Physical Ground State Research Articles

Mode regularization of the supersymmetric sphaleron and kink: Zero modes and discrete gauge symmetry

To obtain the one-loop corrections to the mass of a kink by mode regularization, one may take one-half the result for the mass of a widely separated kink-antikink (or sphaleron) system, where the two bosonic zero modes count as two degrees of freedom, but the two fermionic zero modes as only one degree of freedom in the sums over modes. For a single kink, there is one bosonic zero mode degree of freedom, but it is necessary to average over four sets of fermionic boundary conditions in order (i) to preserve the fermionic Z$_2$ gauge invariance $\psi \to -\psi$, (ii) to satisfy the basic principle of mode regularization that the boundary conditions in the trivial and the kink sector should be the same, (iii) in order that the energy stored at the boundaries cancels and (iv) to avoid obtaining a finite, uniformly distributed energy which would violate cluster decomposition. The average number of fermionic zero-energy degrees of freedom in the presence of the kink is then indeed 1/2. For boundary conditions leading to only one fermionic zero-energy solution, the Z$_2$ gauge invariance identifies two seemingly distinct `vacua' as the same physical ground state, and the single fermionic zero-energy solution does not correspond to a degree of freedom. Other boundary conditions lead to two spatially separated $\omega \sim 0 $ solutions, corresponding to one (spatially delocalized) degree of freedom. This nonlocality is consistent with the principle of cluster decomposition for correlators of observables.

Exponential decay of positivity preserving semigroups on L^p

In recent studies on invariant measures of diffusions on infinite-dimensional spaces, several approaches have been considered. In [6], existence, uniqueness and regularity of invariant measures and asymptotic properties of solutions of stochastic evolution equations were investigated. In purely analytic approaches, a study of invariant measures for Markovian semigroups on L associated with quadratic forms was made in [5] as a generalization of [19, 20, 21, 10], while the regularity of the measures μ solving some elliptic equations L∗μ = 0 was proved in [4, 3], where L is an operator of type Lu = tr(Au′′) +B · ∇u. In this paper, for given Markovian (or more generally, positivity preserving) semigroups {Pt}, we discuss conditions for the existence of invariant measures, and for the exponential decay of {Pt} to a projection operator. We treat this problem in an analytic way, so we always impose the condition that {Pt} is also a strongly continuous semigroup on L for some p ∈ (1,∞). As for the existence of invariant measures, the situation becomes quite simple if {Pt} is eventually compact. But we do not assume this since such compactness seems hard to be expected in the case that the underlying space is infinite dimensional. In Gross’ paper [9] concerning physical ground states for Hermitian operators, similar situations were dealt with and the so-called hyperboundedness of semigroups was used as a replacement of compactness. We apply his idea to our problem; under the condition (I) regarding integrability (see Definition 2.1) for semigroups or resolvents, we prove the existence of invariant measures by approximating the underlying space by a sequence of finite number of sets. The result improves the corresponding ones in [5, 10]. In order to discuss the exponential decay of {Pt} in the L sense, we introduce a kind of ergodicity condition, (E) (see Definition 3.1). We may say that it is a substitution for strict positivity of transition densities; we do not expect the existence of such densities in our concerning infinite-dimensional cases. This type of condition appeared in Kusuoka’s

Supersymmetry and stationary solutions in dilaton-axion gravity.

New stationary solutions of $4$-dimensional dilaton-axion gravity are presented, which correspond to the charged Taub-NUT and Israel-Wilson-Perjes (IWP) solutions of Einstein-Maxwell theory. The charged axion-dilaton Taub-NUT solutions are shown to have a number of interesting properties: i) manifest $SL(2,R)$ symmetry, ii) an infinite throat in an extremal limit, iii) the throat limit coincides with an exact CFT construction. The IWP solutions are shown to admit supersymmetric Killing spinors, when embedded in $d=4,N=4$ supergravity. This poses a problem for the interpretation of supersymmetric rotating solutions as physical ground states. In the context of $10$-dimensional geometry, we show that dimensionally lifted versions of the IWP solutions are dual to certain gravitational waves in string theory.

The polarization approximation and the Amos-Musher intermolecular perturbation theories compared to infinite order at finite separations

We argue that the most common form of intermolecular perturbation theory, the polarization approximation (PA), gives at finite separations, for nearly all systems, no usable, exact relationship between the lowest unperturbed energy and the physical ground state energy. This would be true even if the PA energy expansion converged. In contrast, the Amos-Musher (AM) theory does give an exact relationship between the lowest unperturbed energy and the physical ground state energy. We find no theoretical justification for using the PA theory beyond second order in the energy, but the AM theory may be extended to arbitrarily high order.

Solving Schrödinger's equation for pure SU(3) on a large lattice.

We study the solution of Schr\"odinger's equation for pure SU(3) lattice gauge theory on a 10\ifmmode\times\else\texttimes\fi{}10\ifmmode\times\else\texttimes\fi{}10 spatial lattice. Our basis consists of 9001 states of modified Krylov-type. No gauge fixing is used. Instead we work on an ensemble of basis states each having different color content on the lattice. Physical (colorless) results are obtained by extrapolation in color. We determine the physical ground state energy density from strong to weak coupling at the 10% level of accuracy. We show that physical glueball masses can be measured by our method. Our basis must be improved to see scaling and obtain continuum results.

The problem of unphysical states in the theory of intermolecular interactions

It was shown by Claverie that the interactions between atoms and molecules make unphysical electronic solutions of the Schradinger equation accessible in perturbation calculations of intermolecular interactions, accessible in the sense that the perturbation expansion is likely to converge to an unphysical solution if it converges at all. This is a difficult problem because there are generally an infinite number of unphysical states with energies below that of the physical ground state. We have carried out configuration interaction calculations on LiH of both physical and unphysical states. They show that avoided crossings occur between physical and unphysical energy levels as the interaction between the two atoms is turned on, i.e. as the expansion parameter λ is increased from 0 to 1. The avoided crossing for the lowest energy state occurs for λ < 0.8, implying that the perturbation expansion will diverge for larger values of λ. The behavior of the energy levels as functions of λ. is shown to be understandable in terms of a two-state model. In the remainder of the paper, we concentrate on designing effective Hamiltonians which have physical solutions identical to those of the Schrodinger equation, but which have no unphysical states of lower energy than the physical ground state. We find that we must incorporate ideas from the Hirschfelder-Silbey perturbation theory, as modified by Polymeropoulos and Adams, to arrive finally at an effective Hamiltonian which promises to have the desired properties, namely, that all unphysical states be higher in energy than the physical bound states, that the first and higher order corrections to the energy vanish in the limitR = ∞. that the leading terms of the asymptotic 1/R expansion of the energy be given correctly in second order, and that the overlap between the zeroth order wave function and the corresponding eigenfunction of the effective Hamiltonian be close to one.

ON BOSONIC STRING MODELS IN SEMICLASSICAL APPROXIMATION: THE DERIVATIVE OF THE FUNCTIONAL DETERMINANT OF THE TRANSVERSE FLUCTUATION OPERATOR

Under some restrictions we calculate the functional derivative of the determinant of an operator which is the main ingredient in the semiclassical approximation of the physical ground state of the closed bosonic string. We show that a formula for the determinant itself previously published by other authors is incorrect.

SUPERSYMMETRY AND THE DIRAC OSCILLATOR

We demonstrate the realization of supersymmetric quantum mechanics in the first-order Dirac oscillator equation by associating with it another Dirac equation, which may be considered as its supersymmetric partner. We show that both the particle and the antiparticle spectra, resulting from these two equations after filling the negative-energy states and redefining the physical ground state, indeed present the degeneracy pattern characteristic of unbroken supersymmetry. In addition, we analyze in detail two algebraic structures, each partially explaining the degeneracies present in the Dirac oscillator supersymmetric spectrum in the non-relativistic limit. One of them is the spectrum-generating superalgebra osp(2/2, ℝ), first proposed by Balantekin. We prove that it is closely connected with the supersymmetric structure of the first-order Dirac oscillator equation as its odd generators are the two sets of supercharges respectively associated with the equation and its supersymmetric partner. The other algebraic structure is an so(4)⊕so(3, 1) algebra, which is an extension of a similar algebra first considered by Moshinsky and Quesne. We prove that it is the symmetry algebra of the Dirac oscillator supersymmetric Hamiltonian. Some possible relations between the spectrum-generating superalgebra, the symmetry algebra, and their respective subalgebras are also suggested.

Perturbation theory of intermolecular interactions: Are second-order Rayleigh-Schr�dinger energies meaningful?

The interactions between atoms are so strong that they create an infinite number of unphysical states of lower energy than the physical ground state, thus raising the question of the significance of second and higher order perturbation energies. We have carried out a series of configuration interaction calculations which shed light on this question and other questions regarding the perturbation theory of intermolecular interactions. The calculations used an unconventional set of configuration functions, each being the product of two Slater determinants, one for each atom. This makes it possible to calculate approximations to both unphysical and physical states, and to follow the changes in eigenstate energies as the interaction is turned on between atoms. We report results for LiH which show that eigenvalues of Ĥ0 + λV̂, where Ĥ0 is the sum of atomic Hamiltonians and V̂ the interaction potential, undergo avoided crossings between λ = 0 and 1. The perturbation expansion of the ground state energy must become divergent at the first avoided crossing, λ = 0.76. In addition, we report results of calculations with three effective Hamiltonians which have all unphysical state energies shifted into the physical continuum. They show no avoided crossings between the ground state and other states until λ is almost equal to 1. The perturbation expansion derived from one of these effective Hamiltonians is shown to have several desirable properties which the expansions derived from the other two lack. Its one fault is that it must diverge at λ = 1, where an avoided crossing occurs. This avoided crossing is related to the exchange symmetry of electrons. The implications of our results for the meaning of second-order perturbation energies and for the development of convergent exchange perturbation methods is discussed.

Perturbation theory of intermolecular interactions: What is the problem, are there solutions?

We review the nature of the problem in the framework of Rayleigh–Schrödinger perturbation theory (the polarization approximation) considering explicitly two examples: the interaction of two hydrogen atoms and the interaction of Li with H. We show, in agreement with the work of Claverie and of Morgan and Simon, that the LiH problem is dramatically different from the H2 problem. In particular, the physical states of LiH are higher in energy than an infinite number of discrete, unphysical states and they are buried in a continuum of unbound, unphysical states, which starts well below the lowest physical state. Claverie has shown that the perturbation expansion, under these circumstances, is likely to converge to an unphysical state of lower energy than the physical ground state, if it converges at all. We review, also, the application of two classes of exchange perturbation theory to LiH and larger systems. We show that the spectra of three Eisenschitz–London (EL) class, exchange perturbation theories have no continuum of unphysical states overlaying the physical states and no discrete, unphysical states below the lowest physical state. In contrast, the spectra of two Hirschfelder–Silbey class theories differ hardly at all from that found with the polarization approximation. Not one of the EL class of perturbation theories, however, eliminates all of the discrete unphysical states. The best one establishes a one-to-one correspondence between the lowest energy states of the unperturbed and perturbed Hamiltonians, and a one-to-two correspondence for the higher states. We suggest that the EL class perturbation theories would be good starting points for the development of more effective perturbation theories for intermolecular interactions.

Absolute ground-state (t,p) strengths in germanium isotopes and two-state mixing.

A two-state model that assumed dominant mixing between a ``normal'' basis ground state and an ``intruder'' basis excited state was used earlier to describe the wave functions for the physical ground state and an excited ${0}^{+}$' state in the germanium isotopes. The wave-function amplitudes deduced for the ground state and ${0}^{+}$' state were shown to be consistent with the ${0}^{+}$'/g.s. cross-section ratios, \ensuremath{\sigma}${(0}^{+}$')/\ensuremath{\sigma}(g.s.), in the $^{\mathrm{A}}\mathrm{Ge}$(t,p) and $^{\mathrm{A}+2}\mathrm{Ge}$(p,t) reactions. In this report we investigate whether these same ground-state wave-function amplitudes together with distorted-wave Born-approximation calculations can account for the A dependence in the absolute $^{\mathrm{A}}\mathrm{Ge}$(t,p${)}^{A+2}$Ge(g.s.) cross sections.

Competing coexistence models in medium-mass nuclei.

Simple phenomenological two-state coexistence models have previously been developed to describe the ground state and some low-lying ${0}^{+\mathcal{'}}$ excited state in isotopic chains of even-even medium-mass nuclei. Specifically, these models allow for mixing between a basis ground state and a basis ${0}^{+}$ intruder state to form the wave functions for the physical ground state and ${0}^{+\mathcal{'}}$ excited state. The amount of data accounted for by these models is amazing, considering their simplicity. However, these two-state phenomenological models contain inconsistencies that have previously been overlooked. We point out some of these inconsistencies and emphasize that they can be removed by a slight generalization of the two-state coexistence model, without destroying agreement between data and model predictions.

Vacuum structure in non-abelian compact SU(N) gauge theories

Abstract By constructing an effective non-local action and assuming that the physical ground state is trivial, we found an infinite correlation length associated to all non-abelian compact SU(N) gauge theories.

Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman–Kac formula. II. Applications to simple systems

We have described in part I of this work the theoretical basis of a quantum Monte Carlo method based on the use of a pure diffusion process and of the so-called full generalized Feynman–Kac (FGFK) formula. In this second part, we present a set of applications (one-dimensional oscillator, helium-like systems, hydrogen molecule) with the purpose of illustrating in a systematic way the various aspects pertaining to the practical implementation of this method. We thus show how energy and other observables can be obtained, and we discuss the various sources of biases occurring in the different procedures (notably the so-called short-time approximation pertaining to the generation of the sample trajectories of the diffusion process, and the numerical integration pertaining to the evaluation of the ‘‘Feynman–Kac factor’’). After having thus considered the case of the genuine ‘‘bosonic’’ ground state, we illustrate the various proposals for dealing with some ‘‘relative’’ ground state (namely the lowest state belonging to some prescribed symmetry), one of the most important cases being obviously the physical ground state of many-fermion systems (owing to the Pauli principle requirements). More specifically, we consider the so-called fixed-node approximation (FNA), on one hand, and two variants of a potentially exact procedure, the so-called simple projection (SP) and release-node projection (RNP) methods, on the other hand. Finally, some perspectives concerning future developments are outlined.

Energy and symmetry of states in light nuclei.

The symmetry properties of low-lying eigenstates of s-shell nuclei are examined, and the lowest energy state is shown to be completely antisymmetric, which corresponds to the physical ground state. Green's function Monte Carlo methods may therefore be applied without the rapid increase in noise usually associated with fermion systems.

Confinement and vacuum structure in compact QED on the lattice

Using an effective action and assuming that the physical ground state has no structure, we demonstrate that the spectrum of the compact U(1) gauge model has massless (vector and scalar) bosons.

Spin waves in quantum ferromagnets

Low-temperature properties of Heisenberg quantum ferromagnets (“spin waves”) are derived within aconfiguration space formalism. Most of the work is done without explicitly assuming translational invariance. We provide a general criterion, classical domination, to decide about the nature and uniqueness of ground states for a large class of quantum ferromagnets. We also analyze and clarify the Dyson formalism and indicate why an energy gap between the physical ground state and the improper (unphysical) states does not exist. This is of particular relevance to the kinematical interaction. Using reflection positivity we provide upper and lower bounds to the contribution of the dynamical interaction to the free energy. In a certain approximation, these bounds imply that the dynamical interaction may be dropped if the inverse temperatureβ and the spin quantum numberS are large enough.

ChemInform Abstract: ELECTRONIC GROUND STATE OF IRON(II) PHTHALOCYANINE AS DETERMINED FROM ACCURATE DIFFRACTION DATA

Chemischer InformationsdienstVolume 15, Issue 1 Physical Organic Chemistry ChemInform Abstract: ELECTRONIC GROUND STATE OF IRON(II) PHTHALOCYANINE AS DETERMINED FROM ACCURATE DIFFRACTION DATA P. COPPENS, P. COPPENSSearch for more papers by this authorL. LI, L. LISearch for more papers by this authorN. J. ZHU, N. J. ZHUSearch for more papers by this author P. COPPENS, P. COPPENSSearch for more papers by this authorL. LI, L. LISearch for more papers by this authorN. J. ZHU, N. J. ZHUSearch for more papers by this author First published: January 3, 1984 https://doi.org/10.1002/chin.198401054AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume15, Issue1January 3, 1984 RelatedInformation

ChemInform Abstract: HALOGENATED BENZENE RADICAL CATIONS AND GROUND STATE DEGENERACY SPLITTING BY ASYMMETRIC SUBSTITUTION

Chemischer InformationsdienstVolume 12, Issue 39 Physical Organic Chemistry ChemInform Abstract: HALOGENATED BENZENE RADICAL CATIONS AND GROUND STATE DEGENERACY SPLITTING BY ASYMMETRIC SUBSTITUTION V. E. BONDYBEY, V. E. BONDYBEYSearch for more papers by this authorC. R. VAUGHN, C. R. VAUGHNSearch for more papers by this authorT. A. MILLER, T. A. MILLERSearch for more papers by this authorJ. H. ENGLISH, J. H. ENGLISHSearch for more papers by this authorR. H. SHILEY, R. H. SHILEYSearch for more papers by this author V. E. BONDYBEY, V. E. BONDYBEYSearch for more papers by this authorC. R. VAUGHN, C. R. VAUGHNSearch for more papers by this authorT. A. MILLER, T. A. MILLERSearch for more papers by this authorJ. H. ENGLISH, J. H. ENGLISHSearch for more papers by this authorR. H. SHILEY, R. H. SHILEYSearch for more papers by this author First published: September 29, 1981 https://doi.org/10.1002/chin.198139047AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume12, Issue39September 29, 1981 RelatedInformation

The ‘‘primitive’’ wave function in the theory of intermolecular interactions

The concept of the primitive wave function for a supermolecule consisting of interacting subsystems is critically analyzed. The distinction between formal and genuine primitive functions is stressed. The concept of uniformly complete basis sets as contrasted to simply complete basis is introduced. Primitive basis sets are defined and shown not to be uniformly complete for the expansion of the supersystem wave function while ’full supersystem basis sets’ are. The conditions are specified under which a supersystem wave function can be decomposed into its ’primitive components’ corresponding to different partitions of the electrons among the subsystems. These primitive components satisfy the Schrödinger equation asymptotically. The matrix representation of the Hamiltonian (both the full supersystem Hamiltonian H and the zeroth order Hamiltonian Ho) in terms of these partitions is analyzed. It is shown that in the standard application of RS-perturbation theory to intermolecular forces (the polarization approximation) the limiting processes λ→1 and R→∞ do not commute, that the λ-series is not uniformly convergent with respect to R and that the wave function to any finite order in λ is genuinely primitive. The symmetrized polarization approximation is justified for the ’coasymptotic ground state’ in certain cases and a ’symmetrized polarization approximation with shifted eigenvalues’ is proposed that connects the lowest eigenvalue of Ho with the physical ground state. A justification of simplified schemes in the region of ’small exchange’ is given and alternative perturbation schemes are discussed. Finally the use of the primitive function in variational treatments is outlined. One advantage is that a genuinely (not a formally) primitive function is uniformly expandable in a primitive basis set.