Interacting Quantum Field Theory Research Articles

Gravitational algebras and the generalized second law

We derive the generalized second law (GSL) for arbitrary cuts of Killing horizons from the perspective of crossed-product gravitational algebras, making use of a recent proposal by one of us for the construction of local gravitational algebras. This construction relies on the existence of a state whose modular flow is geometric on the horizon. In both free and interacting quantum field theories, such states are guaranteed to exist by the properties of half-sided translations on the horizon. Using geometric identities derived from the canonical analysis of general relativity on null surfaces, we show that the crossed product entropy agrees with the generalized entropy of the horizon cut in a semiclassical limit, and further reproduce Wall’s result relating the GSL to monotonicity of relative entropy of the quantum field algebras. We also give a novel generalization of the GSL for interacting theories in asymptotically flat spacetimes involving the concept of an algebra at infinity for a half-sided translation, which accounts for triviality of the algebra of fields smeared only on the horizon. Going beyond the semiclassical limit, we compute subleading corrections to the crossed product entropy, but are unable to determine if the GSL continues to hold after accounting for these. We speculate that an improved GSL could follow from a hidden subalgebra structure of the crossed products, assuming the existence of an operator-valued weight between horizon cut algebras.

New purely damped pairs of quasinormal modes in a hot and dense strongly-coupled plasma

Perturbed black holes exhibit damped oscillations whose eigenfrequencies define their quasinormal modes (QNMs). In the case of asymptotically Anti-de Sitter (AdS) black holes, the spectra of QNMs are related to the near-equilibrium behavior of specific strongly interacting quantum field theories via the holographic gauge-gravity duality. In the present work, we numerically obtain the spectra of homogeneous non-hydrodynamic QNMs of a top-down holographic construction called the 2 R-Charge Black Hole (2RCBH) model, which describes a hot and dense strongly-coupled plasma. The main result is the discovery of a new structure of pairs of purely imaginary QNMs. Those new purely damped QNMs dominate the late time equilibration of the strongly-coupled plasma at large values of the chemical potential, while at lower values the fundamental QNMs are instead ordinary poles with imaginary and real parts describing oscillatory decaying perturbations. We also observe a new phenomenon of asymptotic pole fusion for different pairs of purely imaginary QNMs at asymptotically large values of the chemical potential. This phenomenon corresponds to the asymptotic merging of the two poles within each pair of purely imaginary QNMs, with the different pairs of merged poles being evenly spaced by a constant value of 4π in all the different perturbation channels associated to different irreducible representations of the spatial SO(3) rotation symmetry of the medium. In particular, this indicates that characteristic equilibration times for the plasma develop upper bounds that cannot be surpassed by further doping the medium with increasing values of the chemical potential.

Quantum field theories of relativistic Luttinger fermions

We propose relativistic Luttinger fermions as a new ingredient for the construction of fundamental quantum field theories. We construct the corresponding Clifford algebra and the spin metric for relativistic invariance of the action using the spin-base invariant formalism. The corresponding minimal spinor has 32 complex components, matching with the degrees of freedom of a standard-model generation including a right-handed neutrino. The resulting fermion fields exhibit a canonical scaling different from Dirac fermions and thus support the construction of novel relativistic and perturbatively renormalizable, interacting quantum field theories. In particular, new asymptotically free self-interacting field theories can be constructed, representing first examples of high-energy complete quantum field theories based on pure matter degrees of freedom. Gauge theories with relativistic Luttinger fermions exhibit a strong paramagnetic dominance, requiring large non-Abelian gauge groups to maintain asymptotic freedom. We comment on the possibility to use Luttinger fermions for particle physics model building and the expected naturalness properties of such models. Published by the American Physical Society 2024

Local Solutions of RG Flow Equations from the Nash–Moser Theorem

We prove local existence of solutions of a functional renormalisation group equation for the effective action of an interacting quantum field theory, when a suitable local potential approximation is considered. To obtain this equation in a Lorentzian setting a quantum state for the theory is selected and a regulator consisting in a mass is added to the action. The flow equation for mass rescalings is then studied using the renown Nash–Moser theorem.

Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods

As a simple lattice model that exhibits a phase transition, the Ising model plays a fundamental role in statistical and condensed matter physics. The Ising transition is realized by physical systems, such as the liquid-vapor transition. Its continuum limit also furnishes a basic example of interacting quantum field theories and universality classes. Motivated by a recent hybrid bootstrap study of the quantum quartic oscillator, we revisit the conformal bootstrap approach to the 3D Ising model at criticality, without resorting to positivity constraints. We use at most 10 nonperturbative crossing constraints at low derivatives from the Taylor expansion around a crossing symmetric point. The high-lying contributions are approximated by simple analytic formulae deduced from the lightcone singularity structure. Surprisingly, the low-lying properties are determined to good accuracy by this computationally very cheap approach. For instance, the results for the two relevant scaling dimensions (∆σ, ∆ϵ) ≈ (0.518153, 1.41278) are close to the most precise rigorous bounds obtained at a much higher computational cost.

Symmetry restoration and uniformly accelerated observers in Minkowski spacetime

We reassess the problem of symmetry restoration induced by observers’ acceleration within the context of interacting quantum field theories in Minkowski spacetime. We argue that the imposition of a frame-independent renormalization condition negates any observed symmetry restoration by a Rindler observer. Technically, we compute the one-loop effective potential of a λφ4 theory for an accelerated observer, employing a distinct methodology from prior investigations. Emphasizing the intricacies of the model’s renormalization, the analysis offers novel insights into the interplay between acceleration and spontaneous symmetry breaking in quantum field theory.

In-in correlators and scattering amplitudes on a causal set

Causal set theory is an approach to quantum gravity in which spacetime is fundamentally discrete at the Planck scale and takes the form of an irregular Lorentzian lattice, or “causal set,” from which continuum spacetime emerges in a large-scale (low-energy) approximation. In this work, we present new developments in the framework of interacting quantum field theory on causal sets. We derive a diagrammatic expansion for in-in correlators in local scalar field theories with finite polynomial interactions. We outline how these same correlators can be computed using the double-path integral, which acts as a generating functional for the in-in correlators. We modify the in-in generating functional to obtain a generating functional for in-out correlators. We define a notion of scattering amplitudes on causal sets with noninteracting past and future regions and verify that they are given by S-matrix elements (matrix elements of the time-evolution operator). We describe how these formal developments can be implemented to compute early Universe observables under the assumption that spacetime is fundamentally discrete. Published by the American Physical Society 2024

Perturbative construction of coherent states

The perturbative consistency of coherent states within interacting quantum field theory requires them to be altered beyond the simple nonsqueezed form. Building on this point, we perform explicit construction of consistent squeezed coherent states, required by the finiteness of physical quantities at the one-loop order. Extending this analysis to two loops, we demonstrate that a non-Gaussian alteration of squeezed coherent states is necessary. The modifications of the coherent state we propose are perturbative in ℏ and may be an indication that coherence must be viewed through a nonlinearly redefined, background-dependent, degree of freedom. Published by the American Physical Society 2024

Correlation function distributions for O(N) lattice field theories in the disordered phase

Numerical computations in strongly interacting quantum field theories are often performed using Monte Carlo sampling methods. A key task in these calculations is to estimate the value of a given physical quantity from the distribution of stochastic samples that are generated using the Monte Carlo method. Typically, the sample mean and sample variance are used to define the expectation values and uncertainties of computed quantities. However, the Monte Carlo sample distribution contains more information than these basic properties, and it is useful to investigate it more generally. In this work, the exact form of the probability distributions of two-point correlation functions at zero momentum in O(N) lattice field theories in the disordered phase and in infinite volume are determined. These distributions allow for a robust investigation of the efficacy of the Monte Carlo sampling procedure and are shown also to allow for improved estimators of the target physical quantity to be constructed. The theoretical expectations are shown to agree with numerical calculations in the O(2) model. Published by the American Physical Society 2024

FeynMG: A FeynRules extension for scalar-tensor theories of gravity

The ability to represent perturbative expansions of interacting quantum field theories in terms of simple diagrammatic rules has revolutionized calculations in particle physics (and elsewhere). Moreover, these rules are readily automated, a process that has catalyzed the rise of symbolic algebra packages. However, in the case of extended theories of gravity, such as scalar-tensor theories, it is necessary to precondition the Lagrangian to apply this automation or, at the very least, to take advantage of existing software pipelines. We present a Mathematica code FeynMG, which works in conjunction with the well-known package FeynRules, to do just that: FeynMG takes as inputs the FeynRules model file for a non-gravitational theory and a user-supplied gravitational Lagrangian. FeynMG provides functionality that inserts the minimal gravitational couplings of the degrees of freedom specified in the model file, determines the couplings of the additional tensor and scalar degrees of freedom (the metric and the scalar field from the gravitational sector), and preconditions the resulting Lagrangian so that it can be passed to FeynRules, either directly or by outputting an updated FeynRules model file. The Feynman rules can then be determined and output through FeynRules, using existing universal output formats and interfaces to other analysis packages. Program summaryProgram title: FeynMGCPC Library link to program files:https://doi.org/10.17632/bjpfmtt55k.1Developer's repository link:https://gitlab.com/feynmg/FeynMGLicensing provisions: MIT licenseProgramming language:Mathematica 13.2Nature of problem: Determining the additional interactions that scalar-tensor theories of gravity induce between the fields of the Standard Model of particle physics and its extensions is a tedious and time-consuming task that is ripe for automation. FeynMG is a package that provides this automation. It can expand the spacetime metric around a flat background and perform the necessary field redefinitions for a given theory in order to provide a Lagrangian that is in a form that can be processed by FeynRules[1]. The Feynman rules for the theory can then be determined using the existing functionality of FeynRules and output in formats that can be read into other particle-physics analysis packages.Solution method: (1) Load both FeynRules and FeynMG into Mathematica. (2) Load a model file describing a given matter sector and Lagrangian. (3) Use the FeynMG implementation to: (3a) append the gravitational sector and incorporate curvature-dependent objects, such as the metric, curvature scalars and tensors, and covariant derivatives; (3b) effect a Weyl transformation or expand the spacetime metric up to second order in graviton interactions with the matter fields; and (3c) diagonalize mass and kinetic mixings, and canonically normalize dynamical fields. (4) Pass the output directly to FeynRules or output a new model file from FeynMG.

The 𝒫(𝜑)₂ Model on de Sitter Space

In 1975 Figari, Høegh-Krohn and Nappi constructed theP(φ)2{\mathscr P}(\varphi )_2model on the de Sitter space. Here we complement their work with new results, which connect this model to various areas of mathematics. In particular, i.) we discuss the causal structure of de Sitter space and the induces representations of the Lorentz group. We show that the UIRs ofSO0(1,2)SO_0(1,2)for both the principal and the complementary series can be formulated on Hilbert spaces whose functions are supported on a Cauchy surface. We describe the free classical dynamical system in both its covariant and canonical form, and present the associated quantum one-particle KMS structures in the sense of Kay (1985). Furthermore, we discuss the localisation properties of one-particle wave functions and how these properties are inherited by the algebras of local observables.ii.) we describe the relations between the modular objects (in the sense of Tomita-Takesaki theory) associated to wedge algebras and the representations of the Lorentz group. We connect the representations of SO(1,2) to unitary representations ofSO(3)SO(3)on the Euclidean sphere, and discuss how theP(φ)2{\mathscr P}(\varphi )_2interaction can be represented by a rotation invariant vector in the Euclidean Fock space. We present a novel Osterwalder-Schrader reconstruction theorem, which shows that physicalinfrared problemsare absent on de Sitter space. As shown in Figari, Høegh-Krohn, and Nappi (1975), the ultraviolet problems are resolved just like on flat Minkowski space. We state the Haag–Kastler axioms for theP(φ)2{\mathscr P}(\varphi )_2model and we explain how thegeneratorsof the boosts and the rotations for the interacting quantum field theory arise from thestress-energy tensor. Finally, we show that the interacting quantum fields satisfy theequations of motionin their covariant form. In summary, we argue that the de SitterP(φ)2{\mathscr P}(\varphi )_2model is the simplest and most explicit relativistic quantum field theory, which satisfies basic expectations, like covariance, particle creation, stability and finite speed of propagation.

Memories of quenches in operator mixing

We work perturbatively with an interacting quantum field theory comprised of two distinct scalar fields. In this theory, we introduce a sudden quench of the mass of one of the scalars at time $t_0$. Also, the quartic interaction between the two scalars is turned on at time $t_{in}$. These break time-translation invariance. In this setup we examine the effects of the relative ordering of $t_0$ and $t_{in}$ on composite operator mixing. We study how such operator mixing affect features of the scalar potential. We find that the late time effective potential can be sensitive enough to the quenches to trigger phase transitions.

C*-algebraic approach to interacting quantum field theory: inclusion of Fermi fields

We extend the C*-algebraic approach to interacting quantum field theory, proposed recently by Detlev Buchholz and one of us (KF) to Fermi fields. The crucial feature of our approach is the use of auxiliary Grassmann variables in a functorial way.

On generally covariant mathematical formulation of Feynman integral in Lorentz signature

It is widely accepted that the Feynman integral is one of the most promising methodologies for defining a generally covariant formulation of nonperturbative interacting quantum field theories (QFTs) without a fixed prearranged causal background. Recent literature suggests that if the spacetime metric is not fixed, e.g. because it is to be quantized along with the other fields, one may not be able to avoid considering the Feynman integral in the original Lorentz signature, without Wick rotation. Several mathematical phenomena are known, however, which are at some point showstoppers to a mathematically sound definition of Feynman integral in Lorentz signature. The Feynman integral formulation, however, is known to have a differential reformulation, called to be the master Dyson–Schwinger (MDS) equation for the field correlators. In this paper it is shown that a particular presentation of the MDS equation can be cast into a mathematically rigorously defined form: the involved function spaces and operators can be strictly defined and their properties can be established. Therefore, MDS equation can serve as a substitute for the Feynman integral, in a mathematically sound formulation of constructive QFT, in arbitrary signature, without a fixed background causal structure. It is also shown that even in such a generally covariant setting, there is a canonical way to define the Wilsonian regularization of the MDS equation. The main result of the paper is a necessary and sufficient condition for the regularized MDS solution space to be nonempty, for conformally invariant Lagrangians. This theorem also provides an iterative approximation algorithm for obtaining regularized MDS solutions, and is guaranteed to be convergent whenever the solution space is nonempty. The algorithm could eventually serve as a method for putting Lorentz signature QFTs onto lattice, in the original metric signature.

The Unitary Master Ward Identity: Time Slice Axiom, Noether’s Theorem and Anomalies

The C*-algebraic formulation of generic interacting quantum field theories, recently presented by Detlev Buchholz and one of the authors (KF), is enriched by a unitary version of the Master Ward Identity, which was postulated some time ago by Franz Marc Boas, Ferdinand Brennecke and two of us (MD,KF). It is shown that the corresponding axiom implies the validity of the time slice axiom. Moreover, it opens the way for a new approach to Noether’s Theorem where it yields directly the unitaries implementing the symmetries. It also unravels interesting aspects of the role of anomalies in quantum field theory.

Reduced density matrix and entanglement of interacting quantum field theories with Hamiltonian truncation

Entanglement is the fundamental difference between classical and quantum systems and has become one of the guiding principles in the exploration of high- and low-energy physics. The calculation of entanglement entropies in interacting quantum field theories, however, remains challenging. Here, we present the first method for the explicit computation of reduced density matrices of interacting quantum field theories using truncated Hamiltonian methods. The method is based on constructing an isomorphism between the Hilbert space of the full system and the tensor product of Hilbert spaces of subintervals. This naturally enables the computation of the von Neumann and arbitrary Rényi entanglement entropies as well as mutual information, logarithmic negativity, and other measures of entanglement. Our method is applicable to equilibrium states and nonequilibrium evolution in real time. It is model independent and can be applied to any Hamiltonian truncation method that uses a free basis expansion. We benchmark the method on the free Klein-Gordon theory finding excellent agreement with the analytic results. We further demonstrate its potential on the interacting sine-Gordon model, studying the scaling of von Neumann entropy in ground states and real-time dynamics following quenches of the model.5 MoreReceived 17 March 2022Accepted 17 June 2022DOI:https://doi.org/10.1103/PhysRevResearch.4.033039Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasEntanglement entropyQuantum entanglementQuantum field theoryParticles & FieldsQuantum Information

Non-Archimedean statistical field theory

We construct in a rigorous mathematical way interacting quantum field theories on a [Formula: see text]-adic spacetime. The main result is the construction of a measure on a function space which allows a rigorous definition of the partition function. The advantage of the approach presented here is that all the perturbation calculations can be carried out in the standard way using functional derivatives, but in a mathematically rigorous way.

Entanglement of the 3-state Potts model via form factor bootstrap: total and symmetry resolved entropies

In this paper, we apply the form factor bootstrap approach to branch point twist fields in the q-state Potts model for q ≤ 3. For q = 3 this is an integrable interacting quantum field theory with an internal discrete ℤ3 symmetry and therefore provides an ideal starting point for the investigation of the symmetry resolved entanglement entropies. However, more generally, for q ≤ 3 the standard Rényi and entanglement entropies are also accessible through the bootstrap programme. In our work we present form factor solutions both for the standard branch point twist field with q ≤ 3 and for the composite (or symmetry resolved) branch point twist field with q = 3. In both cases, the form factor equations are solved for two particles and the solutions are carefully checked via the ∆-sum rule. Using our analytic predictions, we compute the leading finite-size corrections to the entanglement entropy and entanglement equipartition for a single interval in the ground state.

Tensor Renormalization Group for interacting quantum fields

We present a new tensor network algorithm for calculating the partition function of interacting quantum field theories in 2 dimensions. It is based on the Tensor Renormalization Group (TRG) protocol, adapted to operate entirely at the level of fields. This strategy was applied in Ref.[1] to the much simpler case of a free boson, obtaining an excellent performance. Here we include an arbitrary self-interaction and treat it in the context of perturbation theory. A real space analogue of the Wilsonian effective action and its expansion in Feynman graphs is proposed. Using a λϕ4 theory for benchmark, we evaluate the order λ correction to the free energy. The results show a fast convergence with the bond dimension, implying that our algorithm captures well the effect of interaction on entanglement.

Exact four-point function and OPE for an interacting quantum field theory with space/time anisotropic scale invariance

We identify a nontrivial yet tractable quantum field theory model with space/time anisotropic scale invariance, for which one can exactly compute certain four-point correlation functions and their decompositions via the operator-product expansion(OPE). The model is the Calogero model, non-relativistic particles interacting with a pair potential frac{g}{{left|x-yright|}^2} in one dimension, considered as a quantum field theory in one space and one time dimension via the second quantisation. This model has the anisotropic scale symmetry with the anisotropy exponent z = 2. The symmetry is also enhanced to the Schrödinger symmetry. The model has one coupling constant g and thus provides an example of a fixed line in the renormalisation group flow of anisotropic theories.We exactly compute a nontrivial four-point function of the fundamental fields of the theory. We decompose the four-point function via OPE in two different ways, thereby explicitly verifying the associativity of OPE for the first time for an interacting quantum field theory with anisotropic scale invariance. From the decompositions, one can read off the OPE coefficients and the scaling dimensions of the operators appearing in the intermediate channels. One of the decompositions is given by a convergent series, and only one primary operator and its descendants appear in the OPE. The scaling dimension of the primary operator we computed depends on the coupling constant. The dimension correctly reproduces the value expected from the well-known spectrum of the Calogero model combined with the so-called state-operator map which is valid for theories with the Schrödinger symmetry. The other decomposition is given by an asymptotic series. The asymptotic series comes with exponentially small correction terms, which also have a natural interpretation in terms of OPE.