As explained above, we have a simple, unitary evolution law; however, it only works if all in- and out-particles can be identified exclusively by their momentum distributions, \(p_{\ell m}^\mp \). This will be assumed to be the case, for the time being. Thanks to the spherical harmonics expansion, all calculations can be done explicitly. From here on, we omit the subscripts \(\ell ,\,m\), since the different \(\ell \) and \(m\) values do not mix.Footnote 2
Thus, writing \(u=u^+,\ p=p^-\) for the in-particles, we have a single wave function \(\psi (u)\), where
$$\begin{aligned} \langle u|p\rangle ={\textstyle \frac{1}{\sqrt{2\pi }}}e^{\textstyle ipu}\ .\end{aligned}$$
(7.1)
The dependence on the scaled time parameter \(\tau =t/4GM\) is:
$$\begin{aligned} p(\tau )=p(0)e^\tau \ ,\qquad u(\tau )=u(0)e^{-\tau }\ , \end{aligned}$$
(7.2)
which is opposite for the out-particles. It is then convenient to write both \(u\) and \(p\) in terms of exponentials, since these exponents will grow or shrink linearly in time, while the positions will also vary linearly in time far from the black hole. We are invited to use the tortoise coordinates.
Close to the origin, therefore, we write
$$\begin{aligned} u=\sigma _u\,e^{\textstyle \varrho _u}\ ,\quad p=\sigma _p\,e^{\textstyle \varrho _p}\ , \end{aligned}$$
(7.3)
where we were forced to add explicit variables \(\sigma _u=\pm \,1\) and \(\sigma _p=\pm \,1\), since the exponents themselves would only be positive. Thus, \(\sigma _u=\pm \,1\), depending on whether we are in region \(I\) or region \(II\).
Expressing the wave function in terms of \(\sigma _u\) and \(\varrho _u\), we multiply with a Jacobian factor \(e^{\textstyle {\textstyle \frac{1}{2}}\varrho _u}\) in order to preserve unitarity, so,
$$\begin{aligned} \tilde{\psi }_{\sigma _u}(\varrho _u)\equiv e^{\textstyle {\textstyle \frac{1}{2}}\varrho _u} \psi (\sigma _u\,e^{\textstyle \varrho _u})\ ,\qquad \tilde{\hat{\psi }}_{\sigma _p}(\varrho _p)\equiv e^{\textstyle {\textstyle \frac{1}{2}}\varrho _p}\hat{\psi }(\sigma _p e^{\textstyle \varrho _p})\ ; \end{aligned}$$
(7.4)
Indeed, the norm is preserved:
$$\begin{aligned} |\psi |^2=\sum _{\sigma _u=\pm }\int _{-\infty }^\infty \mathrm{d}\varrho _u|\tilde{\psi }_{\sigma _u}(\varrho _u)|^2=\sum _{\sigma _p=\pm }\int _{-\infty }^\infty \mathrm{d}\varrho _p|\tilde{\hat{\psi }}_{\sigma _p}(\varrho _p)|^2\ . \end{aligned}$$
(7.5)
We write the Fourier transform in terms of a kernel \(K_{\sigma }(\varrho )\), as follows:
$$\begin{aligned} \tilde{\hat{\psi }}_{\sigma _p}(\varrho _p)= & {} \sum _{\sigma _u=\pm 1}\int _{-\infty }^\infty \mathrm{d}\varrho \,K_{\sigma _u\sigma _p}(\varrho )\,\tilde{\psi }_{\sigma _u}(\varrho -\varrho _p)\ , \end{aligned}$$
(7.6)
$$\begin{aligned} \hbox {with }\quad K_\sigma (\varrho )\equiv & {} {\textstyle \frac{1}{\sqrt{2\pi }}}e^{\textstyle {\textstyle \frac{1}{2}}\varrho }\,e^{\textstyle -i\sigma \,e^\varrho }\ . \end{aligned}$$
(7.7)
The integral converges since the kernel becomes rapidly oscillating when \(\varrho \) is large.
We have a symmetry under the transformation
$$\begin{aligned} \varrho _p\rightarrow \varrho _p+\lambda \ ,\quad \varrho _u\rightarrow \varrho _u-\lambda \ , \end{aligned}$$
(7.8)
which is a consequence for the symmetry
$$\begin{aligned} p\rightarrow p\,e^{\textstyle \lambda }\ ,\quad u\rightarrow u\,e^{\textstyle -\lambda }\ \end{aligned}$$
(7.9)
in the Fourier transformations. We can make use of this symmetry by Fourier transforming the wave functions with respect to the variables \(\varrho _u\) and \(\varrho _p\). Write
$$\begin{aligned} \tilde{\psi }_{\sigma _{\! u}}(\varrho _u)\equiv \breve{\psi }_{\sigma _{\! u}}(\kappa )\, e^{\textstyle -i\kappa \varrho _u} \ ;\qquad \tilde{\hat{\psi }}_{\sigma _{\!p}}(\varrho _p)\equiv \breve{{\hat{\psi }}}_{\sigma _{\!p}}(\kappa )\,e^{\textstyle i\kappa \varrho _p}\ . \end{aligned}$$
(7.10)
Thus, \(\kappa \) is the eigen value of the Lorentz transformation at the origin, which is the time boost for the external observer. In short, \(\kappa \) is the energy. It is conserved in the process where an in-particle leaves its footprint in the out-particles. Consequently, the Fourier transform now reduces to the multiplication of the wave function by a factor:
$$\begin{aligned} \breve{\hat{\psi }}_{\sigma _{\! p}}(\kappa )=\sum _{\sigma _p=\pm 1}F_{\sigma _{\!u}\sigma _{\! p}}(\kappa )\breve{\psi }_{\sigma _{\!u}}(\kappa )\ ;\qquad F_\sigma (\kappa )\equiv \int _{-\infty }^\infty K_\sigma (\varrho ) e^{\textstyle -i\kappa \varrho }\mathrm{d}\varrho \ . \end{aligned}$$
(7.11)
The integral can be worked out:
$$\begin{aligned} F_\sigma (\kappa )=\int _0^\infty \frac{\mathrm{d}y}{y} y^{{\scriptstyle \frac{1}{2}}-i\kappa }\,e^{-i\sigma y}\ =\ \Gamma ({\textstyle \frac{1}{2}}-i\kappa )\,e^{\textstyle -{\textstyle \frac{i\sigma \pi }{4}}-{\textstyle \frac{\pi }{2}}\kappa \sigma }\ . \end{aligned}$$
(7.12)
In Eq. (7.11), the matrix \(\left( \begin{array}{ll} F_{+} &{} F_{-}\\ F_{-} &{} F_{+}\\ \end{array}\right) \) is unitary: \(F_+F_-^*=-F_-F_+^*\) and \(|F_+|^2+|F_-|^2=1\) . After adding the factor \(\log (8\pi G/(\ell ^2+\ell +1))\) (see Eq. 5.3), we see that Eq. (7.11) acts as a boundary condition, bouncing the in-going wave back as an out-going wave. During the entire evolution, the Hamiltonian is just the dilation operator [23]:
$$\begin{aligned} H= & {} -{\textstyle \frac{1}{2}}(u^+p^-+p^-u^+)={\textstyle \frac{1}{2}}(u^-p^++p^+u^-)\ \end{aligned}$$
(7.13)
$$\begin{aligned}= & {} i\frac{\partial }{\partial \varrho _{u^+}}\ =\ -i\frac{\partial }{\partial \varrho _{u^-}}= -i\frac{\partial }{\partial \varrho _{p^-}}\ =\ i\frac{\partial }{\partial \varrho _{p^+}}\ =\ \kappa \ . \end{aligned}$$
(7.14)
Again we emphasise the simplicity of these equations, they are merely one-dimensional ordinary differential equations, hiding nothing.
The bounce guarantees that soft particles never become hard, both in the far future and in the far past. Thus, we obtain the complete set of (pure) quantum states of the black hole.
As for the range of allowed \(\ell \) values, there are still some things to be sorted out. In practice, it seems, that the total number of \((\ell ,\,m)\) partial waves that is to be included tends to coincide with the total number of Hawking particles emitted during the black hole lifetime.
An other remark is that, after leaving their footprints in the set of out-particles, an in-particle may be seen to continue its ways in regions \(III\) and/or \(IV\). One might be worried that this would violate the no-quantum-cloning principle. Our best answer to that is that regions \(III\) and \(IV\), in all respects, appear to represent ‘time beyond \(\pm \,\)infinity’. Thus, these particles do not over count the quantum states, but merely extend the time line to beyond infinity, without causing any harm to any of the known physical principles.