Magnetic monopoles: From classical to quantum physics

A modified Metropolis algorithm is developed and used to construct axial-symmetric and multimonopole classical solutions in the (lattice-regularized) SO(3) Georgi-Glashow model outside the BPS limit. As an example one of those solutions is then used as an ansatz to study the quantum stability of the configuration.     

Instanton paths for the problem of coherent quantum tunneling in small ferromagnetic particles

It is shown that the problem of instantons in ferromagnetic materials in a large-spin model is reduced to an exactly integrable dynamical system with a finite number of variables. For a rather wide class of models, there exists a continuum of instanton paths that form a one-parameter family of paths with essentially different shapes but with the same value of the Euclidean action. On the basis of the formalism developed, exact instanton solutions are constructed that describe macroscopic quantum tunneling for a small ferromagnetic particle with uniaxial or biaxial quadratic anisotropy in the presence of a magnetic field applied perpendicularly to the easy axis. These solutions are valid for any relations between the anisotropy parameters and for any magnitude of the magnetic field and its direction in the base plane. Based on the solutions obtained, the principles of macroscopic quantum tunneling in high-spin-molecule-type magnetic particles are described. Tunneling regimes of two types are obtained: (1) regimes that are characterized by destructive interference of instanton trajectories and oscillatory dependence of the transition probability on the magnitude of the magnetic field and (2) regimes in which all instantons have the same purely real value of the Euclidean action and there is no destructive interference.     

Unstable behaviors of classical solutions in spinor-type conformal invariant fermionic models

It is well known that instantons are classical topological solutions existing in the context of quantum field theories that lie behind the standard model of particles. To provide a better understanding for the dynamical nature of spinor-type instanton solutions, conformal invariant pure spinor fermionic models that admit particle-like solutions for the derived classical field equations are studied in this work under cosine wave forcing. For this purpose, the effects of external periodic forcing on two systems that have different dimensions and quantum spinor numbers and have been obtained under the use of Heisenberg ansatz are investigated by constructing their Poincaré sections in phase space. As a result, bifurcations and chaos are observed depending on the excitation amplitude of the external forcing in both pure spinor fermionic models.     

Instantons in supersymmetric theories

Instanton effects are considered for a sample of supersymmetric theories, namely, quantum mechanics, gluodynamics, Higgs model. The problem is how to reconcile the apparent lack of the boson-fermion symmetry in the effective instanton induced interactions with supersymmetry of the corresponding lagrangians. It is shown that in the case of quantum mechanics and the Higgs model there is actually no conflict between supersymmetry and the instanton calculus since the Ward identities, associated with the supersymmetry transformations, are satisfied. In quantum mechanics this is due to spontaneous symmetry breaking, or pole terms in matrix elements of supercharge, while in the case of the supersymmetric Higgs model the effective fermion interaction just reduces to a total derivative. In the case of supersymmetric gluodynamics, however, the standard instanton calculus explicitly violates naive Ward identities.     

Spin-charge separation and the Pauli electron

The separation between the spin and the charge converts the quantum mechanical Pauli Hamiltonian into the Hamiltonian of the non-Abelian Georgi-Glashow model, which is notorious for its magnetic monopoles and confinement. The independent spin and charge fluctuations both lead to the Faddeev model, suggesting the existence of a deep duality structure and indicating that the fundamental carriers of the spin and charge are knotted solitons. The text was submitted by the authors in English.     

Instanton approximation to the graded nonlinear sigma model for the integer quantum hall effect

We construct the multi-instanton solutions for the graded nonlinear σ model with symmetry U(1,1/2)/U(1/1) ⊗ U(1/1), and we calculate the quantum fluctuations around these solutions. The determinant of the fluctuation operator for a fixed multi-instanton solution turns out to be UV finite. However, the integration over instanton parameters contains an integral, ∫d|a| |a|⁻³, over the size, |a|, of each instanton, which is quadratically singular at |a|=0. It is shown that these quadratic divergences cancel exactly in the calculation of all Green functions. The applicability of the present results to the integer quantum Hall effect is discussed.     

Gravitational instantons

The path-integral approach to quantum field theory assigns special importance to finite action Euclidean solutions of classical field equations. In Yang-Mills gauge theories, the instanton solutions of classical field equations with self-dual field strength have given rise to a new, nonperturbative treatment of the quantum field theory and its vacuum state. Since gravitation is also a species of gauge theory, one might think that similar phenomena would occur in gravity. The authors recently sought and found a new self-dual solution to Euclidean gravity which plays a role parallel to that of the Yang-Mills instanton. Gravitational instantons now promise to yield new insights into the nature of quantum gravity.This essay received the second award from the Gravity Research Foundation for the year 1979-Ed.     

The structure of the multi-instanton gas

We parametrize the q-instanton solutions of an SU(2) gauge theory in terms of the positions of 2q constituent particles or “instanton quarks”. Explicit computations of lowest order quantum fluctuations about the q = 1 and q = 2 solutions show that the short-distance interaction between instanton quark pairs is logarithmic. Extending these interactions to arbitrary q, we describe the multi-instanton gas as a plasma of instanton quarks.     

Solitons in models with dynamical spontaneous symmetry breaking

An approximate quasi-classical method for the investigation of solitons appearing at quantum level is described. The existence of the soliton in the model with Yukawa coupling in two-dimensional space-time is established with the help of this method. The existence of solitons in the Georgi-Glashow type model with dynamical spontaneous symmetry breaking is shown.     

Instanton effects in N = 2 supersymmetric quantum mechanics

Supersymmetric quantum mechanics with several bosonic and fermionic dynamic variables is considered. Two different N = 2 supersymmetric models involving instantons are discussed in detail. Instantons fail to break supersymmetry in one of the models considered. The vacuum state is degenerate in this model which generally results in spontaneous breaking of internal left-right symmetry. In another model supersymmetry is destroyed dynamically due to special complex instanton solutions. Possible implications for SUSY field theories are discussed.     

Stability of classical solutions of two-dimensional Grassmannian models

We show that the only finite-action solutions of the two-dimensional Grassmannian σ-model that are stable under small fluctuations are the (anti-)instanton solutions.     

Complex monopoles in the Georgi-Glashow-Chern-Simons model

We investigate the three-dimensional Georgi-Glashow model with a Chern-Simons term. We find that there exist complex monopole solutions of finite action. They dominate the path integral and disorder the Higgs vacuum, but electric charges are not confined. Subtleties in the gauge-fixing procedure in the path integral and issues related to Gribov copies are noted.     

New classical solutions with fermion in conformal invariant field theories

New instanton type solutions for coupled non-linear equations of scalar and fermion are given. Invariance properties of the solutions under the six-dimensional conformal group are studied. Quantum significances are discussed, and the equations of motion for quantum fluctuations turn out to be the eigenvalue equations for the Casimir operators of the O(5) group.     

Fluxon solutions in non-abellian gauge models

The fundamental homotopic charges of the adjoint group in non-abelian gauge theories support fluxons, string-like non-peturbative degrees of freedom conjectured to be relevant to quark confinement and symmetry breaking. Fluxons as localized, static, sourceless solutions are constructed in pure SU(2) Yang-Mills theory and in the Georgi-Glashow model.     

Quantum codimension-two cosmology

We generalize the 2-codimensional quantum cosmological model with the vacuum seed instanton to that with a non-vacuum seed instanton associated with the Gibbons–Townsend monopole. The confusion in the literature that substituting a field configuration into an action and varying that action is not equivalent to substituting it into the field equations is clarified.     

Large orders in the 1/N perturbation theory by inverse scattering in one dimension

When one tries to compute large orders in the 1/N series à la Lipatov a complicated non-linear equation for the instanton is found in ø⁴ or non-linear sigma models.We solve here this equation in the one-dimensional case (quantum mechanics) by inverse scattering techniques. From the instanton solutions we obtain theK ^th order of the 1/N perturbation theory up to 0(K ^–1) for the 0(N) symmetric anharmonic oscillator and up to a factor 0(K ⁰) for a non-symmetric model. In the symmetric case we agree with results recently obtained in quantum mechanics by Hikami and Brézin following a different procedure. For the non-symmetric anharmonic oscillator we believe our formulae are new.     

Magnetic monopole in the general theory of relativity

Equations of motion are obtained for Yang-Mills and Higgs fields using the Georgi-Glashow SO(3)-model in a curved spherically symmetric space-time. Magnetic monopole and dion solutions and the space-time metric are found in the intrinsic gravitational field. A particlelike solution of the magnetic monopole and dion type is constructed in the external field of a static universe in the Bogomol'nyi-Prasad-Sommerfield limit.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 89–93, November, 1982.     

Constrained instanton and black hole creation

It is generally accepted that the Einstein theory of general relativity governs the evolution of the universe. However, the singularity theorem of Hawking and Penrose[1] shows that under the most reasonable physical conditions, a spacetime singularity, where physics laws and even causality are broken down, is unavoidable. One has to impose the boundary or initial conditions at the edge of spacetime for the evolution of the universe. This is the so-called first cause problem. The no-boundary …     

Analytic instanton solutions in two-dimensional field models

Two-dimensional scalar field models with nontrivial potential determined in bounded spatial regions are considered. Exact analytic instanton solutions for these models and soliton solutions for similar three-dimensional systems are obtained.