Semiclassical Amplitude Research Articles

A new 2+1 coherent spin-foam vertex for quantum gravity

Abstract We propose an explicit spin-foam amplitude for Lorentzian gravity in three dimensions, allowing for both space- and time-like boundaries. The model is based on two main requirements: that it should be structurally similar to its well-known Euclidean analog, and that geometricity should be recovered in the semiclassical regime. To this end we introduce new coherent states for space-like boundary edges, derived from the continuous series of unitary SU ( 1 , 1 ) representations. We show that the relevant objects in the amplitude can be written in terms of the defining representation of the group, just as so happens in the Euclidean case. We derive an expression for the semiclassical amplitude at large spins, showing it relates to the Lorentzian Regge action.

Numerical evaluation of spin foam amplitudes beyond simplices

We present the first numerical calculation of the 4D Euclidean spin foam vertex amplitude for vertices with hypercubic combinatorics. Concretely, we compute the amplitude for coherent boundary data peaked on cuboid and frustum shapes. We present the numerical algorithms to explicitly compute the vertex amplitude and compare the results in different cases to the semiclassical approximation of the amplitude. Overall we find good qualitative agreement of the amplitudes and evidence of convergence of the asymptotic formula to the full amplitude already at fairly small spins, yet also differences in the frequency of oscillations and a phase shift absent in the 4-simplex case. However, due to rapidly growing numerical costs, we cannot reach sufficiently high spins to prove agreement of both amplitudes. Lastly, this setup allows us to explore nonuniform vertex amplitudes, where some representations are small while others are large; we find indications that scenarios might exist in which the semiclassical amplitude is a valid approximation even if some spins remain small. This suggests that the transition of the quantum to the semiclassical regime (for a single vertex amplitude) is intricate.

The existence and propagation of dust acoustic waves in quantum four-component plasma

The existence and propagation of dust acoustic waves in quantum four-component plasma are investigated by employing Sagdeev pseudo-potential. The plasma under consideration consists of both quantum electrons and ions beside negative and positive dust particulates. The general determining equation of the Sagdeev pseudo-potential is obtained, and it is found that the equation is very difficult to integrate. Due to the complexity involved in the structure of Sagdeev potential, this equation can only be solved in some limiting cases, such as semi-classical limit and small amplitude approximation. Our findings support the existence of compressive, rarefactive solitary waves and double layers in our plasma system. The influence of the relevant plasma parameters on the existence and profile of the obtained quantum nonlinear waves has been studied. The implications of the obtained results have wide relevance in the study of space and laboratory quantum dusty plasmas situations such as Jupiter’s magnetosphere, upper and lower mesosphere, and comet tails, as well as in condensed matter systems.

Towards many-dimensional real-time quantum theory for heavy-particle dynamics. I. Semiclassics in the Lagrange picture of classical phase flow

We study a quantum theory in terms of action decomposed function (ADF), a class of quantum wave function, towards many-dimensional applications to quantum dynamics of heavy particles as in chemical reactions. The equation of motion for the complex-valued amplitude of ADF represents a coupling between the internal diffusive motion of a wave packet and dynamics of its group velocity in a hierarchical manner ascending from classical to purely quantum mechanics via semiclassical dynamics. We attempt to solve this equation of motion dividing it into two stages: a semiclassical level and beyond. In this paper, as the first stage, we develop a semiclassical approximation in the Lagrange picture of classical phase flow. In the Euler picture (as in the standard WKB picture), continuous integration of the stability matrix along the paths is required. By adopting the Lagrange picture, on the other hand, we represent the semiclassical amplitude in terms of what we call deviation determinant, which can be evaluated readily in many-dimensional systems. Numerical tests show that ADF reproduces quantum wave packets at each space-time point along classical paths very well. However, the ADF in this stage is not free of the semiclassical singularity. In other words, the wave functions diverge at turning points or caustics, depending on the initial conditions chosen. This divergence is known to take place at points where classical paths smoothly distributed in phase space have ``focuses'' in configuration space (or momentum space) and reflects an intrinsic relationship between quantum and classical mechanics. Therefore, it is by studying the mechanism of removing the singularity that the essential feature of quantum mechanics will be clarified. This aspect will be discussed in a companion paper [K. Takatsuka and S. Takahashi, Phys. Rev. A 89, 012109 (2014)] as the second stage of many-body quantum theory.

Low-dimensional nanostructures and a semiclassical approach for teaching Feynman's sum-over-paths quantum theory

An introduction to quantum mechanics based on the sum-over-paths (SOP) method originated by Richard P Feynman and developed by E F Taylor and coworkers is presented. The Einstein–Brillouin–Keller (EBK) semiclassical quantization rules are obtained following the SOP approach for bounded systems, and a general approach to the calculation of propagation amplitude is discussed for unbounded systems. These semiclassical results are obtained when the SOP is limited to the trajectories classically allowed. EBK semiclassical quantization and the topological Maslov index are used to deduce the correct quantum mechanical results for systems which live in a two-dimensional world as quantum dots and quantum rings. In the latter systems, the semiclassical propagation amplitude is used to discuss the Aharonov–Bohm effect. The development involves only elementary calculus and also provides a theoretical introduction to the quantum nature of low-dimensional nanostructures.

Applications of the tunneling method to particle decay and radiation from naked singularities

Following recent literature on dS instability in presence of interactions, we study the decay of massive particles in general FRW models and the emission from naked singularities either associated with 4D charged black holes or with 2D shock waves, by means of the Hamilton--Jacobi tunneling method. It is shown that the two-dimensional semi-classical tunneling amplitude from a naked singularity computed in that way is the same as the one-loop result of quantum field theory.

Phase quantization of chaos in the semiclassical regime

Since the early stage of the study of Hamilton chaos, semiclassical quantization based on the low-order Wentzel-Kramers-Brillouin theory, the primitive semiclassical approximation to the Feynman path integrals (or the so-called Van Vleck propagator), and their variants have been suffering from difficulties such as divergence in the correlation function, nonconvergence in the trace formula, and so on. These difficulties have been hampering the progress of quantum chaos, and it is widely recognized that the essential drawback of these semiclassical theories commonly originates from the erroneous feature of the amplitude factors in their applications to classically chaotic systems. This forms a clear contrast to the success of the Einstein-Brillouin-Keller quantization condition for regular (integrable) systems. We show here that energy quantization of chaos in semiclassical regime is, in principle, possible in terms of constructive and destructive interference of phases alone, and the role of the semiclassical amplitude factor is indeed negligibly small, as long as it is not highly oscillatory. To do so, we first sketch the mechanism of semiclassical quantization of energy spectrum with the Fourier analysis of phase interference in a time correlation function, from which the amplitude factor is practically factored out due to its slowly varying nature. In this argument there is no distinction between integrability and nonintegrability of classical dynamics. Then we present numerical evidence that chaos can be indeed quantized by means of amplitude-free quasicorrelation functions and Heller's frozen Gaussian method. This is called phase quantization. Finally, we revisit the work of Yamashita and Takatsuka [Prog. Theor. Phys. Suppl. 161, 56 (2007)] who have shown explicitly that the semiclassical spectrum is quite insensitive to smooth modification (rescaling) of the amplitude factor. At the same time, we note that the phase quantization naturally breaks down when the oscillatory nature of the amplitude factor is comparable to that of the phases. Such a case generally appears when the Planck constant of a large magnitude pushes the dynamics out of the semiclassical regime.

Phase Quantization of Chaos and the Role of the Semiclassical Amplitude Factor

Phase Quantization of Chaos and the Role of the Semiclassical Amplitude Factor

Quantum amplitudes in black-hole evaporation: Spins 1 and 2

Quantum amplitudes for s = 1 Maxwell fields and for s = 2 linearised gravitational-wave perturbations of a spherically symmetric Einstein/massless scalar background, describing gravitational collapse to a black hole, are treated by analogy with the previous treatment of s = 0 scalar-field perturbations of gravitational collapse at late times. Both the spin-1 and the spin-2 perturbations split into parts with odd and even parity. Their detailed angular behaviour is analysed, as well as their behaviour under infinitesimal coordinate transformations and their linearised field equations. In general, we work in the Regge–Wheeler gauge, except that, at a certain point, it becomes necessary to make a gauge transformation to an asymptotically flat gauge, such that the metric perturbations have the expected fall-off behaviour at large radii. In both the s = 1 and s = 2 cases, we isolate suitable ‘coordinate’ variables which can be taken as boundary data on a final space-like hypersurface Σ F . (For simplicity of exposition, we take the data on the initial surface Σ I to be exactly spherically symmetric.) The (large) Lorentzian proper-time interval between Σ I and Σ F , measured at spatial infinity, is denoted by T. We then consider the classical boundary-value problem and calculate the second-variation classical Lorentzian action S class ( 2 ) , on the assumption that the time interval T has been rotated into the complex: T → | T| exp (−i θ), for 0 < θ ⩽ π/2. This complexified classical boundary-value problem is expected to be well-posed, in contrast to the boundary-value problem in the Lorentzian-signature case ( θ = 0), which is badly posed, since it refers to hyperbolic or wave-like field equations. Following Feynman, we recover the Lorentzian quantum amplitude by taking the limit as θ → 0 + of the semi-classical amplitude exp ( i S class ( 2 ) ) . The boundary data for s = 1 involve the (Maxwell) magnetic field, while the data for s = 2 involve the magnetic part of the Weyl curvature tensor. These relations are also investigated, using 2-component spinor language, in terms of the Maxwell field strength ϕ AB = ϕ ( AB) and the Weyl spinor Ψ ABCD = Ψ ( ABCD) . The magnetic boundary conditions are related to each other and to the natural s = 1 2 boundary conditions by supersymmetry.

Structure of the Semi-Classical Amplitude for General Scattering Relations

ABSTRACT We consider scattering by general compactly supported semi-classical perturbations of the Euclidean Laplace-Beltrami operator. We show that if the suitably cut-off resolvent quantizes a Lagrangian relation on the product cotangent bundle, the scattering amplitude quantizes the natural scattering relation. In the case when the resolvent is tempered, which is true at non-trapping energies or at trapping energies under some non-resonance assumptions, and when we work microlocally near a non-trapped ray, our result implies that the scattering amplitude defines a semiclassical Fourier integral operator associated to the scattering relation in a neighborhood of that ray. Compared to previous work, we allow this relation to have more general geometric structure.

Comparing strings inAdS5×S5to planar diagrams: An example

The correlator of a Wilson loop with a local operator in N=4 SYM theory can be represented by a string amplitude in AdS(5)xS(5). This amplitude describes an overlap of the boundary state, which is associated with the loop, with the string mode, which is dual to the local operator. For chiral primary operators with a large R charge, the amplitude can be calculated by semiclassical techniques. We compare the semiclassical string amplitude to the SYM perturbation theory and find an exact agrement to the first two non-vanishing orders.

Dynamics of black hole formation in an exactly solvable model

We consider the process of black hole formation in particle collisions in the exactly solvable framework of 2+1 dimensional Anti de Sitter gravity. An effective Hamiltonian describing the near horizon dynamics of a head on collision is given. The Hamiltonian exhibits a universal structure, with a formation of a horizon at a critical distance. Based on it we evaluate the action for the process and discuss the semiclassical amplitude for black hole formation. The derived amplitude is seen to contain no exponential suppression or enhancement. Coments on the CFT description of the process are made.

Non-oscillating solutions to uncoupled Ermakov systems and the semi-classical limit

The amplitude-phase formulation of the Schr\"{o}dinger equation is investigated within the context of uncoupled Ermakov systems, whereby the amplitude function is given by the auxiliary nonlinear equation. The classical limit of the amplitude and phase functions is analyzed by setting up a semiclassical Ermakov system. In this limit, it is shown that classical quantities, such as the classical probability amplitude and the reduced action, are obtained only when the semiclassical amplitude and the accumulated phase are non-oscillating functions respectively of the space and energy variables. Conversely, among the infinitely many arbitrary exact quantum amplitude and phase functions corresponding to a given wavefunction, only the non-oscillating ones yield classical quantities in the limit $\hbar \to 0$.

Impact parameter dependence of projectile-electron excitation and loss in relativistic collisions with atomic targets

The semiclassical approximation is extended to relativistic collisions of two composite atomic particles. Electron excitation and loss of relativistic ion-projectiles colliding with atomic targets are studied. A general expression is derived for the semiclassical transition amplitude. In the limit of nonrelativistic collision velocities this expression goes over into the known nonrelativistic formula. The main focus of this study is on the projectile-electron excitation, for which numerical calculations have been performed. Our calculations for ultrarelativistic collisions between a heavy single-electron projectile and light atomic targets show the importance of the shielding effects, which reduce transition probabilities at larger impact parameters, and a considerable contribution from the antiscreening mode in collisions with few-electron atoms. Further, the antiscreening probabilities are shown to extend to much larger impact parameters than do the screening probabilities. An enhancement in probabilities for the dipole-allowed transitions and a reduction of the probabilities for transitions to the 2s state are found compared with relativistic collisions at low . For electronic cross sections the semiclassical treatment is shown to be equivalent to the plane-wave Born approximation.

Gauge-invariant theory for semiclassical magnetotransport through ballistic microstructures

Within the semiclassical theory of magnetotransport through ballistic cavities, fluctuations in the transmission amplitude and in the conductance originate from the Aharonov-Bohm phase of directed areas. We formulate the semiclassical transmission amplitude in gauge-invariant form. The gauge invariant phases can be visualized in terms of areas enclosed by classical paths, which consist of the real path connecting the entrance point to the exit point and a virtual path leading back to the entrance point. We implement this method on different levels of a semiclassical description of magnetotransport with applications to magnetoconductance fluctuations and correlations. The validity of the semiclassical theories is analyzed. {copyright} {ital 1999} {ital The American Physical Society}

Generalized sums over histories for quantum gravity (I). Smooth conifolds

Quantum amplitudes for euclidean gravity constructed by sums over compact manifold histories are a natural arena for the study of topological effects. Such euclidean functional integrals in four dimensions include histories for all boundary topologies. However, a semiclassical evaluation of the integral will yield a semiclassical amplitude for only a small set of these boundaries. Moreover, there are sequences of manifold histories in the space of histories that approach a stationary point of the Einstein action but do not yield a semiclassical amplitude; this occurs because the stationary point is not a compact Einstein manifold. Thus the restriction to manifold histories in the euclidean functional integral eliminates semiclassical amplitudes for certain boundaries even though there is a stationary point for that boundary. In order to incorporate the contributions from such semiclassical histories, this paper proposes to generalize the histories included in euclidean functional integrals a more general set of compact topological spaces. This new set of spaces, called conifolds, includes the nonmanifold stationary points; additionally, it can be proven that sequences of approximately Einstein manifolds and sequences of approximately Einstein conifolds both converge to Einstein conifolds. Consequently, generalized euclidean functional integrals based on these conifold histories yield semiclassical amplitudes for sequences of both manifold and conifold histories that approach a stationary point of the Einstein action. Therefore sums over conifold histories provide a useful and self-consistent starting point for further study of topological effects in quantum gravity.

Semiclassical inelastic scattering with path integrals

An alternative derivation of the Sukumar and Brink formula for the corrections to the primitive semiclassical amplitude for multiple inelastic scattering is presented. The meaning of the correction terms in the context of the semiclassical description of the relative motion is also discussed.

Correlations in α-α scattering and semi-classical optical models

We show the equivalence of semi-classical solutions to optical model coupled-channel equations derived from Watson's form of the nucleus-nucleus multiple-scattering series to the Glauber multiple-scattering series. A second-order solution to the semi-classical coupled-channel elastic amplitude is shown to be nearly equivalent to a second-order optical-phase-shift approximation to the Glauber amplitude if the densities of all nuclear excited states are approximated by the ground-state density. Using the Jastrow method to model the two-body density we find an average excited-state density to be of negligible importance in the double-scattering region of α-α scattering.

Nucleon transfer in heavy-ion reactions: energy dependence of the cross section

The authors derive an analytical formula for the transfer cross section in a heavy-ion reaction at high incident energy (>or=20 MeV/u). A semiclassical transfer amplitude and a sharp cut-off approximation for the absorption from the elastic channel are used. Two reactions are studied and the results compared with finite-range DWBA calculations.

K-shell ionization probability in elastic proton scattering on 138Ba through an f7/2 isobaric-analog resonance.

The bombarding energy dependence of the K-shell ionization probability P/sub K/ was measured in elastic proton scattering on /sup 138/Ba at energies near the 69-keV-wide f/sub 7/2/ isobaric-analog resonance at 9.965 MeV. The best simultaneous fit to P/sub K/ in the scattering angle intervals 45/sup 0/--80/sup 0/ and 100/sup 0/--135/sup 0/ yields R/sub 0/ = -0.30 +- 0.30 for the tangent of the phase angle of the semiclassical monopole ionization amplitude.