A quantum mechanical description of particle propagation on the discrete spacetime of a causal set is presented. The model involves a discrete path integral in which trajectories within the causal set are summed over to obtain a particle propagator. The sum-over-trajectories is achieved by a matrix geometric series. For causal sets generated by sprinkling points into 1+1 and 3+1 dimensional Minkowski spacetime the propagator calculated on the causal set is shown to agree, in a suitable sense, with the causal retarded propagator for the Klein-Gordon equation. The particle propagator described here is a step towards quantum field theory on causal set spacetime.
We consider the covariant graviton propagator in de Sitter spacetime in a gauge with two parameters, alpha and beta, in the Euclidean approach. We give an explicit form of the propagator with a particular choice of beta but with arbitrary value of alpha. We confirm that two-point functions of local gauge-invariant quantities do not increase as the separation of the two points becomes large.
The existence of a minimal and fundamental length scale, say, the Planck length, is a characteristic feature of almost all the models of quantum gravity. The presence of the fundamental length is expected to lead to an improved ultra-violet behavior of the semi-classical propagators. The hypothesis of path integral duality provides a prescription to evaluate the modified propagator of a free, quantum scalar field in a given spacetime, taking into account the existence of the fundamental length in a locally Lorentz invariant manner. We use this prescription to compute the quantum gravitational …
The propagators of the Faddeev-Popov (FP) ghosts for Yang-Mills theories and perturbative quantum gravity in the covariant gauge are infrared (IR) divergent in de Sitter spacetime. We point out, however, that the modes responsible for these divergences will not contribute to loop diagrams in computations of time-ordered products in either Yang-Mills theories or perturbative quantum gravity. Therefore we propose that the IR divergent FP-ghost propagator should be regularized by a small mass term that is sent to zero in the end of any perturbative calculations. This proposal is equivalent to usi…
The basic properties of Poincare gauge invariant Hilbert bundles over Lorentzian manifolds are derived. Quantum connections are introduced in such bundles, which govern a parallel transport that is shown to satisfy the strong equivalence principle in the quantum regime. Path-integral expressions are presented for boson propagators in Hilbert bundles over globally hyperbolic curved spacetimes. Their Poincare gauge covariance is proven, and their special relativistic limit is examined. A method for explicitly computing such propagators is presented for the case of cosmological models with Robert…
The hypothesis of path integral duality provides a prescription to evaluate the propagator of a free, quantum scalar field in a given classical background, taking into account the existence of a fundamental length, say, the Planck length, $\lp$, in a {\it locally Lorentz invariant manner}. We use this prescription to evaluate the duality modified propagators in spacetimes with {\it constant curvature} (exactly in the case of one spacetime, and in the Gaussian approximation for another two), and show that: (i) the modified propagators are ultra violet finite, (ii) the modifications are {\it non…
One of the most challenging technical aspects of the dualities between string theory on anti-de Sitter spaces and conformal field theories is understanding how location in the interior of spacetime is represented in the field theory. It has recently been argued that the interior of the spacetime can be directly probed by using intrinsically non-local quantities in the field theory. In addition, Balasubramanian and Ross [hep-th/9906226] argued that when the spacetime described the formation of an AdS_3 black hole, the propagator in the field theory probed the whole spacetime, including the regi…
The causal set approach to quantum gravity models spacetime as a discrete structure - a causal set. Recent research has led to causal set models for the retarded propagator for the Klein-Gordon equation and the d'Alembertian operator. These models can be compared to their continuum counterparts via a sprinkling process. It has been shown that the models agree exactly with the continuum quantities in the limit of an infinite sprinkling density - the continuum limit. This paper obtains the correction terms for these models for sprinkled causal sets with a finite sprinkling density. These correct…
We study the Moyal anticommutators and their expectation values between vacuum states and non-vacuum states for Dirac fields on noncommutative spacetime. Then we construct the propagators of Moyal star-products for Dirac fields on noncommutative spacetime. We find that the propagators of Moyal star-products for Dirac fields are equal to the propagators of Dirac fields on ordinary commutative spacetime.
The Feynman propagator for a free bosonic scalar field on the discrete spacetime of a causal set is presented. The formalism includes scalar field operators and a vacuum state which define a scalar quantum field theory on a causal set. This work can be viewed as a novel regularisation of quantum field theory based on a Lorentz invariant discretisation of spacetime.
We study the Moyal commutators and their expectation values between vacuum states and non-vacuum states for free scalar field on noncommutative spacetime. Then from the Moyal commutators, we find that the microcausality is satisfied for the linear operators of the free scalar field on noncommutative spacetime. We construct the Feynman propagator of Moyal star-product for noncommutative scalar field theory.
We show that a notion of one-particle state and the corresponding vacuum state exists in general curved backgrounds for spin $\frac{1}{2}$ fields. A curved spacetime can be equipped with a coordinate system in which the metric component $g_{--}=0$. We separate the component of the left-handed massless Dirac field which is annihilated by the null vector $\partial_-$ and compute the corresponding Feynman propagator. We find that the propagating modes are localized on two dimensional subspaces and the Feynman propagator is similar to the Feynman propagator of chiral fermions in two dimensional Mi…
The propagators for the Faddeev-Popov (FP) ghosts in Yang-Mills theory and perturbative gravity in the covariant gauge are infrared (IR) divergent in de Sitter spacetime. An IR cutoff in the momentum space to regularize these divergences breaks the de Sitter invariance. These IR divergences are due to the spatially constant modes in the Yang-Mills case and the modes proportional to the Killing vectors in the case of perturbative gravity. It has been proposed that these IR divergences can be removed, with the de Sitter invariance preserved, by first regularizing them with an additional mass ter…
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 reinterpr…
In this master thesis a configuration space method presented by C. Dullemond and E. van Beveren for computing all propagators of a scalar field (Wightman, Hadamard and Schwinger functions,retarded, advanced and Feynman propagator) is reviewed for four-dimensional Minkowski and Anti de Sitter spacetime AdS_4. This method is then applied for AdS_d as well as de Sitter spacetime dS_d of arbitrary dimension d, obtaining results in agreement with the literature. The advantages of the method are that it needs neither mode summation nor analytic continuation from euclidean time, while delivering the …
Causal set theory provides a model of discrete spacetime in which spacetime events are represented by elements of a causal set---a locally finite, partially ordered set in which the partial order represents the causal relationships between events. The work presented here describes a model for matter on a causal set, specifically a theory of quantum scalar fields on a causal set spacetime background. The work starts with a discrete path integral model for particles on a causal set. Here quantum mechanical amplitudes are assigned to trajectories within the causal set. By summing these over all t…
Graviton propagator diverges in certain gauges in de Sitter spacetime. We address this problem in this work by generalizing the infinitesimal BRST transformations in de Sitter spacetime to finite field-dependent BRST (FFBRST) transformations. These FFBRST transformations are symmetry of the classical action, but do not leave path integral measure invariant for the graviton theory in de Sitter spacetime. Due to the non-trivial Jacobian of such finite transformation the path integral measure changes and hence FFBRST transformation is capable of relating theories in two different gauges. We expli…
It has been often conjectured that the correct theory of quantum gravity will act as a UV regulator in the low energy limit of quantum field theory. Earlier work has shown that if the path integral defining the quantum field theory propagator is modified, so that the amplitude is invariant under the duality transformation l--> 1/l where l is the length of the path, then the propagator is UV-finite and exhibits a ``zero-point length'' of the spacetime. Since string theory uses extended structures and has a T-duality, these results should also emerge directly from string theory. We show, by expl…
The action for a relativistic free particle of mass $m$ receives a contribution $-mds$ from a path segment of infinitesimal length $ds$. Using this action in a path integral, one can obtain the Feynman propagator for a spinless particle of mass $m$. If one of the effects of quantizing gravity is to introduce a minimum length scale $L_P$ in the spacetime, then one would expect the segments of paths with lengths less than $L_P$ to be suppressed in the path integral. Assuming that the path integral amplitude is invariant under the `duality' transformation $ds\to L_P^2/ds$, one can calculate the m…
We continue to derive spacetime quantities and spin 1/2 propagators from rotations. Rotation-invariant projection operators are found for each element of a four element basis, i.e. a basis for four component quantities with specific transformation rules under rotations. With these four projection operators, we make two spacetime invariant projection operators, i.e. once space, time, energy, and momentum are identified. The spacetime invariant operators are propagators for free neutrinos. Except for the substitute basis, the process is the same as the one that gave electron propagators in Part …
We show how quantum fields can be used to measure the curvature of spacetime. In particular, we find that knowledge of the imprint that spacetime curvature leaves in the correlators of quantum fields suffices, in principle, to reconstruct the metric. We then consider the possibility that the quantum fields obey a natural ultraviolet cutoff, for example, at the Planck scale. We investigate how such a cutoff limits the spatial resolution with which curvature can be deduced from the properties of quantum fields. We find that the metric deduced from the quantum correlator exhibits a peculiar scali…
The propagator of a spinless particle is calculated from the quantum mechanical path integral formalism in static curved spacetimes endowed with event-horizons. A toy model, the Gui spacetime, and the 2D and 4D Schwarzschild black holes are considered. The role of the topology of the coordinates configuration space is emphasised in this framework. To cover entirely the above spacetimes with a single set of coordinates, tortoise coordinates are extended to complex values. It is shown that the homotopic properties of the complex tortoise configuration space imply the thermal behaviour of the pro…
In a spacetime with no global timelike Killing vector, we do not have a natural choice for the vacuum state of matter fields, leading to an ambiguity in defining the Feynman propagators. In this paper, taking the vacuum state to be the instantaneous ground state of the Hamiltonian at each moment, we develop a method for calculating wave functions associated with the vacuum and the corresponding in-in and in-out propagators. We apply this method to free scalar field theory in de Sitter space and obtain de Sitter invariant propagators in various coordinate patches. We show that the in-out propag…
The usual (Bunch-Davies) Feynman propagator of a massless field is not well defined in an expanding universe due to the presence of infrared divergences. We propose a new propagator which yields infrared finite answers to any correlation function. The key point is that in a de Sitter spacetime there is an ambiguity in the zero-mode of the propagator. This ambiguity can be used to cancel the apparent divergences which arise in some loop calculations in eternally (or semi-eternally) inflating spacetime. We refer to this process as zero-mode modification. The residual ambiguity is fixed by observ…
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 t…
We construct the quantum mechanical evolution operator in the Functional Schrodinger picture - the kernel - for a scalar field in spatially homogeneous FLRW spacetimes when the field is a) free and b) coupled to a spacetime dependent source term. The essential element in the construction is the causal propagator, linked to the commutator of two Heisenberg picture scalar fields. We show that the kernels can be expressed solely in terms of the causal propagator and derivatives of the causal propagator. Furthermore, we show that our kernel reveals the standard light cone structure in FLRW spaceti…
A quantum mechanical path integral derivation is given of a thermal propagator in non-static Gui spacetime. The thermal nature of the propagator is understood in terms of homotopically non-trivial paths in the configuration space appropriate to tortoise coordinates. The connection to thermal emission from collapsing black holes is discussed.
In Parts I and II we showed that e, $ν$ propagators can be derived from rotation invariant projection operators, thereby providing examples of how quantities with spacetime symmetry can be obtained by constraining rotationally symmetric objects. One constraint is the restriction of the basis; only two kinds of bases were considered, one for the electron and one for the neutrino. In this part, we find that, of a wide range of bases each consistent with the constraint process, only the two kinds of bases considered in Parts I and II give spacetime symmetric propagators. We interpret the result g…
We study the propagator of a non-relativistic, non-interacting particle in any non-relativistic ``time-machine'' spacetime of the type shown in Fig.~1: an external, flat spacetime in which two spatial regions, $V_-$ at time $t_-$ and $V_+$ at time $t_+$, are connected by two temporal wormholes, one leading from the past side of $V_-$ to t the future side of $V_+$ and the other from the past side of $V_+$ to the future side of $V_-$. We express the propagator explicitly in terms of those for ordinary, flat spacetime and for the two wormholes; and from that expression we show that the propagator…
We study free scalar field theory on flat spacetime using a background independent (polymer) quantization procedure. Specifically we compute the propagator using a method that takes the energy spectrum and position matrix elements of the harmonic oscillator as inputs. We obtain closed form results in the infrared and ultraviolet regimes that give Lorentz invariance violating dispersion relations, and show suppression of propagation at sufficiently high energy.
We consider the wave equation on asymptotically Minkowski spacetimes and the Klein-Gordon equation on even asymptotically de Sitter spaces. In both cases we show that the extreme difference of propagators (i.e. retarded propagator minus advanced, or Feynman minus anti-Feynman), defined as Fredholm inverses, induces a symplectic form on the space of solutions with wave front set confined to the radial sets. Furthermore, we construct isomorphisms between the solution spaces and symplectic spaces of asymptotic data. As an application of this result we obtain distinguished Hadamard two-point funct…
We study the graviton propagator in euclidean loop quantum gravity, using the spinfoam formalism. We use boundary-amplitude and group-field-theory techniques, and compute one component of the propagator to first order, under a number of approximations, obtaining the correct spacetime dependence. In the large distance limit, the only term of the vertex amplitude that contributes is the exponential of the Regge action: the other terms, that have raised doubts on the physical viability of the model, are suppressed by the phase of the vacuum state, which is determined by the extrinsic geometry of …
We calculate the fermion propagator in FLRW spacetimes with constant deceleration $q=ε-1$, $ε=-\dot{H}/H^{2}$ for excited states. For fermions whose mass is generated by a scalar field through a Yukawa coupling $m=g_{\mathrm{\scriptscriptstyle{Y}}} ϕ$, we assume $ϕ\propto H$. We first solve for the mode functions by splitting the spinor into a direct product of helicity and chirality spinors. We also allow for non-vacuum states. We normalise the spinors using a consistent canonical quantisation and by requiring orthogonality of particle and anti-particle spinors. We apply our propagator to cal…
We hereby study the properties of a large class of weakly nonlocal gravitational theories around the (anti-) de Sitter spacetime background. In particular, we explicitly prove that the kinetic operator for the graviton field has the same structure as the one in Einstein-Hilbert theory around any maximally symmetric spacetime. Therefore, the perturbative spectrum is the same of standard general relativity, while the propagator on any maximally symmetric spacetime is a mere generalization of the one from Einstein's gravity derived and extensively studied in several previous papers. At quantum le…
The black-hole evaporation implies that the quantum-field propagators in a local Minkowski frame acquire a correction, which gives rise to this process. The modification of the propagators causes, in turn, non-trivial local effects due to the radiative/loop diagrams in non-linear QFTs. In particular, there should be imprints of the evaporation in QED, if one goes beyond the tree-level approximation. Of special interest in this respect is the region near the black-hole horizon, which, already at tree level, appears to show highly non-classical features, e.g., negative energy density and energy …
We study the propagation of quantum fields on $κ$-Minkowsi spacetime. Starting from the non-commutative partition function for a free field written in momentum space we derive the Feynman propagator and analyze the non-trivial singularity structure determined by the group manifold geometry of momentum space. The additional contributions due to such singularity structure result in a deformed field propagation which can be alternatively described in terms of an ordinary field propagation determined by a source with a blurred spacetime profile. We show that the $κ$-deformed Feynman propagator can…
We show that the propagators of gravitons and scalar fields seen by a static patch observer in de Sitter spacetime are controlled by hidden SL(2,R) symmetries, at all frequencies. The retarded Green's function is determined by an SL(2,R) x SL(2,R) action generated by conformal Killing vectors of de Sitter spacetime times a line. This observation uses the fact that the static patch of dS_{d+1} x R is conformal to the hyperbolic patch of AdS_3 x S^{d-1}. The poles of the propagators, the quasinormal frequencies, are generated by associated SL(2,R) actions. The quasinormal mode generating algebra…
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 w…
Closed-form expressions for smeared bi-distributions of a massless scalar field: non-perturbative and asymptotic results in relativistic quantum information
A one-parameter family of random variables, called the Discrete Action, is defined for a 2-dimensional Lorentzian spacetime of finite volume. The single parameter is a discreteness scale. The expectation value of this Discrete Action is calculated for various regions of 2D Minkowski spacetime. When a causally convex region of 2D Minkowski spacetime is divided into subregions using null lines the mean of the Discrete Action is equal to the alternating sum of the numbers of vertices, edges and faces of the null tiling, up to corrections that tend to zero as the discreteness scale is taken to zer…
It is shown that the isometry group of the de Sitter spacetime includes two different three-dimensional Abelian subgroups which transform between themselves through a discrete isometry corresponding to the time reversal in the five-dimensional Minkowski spacetime embedding the de Sitter one. The eigenfunctions of the generators of these Abelian subgroups form two different sets of quantum modes correlated by the mentioned isometry.
In the path integral expression for a Feynman propagator of a spinless particle of mass $m$, the path integral amplitude for a path of proper length ${\cal R}(x,x'| g_{μν})$ connecting events $x$ and $x'$ in a spacetime described by the metric tensor $g_{μν}$ is $\exp-[m {\cal R}(x,x'| g_{μν})]$. In a recent paper, assuming the path integral amplitude to be invariant under the duality transformation ${\cal R} \to (L_P^2/{\cal R})$, Padmanabhan has evaluated the modified Feynman propagator in an arbitrary curved spacetime. He finds that the essential feature of this `principle of path integral …
Exact results, transient generalized Gibbs ensembles, and analytic approximations for spacetime propagators of massive, real scalar fields in one spatial dimension
Starting from Wigner's symmetry representation theorem, we give a general account of discrete symmetries (parity P, charge conjugation C, time-reversal T), focusing on fermions in Quantum Field Theory. We provide the rules of transformation of Weyl spinors, both at the classical level (grassmanian wave functions) and quantum level (operators). Making use of Wightman's definition of invariance, we outline ambiguities linked to the notion of classical fermionic Lagrangian. We then present the general constraints cast by these transformations and their products on the propagator of the simplest a…
We borrow the minisuperspace approximation from Quantum Cosmology and the quenching approximation from QCD in order to derive a new form of the bosonic p-brane propagator. In this new approximation we obtain an exact description of both the collective mode deformation of the brane and the center of mass dynamics in the target spacetime. The collective mode dynamics is a generalization of string dynamics in terms of area variables. The final result is that the evolution of a p-brane in the quenched-minisuperspace approximation is formally equivalent to the effective motion of a particle in a sp…
In this paper, we present a family of regular black hole solutions in the presence of charge and angular momentum. We also discuss the related thermodynamics and we comment about the black hole life cycle during the balding and spin down phases. Interestingly the static solution resembles the Ayón-Beato--García spacetime, provided the T-duality scale is redefined in terms of the electric charge, $l_0\to Q$. The key factor at the basis of our derivation is the employment of Padmanabhan's propagator to calculate static potentials. Such a propagator encodes string T-duality effects. This means th…
We construct quantum effective action in spacetime with branes/boundaries. This construction is based on the reduction of the underlying Neumann type boundary value problem for the propagator of the theory to that of the much more manageable Dirichlet problem. In its turn, this reduction follows from the recently suggested Neumann-Dirichlet duality which we extend beyond the tree level approximation. In the one-loop approximation this duality suggests that the functional determinant of the differential operator subject to Neumann boundary conditions in the bulk factorizes into the product of i…
We consider a quantum scalar field with λϕ^4 interaction in curved spacetimes. The quantum effects are taken into account nonperturbatively using the Hartree approximation to the 2PI effective action. Although this approximation has been considered in many previous works, we reconsider it using a consistent nonperturbative renormalization procedure, which we extend to general curved spacetimes. We obtain the renormalized equations for the mean field and for the propagator of the fluctuations, showing explicitly their independence on the arbitrary scale introduced by the regularization scheme. …
A key incentive of quantum gravity is the removal of spacetime singularities plaguing the classical theory. We compute the non-perturbative momentum-dependence of a specific structure function within the gravitational asymptotic safety program which encodes the quantum corrections to the graviton propagator for momenta above the Planck scale. The resulting quantum corrected Newtonian potential approaches a constant negative value as the distance between the two point masses goes to zero, thereby removing the classical singularity. The generic nature of the underlying mechanism suggests that it…
