Bilocal Operators Research Articles

Algebraic symmetries of the observables on the sky: Variable emitters and observers

In this paper we prove a number of exact relations between optical observables, such as trigonometric parallax, position drift, and the proper motion of a luminous source, in addition to the variations of redshift and the viewing angle. These relations are valid in general relativity for any spacetime, and they are of potential interest for astrometry and precise cosmology. They generalize the well-known Etherington’s reciprocity relation between the angular diameter distance and the luminosity distance. Similar to the Etherington’s relation, they hold independently of the spacetime metric, the positions, and the motions of a light source or an observer. We show that those relations follow from the symplectic property of the bilocal geodesic operator, i.e., the geometric object that describes the light propagation between two distant regions of a spacetime. The set of relations we present is complete in the sense that no other relations between those observables should hold, in general. In the meantime, we develop the mathematical machinery of the bilocal approach to light propagation in general relativity and its corresponding Hamiltonian formalism. Published by the American Physical Society 2024

Effective potential of composite operators in the first order region of the QCD phase transition

We propose a method to determine the effective potential of QCD from the gap equation by introducing the homotopy method between the solutions of the equation of motion. Via this method, the effective potential beyond the bare vertex approximation is obtained, which then generalizes the Cornwall-Jackiw-Tomboulis effective potential for the bilocal composite operators. Moreover, the extended effective potential is set to be a function of self-energy instead of the propagator, which is the key point for the potential to be bounded from below. We then investigate the extended effective potential in the cases of phase transition of the QCD vacuum with a small current quark mass, the first order phase transition of QCD at finite temperature, and high baryon chemical potential. In the former case, the effective potential shows as an inflection point at the critical mass where the multiple solutions of the Dyson-Schwinger equation (DSE) vanishes, which is consistent with that obtained by solving the DSE directly. For the latter case, the in-medium properties, such as the latent heat and the difference of trace anomaly, of QCD is obtained. Published by the American Physical Society 2024

Finite volume form factors in integrable theories

We develop a new method to calculate finite size corrections for form factors in two-dimensional integrable quantum field theories. We extract these corrections from the excited state expectation value of bilocal operators in the limit when the operators are far apart. We elaborate the finite size effects explicitly up to the 3rd Lüscher order and conjecture the structure of the general form. We also fully recover the explicitly known massive fermion finite volume form factors.

Meson spectrum of SU(2) QCD1+1 with quarks in Large representations

We consider SU(2) quantum chromodynamics in 1 + 1 dimensions with a single quark in the spin J representation of the gauge group and study the theory in the large J limit where the gauge coupling g2 → 0 and J → ∞ with λ = g2J2 fixed. We work with a Dirac spinor field for arbitrary J, and with a Majorana spinor for integer J since the integer spin representations of SU(2) are real, and analyze the two cases separately.The theory is reformulated in terms of global colour non-singlet fermion bilocal operators which satisfy a W∞ × U(2J + 1) algebra. In the large J limit, the dynamics of the bilocal fields is captured by fluctuations along a particular coadjoint orbit of the W∞ algebra. We show that the global colour-singlet sector of the bilocal field fluctuations satisfy the same integral equation for meson wavefunctions that appears in the ’t Hooft model. For Majorana spinors in the integer spin J representation, the Majorana condition projects out half of the meson spectrum, as a result of which the linear spacing of the asymptotic meson spectrum for Majorana fermions is double that of Dirac fermions. The Majorana condition also projects out the zero mass bound state that is present for the Dirac quark at zero quark mass.We also consider the formulation of the model in terms of local charge densities and compute the quark spectral function in the large J limit: we see evidence for the absence of a pole in the quark propagator.

Pseudo and quasi quark PDF in the BFKL approximation

I examine the high-energy behavior of the Ioffe-time distribution for the quark bi-local space-like separated operator using the high-energy operator product expansion. These findings have significant implications for lattice calculations, which require extrapolation for large Ioffe-time values. I perform an explicit Fourier transform for both the pseudo-PDF and quasi-PDF, and investigate their behavior within the first two leading twist contributions.I show that the quark pseudo-PDF captures the BFKL resummation (resummation of all twists) and exhibits a rising behavior for small xB values, while the quasi-PDF presents a different behavior. I demonstrate that an appropriate small-xB behavior cannot be achieved solely through DGLAP dynamics, emphasizing the importance of all-twist resummation. This study provides valuable insights into quark non-local operators’ high-energy behavior and the limitations of lattice calculations in this context.

Finite temperature negativity Hamiltonians of the massless Dirac fermion

The negativity Hamiltonian, defined as the logarithm of a partially transposed density matrix, provides an operatorial characterisation of mixed-state entanglement. However, so far, it has only been studied for the mixed-state density matrices corresponding to subsystems of globally pure states. Here, we consider as a genuine example of a mixed state the one-dimensional massless Dirac fermions in a system at finite temperature and size. As subsystems, we consider an arbitrary set of disjoint intervals. The structure of the corresponding negativity Hamiltonian resembles the one for the entanglement Hamiltonian in the same geometry: in addition to a local term proportional to the stress-energy tensor, each point is non-locally coupled to an infinite but discrete set of other points. However, when the lengths of the transposed and non-transposed intervals coincide, the structure remarkably simplifies and we retrieve the mild non-locality of the ground state negativity Hamiltonian. We also conjecture an exact expression for the negativity Hamiltonian associated to the twisted partial transpose, which is a Hermitian fermionic matrix. We finally obtain the continuum limit of both the local and bi-local operators from exact numerical computations in free-fermionic chains.

Virasoro blocks and the reparametrization formalism

An effective theory designed to compute Virasoro identity blocks at large central charge, expressed in terms of the propagation of a reparametrization/shadow mode between bilocal vertices, was recently put forward. In this paper I provide the formal theoretical framework underlying this effective theory by reformulating it in terms of standard concepts: conformal geometry, generating functionals and Feynman diagrams. A key ingredient to this formalism is the bilocal vertex operator, or reparametrized two-point function, which is shown to generate arbitrary stress tensor insertions into a two-point function of reference. I also suggest an extension of the formalism designed to compute generic Virasoro blocks.

Entanglement and negativity Hamiltonians for the massless Dirac field on the half line

We study the ground-state entanglement Hamiltonian of several disjoint intervals for the massless Dirac fermion on the half-line. Its structure consists of a local part and a bi-local term that couples each point to another one in each other interval. The bi-local operator can be either diagonal or mixed in the fermionic chiralities and it is sensitive to the boundary conditions. The knowledge of such entanglement Hamiltonian is the starting point to evaluate the negativity Hamiltonian, i.e. the logarithm of the partially transposed reduced density matrix, which is an operatorial characterisation of entanglement of subsystems in mixed states. We find that the negativity Hamiltonian inherits the structure of the corresponding entanglement Hamiltonian. We finally show how the continuum expressions for both these operators can be recovered from exact numerical computations in free-fermion chains.

Bulk locality and gauge invariance for boundary-bilocal cubic correlators in higher-spin gravity

We consider type-A higher-spin gravity in 4 dimensions, holographically dual to a free O(N) vector model. In this theory, the cubic correlators of higher-spin boundary currents are reproduced in the bulk by the Sleight-Taronna cubic vertex. We extend these cubic correlators from local boundary currents to bilocal boundary operators, which contain the tower of local currents in their Taylor expansion. In the bulk, these boundary bilocals are represented by linearized Didenko-Vasiliev (DV) “black holes”. We argue that the cubic correlators are still described by local bulk structures, which include a new vertex coupling two higher-spin fields to the “worldline” of a DV solution. As an illustration of the general argument, we analyze numerically the correlator of two local scalars and one bilocal. We also prove a gauge-invariance property of the Sleight-Taronna vertex outside its original range of applicability: in the absence of sources, it is invariant not just within transverse-traceless gauge, but rather in general traceless gauge, which in particular includes the DV solution away from its “worldline”.

Higher-spin gravity’s “string”: new gauge and proof of holographic duality for the linearized Didenko-Vasiliev solution

We consider type-A higher-spin gravity in AdS4, holographically dual to a free U(N ) vector model on the boundary. We study the linearized version of the Didenko-Vasiliev “BPS black hole”, which we view as this theory’s equivalent of the fundamental string. The Didenko-Vasiliev solution consists of gauge fields of all spins generated by a particle-like source along a bulk geodesic, and is holographically dual to a bilocal boundary operator at the geodesic’s endpoints. Our first main result is a new gauge for this solution, which makes manifest its behavior under the boundary field equation. It can be viewed as an AdS uplift of flat spacetime’s de Donder gauge, but is not de Donder in AdS. To our knowledge, this gauge is novel even in the spin-2 sector, and thus provides a new expression for the linearized gravitational field of a massive point particle in (A)dS4. Our second main result is a proof of the holographic duality between the mutual bulk action of two Didenko-Vasiliev solutions and the CFT correlator of two boundary bilocals. As an intermediate step, we show that in a bilocal→local limit, the Didenko-Vasiliev solution reproduces the standard boundary-bulk propagators of all spins. We work in the “metric-like” language of Fronsdal fields, and use the embedding-space formalism.

Conformal blocks and bilocal vertex operator transition amplitudes

We revisit the construction of the 2d conformal blocks of primary operator four-point functions as bilocal vertex operator correlators. We find an additional interpretation as a path integral over the reparametrizations of an intermediate cylinder. As a consequence we bridge the gap between the Kähler quantization of virasoro coadjoint orbits, SL(2, ℝ) Chern-Simons theory and the reparametrization formalism of 2d CFT that has made an appearance in recent literature.

Complex saddles and Euclidean wormholes in the Lorentzian path integral

We study complex saddles of the Lorentzian path integral for 4D axion gravity and its dual description in terms of a 3-form flux, which include the Giddings-Strominger Euclidean wormhole. Transition amplitudes are computed using the Lorentzian path integral and with the help of Picard-Lefschetz theory. The number and nature of saddles is shown to qualitatively change in the presence of a bilocal operator that could arise, for example, as a result of considering higher-topology transitions. We also analyze the stability of the Giddings-Strominger wormhole in the 3-form picture, where we find that it represents a perturbatively stable Euclidean saddle of the gravitational path integral. This calls into question the ultimate fate of such solutions in an ultraviolet-complete theory of quantum gravity.

Supergroup structure of Jackiw-Teitelboim supergravity

We develop the gauge theory formulation of mathcal{N} = 1 Jackiw-Teitelboim supergravity in terms of the underlying OSp(1|2, ℝ) supergroup, focusing on boundary dynamics and the exact structure of gravitational amplitudes. We prove that the BF description reduces to a super-Schwarzian quantum mechanics on the holographic boundary, where boundary-anchored Wilson lines map to bilocal operators in the super-Schwarzian theory. A classification of defects in terms of monodromies of OSp(1|2, ℝ) is carried out and interpreted in terms of character insertions in the bulk. From a mathematical perspective, we construct the principal series representations of OSp(1|2, ℝ) and show that whereas the corresponding Plancherel measure does not match the density of states of mathcal{N} = 1 JT supergravity, a restriction to the positive subsemigroup OSp+(1|2, ℝ) yields the correct density of states, mirroring the analogous results for bosonic JT gravity. We illustrate these results with several gravitational applications, in particular computing the late-time complexity growth in JT supergravity.

Bilocal geodesic operators in static spherically-symmetric spacetimes

We present a method to compute exact expressions for optical observables for static spherically symmetric spacetimes in the framework of the bilocal geodesic operator formalism. The expressions are obtained by solving the linear geodesic deviation equations for null geodesics, using the spacetime symmetries and the associated conserved quantities. We solve the equations in two different ways: by varying the geodesics with respect to their initial data and by directly integrating the equation for the geodesic deviation. The results are very general and can be applied to a variety of spacetime models and configurations of the emitter and the observer. We illustrate some of the aspects with an example of Schwarzschild spacetime, focusing on the behaviour of the angular diameter distance, the parallax distance, and the distance slip between the observer and the emitter outside the photon sphere.

Analytic interpolation between the Ji and Jaffe-Manohar definitions of the orbital angular momentum distribution of gluons at small x

In this work, we first introduced a generalized Wilson line gauge link that reproduces both staple and near straight links along a light cone in different parameter limits. We then studied the gauge-invariant bilocal orbital angular momentum operator with such a general gauge link. At the appropriate combination of limits, the operator reproduces both Jaffe-Manohar's and Ji's operator structures and offers a continuous analytical interpolation between the two.

Testing the null energy condition with precise distance measurements

We present an inequality between two types of distance measures to a single source in general relativity. It states that for a given emitter and observer the distance between them measured by the trigonometric parallax is never shorter than the angular diameter distance provided that the null energy condition holds and that there are no focal points in between. This result is independent of the details of the spacetime geometry or the motions of the observer and the source. The proof is based on the geodesic bilocal operator formalism together with well known properties of infinitesimal light ray bundles. Observation of the violation of the distance inequality would mean that on large scales either the null energy condition does not hold or that light does not travel along null geodesics.

Pseudo and quasi gluon PDF in the BFKL approximation

I study the behavior of the gauge-invariant gluon bi-local operator with space- like separation at large longitudinal distances. Performing the Fourier transform, I also calculate the behavior of the pseudo and quasi gluon PDF at low Bjorken x and compare it with the leading and next-to-leading twist approximation. I show that the pseudo-PDF and quasi-PDF are very different at this regime and that the higher twist corrections of the quasi-PDF come in not as inverse powers of P but as inverse powers of xBP.

Short-distance structure of unpolarized gluon pseudodistributions

We present the results that form the basis for calculations of the unpolarized gluon parton distributions (PDFs) using the pseudo-PDF approach. We give the results for the most complicated box diagram both for gluon bilocal operators with arbitrary indices and for combinations of indices corresponding to three matrix elements that are most convenient to extract the twist-2 invariant amplitude. We also present detailed results for the gluon-quark transition diagram. The additional results for the box diagram and the gluon-quark contribution may be used for extractions of the gluon PDF from different matrix elements, with a possible cross-check of the results obtained in this way.

BiGONLight: light propagation with bilocal operators in numerical relativity

BiGONLight, Bilocal Geodesic Operators framework for Numerical Light propagation, is a new tool for light propagation in numerical relativity. The package implements the bilocal geodesic operators formalism, a new framework for light propagation in general relativity. With BiGONLight it is possible to extract observables such as angular diameter distance, luminosity distance, magnification as well as new real-time observables like parallax and redshift drift within the same computation. As a test-bed for our code we consider two exact cosmological models, the ΛCDM and the inhomogeneous Szekeres model, and a simulated dust Universe. All our tests show an excellent agreement with known results.

Isolating nonlinearities of light propagation in inhomogeneous cosmologies

A new formulation for light propagation in geometric optics by means of the Bi-local Geodesic Operators is considered. We develop the BiGONLight Mathematica package, uniquely designed to apply this framework to compute optical observables in Numerical Relativity. Our package can be used for light propagation on a wide range of scales and redshifts and accepts numerical as well as analytical input for the spacetime metric. In this paper we focus on two cosmological observables, the redshift and the angular diameter distance, specializing our analysis to a wall universe modeled within the post-Newtonian approximation. With this choice and the input metric in analytical form, we are able to estimate non-linearities of light propagation by comparing and isolating the contributions coming from Newtonian and post-Newtonian approximations as opposed to linear perturbation theory. We also clarify the role of the dominant post-Newtonian contribution represented by the linear initial seed which, strictly speaking, is absent in the Newtonian treatment. We found that post-Newtonian non-linear corrections are below $1\%$, in agreement with previous results in the literature.