Van Vleck Determinant Research Articles

Relics of the quantum spacetime: from Synge’s world function as the fundamental probe of spacetime architecture to the emergent description of gravity

All our observations that characterise space and time are expressed in terms of non-local, bi-tensorial objects such as geodesic intervals between events and two-point (Green) functions. In this contribution, I highlight the importance of characterising spacetime geome-try in terms of such non-local objects, focusing particularly on two important bi-tensors that play a particular fundamental role – Synge’s World function and the van Vleck determinant. I will first discuss how these bi-tensors help capture information about spacetime geometry, and then describe their role in characterising quantum spacetime endowed with a lower bound, say ℓ 0, on spacetime intervals. Incorporating such a length scale in a Lorentz covariant manner necessitates a description of spacetime geometry in terms of above bi-tensors, and naturally replaces the conventional description based on the metric tensor gab (x) with a description in terms of a non-local bi-tensor qab (x, y). The non-analytic structure of qab (x, y) which renders a perturbative expansion in ℓ 0 meaningless, also generically leaves a non-trivial “relic” in the limit ℓ 0 → 0. I present some results where such a relic term is manifest; specifically, I will discuss how this: (i) suggests a description of gravitational dynamics different from the one based on Einstein-Hilbert lagrangian, (ii) implies dimensional reduction to 2 at small scales, (iii) connects with the notion of cosmological constant itself being a non-local vestige of the small scale structure of spacetime, (iv) helps address the issues of spacetime singularities. I will conclude by discussing the ramifications of these ideas for quantum gravity.

Path-integral treatment of quantum bouncers

The one-sided bouncer (1SB) and the symmetric bouncer (SB) involve a one-dimensional particle in a piecewise linear potential. For such problems, the time-dependent quantum mechanical propagator cannot be found in closed form. The semiclassical Feynman path integral (PI) is a very appealing approach, as it approximates the propagator by a closed-form expression (a sum over a finite number of classical paths). In this paper we solve the classical path enumeration problem. We obtain closed-form expressions for the initial velocity, bounce times, focal times, action, van Vleck determinant, and Morse index for each classical path. We calculate the propagator within the semiclassical approximation. The numerical results agree with eigenfunction expansion results away from caustics. We derive mappings between the 1SB and SB, which explains why each bounce of the 1SB increases the Morse index by 2 and results in a phase change of π. We interpret the semiclassical Feynman PI to obtain visualizations of matter wave propagation based on interference between classical paths, in analogy with the traditional visualization of light wave propagation as interference between classical ray paths.

Einstein–Hilbert action, with quantum corrections, from the Planck scale coarse-graining of the spacetime microstructure

The gravitational dynamics of the coarse-grained spacetime geometry should emerge from extremizing the number of microscopic configurations, [Formula: see text], of the pre-geometric variables corresponding to a given geometry. This [Formula: see text] will be the product over all events [Formula: see text] of the density, [Formula: see text], of microscopic configurations associated with each event [Formula: see text]. I show how [Formula: see text] can be computed, in terms of the van Vleck determinant, and thus obtain directly the gravitational effective action [Formula: see text] at mesoscopic scales. The leading term of this, nonperturbative, effective action gives the Einstein–Hilbert action, thereby providing its microscopic derivation. The higher-order corrections are finite without any need for regularization and I demonstrate how they can be computed in a systematic manner.

Gravitational effective action at mesoscopic scales from the quantum microstructure of spacetime

At mesoscopic scales, the quantum corrected field equations of gravity should arise from extremising, Ω, the number of microscopic configurations of pre-geometric variables consistent with a given geometry. This Ω, in turn, is the product over all events P of the density, ρ(P), of microscopic configurations associated with each event P. One would have expected ρ∝g so that ρd4x scales as the proper volume of a region. On the other hand, at leading order, we would expect the extremum principle to be based on the Hilbert action, suggesting ln⁡ρ∝R. I show how these two apparently contradictory requirements can be reconciled by using the functional dependence of g on curvature, in the Riemann normal coordinates (RNC), and coarse-graining over Planck scales. This leads to the density of microscopic configurations to be ρ=Δ−1=gRNC where Δ is the coarse grained Van-Vleck determinant. The approach also provides: (a) systematic way of computing QG corrections to field equations and (b) a direct link between the effective action for gravity and the kinetic theory of the spacetime fluid.

Small-scale structure of spacetime and its implications

If there exists a lower bound [Formula: see text] to spacetime intervals which is Lorentz-invariant, then the effective description of spacetime that incorporates such a lower bound must necessarily be nonlocal. Such a nonlocal description can be derived using standard tools of differential geometry, but using as basic variables certain bi-tensors instead of the conventional metric tensor [Formula: see text]. This allows one to construct a qmetric [Formula: see text], using the Synge’s world function [Formula: see text] and the van Vleck determinant [Formula: see text], that incorporates the lower bound on spacetime intervals. The same nonanalytic structure of the reconstructed spacetime which renders a perturbative expansion in [Formula: see text] meaningless, will then also generically leave a non-trivial “relic” in the limit [Formula: see text]. We present specific results derived from [Formula: see text] where such a relic term manifests, and discuss several implications of the same. Specifically, we will discuss how these results: (i) suggest a description of gravitational dynamics different from the conventional one based on the Einstein–Hilbert Lagrangian, (ii) imply a dimensional reduction to [Formula: see text] at small scales and (iii) can be significant for the idea that the cosmological constant itself might be related to some nonlocal vestige of the small-scale structure of spacetime. We will conclude by discussing the ramifications of these ideas in the context of quantum gravity.

Perturbative treatment of the luminosity distance

We derive a generalized luminosity distance versus redshift relation for a linearly perturbed FLRW (Friedmann-Lemaitre-Robertson-Walker) metric with two scalar mode excitations. We use two equivalent approaches, based on the Jacobi map and the van Vleck determinant respectively. We apply the resultant formula to two simple models - an exact FLRW universe and an approximate FLRW universe perturbed by a single scalar mode sinusoidally varying with time. For both models we derive a cosmographic expansion for d_L in terms of z. We comment on the interpretation of our results and their possible application to more realistic cosmological models.

Non-perturbative results for the luminosity and area distances

The notion of luminosity distance is most often defined in purely FLRW (Friedmann-Lemaitre-Robertson-Walker) cosmological spacetimes, or small perturbations thereof. However, the abstract notion of luminosity distance is actually much more robust than this, and can be defined non-perturbatively in almost arbitrary spacetimes. Some quite general results are already known, in terms of dAobserver/dΩsource, the cross-sectional area per unit solid angle of a null geodesic spray emitted from some source and subsequently detected by some observer. We shall reformulate these results in terms of a suitably normalized null geodesic affine parameter and the van Vleck determinant, ΔvV. The contribution due to the null geodesic affine parameter is effectively the inverse square law for luminosity, and the van Vleck determinant can be viewed as providing a measure of deviations from the inverse square law. This formulation is closely related to the so-called Jacobi determinant, but the van Vleck determinant has somewhat nicer analytic properties and wider and deeper theoretical base in the general relativity, quantum physics, and quantum field theory communities. In the current article we shall concentrate on non-perturbative results, leaving near-FLRW perturbative investigation for future work.

Prediction and characterization of multiple extremal paths in continuously monitored qubits

We examine most-likely paths between initial and final states for diffusive quantum trajectories in continuously monitored pure-state qubits, obtained as extrema of a stochastic path integral. We demonstrate the possibility of "multipaths" in the dynamics of continuously-monitored qubit systems, wherein multiple most-likely paths travel between the same pre- and post-selected states over the same time interval. Most-likely paths are expressed as solutions to a Hamiltonian dynamical system. The onset of multipaths may be determined by analyzing the evolution of a Lagrange manifold in this phase space, and is mathematically analogous to the formation of caustics in ray optics or semiclassical physics. Additionally, we develop methods for finding optimal traversal times between states, or optimal final states given an initial state and evolution time; both give insight into the measurement dynamics of continuously-monitored quantum states. We apply our methods in two systems: a qubit with two non-commuting observables measured simultaneously, and a qubit measured in one observable while subject to Rabi drive. In the two-observable case we find multipaths due to caustics, bounded by a diverging Van-Vleck determinant, and their onset time. We also find multipaths generated by paths with different "winding numbers" around the Bloch sphere in both systems.

The geometry of gravitational lensing magnification

We present a definition of unsigned magnification in gravitational lensing valid on arbitrary convex normal neighborhoods of time oriented Lorentzian manifolds. This definition is a function defined at any two points along a null geodesic that lie in a convex normal neighborhood, and foregoes the usual notions of lens and source planes in gravitational lensing. Rather, it makes essential use of the van Vleck determinant, which we present via the exponential map, and Etherington’s definition of luminosity distance for arbitrary spacetimes. We then specialize our definition to spacetimes, like Schwarzschild’s, in which the lens is compact and isolated, and show that our magnification function is monotonically increasing along any geodesic contained within a convex normal neighborhood.

Geometric properties of a2Dspacetime arising in4Dblack hole physics

The Schwarzschild exterior space-time is conformally related to a direct product space-time, $\mathcal{M}_2 \times S_2$, where $\mathcal{M}_2$ is a two-dimensional space-time. This direct product structure arises naturally when considering the wave equation on the Schwarzschild background. Motivated by this, we establish some geometrical results relating to $\mathcal{M}_2$ that are useful for black hole physics. We prove that $\mathcal{M}_2$ has the rare property of being a causal domain. Consequently, Synge's world function and the Hadamard form for the Green function on this space-time are well-defined globally. We calculate the world function and the van Vleck determinant on $\mathcal{M}_2$ numerically and point out how these results will be used to establish global properties of Green functions on the Schwarzschild black hole space-time.

Small scale structure of spacetime: The van Vleck determinant and equigeodesic surfaces

It has recently been argued that if spacetime $\mathcal M$ possesses non-trivial structure at small scales, an appropriate semi-classical description of it should be based on non-local bi-tensors instead of local tensors such as the metric $g_{ab}$. Two most relevant bi-tensors in this context are Synge's World function $\Omega(p,p_0)$ and the van Vleck determinant (VVD) $\Delta(p,p_0)$, as they encode the metric properties of spacetime and (de)focussing behaviour of geodesics. They also characterize the leading short distance behavior of two point functions of the d'Alembartian $_{p_0} \square_p$. We begin by discussing the intrinsic and extrinsic geometry of equi-geodesic surfaces $\Sigma_{G,p_0}$ defined by $\Omega(p,p_0)=constant$ in a geodesically convex neighbourhood of an event $p_0$, and highlight some elementary identities relating the VVD with geometry of these surfaces. As an aside, we also comment on the contribution of $\Sigma_{G,p_0}$ to the surface term in the Einstein-Hilbert (EH) action and show that it can be written as a volume integral of $\square \ln \Delta$. We then study the small scale structure of spacetime in presence of a Lorentz invariant short distance cut-off $\ell_0$ using $\Omega$ and $\Delta$, based on some recently developed ideas. We derive a 2nd rank bi-tensor $q_{ab}$ which naturally yields geodesic intervals bounded from below, and present a general and mathematically rigorous analysis of short distance structure of spacetime based on (a) geometry of $\Sigma_{G,p_0}$, (b) structure of the non local d'Alembartian associated with $q_{ab}$, and (c) properties of the VVD. In particular, we show that the Ricci bi-scalar of $q_{ab}$ is completely determined by $\Sigma_{G,p_0}$, the tidal tensor, and first two derivatives of the van Vleck determinant, and has a non-trivial "classical" limit given by (constant) $R_{ab} q^a q^b$ (see text).

On Schrödinger's equation, Hertz's mechanics and Van Vleck's determinant

There has been much research on Schrödinger's route to what we now call Schrödinger's equation. Various authors disagree as to the exact nature of the influence of each of the physicists he cites—and of some that he does not. This paper, intended for graduate students of and researchers in quantum theory, clarifies Schrödinger's original aims in formulating a wave equation for matter, discusses how far he fulfilled his original aspirations and in what respects he fell short of his goal. An analysis of Schrödinger's foundational paper enables us to distinguish between a formal and an epistemological part, and consider the input of the physicists cited on the basis of the part in which each reference occurs. It turns out, for instance, that Hamilton's optical-mechanical analogy belongs entirely to the epistemological part. Indeed, no element of this analogy plays any role in the formal part of Schrödinger's argument. Instead of basing his theory on this analogy, as is often done nowadays in the physics literature and even in the history of science, we maintain that the aim of Schrödinger's project was to represent wave phenomena by a ‘wave’ in configuration- or q-space, paralleling Hertz's treatment in his Mechanics (1894). The influence of this book on Schrödinger's foundational paper is demonstrated by an analysis of his unpublished paper: Hertz's Mechanics and Einstein's Theory of Gravitation. This approach enables us to dispense with the optical-mechanical analogy in tracing the route to Schrödinger's equation. We also discuss the curious role of the Van Vleck determinant as the ‘missing link’ in taking the classical limit of Schrödinger's wavefunction. A concluding section discusses the relation of some of Schrödinger's earlier and later work to the development of quantum field theory.

Transport equation approach to calculations of Hadamard Green functions and non-coincident DeWitt coefficients

Building on an insight due to Avramidi, we provide a system of transport equations for determining key fundamental bi-tensors, including derivatives of the world-function, \sigma(x,x'), the square root of the Van Vleck determinant, \Delta^{1/2}(x,x'), and the tail-term, V(x,x'), appearing in the Hadamard form of the Green function. These bi-tensors are central to a broad range of problems from radiation reaction to quantum field theory in curved spacetime and quantum gravity. Their transport equations may be used either in a semi-recursive approach to determining their covariant Taylor series expansions, or as the basis of numerical calculations. To illustrate the power of the semi-recursive approach, we present an implementation in \textsl{Mathematica} which computes very high order covariant series expansions of these objects. Using this code, a moderate laptop can, for example, calculate the coincidence limit a_7(x,x) and V(x,x') to order (\sigma^a)^{20} in a matter of minutes. Results may be output in either a compact notation or in xTensor form. In a second application of the approach, we present a scheme for numerically integrating the transport equations as a system of coupled ordinary differential equations. As an example application of the scheme, we integrate along null geodesics to solve for V(x,x') in Nariai and Schwarzschild spacetimes.

Semiclassical propagator and van Vleck determinant in a mixed position–momentum space

In this paper a semiclassical propagator in a mixed position–momentum space is derived in the formalism of Maslov's multi-dimensional semiclassical theory. The corresponding mixed van Vleck determinant is also given explicitly. The propagator can be used to locally fix semiclassical divergences in singular regions of configuration space. It is shown that when a semiclassical propagator is transformed from one representation to another, its form is invariant.

Guided Gaussian Wave Packets

We discuss several semiclassical Gaussian wave packet approaches with emphasis on one that is not very well-known, that is, the off-center guiding approach. Off-center guiding of (thawed) Gaussian wave packets uses the same information as other Gaussian propagation schemes, that is, the full van Vleck determinant and multiple trajectories. It retains the well-known caustic smoothing property of Gaussians. The off-center guiding of Gaussians can handle hard chaos and other highly nonlinear situations, where the van Vleck propagator, for example, has over 30,000 separate branches.

Boltzmann equation in classical and quantum field theory

Improving upon the previous treatment by Mueller and Son, we derive the Boltzmann equation that results from a classical scalar field theory. This is obtained by starting from the corresponding quantum field theory and taking the classical limit with particular emphasis on the path integral and perturbation theory. A previously overlooked Van Vleck determinant is shown to control the tadpole type of self-energy that can still appear in the classical perturbation theory. Further comments on the validity of the approximations and possible applications are also given.

Dephasing representation: Employing the shadowing theorem to calculate quantum correlation functions

Due to the Heisenberg uncertainty principle, various classical systems differing only on the scale smaller than Planck's cell correspond to the same quantum system. We use this fact to find a unique semiclassical representation without the Van Vleck determinant, applicable to a large class of correlation functions expressible as quantum fidelity. As in the Feynman path integral formulation of quantum mechanics, all contributing trajectories have the same amplitude: that is why we denote it the "dephasing representation." By relating our approach to the problem of existence of true trajectories near numerically-computed chaotic trajectories, we make the approximation rigorous for any system in which the shadowing theorem holds. Numerical implementation only requires computing actions along the unperturbed trajectories and not finding the shadowing trajectories. While semiclassical linear-response theory was used before in quasi-integrable and chaotic systems, here its validity is justified in the most generic, mixed systems. Dephasing representation appears to be a rare practical method to calculate quantum correlation functions in nonuniversal regimes in many-dimensional systems where exact quantum computations are impossible.

Quantum Caustics for Systems with Quadratic Lagrangians in Multi-dimensions

We study quantum caustics (i.e., the quantum analogue of the classical singularity in the Dirichlet boundary problem) in d-dimensional systems with quadratic Lagrangians of the form L=12Pij(t)xixj+Qij(t)xixj+12Rij(t)xixj+Si(t)xi. Based on Schulman's procedure in the path-integral we derive the transition amplitude on caustics in a closed form for generic multiplicity f, and thereby complete the previous analysis carried out for the maximal multiplicity case (f=d). The unitarity relation, together with the initial condition, fulfilled by the amplitude is found to be a key ingredient for determining the amplitude, which reduces to the well-known expression with Van Vleck determinant for the non-caustics case (f=0). Multiplicity dependence of the caustics phenomena is illustrated by examples of a particle interacting with external electromagnetic fields.

Loop calculations in quantum mechanical non-linear sigma models with fermions and applications to anomalies

We construct the path integral for one-dimensional non-linear sigma models, starting from a given Hamiltonian operator and states in a Hilbert space. By explicit evaluation of the discretized propagators and vertices we find the correct Feynman rules which differ from those often assumed. These rules, which we previously derived in bosonic systems, are now extended to fermionic systems. We then generalize the work of Alvarez-Gaumé and Witten by developing a framework to compute anomalies of an n-dimensional quantum field theory by evaluating perturbatively a corresponding quantum mechanical path integral. Finally, we apply this formalism to various chiral and trace anomalies, and solve a series of technical problems: (i) the correct treatment of Majorana fermions in path integrals with coherent states (the methods of fermion doubling and fermion halving yield equivalent results when used in applications to anomalies), (ii) a complete path integral treatment of the ghost sector of chiral Yang-Mills anomalies, (iii) a complete path integral treatment of trace anomalies, (iv) the supersymmetric extension of the Van Vleck determinant, and (v) a derivation of the spin- 3 2 Jacobian of Alvarez-Gaume and Witten for Lorentz anomalies.

Van Vleck determinants: Traversable wormhole spacetimes.

Prompted by various questions regarding the putative existence, stability, and chronological properties of traversable wormholes, a number of authors have presented calculations of the renormalized stress-energy tensor in wormhole spacetimes. In particular, the use of point-splitting techniques leads to expressions that contain the van Vleck determinant as a common prefactor. Recent technical advances permit one to undertake extensive computations of the van Vleck determinant in traversable wormhole spacetimes, at least in the short-throat flat-space approximation. This paper presents several such computations for various model spacetimes. Implications with regard to Hawking's chronology protection conjecture are discussed. In particular, any attempt to transform a single isolated wormhole into a time machine results in large vacuum polarization effects. These vacuum polarization effects are sufficient to disrupt the internal structure of the wormhole long before the onset of Planck scale physics, and before the onset of time travel. Thus for isolated wormholes, vacuum polarization effects are sufficient to enforce Hawking's chronology protection conjecture. On the other hand, it is possible to conceive of a putative time machine built out of two or more wormholes, each of which taken in isolation is not itself a time machine. Such ``Roman configurations'' are much more subtle to analyze. For ``reasonable'' configurations (traversable by humans) the vacuum polarization effects in such multiple wormhole putative time machines become large long before the onset of Planck scale physics. The disruption scale for would-be ``traversable time machines'' is well above the Planck length. On the other hand, for some particularly bizarre configurations (not traversable by humans) the vacuum polarization effects can be arranged to be arbitrarily small at the onset of Planck scale physics. This indicates that the disruption scale has been pushed down into the Planck slope. This is mildly disturbing. Ultimately, for these bizarre configurations, questions regarding the truth or falsity of Hawking's chronology protection can only be addressed by entering the uncharted wastelands of full fledged quantum gravity.