Algebraic Quantum Field Theory Research Articles

Stress-tensor commutators in conformal field theories near the lightcone

Starting with the general stress-tensor commutation relations consistent with the Poincar\'e algebra in local quantum field theory, we impose the tracelessness condition and focus on the dominating contributions in the lightcone limit. It is shown that, under a certain assumption on the Schwinger term, a Virasoro-algebra-like structure emerges near the lightcone in $d>2$ conformal field theories.

Model-Independent Comparison Between Factorization Algebras and Algebraic Quantum Field Theory on Lorentzian Manifolds

This paper investigates the relationship between algebraic quantum field theories and factorization algebras on globally hyperbolic Lorentzian manifolds. Functorial constructions that map between these two types of theories in both directions are developed under certain natural hypotheses, including suitable variants of the local constancy and descent axioms. The main result is an equivalence theorem between (Cauchy constant and additive) algebraic quantum field theories and (Cauchy constant, additive and time-orderable) prefactorization algebras. A concept of *-involution for the latter class of prefactorization algebras is introduced via transfer. This involves Cauchy constancy explicitly and does not extend to generic (time-orderable) prefactorization algebras.

Deciphering the algebraic CPT theorem

The CPT theorem states that any causal, Lorentz-invariant, thermodynamically well-behaved quantum field theory must also be invariant under a reflection symmetry that reverses the direction of time (T), flips spatial parity (P), and conjugates charge (C). Although its physical basis remains obscure, CPT symmetry appears to be necessary in order to unify quantum mechanics with relativity. This paper attempts to decipher the physical reasoning behind proofs of the CPT theorem in algebraic quantum field theory. Ultimately, CPT symmetry is linked to a reversal of the C*-algebraic Lie product that encodes the generating relationship between observables and symmetries. In any physically reasonable relativistic quantum field theory, it is always possible to systematically flip this generating relationship while preserving the dynamics, spectra, and localization properties of physical systems. Rather than the product of three separate reflections, CPT symmetry is revealed to be a single global reflection of the theory's state space.

Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

The perturbative approach to nonlinear Sigma models and the associated renormalization group flow are discussed within the framework of Euclidean algebraic quantum field theory and of the principle of general local covariance. In particular we show in an Euclidean setting how to define Wick ordered powers of the underlying quantum fields and we classify the freedom in such procedure by extending to this setting a recent construction of Khavkine, Melati and Moretti for vector valued free fields. As a by-product of such classification, we prove that, at first order in perturbation theory, the renormalization group flow of the nonlinear Sigma model is the Ricci flow.

Algebraic field theory operads and linear quantization

We generalize the operadic approach to algebraic quantum field theory (arXiv:1709.08657) to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we study an adjunction whose left adjoint describes the quantization of linear field theories. We also analyze homotopical properties of the linear quantization adjunction for chain complex valued field theories, which leads to a homotopically meaningful quantization prescription for linear gauge theories.

Photon masses in the landscape and the swampland

In effective quantum field theory, a spin-1 vector boson can have a technically natural small mass that does not originate from the Higgs mechanism. For such theories, which may be written in Stückelberg form, there is no point in field space at which the mass is exactly zero. I argue that quantum gravity differs from, and constrains, effective field theory: arbitrarily small Stückelberg masses are forbidden. In particular, the limit in which the mass goes to zero lies at infinite distance in field space, and this distance is correlated with a tower of modes becoming light according to the Swampland Distance Conjecture. Application of Tower or Sublattice variants of the Weak Gravity Conjecture makes this statement more precise: for a spin-1 vector boson with coupling constant e and Stückelberg mass m, local quantum field theory breaks down at energies at or below ΛUV = min((mMPl/e)1/2, e1/3MPl). Combined with phenomenological constraints, this argument implies that the Standard Model photon must be exactly massless. It also implies that much of the parameter space for light dark photons, which are the target of many experimental searches, is compatible only with Higgs and not Stückelberg mass terms. This significantly affects the experimental limits and cosmological histories of such theories. I explain various caveats and weak points of the arguments, including loopholes that could be targets for model-building.

Rigorous lower bound on the scattering amplitude at large angles

We prove a lower bound for the modulus of the amplitude for a two-body process at large scattering angle. This is based on the interplay of the analyticity of the amplitude and the positivity properties of its absorptive part. The assumptions are minimal, namely those of local quantum field theory (in the case when dispersion relations hold). In Appendix A, lower bounds for the forward particle-particle and particle-antiparticle amplitudes are obtained. This is of independent interest.

Free products in AQFT

We apply the free product construction to various local algebras in algebraic quantum field theory. If we take the free product of infinitely many identical half-sided modular inclusions with ergodic canonical endomorphism, we obtain a half-sided modular inclusion with ergodic canonical endomorphism and trivial relative commutant. On the other hand, if we take Mobius covariant nets with trace class property, we are able to construct an inclusion of free product von Neumann algebras with large relative commutant, by considering either a finite family of identical inclusions or an infinite family of inequivalent inclusions. In two dimensional spacetime, we construct Borchers triples with trivial relative commutant by taking free products of infinitely many, identical Borchers triples. Free products of finitely many Borchers triples are possibly associated with Haag-Kastler net having S-matrix which is nontrivial and non asymptotically complete, yet the nontriviality of double cone algebras remains open.

Ghost-free infinite derivative quantum field theory

In this paper we will study Lorentz-invariant, infinite derivative quantum field theories, where infinite derivatives give rise to non-local interactions at the energy scale Ms, beyond the Standard Model. We will study a specific class, where there are no new dynamical degrees of freedom other than the original ones of the corresponding local theory. We will show that the Green functions are modified by a non-local extra term that is responsible for acausal effects, which are confined in the region of non-locality, i.e. Ms−1. The standard time-ordered structure of the causal Feynman propagator is not preserved and the non-local analog of the retarded Green function turns out to be non-vanishing for space-like separations. As a consequence the local commutativity is violated. Formulating such theories in the non-local region with Minkowski signature is not sensible, but they have Euclidean interpretation. We will show how such non-local construction ameliorates ultraviolet/short-distance singularities suffered typically in the local quantum field theory. We will show that non-locality and acausality are inherently off-shell in nature, and only quantum amplitudes are physically meaningful, so that all the perturbative quantum corrections have to be consistently taken into account.

Higher Structures in Algebraic Quantum Field Theory

AbstractA brief overview of the recent developments of operadic and higher categorical techniques in algebraic quantum field theory is given. The relevance of such mathematical structures for the description of gauge theories is discussed.

Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields

We describe the elements of a novel structural approach to classical field theory, inspired by recent developments in perturbative algebraic quantum field theory. This approach is local and focuses mainly on the observables over field configurations, given by certain spaces of functionals which are studied here in depth. The analysis of such functionals is characterized by a combination of geometric, analytic and algebraic elements which (1) make our approach closer to quantum field theory, (2) allow for a rigorous analytic refinement of many computational formulae from the functional formulation of classical field theory and (3) provide a new pathway towards understanding dynamics. Particular attention will be paid to aspects related to nonlinear hyperbolic partial differential equations and their linearizations.

New physics vs new paradigms: distinguishing CPT violation from NSI

Our way of describing Nature is based on local relativistic quantum field theories, and then CPT symmetry, a natural consequence of Lorentz invariance, locality and hermiticity of the Hamiltonian, is one of the few if not the only prediction that all of them share. Therefore, testing CPT invariance does not test a particular model but the whole paradigm. Current and future long baseline experiments will assess the status of CPT in the neutrino sector at an unprecedented level and thus its distinction from similar experimental signatures arising from non-standard interactions is imperative. Whether the whole paradigm is at stake or just the standard model of neutrinos crucially depends on that.

Analyticity and crossing symmetry of superstring loop amplitudes

Bros, Epstein and Glaser proved crossing symmetry of the S-matrix of a theory without massless fields by using certain analyticity properties of the off-shell momentum space Green’s function in the complex momentum plane. The latter properties follow from representing the momentum space Green’s function as Fourier transform of the position space Green’s function, satisfying certain properties implied by the underlying local quantum field theory. We prove the same analyticity properties of the momentum space Green’s functions in superstring field theory by directly working with the momentum space Feynman rules even though the corresponding properties of the position space Green’s function are not known. Our result is valid to all orders in perturbation theory, but requires, as usual, explicitly subtracting / regulating the non-analyticities associated with massless particles. These results can also be used to prove other general analyticity properties of the S-matrix of superstring theory.

Coloring Operads for Algebraic Field Theory

AbstractIn these proceedings we summarize previous work where we formalize a general concept of algebraic field theories using operads. After giving a gentle reminder of algebraic quantum field theory, operads and their algebras, we construct field theory operads, whose algebras are exactly algebraic field theories. Specifically, they satisfy a suitable version of the Einstein causality axiom. From this construction we get adjunctions between different types of field theories, including adjunctions related to local‐to‐global extensions and the time‐slice axiom, and a quantization functor for linear field theories that is compatible with these structures. We also take first steps towards a derived linear quantization functor.

Scalar Models of Formally Interacting Non-Standard Quantum Fields in Minkowski Space-Time

For decades, a lot of work has been devoted to the problem of constructing a non-trivial quantum field theory in four-dimensional space time. This letter addresses the attempts to construct an algebraic quantum field theory in the framework of non-standard theories like hyperfunction or ultra-hyperfunction quantum field theory. For this purpose model theories of formally interacting neutral scalar fields are constructed and some of their characteristic properties like two-point functions are discussed. The formal self-couplings are obtained from local normally-ordered analytic redefinitions of the free scalar quantum field, mimicking a non-trivial structure of the resulting Lagrangians and equations of motion.

Emergent Lorentz symmetry and the Unruh effect in a Lorentzian fermionic tensor network

In this paper we attempt to understand Lorentzian tensor networks, as a preparation for constructing tensor networks that can describe more exotic backgrounds such as black holes. To define notions of reference frames and switching of reference frames on a tensor network, we will borrow ideas from the algebraic quantum field theory literature. With these definitions, we construct simple examples of Lorentzian tensor networks and solve the spectrum for a choice of ``inertial frame'' based on Gaussian models of fermions and integrable models. In particular, the tensor network can be viewed as a periodically driven Floquet system, that by-pass the ``doubling problem'' and gives rise to fermions with exactly linear dispersion relations. We will find that a boost operator connecting different inertial frames, and notions of ``Rindler observers'' can be defined, and that important physics in Lorentz invariant QFT, such as the Unruh effect, can be captured by such skeleton of spacetime. We find interesting subtleties when the same approach is directly applied to bosons -- the operator algebra contains commutators that take the wrong sign -- resembling bosons behind horizons.

Prospects of elementary particle physics

The current situation in high-energy physics is discussed. The review is focused on theoretical ideas underlying new physics beyond the Standard Model (SM) of fundamental interactions: extending the SM symmetry group, adding new particles, increasing the space dimension, and going beyond the limits of local quantum field theory. The priority tasks of contemporary high-energy physics are explored.

Universality of Hawking radiation in non local field theories

Locality plays a key role in the derivation of Hawking radiation. Could relaxing the locality criterion modify Hawking radiation? We consider Hawking radiation in a general class of non local quantum field theories. This includes theories with a minimal length or UV cut-off. We prove that Hawking radiation is unmodified for these theories. Our result establishes the universality of Hawking radiation in a general class of non local theories.

On a Topology and Limits for Inductive Systems of C∗-Algebras over Partially Ordered Sets

Motivated by algebraic quantum field theory and our previous work we study properties of inductive systems of \ $C^*$-algebras over arbitrary partially ordered sets. A partially ordered set can be represented as the union of the family of its maximal upward directed subsets indexed by elements of a certain set. We consider a topology on the set of indices generated by a base of neighbourhoods. Examples of those topologies with different properties are given. An inductive system of $C^*$-algebras and its inductive limit arise naturally over each maximal upward directed subset. Using those inductive limits, we construct different types of $C^*$-algebras. In particular, for neighbourhoods of the topology on the set of indices we deal with the $C^*$-algebras which are the direct products of those inductive limits. The present paper is concerned with the above-mentioned topology and the algebras arising from an inductive system of $C^*$-algebras over a partially ordered set. We show that there exists a connection between properties of that topology and those $C^*$-algebras.

Consistency of Higgsplosion in localizable QFT

We show that large n-particle production rates derived in the semiclassical Higgsplosion limit of scalar field theoretical models with spontaneous symmetry breaking, are consistent with general principles of localizable quantum field theory.The strict localizability criteria of Jaffe defines quantum fields as operator-valued distributions acting on test functions that are localized in finite regions of space–time. The requirement of finite support of test functions in space–time ensures the causality property of QFT. The corresponding localizable fields need not be tempered distributions, and they fit well into the framework of local quantum field theory.