Hamilton–Jacobi Tunneling Formalism

• Hamilton–Jacobi tunneling formalism is a covariant, semiclassical framework that calculates quantum tunneling amplitudes by extracting the imaginary part of the classical action along forbidden trajectories.
• The approach integrates the full Liouville one-form and employs analytic continuation to ensure coordinate invariance and internal consistency even in rotating black hole spacetimes.
• It extends to spinning particles by reducing Dirac and Rarita–Schwinger equations to the eikonal form, thereby unifying and generalizing earlier tunneling methods such as the Parikh–Wilczek approach.

The Hamilton–Jacobi tunneling formalism provides a covariant, semiclassical framework for computing quantum tunneling amplitudes in curved spacetimes—most notably the emission of particles from black hole horizons, commonly associated with Hawking radiation. The key insight is that the imaginary part of the classical action, evaluated along a classically forbidden trajectory that crosses the event horizon, encodes the leading exponential contribution to the tunneling probability. This formalism achieves internal consistency, coordinate covariance, and extensibility to spinning particles by means of analytic continuation in complexified spacetimes and by integrating invariants associated with the Liouville one-form. It generalizes and subsumes earlier approaches based strictly on null-geodesic or radial momentum integrals and establishes equivalence with the Parikh–Wilczek method when properly covariantized.

1. Core Principles and Calculation Schema

The foundation of the Hamilton–Jacobi tunneling formalism is the computation of the classical action $I$ for a particle propagating in the black hole background, obtained by solving the Hamilton–Jacobi equation. For a stationary black hole such as Schwarzschild,

$$ds^2 = -f(r) dt^2 + f(r)^{-1} dr^2 + r^2 d\Omega^2, \quad f(r) = 1 - \frac{2m}{r},$$

the reduced Hamilton–Jacobi equation for a massless, spinless particle with energy $E$ and angular momentum $J$ yields the solution: $I = -Et + \int \frac{E r}{r-2m} dr + J \varphi.$ This integral develops a pole at the event horizon ($r=2m$). The emission probability $P_\text{em}$ is then given by

$P_\text{em} \propto \exp(-2\, \mathrm{Im}\, I).$

Thus, the problem reduces to extracting the imaginary part of the action as the tunneling trajectory traverses the horizon.

2. Covariance, the Feynman Prescription, and Internal Consistency

The appearance of the pole at the horizon raises the issue of consistency and coordinate invariance. The procedure relies on a specific contour integration dictated by the Feynman $i\epsilon$ prescription: $r-2m \to r-2m - i\epsilon,$ so that for the pole,

$$\frac{1}{r-2m-i\epsilon} = \mathcal{P}\left(\frac{1}{r-2m}\right) + i \pi \delta(r-2m).$$

When integrating along the tunneling trajectory, this yields

$\mathrm{Im}\, I = \frac{\pi E}{2\kappa},$

where the surface gravity $\kappa = (4m)^{-1}$ sets the scale. This result holds in both Schwarzschild and Kerr spacetimes, with the relevant invariant combinations $(E - \Omega J)$ for rotating black holes, where $\Omega$ is the angular velocity at the horizon.

Coordinate invariance is guaranteed by working in horizon-regular coordinates (Eddington–Finkelstein, Painlevé–Gullstrand, etc.), and—crucially—by integrating the full Liouville one-form $p_\mu dx^\mu$ rather than coordinate-specific momentum components. The final result, namely

$$2\, \mathrm{Im}(I_{\rm out}) - 2\, \mathrm{Im}(I_{\rm in}) = \frac{2\pi E}{\kappa},$$

is manifestly coordinate invariant.

3. Analytic Continuation and the Role of Complexified Spacetimes

To justify the Feynman prescription covariantly and to uniquely specify the imaginary part, the formalism employs analytic continuation in complexified Kruskal (or analogous) coordinates. Specifically, the ($U, V$) Kruskal coordinates are complexified as

$$\tilde V = V e^{-i\lambda},\quad \tilde U = U e^{i\lambda},\quad 0 \leq \lambda \leq \pi.$$

This rotation is equivalent to a Wick rotation in Schwarzschild time, $t \to t - i \lambda/\kappa$. The differential of the action becomes

$$dI = \partial_U I \, dU + \partial_V I \, dV + (U\partial_U I - V\partial_V I) i d\lambda.$$

Using the horizon generator relation,

$$-U \frac{\partial I}{\partial U} + V \frac{\partial I}{\partial V} = \frac{1}{\kappa} \frac{\partial I}{\partial t} = -\frac{E}{\kappa},$$

the analytic continuation yields

$\mathrm{Im}\, I = \frac{\pi E}{\kappa}$

once both ingoing and outgoing trajectories are taken into account. Analyticity ensures that the prescription is unique and consistent with the causality structure of the semiclassical process.

4. Extension to Spinning Particles and Fermions

The Hamilton–Jacobi equation derived from both the Dirac and Rarita–Schwinger equations (for spin-$1/2$ and spin-$3/2$ particles, respectively) in curved spacetime reduces in the semiclassical limit to the same eikonal equation: $$g^{\mu\nu} \partial_\mu S \partial_\nu S + m^2 = 0.$$ Consequently, the tunneling formalism and its prescription for extracting the imaginary part of the action apply unchanged to fermionic and bosonic particles. The internal structure of the phase, governed by the semiclassical limit, guarantees that the pole at the horizon and the analytic continuation technique remain valid regardless of particle spin.

5. Generalization of the Parikh–Wilczek Method

Traditionally, the Parikh–Wilczek tunneling method considers

$\mathrm{Im} \int p_r dr$

as the key object, where $p_r$ is the radial momentum. However, this is not fully covariant. The generalized formalism advocated analyzes the imaginary part of the full Liouville one-form,

$\omega = p_\mu dx^\mu,$

which, when analytically continued in complexified coordinates, yields

$$\mathrm{Im}\int \omega_{\rm out} - \mathrm{Im}\int \omega_{\rm in} = \frac{\pi E}{\kappa},$$

in direct correspondence with the Hamilton–Jacobi result. Thus, the null-geodesic (Parikh–Wilczek) and HJ tunneling approaches are equivalent upon proper covariantization, linking the coordinate-invariant structure of the tunneling rate to the underlying geometry.

Approach	Action Expression	Covariant?	Imaginary Part at Horizon
Original HJ	$I = -Et + \int p_r dr + J\varphi$	Only with care	$\,\mathrm{Im}\,I = \frac{\pi E}{2\kappa}$
Parikh–Wilczek radial	$\mathrm{Im} \int p_r dr$	No	Matching only after proper extension
Generalized HJ (Liouville)	$\int p_\mu dx^\mu$	Yes	$\,\mathrm{Im}\,I = \frac{\pi E}{\kappa}$

6. Covariance, Gauge Invariance, and Observational Consequences

The formalism's reliance on invariants—Liouville one-form, analytic continuation, and surface gravity—establishes the covariance and gauge invariance of the final result. For the Schwarzschild case,

$T_H = \frac{\kappa}{2\pi} = \frac{1}{8\pi m}$

follows directly from the emission probability, with the same form emerging for Kerr, Kerr–Newman, and dynamical spherically symmetric backgrounds provided the correct invariant combinations are used.

The analysis clarifies ambiguities present in earlier methods, particularly regarding the selection of contour and the possible dependence on coordinate choices or gauge fixing near the horizon.

7. Extensions and Broader Significance

By demanding analyticity in complexified spacetime and making use of covariant objects, the Hamilton–Jacobi tunneling formalism accommodates backgrounds with rotation, charge, and arbitrary spin. It is necessary neither to solve the full HJ PDE explicitly nor to adopt a coordinate-dependent gauge-fixing. The method thus permits efficient calculation of the leading semiclassical contribution to black hole emission, supports the universality of the Hawking temperature, and provides a platform for generalizations to time-dependent or higher-spin backgrounds.

The approach cements the direct correspondence between tunneling rates computed via the imaginary part of the action and the thermal spectrum of black hole radiation, affirming and extending the link between horizon geometry and quantum field theoretic emission properties.

• S. A. Hayward, R. Di Criscienzo, L. Vanzo, M. Nadalini, and G. Zerbini, "On tunneling across horizons" (Vanzo, 2011).