arXiv subjects

T. Padmanabhan

Publications and source records attributed to T. Padmanabhan.

At least 19 recordsLinked to original sources

arXiv2021

Microscopic origin of Einstein's field equations and the raison d'être for a positive cosmological constant

In the paradigm of effective field theory, one hierarchically obtains the effective action $\mathcal{A}_{\rm eff}[q, \cdots]$ for some low(er) energy degrees of freedom $q$, by integrating out the high(er) energy degrees of freedom $ξ$, in a path integral, based on an action $\mathcal{A}[q,ξ, \cdots]$. We show how one can integrate out a vector field $v^a$ in an action $\mathcal{A}[Γ,v,\cdots ]$ and obtain an effective action $\mathcal{A}_{\rm eff}[Γ, \cdots]$ which, on variation with respect to the connection $Γ$, leads to the Einstein's field equations and a metric compatible with the connection. The derivation \textit{predicts} a non-zero, positive, \cc, which arises as an integration constant. The Euclidean action $\mathcal{A}[Γ,v, \cdots]$, has an interpretation as the heat density of null surfaces, when translated into the Lorentzian spacetime. The vector field $v^a$ can be interpreted as the Euclidean analogue of the microscopic degrees of freedom hosted by any null surface. Several implications of this approach are discussed.

gr-qc↗

arXiv2021

Thermal nature of a generic null surface

Dynamical properties of a generic null surface are known to have a thermodynamic interpretation. Such an interpretation is completely based on an analogy between the usual law of thermodynamics and structure of gravitational field equation on the surface. Here we materialise this analogy and show that assigning a temperature on the null surface for a local observer is indeed physically relevant. We find that for a local frame, chosen as outgoing massless chargeless particle (or field mode), perceives a "{\it local unstable Hamiltonian}" very near to the surface. Due to this it has finite quantum probability to escape through acausal null path which is given by Maxwell-Boltzmann like distribution, thereby providing a temperature on the surface.

gr-qc↗

arXiv2021

A nested sequence of inequivalent Rindler vacua : Universal Relic Thermality of Planckian origin

The Bogoliubov transformation connecting the standard inertial frame mode functions to the standard mode functions defined in the Rindler frame $R_0$, leads to the result that the inertial vacuum appears as a thermal state with temperature $T_0=a_0/2\pi$ where $a_0$ is the acceleration parameter of $R_0$. We construct an infinite family of nested Rindler-like coordinate systems $R_1, R_2, ...$ within the right Rindler wedge, with time coordinates $\tau_1, \tau_2, ...,$ and acceleration parameters $a_1, a_2, ...$ by shifting the origin along the inertial $x$-axis by amounts $\ell_1, \ell_2,\cdots$. We show that, apart from the inertial vacuum, the \textit{Rindler vacuum} of the frame $R_n$ also appears to be a thermal state in the frame $R_{n+1}$ with the temperature $a_{n+1}/2\pi$. \textit{In fact, the Rindler frame $R_{n+1}$ attributes to all the Rindler vacuum states of $R_1, R_2, ... R_n$, as well as to the inertial vacuum state, the same temperature $a_{n+1}/2\pi$.} We further show that our result is discontinuous in an essential way in the coordinate shift parameters. For a Rindler frame $R_i$, this thermality {\it turns on} with smallest non-zero $\ell_i$ allowed in the semiclassical framework and remains insensitive to $(\ell_i,a_{i-1})$ thereafter, indicating its universal Planckian origin. Similar structures can be introduced in the right wedge of any spacetime with bifurcate Killing horizon, like, for e.g., Schwarzschild spacetime. Apart from providing unsuppressed observables capturing Planck scale effects, these results have important implications for quantum gravity when flat spacetime is treated as the ground state of quantum gravity. (Truncated Abstract)

gr-qc↗

arXiv2021

World-line Path integral for the Propagator expressed as an ordinary integral: Concept and Applications

The (Feynman) propagator $G(x_2,x_1)$ encodes the entire dynamics of a massive, free scalar field propagating in an arbitrary curved spacetime. The usual procedures for computing the propagator -- either as a time ordered correlator or from a partition function defined through a path integral -- requires introduction of a field $ϕ(x)$ and its action functional $A[ϕ(x)]$. An alternative, more geometrical, procedure is to define a propagator in terms of the world-line path integral which only uses curves, $x^i(s)$, defined on the manifold. I show how the world-line path integral can be reinterpreted as an ordinary integral by introducing the concept of effective number of quantum paths of a given length. Several manipulations of the world-line path integral become algebraically tractable in this approach. In particular, I derive an explicit expression for the propagator $G_{\rm QG}(x_2,x_1)$, which incorporates the quantum structure of spacetime through a zero-point-length, in terms of the standard propagator $G_{\rm std}(x_2,x_1)$, in an arbitrary curved spacetime. This approach also helps to clarify the interplay between the path integral amplitude and the path integral measure in determining the form of the propagator. This is illustrated with several explicit examples.

gr-qc↗

arXiv2021

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 extremizing, $Ω$, 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 $ρ\propto\sqrt{g}$ so that $ρd^4x$ 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 $\lnρ\propto R$. I show how these two apparently contradictory requirements can be reconciled by using the functional dependence of $\sqrt{g}$ on curvature, in the Riemann normal coordinates (RNC), and coarse-graining over Planck scales. This leads to the density of microscopic configurations to be $ρ= Δ^{-1} = \sqrt{g}_{RNC}$ 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.

gr-qc↗

arXiv2020

Eddington gravity with matter: An emergent perspective

We describe an action principle, within the framework of the Eddington gravity, which incorporates the matter fields in a simple manner. Interestingly, the gravitational field equations derived from this action is identical to the Einstein's equations, in contrast with the earlier attempts in the literature. The cosmological constant arises as an integration constant in this approach. In fact, the derivation of the field equations demands the existence of a non-zero cosmological constant, thereby providing the raison d'être for a non-zero cosmological constant, implied by the current observations. Several features of our approach strongly support the paradigm that gravity is an emergent phenomenon and, in this perspective, our action principle could have a possible origin in the microstructure of the spacetime. We also discuss several extensions of the action principle, including the one which can incorporate torsion in the spacetime. We also show that an Eddington-like action can be constructed to obtain the field equations of the Lanczos-Lovelock gravity.

gr-qc↗

arXiv2020

A class of QFTs with higher derivative field equations leading to standard dispersion relation for the particle excitations

Given any (Feynman) propagator which is Lorentz and translation invariant, it is possible to construct an action functional for a scalar field such that the quantum field theory, obtained by path integral quantization, leads to this propagator. In general, such a theory will involve derivatives of the field higher than two and can even involve derivatives of infinite order. The poles of the given propagator determine the dispersion relation for the excitations of this field. I show that it is possible to construct field theories in which the dispersion relation is the same as that of standard Klein-Gordan field, even though the Lagrangian contains derivatives of infinite order. I provide a concrete example of this situation starting from a propagator which incorporates the effects of the zero-point-length of the spacetime. I compare the path integral approach with an alternative, operator-based approach, and highlight the advantages of using the former.

hep-th↗

arXiv2020

Probing the Planck scale: The modification of the time evolution operator due to the quantum structure of spacetime

The propagator which evolves the wave-function in NRQM, can be expressed as a matrix element of a time evolution operator: i.e $ G_{\rm NR}(x)= \langle{\mathbf{x}_2}|{U_{\rm NR}(t)}|{\mathbf{x}_1}\rangle$ in terms of the orthonormal eigenkets $|{\mathbf{x}}\rangle$ of the position operator. In QFT, it is not possible to define a conceptually useful single-particle position operator or its eigenkets. It is also not possible to interpret the relativistic (Feynman) propagator $G_R(x)$ as evolving any kind of single-particle wave-functions. In spite of all these, it is indeed possible to express the propagator of a free spinless particle, in QFT, as a matrix element $\langle{\mathbf{x}_2}|{U_{\rm R}(t)}|{\mathbf{x}_1}\rangle$ for a suitably defined time evolution operator and (non-orthonormal) kets $|{\mathbf{x}}\rangle$ labeled by spatial coordinates. At mesoscopic scales, which are close but not too close to Planck scale, one can incorporate quantum gravitational corrections to the propagator by introducing a zero-point-length. It turns out that even this QG corrected propagator can be expressed as a matrix element $\langle{\mathbf{x}_2}|{U_{\rm QG}(t)}|{\mathbf{x}_1}\rangle$. I describe these results and explore several consequences. It turns out that the evolution operator $U_{\rm QG}(t)$ becomes non-unitary for sub-Planckian time intervals while remaining unitary for time interval is larger than Planck time. The results can be generalised to any ultrastatic curved spacetime.

gr-qc↗

arXiv2020

Exploring the Rindler vacuum and the Euclidean Plane

In flat spacetime, two inequivalent vacuum states which arise rather naturally are the Rindler vacuum (R) and the Minkowski vacuum (M). We disuss several aspects of the Rindler vacuum, concentrating on the propagator and Schwinger (heat) kernel defined using R, both in the Lorentzian and Euclidean sectors. We start by exploring an intriguing result due to Candelas and Raine , viz., that $G_{R}$, the Feynman propagator corresponding to R, can be expressed as a curious integral transform of $G_{M}$, the Feynman propagator in M. We show that, this relation actually follows from the well known result that, $G_{M}$ can be written as a periodic sum of $G_{R}$, in the Rindler time $τ$, with the period $2πi$. We further show that, the integral transform result holds for a wide class of pairs of bi-scalars $(F_{M},F_{R})$, provided $F_{M}$ can be represented as a periodic sum of $F_{R}$ with period $2πi$. We provide an explicit procedure to retrieve $F_{R}$ from its periodic sum $F_{M}$, for a wide class of functions. An example of particular interest is the pair of Schwinger kernels $(K_{M},K_{R})$, corresponding to the Minkowski and the Rindler vacua. We obtain explicit expression for $K_{R}$ and clarify several conceptual and technical issues related to these biscalars both in the Euclidean and Lorentzian sector. In particular we address the issue of retrieving the information contained in all the four wedges of the Rindler frame in the Lorentzian sector, starting from the Euclidean Rindler (polar) coordinates. This is possible but require four different types of analytic continuations, based on one unifying principle. Our procedure allows generalisation of these results to any (bifurcate Killing) horizon in curved spacetime.

gr-qc↗

arXiv2020

Principle of Equivalence at Planck scales, QG in locally inertial frames and the zero-point-length of spacetime

Principle of Equivalence makes effects of classical gravity vanish in local inertial frames. What role does the Principle of Equivalence play as regards quantum gravitational effects in the local inertial frames? I address this question here from a specific perspective. At mesoscopic scales close to, but somewhat larger than, Planck length one could describe quantum spacetime and matter in terms of an effective geometry. The key feature of such an effective quantum geometry is the existence of a zero-point-length. When we proceed from quantum geometry to quantum matter, the zero-point-length will introduce corrections in the propagator for matter fields in a specific manner. On the other hand, one cannot ignore the self-gravity of matter fields at the mesoscopic scales and this will also modify the form of the propagator. Consistency demands that, these two modifications - coming from two different directions - are the same. I show that this non-trivial demand is actually satisfied. Surprisingly, the Principle of Equivalence, operating at sub-Planck scales, ensures this consistency in a subtle manner.

gr-qc↗

arXiv2020

Boundary Term in the Gravitational Action is the Heat Content of the Null surfaces

The Einstein-Hilbert Lagrangian has no well-defined variational derivative with respect to the metric. This issue has to be tackled by adding a suitable surface term to the action, which is a peculiar feature of gravity. We also know that null surfaces in spacetime exhibit (observer-dependent) thermodynamic features. This suggests a possible thermodynamic interpretation of the boundary term when the boundary is a null surface. For timelike/spacelike surfaces it is easy to construct the boundary term but there are some subtleties in the case of the null surface. The correct form of boundary term for null surfaces was obtained recently from first principles. We show that this surface term, as well as its variation, have direct thermodynamic interpretation in terms of a heat density of null surfaces. The implications of the result are discussed.

gr-qc↗

arXiv2020

Geodesic distance: A descriptor of geometry and correlator of pre-geometric density of spacetime events

Classical geometry can be described either in terms of a metric tensor $g_{ab}(x)$ or in terms of the geodesic distance $σ^2(x,x')$. Recent work, however, has shown that the geodesic distance is better suited to describe the quantum structure of spacetime. This is because one can incorporate some of the key quantum effects by replacing $σ^2$ by another function $S[σ^2]$ such that $S[0]=L_0^2$ is non-zero. This allows one to introduce a zero-point-length in the spacetime. I show that the geodesic distance can be an emergent construct, arising in the form of a correlator $S[σ^2(x,y)]=\langle J(x)J(y)\rangle$, of a pregeometric variable $J(x)$, which, in turn, can be interpreted as the quantum density of spacetime events. This approach also shows why null surfaces play a special role in the interface of quantum theory and gravity. I describe several technical and conceptual aspects of this construction and discuss some of its implications.

gr-qc↗

arXiv2019

Gravity and Quantum Theory: Domains of Conflict and Contact

There are two strong clues about the quantum structure of spacetime and the gravitational dynamics, which are almost universally ignored in the conventional approaches to quantize gravity. The first clue is that null surfaces exhibit (observer dependent) thermal properties and possess a heat density. This suggests that spacetime, like matter, has microscopic degrees of freedom and its long wavelength limit should be described in thermodynamic language and not in a geometric language. Second clue is related to the existence of the cosmological constant. Its understanding from first principles will require the dynamical principles of the theory to be invariant under the shift $T^a_b \to T^a_b + (constant) δ^a_b$. This puts strong constraints on the nature of gravitational dynamics and excludes metric tensor as a fundamental dynamical variable. In fact, these two clues are closely related to each other. When the dynamical principles are recast, respecting the symmetry $T^a_b \to T^a_b + (constant) δ^a_b$, they automatically acquire a thermodynamic interpretation related to the first clue. The first part of this review provides a pedagogical introduction to thermal properties of the horizons, including some novel derivations. The second part describes some aspects of cosmological constant problem and the last part provides a perspective on gravity which takes into account these principles.

gr-qc↗

arXiv2019

Thermality of the Rindler horizon: A simple derivation from the structure of the inertial propagator

The Feynman propagator encodes all the physics contained in a free field and transforms as a covariant bi-scalar. Therefore, we should be able to discover the thermality of the Rindler horizon, just by probing the structure of the propagator, expressed in the Rindler coordinates. I show that the thermal nature of the Rindler horizon is indeed contained --- though hidden --- in the standard, inertial, Feynman propagator. The probability $P(E)$ for a particle to propagate between two events, with energy $E$, can be related to the temporal Fourier transform of the propagator. A strikingly simple computation reveals that: (i) $P(E)$ is equal to $P(-E)$ if the propagation is between two events in the same Rindler wedge while (ii) they are related by a Boltzmann factor with temperature $T=g/2π$, if the two events are separated by a horizon. A more detailed computation reveals that the propagator itself can be expressed as a sum of two terms, governing absorption and emission, weighted correctly by the factors $(1+n_ν)$ and $n_ν$ where $n_ν$ is a Planck distribution at the temperature $T=g/2π$. In fact, one can discover the Rindler vacuum and the alternative (Rindler) quantization, just by probing the structure of the inertial propagator. These results can be extended to local Rindler horizons around any event in a curved spacetime. The implications are discussed.

gr-qc↗

arXiv2019

A Measure for Quantum Paths, Gravity and Spacetime Microstructure

The number of classical paths of a given length, connecting any two events in a (pseudo) Riemannian spacetime is, of course, infinite. It is, however, possible to define a useful, finite, measure $N(x_2,x_1;σ)$ for the effective number of quantum paths [of length $σ$ connecting two events $(x_1,x_2)$] in an arbitrary spacetime. When $x_2=x_1$, this reduces to $C(x,σ)$ giving the measure for closed quantum loops of length $σ$ containing an event $x$. Both $N(x_2,x_1;σ)$ and $C(x,σ)$ are well-defined and depend only on the geometry of the spacetime. Various other physical quantities like, for e.g., the effective Lagrangian, can be expressed in terms of $N(x_2,x_1;σ)$. The corresponding measure for the total path length contributed by the closed loops, in a spacetime region $\mathcal{V}$, is given by the integral of $L(σ;x) \equivσC(σ;x)$ over $\mathcal{V}$. Remarkably enough $L(0;x) \propto R(x)$, the Ricci scalar; i.e, the measure for the total length contributed by infinitesimal closed loops in a region of spacetime gives us the Einstein-Hilbert action. Its variation, when we vary the metric, can provide a new route towards induced/emergent gravity descriptions. In the presence of a background electromagnetic field, the corresponding expressions for $N(x_2,x_1;σ)$ and $C(x,σ)$ can be related to the holonomies of the field. The measure $N(x_2,x_1;σ)$ can also be used to evaluate a wide class of path integrals for which the action and the measure are arbitrary functions of the path length. As an example, I compute a modified path integral which incorporates the zero-point-length in the spacetime. I also describe several other properties of $N(x_2,x_1;σ)$ and outline a few simple applications.

gr-qc↗

arXiv2019

Generalized Schwinger effect and particle production in an expanding universe

We discuss several aspects of particle production in: (a) time dependent electric field and (b) expanding Friedmann background. In the first part of the paper, we provide an algebraic mapping between the differential equations describing these two phenomena. This mapping allows a direct comparison between (a) and (b) and we highlight several interesting features of both cases using this approach. We determine the form of the (equivalent) electric field corresponding to different Friedmann spacetimes and discover, for example, a time-dependent electric field which, in a specific limit, leads to a Planck's spectrum of particles. We also discuss the conditions under which the particle production in an expanding background will be non-analytic in the parameter which encodes the coupling to the curved spacetime, in close analogy with the generalized Schwinger effect. In the second part of the paper, we study the situation in which both time dependent electric field and an expanding background are simultaneously present. We compute particle production rate in this context by several different methods paying special attention to its limiting forms and possible non-analytic behaviour. We also clarify several conceptual issues related to definitions of in-vacuum and out-vacuum in these systems.

gr-qc↗

arXiv2019

Quantum Correlators in Friedmann Spacetimes -The omnipresent de Sitter and the invariant vacuum noise

We discuss several aspects of quantum field theory of a scalar field in a Friedmann universe, clarifying and highlighting several conceptual and technical issues. (A) We show that one can map the dynamics of (1) a massless scalar field in a universe with power law expansion to (2) a massive scalar field in the de Sitter spacetime, which allows us to understand several features of either system and clarifies several issues related to the massless limit. (B) We obtain a useful integral representation for the Euclidean Green's function for the de Sitter spacetime, by relating it to the solution of a hypothetical electrostatic problem in five dimensions. This is helpful in the study of several relevant limits. (C) We recover that in any Friedmann universe, sourced by a negative pressure fluid, the Wightman function for a massless scalar field is divergent. This shows that the divergence of Wightman function for the massless field in the de Sitter spacetime is just a special, limiting, case of this general phenomenon. (D) We provide a generally covariant procedure for defining the power spectrum of vacuum fluctuations in terms of the different Killing vectors present in the spacetime. This allows one to study the interplay of the choice of vacuum state and the nature of the power spectrum in different coordinate systems, in the de Sitter universe, in a unified manner. (Truncated Abstract; see the paper for full Abstract.)

gr-qc↗

arXiv2019

Path integrals for the relativistic particle: Some conceptual and pedagogical comments

In my textbook on Quantum Field Theory \cite{tpqft} and in a recent paper \cite{tpejc2018}, I advocated a lattice regularization procedure for defining the path integral for the relativistic particle, using the non-quadratic action containing a square root. I also provided an interpretation of this result in terms of the Jacobi action principle. This note clarifies several conceptual and pedagogical issues related to this approach and highlights some interesting open questions which this result leads to.

hep-th↗