j\right) \right] , \end{aligned}$$
(120)
where \(f_1,\, f_2,\, f_3\) and \(g\) are again arbitrary analytic functions satisfying Eqs. (100–102). Thereby, one should notice, that
$$\begin{aligned} \min \big \{\left\{ \alpha ,\beta ,\rho ,\nu \right\} \setminus \left\{ \min \left( \alpha ,\beta ,\rho ,\nu \right) \right\} \big \}\ \end{aligned}$$
is the second smallest number out of the set \(\left\{ \alpha ,\beta ,\rho ,\nu \right\}\) and
$$\begin{aligned} \max \big \{\left\{ \alpha ,\beta ,\rho ,\nu \right\} \setminus \left\{ \max \left( \alpha ,\beta ,\rho ,\nu \right) \right\} \big \} \end{aligned}$$
the second largest number out of the set \(\left\{ \alpha ,\beta ,\rho ,\nu \right\}\).
1.2 Appendix 3.2: Classical Hamiltonians in the hole picture
The cases considered above, we call particle picture, since the boundary conditions are such, that \(\left| \phi _j\right| ^2=1\) corresponds to the \(j\)-th single-particle state being occupied, while \(\left| \phi _j\right| ^2=0\) corresponds to the \(j\)-th single-particle state being empty. However, the role of occupied and unoccupied states can be reversed, if Eqs. (88) are replaced by
$$\begin{aligned}&\int \mathrm{d}^{2J}\zeta ^{(0)}_{}\exp \left( -\left. {\varvec{\zeta }}^{(0)}\right. ^*\cdot {\varvec{\zeta }}^{(0)}\right) \left[ \prod \limits _{j=1}^{J}\left( 1+\left. \chi ^{(0)}_{j}\right. ^*\zeta ^{(0)}_{j}\right) \right] \prod \limits _{j=1}^{J}\left( \left. \zeta ^{(0)}_{j}\right. ^*\right) ^{n^{(i)}_{j}} \\&= \int \frac{\mathrm{d}^{2\left( J-N_i\right) }\phi ^{(0)}_{}}{\pi ^{J-N_i}}\int \limits _{0}^{2\pi }\frac{\mathrm{d}^{J-N_i}\theta ^{(i)}_{}}{\left( 2\pi \right) ^{J-N_i}}\int \mathrm{d}^{2J}\zeta ^{(0)}_{} \\&\exp \left( -\left. {\varvec{\zeta }}^{(0)}\right. ^*\cdot {\varvec{\zeta }}^{(0)}-\left| {{\phi }}^{(0)}\right| ^2+\left. {{\phi }}^{(0)}\right. ^*\cdot {{\mu }}^{(0)}\right) \\&\times \left[ \prod \limits _{j=1}^{J}\left( \phi ^{(0)}_{j}+\left. \chi ^{(0)}_{j}\right. ^*\right) \right] \left[ \prod \limits _{j=0}^{J-1}\left( \left. \mu ^{(0)}_{J-j}\right. ^*+\zeta ^{(0)}_{J-j}\right) \right] \prod \limits _{j=1}^{J}\left( \left. \zeta ^{(0)}_{j}\right. ^*\right) ^{n^{(i)}_{j}}, \end{aligned}$$
(121)
where the integrations over \({\varvec{\theta }}^{(i)}\) and \({{\phi }}^{(0)}\) run over those components, which are initially empty \(\mu ^{(0)}_{j}=( 1-n^{(i)}_{j}) \exp ( \mathrm{i}\theta ^{(i)}_{j})\), as well as
$$\begin{aligned} a^{(m)}&= \int \frac{\mathrm{d}^{2J}\mu ^{(m)}_{}}{\pi ^J}\int \frac{\mathrm{d}^{2J}\phi ^{(m)}_{}}{\pi ^J}\left[ \prod \limits _{j=1}^{J}\left( \phi ^{(m)}_{j}\right) +\left. \chi ^{(m)}_{j}\right. ^*\right] \\&\quad \times \exp \left( -\left| {{\phi }}^{(m)}\right| ^2-\left| {{\mu }}^{(m)}\right| ^2+\left. {{\phi }}^{(m)}\right. ^*\cdot {{\mu }}^{(m)}\right) \prod \limits _{j=0}^{J-1}\left[ \sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}+\zeta ^{(m)}_{J-j}\right] , \end{aligned}$$
(122)
$$\begin{aligned} b_{\alpha }^{(m)}&= \int \frac{\mathrm{d}^{2J}\mu ^{(m)}_{}}{\pi ^J}\int \frac{\mathrm{d}^{2J}\phi ^{(m)}_{}}{\pi ^J} \\&\quad\times\exp \left( -\left| {{\phi }}^{(m)}\right| ^2-\left| {{\mu }}^{(m)}\right| ^2+\left. {{\phi }}^{(m)}\right. ^*\cdot {{\mu }}^{(m)}\right) \left. \zeta ^{(m)}_{\alpha }\right. ^*\chi ^{(m-1)}_{\alpha }\left\{ \prod \limits _{j=0}^{J-\alpha -1}\left[ \sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}+\zeta ^{(m)}_{J-j}\right] \right\} \\&\quad \times \left[ \sum \limits _{k=1}^{\infty }c_k^{(1)}\left( \phi ^{(m-1)}_{\alpha }\right) \left( \left. \mu ^{(m)}_{\alpha }\right. ^*\right) ^{k}+\zeta ^{(m)}_{\alpha }\right] \left\{ \prod \limits _{j=J-\alpha +1}^{J-1}\left[ \sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}+\zeta ^{(m)}_{J-j}\right] \right\} \\&\quad \times \left[ \prod \limits _{j=1}^{J}\left( \phi ^{(m)}_{j}+\left. \chi ^{(m)}_{j}\right. ^*\right) \right] , \end{aligned}$$
(123)
$$\begin{aligned} c_{\alpha \beta }^{(m)}&= \int \frac{\mathrm{d}^{2J}\mu ^{(m)}_{}}{\pi ^J}\int \frac{\mathrm{d}^{2J}\phi ^{(m)}_{}}{\pi ^J}\exp \left( -\left| {{\phi }}^{(m)}\right| ^2-\left| {{\mu }}^{(m)}\right| ^2+\left. {{\phi }}^{(m)}\right. ^*\cdot {{\mu }}^{(m)}\right) \left. \zeta ^{(m)}_{\alpha }\right. ^*\chi ^{(m-1)}_{\beta } \\&\quad\times\left\{ \prod \limits _{j=0}^{J-\max \left( \alpha ,\beta \right) -1}\left[ \sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}+\zeta ^{(m)}_{J-j}\right] \right\} \\&\quad \times \left[ \sum \limits _{k=1}^{\infty }c_k^{(2)}\left( \phi ^{(m-1)}_{\max \left( \alpha ,\beta \right) }\right) \left( \left. \mu ^{(m)}_{\max \left( \alpha ,\beta \right) }\right. ^*\right) ^k+\zeta ^{(m)}_{\max \left( \alpha ,\beta \right) }\right] \\&\quad \times \left\{ \prod \limits _{j=J-\max \left( \alpha ,\beta \right) +1}^{J-\min \left( \alpha ,\beta \right) -1}\left[ \sum \limits _{k=1}^{\infty }c_k^{(3)}\left( \phi ^{(m-1)}_{J-j}\right) \left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}+\zeta ^{(m)}_{J-j}\right] \right\} \\&\quad \times \left[ \sum \limits _{k=1}^{\infty }c_k^{(2)}\left( \phi ^{(m-1)}_{\min \left( \alpha ,\beta \right) }\right) \left( \left. \mu ^{(m)}_{\min \left( \alpha ,\beta \right) }\right. ^*\right) ^k+\zeta ^{(m)}_{\min \left( \alpha ,\beta \right) }\right] \\&\quad \times \left\{ \prod \limits _{j=J-\min \left( \alpha ,\beta \right) +1}^{J-1}\left[ \sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}+\zeta ^{(m)}_{J-j}\right] \right\} \\&\quad \times \left[ \prod \limits _{j=1}^{J}\left( \phi ^{(m)}_{j}+\left. \chi ^{(m)}_{j}\right. ^*\right) \right] \end{aligned}$$
(124)
$$\begin{aligned} d_{\alpha \beta }^{(m)}&= \int \frac{\mathrm{d}^{2J}\mu ^{(m)}_{}}{\pi ^J}\int \frac{\mathrm{d}^{2J}\phi ^{(m)}_{}}{\pi ^J}\exp \left( -\left| {{\phi }}^{(m)}\right| ^2-\left| {{\mu }}^{(m)}\right| ^2+\left. {{\phi }}^{(m)}\right. ^*\cdot {{\mu }}^{(m)}\right) \\&\quad \times\left[ \prod \limits _{j=1}^{J}\left( \phi ^{(m)}_{j}+\left. \chi ^{(m)}_{j}\right. ^*\right) \right] \left. \zeta ^{(m)}_{\alpha }\right. ^*\left. \zeta ^{(m)}_{\beta }\right. ^*\chi ^{(m-1)}_{\beta }\chi ^{(m-1)}_{\alpha } \\&\quad \times \left\{ \prod \limits _{j=0}^{J-\max \left( \alpha ,\beta \right) -1}\left[ \sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}+\zeta ^{(m)}_{J-j}\right] \right\} \\&\quad\times \left[ \sum \limits _{k=1}^{\infty }c_k^{(4)}\left( \phi ^{(m-1)}_{\max \left( \alpha ,\beta \right) }\right) \left( \left. \mu ^{(m)}_{\max \left( \alpha ,\beta \right) }\right. ^*\right) ^k+\zeta ^{(m)}_{\max \left( \alpha ,\beta \right) }\right] \\&\quad \times \left\{ \prod \limits _{j=J-\max \left( \alpha ,\beta \right) +1}^{J-\min \left( \alpha ,\beta \right) -1}\left[ \sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}+\zeta ^{(m)}_{J-j}\right] \right\} \\&\quad\times \left[ \sum \limits _{k=1}^{\infty }c_k^{(4)}\left( \phi ^{(m-1)}_{\min \left( \alpha ,\beta \right) }\right) \left( \left. \mu ^{(m)}_{\min \left( \alpha ,\beta \right) }\right. ^*\right) ^k+\zeta ^{(m)}_{\min \left( \alpha ,\beta \right) }\right] \\&\quad \times \left\{ \prod \limits _{j=J-\min \left( \alpha ,\beta \right) +1}^{J-1}\left[ \sum \limits _{k=1}^{\infty }\frac{1}{k!}\left( \left. \mu ^{(m)}_{J-j}\right. ^*\right) ^{k}\left( \phi ^{(m-1)}_{J-j}\right) ^{k-1}+\zeta ^{(m)}_{J-j}\right] \right\} , \end{aligned}$$
(125)
Inserting the integrals like this results in the following path integral:
$$\begin{aligned} K\left( \mathbf{n}^{(f)},\mathbf{n}^{(i)};t_f\right)&= \left[ \prod \limits _{j:n_j^{(i)}=0}^{}\int \limits _{0}^{2\pi }\frac{\mathrm{d}\theta _j^{(0)}}{2\pi }\exp \left( -\mathrm{i}\theta _j^{(0)}\right) \right] \left[ \prod \limits _{m=1}^{M-1}\prod \limits _{j}^{}\int \limits _{\mathbb {C}}^{}\frac{\mathrm{d}\phi _j^{(m)}}{\pi }\exp \left( -\left| \phi _j^{(m)}\right| ^2\right) \right] \\&\quad \times \left[ \prod \limits _{j:n_j^{(f)}=0}^{}\int \limits _{\mathbb {C}}^{}\frac{\mathrm{d}\phi _j^{(M)}}{\pi }\phi _j^{(M)}\right. \left. \exp \left( -\left| \phi _j^{(M)}\right| ^2\right) \right] \\&\quad \times\exp \left\{ \sum \limits _{m=1}^{M}\left[ {{\phi }^{(m)}}^*\cdot {\phi }^{(m-1)}\right. \right. \left. \left. -\frac{\mathrm{i}\tau }{\hbar }H_{cl}\left( {{\phi }^{(m)}}^{*},{\phi }^{(m-1)}\right) \right] \right\} , \end{aligned}$$
(126)
with the classical hamiltonian
$$\begin{aligned}&{H^{(cl)}}^{(m)}\left( \left. {\mu }\right. ^*,{\phi }\right) \\&\quad= \sum \limits _{\alpha =1}^{J}h_{\alpha \alpha }^{(m)}\exp \left( -\left. \mu _\alpha \right. ^*\phi _\alpha \right) \\&\quad +{\mathop {\mathop {\sum }\limits _{\alpha ,\beta =1 }}\limits _{\alpha \ne \beta }^{J}}U_{\alpha \beta }^{(m)}\exp \left( -\left. \mu _\alpha \right. ^*\phi _\alpha -\left. \mu _\beta \right. ^*\phi _\beta \right) \\&\quad +{\mathop {\mathop {\sum }\limits _{\alpha ,\beta =1 }}\limits _{\alpha \ne \beta }^{J}}h_{\alpha \beta }^{(m)}\left. \mu _\beta
\right. ^*\phi _\alpha \exp \left( -\left. \mu _\alpha \right. ^*\phi _\alpha \right) f\left( \left. \mu _\beta \right. ^*,\phi _\beta \right) \prod \limits _{j=\min \left( \alpha ,\beta \right) +1}^{\max \left( \alpha ,\beta \right) -1}g\left( \left. \mu _j\right. ^*,\phi _j\right) , \end{aligned}$$
(127)
where \(f\) and \(g\) are arbitrary analytical functions satisfying
$$\begin{aligned}&f(0,\phi )=1 \end{aligned}$$
(128)
$$\begin{aligned}&g(0,\phi )=-1 \end{aligned}$$
(129)
$$\begin{aligned}&\left. \frac{\partial }{\partial \left. \mu \right. ^*}g\left( \left. \mu \right. ^*,\phi \right) \right| _{\left. \mu \right. ^*=0}=2\phi . \end{aligned}$$
(130)
