Quantum Gauge Field Theory Research Articles

William Edward Caswell

William Edward Caswell

MINIMAL EXTENDED FLAVOR GROUPS, MATTER FIELDS CHIRAL REPRESENTATIONS, AND THE FLAVOR QUESTION

We show the specific unusual features on chiral gauge anomalies cancellation in the minimal, necessarily 3-3-1, and the largest 3-4-1 weak isospin chiral gauge semisimple group leptoquark–bilepton extensions of the 3-2-1 conventional standard model of nuclear and electromagnetic interactions. In such models a natural answer for the fundamental question of fermion generation replication arises directly from the self-consistency of a local gauge quantum field theory, which constrains the number of the QFD fermion families to the QCD color charges.

Prescriptionless light-cone integrals

Perturbative quantum gauge field theory seen within the perspective of physical gauge choices such as the light-cone entails the emergence of troublesome poles of the type $(k\cdot n)^{-\alpha}$ in the Feynman integrals, and these come from the boson field propagator, where $\alpha = 1,2,...$ and $n^{\mu}$ is the external arbitrary four-vector that defines the gauge proper. This becomes an additional hurdle to overcome in the computation of Feynman diagrams, since any graph containing internal boson lines will inevitably produce integrands with denominators bearing the characteristic gauge-fixing factor. How one deals with them has been the subject of research for over decades, and several prescriptions have been suggested and tried in the course of time, with failures and successes. However, a more recent development in this front which applies the negative dimensional technique to compute light-cone Feynman integrals shows that we can altogether dispense with prescriptions to perform the calculations. An additional bonus comes attached to this new technique in that not only it renders the light-cone prescriptionless, but by the very nature of it, can also dispense with decomposition formulas or partial fractioning tricks used in the standard approach to separate pole products of the type $(k\cdot n)^{-\alpha}[(k-p)\cdot n]^{-\beta}$, $(\beta = 1,2,...)$. In this work we demonstrate how all this can be done.

Simulating hot Abelian gauge dynamics

The time evolution of soft modes in a quantum gauge field theory is to first approximation classical, but the equations of motion are non-local. We show how they can be written in a local and Hamiltonian way in an Abelian theory, and that this formulation is particularly suitable for numerical simulations. This makes it possible to simulate numerically non-equilibrium processes such as the phase transition in the Abelian Higgs model and to study, for instance, bubble nucleation and defect formation. Such simulations would also help to understand phase transitions in more complicated gauge theories. Moreover, we show that the existing analytical results for the time evolution in a pure-gauge theory correspond to a special class of initial conditions and that different initial conditions can lead to qualitatively different behavior. We compare the results of the simulations to analytical calculations and find an excellent agreement.

HOW TO TREAT γ5

In the past two decades, Dyson's formalism of renormalization has been mostly superceded by dimensional regularization, particularly in the treatment of quantum gauge field theories with spontaneous symmetry breaking or those with chiral fermions. In this paper, we shall carry out explicitly Dyson's subtraction program, making it applicable to such field theories. In particular, we show with the example of the Abelian–Higgs theory how to handle amplitudes of chiral fermions. We show that these amplitudes which involve the γ5 matrix can be calculated in an unambiguous and gauge invariant way. This is done by establishing the subtraction conditions for the propagator of a chiral fermion as well as those for the VVV amplitude, when V denotes the vector meson. The renormalized constants are chosen to satisfy the Ward–Takahashi identities. As a demonstration, we calculate the next-lowest order correction of the anomaly in the Abelian–Higgs model and find that it vanishes.

Magnetic Screening at Finite Temperature

It is shown that at finite temperature and in the presence of magnetic sources magnetic fields are screened. This is proven within the framework of classical transport theory both for the Abelian and the non-Abelian plasmas. Magnetic screening arises in this formalism as a consequence of polarization effects occurring in the plasmas, and it is proportional to the inverse of the gauge coupling constant. It is then discussed whether this mechanism could be relevant in realistic quantum gauge field theories, such as QCD.

The inverse loop transform

The loop transform in quantum gauge field theory can be recognized as the Fourier transform (or characteristic functional) of a measure on the space of generalized connections modulo gauge transformations. Since this space is a compact Hausdorff space, conversely, we know from the Riesz–Markov theorem that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform of a regular Borel measure on the moduli space. In the present article we show how one can compute the finite joint distributions of a given characteristic functional, that is, we derive the inverse loop transform.

Kaluza-Klein or fiber bundles?

Two basic geometric approaches to the modern theory of gauge fields are analyzed and compared. The first approach is an extension of the Kaluza-Klein unified theory of gravity and electromagnetism (1921). The second approach generalizes Cartan's formulation (1925) of Riemannian geometry and GR which now is transformed to fiber bundle theory. The goal of this paper is to show that the above-mentioned geometric approaches to the classical gauge field theory are nonequivalent and lead to different forms of quantum gauge field theory. Bibliography:12 titles.

Effective Regge QCD

A new framework for a high energy limit of quantum gauge field theories is introduced. Its potency is illustrated on a new derivation of the reggeization of the gluon.

An axiomatic approach to Quantum Gauge Field Theory

In the present article we display a new constructive quantum field theory approach to quantum gauge field theory, utilizing the recent progress in the integration theory on the moduli space of generalized connections modulo gauge transformations. That is, we propose a new set of Osterwalder Schrader like axioms for the characteristic functional of a measure on the space of generalized connections modulo gauge transformations rather than for the associated Schwinger distributions. We show non-triviality of our axioms by demonstrating that they are satisfied for twodimensional Yang-Mills theory on the plane and the cylinder. As a side result we derive a closed and analytical expression for the vacuum expectation value of an arbitrary product of Wilson-loop functionals from which we derive the quantum theory along the Glimm and Jaffe algorithm which agrees exactly with the one as obtained by canonical methods.

A Quantum Mechanical Gauge Model and a Possible Dynamical Solution of the StrongCPProblem

A model describing a quantum mechanical particle on a circle with minimal electromagnetic interaction with a space independent vector potential, and with a potential −Mcos(ϕ−θM) so that it mimics the massive Schwinger model, is discussed as a prototype of mechanisms and infrared structures which characterize gauge quantum field theories in positive gauges and QCD in particular. The functional integral representation in terms of the field variables which enter in the Lagrangean displays non-standard features, like a complex functional measure (failure of Nelson positivity), a crucial rôle of the boundary conditions, and the decomposition intoθsectors already in finite volume. In the infinite volume limit, one essentially recovers the standard picture whenM=0 (“massless fermions”), but one meets substantial differences forM≠0: for generic boundary conditions, independently of the Lagrangean angle of the topological term, the infinite volume limit selects the sector withθ=θMand provides a natural “dynamical” solution of the strong CP problem. In comparison with previous approaches, the strategy discussed here allows us to exploit the consequences of theθdependence of the free energy density, with a unique minimum atθ=θM.

Reality conditions inducing transforms for quantum gauge field theory and quantum gravity

The algebraic form of the Hamiltonian or Hamiltonian constraint of various (field) theories simplifies considerably if one uses certain complex-valued phase space variables. We show, for a general theory, that if we prescribe first a canonical complexification and second representation of the canonical commutation relations in which the real connection is diagonal, then there is only one choice of a holomorphic representation which incorporates the correct reality conditions and keeps the Hamiltonian (constraint) algebraically simple. We derive a canonical algorithm to obtain this holomorphic representation and in particular explicitly compute it for quantum gravity in terms of a Wick rotation transform.

An Extension of Distribution Theory and of the Paley-Wiener-Schwartz Theorem Related to Quantum Gauge Theory

We show that a considerable part of the theory of (ultra)distributions and hyperfunctions can be extended to more singular generalized functions, starting from an angular localizability notion introduced previously. Such an extension is needed to treat gauge quantum field theories with indefinite metric in a generic covariant gauge. Prime attention is paid to the generalized functions defined on the Gelfand-Shilov spaces $S_\alpha^0$ which gives the widest framework for construction of gauge-like models. We associate a similar test function space with every open and every closed cone, show that these spaces are nuclear and obtain the required formulas for their tensor products. The main results include the generalization of the Paley--Wiener--Schwartz theorem to the case of arbitrary singularity and the derivation of the relevant theorem on holomorphic approximation.

More on the subtraction algorithm

We continue the programme of investigating the removal of divergences of a generic quantum gauge field theory, in the context of the Batalin--Vilkovisky formalism. We extend to open gauge-algebras a recently formulated algorithm, based on redefinitions of the parameters of the classical Lagrangian and canonical transformations. The key point is to generalize a well known conjecture on the form of the divergent terms to the case of open gauge-algebras. We also show that it is possible to obtain complete control over the effects of the subtraction algorithm on the space of the gauge-fixing parameters. We develop a differential calculus on providing an intuitive geometrical description of the fact that the on-shell physical amplitudes cannot depend on . A principal fibre bundle with a connection is defined such that the canonical transformations are gauge transformations for . A geometrical description of the effect of the subtraction algorithm on the space of the physical parameters is also proposed. Finally, the full subtraction algorithm is described as a series of diffeomorphisms on , orthogonal to (under which the action transforms as a scalar), and gauge transformations on . In this geometrical context, a suitable concept of predictivity is formulated. We give some examples of (unphysical) toy models that satisfy this requirement, though being neither power counting renormalizable, nor finite.

The Wick-ordered exponential of the dipole field as a field of type 𝒮

We construct the Wick-ordered exponential of the dipole field, an infrared singular field. We show that the field operator is an object much more singular than its own vacuum expectation values. The actual construction involves the spaces of type 𝒮 by Gelfand and Shilov and seems to provide the most general distributional scheme for the choice of test function in (gauge) quantum field theories.

The choice of test functions in gauge quantum field theories

We discuss the problem of the choice of test functions in gauge quantum field theories. Analysis of explicitly soluble models suggests that the test function spaces which are suitable for local and covariant formulation of gauge theories are the Gelfand and Shilov spaces ℒ α β , α+β>1. We also discuss a possible generalization of the spectral condition.

Nonlocal renormalization “stopping” the running gauge

A general modification of renormalization schemes in gauge quantum field theories, that fixes a running gauge parameter a(Q 2) → a = const is proposed. It is based on nonlocal counter-terms ΔL = {ƒ(□)(∂B)} 2 2a . The resulting simplification of the renorm-group analysis in gauge-dependent QCD destroys the threats to the asymptotic freedom that have recently been observed in several MOM schemes. The nonlocality of the counter-terms used is of a formal type as these can be absorbed by a gradient transformation of the vector potential B μ . Analysis of this transformation reveals a singularity at the point a = 0.

Asymptotic analysis of fundamental solutions of Dirac operators on even dimensional Euclidean spaces

We analyze the short distance asymptotic behavior of some quantities formed out of fundamental solutions of Dirac operators minimally coupled to finite dimensional matrix-valued gauge potentials on even dimensional Euclidean spaces. The analysis gives a mathematically rigorous proof as well as a mathematically self-consistent description to the so-called “(Abelian) chiral anomaly” in quantum gauge field theories.

Correspondence between quantum gauge theories without ghost fields and their covariantly quantized theories with ghost fields

We give a correspondence formula which equates transition amplitudes in a quantum gauge field theory without ghost fields to those in a quantum theory with the gauge fields covariantly quantized and coupled to ghost fields.

Unbounded representations of symmetry groups in gauge quantum field theory. II. Integration

Within the gauge quantum field theory of the Wightman–Gårding type, the integration of representations of Lie algebras is investigated. By means of the covariance condition (substitution rules) for the basic fields, it is shown that a form skew-symmetric representation of a Lie algebra can be integrated to a form isometric and in general unbounded representation of the universal covering group of a corresponding Lie group provided the conditions (Nelson, Sternheimer, etc.), which are well known for the case of Hilbert or Banach representations, hold. If a form isometric representation leaves the subspace from which the physical Hilbert space is obtained via factorization and completion invariant, then the same is proved to be true for its differential. Conversely, a necessary and sufficient condition is derived for the transmission of the invariance of this subspace under a form skew-symmetric representation of a Lie algebra to its integral.