Michael O’Carroll

Ground state properties and lower bounds for energy levels of a particle in a uniform magnetic field and external potential

The Hamiltonian H (B), for a particle of mass μ and charge e in a uniform magnetic field of strength B in the z direction and an external axially symmetric potential V, is a direct sum of operators H (m,B) acting in the subspace of eigenvalue m of the z component of angular momentum Lz. Let λ (B) [λ (m,B)] denote the smallest eigenvalue of H (B) [H (m,B)]. For V=−e2/r (r=‖x‖), the attractive Coulomb potential, we obtain lower bounds l (m,B), for the spectrum of H (m,B) such that l (0,B) ≳l (0,0) =λ (0,0) for B≳0, l (m,B) ≳l (0,B) for m≠0, and l (‖m‖,B)−l (−‖m‖,B) =eB‖m‖/μ, l (m′,B) ≳l (m,B) if m′<m?0. We show at least for an interval [0,B′≳0] of B that the ground state of H (B) is the lowest eigenvalue, λ (0,B), of H (0,B) and is an almost everywhere positive function. If V=A/r2+r2 for B=0, λ (0) =λ (0,0) and the ground state wavefunction is an almost everywhere positive function with Lz eigenvalue zero. However, for large A, we prove that for an interval of B away from B=0, the lowest eigenvalue, λ (−1,B...

Eightfold way from dynamical first principles in strongly coupled lattice quantum chromodynamics

We obtain from first principles, i.e., from the quark-gluon dynamics, the Gell’Mann-Ne’eman baryonic eightfold way energy momentum spectrum exactly in an imaginary-time functional integral formulation of strongly coupled lattice quantum chromodynamics in 3+1 dimensions, with local SU(3)c gauge and global SU(3)f flavor symmetries. We take the hopping parameter κ and the pure gauge coupling β satisfying the strong coupling regime condition 0⩽β⪡κ⪡1. The form of the 56 baryon fields emerges naturally from the dynamics and is unveiled using the hyperplane decoupling method. There is no a priori guesswork. In the associated physical quantum mechanical Hilbert space H, spectral representations are derived for the two-baryon functions, which are used to rigorously detect the particles in the energy-momentum spectrum. Using the SU(3)f symmetry, the 56 baryon states admit a decomposition into 8×2 states associated with a spin 1∕2 octet and 10×4 states associated with a spin 3∕2 decuplet. The states are labeled by t...

Existence of mesons and mass splitting in strong coupling lattice quantum chromodynamics

We consider one flavor lattice quantum chromodynamics in the imaginary time functional integral formulation for space dimensions d=2, 3 with 4×4 Dirac spin matrices, small hopping parameter κ, 0<κ≪1, and zero plaquette coupling. We determine the energy-momentum spectrum associated with four-component gauge invariant local meson fields which are composites of a quark and an antiquark field. For the associated correlation functions, we establish a Feynman–Kac formula and a spectral representation. Using this representation, we show that the mass spectrum consists of two distinct masses ma and mb, given by mc=−2 ln κ+rc(κ), c=a,b, where rc is real analytic. For d=2, ma and mb have multiplicity two and the mass splitting is κ4+O(κ6); for d=3, one mass has multiplicity one and the other three, with mass splitting 2κ4+O(κ6). In the subspace of the Hilbert space generated by an even number of fermion fields the dispersion curves are isolated (upper gap property) up to near the two-meson threshold of asymptotic m...

Mesonic eightfold way from dynamics and confinement in strongly coupled lattice quantum chromodynamics

We show the existence of all the 36 eightfold way mesons and determine their masses and dispersion curves exactly, from dynamical first principles such as directly from the quark-fluon dynamics. We also give a proof of confinement below the two-meson energy threshold. For this purpose, we consider an imaginary time functional integral representation of a 3+1 dimensional lattice QCD model with Wilson action, SU(3)f global and SU(3)c local symmetries. We work in the strong coupling regime, such that the hopping parameter κ>0 is small and much larger than the plaquette coupling β>1/g02⩾0 (β⪡κ⪡1). In the quantum mechanical physical Hilbert space H, a Feynman-Kac type representation for the two-meson correlation and its spectral representation are used to establish an exact rigorous connection between the complex momentum singularities of the two-meson truncated correlation and the energy-momentum spectrum of the model. The total spin operator J and its z-component Jz are defined by using π∕2 rotations about t...

The Existence and Completeness of Generalized Wave Operators for Spherically Symmetric, Asymptotically Coulomb Potentials

By applying a theorem of Kuroda, we prove the existence and continuous completeness of the generalized wave operators W±(H2, H1), W±(H1, H2), where W±(Hj, Hi) = s — limeiHjte−iHit × Pi, in the Hilbert space L2(R3). Pi is the projection operator on the subspace of absolute continuity of Hi. H1 is the self‐adjoint Hamiltonian for a particle in a pure Coulomb potential Vc = ze2/|x|, and H2 is the self‐adjoint Hamiltonian for a system described by the potential function Vc + V, where V is a real‐valued, measurable function of x e R3, spherically symmetric [V(x) = V(r = |x|)], satisfies the condition ∫ 0Rr2|V(r)|2dr+ ∫ 0∞(1+r)δ|V(r)|idr<∞ for some R(0 ≤ R < ∞), some 0 < δ < 1 with i = 1, 2, and is continuous except at r = 0. In conjunction with our result, we obtain a bound for the radial Coulomb Greens function. Dropping the continuity assumption on V, we have absolutely continuous completeness of the wave operators.

One-baryon spectrum and analytical properties of one-baryon dispersion curves in 3 + 1 dimensional strongly coupled lattice QCD with three flavors

Considering a 3 + 1 dimensional lattice quantum chromodynamics (QCD) model defined with the improved Wilson action, three flavors, and 4 × 4 Dirac spin matrices, in the strong coupling regime, we reanalyze the question of the existence of the eightfold way baryons and complete our previous work where the existence of isospin octet baryons was rigorously solved. Here, we show the existence of isospin decuplet baryons which are associated with isolated dispersion curves in the subspace of the underlying quantum mechanical Hilbert space with vectors constructed with an odd number of fermion and antifermion basic quark and antiquark fields. Moreover, smoothness properties for these curves are obtained. The present work deals with a case for which the traditional method to solve the implicit equation for the dispersion curves, based on the use of the analytic implicit function theorem, cannot be applied. We do not have only one but two solutions for each one-baryon decuplet sector with fixed spin third compone...

Bounds on the number of bound states in the transfer matrix spectrum for some weakly correlated lattice models

We consider the interaction of particles in weakly correlated lattice quantum field theories. In the imaginary time functional integral formulation of these theories there is a relative coordinate lattice Schroedinger operator H which approximately describes the interaction of these particles. Scalar and vector spin, QCD and Gross-Neveu models are included in these theories. In the weakly correlated regime H = Ho + W where Ho = −γΔl, 0 < γ ≪ 1 and Δl is the d-dimensional lattice Laplacian: γ = β, the inverse temperature for spin systems and γ = κ3 where κ is the hopping parameter for QCD. W is a self-adjoint potential operator which may have non-local contributions but obeys the bound |W(x, y)| ⩽ cexp ( − a(|x| + |y|)), a large: exp−a=β/βo12(κ/κo) for spin (QCD) models. Ho, W, and H act in l2(Zd), d ⩾ 1. The spectrum of H below zero is known to be discrete and we obtain bounds on the number of states below zero. This number depends on the short range properties of W, i.e., the long range tail does not inc...

Scaled lattice fermion fields, stability bounds, and regularity

We consider locally gauge-invariant lattice quantum field theory models with locally scaled Wilson-Fermi fields in d = 1, 2, 3, 4 spacetime dimensions. The use of scaled fermions preserves Osterwalder-Seiler positivity and the spectral content of the models (the decay rates of correlations are unchanged in the infinite lattice). In addition, it also results in less singular, more regular behavior in the continuum limit. Precisely, we treat general fermionic gauge and purely fermionic lattice models in an imaginary-time functional integral formulation. Starting with a hypercubic finite lattice Λ⊂(aZ)d, a ∈ (0, 1], and considering the partition function of non-Abelian and Abelian gauge models (the free fermion case is included) neglecting the pure gauge interactions, we obtain stability bounds uniformly in the lattice spacing a ∈ (0, 1]. These bounds imply, at least in the subsequential sense, the existence of the thermodynamic (Λ ↗ (aZ)d) and the continuum (a ↘ 0) limits. Specializing to the U(1) gauge gro...

Hadron‐Hadron Bound States From First Principles

We present a summary of our results on the hadron spectrum and hadron‐hadron bound state spectrum which were obtained as part of a program which tries to bridge the gap between QCD and Nuclear Physics. We analyzed SU(3) lattice QCD models in the strong coupling regime (small hopping parameter κ > 0 and large glueball mass). We considered an imaginary time formulation for 2 + 1 and 3 + 1 dimensions models with one and two flavors. For the more algebraically complex and more realistic model in 3 + 1 dimensions, 4 × 4 spin matrices, and two flavors, we show the existence of three‐quark isospin 1/2 particles (proton and neutron) and isospin 3/2 baryons (delta particles), with asymptotic masses −3ln κ and isolated dispersion curves. Baryon‐baryon bound states of isospin zero are found with binding energy of order κ2. The two‐particle analysis is performed using a ladder approximation to a lattice Bethe‐Salpeter equation written with the help of relative coordinates adapted to the lattice. The dominant baryon‐baryon interaction is an energy‐independent spatial range‐one attractive potential with an O(κ2) strength. There is also attraction arising from gauge field correlations associated with six overlapping bonds, but it is counterbalanced by Pauli repulsion to give a vanishing zero‐range potential. The overall range‐one potential results from a quark, antiquark exchange with no meson exchange interpretation since the spin indices are not of a meson particle; we call it a quasimeson exchange. The repulsive or attractive nature of the interaction depends on the isospin and spin of the two‐baryon state.We present a summary of our results on the hadron spectrum and hadron‐hadron bound state spectrum which were obtained as part of a program which tries to bridge the gap between QCD and Nuclear Physics. We analyzed SU(3) lattice QCD models in the strong coupling regime (small hopping parameter κ > 0 and large glueball mass). We considered an imaginary time formulation for 2 + 1 and 3 + 1 dimensions models with one and two flavors. For the more algebraically complex and more realistic model in 3 + 1 dimensions, 4 × 4 spin matrices, and two flavors, we show the existence of three‐quark isospin 1/2 particles (proton and neutron) and isospin 3/2 baryons (delta particles), with asymptotic masses −3ln κ and isolated dispersion curves. Baryon‐baryon bound states of isospin zero are found with binding energy of order κ2. The two‐particle analysis is performed using a ladder approximation to a lattice Bethe‐Salpeter equation written with the help of relative coordinates adapted to the lattice. The dominant baryon‐b...