Axiomatic Quantum Field Theory Research Articles

The Quantum Yang–Mills Theory

In axiomatic quantum field theory, the postulate of the uniqueness of the vacuum (a pure vacuum state) is independent from the other axioms and equivalent to the cluster decomposition property. The latter, however, implies a Coulomb or Yukawa attenuation of the interactions at growing distances and hence cannot accommodate the confining properties of the strong interaction. Thesolution of the Yang–Mills quantum theory given previously uses an auxiliary field to incorporate Gauss’s law and demonstrates the existence of two separate vacua, the perturbative and the confining vacuum, therefore resulting in a mixed vacuum state, deriving confinement, as well as the related, expected properties of the strong interaction. The existence of multiple vacua is, in fact, expected by the axiomatic, algebraic quantum field theory, via the decomposition of the vacuum state to eigenspaces of the auxiliary field. The general vacuum state is a mixed quantum state, and the cluster decomposition property does not hold. Because of the energy density difference between the two vacua, the physics of the strong interactions does not admit a Lagrangian description. I clarify the above remarks related to the previous solution of the Yang–Mills interaction and conclude with some discussion a criticism of a related mathematical problem and some tentative comments regarding the spin-2 case.

Hopf monoids, permutohedral cones, and generalized retarded functions

The commutative Hopf monoid of set compositions is a fundamental Hopf monoid internal to vector species, having undecorated bosonic Fock space the combinatorial Hopf algebra of quasisymmetric functions. We construct a geometric realization of this Hopf monoid over the adjoint of the (essentialized) braid hyperplane arrangement, which identifies the monomial basis with signed characteristic functions of the interiors of permutohedral tangent cones. We show that the indecomposable quotient Lie coalgebra is obtained by restricting functions to chambers of the adjoint arrangement, i.e., by quotienting out the higher codimensions. The resulting functions are characterized by the Steinmann relations of axiomatic quantum field theory, demonstrating an equivalence between the Steinmann relations, tangent cones to (generalized) permutohedra, and having algebraic structure internal to species. Our results give a new interpretation of a construction appearing in the mathematically rigorous formulation of renormalization by Epstein–Glaser, called causal perturbation theory. In particular, we show that operator products of time-ordered products correspond to the H-basis of the cocommutative Hopf monoid of set compositions, and generalized retarded products correspond to a spanning set of its primitive part Lie algebra.

Axiomatic de Sitter quantum Yang-Mills theory with color confinement and mass gap

The analyticity property of de Sitter's quantum Yang-Mills theory in the framework of Krein space quantization, including quantum metric fluctuation, is demonstrated. This property completes our previous work regarding quantum Yang-Mills theory in de Sitter's ambient space formalism, and we can construct an axiomatic quantum field theory similar to Wightman's axioms. The color confinement is proven for the general case, which was previously approved in the early universe. It is shown by using the interaction between gluon fields and the conformal sector of the gravitational field, which is a massless minimally coupled scalar gauge field. The gluon mass results from the interaction between the gluon fields and the massless minimally coupled scalar field as a conformal sector of the gravitational field and then the symmetry-breaking setting due to the vacuum expectation value of the scalar field.

Positivity in electron-positron scattering: testing the axiomatic quantum field theory principles and probing the existence of UV states * *CZ is supported by IHEP (Y7515540U1) and National Natural Science Foundation of China (NSFC) (12035008). SYZ acknowledges support from the starting grants from University of Science and Technology of China (KY2030000089, GG2030040375) and is also supported by NSFC (12075233, 11947301, 12047502)

We consider the positivity bounds on dimension-8 four-electron operators and study two related phenomenological aspects at future lepton colliders. First, if positivity is violated, probing such violations will revolutionize our understanding of the fundamental pillars of quantum field theory and the S-matrix theory. We observe that positivity violation at scales of 1-10 TeV can potentially be probed at future lepton colliders even if one assumes that dimension-6 operators are also present. Second, the positive nature of the dimension-8 parameter space often allows us to either directly infer the existence of UV-scale particles together with their quantum numbers or exclude them up to certain scales in a model-independent way. In particular, dimension-8 positivity plays an important role in the test of the Standard Model. If no deviations from the Standard Model are observed, it allows for simultaneous exclusion limits on all kinds of potential UV-complete models. Unlike the dimension-6 case, these limits apply regardless of the UV model setup and cannot be removed by possible cancellations among various UV contributions. This thus consists of a novel and universal test to confirm the Standard Model. We demonstrate with realistic examples how all the previously mentioned possibilities, including the test of positivity violation, can be achieved. Hence, we provide an important motivation for studying dimension-8 operators more comprehensively.

Towards an axiomatic formulation of noncommutative quantum field theory. II

Classical results of the axiomatic quantum field theory – irreducibility of the set of field operators, Reeh and Schlieder's theorems and generalized Haag's theorem are proven in SO(1,1) invariant quantum field theory, of which an important example is noncommutative quantum field theory. In SO(1,3) invariant theory new consequences of generalized Haag's theorem are obtained. It has been proven that the equality of four-point Wightman functions in two theories leads to the equality of elastic scattering amplitudes and thus the total cross-sections in these theories.

The Solution Cosmological Constant Problem

We introduce Hausdorff-Colombeau measure in respect with negative fractal dimensions. Axiomatic quantum field theory in spacetime with negative fractal dimensions is proposed.Spacetime is modelled as a multifractal subset of $R^{4}$ with positive and negative fractal dimensions.The cosmological constant problem arises because the magnitude of vacuum energy density predicted by quantum field theory is about 120 orders of magnitude larger than the value implied by cosmological observations of accelerating cosmic expansion. We pointed out that the fractal nature of the quantum space-time with negative Hausdorff-Colombeau dimensions can resolve this tension. The canonical Quantum Field Theory is widely believed to break down at some fundamental high-energy cutoff $E$ and therefore the quantum fluctuations in the vacuum can be treated classically seriously only up to this high-energy cutoff. In this paper we argue that Quantum Field Theory in fractal space-time with negative Hausdorff-Colombeau dimensions gives high-energy cutoff on natural way. In order to obtain disered physical result we apply the canonical Pauli-Villars regularization up to $E$. It means that there exist the ghost-driven acceleration of the univers hidden in cosmological constant.

CPT symmetry and some theorems

In this brief paper, we endeavor to prove two theorems that are connected with the CPT symmetry in the framework of axiomatic quantum field theory.

Experimental Results on Ultrahigh-Energy Cosmic Rays and Asymptotic Behavior of the Total Cross Section for Nucleon–Nucleon Interaction

Results obtained by studying the energy dependence of the total cross section for nucleon–nucleon interaction are presented. The analytic parametrization proposed within axiomatic quantum field theory describes quantitatively a unified set of experimental data on proton–proton and antiproton–proton scattering. At collision energies above 86 GeV, the parameter values fitted to subsets of all available data and accelerator data alone were found to deviate from the respective asymptotic values. This indicates that the Froissart–Martin limit for the total cross section has not yet been reached in the case of these subsets. An approximation of the total nucleon–nucleon cross section measured in cosmic rays leads for some parameters to fitted values that, within 1.0 to 1.3 standard deviations, agree with the respective asymptotic values. This resultmay be viewed as an indication of the beginning of the asymptotic functional behavior of the total nucleon–nucleon cross section measured in cosmic rays of ultrahigh energy O(100 TeV). This is confirmed within the color glass condensate approach.

Quantum simulation and spectroscopy of entanglement Hamiltonians

Entanglement is central to our understanding of many-body quantum matter. In particular, the entanglement spectrum, as eigenvalues of the reduced density matrix of a subsystem, provides a unique footprint of properties of strongly correlated quantum matter from detection of topological order to characterisation of quantum critical systems. However, direct experimental measurement of the entanglement spectrum has so far remained elusive due to lack of direct experimental probes. Here we show that the entanglement spectrum of the ground state of a broad class of Hamiltonians becomes directly accessible as quantum simulation and spectroscopy of an entanglement Hamil- tonian, building on the Bisognano-Wichmann (BW) theorem of axiomatic quantum field theory. Remarkably, this theorem gives an explicit physical construction of the entanglement Hamiltonian, identified as Hamiltonian of the many-body system of interest with spatially varying couplings. Building on this, we propose an immediate, scalable recipe for implementation of the entanglement Hamiltonian, and measurement of the corresponding entanglement spectrum as spectroscopy of the Bisognano-Wichmann Hamiltonian with synthetic quantum systems, including atoms in optical lat- tices and trapped ions. We illustrate and benchmark this scenario on a variety of models, spanning phenomena as diverse as conformal field theories, topological order, and quantum phase transitions.

Analysis of pp and p̄p elastic scatterings based on theoretical bounds in high-energy physics: An update

In a previous work a novel parametrization was proposed for the [Formula: see text] and [Formula: see text] total cross-sections. Here, results are presented for the updated analysis with taking into account the recent data from accelerator experiments as well as from cosmic ray measurements. The analytic parametrizations suggested within axiomatic quantum field theory (AQFT) provide the quantitative description of energy dependence of global scattering observables with robust values of fit parameters. Based on the fit results the estimations are derived for the total cross-section and the [Formula: see text] parameter in elastic [Formula: see text] scattering at various [Formula: see text] up to energy frontier [Formula: see text] PeV which can be useful for present and future hadron colliders as well as for cosmic ray measurements at ultra-high energies.

CPT Symmetry and Its Violation

One of the most fundamental symmetries in physics is CPT invariance. This article reviews the conditions under which CPT symmetry holds by recalling two proofs of the CPT theorem: The original Lagrangian-based analysis and the more rigorous one in the context of axiomatic quantum field theory. The presentation of the proofs is followed by a discussion of the major physical implications that arise from CPT symmetry. Motivated by recent theoretical and experimental interest in CPT tests, various approaches to the violation of CPT symmetry are mentioned, and it is briefly discussed how they evade the CPT theorem. An attempt has been made to keep this work self-contained and at a level suitable for a wider readership by excising as many technical aspects as possible.

Haag's theorem in renormalisable quantum field theories

We review a package of no-go results in axiomatic quantum field theory with Haag's theorem at its centre. Since the concept of operator-valued distributions in this framework comes very close to what we believe canonical quantum fields are about, these results are of consequence to quantum field theory: they suggest the seeming absurdity that this highly victorious theory is incapable of describing interactions. We single out unitarity of the interaction picture's intertwiner as the most salient provision of Haag's theorem and critique canonical perturbation theory to argue that renormalisation bypasses Haag's theorem by violating this very assumption.

Coarse-Graining as a Route to Microscopic Physics: The Renormalization Group in Quantum Field Theory

The renormalization group (RG) has been characterized as merely a coarse-graining procedure that does not illuminate the microscopic content of quantum field theory (QFT) but merely gets us from that content, as given by axiomatic QFT, to macroscopic predictions. I argue that in the constructive field theory tradition, RG techniques do illuminate the microscopic dynamics of a QFT, which are not automatically given by axiomatic QFT. RG techniques in constructive field theory are also rigorous, so one cannot object to their foundational import on grounds of lack of rigor.

Hilbert's 6th Problem and Axiomatic Quantum Field Theory

��� 1. The Main Claims This paper has two parts, a historical and a systematic. In the historical part it is argued that the two major axiomatic approaches to relativistic quantum aeld theory, the Wightman and Haag-Kastler axiomatizations, are realizations of the program of axiomatization of physical theories announced by Hilbert in his 6th of the 23 problems discussed in his famous 1900 Paris lecture on open problems in mathematics, if axiomatizing physical theories is interpreted in a soft and opportunistic sense suggested in 1927 by Hilbert, Nordheim, and von Neumann. To show this, Section 2 recalls arst some of Hilbert’s views about the axiomatic approach to physical theories as these were formulated in his 6th problem. It will be seen that Hilbert did not specify in his Paris lecture what precisely the axiomatic approach in physics is supposed to be. It will be seen in Section 3 that he was more explicit in 1927 however: The joint paper (Hilbert et al. 1927) addresses the issue of the axiomatic approach in physics explicitly and embraces a soft notion of axiomatization, which is assumed to be carried out in an opportunistic way only in physics. Axiomatic aeld theories are axiomatizations in this sense. The systematic part of the paper isolates and comments on some characteristic features of the axiomatic quantum aeld theories, elements of which are recalled brieoy in Section 4. The features will be of two types: general-methodological and speciac-physical. The arst category includes what we call here ‘intended non-categoricity” and “large internal room,” discussed in Section 5; the speciac-physical we discuss (in Section 6) are “ontological silence” and “causal completeness.” “Intended Research supported in part by the Hungarian Scientiac Research Found (OTKA). Contract number: 100715

Gröbner bases for finite-temperature quantum computing and their complexity

Following the recent approach of using order domains to construct Gröbner bases from general projective varieties, we examine the parity and time-reversal arguments relating to the Wightman axioms of quantum field theory and propose that the definition of associativity in these axioms should be introduced a posteriori to the cluster property in order to generalize the anyon conjecture for quantum computing to indefinite metrics. We then show that this modification, which we define via ideal quotients, does not admit a faithful representation of the Braid group, because the generalized twisted inner automorphisms that we use to reintroduce associativity are only parity invariant for the prime spectra of the exterior algebra. We then use a coordinate prescription for the quantum deformations of toric varieties to show how a faithful representation of the Braid group can be reconstructed and argue that for a degree reverse lexicographic (monomial) ordered Gröbner basis, the complexity class of this problem is bounded quantum polynomial.

How to take particle physics seriously: A further defence of axiomatic quantum field theory

Further arguments are offered in defence of the position that the variant of quantum field theory (QFT) that should be subject to interpretation and foundational analysis is axiomatic quantum field theory. I argue that the successful application of renormalization group (RG) methods within alternative formulations of QFT illuminates the empirical content of QFT, but not the theoretical content. RG methods corroborate the point of view that QFT is a case of the underdetermination of theory by empirical evidence. I also urge caution in extrapolating interpretive conclusions about QFT from the application of RG methods in other contexts (e.g., condensed matter physics). This paper replies to criticisms advanced by David Wallace, but aims to be self-contained.

Axiomatic Quantum Field Theory in Terms of Operator Product Expansions: General Framework, and Perturbation Theory via Hochschild Cohomology

In this paper, we propose a new framework for quantum field theory in terms of consistency conditions. The consistency conditions that we consider are associativity or factorization conditions on the operator product expansion (OPE) of the theory, and are proposed to be the defining property of any quantum field theory. Our framework is presented in the Euclidean setting, and is applicable in principle to any quantum field theory, including non-conformal ones. In our framework, we obtain a characterization of perturbations of a given quantum field theory in terms of a certain cohomology ring of Hochschild-type. We illustrate our framework by the free field, but our constructions are general and apply also to interacting quantum field theories. For such theories, we propose a new scheme to construct the OPE which is based on the use of non-linear quantized field equations.

Axiomatic Quantum Field Theory in Curved Spacetime

The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincaré invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globally hyperbolic curved spacetimes, it is essential that the theory be formulated in an entirely local and covariant manner, without assuming the presence of a preferred state. We propose a new framework for quantum field theory, in which the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and, in essence, all of the properties of the quantum field theory are determined by its OPE. We provide general axioms for the OPE coefficients of a quantum field theory. These include a local and covariance assumption (implying that the quantum field theory is constructed in a local and covariant manner from the spacetime metric and other background structure, such as time and space orientations), a microlocal spectrum condition, an “associativity” condition, and the requirement that the coefficient of the identity in the OPE of the product of a field with its adjoint have positive scaling degree. We prove curved spacetime versions of the spin-statistics theorem and the PCT theorem. Some potentially significant further implications of our new viewpoint on quantum field theory are discussed.

Unified model for small-t and high-t scattering at high energies: predictions at RHIC and LHC

The urgency of predictions in large-t region at LHC stimulated us to present a unified model of small and high t scattering at high energies. Our model is based upon a safe theoretical ground: analyticity, unitarity, Regge behavior, gluon exchange and saturation of bounds established in axiomatic quantum field theory. We make precise predictions for the behavior of the differential cross sections at high t, the evolution of the dip-shoulder structure localized in the region of -t between 0.5 and 0.8 GeV**2 and the radical violation of the exponential behavior of the first diffraction cone at small t.

On model-independent analyses of elastic hadron scattering

By means of an almost model-independent parametrization for the elastic hadron-hadron amplitude, as a function of the energy and the momentum transfer, we obtain good descriptions of the physical quantities that characterize elastic proton-proton and antiproton-proton scattering (total cross section, r parameter and differential cross section). The parametrization is inferred on empirical grounds and selected according to high energy theorems and limits from axiomatic quantum field theory. Based on the predictive character of the approach we present predictions for the above physical quantities at the Brookhaven RHIC, Fermilab Tevatron and CERN LHC energies.