Quantum Amplitudes in Black–Hole Evaporation with Local Supersymmetry

Abstract

In 1974, Hawking showed that a Schwarzschild black hole (mass $M$) radiates all dynamical fields out to infinity, at a calculable rate. If one changes the system by including a heat bath at a suitable temperature, ${\theta }={\hbar }c^{3}/8{\pi }GMk$, where $k$ is Boltzmann’s constant, then the entire system will be in thermal balance. One says that ${\theta }$ is the temperature of the black hole. The analogy between the black hole and a thermal body is more than formal: for example, one can assign an entropy to the black hole, $Ac^{3}/4{\hbar }G$, where $A=16{\pi }M^{2}$ is the area of the black hole, and formulate a generalised second law of thermodynamics. The rate of loss of mass by radiation implies that the black hole will have lost all its mass in a time which scales as $M^{3}$. The quasi–classical approximations involved in the above derivation are expected to have broken down well before this time. These difficulties should be reflected in a full quantum treatment. In 1975/1976, Hawking described black–hole evaporation in the language of density matrices: one only attempts to calculate probabilities for quantum processes, not quantum amplitudes with their extra phase information. It will be seen below that in the case that the action functional is invariant under local supersymmetry (that is, for supergravity + supermatter), quantum amplitudes involving black holes can be calculated. Indeed, in a certain sense (below), such an amplitude is semi-classical. The first three to four pages of this chapter will expand on this background description.

7.1 Introduction

Since 1974, it has been known that a Schwarzschild–like black hole (mass $M$), formed from gravitational collapse, radiates dynamical fields thermally to infinity at temperature, ${\theta }={\hbar }c^3/8{\pi }GMk$, where $k$ is Boltzmann’s constant and ${\hbar } =1$ [1, 2]. The analogy between a black hole and a thermal body is more than formal: one assigns an entropy, $Akc^3/4G$, to the black hole, where $A =16{\pi }GM^2/c^3$ is the area of the black hole, with a generalised second law of thermodynamics [3, 4]. The rate of loss of mass by radiation implies that the black hole will have lost all its mass in a time proportional to $G^{2}M^3/c^4$. The quasi–classical approximations involved above are expected to have broken down well before this time. These features should be reflected in a full quantum treatment. However, since 1975/1976 [5], black–hole evaporation has been mistakenly described in the language of density matrices: assuming that one can only calculate probabilities for quantum processes, instead of quantum amplitudes, with their extra phase information. Since 2001, it became clear that, provided the action is invariant under local supersymmetry (that is, for supergravity $+$ supermatter [6]), quantum amplitudes involving black holes can be calculated [7]. Indeed, in a certain sense (below), such amplitudes are semi–classical.

This Chapter is concerned with the quantum–mechanical decay of a four-dimensional Schwarzschild-like black hole, formed by gravitational collapse, into almost–flat space–time and weak radiation at late times, within the framework of quantum field theory. We consider quantum amplitudes (not just probabilities) for transitions from initial to final configurations, allowing for possible formation and evaporation of (one or more) black hole(s). Such quantum amplitudes are indeed found when one takes a locally–supersymmetric generalisation of Einstein gravity, such as $N = 1$ supergravity, plus possible supermatter [7–10]. We follow Dirac’s approach to the quantisation of constrained Hamiltonian systems (systems having local invariances, as in general relativity, supergravity or gauge theory) [11]. Boundary data for a quantum amplitude are set on asymptotically–flat initial and final space–like hypersurfaces, ${\Sigma }_{I, F}$, separated by a time–interval, $T$, at spatial infinity. Below, we take $T$ slightly complex, with $\mathrm{Im}(T)<0$. One can have (for example) in a Hamiltonian treatment, as bosonic ‘co–ordinate boundary data’ on ${\Sigma }_{I,F}$, the intrinsic spatial three–metric, $h_{ij}(x)$, and a scalar field, ${\phi }(x)$. Suitable components of the spin–${1\over 2}$ and spin–${3\over 2}$ potentials give fermionic data on ${\Sigma }_{I,F}$. One asks for the quantum amplitude to go from initial to final data in time, $T$. Such amplitudes, in the ‘field representation’, are not, in a simple way, related to the familiar scattering amplitudes of quantum field theory, defined in a particle representation, via harmonic–oscillator particle eigenstates for asymptotic in– or out–modes. Following Dirac’s approach, one finds, remarkably, that for a field theory with local supersymmetry in four dimensions and boundary data as above, quantum amplitudes have a ‘semi–classical’ form, ${\delta }{\delta } \exp \bigl (-I_\mathrm{class}\bigr )$ [8]. $I_\mathrm{class}$ denotes the Euclidean action of the classical solution joining initial to final data; such a solution is expected to exist, provided $\mathrm{Im}(T)<0$. The terms, ${\delta }{\delta }$, denote a fermionic delta–functional—a product of the fermionic supersymmetry constraints (the normal components of the gravitino field equation) at the boundaries. For other field–theory Lagrangians containing Einstein gravity plus possible matter (but which are not locally supersymmetric) in four dimensions, quantum amplitudes (and, hence, the model itself) are typically infinite and meaningless.

In the black–hole example, one can take the time–interval, $T$, sufficiently large that (for preference) ${\Sigma }_{F}$ catches all the quantum radiation. One deforms $T$ into the lower–half complex plane: $T\longrightarrow {\mid }T{\mid } \exp (-i{\theta }) (0<{\theta }\le {\pi }/2)$. This corresponds to Feynman’s $+i{\epsilon }$ prescription [12, 13]. First, for simplicity, consider a ‘background’ spherically–symmetric classical bosonic solution, with gravitational data, $h_{ijI,F}$, on ${\Sigma }_{I,F}$, and scalar data, ${\phi }(r)_{I,F}$. Then, take non–spherical perturbations of the four–metric, $g_{\mu \nu }$, and ${\phi }$, at linear order, obeying the classical field equations linearised about the spherically–symmetric background, subject to boundary data on ${\Sigma }_{I,F}$. Calculate the second–variation classical Euclidean action, $I^{(2)}_\mathrm{class}$, appearing in the amplitude, $\propto \exp (-I_\mathrm{class})$, above, as a functional of the boundary data. This leads (inter alia) to the quantum amplitude (real and imaginary parts) for weak final scalar fields [10]. The treatment involves adiabatic solutions of the scalar wave equation. In such locally–supersymmetric examples, no information is lost because of the black hole.

7.2 ‘Semi–Classical’ Amplitudes

7.2.1 Locally–Supersymmetric Quantum Mechanics

Work by Bern et al. [14] indicates that Feynman–diagram amplitudes in $N = 8$ supergravity might be finite. An alternative approach to quantum amplitudes in supergravity, based on Dirac’s canonical quantisation of constrained Hamiltonian systems, has been pursued since 1981 [15–17]. In this Section, Dirac’s approach is outlined (i) for locally–supersymmetric quantum mechanics (QM) [18, 19] and (ii) for $N = 1$ supergravity [8, 15–17]. The main result is that, in both cases, (i) and (ii), with suitable local boundary data on initial and final asymptotically–flat hypersurfaces, ${\Sigma }_{I}, {\Sigma }_{F}$, the quantum amplitude has a ‘semi–classical’ form, in a precise sense [Eq. (7.2.20), below]. The amplitude involves the semi–classical exponential, $\exp \bigl (-I_\mathrm{class}\bigr )$, where $I_\mathrm{class}$ is the Euclidean action of a classical solution joining initial and final data, times a product of delta–functionals of the classical fermionic supersymmetry constraints at ${\Sigma }_{I}, {\Sigma }_{F}$ (in $N = 1$ supergravity, of the normal components of the gravitino field equation.)

A simple example, showing how such ‘semi–classical’ amplitudes arise in locally–supersymmetric QM, follows from Witten [18] and Alvarez [19]. Standard non–relativistic QM, with bosonic spatial co–ordinate, $q$, is extended to a QM model invariant under local supersymmetry [19] by adjoining the odd (anti–commuting) co–ordinates, $({\xi }, {\eta })$, with odd gravitino variables, $({\psi }_{1}, {\psi }_{2})$, and a bosonic ‘vierbein’, $e$. The last three variables appear algebraically in the Lagrangian [19], with no time– or space–derivatives—such variables are not dynamical, nor part of the boundary data.

In Alvarez’ notation, setting his constant, $g$, to $1$ (without loss of generality), the Witten model with rigid N = 2 supersymmetry has Lagrangian,

$$\begin{aligned} L = {{1}\over {2}} {\dot{q}}^{2} + i \Bigl ({\xi }{\dot{\xi }}+{\eta }{\dot{\eta }}\Bigr ) - {{1}\over {2}} \Bigl (W^{\prime }\Bigr )^{2} - 2i W^{\prime \prime } {\xi } {\eta }, \end{aligned}$$

(7.2.1)

where $W = W(q)$ is a smooth function of $q$, otherwise unrestricted. A dot ${\dot{}}$ denotes a $t$–derivative, and a prime $^{\prime }$ a $q$–derivative. The action is invariant, modulo boundary terms, under rigid $N = 2$ supersymmetry transformations, with constant anti–commuting parameters, ${\epsilon }_{1}, {\epsilon }_{2}$:

$$\begin{aligned} {\delta }q = - i \sqrt{2} \Bigl ({\epsilon }_{1}{\xi }+{\epsilon }_{2}{\eta }\Bigr ),{\quad } {\delta }{\xi } = {{{\epsilon }_{1}}\over {\sqrt{2}}} {\dot{q}} + {{{\epsilon }_{2}}\over {\sqrt{2}}} \Bigl (W^{\prime }\Bigr ), \end{aligned}$$

$$\begin{aligned} {\delta }{\eta } = - {{{\epsilon }_{1}}\over {\sqrt{2}}} \Bigl (W^{\prime }\Bigr ) + {{{\epsilon }_{2}}\over {\sqrt{2}}} {\dot{q}}. \end{aligned}$$

(7.2.2)

The generators of rigid supersymmetry transformations are

$$\begin{aligned} Q_{1} = - i \sqrt{2} \Bigl ({\xi }{\dot{q}}-{\eta } W^{\prime }\Bigr ), \end{aligned}$$

(7.2.3a)

$$\begin{aligned} Q_{2} = - i \sqrt{2} \Bigl ({\eta }{\dot{q}}+{\xi } W^{\prime }\Bigr ). \end{aligned}$$

(7.2.3b)

Alvarez’ model, with local N = 2 supersymmetry, includes $\bigl ({\psi }_{1}(t), {\psi }_{2}(t)\bigr )$ and $e(t)$. This model has a (bosonic) local invariance under re–parametrisations:

$$\begin{aligned} t \longrightarrow {\tau }(t). \end{aligned}$$

(7.2.4)

The corresponding generator is the Hamiltonian density,

$$\begin{aligned} \mathcal{{H}} = e \Bigl [\bigl (Q_{1}\bigr )^{2} + \bigl (Q_{2}\bigr )^{2} + i{\psi }_{1}Q_{1} + i{\psi }_{2}Q_{2}\Bigr ]. \end{aligned}$$

(7.2.5)

Alvarez’ model also has (fermionic) local supersymmetry invariances. Local supersymmetry transformations are parametrised by independent (odd) fermionic functions, ${\epsilon }_{1}(t), {\epsilon }_{2}(t)$. The local (fermionic) classical supersymmetry generators, related to Eqs. (7.2.2, 7.2.3a, 7.2.3b), vanish at a classical solution:

$$\begin{aligned} Q_{1} = 0, {\quad }{\quad } Q_{2} = 0. \end{aligned}$$

(7.2.6)

These local supersymmetry constraints are preserved in classical evolution of the model. The bosonic constraint,

$$\begin{aligned} \mathcal{{H}} = 0, \end{aligned}$$

(7.2.7)

corresponds to invariance under local re–parametrisations, Eq. (7.2.4).

In Dirac quantisation, constraint generators are promoted to operators, ${\hat{Q}}_{1}, {\hat{Q}}_{2}$ and ${\mathcal{{\hat{H}}}}$, which annihilate any physical wave–functional, ${\Psi }$:

$$\begin{aligned} {\hat{Q}}_{1}{\Psi } = 0, \end{aligned}$$

(7.2.8a)

$$\begin{aligned} {\hat{Q}}_{2}{\Psi } = 0, \end{aligned}$$

(7.2.8b)

$$\begin{aligned} \mathcal {\hat{H}} {\Psi } = 0. \end{aligned}$$

(7.2.8c)

Since the anti–commutator of ${\hat{Q}}_{1}$ and ${\hat{Q}}_{2}$ involves $\mathcal {\hat{H}}$ [18, 19], it is sufficient to solve only the quantum supersymmetry constraints, ${\hat{Q}}_{1}{\Psi } = 0, {\hat{Q}}_{2}{\Psi } = 0$. These are simplified by the fermionic re–definition:

$$\begin{aligned} {\phi } = {\xi } + i{\eta }, {\quad }{\quad } {\overline{\phi }} = {\xi } - i{\eta }. \end{aligned}$$

(7.2.9)

In the Dirac procedure, one can take (say) $q$ and ${\phi }$ as bosonic and fermionic co–ordinates for (or arguments of) ${\Psi }$. In the quantum theory, $p$, the momentum conjugate to $q$, corresponds to $-i \partial /\partial {q}$. By anti–commutation, ${\overline{\phi }}$ acts as a momentum conjugate to ${\phi }$; more precisely, ${\overline{\phi }}$ corresponds to the left fermionic derivative, $\partial /\partial {\phi }$ [6]. To calculate a left derivative, ${\partial }/{\partial {\phi }}$, acting on a quantity, ${\alpha }$, one moves each appearance of ${\phi }$ in ${\alpha }$ to the left, using anti–commutation, before differentiating [6].

In the simplest gauge [19], ${\psi }_{1} = {\psi }_{2} = 0 , e = 1$, the quantum supersymmetry constraints, Eqs. (7.2.8a, 7.2.8b, 7.2.8c), read:

$$\begin{aligned} \biggl ({\phi }+{{\partial }\over {\partial {\phi }}}\biggr ) {{\partial {\Psi }}\over {\partial {q}}} - W^{\prime }(q) \biggl ({\phi }-{{\partial }\over {\partial {\phi }}}\biggr ){\Psi } = 0, \end{aligned}$$

(7.2.10)

$$\begin{aligned} \biggl ({\phi }-{{\partial }\over {\partial {\phi }}}\biggr ) {{\partial {\Psi }}\over {\partial {q}}} - W^{\prime }(q) \biggl ({\phi }+{{\partial }\over {\partial {\phi }}}\biggr ){\Psi } = 0. \end{aligned}$$

(7.2.11)

Since ${\phi }^{2} = 0$ (anti–commutation), the quantum constraints, Eqs. (7.2.10, 7.2.11), imply:

$$\begin{aligned} {\Psi }(q, {\phi }) = A(q) + {\phi } B(q), \end{aligned}$$

(7.2.12)

for functions, $A(q), B(q)$. The general solution of Eqs. (7.2.10, 7.2.11) is:

$$\begin{aligned} A(q) = c_{1} \exp \Bigl (W(q)\Bigr ), \end{aligned}$$

(7.2.13a)

$$\begin{aligned} B(q) = c_{2} \exp \Bigl (-W(q)\Bigr ), \end{aligned}$$

(7.2.13b)

where $c_{1}, c_{2}$ are (independent) constants and $W(q)$ appears in Eq. (7.2.1). In Dirac’s approach, the arguments of ${\Psi }$ are ‘co–ordinate boundary data’ (below).

For comparison with the four–dimensional case [Sects. (7.2.2–7.2.4)], consider classical bosonic/fermionic solutions in the local QM theory. The classical constraints, $Q_{1} = 0$, $Q_{2} = 0$, imply that ${\xi }$ and ${\eta }$ are proportional. For non–trivial $({\xi }, {\eta })$, the determinant,

$$\begin{aligned} {\dot{q}}^{2} + {\Bigl (W^{\prime }(q)\Bigr )}^{2} = 0, \end{aligned}$$

(7.2.14)

vanishes. Real classical motion occurs only in the ‘Euclidean’ régime, ${\tau } = \pm it$, with ${\tau }$ real, whence,

$$\begin{aligned} \biggl ({{dq}\over {d{\tau }}}\biggr )^{2} = \biggl ({{dW}\over {dq}}\biggr )^{2}, \end{aligned}$$

(7.2.15)

with two roots:

$$\begin{aligned} \biggl ({{dq}\over {d{\tau }}}\biggr ) = {\pm } W^{\prime }(q). \end{aligned}$$

(7.2.16)

The motion is integrable, with two classical solutions. The exponential terms in the amplitude, Eqs. (7.2.13a, 7.2.13b), correspond to $\exp (-I_\mathrm{class})$ for one or other classical solution, where $I_\mathrm{class}$ is the Euclidean action of the classical solution joining initial to final data.

Subject to the boundary data, this leads to Eqs. (7.2.17a, 7.2.17b), below, for the quantum amplitude to go from initial position, $q_{a}$, and initial fermionic data, to analogous final data, in time–interval, $T$. [In standard QM, following Feynman and Hibbs [12], the amplitude would be denoted by $K(q_{b},t_{b}; q_{a},t_{a})$, where $T = (t_{b}-t_{a})$.]

A classical solution must have either:

• (i) ${\widetilde{\phi }} = 0$ in the interior, with $(dq/d{\tau }) = e({\tau }) W^{\prime }(q)$, or

• (ii) ${\phi } = 0$ in the interior, with $(dq/d{\tau }) = - e({\tau }) W^{\prime }(q)$.

In cases (i) and (ii), respectively, from the classical and quantum constraints, one finds the boundary data and amplitude:

$$\begin{aligned} \begin{array}{ll} K\bigl (q_{b},{\phi }_{b}; &{}q_{a},{\phi }_{a}; T = -i {\tau }_{0}\bigr )\\ &{} = \mathrm{const.} \delta \bigl (Q_{1}+iQ_{2}\bigr )_{a} \delta \bigl (Q_{1}+iQ_{2}\bigr )_{b} \exp \bigl (-I_\mathrm{class}\bigr ), \end{array} \end{aligned}$$

(7.2.17a)

$$\begin{aligned} \begin{array}{ll} K\bigl (q_{b},{\widetilde{\phi }}_{b}; &{}q_{a},{\widetilde{\phi }}_{a}; T = -i {\tau }_{0}\bigr )\\ &{} = \mathrm{const.} \delta \bigl (Q_{1}-iQ_{2}\bigr )_{a} \delta \bigl (Q_{1}-iQ_{2}\bigr )_{b} \exp \bigl (-I_\mathrm{class}\bigr ). \end{array} \end{aligned}$$

(7.2.17b)

Equations (7.2.17a, 7.2.17b) contain semi–classical exponentials, multiplied by fermionic delta–functions. The delta–functions (below) impose the classical supersymmetry constraints, Eq. (7.2.6), at the boundaries. These constraints are part of the classical field equations. The ‘vierbein’, $e(t)$, must be chosen such that $q$ changes from $q_{a}$ to $q_{b}$ along the classical path, during the time–interval, $T = - i {\tau }_{0}$, with ${\tau }_{0}$ real.

7.2.2 N $=$ 1 Supergravity: Dirac Approach

Turn to N $\varvec{=}$1 supergravity in four space–time dimensions, with ‘co–ordinate’ data on ${\Sigma }_{I}$ and ${\Sigma }_{F}$, via the Dirac approach. (Hence, ultimately, to questions concerning black–hole evaporation.) The quantum amplitude in $N = 1$ supergravity is found (below) to have the ‘semi–classical’ form, Eq. (7.2.20), analogous to Eqs. (7.2.17a, 7.2.17b) for locally–supersymmetric QM. As in Eqs. (7.2.17a, 7.2.17b), the quantities, ${\delta }$, in Eq. (7.2.20), are delta–functionals of the supersymmetry constraints at the bounding surfaces. In calculating such quantum amplitudes, one is led to study classical N $\varvec{=}$1 supergravity (generalising classical general relativity), as a boundary–value problem. This is only well posed if the time–interval, $T$, at spatial infinity, is complex, with $\mathrm{Im}(T)<0$ [16].

As a simple example of the need for the condition, $\mathrm{Im}(T)<0$, in a boundary–value problem, consider the scalar wave equation, $\bigl ({\partial }^2{\phi }/{\partial }t^2\bigr ) -\bigl ({\partial }^2{\phi }/{\partial }x^2\bigr ) = 0$, in Minkowski space–time, with co–ordinates, $(t,x)$. The rôles of ${\Sigma }_{I}$ and ${\Sigma }_{F}$ are taken by the lines, $t=0$ and $t=T$. For simplicity, choose boundary data with ${\phi }(0,x) =0$, ${\phi }(T,x) ={\phi }_{1}(x)$, where ${\phi }_{1}$ is a smooth function of rapid decrease as ${\mid }x{\mid }\longrightarrow {\infty }$. Taking the Fourier transform, ${\widetilde{\phi }}(t,k)$, with respect to $x$, one has the (formal) solution:

$$\begin{aligned} {\phi }(t,x) = \mathrm{const.} \int d^{3}k {{{\widetilde{\phi }}_{1}(T,k)}\over {\sin (kT)}} e^{ikx} \sin (kt). \end{aligned}$$

(7.2.18)

For real $T$ and generic data, ${\phi }_{1}(x)$, there is no classical solution obeying the boundary conditions. With real $T, {\widetilde{\phi }}_{1}(T,k)$ would need to obey the very restrictive conditions, ${\widetilde{\phi }}_{1}(T,n{\pi })=0, \forall {n}\in {\mathbb {Z}}$, to provide a solution. When $\mathrm{Im}(T)<0$, however, Eq. (7.2.18) defines a (complex) solution, for any smooth data, ${\phi }_{1}(x)$, of rapid decrease.

As in locally–supersymmetric QM, a classical solution to the boundary–value problem in $N=1$ supergravity exists only when the classical supersymmetry constraints hold at the boundaries [15, 16]. It was recognised in [21] that the term, $\exp \bigl (-I_\mathrm{class}\bigr )$, in the supergravity amplitude, should be multiplied by an infinite product of fermionic terms. These fermionic factors are, in fact, those in Eq. (7.2.20). Similarly, in locally–supersymmetric QM, the arguments of the delta–functions in Eqs. (7.2.17a, 7.2.17b) are the classical supersymmetry constraints [Eq. (7.2.6)] at the boundaries.

As stated in Sect. 7.1, the boundary conditions taken in Eqs. (7.2.20, 7.2.30), below, for $N = 1$ supergravity, are not the same as those used in scattering theory (via Feynman diagrams). The relation between the two types of boundary condition or amplitude is quite complicated (below). As with Eq. (7.2.17a, 7.2.17b), the boundary conditions in the Dirac approach involve suitable ‘co–ordinate’ fields at ${\Sigma }_{I}, {\Sigma }_{F}$, analogous to those in the amplitude, $K(q_{b},t_{b}; q_{a},t_{a})$, for non–relativistic QM [12].

In N $\varvec{=}$1 supergravity or its generalisations, there are necessarily spin–${{3}\over {2}}$ gravitino fields (possibly spin–${{1}\over {2}}$ fields). Spinor fields can only be defined on a curved space–time admitting a pseudo–orthonormal tetrad field [22], denoted by $e^{a}_{{\mu }}(x) (a=0,1,2,3)$. Here, $a$ is a tetrad index, and ${\mu }$ is a space–time or one–form index.

In the classical or quantum boundary–value problem for supergravity, one natural choice of bosonic boundary data (‘field co–ordinates’) is to specify the spatial tetrad (asymptotically–flat), $e^{a}_{i}(x) (a=0,1,2,3)$, on ${\Sigma }_{I}$ and ${\Sigma }_{F}$ [15, 16]. Here, $i = 1,2,3$ is a spatial one–form index. The intrinsic spatial three–metric is:

$$\begin{aligned} h_{ij}(x) = e^{a}_{i}(x) e_{aj}(x). \end{aligned}$$

(7.2.19)

For fermionic co–ordinates, one can take the primed spatial gravitino field, ${\widetilde{\psi }}^{A^{\prime }}_{i}(x)$, on ${\Sigma }_{I}$, and unprimed field, ${\psi }^{A}_{i}(x)$, on ${\Sigma }_{F}$ [15, 16]. [Or, the rôles of ${\Sigma }_{I}$ and ${\Sigma }_{F}$ can be reversed.] $A,A^{\prime }$ are two–component spinor indices, streamlined for four space–time dimensions [23]. For harmony of language, with fermionic data, ${\widetilde{\psi }}^{A^{\prime }}_{i}(x)$ or ${\psi }^{A}_{i}(x)$, describe the spatial geometry via the spinor–valued one–form, $e^{AA^{\prime }}_{i}(x)$ [16], rather than the equivalent spatial tetrad, $e^{a}_{i}(x)$. Specify the time interval, $T$, at spatial infinity, between ${\Sigma }_{I}$ and ${\Sigma }_{F}$ (with $\mathrm{Im}(T)<0$, for a well–posed classical boundary–value problem).

In the Dirac approach, one has a field representation with ‘co–ordinate’ boundary data. For scattering theory, with Feynman diagrams, one instead has a particle representation with ‘particle’ in– and out–data.

(In $N = 1$ supergravity, Feynman diagrams give finite scattering amplitudes at 1–, 2–loop orders [24, 25]. Scattering amplitudes at $3$–, $4$–, ${\ldots }$–loop order (if meaningful) are not known. Feynman diagrams in $N=8$ supergravity give indications of finiteness [14].)

With this choice of data for $N = 1$ supergravity, and $\mathrm{Im}(T)<0$, the quantum amplitude has the ‘semi–classical’ form:

$$\begin{aligned} K = \mathrm{const.} {\delta }\bigl (S_{A}(x_{I})\bigr ) {\delta }\bigl ({\widetilde{S}}_{A^{\prime }}(x_{F})\bigr ) \exp \bigl (-I_\mathrm{class}\bigr ), \end{aligned}$$

(7.2.20)

analogous to Eqs. (7.2.17a, 7.2.17b) for locally–supersymmetric QM [Sect. (7.2.3)].

In Eq. (7.2.20), the ‘Euclidean’ action, $I_\mathrm{class}$, includes fermionic contributions [16]. Here, ${\widetilde{S}}_{A^{\prime }}(x_{F})$ denotes the primed classical supersymmetry constraint [Eq. (7.2.24), below] at ${\Sigma }_{F}$, and $S_{A}(x_{I})$ the unprimed constraint at ${\Sigma }_{I}$ [16].

The delta–functional of the odd (anti–commuting) spinor field, $S_{A}(x)$, in Eq. (7.2.20), is an (infinite) product of $S_{A}(x_{I})$ over all points, $x_{I}$, and spinor indices, $A = 0, 1$, on ${\Sigma }_{I}$. For given $x_{I}$, each factor is a fermionic delta–function, ${\delta }(S_{A}(x_{I}))$, defined in the holomorphic representation for fermions [13, 26, 27]. In this representation, fermionic and bosonic fields are treated on an equal footing. Fermionic quantities anti–commute among themselves. In the finite–dimensional case, one has variables $a_{n} (n=1,2,{\ldots },Q)$ (say) and their conjugates, $a_{n}^{*}$. Functions of the form, $f(a_{n})$, are holomorphic functions, describing state vectors of the system. Berezin integration [26, 27] defines the integral of a function involving both fermions and bosons. As in [26, 27], the (fermionic) delta–function of $(a_{n})$ can be identified with the product, ${\prod }_{n=1}^{Q} a_{n}$.

The explicit form, Eq. (7.2.20), for a quantum amplitude in four–dimensional $N=1$ supergravity, via the Dirac approach, highlights the question of different boundary conditions and amplitudes, relevant to the Dirac and Feynman approaches, respectively. Different, but (a priori) equally valid types of boundary condition are natural to one or the other approach. As in a first course on quantum field theory, scattering (Feynman–diagram) boundary conditions are included via the LSZ (Lehmann, Symanzik, Zimmermann) description [13]. One studies quantum states which, at early or late times, resemble a product of 1–particle states (one for each particle) and of no–particle states. Each 1–particle state involves a first excited state of a harmonic oscillator, for a given momentum, k. A finite product of such 1–particle states multiplies an infinite product of the remaining ground states. Is there a unitary transformation, linking the amplitudes of Eq. (7.2.20) to scattering amplitudes, which refer to an asymptotic particle basis?

Consider the description of in– and out–states, but in language natural to the Dirac approach. The transcription, implicit above, between harmonic–oscillator and ‘co–ordinate’ descriptions, is made at fixed, finite time–separation, $T$, between ${\Sigma }_{I}$ and ${\Sigma }_{F}$. Only after the transcription does one take the limit, ${\mid }T{\mid }\longrightarrow {\infty }$. One must allow for possible large deviations from flat or trivial ‘co–ordinate’ data on each boundary. These deviations arise from large excursions allowed (with low probability) in the co–ordinates (or arguments), $x_{kI}, x_{{\ell }F}$, of harmonic–oscillator wave–functions, corresponding, respectively, to spatial momenta, k, l. Roughly speaking, the exponential factor of each harmonic oscillator is multiplied by $\exp (-I_\mathrm{class})$, where $I_\mathrm{class}$ is the classical action for the corresponding non–linear boundary–value problem in classical supergravity, with boundary data possibly far from flatness, and $T$ fixed. The result must then be integrated over all excursions. Finally, take $T\longrightarrow {\infty }$, to obtain a (putative) Feynman diagram, via this elaborate construction. Studying the behaviour of $I_\mathrm{class}$ for the classical boundary–value problem in general relativity or supergravity, as a functional of the boundary data, involves very strong non–linear gravitational (and other) fields. This problem is not understood even qualitatively, although one might attempt some form of ‘large–excursion’ classical perturbative approach. At present, it is very complicated to compare Feynman–diagram results with results from the Dirac approach, via the algorithm just suggested. The main difficulty resides in the classical boundary–value problem.

The Dirac approach is intrinsically Hamiltonian, with its one–parameter family of space–like hypersurfaces [11, 28]. Dirac’s approach is well adapted to $N = 1$ supergravity and its gauge–invariant version [6, 16]—both theories have a large amount of local symmetry. The continuous invariances comprise:

• (i) the co–ordinate transformations of general relativity;

• (ii) the gauge symmetries of particle physics (if appropriate) [6];

• (iii) local Lorentz transformations, acting on the indices, $a, b, {\ldots }$, of a pseudo–orthonormal tetrad in curved space–time, $e^{a}_{{\mu }}(x)$.

• (iv) local supersymmetry transformations [6, 15, 16, 24].

These invariances provide a large number of local–symmetry generators in the Hamiltonian approach. The most useful involve one power of momentum, namely,

• (i) three generators of spatial co–ordinate transformations,

• (ii) generators of gauge symmetries (if appropriate),

• (iii) generators of local Lorentz transformations,

• (iv) half of the local supersymmetry generators—the relevant splitting (into primed and unprimed spinors) is explicit with two–component spinors [16, 23]. For the above ‘co–ordinate’ boundary data on ${\Sigma }_{I}, {\Sigma }_{F}$, the primed quantum supersymmetry operator, ${\hat{\overline{S}}}_{A^{\prime }}$, involves a first–order functional derivative at ${\Sigma }_{F}$ [Eq. (7.2.29), below]. The unprimed supersymmetry operator, ${\hat{S}}_{A}$, is of second order at ${\Sigma }_{F}$ [16], though of first order at ${\Sigma }_{I}$.

To summarise: The perpective gained from Dirac’s approach to quantum theories with local symmetries, such as $N = 1$ supergravity, may be quite different from a view based only on scattering theory and Feynman diagrams.

7.2.3 The Quantum Constraints

As in [15, 16], a simple choice of canonical variables for $N = 1$ supergravity is:

• (i) Bosonic variables: the spinor–valued spatial one–forms, $e^{AA^{\prime }}_{i}(x)$, and conjugate momenta, $p_{AA^{\prime }}^{i}(x)$. For later use, the unit (Lorentzian) normal vector, $n^{AA^{\prime }}$, at each spatial point, $x$, is defined (up to a choice between future and past directions) by [15, 16]:

  $$\begin{aligned} n_{AA^{\prime }} n^{AA^{\prime }} = 1, {\quad }{\quad } n_{AA^{\prime }} e^{AA^{\prime }}_{i} = 0 {\quad }{\quad } (i = 1,2,3). \end{aligned}$$

  (7.2.21)

• (ii) Fermionic variables: the odd (anti–commuting) spatial spinor–valued one–forms, ${\psi }^{A}_{i}(x), {\widetilde{\psi }}^{A^{\prime }}_{i}(x)$ (spatial projections of the spinor–valued one–forms, ${\psi }^{A}_{{\mu }}(x), {\widetilde{\psi }}^{A^{\prime }}_{{\mu }}(x)$). Here, ${\psi }^{A}_{i}(x), {\widetilde{\psi }}^{A^{\prime }}_{i}(x)$, are liberated from being hermitian conjugates of each other (as holds in Lorentzian $N = 1$ supergravity [6, 16, 24]). This liberation is inevitable in Riemannian (or complexified) supergravity [29]. The (odd) classical momentum, conjugate to ${\psi }^{A}_{i}(x)$,

  $$\begin{aligned} {\pi }_{A}^{i}(x) = - {{1}\over {2}} {\epsilon }^{ijk} {\widetilde{\psi }}^{A^{\prime }}_{j}(x) e_{AA^{\prime }k}(x), \end{aligned}$$

  (7.2.22)

  involves the gravitino variable of opposite chirality, ${\widetilde{\psi }}^{A^{\prime }}_{i}(x)$ [16]. Equation (7.2.22) can be inverted to give ${\widetilde{\psi }}^{A^{\prime }}_{j}(x)$ linearly in terms of ${\pi }_{A}^{i}(x)$ [15, 16]. A complete treatment of this matter involves the use of Dirac brackets [11, 15, 16, 28].

The classical Hamiltonian takes a standard form for a theory with gauge–like invariances—a finite sum over generators, $G_{s}(x)$, one for each local symmetry, with Lagrange multiplier, $L_{s}(x)$ [11, 15, 16, 28]:

$$\begin{aligned} H = {\int }d^{3}x L_{s}(x) G_{s}(x), \end{aligned}$$

(7.2.23)

summed over $s$. In our case, the generators, $G_{s}$, are:

• (i) the generator, $\mathcal{{H}}_{AA^{\prime }}(x)$, of infinitesimal space–time co–ordinate transformations,

• (ii) the independent odd generators, $S_{A}(x)$ and ${\widetilde{S}}_{A^{\prime }}(x)$, of unprimed and primed local supersymmetry transformations,

• (iii) the generators, $J^{AB} = J^{(AB)}, {\widetilde{J}}^{A^{\prime }B^{\prime }} = {\widetilde{J}}^{(A^{\prime }B^{\prime })}$, of local Lorentz transformations.

Each $G_{s}$ is a function of the canonical variables, $(e^{AA^{\prime }}_{i}(x), p_{AA^{\prime }}^{i}(x); {\psi }^{A}_{i}(x), {\widetilde{\psi }}^{A^{\prime }}_{i}(x))$, and of their first or second spatial derivatives.

Remarkably, the classical generators, $S_{A}$ and ${\widetilde{S}}_{A^{\prime }}$, become simple when expressed in terms of the standard (torsion–free) covariant derivative, ${}^{3s}D_{j}$, on spinors [15, 16, 28]:

$$\begin{aligned} {\widetilde{S}}_{A^{\prime }} = {\epsilon }^{ijk} e_{AA^{\prime }i} \Bigl ({}^{3s}D_{j}{\psi }^{A}_{k}\Bigr ) + {{1}\over {2}} i {\kappa }^{2} p_{AA^{\prime }}^{i} {\psi }^{A}_{i}, \end{aligned}$$

(7.2.24)

with $S_{A}$ given by the ‘conjugate’ expression, with ${\psi }^{A}_{k}$ replaced by ${\widetilde{\psi }}^{A^{\prime }}_{k}$ and $i$ by $-i$. Define ${\kappa }^{2} = 8{\pi }$ in geometrical units, where $c = G = 1$. Analogous expressions hold for the other generators [16].

Following the Dirac approach [11], take, say, $(e^{AA^{\prime }}_{i}(x), {\psi }^{A}_{i}(x))$ as bosonic and fermionic co–ordinates [16]. In the (natural) holomorphic description, the momentum variables, $(p_{AA^{\prime }}^{i}(x), {\pi }_{A}^{i}(x))$, are represented by first–order derivative operators on the wave–functional, ${\Psi }[e^{AA^{\prime }}_{i}(x), {\psi }^{A}_{i}(x)]$, which lives in a Grassmann algebra [16, 26, 27]. From the classical (Dirac) brackets between the momentum, ${\pi }_{A}^{i}(x)$ [Eq. (7.2.22)] and co–ordinate, ${\psi }^{A}_{i}(x)$, the quantum version of ${\widetilde{\psi }}_{A^{\prime }i}(x)$ is [16]:

$$\begin{aligned} \hat{\overline{\psi }}_{A^{\prime }i}(x) = - i D^{A}_{A^{\prime }ji}(x) {\delta }/{\delta }{\psi }^{A}_{j}(x), \end{aligned}$$

(7.2.25)

$$\begin{aligned} D^{A}_{A^{\prime }ji} = - 2i h^{-{{1}\over {2}}} e^{AB^{\prime }}_{i} e_{BB^{\prime }j} n^{B}_{A^{\prime }}, \end{aligned}$$

(7.2.26a)

$$\begin{aligned} h = \det (h_{ij}) . \end{aligned}$$

(7.2.26b)

In Dirac’s approach [11, 15, 16, 28], the classical constraint generators become quantum operators: ${\hat{\mathcal{H}}}_{AA^{\prime }}, {\hat{S}}_{A}, {\hat{{\overline{S}}}}_{A^{\prime }}, {\hat{J}}_{AB}, {\hat{\overline{J}}}_{A^{\prime }B^{\prime }}$. The wave–functional of a physically-allowed quantum state, ${\Psi }[e^{AA^{\prime }}_{i}(x), {\psi }^{A}_{i}(x)]$, is (by definition) annihilated by the quantum constraint operators (giving the quantum constraint equations):

$$\begin{aligned} {\hat{\overline{S}}}_{A^{\prime }}{\Psi } = 0, \end{aligned}$$

(7.2.27a)

$$\begin{aligned} {\hat{S}}_{A}{\Psi } = 0, \end{aligned}$$

(7.2.27b)

$$\begin{aligned} {\hat{J}}_{AB}{\Psi } = 0, {\quad } \end{aligned}$$

(7.2.27c)

$$\begin{aligned} {\hat{\overline{J}}}_{A^{\prime }B^{\prime }}{\Psi } = 0. \end{aligned}$$

(7.2.27d)

The (anti–)commutation relations among these operators imply the remaining quantum constraint:

$$\begin{aligned} \mathcal{{\hat{H}}}_{AA^{\prime }} {\Psi } = 0, \end{aligned}$$

(7.2.28)

with a suitable definition of $\mathcal{{\hat{H}}}_{AA^{\prime }}$ [16].

Equation (7.2.27c, 7.2.27d) describe the invariance of ${\Psi }$ under local Lorentz transformations, applied to its arguments.

The constraint, Eq. (7.2.27a), involves only first–order functional derivatives, and leads to a simple expression for the transformation of ${\Psi }[e^{AA^{\prime }}_{i}(x), {\psi }^{A}_{i}(x)]$ under an infinitesimal primed local supersymmetry transformation, parametrised by an (odd) spinor field, ${\widetilde{\epsilon }}^{A^{\prime }}(x)$ [6, 15, 16]. In this representation, the classical generator, ${\widetilde{S}}_{A^{\prime }}$, of Eq. (7.2.24), becomes the quantum operator [15, 16]:

$$\begin{aligned} {\hat{\overline{S}}}_{A^{\prime }} = {\epsilon }^{ijk} e_{AA^{\prime }i} \Bigl ({}^{3s}D_{j}{\psi }^{A}_{k}\Bigr ) + {{1}\over {2}} {\kappa }^{2} {\psi }^{A}_{i} {{\delta }\over {{\delta }e^{AA^{\prime }}_{i}}}. \end{aligned}$$

(7.2.29)

A corresponding property of the unprimed quantum constraint, Eq. (7.2.27b), holds with respect to infinitesimal unprimed local supersymmetry transformations. To express this so as to treat primed and unprimed quantities symmetrically, define a ‘dual’ wave–functional, ${\widetilde{\Psi }}[e^{AA^{\prime }}_{i}(x), {\widetilde{\psi }}^{A^{\prime }}_{i}(x)]$, as in Eqs. (7.2.36–7.2.38), below—the fermionic Fourier transform of the original wave–functional, ${\Psi }[e^{AA^{\prime }}_{i}(x), {\psi }^{A}_{i}(x)]$. The first sentence of the paragraph above can be repeated, on interchanging:

(i) ‘primed’ $\leftrightarrow$ ‘unprimed’, (ii) ${\psi }^A_{i}(x) \leftrightarrow {\widetilde{\psi }}^{A^{\prime }}_{i}(x)$, (iii) ${\Psi } \leftrightarrow {\widetilde{\Psi }},$ (iv) ${\epsilon }^A(x) \leftrightarrow {\widetilde{\epsilon }}^{A^{\prime }}(x)$.

In applying the Dirac approach to $N = 1$ supergravity, assume provisionally that there is a classical solution joining asymptotically–flat initial and final data on ${\Sigma }_{I}$ and ${\Sigma }_{F}$, with $\mathrm{Im}(T)<0$ (up to gauge). One builds up a bosonic/fermionic classical solution of the supergravity field equations, starting from the (complex–valued) bosonic solution of the classical Einstein boundary–value problem—the solution which remains (up to gauge) when fermionic fields are set to zero. Given a bosonic solution, one solves the classical supergravity field equations perturbatively in powers of fermionic data, ${\psi }$ or ${\widetilde{\psi }}$, about the bosonic solution [16]. The classical fields and action live in appropriate Grassmann algebras. The question of building up non–linear bosonic/fermionic solutions of the $N = 1$ supergravity field equations was addressed rigorously in [30], for the Cauchy problem in Lorentzian signature. We follow [30], but for a boundary–value problem (Riemannian or complexified). Since $\mathrm{Im}(T)<0$, one expects, in general, a complex strongly elliptic boundary–value problem [31]. Such problems have many of the good properties of real elliptic boundary–value problems, with existence and uniqueness. Riemannian Einstein boundary–value problems are, in general, expected to be elliptic.

This approach, based on rotation of $T$ into the complex (or Feynman’s $+i{\epsilon }$ prescription [12, 13]), has led to the calculation of quantum amplitudes, not just probabilities, concerning gravitational collapse to a black hole, for energetically–possible many–particle outcomes at late times, in the quantum state which continues to exist after collapse and during subsequent quantum evaporation [4, 7, 9, 10, 20, 32–38].

For such boundary data, the quantum amplitude is written, after Feynman and Hibbs [12], as:

$$\begin{aligned} K = K\bigl (e^{AA^{\prime }}_{i}(x_{I}), {\widetilde{\psi }}^{A^{\prime }}_{i}(x_{I}); e^{AA^{\prime }}_{i}(x_{F}), {\psi }^{A}_{i}(x_{F}); T\bigr ). \end{aligned}$$

(7.2.30)

Here, $K$ lives in an infinite–dimensional Grassmann algebra over the complex numbers, ${\mathbb C}$, in the holomorphic representation above.

The classical supersymmetry constraint equations imply that $K$ is proportional to a (fermionic) delta–functional of ${\widetilde{S}}_{A^{\prime }}(x)$ at ${\Sigma }_{F}$, and to a delta–functional of $S_{A}(x)$ at ${\Sigma }_{I}$. Further investigation (below) leads to the semi–classical form, Eq. (7.2.20), for $K$.

7.2.4 ‘Semi–Classical’ Amplitude in N $=$ 1 Supergravity

In the locally–supersymmetric Witten/Alvarez QM model [18, 19], one can verify that the wave function, $K$, of Eqs. (7.2.17a, 7.2.17b), is the quantum amplitude for the given boundary data, by checking that:

• (i) for each local invariance of the classical model, the corresponding Dirac quantum constraint holds;

• (ii) the Schrödinger equation holds, for all $T>0$ (or for all ${\tau }>{0}$, with ${\tau }= it$) [this is automatic, given (i)]; and

• (iii) $K$ tends to a delta–function(al) of the boundary data, as (say) ${\tau }\longrightarrow {0}_{+}$, just as the amplitude in non–relativistic QM [12] tends to ${\delta }(q_{a}, q_{b})$, as ${\tau }\longrightarrow {0}_{+}$.

We test in a similar way the proposed expression, Eq. (7.2.20), for the quantum amplitude, $K$, in $N = 1$ supergravity, given the boundary data of Eq. (7.2.30). Write Eq. (7.2.20) as:

$$\begin{aligned} K = \Bigl ({\prod }_{x_{I},B}S_{B}(x_{I})\Bigr ) \Bigl ({\prod }_{x_{F},B^{\prime }}{\widetilde{S}}_{B^{\prime }}(x_{F})\Bigr ) \exp \bigl (-I_\mathrm{class}/{\hbar }\bigr ). \end{aligned}$$

(7.2.31)

For (i), $K$ must obey the quantum constraints, Eqs. (7.2.27a, 7.2.27b, 7.2.27c, 7.2.27d), at the boundaries, ${\Sigma }_{I}, {\Sigma }_{F}$. The local–Lorentz constraints, Eqs. (7.2.27c, 7.2.27d), hold automatically, since the right–hand side of Eq. (7.2.20) is invariant under local Lorentz transformations.

One can verify that the primed constraint operator, ${\hat{\overline{S}}}_{A^{\prime }}(x)$, at ${\Sigma }_{F}$, annihilates the amplitude, $K$:

$$\begin{aligned} {\hat{\overline{S}}}_{A^{\prime }}(x_{F}) K = 0, \end{aligned}$$

(7.2.32)

as follows: From the first–order expression, Eq. (7.2.29), for ${\hat{\overline{S}}}_{A^{\prime }}(x_F)$;

$$\begin{aligned} {\hat{\overline{S}}}_{A^{\prime }}(x_{F}) \exp \bigl (-I_\mathrm{class}/{\hbar }\bigr ) = {\widetilde{S}}_{A^{\prime }}(x_{F}) \exp \bigl (-I_\mathrm{class}/{\hbar }\bigr ). \end{aligned}$$

(7.2.33)

Since operators such as ${\hat{\overline{S}}}_{A^{\prime }}(x_{1F}), {\hat{\overline{S}}}_{B^{\prime }}(x_{2F})$ anti–commute [15, 16], $K$ can be written as:

$$\begin{aligned} K = \Bigl ({\prod }_{x_{I},B}S_{B}(x_{I})\Bigr ) \Bigl ({\prod }_{x_{F},B^{\prime }}{\hat{\overline{S}}}_{B^{\prime }}(x_{F})\Bigr ) \exp \bigl (-I_\mathrm{class}/{\hbar }\bigr ). \end{aligned}$$

(7.2.34)

Apply ${\hat{\overline{S}}}_{A^{\prime }}(x)$ to $K$. Note the commutation relation, $\bigl [{\hat{\overline{S}}}_{A^{\prime }}(x_{F}), K\bigr ] = 0$. Hence, for $x\in {\Sigma }_{F}$:

$$\begin{aligned} \begin{array}{ll} &{}{\hat{\overline{S}}}_{A^{\prime }}(x) \Bigl ({\prod }_{x_{I},B}S_{B}(x_{I})\Bigr ) \Bigl ({\prod }_{x_{F},B^{\prime }}{\hat{\overline{S}}}_{B^{\prime }}(x_{F})\Bigr ) \exp \bigl (-I_\mathrm{class}/{\hbar }\bigr )\\ &{}\qquad = \Bigl ({\prod }_{x_{I},B}S_{B}(x_{I})\Bigr ) {\hat{\overline{S}}}_{A^{\prime }}(x) \Bigl ({\prod }_{x_{F},B^{\prime }}{\hat{\overline{S}}}_{B^{\prime }}(x_{F})\Bigr ) \exp \bigl (-I_\mathrm{class}/{\hbar }\bigr ). \end{array} \end{aligned}$$

(7.2.35)

The right–hand side of Eq. (7.2.35) includes one repeated pair of factors, among the terms, ${\hat{\overline{S}}}_{A^{\prime }}(x) \bigl ({\prod }_{x_{F},B^{\prime }} {\hat{\overline{S}}}_{B^{\prime }}(x_{F})\bigr )$, for which $x = x_{F}$ and $A^{\prime } = B^{\prime }$. By anti–commutation among operators ${\hat{\overline{S}}}_{A^{\prime }}(x), {\hat{\overline{S}}}_{B^{\prime }}(y)$, this product of two identical fermionic expressions gives zero. Hence, the primed constraint equation, (7.2.32), holds at ${\Sigma }_{F}$.

The unprimed supersymmetry constraint operator, ${\hat{S}}_{A}$, is of second order at ${\Sigma }_{F}$, involving mixed functional derivatives, ${\delta }^{2}/{\delta }e{\delta }{\psi }$, with respect to $(e^{AA^{\prime }}_{i}(x_F), {\psi }^{A}_{i}(x_F))$ (Sect. 3.4 of [16]). One might, therefore, expect to need a more difficult argument to establish that ${\hat{S}}_{A}K = 0$ at ${\Sigma }_{F}$. This is simplified via symmetry properties of $K$:

• (a) interchange of the boundary surfaces, ${\Sigma }_{I} , {\Sigma }_{F}$, and

• (b) interchange of the rôles of the unprimed and primed fermionic arguments, ${\psi }^{A}_{i}(x)$ and ${\widetilde{\psi }}^{A^{\prime }}_{i}(x)$, on a given hypersurface, ${\Sigma }$ ($e^{AA^{\prime }}_{i}(x)$ remaining unchanged).

Given the expression, Eq. (7.2.22), for the fermionic momentum, ${\pi }_{A}^{i}(x)$, (b) corresponds to interchanging fermionic co–ordinates and momenta. We proceed (Eqs. (3.2.35), (3.3.5–16) of [16]) by defining the fermionic Fourier transform, which maps a wave–functional, schematically, $f(e, {\psi })$, to its transform, ${\widetilde{f}}(e, {\widetilde{\phi }})$. Sequentially:

$$\begin{aligned} C_{AA^{\prime }}^{ij}(x) = - i {\epsilon }^{ijk} e_{AA^{\prime }k}(x), \end{aligned}$$

(7.2.36)

$$\begin{aligned} \exp \Bigl [-iC{\psi }{\widetilde{\phi }}\Bigr ] = \exp \Bigl [-i{\int }d^{3}x^{\prime }C_{AA^{\prime }}^{ij}(x^{\prime }) {\psi }^{A}_{i}(x^{\prime }){\widetilde{\phi }}^{A^{\prime }}_{j}(x^{\prime })\Bigr ], \end{aligned}$$

(7.2.37)

$$\begin{aligned} {\widetilde{f}}\bigl (e,{\widetilde{\phi }}\bigr ) = D^{-1}(e)\int \mathcal{{D}}{\psi } f(e,{\psi }) \exp \Bigl [-iC{\psi }{\widetilde{\phi }}\Bigr ]. \end{aligned}$$

(7.2.38)

Berezin integration is used [26, 27]. The determinant, $D(e)$, is defined in Eq. (3.3.7) of [16].

The inverse transform is defined similarly, the factor, $+i$, replacing $-i$. These Fourier transforms have their finite–dimensional analogues in the holomorphic description of QM [26, 27]; they map between holomorphic and anti–holomorphic representations.

Consider next the unprimed quantum constraint, ${\hat{S}}_{A}K = 0$, at ${\Sigma }_{F}$. Since the classical constraints, $S_{A} = 0, {\widetilde{S}}_{A^{\prime }} = 0$, are conserved in Hamiltonian evolution, Eq. (7.2.20) can equivalently be written:

$$\begin{aligned} K = \mathrm{const.} {\delta }\bigl (S_{A}(x_{F})\bigr ) {\delta }\bigl ({\widetilde{S}}_{A^{\prime }}(x_{I})\bigr ) \exp \bigl (-I_\mathrm{class}\bigr ), \end{aligned}$$

(7.2.39)

the unprimed constraint now taken at ${\Sigma }_{F}$, the primed constraint at ${\Sigma }_{I}$. [The boundary data are still as in Eq. (7.2.30).] Make a functional Fourier transform at ${\Sigma }_{F}$, as in Eqs. (7.2.36–7.2.38), changing the final fermionic argument from ${\psi }^{A}_{i}$ to a new primed quantity, ${\widetilde{\phi }}^{A^{\prime }}_{i}$, but leaving invariant the final bosonic data. At ${\Sigma }_{I}$, make a reverse transform, replacing the original, ${\widetilde{\psi }}^{A^{\prime }}_{i}$, with new unprimed data, ${\phi }^{A}_{i}$. The new data on ${\Sigma }_{I}, {\Sigma }_{F}$, should again give a well–posed boundary–value problem, leading to a ‘new’ classical solution. The action, $I_\mathrm{class}$, at a classical solution, reduces to a sum of boundary contributions [16]—a gravitational part and a gravitino part. The sign of a gravitino boundary contribution depends on whether ${\psi }^{A}_{i}$ or ${\widetilde{\psi }}^{A^{\prime }}_{i}$ is held fixed there (see Sects. 3.3–5 of [16]). Correspondingly, the above pair of ‘Fourier transforms’ change the ‘old’ classical action with original boundary data into the ‘new’ classical action for the ‘new’ data, $(e^{AA^{\prime }}_{i}(x_{I}), {\phi }^{A}_{i}(x_{I}))$ and $(e^{AA^{\prime }}_{i}(x_{F}), {\widetilde{\phi }}^{A^{\prime }}_{i}(x_{F}))$. This gives the transformation of the semi–classical factor, $\exp (-I_\mathrm{class})$, under the change to ‘new’ boundary data.

These transforms yield a simple form of ${\hat{S}}_{A}(x_{F})$, with respect to the ‘new’ variables on ${\Sigma }_{F}$; given Eq. (7.2.29) for ${\hat{\overline{S}}}_{A^{\prime }}(x_{I})$, one replaces ${\psi }^{A}_{k}(x)$ by ${\widetilde{\psi }}^{A^{\prime }}_{k}(x)$. Analogously, the ‘new’ quantum constraint, ${\hat{S}}_{A}(x_{F}){\Psi } = 0$, is related to the transformation of the ‘new’ amplitude under an unprimed local supersymmetry transformation at ${\Sigma }_{F}$. (Similarly, at ${\Sigma }_{I}$, the primed operator, ${\hat{\overline{S}}}_{A^{\prime }}(x_{I})$, becomes simple with respect to the new variables.) At ${\Sigma }_{F}$, the interchange (b) of primed and unprimed fermionic data shows that ${\hat{S}}_{A}(x_{F})K =0$, following Eqs. (7.2.33–7.2.35) for ${\hat{\overline{S}}}_{A^{\prime }}(x_{F})K =0$. Thus, $K$ obeys the quantum constraints, Eqs. (7.2.27a, 7.2.27b, 7.2.27c, 7.2.27d).

For (ii) and (iii), consider the Schrödinger equation (Lorentzian) or heat equation (Riemannian), for the evolution of $K$ with $T$ or ${\tau }$ [the remaining data of Eq. (7.2.30) being fixed]. In particular, examine the property that $K$ tends to a delta–function(al) of the boundary data, analogous to ${\delta }(q_{a}, q_{b})$, as (say) ${\tau }\longrightarrow {0}_{+}$. In supergravity, these conditions are most simply analysed in the Riemannian case, ${\tau }= iT$, real.

Property (ii) holds since $K$, the right–hand side of Eq. (7.2.30), obeys the equations,

$$\begin{aligned} ({\partial }K)/({\partial }{\tau }) + [({\partial }I_\mathrm{class})/({\partial }{\tau })] K = 0, \end{aligned}$$

(7.2.40a)

$$\begin{aligned} ({\partial } I_\mathrm{class})/({\partial }{\tau }) = M. \end{aligned}$$

(7.2.40b)

Equation (7.2.40b) is the Riemannian version of the Hamilton–Jacobi equation [39]. The mass, $M$, is a functional of the boundary data, including ${\tau }$. As in [16], this gives the evolution equation of the amplitude.

For (iii), examine in more detail the QM condition [12],

$$\begin{aligned} K(q_{b},q_{a};{\tau }) \longrightarrow {\delta }(q_{b},q_{a}), \end{aligned}$$

(7.2.41)

as ${\tau }\longrightarrow {0}_{+}$. In this limit, further information is available from an asymptotic expansion of $K$, as ${\tau }\longrightarrow {0}_{+}$, other variables being fixed. First, note the exact solution for $K$ for a free particle of mass $m$ [12]:

$$\begin{aligned} K(q_{b},q_{a};{\tau }) = \biggl ({{m}\over {2{\pi }{\tau }}}\biggr )^{1/2} \exp \biggl ({{-m (q_{b}-q_{a})^2}\over {2{\tau }}} \biggr ). \end{aligned}$$

(7.2.42)

When a potential, $V(q)$, is included in this QM model, Eq. (7.2.42) still gives the leading term in an asymptotic expansion of $K(q_{a},q_{b};{\tau })$, as ${\tau }\longrightarrow {0}_{+}$, with $q_{a} , q_{b}$, held fixed (not necessarily equal to each other) [12].

Correspondingly, for quantum amplitudes in $N = 1$ supergravity, with data as in Eq. (7.2.30) and $T = - i{\tau } ({\tau }>0)$, one can study the proposed amplitude, Eq. (7.2.20), defined modulo solution of the classical boundary–value problem. Given the non–linearity of this problem, one cannot expect classical solutions (or amplitude, $K$) to arise in closed form. Rather, following [40], one can make an asymptotic expansion of $K$ for $N = 1$ supergravity, as ${\tau }\longrightarrow {0}_{+}$, other data being held fixed, analogous to Eq. (7.2.42). This is discussed for Riemannian signature in Sect. 4.4 of [16]. For the bosonic (Einstein) part of the Lagrangian, one finds:

$$\begin{aligned} I_\mathrm{class} (h_{ijF}, h_{ijI};{\tau }) \sim {\tau }^{-1} {\nu }(h_{ijF}, h_{ijI}), \end{aligned}$$

(7.2.43)

where ${\nu }$ is a non–negative functional of the bounding three–geometries, plus smaller corrections, as ${\tau }\longrightarrow {0}_{+}$.

The limiting behaviour of the fermionic part of $I_\mathrm{class}$ gives the appropriate fermionic behaviour when ${\tau }$ is small. Following Eq. (3.5.5) of [16] and Eqs. (7.2.36–7.2.38), the fermionic contribution to $I_\mathrm{class}$, for small ${\tau }$, is proportional to ${\int }d^{3}x C_{AA^{\prime }}^{ij} (x) {\psi }^{A}_{i}(x) {\widetilde{\psi }}^{A^{\prime }}_{j}(x)$. Integrated against a wave–functional, $f(e,{\psi })$, as in Eq. (7.2.38), this gives the transformed wave–functional, ${\widetilde{f}}(e,{\widetilde{\psi }})$, at the other surface, whence, the correct fermionic behaviour as ${\tau }\longrightarrow {0}_{+}$. The two delta–functionals in Eq. (7.2.20), present for all interior ${\tau }$, require that the boundary data obey the classical constraints, $S_{A}(x_{I}) =0, {\widetilde{S}}_{A^{\prime }}(x_{F}) =0$.

The asymptotic expansion of $K$ in $N = 1$ supergravity, via estimates of $I_\mathrm{class}$ and $M_\mathrm{class}$ for the infilling classical Einstein geometry, as ${\tau }\longrightarrow {0}_{+}$ [16, 40], thus agrees (up to a constant factor) with the small–${\tau }$ expansion of $K$ in Eq. (7.2.20) for fixed boundary data. Hence, as in the asymptotic version of Eq. (7.2.42), Eq. (7.2.20) satisfies the asymptotic version of (iii).

The quantum amplitude, $K$, with boundary data, Eq. (7.2.30), is given by Eq. (7.2.20). Apart from the fermionic delta–functionals in Eq. (7.2.20), associated with the classical supersymmetry constraints, $S_{B}(x_{I}) =0, {\widetilde{S}}_{B^{\prime }}(x_{F}) =0$, the amplitude, $K$, is exactly semi–classical, proportional to $\exp (-I_\mathrm{class}/{\hbar })$. Emphasis is thrown back entirely onto the classical bosonic/fermionic N $\varvec{=}$1 supergravity boundary–value problem.

7.3 Quantum Amplitudes in Black–Hole Evaporation

7.3.1 Introduction

This Section is concerned with the simplest example of quantum radiation following gravitational collapse to a black hole—we take the bosonic part of the Lagrangian to describe Einstein gravity coupled minimally to a massless scalar field, ${\phi }$ [7, 9, 37]. The spin–$1$, spin–$2$ (graviton) and fermionic spin–$1 \over 2$ cases are treated similarly [7, 33, 36].

Given the results of Sect. 7.2, it will be assumed that the full Lagrangian is locally supersymmetric, describing $N = 1$ supergravity + supermatter. The simplest possibility is the model of [41], with $m=g=0$. This has a complex massless scalar field coupled to Einstein gravity, with spin–${1}\over {2}$ and spin–${3}\over {2}$ partners. In the present context, we (consistently) truncate the scalar field to be real.

As in Sects. (7.2.2–7.2.4), pose boundary data on ${\Sigma }_{I,F}$ and require

$$\begin{aligned} T ={\mid }T{\mid }\exp (-i{\theta }) {\quad }(0<{\theta }\le {\pi /2}), \end{aligned}$$

(7.3.1)

for a well–posed classical boundary–value problem. A small negative imaginary part in $T$ induces an imaginary part in the total Lorentzian action—crucial in calculating the quantum amplitude. Conversely, even for small values of ${\theta }$, solution of the rotated classical boundary–value problem is expected to smooth variations or oscillations of the boundary data, as one moves from the boundary, ${\Sigma }_{I}$ or ${\Sigma }_{F}$, into the interior by a few multiples of the relevant wavelength.

To fix physical intuition, imagine that the initial scalar field, ${\phi }_{I}$, on ${\Sigma }_{I}$, is nearly spherically symmetric and very diffuse, with most of the mass distributed over radii much greater than the ‘Schwarzschild radius’, $2M_{0}$. Here, $M_{0}$ is the Arnowitt–Deser–Misner ($ADM$) mass, defined via the $r^{-1}$ fall–off of initial data, at large radii on a spacelike hypersurface [42]. The three–metric, $h_{ijI}$, on ${\Sigma }_{I}$, is almost spherically symmetric, and varies slowly with radius. The value of $T$ at ${\Sigma }_{F}$ is chosen large and positive, to register all the evaporated radiation. The total $ADM$ mass on ${\Sigma }_{F}$ must also equal $M_{0}$, since otherwise the classical boundary–value problem with time–interval, $T$, at spatial infinity, will have no solution [40, 43]. Regarding the classical solution in the interior $(0<t<T)$, the geometry is well approximated at late times by the radiating Schwarzschild–like Vaidya metric [34, 44, 45]. The classical scalar–field solution will depend on the enormous amount of detail which, in general, is present in the prescribed final data, ${\phi }_{F}$, on ${\Sigma }_{F}$.

In Sect. 7.3.3, we discuss the ‘background’ spherically–symmetric four–metric, ${\gamma }_{\mu \nu }$, and scalar field, ${\Phi }$, as part of a self–consistent treatment of the classical field equations. The Einstein field equations give, at lowest order, a ‘source’ for ${\gamma }_{{\mu }{\nu }}$ which includes the energy–momentum tensor of the scalar field, ${\phi }$, and a source quadratic in graviton perturbations (and for other matter fields). In Sect. 7.3.3, we treat the decomposition of scalar perturbations in spherical harmonics. We also describe the classical action functionals, $S_\mathrm{class}$ (Lorentzian action), or $I_\mathrm{class}$ (Euclidean or Riemannian action), related by

$$\begin{aligned} iS_\mathrm{class} = -I_\mathrm{class}, \end{aligned}$$

(7.3.2)

for the Einstein/scalar system. Since $S_\mathrm{class}$ or $I_\mathrm{class}$ are evaluated at a solution of the classical field equations, they reduce to a sum of boundary terms. Section 7.3.3 also treats adiabatic radial equations for the (linearised) scalar field. For adiabatic perturbations (high frequencies), the time–dependence is approximately harmonic and can be factored out, leading to a second–order radial equation for given frequency, ${\omega }$, and angular quantum numbers, ${\ell }, m$. Further, ‘co–ordinates’ are described which are suited to specifying final data, ${\phi }_{F}$, on ${\Sigma }_{F}$, and a suitable basis of radial eigenfunctions on ${\Sigma }_{F}$ is discussed. The Lorentzian quantum amplitude is defined as the limit, ${\theta }\longrightarrow {0}_{+}$, of the amplitude for complex $T$, following Feynman [12, 13]. By these methods, one can evaluate the real and imaginary parts of the lowest–order perturbative classical action, $S^{(2)}_\mathrm{class}$, and, hence, of the semi–classical amplitude, $\exp (iS^{(2)}_\mathrm{class})$. Section 7.3.4 contains the Conclusion.

7.3.2 The Quantum Amplitude for Bosonic Boundary Data

From one point of view, this amplitude can be regarded as a Feynman path integral over all Riemannian infilling four–geometries and other fields. Each configuration is weighted by $\exp (-I)$, where $I$ is the Euclidean action. The ‘differential’ Dirac approach of Sect. 7.2 is dual to the ‘integral’ Feynman approach. In the present boundary–value context, the Dirac approach appears to give the quantum amplitude more directly and explicitly than does the path–integral approach. Following the Dirac approach, given local supersymmetry, the amplitude has the ‘semi–classical’ form, Eq. (7.2.20). Related semi–classical behaviour holds for $N = 1$ supergravity with (gauge–invariant) supermatter [8, 17].

We assume, as in Sect. 7.2, the existence of a (complex) classical solution to the Dirichlet boundary–value problem, posed in Sect. 7.2, given Eq. (7.3.1). The solution should vary smoothly with ${\theta }({\epsilon }\le {\theta }\le {\pi }/2)$, and the expression, Eq. (7.2.20), should continue to give the quantum amplitude as ${\theta }$ varies. In particular, this would occur if strong ellipticity [31] held for the coupled Einstein/bosonic–matter field equations, up to gauge. As above, we consider locally supersymmetric models. For simplicity, take only gravitational and scalar quantities to be present in the data and solution, but no fermions.

Consider the division, above, of $(g_{\mu \nu },{\phi })$ into ‘background’ and perturbative parts. Classical solutions, $(g_{{\mu }{\nu }},{\phi })$, of the coupled Einstein/scalar field equations are taken to have a ‘background’ time–dependent spherically–symmetric part, $({\gamma }_{{\mu }{\nu }}, {\Phi })$, together with a ‘small’ perturbative part, $({\beta }_{{\mu }{\nu }}, {\phi }_\mathrm{pert})$. The fields, (${\beta }_{{\mu }{\nu }}, {\phi }_\mathrm{pert}$), live (mathematically speaking) on the background four–geometry, with metric, ${\gamma }_{{\mu }{\nu }}$. (${\beta }_{{\mu }{\nu }}, {\phi }_\mathrm{pert}$) can be expanded in terms of sums over tensor (spin–2), vector (spin–1) and scalar harmonics [46, 47]. Each harmonic is weighted by a function of time and radius, $(t,r)$.

The Einstein field equations are:

$$\begin{aligned} G_{{\mu }{\nu }} \equiv R_{{\mu }{\nu }}-{{1}\over {2}}Rg_{{\mu }{\nu }} = 8{\pi }T_{{\mu }{\nu }}, \end{aligned}$$

(7.3.3)

$$\begin{aligned} T_{{\mu }{\nu }} = {\phi },_{\mu }{\phi },_{\nu } -{{1}\over {2}}g_{{\mu }{\nu }} \bigl ({\phi },_{\alpha }{\phi },_{\beta }g^{{\alpha }{\beta }}\bigr ). \end{aligned}$$

(7.3.4)

The scalar field equation is:

$$\begin{aligned} \partial _\mu \bigl ((-g)^{{1}\over {2}}g^{{\mu }{\nu }}{\phi },_{\nu }\bigr ) = 0, \end{aligned}$$

(7.3.5)

where $g$ denotes $\det (g_{\mu \nu })$, and $g_{{\mu }{\nu }}$ has Lorentzian signature, whence, $g<0$.

The corresponding Lorentzian action is [16]

$$\begin{aligned} S = {{1}\over {16{\pi }}} {\int }d^{4}x\bigl (-g\bigr )^{{1}\over {2}}R -{{1}\over {2}} {\int }d^{4}x\bigl (-g\bigr )^{{1}\over {2}}\bigl (\nabla {\phi }\bigr )^{2} +{\mathrm{{boundary}} \mathrm{{contributions}}}, \end{aligned}$$

(7.3.6)

in geometrical units. Boundary terms will be discussed in Sect. 7.3.3.

In Lorentzian signature, the spherically–symmetric ‘background’ four–metric, ${\gamma }_{\mu \nu }$, and scalar field can be written via:

$$\begin{aligned} ds^{2} = -e^{b}dt^{2} +e^{a}dr^{2} +r^{2}\bigl (d{\theta }^{2} +{\sin }^{2}{\theta }d{\phi }^{2}\bigr ), \end{aligned}$$

$$\begin{aligned} b = b(t,r), {\quad } a = a(t,r),{\quad }{\phi }={\Phi }(t,r). \end{aligned}$$

(7.3.7)

The classical field equations, assuming that one had exact spherical symmetry in Lorentzian signature, are given in [48, 49].

In a typical bosonic black–hole evaporation problem, the classical metric and scalar field are not spherically symmetric. Similarly, for non–zero spin–${{1}\over {2}}$ and spin–${{3}\over {2}}$ classical (odd Grassmann–algebra–valued [16]) fermionic solutions in a locally–supersymmetric generalisation [6]. In particle language, rather than the field language of this Chapter, huge numbers of gravitons and scalar particles are continually given off by the black hole (together with any fermions allowed), leading effectively to a stochastic distribution, in which, for given spin, $s$, the field fluctuates around a spherically–symmetric reference field.

Consider, say, a one–parameter family of gravitational and scalar perturbations about a spherically–symmetric reference four–metric, ${\gamma }_{{\mu }{\nu }}$, and field, ${\Phi }$:

$$\begin{aligned} g_{{\mu }{\nu }}(x,{\epsilon }) \sim {\gamma }_{{\mu }{\nu }}(x) +\epsilon {\beta }^{(1)}_{{\mu }{\nu }}(x) +{\epsilon }^{2}{\beta }^{(2)}_{{\mu }{\nu }}(x) +\cdots , \end{aligned}$$

(7.3.8)

$$\begin{aligned} {\phi }(x,{\epsilon }) \sim {\Phi }(t,r) +{\epsilon }{\phi }^{(1)}(x) +{\epsilon }^{2}{\phi }^{(2)}(x) +\cdots . \end{aligned}$$

(7.3.9)

The background field, ${\Phi }$, will be non–zero if the scalar data, ${\phi }$, at early and late times, $t$, contain a non–trivial spherical component. The perturbation fields, ${\phi }^{(1)}(x), {\phi }^{(2)}(x), \ldots$ typically contain all non–spherical angular harmonics. These fields must be chosen such that the entire system obeys the classical coupled Einstein/scalar field equations, and agrees with the (prescribed) small non–spherical perturbations in the initial and final data, gravitational and scalar. The effective energy–momentum source for the background part of the metric, ${\gamma }_{\mu \nu }$, includes contributions quadratic in the non–spherical gravitational and scalar quantities, $\bigl ({\beta }^{(1)}_{{\mu }{\nu }}, {\phi }^{(1)}\bigr )$—see [7, 10] for further detail.

The linearised $({\epsilon }^{1})$ part of the Einstein equations reads (Sect. 35.13 of [42]):

$$\begin{aligned} \begin{array}{ll} &{} {\bar{\beta }}^{(1);{\sigma }}_{{\mu }{\nu };{\sigma }} -2 {\bar{\beta }}^{(1);{\sigma }}_{{\sigma }{(}{\mu };{\nu }{)}} -2R^{(0)}_{{\sigma }{\mu }{\nu }{\alpha }} {\bar{\beta }}^{(1){\sigma }{\alpha }} -2R^{(0){\alpha }}_{{(}{\mu }} {\bar{\beta }}^{(1)}_{{\nu }{)}{{\alpha }}}\\ &{}{\quad } +{\gamma }_{{\mu }{\nu }} \Bigl ({\bar{\beta }}^{(1);{\alpha }{\beta }}_{{\alpha }{\beta }} -{\bar{\beta }}^{(1){\alpha }{\beta }} R^{(0)}_{{\alpha }{\beta }}\Bigr ) +{\bar{\beta }}^{(1)}_{{\mu }{\nu }} R^{(0)} = -16{\pi }T^{(1)}_{{\mu }{\nu }}. \end{array} \end{aligned}$$

(7.3.10)

Here, ${\bar{\beta }}^{(1)}_{{\mu }{\nu }}$ is defined by

$$\begin{aligned} {\bar{\beta }}^{(1)}_{{\mu }{\nu }} = {\beta }^{(1)}_{{\mu }{\nu }} -{{1}\over {2}}{\gamma }_{{\mu }{\nu }}{\beta }^{(1)}, \end{aligned}$$

(7.3.11a)

$$\begin{aligned} {\beta }^{(1)} = {\beta }^{(1){\mu }}_{{\mu }}. \end{aligned}$$

(7.3.11b)

The Riemann tensor of the background geometry, ${\gamma }_{{\mu }{\nu }}$, is denoted $R^{(0)}_{{\sigma }{\mu }{\nu }{\alpha }}$. Further, $T^{(1)}_{{\mu }{\nu }}$ denotes the linearisation or $O({\epsilon }^{1})$ part of the energy–momentum tensor, $T_{{\mu }{\nu }}(x,{\epsilon })$:

$$\begin{aligned} \begin{array}{ll} T^{(1)}_{{\mu }{\nu }} &{} = 2\nabla _{(\mu }{\phi }^{(1)}\nabla _{\nu )}{\Phi } -{\gamma }_{{\mu }{\nu }}\bigl (\nabla _{\alpha }{\Phi }\bigr ) \bigl (\nabla ^{\alpha }{\phi }^{(1)}\bigr )\\ &{}{\quad }+{{1}\over {2}} \Bigl ({\gamma }_{{\mu }{\nu }}{\beta }^{(1){\sigma }{\rho }} -{\beta }^{(1)}_{{\mu }{\nu }}{\gamma }^{{\sigma }{\rho }}\Bigr ) \bigl (\nabla _{\sigma }{\Phi }\bigr ) \bigl (\nabla _{\rho }{\Phi }\bigr ). \end{array} \end{aligned}$$

(7.3.12)

The linearised Einstein equations, (7.3.10–7.3.12), are most easily studied in a ‘linearised harmonic gauge’ [42, 50], in which, following an infinitesimal co–ordinate transformation,

$$\begin{aligned} {\bar{h}}^{(1);{\alpha }}_{{\mu }{\alpha }} = 0. \end{aligned}$$

(7.3.13)

The Einstein equations, (7.3.10–7.3.12), linearised about spherical symmetry, can be decomposed into three independent sets. These describe, respectively, scalar (spin–0) perturbations associated with matter–density changes, $(T^{(1)}_{{\tau }{\tau }})$, vector (spin–1) perturbations for matter–velocity changes, $(T^{(1)}_{{\tau }i})$, and gravitational radiation (spin–2) for anisotropic stresses, $(T^{(1)}_{ij})$ [51]. The resulting equations and their solutions are described in [36].

The linearised $({\epsilon }^{1})$ part of the scalar–field equation, (7.3.5), reads:

$$\begin{aligned} {\gamma }^{{\mu }{\nu }}{\phi }^{(1)}_{;{\mu }{\nu }} -\Bigl ({\bar{\beta }}^{(1){\mu }{\nu }}{\Phi },_{\nu }\Bigr )_{;{\mu }} = 0. \end{aligned}$$

(7.3.14)

The linearised Einstein and scalar–field equations, (7.3.10–7.3.12, 7.3.14), are coupled. The remaining perturbative field equations, needed to complete this calculation, are in [7, 10].

These field equations arise, for example, in studying the Vaidya metric [44]; as shown in [34], this describes approximately the late–time region of the geometry following gravitational collapse to a black hole, containing a nearly–steady outgoing flux of radiation. The Einstein field equations, averaged over small regions, give the contribution of massless scalar particles, gravitons, etc., to the nearly–isotropic flux.

As above, for perturbed boundary data which contain numerous high harmonics, the interior perturbations can be regarded as effectively stochastic in nature. Averaged over a number of wavelengths, the effective perturbative energy–momentum tensor, $T^{EFF}_{{\mu }{\nu }}$, yields a spherically–symmetric smoothed–out quantity, $<T^{EFF}_{{\mu }{\nu }}>$ [52, 53].

The averaged form of $T^{EFF}_{{\mu }{\nu }}$ accounts for the gradual loss of mass of a black hole by radiation, in a slightly–complexified (nearly–Lorentzian) ‘space–time’, with ${\theta } = {\delta } {\ll } 1$ in Eq. (7.3.1). Although $<T^{EFF}_{{\mu }{\nu }}>$ is small [$O({\epsilon }^{2})$], its effects on the black–hole geometry, including those on the mass, build up in secular fashion, over a time–scale of order $O({\epsilon }^{-2})$. Secular behaviour appears often in perturbation problems [54, 55]—for example, in the perihelion precession of nearly–circular orbits in the Schwarzschild geometry [56]. In our boundary–value problem, classical or quantum, the initial boundary data are spread over a ‘background’ extent of $O(1)$, measured by the co–ordinate, $r$, on ${\Sigma }_{I}$. Corresponding to the $O({\epsilon }^{-2})$ time–scale for the black hole to radiate, the data on ${\Sigma }_{F}$ will be spread over a radial–coordinate scale of $O({\epsilon }^{-2})$. Thus, even the classical boundary–value problem is an example of singular perturbation theory [54, 55].

High–frequency averaging in general relativity was treated by Brill and Hartle [52], Isaacson [53]. Let $<>$ denote an average over a time, $T_{0}$, longer than typical wave periods, and a spatial average over several wavelengths, ${\bar{\lambda }}$. Rules for manipulating these averages in the high–frequency aproximation are in [53]. In particular, one finds [7, 10]:

$$\begin{aligned} <T^{(1)}_{{\mu }{\nu }}> \,=\, 0, \end{aligned}$$

(7.3.15)

$$\begin{aligned} <T^{(2)}_{{\mu }{\nu }}> \,=\, <{\nabla }_{(\mu }{\phi }^{(1)}{\nabla }_{\nu )}{\phi }^{(1)} -{1\over 2} {\gamma }_{{\mu }{\nu }} \nabla _{\alpha }{\phi }^{(1)}\nabla ^{\alpha }{\phi }^{(1)}>. \end{aligned}$$

(7.3.16)

The background field equations can be re–written in a form smoothed by averaging [52, 53].

7.3.3 Classical Action and Amplitude for Weak Perturbations

Consider small bosonic perturbations, $\bigl ({\beta }^{(1)}_{{\mu }{\nu }},{\phi }^{(1)}\bigr )$, obeying the linearised classical field equations, (7.3.10, 7.3.14), about a spherically–symmetric classical solution, $({\gamma }_{{\mu }{\nu }},{\Phi })$ [10]. The background data for ${\gamma }_{{\mu }{\nu }}$ and ${\Phi }$ are posed, as in Sects. 7.1 and 7.2, on ${\Sigma }_{I,F}$, separated at spatial infinity by time, $T$, with $\mathrm {Im}(T)<0$. Similarly, the linearised classical perturbations, $\bigl ({\beta }^{(1)}_{{\mu }{\nu }},{\phi }^{(1)}\bigr )$, are solutions of a (complex) linear strongly–elliptic problem, with prescribed perturbations, $\bigl ({\beta }^{(1)}_{ij},{\phi }^{(1)}\bigr )$, on ${\Sigma }_{I,F}$.

Given the background spherical symmetry, one can expand ${\phi }^{(1)}$ as:

$$\begin{aligned} {\phi }^{(1)}(t,r,{\theta },{\phi }) = {1\over r}\sum ^{\infty }_{{\ell }=0} \sum ^{\ell }_{m=-{\ell }} Y_{{\ell }m}({\Omega })R_{{\ell }m}(t,r), \end{aligned}$$

(7.3.17)

where $Y_{{\ell }m}({\Omega })$ is the $({\ell },m)$ scalar spherical harmonic of [57].

Similarly, a metric perturbation, ${\beta }^{(1)}_{{\mu }{\nu }}$, can be expanded as a sum over tensor, vector and scalar $({\ell }, m)$ harmonics, weighted by functions of $t$ and $r$ [46, 47, 58–61]. Because of the coupling in the linearised field equations, (7.3.10, 7.3.14), for $\bigl ({\beta }^{(1)}_{{\mu }{\nu }},{\phi }^{(1)}\bigr )$, the linear field equations for $R_{{\ell }m}(t,r)$ in Eq. (7.3.17) and its gravitational analogues are also coupled in the strong–field ‘collapse’ region of the space–time.

The boundary conditions on $R_{{\ell }m}(t,r)$, as $r\longrightarrow {0}$, follow from regularity of the solution, $\bigl (g_{{\mu }{\nu }},{\phi }\bigr )$, viewed in ‘nearly–Cartesian co–ordinates’ near $r =0$. Regularity follows since the coupled field equations are ‘strongly–elliptic modulo gauge’. The asymptotically–flat boundary data on ${\Sigma }_{I,F}$, are chosen smooth over ${\mathbb R}^{3}$. The field equations (up to gauge) should be strongly elliptic [31], giving analytic classical fields in the interior.

Take the final boundary data to describe very weak, diffuse scalar and gravitational fields, viewed as perturbations of nearly–flat space–time. (As in Sect. 7.3.1, the $ADM$ mass of $h_{ijF}$ must equal the $ADM$ mass of ${\Sigma }_{I}$.) Physically, such final data may be imagined as a possible late–time remnant of gravitational collapse, namely, a snap–shot of a large number of scalar particles and gravitons, as they make their way out to infinity. Near ${\Sigma }_{F}$, the coupling in Eqs. (7.3.10, 7.3.14) between ${\beta }^{(1)}_{{\mu }{\nu }}$ and ${\phi }^{(1)}$ will almost disappear. The perturbed scalar field equation is then:

$$\begin{aligned} {\nabla }^{\mu }{\nabla }_{\mu }{\phi }^{(1)} = 0, \end{aligned}$$

(7.3.18)

with respect to the spherically–symmetric geometry, ${\gamma }_{{\mu }{\nu }}$.

From the decomposition, Eq. (7.3.17), of ${\phi }^{(1)}$, one finds the $({\ell },m)$ mode equation:

$$\begin{aligned} \Bigl (e^{(b-a)/2}\partial _{r}\Bigr )^{2}R_{{\ell }m} -\bigl (\partial _{t}\bigr )^{2}R_{{\ell }m} -{{1}\over {2}}\Bigl (\partial _{t}\bigl (a-b\bigr )\Bigr ) \bigl (\partial _{t}R_{{\ell }m}\bigr ) -V_{\ell }\bigl (t,r\bigr )R_{{\ell }m} = 0.\quad \end{aligned}$$

(7.3.19)

The potential, $V_{\ell }(t,r)$, and function, $m(t,r)$, are defined by:

$$\begin{aligned} V_{\ell }(t,r) = {{e^{b(t,r)}\over {r^{2}}}} {\Bigl ({\ell }({\ell }+1)+{{2{\,}m(t,r)}\over {r}}\Bigr )}, \end{aligned}$$

(7.3.20)

$$\begin{aligned} \exp \Bigl (-a(t,r)\Bigr ) = 1 - {{2m(t,r)}\over r}. \end{aligned}$$

(7.3.21)

In an exact Schwarzschild solution with no scalar field, one has $1-(2M/r) = e^{-a} = e^{b}$, with $M$ the Schwarzschild mass; in that case, $m(t,r)$ is identically $M$. The potential, $V_{\ell }(t,r)$, generalises the well–known massless–scalar effective potential in the Schwarzschild geometry [42], which vanishes at the event horizon, $\{r=2M\}$, and at spatial infinity, and has a peak near $\{r=3M\}$.

The definition, Eq. (7.3.21), of $m(t,r)$, is consistent with the usual description in Lorentzian signature of the Vaidya metric [44, 45]. In terms of a null co–ordinate, $u$, and intrinsic radial co–ordinate, $r$, the Vaidya metric reads:

$$\begin{aligned} ds^{2} = -2dudr -\biggl (1-{{2m(u)}\over {r}}\biggr )du^{2} +r^{2} \bigl (d{\theta }^{2}+\sin ^{2}{\theta }d{\phi }^{2}\bigr ). \end{aligned}$$

(7.3.22)

Here, $m(u)$ is a monotonic–decreasing smooth function of $u$, corresponding to a spherically–symmetric outflow of null particles—for example, from the energy–momentum tensor of a black hole evaporating via emission of scalar particles at the speed of light. The Vaidya metric has been used often to give an approximate gravitational background for black–hole evaporation at late times [62–64]. In connection with the present work, see [34].

An analogous decoupled harmonic decomposition is valid near ${\Sigma }_{F}$, for late–time gravitational–wave perturbations of the spherically–symmetric background—again as in [36]. To simplify the exposition, restrict attention to non–zero weak–field final configurations with spin 0 , and calculate their quantum amplitudes (below). Make the further assumption that the final three–metric, $h_{ijF}$, is exactly spherically symmetric (in addition to the assumed spherical symmetry of the initial data, ($h_{ijI},{\phi }_{I}$)).

Consider an asymptotically–flat classical solution, $(g_{{\mu }{\nu }}, {\phi })$, of the coupled Einstein/massless–scalar field equations, between ${\Sigma }_{I}$ and ${\Sigma }_{F}$, with $\mathrm {Im}(T)<0$. The classical Riemannian and Lorentzian actions, for fixed boundary data, $(h_{ij}, {\phi })_{I}, (h_{ij}, {\phi })_{F}$, are related by Eq. (7.3.2). At a solution [16, 65], the classical action is:

$$\begin{aligned} \begin{array}{ll} &{} S_\mathrm{class}\bigl [(h_{ij},{\phi })_{I}; (h_{ij},{\phi })_{F}; T\bigr ]\\ &{} {\quad } = {{1}\over {32{\pi }}} \biggl (\int _{{\Sigma }_{F}}-\int _{{\Sigma }_{I}}\biggr ) d^{3}x{\pi }^{ij}h_{ij} +{{1}\over {2}} \biggl (\int _{{\Sigma }_{F}}-\int _{{\Sigma }_{I}}\biggr ) d^{3}x{\pi }_{\phi }{\phi } -MT. \end{array} \end{aligned}$$

(7.3.23)

Here, ${\pi }^{ij} = {\pi }^{ji}$ is $16{\pi }$ times the (Lorentzian) momentum conjugate to the ‘co–ordinate’ variable, $h_{ij}$, on a space–like hypersurface, in a $3+1$ Hamiltonian decomposition of the Einstein/massless–scalar theory. In terms of the second fundamental form, $K_{ij} = K_{(ij)}$, of the hypersurface [16, 50], ${\pi }^{ij}$ is

$$\begin{aligned} {\pi }^{ij} = h^{{1}\over {2}}\bigl (K^{ij}-Kh^{ij}\bigr ), \end{aligned}$$

(7.3.24)

with $h = \mathrm{det}(h_{ij}), K = h^{ij}K_{ij}$. Also, ${\pi }_{\phi }$ is the momentum conjugate to the ‘co–ordinate’ variable, ${\phi }$. Explicitly,

$$\begin{aligned} {\pi }_{\phi } = h^{{1}\over {2}}n^{\mu }\nabla _{\mu }{\phi }. \end{aligned}$$

(7.3.25)

The quantity, $M$, in Eq. (7.3.23) is the $ADM$ mass of the space–time. As above, it is essential, for a well–posed asymptotically–flat boundary–value problem, that the intrinsic metrics, $h_{ijI}$ and $h_{ijF}$, have the same value of $M$. Otherwise, if ${M}_{I}\ne {M}_{F}$, a classical infilling space–time will have ${\Sigma }_{I}$ and ${\Sigma }_{F}$ badly embedded near spatial infinity, and the four–metric, $g_{{\mu }{\nu }}$, will not fall off to flatness at the standard $1/r$ rate, as $r\longrightarrow {\infty }$ [43].

Without loss of generality, assume that any spherically–symmetric ${\ell } =0$ linear–order perturbation modes have been absorbed into the spherically–symmetric background, $({\gamma }_{{\mu }{\nu }}, {\Phi })$. The Lorentzian classical action, $S_\mathrm{class}$, of Eq. (7.3.23), may be split as

$$\begin{aligned} S_\mathrm{class} = S^{(0)}_\mathrm{class} +S^{(2)}_\mathrm{class} +S^{(3)}_\mathrm{class} +\cdots . \end{aligned}$$

(7.3.26)

(The formal device of including a small parameter, ${\epsilon }$, has been relaxed; we now set ${\epsilon } =1$.)

Here, $S^{(0)}_\mathrm{class}$ is the background action, evaluated at the spherically–symmetric solution, $({\gamma }_{{\mu }{\nu }}, {\Phi })$. The mass, $M$, appearing in $S^{(0)}_\mathrm{class}$, is that determined from (either of) ${\gamma }_{ijI}$ or ${\gamma }_{ijF}$. The next term, $S^{(2)}_\mathrm{class}$, is formed quadratically from the linear–order perturbations. The linear–order term, $S^{(1)}_\mathrm{class}$, is zero, since one is perturbing around a classical solution. In an obvious notation:

$$\begin{aligned} S^{(2)}_\mathrm{class} = {{1}\over {32{\pi }}} \biggl (\int _{{\Sigma }_{F}}-\int _{{\Sigma }_{I}}\biggr ) d^{3}x{\pi }^{(1)ij}\beta ^{(1)}_{ij} +{{1}\over {2}} \biggl (\int _{{\Sigma }_{F}}-\int _{{\Sigma }_{I}}\biggr ) d^{3}x{\pi }^{(1)}_{\phi }{\phi }^{(1)}.\quad \end{aligned}$$

(7.3.27)

Thus, $S^{(2)}_\mathrm{class}$ is contructed only from first–order quantities on ${\Sigma }_{I}$ and ${\Sigma }_{F}$. Note that there is no contribution to $S^{(2)}_\mathrm{class}$ from the $-MT$ term in Eq. (7.3.23), since, from the above definitions, both $M$ and $T$ are known (zeroth–order) quantities.

The expression, Eq. (7.3.23), for $S_\mathrm{class}[(h_{ij},{\phi })_{I}; (h_{ij},{\phi })_{F}; T]$, together with the asymptotic series, Eqs. (7.3.11a, 7.3.11b), for the classical action, and Eq. (7.3.27) for $S^{(2)}_\mathrm{class}$, are basic to the calculations below, leading to explicit forms for quantum amplitudes.

Return to the evolution of linearised scalar perturbations, ${\phi }^{(1)}$, via the mode sum of Eq. (7.3.17). For quantum amplitudes of interest, one computes expressions of the form,

$$\begin{aligned} \mathrm{Amplitude} = \mathrm{const.}{\times } \exp \Bigl \{iS_\mathrm{class} \bigl [\bigl (h_{ij},{\phi }\bigr )_{I}; \bigl (h_{ij},{\phi }\bigr )_{F}; T\bigr ]\Bigr \}, \end{aligned}$$

(7.3.28)

As above, take $T$ as in Eq. (7.3.1), to give a strongly–elliptic boundary–value problem. The classical action, $S_\mathrm{class}$, is that of Eq. (7.3.23), subject to the classical field equations.

Consider the perturbative amplitude corresponding to weak–field non–spherical data, $\bigl (h^{(1)}_{ijF}, {\phi }^{(1)}\bigr )_{F}$, on ${\Sigma }_{F}$, given at lowest order by $\exp \bigl (iS^{(2)}_\mathrm{class}\bigr )$. For simplicity, take the initial data to be exactly spherically symmetric, $({\gamma }_{ij}, {\Phi })_{I}$. Equivalently,

$$\begin{aligned} {\beta }_{ijI}^{(1)} = 0, {\quad } {\phi }^{(1)}_{I} = 0. \end{aligned}$$

(7.3.29)

The amplitude, $\exp \bigl (iS_\mathrm{class}\bigr )$, depends only on the contributions at ${\Sigma }_{F}$, in Eq. (7.3.27) [which themselves depend on $({\beta }^{(1)}_{ijF}, {\phi }^{(1)}_{F}; T)$]. As a practical matter, one could put non–zero $({\beta }^{(1)}_{ij}, {\phi }^{(1)})_{I}$ back into the calculations that follow. Physically, the analogous step of ‘turning back on the early–time perturbations’ corresponds, in ‘particle language’ rather than ‘field language’, to the inclusion of extra particles in the in–states, together with the original spherical collapsing matter, and asking for the late–time consequences. This was first carried out by Wald [66].

At present, study the scalar–field contribution to the quantum amplitude, $\exp (iS_\mathrm{class})$: consider

$$\begin{aligned} S^{(2)}_\mathrm{class,scalar} = {{1}\over {2}} \int _{{\Sigma }_{F}}d^{3}x {\pi }^{(1)}_{\phi }{\phi }^{(1)}, \end{aligned}$$

(7.3.30)

where $({\beta }^{(1)}_{{\mu }{\nu }}, {\phi }^{(1)})$ obey the linearised field equations, (7.3.10–7.3.12, 7.3.14), about the spherically–symmetric background, $({\gamma }_{{\mu }{\nu }}, {\Phi })$. Here, $({\beta }^{(1)}_{{\mu }{\nu }}, {\phi }^{(1)})$ must agree with the prescribed final data, $({\beta }^{(1)}_{ij}, {\phi }^{(1)})_{F}$, at ${\Sigma }_{F}$, and be zero at ${\Sigma }_{I}$. In the present case, with complex time–interval, $T$ [Eq. (7.3.1)], one expects the linear boundary–value problem to be well–posed. The other, gravitational, contribution,

$$\begin{aligned} S^{(2)}_\mathrm{class,grav} = {1\over {32{\pi }}} \int _{{\Sigma }_{F}}d^{3}x {\pi }^{(1)ij}{\beta }^{(1)}_{ij}, \end{aligned}$$

(7.3.31)

to $S^{(2)}_\mathrm{class}$ in Eq. (7.3.37), is studied in [36].

Consider late–time high–frequency oscillations of the scalar field in the nearly–Lorentzian case, with ${\theta }= {\delta }{\ll } 1$ in Eq. (7.3.1), and a solution, $R_{\ell m}(t,r)$, of the form

$$\begin{aligned} R_{\ell m}(t,r) \,{\sim }\, \exp (ikt){\xi }_{k\ell m}(t,r), \end{aligned}$$

(7.3.32)

as $r\longrightarrow {\infty }$, where $k$ is a ‘large’ frequency, but ${\xi }_{k{\ell }m}(t,r)$ varies ‘slowly’ with $t$. In particular, require that, as $r\longrightarrow {\infty }$, $R_{\ell m}(t,r)$ reduces to a flat space–time separated solution, in which ${\xi }_{k{\ell }m}(t,r)$ loses its $t$–dependence [Eqs. (7.3.34, 7.3.36) below].

Our boundary–value problem is for scalar perturbations, ${\phi }^{(1)}(t,r,{\theta },{\phi })$, or, equivalently, functions, $R_{{\ell }m}(t,r)$, as in Eqs. (7.3.17, 7.3.19), subject to the initial condition, ${\phi }^{(1)}{\bigl \vert }_{t=0} = 0$, and to prescribed real final data, ${\phi }^{(1)}{\bigl \vert }_{t=T}$. Were the propagation simply in flat space–time, the solution would be of the form:

$$\begin{aligned} {\phi }^{(1)} = {{1}\over {r}} \sum ^{\infty }_{\ell =0} \sum ^{\ell }_{m=-\ell } \int ^{\infty }_{-\infty }dk a_{k{\ell }m} {\xi }_{k{\ell }m}(r){{\sin (kt)}\over {\sin (kT)}} Y_{{\ell }m}(\Omega ), \end{aligned}$$

(7.3.33)

where the $\{a_{k{\ell }m}\}$ are real coefficients and each function, ${\xi }_{k{\ell }m}(r)$, is proportional (up to a factor of $r$) to a spherical Bessel function, $j_{\ell }(kr)$ [67]. In our gravitational–collapse case, ${\xi }_{k{\ell }m}$ becomes a function of $t$ as well as $r$, but the pattern remains:

$$\begin{aligned} {\phi }^{(1)} = {{1}\over {r}} \sum ^{\infty }_{{\ell }=0} \sum ^{\ell }_{m=-{\ell }} \int ^{\infty }_{-\infty }dka_{k{\ell }m} {\xi }_{k{\ell }m}(t,r){{\sin (kt)}\over {\sin (kT)}} Y_{{\ell }m}({\Omega }). \end{aligned}$$

(7.3.34)

The $\{a_{k{\ell }m}\}$ characterise the final data: they can be constructed from the given ${\phi }^{(1)}{\bigl \vert }_{t=T}$ by inverting Eq. (7.3.34). The functions, ${\xi }_{k{\ell }m}(t,r)$, are defined in the adiabatic or large–${\mid }k{\mid }$ limit, as in the previous paragraph, via Eq. (7.3.32), where $R_{{\ell }m}(t,r)$ obeys the mode equation, (7.3.19).

More precisely, given that $k$ is large, in that the adiabatic approximation,

$$\begin{aligned} {\bigl \vert }k{\bigl \vert } \gg {{1}\over {2}}{\bigl \vert }{\dot{a}}-{\dot{b}}{\bigl \vert }, \end{aligned}$$

(7.3.35a)

$$\begin{aligned} {\bigl \vert }k{\bigl \vert } \gg {\biggl \arrowvert } {{{\dot{\xi }}_{k{\ell }m}}\over {{\xi }_{k{\ell }m}}} {\biggl \arrowvert }, \end{aligned}$$

(7.3.35b)

$$\begin{aligned} k^{2} \gg {\biggl \arrowvert } {{{\ddot{\xi }}_{k{\ell }m}}\over {{\xi }_{k{\ell }m}}} {\biggl \arrowvert }, \end{aligned}$$

(7.3.35c)

holds, the mode equation reduces approximately to

$$\begin{aligned} e^{(b-a)/2}{{\partial }\over {\partial r}} \Bigl (e^{(b-a)/2}{{\partial {\xi }_{k{\ell }m}}\over {\partial {r}}}\Bigr ) +\bigl (k^{2}-V_{\ell }\bigr ){\xi }_{k{\ell }m} = 0 . \end{aligned}$$

(7.3.36)

Of course, the functions, $e^{(b-a)/2}$ and $V_{\ell }$, do still vary with the time–coordinate, $t$, but only adiabatically or ‘slowly’.

As described in [7, 22, 44], one expects the geometry in the radiative region of the space–time to be approximated accurately by a spherically–symmetric Vaidya metric [44, 45], with a luminosity in the radiated particles which varies slowly with time. Such a metric can be put in diagonal form:

$$\begin{aligned} e^{-a} = 1-{{2m(t,r)}\over r}, {\quad }{\quad } e^{b} = \biggl ({{\dot{m}}\over {f(m)}}\biggr )^{2}e^{-a}, \end{aligned}$$

(7.3.37)

where $m(t,r)$ is a slowly–varying function, with ${\dot{m}} = (\partial m/\partial {t})$. The function, $f(m)$, depends on the details of the radiation. The adiabatic condition is then:

$$\begin{aligned} {\bigl \vert }k{\bigl \vert } \gg {\Bigl \vert }{{\dot{m}}\over {m}}{\Bigl \vert }, \end{aligned}$$

(7.3.38)

provided that $2m(t,r)<r<4m(t,r)$. Then, the rate of change of the metric with time is slow compared to typical radiation frequencies. Equivalently, the time–scale of variations of the background metric, ${\gamma }_{{\mu }{\nu }}$, is much greater than the period of the waves. With frequencies of magnitude, ${\bigl \vert }k{\bigl \vert }{\sim }{m}^{-1}$, dominating the radiation, and with ${\bigl \vert }{\dot{m}}{\bigl \vert }$ of order $m^{-2}$ [5], the adiabatic approximation is equivalent to $m^{2}\gg {1}$, corresponding to the semi–classical approximation.

It is natural to define a generalisation, $r^*$, of the standard Regge–Wheeler co–ordinate, $r^{*}_{s}$, for the Schwarzschild geometry [42, 58], according to

$$\begin{aligned} {{\partial }\over {\partial {r^{*}}}} = e^{(b-a)/2}{{\partial }\over {\partial {r}}}. \end{aligned}$$

(7.3.39)

Under adiabatic conditions, the time–dependence of $r^{*}(t,r)$ is negligibly small, and $r^{*}{\sim }{r}^{*}_{s}$ for large $r$. By definition,

$$\begin{aligned} r^{*}_{s} = r +2M\log \Bigl (\bigl (r/2M\bigr )-1\Bigr ), \end{aligned}$$

(7.3.40)

with $r$ the Schwarzschild radial co–ordinate. In terms of $r^{*}$, the approximate (adiabatic) mode equation, (7.3.36), reads

$$\begin{aligned} {{\partial ^{2}{\xi }_{k{\ell }m}}\over {\partial {r^{*2}}}} +\bigl (k^{2}-V_{\ell }\bigr ){\xi }_{k{\ell }m} = 0. \end{aligned}$$

(7.3.41)

We consider, in more detail, a set of suitable radial functions, $\{{\xi }_{k{\ell }m}(r)\}$, on ${\Sigma }_{F}$. Since the mode equation, (7.3.19), does not depend on the quantum number, $m$, we choose ${\xi }_{k{\ell }m}(r) = {\xi }_{k{\ell }}(r), \forall m$.

We seek a complete set, such that any smooth perturbation field, ${\phi }^{(1)}(T,r,{\theta },{\phi })$, of rapid decay near spatial infinity, when restricted to the final surface, $\{t=T\}$, can be expanded in terms of the ${\xi }_{k{\ell }m}(r)$. The ‘left’ boundary condition on the radial functions, $\{{\xi }_{k{\ell }}(r)\}$, is that of regularity at the origin, $r = 0$:

$$\begin{aligned} {\xi }_{k{\ell }}(0) = 0. \end{aligned}$$

(7.3.42)

The solution to the radial equation, regular near the origin, is:

$$\begin{aligned} {\xi }_{k{\ell }}(r) = r{\phi }_{k{\ell }}(r) \sim \Bigl \{(\mathrm{const.}) \times \bigl [rj_{\ell }(kr)\bigr ]\Bigr \} \end{aligned}$$

$$\begin{aligned} \propto \Bigl \{\bigl (\mathrm{const.}\times (kr)^{{\ell }+1}\bigr ) +O\bigl ((kr)^{{\ell }+3}\bigr )\Bigr \}. \end{aligned}$$

(7.3.43)

Again, $j_{\ell }$ denotes a spherical Bessel function [67]; we have assumed that $m(r){\propto }{r}^{3}$ for small $r$, and neglected $O(r^{2})$ terms. These radial functions are real, for real $k$ and $r$. For $k$ real and positive, the radial functions describe standing waves, which, for mode time–dependence, $e^{{\pm }ikt}$, have equal amounts of ‘ingoing’ and ‘outgoing’ radiation.

For the ‘right’ boundary condition, the potential, $V_{\ell }(r)$, of Eq. (7.3.20), vanishes rapidly as $r\longrightarrow {\infty }$, such that a real solution to Eq. (7.3.41) obeys

$$\begin{aligned} {\xi }_{k{\ell }}(r) {\sim } \Bigl (z_{k{\ell }}\exp \bigl (ikr^{*}_{s}\bigr ) +z^{*}_{k{\ell }}\exp \bigl (-ikr^{*}_{s}\bigr )\Bigr ), \end{aligned}$$

(7.3.44)

as $r\longrightarrow {\infty }$. The $z_{k{\ell }}$ are dimensionless complex coefficients, which can be determined via the differential equation, using regularity at $r = 0$. Note that the radial functions, $\{{\xi }_{k{\ell }}\}$, form a complete set only for $k>0$, as a result of the boundary conditions [7, 10].

This makes it possible to evaluate the contribution, $S^{(2)}_\mathrm{class,scalar}$ [Eq. (7.3.30)], to $S^{(2)}_\mathrm{class}$ [Eq. (7.3.27)] in the classical action, $S_\mathrm{class} = S^{(0)}_\mathrm{class} +S^{(2)}_\mathrm{class} +{\cdots }$, [Eq. (7.3.26)]:

$$\begin{aligned} S^{(2)}_\mathrm{class}\bigl [{\phi }^{(1)};T\bigr ] = {{1}\over {2}} \sum ^{\infty }_{{\ell }=0} \sum ^{\ell }_{m=-{\ell }} \int ^{R_{\infty }}_{0}dre^{(a-b)/2}R_{{\ell }m} \bigl (\partial _{t}R^{*}_{{\ell }m}\bigr )\bigl \arrowvert _{T} . \end{aligned}$$

(7.3.45)

With the adiabatic approximation above, this gives the frequency–space expression [10]:

$$\begin{aligned} S^{(2)}_\mathrm{class}\bigl [\{a_{k{\ell }m}\}; T\bigr ] = {\pi }\sum ^{\infty }_{{\ell }=0} \sum ^{\ell }_{m=-{\ell }} \int ^{\infty }_{0}dkk {\bigl \vert }z_{k\ell }{\bigl \vert }^{2} {\bigl \vert }a_{k{\ell }m}+a_{-k{\ell }m}{\bigl \vert }^{2} \cot (kT), \end{aligned}$$

(7.3.46)

in terms of the final data, $\{a_{k{\ell }m}\}$.

Define

$$\begin{aligned} {\psi }_{{\ell }m}(r) = r{\int }d{\Omega }Y_{{\ell }m}({\Omega }) {\phi }^{(1)}(t,r,{\Omega })\bigl \arrowvert _{t = T}. \end{aligned}$$

(7.3.47)

The inverse of Eq. (7.3.34) can be shown to be [7]:

$$\begin{aligned} a_{k{\ell }m}+a_{-k{\ell }m} = {{1}\over {2{\pi }{\bigl \vert }z_{k{\ell }}{\bigl \vert }^{2}}} \int ^{R_{\infty }}_{0}dre^{(a-b)/2} {\xi }_{k{\ell }}(r){\psi }_{{\ell }m}(r). \end{aligned}$$

(7.3.48)

The perturbative classical scalar action, $S^{(2)}_\mathrm{class}$, of Eq. (7.3.46), was derived subject to the adiabatic approximation and to the requirement, Eq. (7.3.1). In this case, the term, $k\cot (kT)$, in the integrand of Eq. (7.3.46) remains bounded near $k =0$, and one expects to obtain a finite complex–valued action, $S^{(2)}_\mathrm{class}[\{a_{k{\ell }m}\}; T]$, given square–integrable data, ${\phi }^{(1)}$, on ${\Sigma }_{F}$. The dependence of the complex function, $S^{(2)}_\mathrm{class}[\{a_{k{\ell }m}\}; T]$, on the complex variable, $T$, should be analytic in this domain $(0<{\theta }\le {\pi }/2)$. Following Feynman [12, 13], Lorentzian–signature quantum amplitudes are the limit of $\exp (iS_\mathrm{class})$, as ${\theta }\longrightarrow {0}_{+}$.

If, on the other hand, one takes real Lorentzian geometries $({\theta }=0)$, the integral in Eq. (7.3.46) will typically diverge, due to the simple poles on the real–frequency axis at

$$\begin{aligned} k = k_{n} = {{n{\pi }}\over {T}} {\quad } (n=1,2,3,\ldots ). \end{aligned}$$

(7.3.49)

Consider an integral such as Eq. (7.3.46) for $S^{(2)}_\mathrm{class}[\{a_{k{\ell }m}\}; T]$, in the form,

$$\begin{aligned} J = \sum _{{\ell }m} \int ^{\infty }_{0}dkf_{{\ell }m}(k)\cot (kT), \end{aligned}$$

(7.3.50)

$$\begin{aligned} f_{{\ell }m}(k) = {\pi }k{\bigl \vert }z_{k{\ell }}{\bigl \vert }^{2} {\bigl \vert }a_{k{\ell }m}+a_{-k{\ell }m}{\bigl \vert }^{2}. \end{aligned}$$

(7.3.51)

There are infinitely many simple poles of the integrand at $k = k_{n} (n=1,2,{\ldots })$, just above the positive real $k$–axis. One deforms the original contour, $C$, along the positive real $k$–axis into three parts, $C_{\epsilon },C_{R}$, and $C_{\alpha }$, where $0<{\alpha }\ll {1}$. The contour, $C_{\epsilon }$, lies in the lower half–plane, half–encircling each of the simple poles near the positive real $k$–axis, with radius ${\epsilon }$. The curve, $C_{R}$, also in the lower half–plane, is an arc of a circle, ${\mid }k{\mid } = R$, of large radius. The curve, $C_{\alpha }$, is part of the radial line, $\arg (k) = -{\alpha }$.

In studying these integrals, one needs an estimate of the rate of decay of $f_{\ell {m}}(k)$, as ${\mid }k{\mid }\longrightarrow \infty$. On dimensional grounds, one expects:

$$\begin{aligned} {\bigl \vert }f_{{\ell }m}(k){\bigl \vert } \,{\sim }\, \mathrm{const.}{\times }{\bigl \vert }k{\bigl \vert }^{-3}, \end{aligned}$$

(7.3.52)

as ${\mid }k{\mid }\longrightarrow {\infty }$. To see this, re–write the radial equation, (7.3.41), in terms of the operator,

$$\begin{aligned} \mathcal{{L}}_{\ell } = e^{(b-a)/2} {{d}\over {dr}} \Bigl (e^{(b-a)/2}{{d}\over {dr}}\bigl (\bigr )\Bigr ) -V_{\ell }(r), \end{aligned}$$

(7.3.53)

self–adjoint with respect to the relevant inner product [7, 10]. Note that Eq. (7.3.48) can be re–written as:

$$\begin{aligned} a_{k{\ell }m}+a_{-k{\ell }m} = {{-1}\over {2{\pi }k^{2} {\bigl \vert }z_{k{\ell }}{\bigl \vert }^{2}}} \int ^{R_{\infty }}_{0}dr e^{(a-b)/2}{\xi }_{k{\ell }}(r) \mathcal{L}_{\ell }{\psi }_{{\ell }m}(r). \end{aligned}$$

(7.3.54)

We have used the boundary condition, Eq. (7.3.42), and assumed that ${\psi }_{{\ell }m}(r)$ decays at large $r$. The form, Eq. (7.3.54), is an expression of self–adjointness of the radial equation. One finds that ${\psi }_{{\ell }m}(r)$ has dimensions of length and that ${\mid }z_{k\ell }{\mid }^{2}$ is dimensionless [10]. In the limit, $R_{\infty }\longrightarrow {\infty }$, and for large $k$ (hence, a WKB approximation for the radial functions), the integral in Eq. (7.3.54) can only involve the dimensionless frequency, $2Mk$, where $M$ is the mass. This gives the desired behaviour, Eq. (7.3.52), at large ${\mid }k{\mid }$.

One further definition is needed:

$$\begin{array}{c}\hfill {\mathit{\sigma }}_n=\frac{n\mathit{\pi }}{\mid T\mid }(n=1,2,3,\dots ).\end{array}$$

(7.3.55)

Note the difference between the definitions, Eq. (7.3.49) of $k_{n}$ and Eq. (7.3.55) of ${\sigma }_{n}$.

The classical action for massless scalar–field perturbations, with ${\theta } = {\delta }{\ll } 1$ in Eq. (7.3.1), in terms of boundary data on ${\Sigma }_{F}$, is:

$$\begin{aligned} \begin{array}{ll} S^{(2)}_\mathrm{class} \bigl [\{a_{k{\ell }m}\}; {\mid }T{\mid }\bigr ] &{}= \mathrm{realpart} +{{2i{\pi }}\over {{\mid }T{\mid }}} \sum \limits ^{\infty }_{{\ell }=0} \sum \limits ^{\ell }_{m=-{\ell }} \sum \limits ^{\infty }_{n=1} f_{{\ell }m}({\sigma }_{n})\\ &{} = \mathrm{realpart} +{{2i{\pi }^{2}}\over {{\mid }T{\mid }}} \sum \limits _{{\ell }mn} {\sigma }_{n} {\bigl \vert }z_{n{\ell }}{\bigl \vert }^{2} {\bigl \vert }a_{n{\ell }m}+a_{-n{\ell }m}{\bigl \vert }^{2}. \end{array} \end{aligned}$$

(7.3.56)

The real part of $S^{(2)}_\mathrm{class}$ is also calculable.

The main, semi–classical, contribution to the quantum amplitude is $\exp (iS^{(2)}_\mathrm{class} [\{a_{k{\ell }m}\};{\mid }T{\mid }])$. The probability distribution for final configurations involves $\mathrm{Im}(S^{(2)}_\mathrm{class})$; the more probable configurations have $S^{(2)}_\mathrm{class}$ lying only infinitesimally in the upper half–plane. Probable or not, those final configurations, $\{a_{k{\ell }m}\}$, which contribute to the probability distribution must yield finite expressions in the infinite sums over $(n,{\ell })$ in Eq. (7.3.56). There will be a corresponding restriction when the data are instead described in terms of the spatial configurations, $\{{\psi }_{{\ell }m}(r)\}$. Also (see [37]), the complex quantities, $z_{n{\ell }}(a_{n{\ell }m}+a_{-n{\ell }m})$, appearing in Eq. (7.3.56), are related to Bogoliubov transformations between initial and final states, providing a further characterisation of the finiteness of $\mathrm{Im}(S^{(2)}_\mathrm{class})$ in Eq. (7.3.56).

With regard to the sum over ${\ell }$ in Eq. (7.3.56), one expects that a cut–off, ${\ell }_{\mathrm{max}}$, can be provided by the radial equation, (7.3.41). In the region, $(V_{\ell }(r)-k^{2})>0$, one has exponentially–growing radial functions; for $(V_{\ell }(r)-k^{2})<0$, one has oscillatory radial functions. One defines ${\ell }_\mathrm{max}$ by $(V_{{\ell }\mathrm{max}}(r)-k^{2}) = 0$, and restricts attention mainly to oscillatory solutions.

Given initial and final non–zero Dirichlet data labelled by ‘co–ordinates’, $\{a^{(I)}_{k{\ell }m}\}, \{a^{(F)}_{k{\ell }m}\}$, the perturbative classical scalar action, $S^{(2)}_\mathrm{class}$, includes separate terms of the form, Eq. (7.3.56), from $\Sigma _{I},\Sigma _{F}$. But $S^{(2)}_\mathrm{class}$ also includes a cross–term between $a^{(I)}_{k{\ell }m}$ and $a^{(F)}_{k{\ell }m}$, representing the correlation or mixing between initial and final data. The total action is symmetric in $a^{(I)}_{k{\ell }m}$ and $a^{(F)}_{k{\ell }m}$, and the coefficients, $z_{n{\ell }}$, are the same (they are time–independent) up to a phase. For large ${\mid }T{\mid }$, the cross–term becomes negligible, and one has two independent contributions to the classical action, from $\{a^{(I)}_{k{\ell }m}\}$ and $\{a^{(F)}_{k{\ell }m}\}$.

7.3.4 Comments

The work in Sect. 7.3 began as a doctoral dissertation at Cambridge by one of the present authors [7] in 1997–2001. The thesis being approved in summer 2002, the results were published in the joint papers [4, 9, 10, 20, 32–38]. This work used local supersymmetry in an essential way to find quantum amplitudes in explicit form. Unsubstantiated opposition (since 1975) to the idea that one could calculate amplitudes for quantum processes involving gravitational collapse to a black hole continued until our 2004 (and following) publications. In response to our work, the paper [68] appeared in 2005. The author, despite earlier, strongly–held views (1975–2004), finally admitted that, for a particular bosonic field theory in four space–time dimensions, quantum amplitudes did, after all, exist [68]. That argument was based on quantum field theory, though not on local supersymmetry. Further, it applied only to the case of asymptotically–anti–deSitter space–time, but not to asymptotically–flat space–times. Moreover, by working with a Lagrangian that was not locally supersymmetric, it failed to deal with the problem of infinities in field–theoretic quantum gravity.

We have arrived at a quantum amplitude (not just a probability distribution) for such processes, simply by following Feynman’s $+i{\epsilon }$ prescription, applied to the ‘semi–classical’ expression, Eq. (7.2.20), for the quantum amplitude. This, in turn, was derived via Dirac’s canonical–quantisation approach for a locally–supersymmetric Lagrangian, outlined in Sect. 7.2. We studied the classical and quantum–mechanical boundary–value problems, before rotating ${\theta }$ [of Eq. (7.3.1)] back towards zero.

These ideas have also been applied to black–hole evaporation for particles of spin $1$ and $2$ [36], and to the fermionic spin–${1\over 2}$ case [33]. In [33, 36], the form of the complex quantum amplitudes was not derived in the greater detail obtained in [10] for spin–0 amplitudes.

In [20, 37], we described a relation between the present approach and the familiar description in terms of Bogoliubov coefficients [2, 3, 69]. A more general conceptual framework has been provided in the language of coherent and squeezed states [35, 38].