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Semiclassical theory and the Koopman-van Hove equation *
Abstract The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton–Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton–Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys–Wentzel–Kramers–Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein–Brillouin–Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.
Semiclassical dynamics of Hawking radiation
Here, we consider gravity in 3+1 spacetime dimensions coupled to N scalar matter fields in a semiclassical limit where $N\to\infty$. The dynamical evolution of a black hole including the back-reaction of the Hawking radiation on the metric is formulated as an initial-value problem. The quantum stress-energy tensor is evaluated using a point-splitting regularization along spacelike geodesics. To account for the quantum entanglement of the matter fields, they are treated as a set of bilocal collective fields defined on spacelike hypersurfaces. The resulting semiclassical field equations include terms up to fourth order in derivatives that can be treated in a perturbative $\hbar$ expansion. The formulation we arrive at should be amenable to numerical simulation of time-dependent semiclassical spacetime.
Quarkonium semiclassical transport in quark-gluon plasma: factorization and quantum correction
We study quarkonium transport in the quark-gluon plasma by using the potential nonrelativistic QCD (pNRQCD) effective field theory and the framework of open quantum systems. We argue that the coupling between quarkonium and the thermal bath is weak using separation of scales, so the initial density matrix of the total system factorizes and the time evolution of the subsystem is Markovian. We derive the semiclassical Boltzmann equation for quarkonium by applying a Wigner transform to the Lindblad equation and carrying out a semiclassical expansion. We resum relevant interactions to all orders in the coupling constant at leading power of the nonrelativistic and multipole expansions. The derivation is valid for both weakly coupled and strongly coupled quark-gluon plasmas. We find reaction rates in the transport equation factorize into a quarkonium dipole transition function and a chromoelectric gluon distribution function. For the differential reaction rate, the definition of the momentum dependent chromoelectric gluon distribution function involves staple-shaped Wilson lines. For the inclusive reaction rate, the Wilson lines collapse into a straight line along the real time axis and the distribution becomes momentum independent. The relation between the two Wilson lines is analogous to the relation between the Wilson lines appearing in the gluon parton distribution function (PDF) and the gluon transverse momentum dependent parton distribution function (TMDPDF). The centrality dependence of the quarkonium nuclear modification factor measured by experiments probes the momentum independent distribution while the transverse momentum dependence and measurements of the azimuthal angular anisotropy may be able to probe the momentum dependent one. We discuss one way to indirectly constrain the quarkonium in-medium real potential by using the factorization formula and lattice calculations. The leading quantum correction to the semiclassical transport equation of quarkonium is also worked out. The study can be easily generalized to quarkonium transport in cold nuclear matter, which is relevant for quarkonium production in eA collisions in the future Electron-Ion Collider.
A principle of maximum ignorance for semiclassical gravity
The principle of maximum ignorance posits that the coarse-grained description of a system is maximally agnostic about its underlying microscopic structure. We briefly review this principle for random matrix theory and for the eigenstate thermalization hypothesis. We then apply this principle in holography to construct ensembles of random mixed states. This leads to an ensemble of microstates which models our microscopic ignorance, and which on average reproduces the effective semiclassical physics of a given bulk state. We call this ensemble the state-averaging ansatz. The output of our model is a prediction for semiclassical contributions to variances and higher statistical moments over the ensemble of microstates. The statistical moments provide coarse-grained — yet gravitationally non-perturbative — information about the microstructure of the individual states of the ensemble. We show that these contributions exactly match the on-shell action of known wormhole configurations of the gravitational path integral. These results strengthen the view that wormholes simply parametrize the ignorance of the microstructure of a fundamental state, given a fixed semiclassical bulk description.
On the connection between perturbation theory and new semiclassical expansion in quantum mechanics
Here, it is shown that for the one-dimensional anharmonic oscillator with potential V(x) = ax 2 + bgx 3 + ... = $\frac{1}{g^2}$ $\hat{V}$ (gx), as well as for the radial oscillator V(r) = $\frac{1}{g^2}$ $\hat{V}$ (gr) and for the perturbed Coulomb problem V(r) = $\frac{a}{r}$ + βgr + ... = g $\tilde{V}$ (gr), the Perturbation Theory in powers of the coupling constant g (weak coupling regime) and the semiclassical expansion in powers of $\hbar$ 1/2 for the energies coincide. This is related to the fact that the dynamics developed in two spaces: x (r)-space and gx (gr)-space, lead to the same energy spectra. The equations which govern dynamics in these two spaces, the Riccati-Bloch equation and the Generalized Bloch equation, respectively, are presented. It is shown that the perturbation theory for the logarithmic derivative of the wavefunction in - space leads to (true) semiclassical expansion in powers of $\hbar$ 1/2 ; for the one-dimensional case this corresponds to the flucton calculus for the density matrix in the path integral formalism in Euclidean (imaginary) time proposed by one of the authors. Matching the perturbation theory in powers of g and the semiclassical expansion in powers of $\hbar$ 1/2 for the wavefunction leads to a highly accurate local approximation in the entire coordinate space, its expectation value for the Hamiltonian provides a prescription for the summation of the perturbative (trans)-series.
Generalized quantum master equations can improve the accuracy of semiclassical predictions of multitime correlation functions
Multitime quantum correlation functions are central objects in physical science, offering a direct link between the experimental observables and the dynamics of an underlying model. While experiments such as 2D spectroscopy and quantum control can now measure such quantities, the accurate simulation of such responses remains computationally expensive and sometimes impossible, depending on the system’s complexity. A natural tool to employ is the generalized quantum master equation (GQME), which can offer computational savings by extending reference dynamics at a comparatively trivial cost. However, dynamical methods that can tackle chemical systems with atomistic resolution, such as those in the semiclassical hierarchy, often suffer from poor accuracy, limiting the credence one might lend to their results. By combining work on the accuracy-boosting formulation of semiclassical memory kernels with recent work on the multitime GQME, here we show for the first time that one can exploit a multitime semiclassical GQME to dramatically improve both the accuracy of coarse mean-field Ehrenfest dynamics and obtain orders of magnitude efficiency gains.
Semiclassical treatment of bottomonium suppression and regeneration in 𝑝 + Pb collisions
Here, we study bottomonium suppression in 𝑝 + Pb relative to 𝑝 + 𝑝 collisions at center-of-mass energies of $\sqrt{s_{NN}}$ = 5.02 and 8.16 TeV. Specifically, we combine cold nuclear matter effects (nuclear modifications of the parton densities, energy loss, and momentum broadening) with those from hot nuclear matter (suppression and regeneration) by implementing the formation of a quark-gluon plasma in hydrodynamic simulations. Bottomonium transport in the quark-gluon plasma is evaluated semiclassically, employing two different reaction rates. The first includes quasifree inelastic scattering and gluodissociation employing a perturbative coupling to the medium. The second is based on in-medium 𝑇-matrix calculations where the input potential is constrained by lattice quantum chromodynamics to extract the bottomonium masses and dissociation rates. These semiclassical results are compared to previous calculations in an open quantum system approach and to the experimental data. Predictions for 𝜒 𝑏 suppression at $\sqrt{s_{NN}}$ = 8.16 TeV are also presented.
Semiclassical Wormholes toward Typical Entangled States
What do the typical entangled states of two black holes look like? Do they contain semiclassical interiors? We approach these questions constructively, providing ensembles of states that densely explore the black hole Hilbert space. The states contain very long Einstein-Rosen caterpillars : semiclassical wormholes with large numbers of matter inhomogeneities. Distinguishing these ensembles from the typical entangled states of the black holes is hard. We quantify this by deriving the correspondence between a microscopic notion of quantum randomness and the geometric length of the wormhole. This formalizes a “complexity = geometry” relation.
Linear magnetoresistance induced by intra-scattering semiclassics of Bloch electrons
Weak-field magnetoresistance has seen a revived interest due to the distinct role played by the momentum-space Berry curvature of Bloch electrons. While most previous studies in this regard focus on the inter-scattering motion of semiclassical Bloch electrons in electromagnetic fields, the intra-scattering effects of the semiclassical dynamics augmented by the Berry curvature, magnetic moment, and shift vector on the magnetoresistance have been largely overlooked. Here, we uncover that these intra-scattering effects, which are neglected in the field-independent relaxation time approximation to the Boltzmann collision integral, can be as important as the inter-scattering ones. In this work, concrete calculations on the two-dimensional gapped Dirac model show that the sign of the negative linear magnetoresistance given by the Berry curvature alone is reversed when one considers the magnetic moment and shift vector.
Do holographic CFT states have unique semiclassical bulk duals?
Here, we discuss a situation where a holographic CFT state has multiple semiclassical bulk duals. In our example, a given holographic state has two simple, semiclassical descriptions, one with a closed universe, constructed using the gravitational path integral, and one without a closed universe, constructed using the extrapolate dictionary. This highlights an ambiguity in the AdS/CFT dictionary. We propose various options for resolving this tension although none is perfectly satisfactory. We also discuss what this implies for the black hole interior and the gravitational path integral.
Restricted Partition Function Semiclassical Transition State Theory (RPF-SCTST): Applications to Reactions Involving Hydrogen Cyanide, Hydrogen Peroxide, and Formaldehyde Oxide
We demonstrate the efficiency of the numerical calculation of thermal semiclassical transition state theory (SCTST) rates across several representative chemical reactions using the restricted partition function (RPF)─viz a function that depends only on the imaginary action associated with a reaction. Here, we treat the potential energy surfaces (PESs) through fourth order expansions around the saddle point and integrate the Hamiltonian up to second order in employing vibrational perturbation theory. We apply this formalism to uni- and bimolecular reactions with different─though relatively high─barrier heights to uncover the influence of the barrier properties, minimal energies, and separability of the rotational motion on rate constants and tunneling corrections. Although all modes are coupled within the RPF, we found in our numerical examples that the rotational component in the absence of significant rotational distortions can be separated from the vibrational contributions. Moreover, a classical treatment of the rotational contribution is adequate over the temperature range from approximately 100 to 1000 K. We also found that the choice of DFT basis set can lead to variations in rate constants of up to an order of magnitude. Lastly, different schemes for counting eligible energy levels result in rate constants that differ by no more than a factor of 2, with the remaining discrepancies attributed to the treatment of near-convergent levels in perturbation theory.
Semiclassical transition state theory through the lens of the restricted partition function
The wide adoption of transition state theory in resolving the rates of molecular processes relies on the simplification from reducing the formal and numerical expense of dynamics by a geometric constraint. Such a reduction is at odds with the uncertainty in localization that the uncertainty principle requires. While many forms of semiclassical transition state theory (SCTST) have been aimed at addressing this challenge, a popular approach has relied on resolving the underlying phase space structure of the exact rate formula to leverage Bohr–Sommerfeld quantization. Here, the Hernandez–Miller SCTST reframed the thermal rate formula into an integral of the so-called restricted partition function (RPF) over the action associated with the reaction. The density-of-state SCTST has reframed the rate formula in terms of the instanton’s density of states (DoS). Here, we show the relationship between the RPF-SCTST and the DoS-SCTST and derive the latter from the former. In this way, we help unify these branches of SCTST and provide a clearer formalism for future advances.
Semiclassical shell-structure micro-macroscopic approach for the level density
Level density ρ(E,A) is derived for a one-component nucleon system with a given energy E and particle number A within the mean-field semiclassical periodic-orbit theory beyond the saddle-point method of the Fermi gas model. We obtain ρ∝I ν (S)/S ν , with I ν (S) being the modified Bessel function of the entropy S. Within the micro-macro-canonical approximation (MMA), for a small thermal excitation energy U, with respect to rotational excitations E rot , one obtains ν = 3/2 for ρ(E,A). In the case of excitation energy U larger than E rot but smaller than the neutron separation energy, one finds a larger value of ν = 5/2. A role of the fixed spin variables for rotating nuclei is discussed. The MMA level density ρ reaches the well-known grand-canonical ensemble limit (Fermi gas asymptote) for large S related to large excitation energies, and also reaches the finite micro-canonical limit for small combinatorial entropy S at low excitation energies (the constant “temperature” model). Fitting the ρ(E,A) of the MMA to the experimental data for low excitation energies, taking into account shell and, qualitatively, pairing effects, one obtains for the inverse level density parameter K a value which differs essentially from that parameter derived from data on neutron resonances.
Semiclassical Trajectory Studies of Reactive and Nonreactive Scattering of OH( 2 Σ + ) by H 2 Based on an Improved Full-Dimensional Ab Initio Diabatic Potential Energy Matrix
Here we present a new full-dimensional diabatic potential energy matrix (DPEM) for electronically nonadiabatic collisions of OH(A 2 Σ + ) with H 2 , and we calculate the probabilities of electronically adiabatic inelastic collisions, nonreactive quenching, and reactive quenching to form H 2 O+H. The DPEM was fitted using a many-body expansion with permutationally invariant polynomials in bond-order functions to represent the many-body part. The dynamics calculations were carried out with the fewest-switches with time uncertainty and stochastic decoherence (FSTU/SD) semiclassical trajectory method. We present results both for head-on collisions (impact parameter b equal to zero) and for a full range of impact parameters. The results are compared to experiment and to earlier FSTU/SD and quantum dynamics calculations with a previously published DPEM. The various theoretical results all agree that nonreactive quenching dominates reactive quenching, but there are quantitative differences between the two DPEMs and between the b=0 results and the all-b results, especially for the probability of reactive quenching.
Realistic magnetic thermodynamics by local quantization of a semiclassical Heisenberg model
Abstract Classical Monte Carlo simulation of the Heisenberg model poorly describes many thermodynamic phenomena due to its neglect of the quantum nature of spins. Alternatively, we discuss how to semiclassically approach the quantum problem and demonstrate a simple method for introducing a locally approximate form of spin quantization. While the procedure underestimates magnetic short-range order, our results suggest a simple correction for recovering realistic spin–spin correlations above the critical temperature. Moreover, ensemble fluctuations are found to provide reasonably accurate thermodynamics, largely reproducing quantum mechanically calculated heat capacities and experimental magnetometry for ferromagnetic Fe and antiferromagnetic RbMnF 3 . Extensions of the method are proposed to address remaining inaccuracies.
Charge transfer rate constants for the carotenoid-porphyrin-C 60 molecular triad dissolved in tetrahydrofuran: The spin-boson model vs the linearized semiclassical approximation
Charge transfer rate constants were calculated for the carotenoid-porphyrin-C 60 (CPC 60 ) molecular triad dissolved in explicit tetrahydrofuran. The calculation was based on mapping the all-atom anharmonic Hamiltonian of this system onto the spin-boson Hamiltonian. The mapping was based on discretizing the spectral density from the time correlation function of the donor–acceptor potential energy gap, as obtained from all-atom molecular dynamics simulations. Different spin-boson Hamiltonians were constructed for each of the possible transitions between the three excited electronic states in two different triad conformations. The rate constants of three possible transitions were calculated via the quantum-mechanically exact Fermi’s golden rule (FGR), as well as a progression of more approximate expressions that lead to the classical Marcus expression. The advantage of the spin-boson approach is that once the mapping is established, the quantum-mechanically exact FGR and the hierarchy of approximations are known in closed form. The classical Marcus charge transfer rate constants obtained with the spin-boson Hamiltonians were found to reproduce those obtained from all-atom simulations with the linearized semiclassical approximation, thereby confirming the equivalence of the two approaches for this system. Furthermore, within the spin-boson Hamiltonian, we also found that the quantum-mechanically exact FGR rate constants were significantly enhanced compared to the classical Marcus theory rate constants for two out of three transitions in one of the two conformations under consideration. The results confirm that mapping to the spin-boson model can yield accurate predictions for charge transfer rate constants in a system as complex as CPC 60 dissolved in tetrahydrofuran.
Unified bulk semiclassical theory for intrinsic thermal transport and magnetization currents
In this work, we reveal the unexpected role of the material inhomogeneity in unifying the formulation of intrinsic thermal and thermoelectric transport as well as magnetization currents. The smooth inhomogeneity leads to the position dependent local band dispersion and phase-space Berry curvature, enabling a general and rapid access to transport and magnetization currents displaying the momentum-space Berry curvature physics. Our theory does not invoke the boundary current, the thermodynamic approach to magnetization or any mechanical counterpart of statistical forces. By introducing a fictitious inhomogeneity, it applies to homogeneous samples as well, promoting the inhomogeneity to be a basic trick in semiclassical transport theories. Such a trick works regardless of the driving force of transport, e.g., temperature gradient, in contrast to the trick of fictitious gravitational field in quantum transport theories. We thus include more general mechanical driving forces and establish the Mott relation between the resulting transport thermal and electric currents, whereas this relation for these two currents was previously only known when an electric field is the driving force.