Stability and decay of the Dirac vacuum in external gauge fields

We consider various gauge fields coupled to the free Dirac equation according to symmetry principles. The gauge fields are treated as classical, unquantized fields. Sufficiently strong time-independent fields may give rise to spontaneous particle creation and to the decay of the symmetric Dirac vacuum into a new ground state with broken symmetry. The vacuum stability of the Dirac field is studied for the cases of external electromagnetic (U(1)), gravitational (Poincaré group including torsion) and Yang-Mills (SU(2)) potentials.     

Quantenphysikalischer Ursprung der Eichidee

The Quantum Physical Origin of the Gauge Idea To consider quantum physics as an interplay of creation and annihilation processes has the consequence that gauge field theories are not only possible but necessary. Since the complex conjugate phase factors of each pair of fermion creators and annihilators can be arbitrary chosen, quantum field theories must be completely phase invariant. Unfortunately, even globally the Dirac equation for systems of free fermions is not phase invariant. The Dirac matrices are namely transformed, if we multiply the spinor components by different constant phase factors. The Dirac equations before and after the transformation are however physically equivalent. We may therefore say: Systems of free fermions will be completely described, only if we consider the class of all equivalent Dirac equations. Since Dirac's commutation relations are unitarily invariant, the class equivalent Dirac equations is invariant under all transformations of the group U 4. Unitary diagonal matrices yield arbitrary phase transformations. Hence, gauge fields of the group U 4 are compatible with the postulate of general phase invariance. These gauge file are so similar to the QED that we may speak of an “extended quantum electrodynamics”, EQE. Here, we will show that EQE exists. The invariant subgroup U 1 U 4 yields QED. The complementary subgroup SU 4 includes four subgroups SU 3, there subgroups O 4, and six subgroups SU 2. The latter ones may yield three pairs of quarks and three pairs of leptons, where the quarks form a group SU 3. More than two times three pairs of elementary fermions does not exist in in EQE Probably, EQE is different from the United EQD and QCD. However, it should be a promising version of a field theory in elementary particle physics, because it follows from an existing symmetry of the empirically wel founded Dirac theory. EQE is therefore free from hypothesis in the Newtonian sense of the word. Whatever it will finally mean, it cannot be rejected, since phase invariance must be required. The invention of new symmetries and the acception of a bie number of independent spinor components is dispensable or must be postponed at least.     

Nonrelativistic Wave Equations with Gauge Fields

We illustrate a metric formulation of Galilean invariance by constructing wave equations with gauge fields. It consists of expressing nonrelativistic equations in a covariant form, but with a five-dimensional Riemannian manifold. First we use the tensorial expressions of electromagnetism to obtain the two Galilean limits of electromagnetism found previously by Le Bellac and Lévy-Leblond. Then we examine the nonrelativistic version of the linear Dirac wave equation. With an Abelian gauge field we find, in a weak field approximation, the Pauli equation as well as the spin—orbit interaction and a part reminiscent of the Darwin term. We also propose a generalized model involving the interaction of the Dirac field with a non-Abelian gauge field; the SU(2) Hamiltonian is given as an example.     

Simulation of 4d \mathcal{N}=1 supersymmetric Yang–Mills theory with Symanzik improved gauge action and stout smearing

We report on the results of a numerical simulation concerning the low-lying spectrum of four-dimensional $\mathcal{N}=1$ SU(2) Supersymmetric Yang–Mills (SYM) theory on the lattice with light dynamical gluinos. In the gauge sector the tree-level Symanzik improved gauge action is used, while we use the Wilson formulation in the fermion sector with stout smearing of the gauge links in the Wilson–Dirac operator. The ensembles of gauge configurations were produced with the Two-Step Polynomial Hybrid Monte Carlo (TS-PHMC) updating algorithm. We performed simulations on large lattices up to a size of 24³?48 at β=1.6. Using QCD units with the Sommer scale being set to r ₀=0.5 fm, the lattice spacing is about a?0.09 fm, and the spatial extent of the lattice corresponds to 2.1 fm. At the lightest simulated gluino mass the spin-1/2 gluino–glue bound state appeared to be considerably heavier than its expected super-partner, the pseudoscalar bound state. Whether supermultiplets are formed remains to be studied in upcoming simulations.     

Topological Lifshitz transition and novel edge states induced by non-Abelian SU(2) gauge field on bilayer honeycomb lattice

We investigate the SU(2) gauge effects on bilayer honeycomb lattice thoroughly. We discover a topological Lifshitz transition induced by the non-Abelian gauge potential. Topological Lifshitz transitions are determined by topologies of Fermi surfaces in the momentum space. Fermi surface consists of N = 8 Dirac points at π-flux point instead of N = 4 in the trivial Abelian regimes. A local winding number is defined to classify the universality class of the gapless excitations. We also obtain the phase diagram of gauge fluxes by solving the secular equation. Furthermore, the novel edge states of biased bilayer nanoribbon with gauge fluxes are also investigated.     

Solution of the Dirac equation by separation of variables in spherical coordinates for a large class of non-central electromagnetic potentials

A systematic and intuitive approach for the separation of variables of the three-dimensional Dirac equation in spherical coordinates is presented. Using this approach, we consider coupling of the Dirac spinor to electromagnetic four-vector potential that satisfies the Lorentz gauge. The space components of the potential have angular (non-central) dependence such that the Dirac equation becomes separable in all coordinates. We obtain exact solutions for a class of three-parameter static electromagnetic potential whose time component is the Coulomb potential. The relativistic energy spectrum and corresponding spinor wave functions are obtained. The Aharonov–Bohm and magnetic monopole potentials are included in these solutions.     

Minimal technicolor on the lattice

We present results from a lattice study of SU(2) gauge theory with 2 flavors of Dirac fermions in adjoint representation. This is a candidate for a minimal (simplest) walking technicolor theory, and has been predicted to possess either an IR fixed point (where the physics becomes conformal) or a coupling which evolves very slowly, so-called walking coupling. In this initial part of the study we investigate the lattice phase diagram and the excitation spectrum of the theory.     

On superstring-inspired SU(3) × SU(2) × (U(1))2 models with SO(10) unification

Alternative compactification of the heterotic superstring can give rise to a residual four-dimensional SO(10) gauge symmetry for the observable sector. We consider a model with gauge group SU(3) × SU(2) × (U(1))², resulting from the Hosotani breaking of SO(10), study its further gauge symmetry breaking and calculate the particle spectrum. We find that all breakings can occur close to the weak scale; the top mass lies preferentially between 40 and 60 GeV, the second Z boson mass is O(200–400) GeV. The sparticle spectrum is quite heavy, apart possibly from a light chargino, whose mass can be as light as O(30) GeV.     

Fermions and gauge fields in the Einstein static universe

Exact solutions of the massive Dirac equation are obtained in an SU(2) gauge field background in the Einstein static universe. The static, finite-energy gauge field used as background is the one obtained by continuing the meron-antimeron solution to this base space. The regular spinor solutions lead to a quantization condition.     

Spectra of massive and massless QCD dirac operators: A novel link

We show that integrable structure of chiral random matrix models incorporating global symmetries of QCD Dirac operators (labeled by the Dyson index beta = 1,2, and 4) leads to emergence of a connection relation between the spectral statistics of massive and massless Dirac operators. This novel link established for beta-fold degenerate massive fermions is used to explicitly derive (and prove the random matrix universality of) statistics of low-lying spectra of QCD Dirac operators in the presence of SU(2) massive fermions in the fundamental representation ( beta = 1) and SU(N(c)>/=2) massive adjoint fermions ( beta = 4). Comparison with available lattice data for SU(2) dynamical staggered fermions reveals a good agreement.     

在SU(2)模型中费密子的质量与Rubakov效应

在自发破缺的具有Higgs三重态的SU(2)规范理论中,讨论了同位旋1/2的费密子的质量对费密子-磁单极束缚态的影响。结果表明,当费密子与Higgs场之间的直接耦合趋于零,但狄拉克质量保持固定时,费密子-磁单极束缚态的必要条件必不被满足。这个结果意味着对于SU(2)模型由于狄拉克质量效应,Rubakov效应不存在。 关键词：     

DOES THE RUBAKOV EFFECT EXIST IN AN SU(2) MODEL

In an SU(2) spontaneously broken gauge theory with a Higgs triplet, the mass effect of an iosspin 1/2 fermion on the fernlion-nlonopole bound states is discussed. It is shown that when the direct coupling betrveen fermion and Higgs field approaches zero, but the Dirac mass remalns finiteness, the necessary condition of the fermion-monopole bound state cannot be satisfied. This result means that the Rubakov effect is absent for SU(2) monopole because of the Dirac mass.     

Non-abelian Gauge Symmetry for Fields in Phase Space: a Realization of the Seiberg-Witten Non-abelian Gauge Theory

The Seiberg-Witten formalism has been realized as an electrodynamics in phase space (associated to the Dirac equation written in phase space) and this fact is explored here with non-abelian gauge group. First, a physically heuristic presentation of the Seiberg-Witten approach is carried out for non-abelian gauge in order to guide the calculation procedures. These results are realized by starting with the Lagrangian density for the free Dirac field in phase space. Then a field strength is derived, where the non-abelian gauge group is the SU(2), corresponding to an isospin (non-abelian) field theory in phase space. An application to nucleon is then discussed.

     

From SU(2) Gauge Theory to Qubits on the Fuzzy Sphere

We consider a classical pure SU(2) gauge theory, and make an ansatz, which separates the spatio-temporal degrees of freedom from the internal ones. This ansatz is gauge-invariant but not Lorentz invariant. In a limit case of the ansatz, obtained through a contraction map, and corresponding to a vacuum solution, the SU(2) gauge field reduces to an operator, which is the product of the generator of a global U(1) group times a Pauli matrix. We give a geometrical interpretation of the ansatz and of the contraction map in the framework of principal fiber bundles. Then, we identify the internal degrees of freedom of the gauge field with the non-commutative coordinates of the fuzzy sphere in the fundamental representation. In this way we obtain a qubit state.     

Construction ofYM 4 with an infrared cutoff

We provide the basis for a rigorous construction of the Schwinger functions of the pure SU(2) Yang-Mills field theory in four dimensions (in the trivial topological sector) with a fixed infrared cutoff but no ultraviolet cutoff, in a regularized axial gauge. The construction exploits the positivity of the axial gauge at large field. For small fields, a different gauge, more suited to perturbative computations is used; this gauge and the corresponding propagator depends on large background fields of lower momenta. The crucial point is to control (in a non-perturbative way) the combined effect of the functional integrals over small field regions associated to a large background field and of the counterterms which restore the gauge invariance broken by the cutoff. We prove that this combined effect is stabilizing if we use cutoffs of a certain type in momentum space. We check the validity of the construction by showing that Slavnov identities (which express infinitesimal gauge invariance) do hold non-perturbatively.     

Unification of effective field theories

We consider the renormalization of the coupling constants in theories with extended gauge hierarchies. An effective field theory approach is used to include an interesting class of higher-order effects in the renormalization group formulas. We calculate these corrections for all possible breakdowns of O(10) to SU(3)×SU(2)×U(1).     

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.     

Axial-vector anomaly for Dirac-Kähler fermions on the lattice

The axial-vector current of Dirac-Kähler fermions on the lattice is studied. We consider a U(1) gauge theory in two dimensions as well as an SU(N) gauge theory in four dimensions. Using a short-distance expansion of the fermion propagator in an external gauge field, we show that the correct anomaly is reproduced in the continuum limit.