Quantization Of Gauge Theories Research Articles

Electroweak Parameters from Mixed SU(2) Yang–Mills Thermodynamics

Based on the thermal phase structure of pure SU(2) quantum Yang–Mills theory, we describe the electron at rest as an extended particle, a droplet of radius r0∼a0, where a0 is the Bohr radius. This droplet is of vanishing pressure and traps a monopole within its bulk at a temperature of Tc=7.95 keV. The monopole is in the Bogomolny–Prasad–Sommerfield (BPS) limit. It is interpreted in an electric–magnetically dual way. Utilizing a spherical mirror-charge construction, we approximate the droplet’s charge at a value of the electromagnetic fine-structure constant α of α−1∼134 for soft external probes. It is shown that the droplet does not exhibit an electric dipole or quadrupole moment due to averages of its far-field electric potential over monopole positions. We also calculate the mixing angle θW∼30°, which belongs to deconfining phases of two SU(2) gauge theories of very distinct Yang–Mills scales (Λe=3.6 keV and ΛCMB∼10−4 eV). Here, the condition that the droplet’s bulk thermodynamics is stable determines the value of θW. The core radius of the monopole, whose inverse equals the droplet’s mass in natural units, is about 1% of r0.

Bosonic quantum integrable systems and gauge theories

Bosonic quantum integrable systems and gauge theories

Hyperbolic Lattices and Two-Dimensional Yang-Mills Theory.

Hyperbolic lattices are a new type of synthetic quantum matter emulated in circuit quantum electrodynamics and electric-circuit networks, where particles coherently hop on a discrete tessellation of two-dimensional negatively curved space. While real-space methods and a reciprocal-space hyperbolic band theory have been recently proposed to analyze the energy spectra of those systems, discrepancies between the two sets of approaches remain. In this work, we reconcile those approaches by first establishing an equivalence between hyperbolic band theory and U(N) topological Yang-Mills theory on higher-genus Riemann surfaces. We then show that moments of the density of states of hyperbolic tight-binding models correspond to expectation values of Wilson loops in the quantum gauge theory and become exact in the large-N limit.

Relaxation of first-class constraints and the quantization of gauge theories: From “matter without matter” to the reappearance of time in quantum gravity

We make a conceptual overview of a particular approach to the initial-value problem in canonical gauge theories. We stress how the first-class phase-space constraints may be relaxed if we interpret them as fixing the values of new degrees of freedom. This idea goes back to Fock and Stueckelberg, leading to restrictions of the gauge symmetry of a theory, and it corresponds, in certain cases, to promoting constants of Nature to physical fields. Recently, different versions of this formulation have gained considerable attention in the literature, with several independent iterations, particularly in classical and quantum descriptions of gravity, cosmology, and electromagnetism. In particular, in the case of canonical quantum gravity, the Fock–Stueckelberg approach is relevant to the so-called problem of time. Our overview recalls and generalizes the work of Fock and Stueckelberg and its physical interpretation with the aim of conceptually unifying the different iterations of the idea that appear in the literature and of motivating further research.

Neutrino Masses from Generalized Symmetry Breaking

We explore generalized global symmetries in theories of physics beyond the standard model. Theories of Z′ bosons generically contain “noninvertible” chiral symmetries, whose presence indicates a natural paradigm to break this symmetry by an exponentially small amount in an ultraviolet completion. For example, in models of gauged lepton family difference such as the phenomenologically well motivated U(1)Lμ−Lτ, there is a noninvertible lepton number symmetry which protects neutrino masses. We embed these theories in gauged non-Abelian horizontal lepton symmetries, e.g., U(1)Lμ−Lτ⊂SU(3)H, where the generalized symmetries are broken nonperturbatively by the existence of lepton family magnetic monopoles. In such theories, either Majorana or Dirac neutrino masses may be generated through quantum gauge theory effects from the charged lepton Yukawas, e.g., yν∼yτexp(−Sinst). These theories require no bevy of new fields nor additional global symmetries but are instead simple, natural, and predictive: The discovery of a lepton family Z′ at low energies will reveal the scale at which Lμ−Lτ emerges from a larger gauge symmetry. Published by the American Physical Society 2024

Real-space blocking of qubit variables on parallel lattice gauge theory links for quantum simulation

One of the methods proposed in recent years for studying nonperturbative gauge theory physics is quantum simulation, where lattice gauge theories are mapped onto quantum devices which can be built in the laboratory, or quantum computers. While being very promising and already showing some experimental results, these methods still face several challenges related to the interface between the technological capabilities and the demands of the simulated models; in particular, one such challenge is the need to simulate infinitely dimensional local Hilbert spaces, describing the gauge fields on the links in the case of compact Lie gauge groups, requiring some truncations and approximations which are not completely understood or controllable in the general case. This work proposes a way to obtain arbitrarily large such local Hilbert spaces by using coarse graining of simple, low-dimensional qubit systems, made of components available on most quantum simulation platforms, and thus opening the way to new types of lattice gauge theory quantum simulations. Published by the American Physical Society 2024

Compositional quantum field theory: An axiomatic presentation

We introduce Compositional Quantum Field Theory (CQFT) as an axiomatic model of quantum field theory, based on the principles of locality and compositionality. Our model is a refinement of the axioms of general boundary quantum field theory, and is phrased in terms of correspondences between certain commuting diagrams of gluing identifications between manifolds and corresponding commuting diagrams of state-spaces and linear maps, thus making it amenable to formalization in terms of involutive symmetric monoidal functors and operad algebras. The underlying novel framework for gluing leads to equivariance of CQFT. We study CQFTs in dimension 2 and the algebraic structure they define on open and closed strings. We also consider, as a particular case, the compositional structure of 2-dimensional pure quantum Yang–Mills theory.

Identity, individuality and indistinguishability in physics and mathematics.

In this brief survey, we discuss some of the scientific and philosophical problems and debates that underlie the notions of identity, individuality and indistinguishability in physics and mathematics. We critically analyse the different positions for or against the existence of indistinguishable objects in different scientific theories, notably quantum mechanics and gauge theories in physics and homotopy type theory in mathematics. We argue that the different forms of indistinguishability that occur in many areas of physics and mathematics-far from being a problem to be eradicated-exhibit a rich formal structure that plays a key role in the corresponding theories that needs to be properly understood. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.

Quantum Holonomy Fields

We investigate lattice and continuous quantum gauge theories on the Euclidean plane with a structure group that is replaced by a H-algebra. H-algebras are non-commutative analogues of groups and contain the class of Voiculescu's dual groups. We are interested in non-commutative analogues of random gauge fields, which we describe through the random Holonomy that they induce. We propose a general definition of a Holonomy Field with symmetries displaying the structure of a H-algebra and construct such a field starting from a quantum Lévy process on a H-algebra in the category of probability spaces. We call them Quantum Holonomy Fields. We also consider the more abstract case of an H-algebra in a given algebraic category. This yields the notion of a Categorical Holonomy Field. As an application, we define higher dimensional generalizations of the so-called Master Field on the plane.

Inherent color symmetry in quantum Yang-Mills theory

We present the basic non-perturbative structure of the space of classical dynamical solutions and corresponding one particle quantum states in SU(3) Yang-Mills theory. It has been demonstrated that the Weyl group of su(3) algebra plays an important role in constructing non-perturbative solutions and leads to profound changes in the structure of the classical and quantum Yang-Mills theory. We show that the Weyl group as a non-trivial color subgroup of SU(3) admits singlet irreducible representations on a space of classical dynamical solutions which lead to strict concepts of one particle quantum states for gluons and quarks. The Yang-Mills theory is a non-linear theory and, in general, it is not possible to construct a Hilbert space of classical solutions and quantum states as a linear vector space, so, usually, a perturbative approach is applied. We propose a non-perturbative approach based on Weyl symmetric solutions to full non-linear equations of motion and construct a full space of dynamical solutions representing an infinite but countable solution space classified by a finite set of integer numbers. It has been proved that the Weyl singlet structure of classical solutions provides the existence of a stable non-degenerate vacuum which serves as a main precondition of the color confinement phenomenon. Some physical implications in quantum chromodynamics are considered.

Quantum field theory with ghost pairs

We explicitly show that general local higher-derivative theories with only complex conjugate ghosts and normal real particles are unitary at any perturbative order in the loop expansion. The proof presented here relies on integrating the loop energies on complex paths resulting from the deformation of the purely imaginary paths, when the external energies are continued from imaginary to real values. Contrary to the case of nonlocal theories, where the same integration path was first proposed, for the classes of theories studied here the same procedure is not analytic, but the resulting theory is unitary and unique when the complex ghosts are present in pairs. As an explicit application, a special class of higher-derivative super-renormalizable or finite gravitational and gauge theories turns out to be unitary at any perturbative order if we exclude the complex ghosts from the spectrum of the theory, as it is normally accepted for Becchi-Rouet-Stora-Tyutin (BRST) ghosts. Finally, we propose an analogy between confined gluons in quantum Yang-Mills theory and classical complex pairs in local higher-derivative theories. According to such interpretation, complex ghosts will not appear on shell as asymptotic states because confined in what is natural to name “ghostballs.”

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.

Symmetry in Quantum Mechanics and Gauge Theory for Gravity

To unify gravitational interaction with other three kinds of interaction are a work hadn’t been finished by Albert Einstein and a big problem that human must solve to continue living in the universe.

Noether’s theorems and the energy-momentum tensor in quantum gauge theories

Noether's first and second theorems both imply conserved currents that can be identified as an energy-momentum tensor (EMT). The first theorem identifies the EMT as the conserved current associated with global spacetime translations, while the second theorem identifies it as a conserved current associated with local spacetime translations. This work obtains an EMT for quantum electrodynamics and quantum chromodynamics through the second theorem, which is automatically symmetric in its indices and invariant under the expected symmetries (e.g., BRST invariance) without the need for introducing an ad hoc improvement procedure.

Entanglement-Asymmetry Correspondence for Internal Quantum Reference Frames.

In the quantization of gauge theories and quantum gravity, it is crucial to treat reference frames such as rods or clocks not as idealized external classical relata, but as internal quantum subsystems. In the Page-Wootters formalism, for example, evolution of a quantum system S is described by a stationary joint state of S and a quantum clock, where time dependence of S arises from conditioning on the value of the clock. Here, we consider (possibly imperfect) internal quantum reference frames R for arbitrary compact symmetry groups, and show that there is an exact quantitative correspondence between the amount of entanglement in the invariant state on RS and the amount of asymmetry in the corresponding conditional state on S. Surprisingly, this duality holds exactly regardless of the choice of coherent state system used to condition on the reference frame. Averaging asymmetry over all conditional states, we obtain a simple representation-theoretic expression that admits the study of the quality of imperfect quantum reference frames, quantum speed limits for imperfect clocks, and typicality of asymmetry in a unified way. Our results shed light on the role of entanglement for establishing asymmetry in a fully symmetric quantum world.

Formula for two-loop divergent part of 4-D Yang–Mills effective action

In the paper, we study the two-loop contribution to the effective action of the four-dimensional quantum Yang–Mills theory. We derive a new formula for the contribution in terms of three functions, formed from the Green’s function expansion near the diagonal. This result can be applied to different types of regularization. Therefore, we test it by using the dimensional regularization and cutoff ones and show the consistence with the results, obtained in other works.

Some Classical Models of Particles and Quantum Gauge Theories

The article contains a review and new results of some mathematical models relevant to the interpretation of quantum mechanics and emulating well-known quantum gauge theories, such as scalar electrodynamics (Klein–Gordon–Maxwell electrodynamics), spinor electrodynamics (Dirac–Maxwell electrodynamics), etc. In these models, evolution is typically described by modified Maxwell equations. In the case of scalar electrodynamics, the scalar complex wave function can be made real by a gauge transformation, the wave function can be algebraically eliminated from the equations of scalar electrodynamics, and the resulting modified Maxwell equations describe the independent evolution of the electromagnetic field. Similar results were obtained for spinor electrodynamics. Three out of four components of the Dirac spinor can be algebraically eliminated from the Dirac equation, and the remaining component can be made real by a gauge transformation. A similar result was obtained for the Dirac equation in the Yang–Mills field. As quantum gauge theories play a central role in modern physics, the approach of this article may be sufficiently general. One-particle wave functions can be modeled as plasma-like collections of a large number of particles and antiparticles. This seems to enable the simulation of quantum phase-space distribution functions, such as the Wigner distribution function, which are not necessarily non-negative.

Internal quantum reference frames for finite Abelian groups

Employing internal quantum systems as reference frames is a crucial concept in quantum gravity, gauge theories, and quantum foundations whenever external relata are unavailable. In this work, we give a comprehensive and self-contained treatment of such quantum reference frames (QRFs) for the case when the underlying configuration space is a finite Abelian group, significantly extending our previous work [M. Krumm, P. A. Höhn, and M. P. Müller, Quantum 5, 530 (2021)]. The simplicity of this setup admits a fully rigorous quantum information–theoretic analysis, while maintaining sufficient structure for exploring many of the conceptual and structural questions also pertinent to more complicated setups. We exploit this to derive several important structures of constraint quantization with quantum information–theoretic methods and to reveal the relation between different approaches to QRF covariance. In particular, we characterize the “physical Hilbert space”—the arena of the “perspective-neutral” approach—as the maximal subspace that admits frame-independent descriptions of purifications of states. We then demonstrate the kinematical equivalence and, surprising, dynamical inequivalence of the “perspective-neutral” and the “alignability” approach to QRFs. While the former admits unitaries generating transitions between arbitrary subsystem relations, the latter, remarkably, admits no such dynamics when requiring symmetry-preservation. We illustrate these findings by example of interacting discrete particles, including how dynamics can be described “relative to one of the subystems.”

Ludwig Dmitrievich Faddeev. 23 March 1934—26 February 2017

Ludwig Faddeev was a preeminent mathematical physicist of the second half of the twentieth and early twenty-first century. He made fundamental contributions to mathematics and theoretical physics. These included the solution of the quantum three-body scattering problem, groundbreaking work on the quantization of gauge theories, the formulation of Hamiltonian methods for classical integral models, and the creation, with colleagues, of the quantum inverse scattering method which led to the discovery of quantum groups. An incomplete list of important concepts and results named after him include the Faddeev equations, the Faddeev–Popov determinant, Faddeev–Popov ghosts, the Faddeev–Green function, the Gardner–Faddeev–Zakharov bracket, Faddeev–Zamolodchikov algebra and the Faddeev quantum dilogarithm. Faddeev was a charismatic teacher and a natural leader at the Steklov Institute of the Russian Academy of Sciences and the International Mathematical Union.

Gauge Symmetries and Renormalization

We study the perturbative renormalization of quantum gauge theories in the Hopf algebra setup of Connes and Kreimer. It was shown by van Suijlekom (Commun Math Phys 276:773–798, 2007) that the quantum counterparts of gauge symmetries—the so-called Ward–Takahashi and Slavnov–Taylor identities—correspond to Hopf ideals in the respective renormalization Hopf algebra. We generalize this correspondence to super- and non-renormalizable Quantum Field Theories, extend it to theories with multiple coupling constants and add a discussion on transversality. In particular, this allows us to apply our results to (effective) Quantum General Relativity, possibly coupled to matter from the Standard Model, as was suggested by Kreimer (Ann Phys 323:49–60, 2008). To this end, we introduce different gradings on the renormalization Hopf algebra and study combinatorial properties of the superficial degree of divergence. Then we generalize known coproduct and antipode identities to the super- and non-renormalizable cases and to theories with multiple vertex residues. Building upon our main result, we provide criteria for the compatibility of these Hopf ideals with the corresponding renormalized Feynman rules. A direct consequence of our findings is the well-definedness of the Corolla polynomial for Quantum Yang–Mills theory without reference to a particular renormalization scheme.