External Yang-Mills Field Research Articles

Classical limits of gauge-invariant states and the choice of algebra for strict quantization

We analyze the quantization of a system consisting of a particle in an external Yang–Mills field within a C*-algebraic framework. We show that in both the classical and quantum theories of such a system, the kinematical algebra of physical quantities can be obtained by restricting attention to symmetry-invariant states on a C*-algebra. We use this to show that symmetry-invariant quantum states correspond to symmetry-invariant classical states in the classical limit.

Wong's Equations and Charged Relativistic Particles in Non-Commutative Space

In analogy to Wong's equations describing the motion of a charged relativistic point particle in the presence of an external Yang-Mills field, we discuss the motion of such a particle in non-commutative space subject to an external $U_\star(1)$ gauge field. We conclude that the latter equations are only consistent in the case of a constant field strength. This formulation, which is based on an action written in Moyal space, provides a coarser level of description than full QED on non-commutative space. The results are compared with those obtained from the different Hamiltonian approaches. Furthermore, a continuum version for Wong's equations and for the motion of a particle in non-commutative space is derived.

Inverse Boundary Problems for Systems in Two Dimensions

We prove identification of coefficients up to gauge equivalence by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of \({\mathbb{C}}\) . In the geometric setting, we fix a Riemann surface with boundary and consider both a Dirac-type operator plus potential acting on sections of a Clifford module and a connection Laplacian plus potential (i.e. Schrodinger Laplacian with external Yang–Mills field) acting on sections of a Hermitian bundle. In either case we show that the Cauchy data determine both the connection and the potential up to a natural gauge transformation: conjugation by an endomorphism of the bundle which is the identity at the boundary. For domains of \({\mathbb{C}}\) , we recover zeroth order terms up to gauge from Cauchy data at the boundary in first order elliptic systems.

Nonlinear Schroedinger Solitons in Massive Yang-Mills Theory and Partial Localization of Dirac Matter

We investigate the classical dynamics of the massive SU(2) Yang-Mills field in the framework of multiple scale perturbation theory. We show analytically that there exists a subset of solutions having the form of a kink soliton, modulated by a plane wave, in a linear subspace transverse to the direction of free propagation. Subsequently, we explore how these solutions affect the dynamics of a Dirac field possessing an SU(2) charge. We find that this class of Yang-Mills configurations, when regarded as an external field, leads to the localization of the fermion along a line in the transverse space. Our analysis reveals a mechanism for trapping SU(2) charged fermions in the presence of an external Yang-Mills field indicating the non-abelian analogue of Landau localization in electrodynamics.

The long wavelength limit of hard thermal loop effective actions

We derive a closed form expression for the long wavelength limit of the effective action for hard thermal loops in an external gravitational field. It is a function of the metric, independent of time derivatives. It is compared and contrasted with the static limit, and with the corresponding limits in an external Yang–Mills field.

Rashba interaction as an external Yang–Mills field applied to one-dimensional system

The Rashba interaction in the low-dimensional semiconductor systems, which plays an important role in the spintronics, can be regarded as an external Yang–Mills field. With this viewpoint, the gauge field in a one dimensional (1D)-ring in the presence of the Rashba interaction can be reduced into two independent U(1) fields through an SU(2) transformation, which can also be extended to arbitrary 1D Rashba system. As an instance, a two-arms spin interferometer with a quantum dot embedded in one of its arms is considered. By tuning the Rashba interaction, the oscillation of the spin polarization of the current is predicted. Another application is about the persistent spin current in a Rashba ring. Keep the SU(2) gauge field in mind, it is straightforward to find that the persistent spin current may result from the Rashba interaction.

Yang–Mills action from minimally coupled bosons on R4 and on the four-dimensional Moyal plane

We consider bosons on (Euclidean) R4 that are minimally coupled to an external Yang–Mills field. We compute the logarithmically divergent part of the cutoff regularized quantum effective action of this system. We confirm the known result that this term is proportional to the Yang–Mills action. We use pseudodifferential operator methods throughout to prepare the ground for a generalization of our calculation to the noncommutative four-dimensional Moyal plane Rθ4. We also include a detailed comparison of our cutoff regularization to heat kernel techniques. In the case of the noncommutative space, we complement the usual technique of asymptotic expansion in the momentum variable with operator theoretic arguments in order to keep separated quantum from noncommutativity effects. We show that the result from the commutative space R4 still holds if one replaces all pointwise products by the noncommutative Moyal product.

Lie Groupoid C * -Algebras and Weyl Quantization

For any Lie groupoid $G$, the vector bundle $g^*$ dual to the associated Lie algebroid $g$ is canonically a Poisson manifold. The (reduced) C*-algebra of $G$ (as defined by A. Connes) is shown to be a strict quantization (in the sense of M. Rieffel) of $g^*$. This is proved using a generalization of Weyl's quantization prescription on flat space. Many other known strict quantizations are a special case of this procedure; on a Riemannian manifold, one recovers Connes' tangent groupoid as well as a recent generalization of Weyl's prescription. When $G$ is the gauge groupoid of a principal bundle one is led to the Weyl quantization of a particle moving in an external Yang-Mills field. In case that $G$ is a Lie group (with Lie algebra $g$) one recovers Rieffel's quantization of the Lie-Poisson structure on $g^*$. A transformation group C*-algebra defined by a smooth action of a Lie group on a manifold $Q$ turns out to be the quantization of the semidirect product Poisson manifold $g^*x Q$ defined by this action.

Geometric quantization on homogeneous spaces and the meaning of “inequivalent” quantizations

Consideration of the geometric quantization of the phase space of a particle in an external Yang-Mills field allows the results of the Mackey-Isham quantization procedure for homogeneous configuration spaces to be reinterpreted. In particular, a clear physical interpretation of the “inequivalent” quantizations occuring in that procedure is given.

Wigner function and quantum kinetic theory in curved space–time and external fields

A new definition of the Wigner function for quantum fields coupled to curved space–time and an external Yang–Mills field is studied on the example of scalar and Dirac fields. The definition uses the formalism of the tangent bundles and is explicitly covariant and gauge invariant. Derivation of collisionless quantum kinetic equations is carried out for both quantum fields by using the first order formalism of Duffin and Kemmer. The evolution of the Wigner function is governed by the quantum corrected Liouville–Vlasov equation supplemented by the generalized mass–shell constraint. The structure of the quantum corrections is perturbatively found in all adiabatic orders. The lowest order quantum–curvature corrections coincide with the ones found by Winter.

DEFORMATIONS OF ALGEBRAS OF OBSERVABLES AND THE CLASSICAL LIMIT OF QUANTUM MECHANICS

The quantum algebra of observables of a particle moving on a homogeneous configuration space Q = G/H, the transformation group C*-algebra C* (G, G/H), is deformed into its classical counterpart C0 ((T*G)/H). The Poisson structure of the latter is obtained as the classical limit of the quantum commutator. The superselection sectors of both algebras describe the particle moving in an external Yang–Mills field. Analytical aspects of deformation theory, such as the nature of the limit ħ → 0, are studied in detail. A physically motivated convergence criterion in ħ is introduced. The Weyl–Moyal quantization formalism, and the associated use of Wigner distribution functions, is generalized from flat phase spaces T*ℝn to Poisson manifolds of the form (T*G)/H. The classical limit of quantum states as well as of superselection sectors is investigated. The former is handled by introducing the notion of a classical germ, generalizing coherent states. The latter is analyzed by studying the Jacobson topology on the primitive ideal space of a certain continuous field of C*-algebras, constructed from the classical and the quantum algebras of observables. The symplectic leaves of (T*G)/H are confirmed to be the correct classical analogue of the quantum superselection sectors of C*(G, G/H).

Fermion current algebras and Schwinger terms in (3+1)-dimensions

We discuss the restricted linear group in infinite dimensions modeled by the Schatten class of rank $2p=4$ which contains $(3+1)$-dimensional analog of the loop groups and is closely related to Yang-Mills theory with fermions in $(3+1)$-dimensions. We give an alternative to the construction of the ``highest weight'' representation of this group found by Mickelsson and Rajeev. Our approach is close to quantum field theory, with the elements of this group regarded as Bogoliubov transformations for fermions in an external Yang-Mills field. Though these cannot be unitarily implemented in the physically relevant representation of the fermion field algebra, we argue that they can be implemented by sesquilinear forms, and that there is a (regularized) product of forms providing an appropriate group structure. On the Lie algebra level, this gives an explicit, non-perturbative construction of fermion current algebras in $(3+1)$ space-time dimensions which explicitly shows that the ``wave function renormalization'' required for a consistent definition of the currents and their Lie bracket naturally leads to the Schwinger term identical with the Mickelsson-Rajeev cocycle. Though the explicit form of the Schwinger term is given only for the case $p=2$, our arguments apply also to the restricted linear groups modeled by Schatten classes of rank $2p=6,8,\ldots$ corresponding to current algebras in $(d+1)$- dimensions, $d=5,7,\ldots$.

Yang-Mills fields permitted by abelian complete sets of first-order symmetry operators of the Dirac equation

Nonequivalent complete sets of first-order symmetry operators of the Dirac free equation determine the Yang-Mills field, permitting complete variable separation in the Dirac equations with an external Yang-Mills field. Typical representatives of the classes of permissible fields are considered.

Current algebra representation for the 3+1 dimensional Dirac-Yang-Mills theory

The structure of the current algebra representation in the state space of fermions in an external Yang-Mills field in 3+1 space-time dimensions is analyzed; the topology of the vector space is determined by a countable family of semi-definite inner products. We show that there is no hermitian non-trivial Hilbert space representation such that the energy is bounded from below. The structure of the Hilbert space for the quantized coupled Dirac-Yang-Mills system is discussed and the existence of the vacuum vector and the cancellation of commutator anomalies is described in terms of complex line bundles over infinite-dimensional Grassmannians.

Reduction, the trace formula, and semiclassical asymptotics

We state a theorem that relates the theory of dimensional reduction in Hamiltonian mechanics to the spectral properties of elliptic operators with symmetries on compact manifolds. As an application, we show that the spectrum of the Schrödinger operator, -[unk]hDelta + V, as [unk]h --> 0, contains geometric information about the closed trajectories of a classical particle with Hamiltonian p(2) + V(q). More generally, we show that this is true for particles with internal degrees of freedom and subject to an external Yang-Mills field, the classical limit being the Wong-Sternberg-Weinstein system for such particles.

Classical particle in external Yang-Mills field and SU(2) coherent states

The relationship is indicated between a classical isospin vector for a classical particle in an external Yang-Mills field and SU(2) coherent states. The recipe for obtaining approximate isospin amplitudes from simple classical equations of motion is given.

Classical scalar and spinning particles interacting with external Yang-Mills fields

We derive the classical equations of motion for scalar and spinning particles interacting with an external Yang-Mills field. The internal degrees of freedom are described by a set of Grassmann variables; in this way, after quantization, we get automatically finite-dimensional representations of the internal symmetry group. By requiring that the Lagrangian is gauge invariant we determine the general form for the coupling terms. In particular we obtain a generalized version of the Bargmann-Michel-Telegdi equation, which is valid for a spin- 1 2 particle in an arbitrary external field.

Definition of the Dirac isotopic current operator in Yang-Mills theory

The isotopic current operator for a Dirac field in the presence of an external Yang-Mills field is examined. The current is viewed as the limit of a well-defined operator where the points at which the Dirac fields are evaluated have been separated infinitesimally and appropriate line integrals inserted to maintain the correct gauge transformation properties. It is shown that the matrix elements of the current operator then obey the correct conservation law to arbitrary order in the Yang-Mills field as the separation between the Dirac field points limits to zero.