Worldline Path Integral Research Articles

Pathwise differentiation of worldline path integrals

Pathwise differentiation of worldline path integrals

Worldline path integrals for the graviton

We present an extension to arbitrary dimensions of a worldline path integral approach to one-loop quantum gravity, which was previously formulated in four spacetime dimensions. By utilizing this method, we recalculate gauge invariant coefficients related to the UV divergences of quantum gravity. These gauge invariant coefficients were previously obtained in arbitrary dimensions through two alternative techniques: the quantization of the N=4 spinning particle that propagates the graviton on Einstein spaces and the more conventional heat kernel approach. Our worldline path integrals are closer to the latter method and are employed to compute the trace of the heat kernel.

Light bending from eikonal in worldline quantum field theory

Using the worldline quantum field theory (WQFT) formalism for classical scattering, we study the deflection of light by a heavy massive spinless/spinning object. WQFT requires the use of the worldline dressed propagator of a photon in a gravitational background, which we construct from first principles. The action required to set up the worldline path integral is constructed using auxiliary variables, which describe dynamically the spin degrees of freedom of the photon and take care of path ordering. We test the fully regulated path integral by recovering the photon-photon-graviton vertex. With the dressed propagator at hand, we follow the WQFT procedure by setting up the partition function and deriving the Feynman rules which can be used to evaluate it perturbatively. These rules depend on the auxiliary variables. The latter ultimately do not contribute in the geometric-optics regime, which realizes the equivalence between the scattering of a photon and a massive scalar with that of a massless and a massive scalar. Then, the calculation of the eikonal phase and the deflection angle simplifies considerably. Using the eikonal phase defined in terms of the partition function, we calculate explicitly the deflection angle at NLO in the spinless case, and at LO in the spinning case up to quadratic order in spin.

N-photon amplitudes in a plane-wave background

We use the worldline formalism to derive master formulas for the one-loop N-photon amplitudes in a plane-wave background, for both scalar and spinor quantum electrodynamics. This generalises previous work by Ilderton and Torgrimsson for the vacuum polarisation case, although with some change in methodology since, instead of evaluating the path integral on the semi-classical trajectory, we use the special kinematics of the plane-wave background to uncover the crypto-gaussian character of this type of worldline path integral.

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 $\phi(x)$ and its action functional $A[\phi(x)]$. 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.

A Monte Carlo approach to the worldline formalism in curved space

We present a numerical method to evaluate worldline (WL) path integrals defined on a curved Euclidean space, sampled with Monte Carlo (MC) techniques. In particular, we adopt an algorithm known as YLOOPS with a slight modification due to the introduction of a quadratic term which has the function of stabilizing and speeding up the convergence. Our method, as the perturbative counterparts, treats the non-trivial measure and deviation of the kinetic term from flat, as interaction terms. Moreover, the numerical discretization adopted in the present WLMC is realized with respect to the proper time of the associated bosonic point-particle, hence such procedure may be seen as an analogue of the time-slicing (TS) discretization already introduced to construct quantum path integrals in curved space. As a result, a TS counter-term is taken into account during the computation. The method is tested against existing analytic calculations of the heat kernel for a free bosonic point-particle in a D-dimensional maximally symmetric space.

String-inspired methods and the worldline formalism in curved space

.The worldline approach to Quantum Field Theory (QFT) allows to efficiently compute several quantities, such as one-loop effective actions, scattering amplitudes and anomalies, which are linked to particle path integrals on the circle. A helpful tool in the worldline formalism on the circle are string-inspired (SI) Feynman rules, which correspond to a specific way of factoring out a zero mode. In flat space this is known to generate no difficulties. In curved space, it was shown how to correctly achieve the zero mode factorization by applying BRST techniques to fix a shift symmetry. Using special coordinate systems, such as Riemann Normal Coordinates, implies the appearance of a non-linear map --originally introduced by Friedan-- which must be taken care of in order to obtain the correct results. In particular, employing SI Feynman rules, the map introduces further interactions in the worldline path integrals. In the present paper, we compute in closed form Friedan’s map for RNC coordinates in maximally symmetric spaces, and test the path integral model by computing trace anomalies. Our findings match known results.

Discrete worldline instantons

The semiclassical approximation of the worldline path integral is a powerful tool to study nonperturbative electron-positron pair creation in spacetime-dependent background fields. Finding solutions of the classical equations of motion, i.e., worldline instantons, is possible analytically only in special cases, and a numerical treatment is nontrivial as well. We introduce a completely general numerical approach based on an approximate evaluation of the discretized path integral that easily and robustly gives the full semiclassical pair production rate in nontrivial multidimensional fields, and apply it to some example cases.

New perspective on the Unruh effect

In this work, based on the worldline path integral representation of the vacuum energy in spacetime with a Lorentzian metric, we provide a new but complementary interpretation of the Unruh effect. We perform the quantization of the massless free scalar field in Rindler space specifying initial and final conditions. After quantization, the final outcome for the vacuum energy is interpreted as world line path integrals. In this picture we find that the Unruh radiation is made of real particles as well as real antiparticles. The prediction regarding the presence of antiparticles in the radiation might open new lines for experimental detection of the effect. We present a thought experiment which offers a clear picture and supports the new interpretation.

Chiral anomaly, Berry phase, and chiral kinetic theory from worldlines in quantum field theory

We outline a novel chiral kinetic theory framework for systematic computations of the Chiral Magnetic Effect (CME) in ultrarelativistic heavy-ion collisions. The real part of the fermion determinant in the QCD effective action is expressed as a supersymmetric world-line action of spinning, colored, Grassmanian point particles in background gauge fields, with equations of motion that are covariant generalizations of the Bargmann-Michel-Telegdi and Wong equations. Berry's phase is obtained in a consistent non-relativistic adiabatic limit. The chiral anomaly, in contrast, arises from the phase of the fermion determinant; its topological properties are therefore distinct from those of the Berry phase. We show that the imaginary contribution to the fermion determinant too can be expressed as a point particle world-line path integral and derive the corresponding anomalous axial vector current. Our results can be used to derive a covariant relativistic chiral kinetic theory including the effects of topological fluctuations that has overlap with classical-statistical simulations of the CME at early times and anomalous hydrodynamics at late times.

Induced Action for Conformal Higher Spins from Worldline Path Integrals

Conformal higher spin (CHS) fields, yet being non unitary, provide a remarkable example of a consistent interacting higher spin theory in flat space background, that is local to all orders. The non-linear action is defined as the logarithmically UV divergent part of a one-loop scalar effective action. In this paper we take a particle model, that describes the interaction of a scalar particle to the CHS background, and compute its path integral on the circle. We thus provide a worldline representation for the CHS action, and rederive its quadratic part. We plan to come back to the subject, to compute cubic and higher vertices, in a future work.

Schwinger pair production at finite temperature

Thermal corrections to Schwinger pair production are potentially important in particle physics, nuclear physics and cosmology. However, the lowest-order contribution, arising at one loop, has proved difficult to calculate unambiguously. We show that this thermal correction may be calculated for charged scalars using the worldline formalism, where each term in the decay rate is associated with a worldline instanton. We calculate all finite-temperature worldline instantons, their actions and fluctuations prefactors, thus determining the complete one-loop decay rate at finite temperature. The thermal contribution to the decay rate becomes nonzero at a threshold temperature $T=eE/2m$, above which it dominates the zero temperature result. This is the lowest of an infinite set of thresholds at $T=neE/2m$. The decay rate is singular at each threshold as a consequence of the failure of the quadratic approximation to the worldline path integral. We argue that that higher-order effects will make the decay rates finite everywhere, and model those effects by the inclusion of hard thermal loop damping rates. We also demonstrate that the formalism developed here generalizes to the case of finite-temperature pair production in inhomogeneous fields.

Worldline Path-Integral Representations for Standard Model Propagators and Effective Actions

We develop the formalism to do worldline calculations relevant for the Standard Model. For that, we first figure out the worldline representations for the free propagators of massless chiral fermions of a single generation and gauge bosons of the Standard Model. Then we extend the formalism to the massive and dressed cases for the fermions and compute the QED vertex. We then go over fermionic one-loop effective actions and anomalies. To our knowledge, in the places where there has been an attempt at deriving the gauge boson propagator, the derivation is somewhat contrived, and we believe our derivation is more straightforward. Moreover, our incorporation of internal degrees of freedom is novel and sports some new features. The derivation of the QED vertex is also new. The treatment of the fermionic one-loop effective actions leads to a particularly economical derivation of chiral anomalies and the gauge anomaly freedom in the Standard Model, improving upon the state of the art in the literature. The appropriate worldline formalism developed thus sets the stage for Standard Model calculations beyond the tree and one-loop cases that incorporate Bern-Kosower type formulae for multiloop amplitudes, relevant for processes at the LHC.

Worldline approach for numerical computation of electromagnetic Casimir energies: Scalar field coupled to magnetodielectric media

We present a worldline method for the calculation of Casimir energies for scalar fields coupled to magnetodielectric media. The scalar model we consider may be applied in arbitrary geometries, and it corresponds exactly to one polarization of the electromagnetic field in planar layered media. Starting from the field theory for electromagnetism, we work with the two decoupled polarizations in planar media and develop worldline path integrals, which represent the two polarizations separately, for computing both Casimir and Casimir-Polder potentials. We then show analytically that the path integrals for the transverse-electric (TE) polarization coupled to a dielectric medium converge to the proper solutions in certain special cases, including the Casimir-Polder potential of an atom near a planar interface, and the Casimir energy due to two planar interfaces. We also evaluate the path integrals numerically via Monte-Carlo path-averaging for these cases, studying the convergence and performance of the resulting computational techniques. While these scalar methods are only exact in particular geometries, they may serve as an approximation for Casimir energies for the vector electromagnetic field in other geometries.

Noncommutative U(1) gauge theory from a worldline perspective

We study pure noncommutative U(1) gauge theory representing its one-loop effective action in terms of a phase space worldline path integral. We write the quadratic action using the background field method to keep explicit gauge invariance, and then employ the worldline formalism to write the one-loop effective action, singling out UV-divergent parts and finite (planar and non-planar) parts, and study renormalization properties of the theory. This amounts to employ worldline Feynman rules for the phase space path integral, that nicely incorporate the Fadeev-Popov ghost contribution and efficiently separate planar and non-planar contributions. We also show that the effective action calculation is independent of the choice of the worldline Green's function, that corresponds to a particular way of factoring out a particle zero-mode. This allows to employ homogeneous string-inspired Feynman rules that greatly simplify the computation.

World-line instantons and the Schwinger effect as a Wentzel–Kramers–Brillouin exact path integral

A detailed study of the semiclassical expansion of the world line path integral for a charged relativistic particle in a constant external electric field is presented. We show that the Schwinger formula for charged particle pair production is reproduced exactly by the semiclassical expansion around classical instanton solutions when the leading order of fluctuations is taken into account. We prove that all corrections to this leading approximation vanish and that the WKB approximation to the world line path integral is exact.

Effective action for higher spin fields on (A)dS backgrounds

We study the one loop effective action for a class of higher spin fields by using a first-quantized description. The latter is obtained by considering spinning particles, characterized by an extended local supersymmetry on the worldline, that can propagate consistently on conformally flat spaces. The gauge fixing procedure for calculating the worldline path integral on a loop is delicate, as the gauge algebra contains nontrivial structure functions. Restricting the analysis on (A)dS backgrounds simplifies the gauge fixing procedure, and allows us to produce a useful representation of the one loop effective action. In particular, we extract the first few heat kernel coefficients for arbitrary even spacetime dimension D and for spin S identified by a curvature tensor with the symmetries of a rectangular Young tableau of D/2 rows and [S] columns.

On the Yang–Mills two-loop effective action with worldline methods

We derive the two-loop effective action for covariantly constant field strength of pure Yang–Mills theory in the presence of an infrared scale. The computation is done in the framework of the worldline formalism, based on a generalization procedure of constructing multiloop effective actions in terms of the bosonic worldline path integral. The two-loop β-function is correctly reproduced. This is the first derivation in the worldline formulation, and serves as a non-trivial check on the consistency of the multiloop generalization procedure in the worldline formalism.

Mode regularization forN= 1, 2 SUSY Sigma model

Worldline N=1 and N=2 supersymmetric sigma models in curved background are useful to describe spin one-half and spin one particles coupled to external gravity, respectively. It is well known that worldline path integrals in curved space require regularization: we present here the mode-regularization for these models, finding in particular the corresponding counterterms, both in the case of flat and curved indices for worldline fermions. For N=1, using curved indices we find a contribution to the counterterm from the fermions that cancels the contribution of the bosons, leading to a vanishing total counterterm and thus preserving the covariance and supersymmetry of the classical action. Conversely in the case of N=2 supersymmetries we obtain a non-covariant counterterm with both curved and flat indices. This work completes the analysis of the known regularization schemes for N=1,2 nonlinear sigma models in one dimension.

Worldline path integrals for a Dirac particle in a weak gravitational plane wave

The problem of a relativistic spinning particle interacting with a weak gravitational plane wave in (3+1) dimensions is formulated in the frame work of covariant supersymmetric path integrals. The relative Green function is expressed through a functional integral over bosonic trajectories that describe the external motion and fermionic variables that describe the spin degrees of freedom. The (3+1) dimensional problem is reduced to the (1+1) dimensional one by using an identity. Next, the relative propagator is exactly calculated and the wave functions are extracted.