Formulation Of Quantum Field Theory Research Articles

A concrete construction of a topological operator in factorization algebras

Abstract Factorization algebras play a central role in the formulation of quantum field theories given by Kevin Costello and Owen Gwilliam. In this paper, we propose a concrete construction of a topological operator in their formulation. We focus on a shift symmetry of a one-dimensional massless scalar theory. And we give some discussions of $\mathbb {Z}$-gauging of it, i.e., compact scalar theory and its θ-vacuum.

Ftint: Calculating gradient-flow integrals with pySecDec

The program ftint is introduced which numerically evaluates dimensionally regularized integrals as they occur in the perturbative approach to the gradient-flow formalism in quantum field theory. It relies on sector decomposition in order to determine the coefficients of the individual orders in ϵ=(4−D)/2, where D is the space-time dimension. For that purpose, it implements an interface to the public library pySecDec. The current version works for massive and massless integrals up to three-loop level with vanishing external momenta, but the underlying method is extendable to more general cases.

Conservative Black Hole Scattering at Fifth Post-Minkowskian and First Self-Force Order

We compute the fifth post-Minkowskian (5PM) order contributions to the scattering angle and impulse of classical black hole scattering in the conservative sector at first self-force order using the worldline quantum field theory formalism. This challenging four-loop computation required the use of advanced integration-by-parts and differential equation technology implemented on high-performance computing systems. Use of partial fraction identities allowed us to render the complete integrand in a fully planar form. The resulting function space is simpler than expected: In the scattering angle, we see only multiple polylogarithms up to weight three and a total absence of the elliptic integrals that appeared at 4PM order. All checks on our result, both internal—cancellation of dimensional regularization poles and preservation of the on-shell condition—and external—matching the slow-velocity limit with the post-Newtonian (PN) literature up to 5PN order and matching the tail terms to the 4PM loss of energy—are passed. Published by the American Physical Society 2024

On the running and the UV limit of Wilsonian renormalization group flows

In nonperturbative formulation of quantum field theory, the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman type field correlators. Such a flow is a parametric family of ultraviolet (UV) regularized field correlators, the parameter being the strength of the UV regularization, and the instances with different strength of UV regularizations are linked by the renormalization group equation. Important RG flows are those which reach out to any UV regularization strengths. In this paper it is shown that for these flows a natural, mathematically rigorous generally covariant definition can be given, and that they form a topological vector space which is Hausdorff, locally convex, complete, nuclear, semi-Montel, Schwartz. That is, they form a generalized function space having favorable properties, similar to multivariate distributions. The other theorem proved in the paper is that for Wilsonian RG flows reaching out to all UV regularization strengths, a simple factorization formula holds in case of bosonic fields over flat (affine) spacetime: the flow always originates from a regularization-independent distributional correlator, and its running satisfies an algebraic ansatz. The conjecture is that this factorization theorem should generically hold, which is worth future investigations.

Spin and Susceptibility Effects of Electromagnetic Self-Force in Effective Field Theory.

The classic Abraham-Lorentz-Dirac self-force of pointlike particles is generalized within an effective field theory setup to include linear spin and susceptibility effects described perturbatively, in that setup, by effective couplings in the action. Electromagnetic self-interactions of the pointlike particle are integrated out using the in-in supersymmetric worldline quantum field theory formalism. Divergences are regularized with dimensional regularization, and the resulting equations of motion are in terms only of an external electromagnetic field and the particle degrees of freedom.

Light-front puzzles

Light-front formulations of quantum field theories have many advantages for computing electroweak matrix elements of strongly interacting systems and other quantities that are used to study hadronic structure. The theory can be formulated in Hamiltonian form so non-perturbative calculations of the strongly interacting initial and final states are in principle reduced to linear algebra. These states are needed for calculating parton distribution functions and other types of distribution amplitudes that are used to understand the structure of hadrons. Light-front boosts are kinematic transformations so the strongly interacting states can be computed in any frame. This is useful for computing current matrix elements involving electroweak probes where the initial and final hadronic states are in different frames related by the momentum transferred by the probe. Finally in many calculations the vacuum is trivial so the calculations can be formulated in Fock space. The advantages of light front-field theory would not be interesting if the light-front formulation was not equivalent to the covariant or canonical formulations of quantum field theory. Many of the distinguishing properties of light-front quantum field theory are difficult to reconcile with canonical or covariant formulations of quantum field theory. This paper discusses the resolution of some of the apparent inconsistencies in canonical, covariant and light-front formulations of quantum field theory. The puzzles that will be discussed are (1) the problem of inequivalent representations (2) the problem of the trivial vacuum (3) the problem of ill-posed initial value problems (4) the problem of rotational covariance (5) the problem of zero modes and (6) the problem of spontaneously broken symmetries.

Postawić kwantową teorię pola z głowy na nogi część 2

Despite the impression that can be gained from most of the standard textbooks, Dirac, Klein–Gordon and other wave equations do not constitute the basis of relativistic quantum field theory. In this article I attempt to show how it should be formulated properly and discuss some of its aspects which usually are presented unsatisfactorily. My aim is to cause the change in the way quantum field theory is taught.The text is split into three parts. In the second part I discuss the formulation of quantum field theory as a theory of interacting fields as well as the physical sense of the renormalization procedure.

Dissipative Scattering of Spinning Black Holes at Fourth Post-Minkowskian Order.

We compute the radiation reacted momentum impulse Δp_{i}^{μ}, spin kick ΔS_{i}^{μ}, and scattering angle θ between two scattered spinning massive bodies (black holes or neutron stars) using the N=1 supersymmetric worldline quantum field theory formalism up to fourth post-Minkowskian (4PM) order. Our calculation confirms the state-of-the-art nonspinning results, and extends them to include spin-orbit effects. Advanced multiloop Feynman integral technology including differential equations and the method of regions are applied and extended to deal with the retarded propagators arising in a causal description of the scattering dynamics. From these results we determine a complete set of radiative fluxes at subleading PM order: the 4PM radiated four-momentum and, via linear response, the 3PM radiated angular momentum, both again including spin-orbit effects.

Unitarity violation in field theories of Lee–Wick’s complex ghost

Abstract Theories with fourth-order derivatives, including the Lee–Wick finite QED model and quadratic gravity, have a better UV behavior, but the presence of negative metric ghost modes endangers unitarity. Noticing that the ghost acquires a complex mass by radiative corrections, Lee and Wick, in particular, claimed that such complex ghosts would never be created by collisions of physical particles because of energy conservation, so that the physical S-matrix unitarity must hold. We investigate the unitarity problem faithfully, working in the operator formalism of quantum field theory. When complex ghosts participate, a complex delta function (a generalization of the Dirac delta function) appears at each interaction vertex, which enforces a specific conservation law of complex energy. Its particular property implies that the naive Feynman rule is wrong if the four-momenta are assigned to the internal lines after taking account of the conservation law in advance. We show that complex ghosts are actually created and unitarity is violated in such fourth-order derivative theories. We also find a definite energy threshold below which ghosts cannot be created: The theories are unitary and renormalizable below the threshold.

Structural Algebraic Quantum Field Theory: Particles with Structure

Conventional quantum field theory is a method for studying structureless elementary particles. Non-elementary particles, on the other hand, are those with internal structure or particles that are made up of elementary constituents like the nucleons, which contain quarks and gluons. We introduce a structure-inclusive algebraic formulation of quantum field theory that could handle such particles and in which orthogonal polynomials play a central role. For simplicity, we consider non-elementary scalar particles in 3 + 1 Minkowski space-time and, in an appendix, we treat spinors having structure but in 1 + 1 space-time. We show how scattering calculations are done in this theory for an illustrative example with nonlinear coupling. The aim of this short exposé is to motivate further studies and research using this approach.

Conservative Scattering of Spinning Black Holes at Fourth Post-Minkowskian Order.

Using the N=1 supersymmetric, spinning worldline quantum field theory formalism, we compute the conservative spin-orbit part of the momentum impulse Δp_{i}^{μ}, spin kick ΔS_{i}^{μ}, and scattering angle θ from the scattering of two spinning massive bodies (black holes or neutron stars) up to fourth post-Minkowskian (PM) order. These three-loop results extend the state of the art for generically spinning binaries from 3PM to 4PM. They are obtained by employing recursion relations for the integrand construction and advanced multiloop Feynman integral technology in the causal (in-in) worldline quantum field theory framework to directly produce classical observables. We focus on the conservative contribution (including tail effects) and outline the computations for the dissipative contributions as well. Our spin-orbit results agree with next-to-next-to-next-to-leading-order post-Newtonian and test-body data in the respective limits. We also reconfirm the conservative 4PM nonspinning results.

Time-symmetric quantization of relativistic fields. 2. Electroweak theory. Observable effects of TSQ

In the method of time-symmetric quantization (TSQ) of relativistic systems, based on the Stückelberg-Feynman interpretation, the creation-annihilation operators of quanta of complex fields and massless gauge fields, are automatically normally ordered, and there is no vacuum energy and charge (Z. Zakir 2023, article 1). In this second article the application of TSQ to massive bosonic fields of electroweak theory and the observational consequences of TSQ are considered. It is shown that the vacuum of these fields is free of zero-point energy and zero-point charge, and thus a contribution of these fields to the cosmological constant is absent. A direct observational consequence of TSQ is crossing symmetry in particle physics. The observable effects, which were attributed to the zero-point energy of the vacuum, are actually generated by the fields of real charges, and there is no evidence of the existence of zero-point energy of fundamental fields. This fact contradicts the prediction of the standard formulation of quantum field theory, but indirectly confirms TSQ.

A Rigorous Derivation of the Functional Renormalisation Group Equation

The functional renormalisation group equation is derived in a mathematically rigorous fashion in a framework suitable for the Osterwalder–Schrader formulation of quantum field theory. To this end, we devise a very general regularisation scheme which retains reflection positivity as well as the infinite degrees of freedom including smoothness. Furthermore, it is shown how the classical limit is altered by the regularisation process leading to an inevitable breaking of translation invariance. We also give precise conditions for the convergence of the obtained theories upon removal of the regularisation.

Leptonic Decay Widths for the Composite System of Two Relativistic Fermions

The new relativistic semiclassical leptonic decay widths of vector mesons as the relativistic systems of two quarks with arbitrary masses interacting by means of the funnel-type potentials with Coulomb interaction are obtained. The behavior of the relativistic leptonic decay widths of vector mesons is investigated. Comparison of the behavior for new expression with its relativistic spinless analog is given. Consideration is conducted within the framework of completely covariant quasipotential approach in the Hamiltonian formulation of quantum field theory via a transition to the relativistic configurational representation in the case of two relativistic spin particles of arbitrary masses.

What quantum matter tells about quantisation of gravity in a statistical mechanics context

We propose a “guide” towards quantisation of gravity based on quantum matter in a statistical mechanics context. On one hand, a statistical mechanics model naturally arises from the thermodynamic interpretation of horizons in Rindler space. On the other hand, the path integral formulation of quantum field theory can be interpreted from the point of view of statistical mechanics. From these perspectives, gravity and matter are related to each other in the same way as a gas and its chemical potential are. This statistical mechanics interpretation of gravity and matter suggests that gravity should be quantised in a precise way which is determined by the quantisation of matter. Although, in a first step, quantisation of gravity applies for small perturbations of the metric with respect to the vacuum, the most central and general features of quantisation (quantisation prescription, quantum space dimensions) are supported from statistical mechanics and remain valid non-perturbatively.

Non-Abelian gauge structure and flavor mixing in quantum field theory1

The quantum field theory formalism describing the phenomenon of neutrino mixing and oscillations is reviewed in its essential aspects. The condensate structure of the flavor vacuum state is considered and its non-perturbative nature is discussed within the existence in the quantum field theory of the manifold of unitarily inequivalent representations of the anticommutation relations. The Poincaré structure of the theory is discussed in connection with the gauge theory features of neutrino mixing, where the possibility arises to describe flavored neutrinos as on-shell fields with definite masses. The flavor vacuum state may be thought of as a refractive medium where different refraction indexes are related to the different neutrino masses. Neglecting the condensate structure of the flavor vacuum implies neglecting second-order contributions in the gauge field coupling and leads to experimentally inconsistent results.

Time-symmetric quantization of relativistic fields. 1. Complex fields, massless gauge fields and gravitons

In the standard formulation of quantum field theory (QFT), where there are only positive energy particles and antiparticles, the energy and charge of the vacuum diverge, which, due to the existence of gravity, leads to the inconsistency of the theory (cosmological constant problem). In the article, it is shown that in the Stueckelberg-Feynman (SF) interpretation, where antiparticles are described as negative energy particles moving backward in time, the zero-point energy and zero-point charge of vacuum of complex fields are absent and there is no cosmological constant problem. However, until now it was believed that the SF interpretation leads to negative probabilities and incompatible with QFT. In the article, it is presented a new formulation of QFT on the basis of the SF interpretation in the form of time-symmetric quantization (TSQ), where the probability of states is positive. In TSQ, the consequences of the SF interpretation are taken into account consecutively and it is shown that: a) the negative sign of the norm of states only changes the sign of the wave function, and not the probabilities; b) the expression of backward in time integrals through the forward in time integrals changes sign; c) the time ordering of the operators is symmetric in time and writing them through the usual ordering leads to the standard diagram technique. For this reason, TSQ correctly describes the known observable effects. In TSQ, the results of unification models change, in particular, a) there is no zero-point energy even with broken supersymmetry between complex fields; b) there is no zero-point energy of modes in string theories, which allows to include gravity, but there is no a conformal anomaly and the dimension of space can be arbitrary.

Addressing energy density functionals in the language of path-integrals I: comparative study of diagrammatic techniques applied to the (0 + 0)-D O(N)-symmetric $$\varphi ^{4}$$-theory

The energy density functional (EDF) method is currently the only microscopic theoretical approach able to tackle the entire nuclear chart. Nevertheless, it suffers from limitations resulting from its empirical character and deteriorating its reliability. This paper is part of a larger program that aims at formulating the EDF approach as an effective field theory (EFT) in order to overcome these limitations. A relevant framework to achieve this is the path-integral formulation of quantum field theory. The latter indeed provides a wide variety of treatments of the many-body problem well suited to deal with non-perturbative interactions and to exploit a Lagrangian resulting from an EFT as a starting point. While developing the formalism in a general setting, we present below a comparative study of such techniques applied to a toy model, i.e. the (0 + 0)-D O(N)-symmetric $$\varphi ^{4}$$ -theory. More specifically, our focus will be on the following diagrammatic techniques: loop expansion, optimized perturbation theory and self-consistent perturbation theory. With these methods, we notably address the spontaneous breakdown of the O(N) symmetry with care especially since spontaneous symmetry breakings play a paramount role in current implementations of the EDF approach.

A consistent quantum field theory from dimensional reduction

We incorporate the concept of dimensional reduction at high energies within the perturbative formulation of quantum field theory (QFT). In this new framework, space and momentum integrations are modified by a weighting function incorporating an effective mass energy associated with the dimensional reduction scale. We quantize the theory within canonical formalism. We then show that it can be made finite in perturbation theory, free of renormalon ambiguities, and with better analytic behavior for infinitesimal coupling constant compared to standard QFT. The new approach reproduces the known results at low energies. One key feature of this class of models is that the coupling constant always reaches a fixed point in the ultraviolet (UV) region, making the models UV complete.

Linear response, Hamiltonian, and radiative spinning two-body dynamics

Using the spinning, supersymmetric worldline quantum field theory formalism we compute the momentum impulse and spin kick from a scattering of two spinning black holes or neutron stars up to quadratic order in spin at third post-Minkowskian (PM) order, including radiation-reaction effects and with arbitrarily misaligned spin directions. Parts of these observables, both conservative and radiative, are also inferred from lower-PM scattering data by extending Bini and Damour's linear response formula to include misaligned spins. By solving Hamilton's equations of motion we also use a conservative scattering angle to infer a complete 3PM two-body Hamiltonian including finite-size corrections and misaligned spin-spin interactions. Finally, we describe mappings to the bound two-body dynamics for aligned spin vectors: including a numerical plot of the binding energy for circular orbits compared with numerical relativity, analytic confirmation of the NNLO PN binding energy, and the energy loss over successive orbits.