Path integrals with discarded degrees of freedom

This technical paper of mathematical physics arose as an aftermath of the 2002 Cassini experiment (Bertotti et al 2003 Nature 425 374–6), in which the PPN parameter γ was measured with an accuracy σγ = 2.3 × 10−5 and found consistent with the prediction γ = 1 of general relativity. The Orbit Determination Program (ODP) of NASA's Jet Propulsion Laboratory, which was used in the data analysis, is based on an expression (8) for the gravitational delay Δt that differs from the standard formula (2); this difference is of second order in powers of m—the gravitational radius of the Sun—but in Cassini's case it was much larger than the expected order of magnitude m2/b, where b is the distance of the closest approach of the ray. Since the ODP does not take into account any other second-order terms, it is necessary, also in view of future more accurate experiments, to revisit the whole problem, to systematically evaluate higher order corrections and to determine which terms, and why, are larger than the expected value. We note that light propagation in a static spacetime is equivalent to a problem in ordinary geometrical optics; Fermat's action functional at its minimum is just the light-time between the two end points A and B. A new and powerful formulation is thus obtained. This method is closely connected with the much more general approach of Le Poncin-Lafitte et al (2004 Class. Quantum Grav. 21 4463–83), which is based on Synge's world function. Asymptotic power series are necessary to provide a safe and automatic way of selecting which terms to keep at each order. Higher order approximations to the required quantities, in particular the delay and the deflection, are easily obtained. We also show that in a close superior conjunction, when b is much smaller than the distances of A and B from the Sun, say of order R, the second-order correction has an enhanced part of order m2R/b2, which corresponds just to the second-order terms introduced in the ODP. Gravitational deflection of the image of a far away source when observed from a finite distance from the mass is obtained up to O(m2).

Recent works by Chandrasekaran, Penington, and Witten have shown in various special contexts that the quantum-corrected Ryu-Takayanagi (RT) entropy (or its covariant Hubeny-Rangamani-Takayanagi (HRT) generalization) can be understood as computing an entropy on an algebra of bulk observables. These arguments do not rely on the existence of a local holographic dual field theory. We show that analogous-but-stronger results hold in any UV-completion of asymptotically anti-de Sitter quantum gravity with a Euclidean path integral satisfying a simple and (largely) familiar set of axioms. We consider a quantum context in which a standard Lorentz-signature classical bulk limit would have Cauchy slices with asymptotic boundaries BL ⊔ BR where both BL and BR are compact manifolds without boundary. Our main result is then that (the UV-completion of) the quantum gravity path integral defines type I von Neumann algebras ALBL, ARBR of observables acting respectively at BL, BR such that ALBL, ARBR are commutants. The path integral also defines entropies on ALBL, ARBR. Positivity of the Hilbert space inner product then turns out to require the entropy of any projection operator to be quantized in the form ln N for some N ∈ ℤ+ (unless it is infinite). As a result, our entropies can be written in terms of standard density matrices and standard Hilbert space traces. Furthermore, in appropriate semiclassical limits our entropies are computed by the RT-formula with quantum corrections. Our work thus provides a Hilbert space interpretation of the Ryu-Takayanagi entropy. Since our axioms do not severely constrain UV bulk structures, it is plausible that they hold equally well for successful formulations of string field theory, spin-foam models, or any other approach to constructing a UV-complete theory of gravity.

Path integrals offer a robust approach for simulating open quantum dynamics with advancements transcending initial system size limitations. However, accurately modeling systems governed by mechanisms that do not conserve the number of quantum particles, such as lossy cavity modes, remains a challenge. We present a method to incorporate such empirical source and drain mechanisms within a path integral framework using quantum master equations. This technique facilitates rigorous inclusion of bath degrees of freedom while accommodating empirical time scales via Lindbladian dynamics. Computational costs are primarily driven by the path integral method with minimal overhead from Lindbladian terms. We use it to study exciton transport in a four-site Fenna-Matthews-Olson model, examining the potential loss of the exciton to the reaction center. This path integral Lindblad method promises an enhanced ability to simulate dynamics and will be fundamental to simulation of spectra in diverse quantum processes in open systems.

The identification of physical degrees of freedom is sometimes obscured in the path-integral formalism, and this makes it difficult to impose some constraints or to do some approximations. I review a number of cases where the difficulty is overcome by deriving the path integral from the operator form of the partition function after such identification has been made.

Theoretical methods are developed for studying the dynamics of chemical reactions. Many chemical processes can be adequately described as a system'' of one or two reaction coordinates coupled to a bath'' of harmonic oscillators. Such a formulation of the problem has obvious advantages as it is often possible to integrate out the harmonic bath, thus reducing the many degree of freedom problem to a one or two dimensional calculation. A simple semiclassical model for incorporating tunneling effects in classical trajectory simulations is constructed and shown to give excellent results over a variety of conditions. Feynman path integration provides an exact way of calculating quantum, dynamical properties and is in principle applicable to problems of involving many degrees of freedom. However, the standard Monte Carlo methodology which must be used to evaluate the multidimensional integrals that occur, is not directly applicable to the path integral expression for the real time propagator, because the integrand of the latter is highly, oscillatory. A technique which helps circumvent this problem by sampling about the stationary phase points of the integrand is developed and illustrated with several numerical applications. A new effective real time propagator is also constructed, which is localized and devoid of rapid oscillations. Finally, accurate approximations to the short time propagator are derived, which drastically improve the convergence characteristics of path integrals. These techniques are expected to extend the applicability of path integral calculations to more complex chemical problems. 258 refs., 63 figs., 3 tabs.

We demonstrate that the Lévy-Ciesielski implementation of Lie-Trotter products enjoys several properties that make it extremely suitable for path-integral Monte Carlo simulations: fast computation of paths, fast Monte Carlo sampling, and the ability to use different numbers of time slices for the different degrees of freedom, commensurate with the quantum effects. It is demonstrated that a Monte Carlo simulation for which particles or small groups of variables are updated in a sequential fashion has a statistical efficiency that is always comparable to or better than that of an all-particle or all-variable update sampler. The sequential sampler results in significant computational savings if updating a variable costs only a fraction of the cost for updating all variables simultaneously or if the variables are independent. In the Lévy-Ciesielski representation, the path variables are grouped in a small number of layers, with the variables from the same layer being statistically independent. The superior performance of the fast sampling algorithm is shown to be a consequence of these observations. Both mathematical arguments and numerical simulations are employed in order to quantify the computational advantages of the sequential sampler, the Lévy-Ciesielski implementation of path integrals, and the fast sampling algorithm.

Investigation on bit stick-slip vibration with random friction coefficients

In this paper, the quantum corrections to the kinematics of geometry, specifically geodesics, are presented. This is done by employing the path integral over the geodesics. Interestingly, the geodesics do not see any modifications in this framework. However for the distances, it is demonstrated that these quantum corrections exhibit distinct behaviors for time-like, light-like, and space-like geodesics. For time-like geodesics, the maximum correction is the Planck length, which disappears when the classical separation vanishes. The light-like geodesics do not exhibit quantum corrections, meaning that the causal light cone remains the same in both classical and quantum frameworks under certain conditions. The quantum corrections for space-like geodesics impose a minimum on space-like separation, potentially playing a role in removing singularities by preventing null congruences from being closer than the Planck length. This framework also explores the correspondence between space-like/time-like geodesics and quantum/statistical physics.

The numerical advantage of quantum Monte Carlo simulations of rigid bodies relative to the flexible simulations is investigated for some simple systems. The results show that if high frequency modes in molecular condensed matter are predominantly in the ground state, the convergence of path integral simulations becomes nonuniform. Rigid body quantum parallel tempering simulations are necessary to accurately capture thermodynamic phenomena in the temperature range where the dynamics are influenced by intermolecular degrees of freedom; the stereographic projection path integral adapted for quantum simulations of asymmetric tops is a significantly more efficient strategy compared with Cartesian coordinate simulations for molecular condensed matter under these conditions. The reweighted random series approach for stereographic path integral Monte Carlo is refined and implemented for the quantum simulation of water clusters treated as an assembly of rigid asymmetric tops.

Describing the spin degree of freedom in terms of Grassmann variables, we study the quantization problem of nonrelativistic and relativistic spinning particles interacting with crossed electric and magnetic fields by using the path integral method. The problem presents some difficulties due to the odd-dimensional nature of the phase space for the spin variables. We show how to deal with these difficulties by bringing back all the problems to the study of the quantum mechanics of a single Grassmann variable. Then it is straightforward to perform the path integrals in all the cases considered here, simply by solving the classical equations of motion.

Recently, we introduced a symmetry on the structure of angular momentum that interchanges internal and external degrees of freedom. The spin-orbit duality is a holographic map that projects a massive theory in four-dimensional flat spacetime onto the three-dimensional S2×R null infinity. This cylinder has radius R∼1/m and, quantum mechanically, its vacuum state is a fuzzy sphere. Progress shows that, first, this duality realizes the Hopf map, a fact manifest on the superparticle. Second, the bulk Poincarè group transforms into the conformal group on the cylinder. In fact, the general version of the duality yields that the dual symmetries include the Bondi-Metzner-Sachs group, as is appropriate at null infinity. As an example, the Landau levels in R3 are shown to match those of a Dirac monopole on the dual S2, in the thermodynamic limit. This dual system is actually identified with a three-dimensional critical Ising model. The map is then realized on Nf massive fermions in flat space which, indeed, are the hologram of 2Nf massless fermions on the cylinder. However, the dual space is really the conformal class of S2×R, naturally enclosing the universal cover of a conformally compactified four-dimensional anti–de Sitter (AdS4) spacetime. We argue that, in the absence of interactions, the massless fermions on the conformal boundary are, in turn, dual to Nf massive fermions in AdS4. For free fermions, all path integrals—the ones in R4 and S2×R and AdS4—are shown to match. Hence, AdS/CFT duality emerges into a larger context, where one holography nests inside the other, suggesting a complete holographic bridge between fields in flat space and the AdS superstring. Published by the American Physical Society 2025

The time evolution of a system of bilinearly coupled bosonic modes is investigated using the real-time path integral technique in the coherent-state representation. In order to get the stationary trajectories, the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The present treatment is applied to a dissipative harmonic oscillator within the Caldeira–Leggett model. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system–bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In contrast to many perturbative approaches, the bath degrees of freedom are treated explicitly. In addition, it is shown that systems without initial system–bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations can the classical trajectories for the system coordinate be recovered.

Neuronal representations of external events are often distributed across large populations of cells. We study the effect of correlated noise on the accuracy of these neuronal population codes. Our main question is whether the inherent error in the population code can be suppressed by increasing the size of the population N in the presence of correlated noise. We address this issue using a model of a population of neurons that are broadly tuned to an angular variable in two dimensions. The fluctuations in the neuronal activities are modeled as Gaussian noises with pairwise correlations that decay exponentially with the difference between the preferred angles of the correlated cells. We assume that the system is broadly tuned, which means that both the correlation length and the width of the tuning curves of the mean responses span a substantial fraction of the entire system length. The performance of the system is measured by the Fisher information (FI), which bounds its estimation error. By calculating the FI in the limit of a large N, we show that positive correlations decrease the estimation capability of the network, relative to the uncorrelated population. The information capacity saturates to a finite value as the number of cells in the population grows. In contrast, negative correlations substantially increase the information capacity of the neuronal population. These results are supplemented by the effect of correlations on the mutual information of the system. Our analysis provides an estimate of the effective number of statistically independent degrees of freedom, denoted N(eff), that a large correlated system can have. According to our theory N(eff) remains finite in the limit of a large N. Estimating the parameters of the correlations and tuning curves from experimental data in some cortical areas that code for angles, we predict that the number of effective degrees of freedom embedded in localized populations in these areas is less than or of the order of approximately 10(2).

If the diffeomorphism symmetry of general relativity is fully implemented into a path integral quantum theory, the path integral leads to a partition function which is an invariant of smooth manifolds. We comment on the physical implications of results on the classification of smooth and piecewise-linear 4-manifolds which show that the partition function can already be computed from a triangulation of space–time. Such a triangulation characterizes the topology and the differentiable structure, but is completely unrelated to any physical cut-off. It can be arbitrarily refined without affecting the physical predictions and without increasing the number of degrees of freedom proportionally to the volume. Only refinements at the boundary have a physical significance as long as the experimenters who observe through this boundary, can increase the resolution of their measurements. All these are consequences of the symmetries. The Planck scale cut-off expected in quantum gravity is rather a dynamical effect.

Testing Monte Carlo methods for path integrals in some quantum mechanical problems