Online versions of mathematical publications

Complete list of publications

Below are abstracts and downloadable preprints of my recent papers in physics and chemistry. For the published version see the references given.

I do

For manuscripts with an e-print number, you can also get the latex source (of some version of the paper) from an e-print archive such as http://xxx.uni-augsburg.de

P000.

Currently (January 1, 2012) contains 218 sections with explanations answering questions from theoretical physics, collected from my answers to postings to various physics discussion groups. Most topics are related to quantum mechanics, quantum field theory, renormalization, the measurement problem, randomness, and philosophical issues in physics.

AN000.

pdf file (460K)

We propose a new form for equations of state (EOS) of thermodynamic systems in the Ising universality class. The new EOS guarantees the correct universality and scaling behavior close to critical points and is formulated in terms of the scaling fields only -- unlike the traditional Schofield representation, which uses a parametric form.

Close to a critical point, the new EOS expresses the square of the strong scaling field as an explicit function of the thermal scaling field and the dependent scaling field. A numerical expression is derived, valid close to critical points.

As a consequence of the construction it is shown that the dependent scaling field can be written as an explicit function of the relevant scaling fields without causing strongly singular behavior of the thermodynamic potential in the one-phase region.

Augmented by additional scaling correction fields, the new EOS also describes the state space further away from critical points. It is indicated how to use the new EOS to model multiphase fluid mixtures, in particular for vapor-liquid-liquid equilibrium (VLLE) where the traditional revised scaling approach fails.

AN000.

pdf file (193K)

Realistic equations of state valid in the whole state space of a multi-component mixture should satisfy at least three important constraints:

(i) The Gibbs phase rule holds.

(ii) At low densities, one can deduce a virial equation of state with the correct multicomponent structure.

(iii) Close to critical points, plait points, and consolute points, the correct universality and scaling behavior is guaranteed.

This paper discusses semiempirical equations of state for mixtures that express the pressure as an explicit function of temperature and the chemical potentials. In the first part, expressions are derived for the most important thermodynamic quantities. The main result of the second part is the construction of a large family of equations of state with the properties (i)--(iii).

P000.

phenTherm.pdf (207K)

This paper gives a concise, mathematically rigorous description of phenomenological equilibrium thermodynamics for single-phase systems in the absence of chemical reactions and external forces. From the formulas provided, it is an easy step to go to various examples and applications discussed in standard textbooks (such as those by Callen or Reichl). A full discussion of global equilibrium would also involve the equilibrium treatment of multiple phases and chemical reactions. Since their discussion offers no new aspects compared with traditional textbook treatments, they are not treated here.

The present phenomenological approach is similar to that of Callen, who introduces in his well-known thermodynamics book the basic concepts by means of a few postulates from which everything else follows. The present setting is a modified version designed to match the more fundamental approach based on statistical mechanics. By specifying the kinematical properties of states outside equilibrium, his informal thermodynamic stability arguments (which depend on a dynamical assumption close to equilibrium) can be replaced by rigorous mathematical arguments.

P000.
**
A. Neumaier and D. Westra,
Classical and Quantum Mechanics via Lie algebras**.
Manuscript (2008,2011)

arXiv:0810.1019

pdf file (3165K)

The goal of this book is to present classical mechanics, quantum
mechanics, and statistical mechanics in an almost completely algebraic
setting, thereby introducing mathematicians, physicists, and
engineers to the ideas relating classical and quantum mechanics with
Lie algebras and Lie groups. The book emphasizes the
closeness of classical and quantum mechanics, and the material is
selected in a way to make this closeness as apparent as possible.

Much of the material covered here is not part of standard
textbook treatments of classical or quantum mechanics (or is only
superficially treated there). For physics students who want to
get a broader view of the subject, this book may therefore serve
as a useful complement to standard treatments of quantum mechanics.

Almost without exception, this book is about precise concepts and
exact results in classical mechanics, quantum mechanics, and
statistical mechanics. The structural properties of
mechanics are discussed independent of computational techniques for
obtaining quantitatively correct numbers from the assumptions made.
The standard approximation machinery for calculating from first
principles explicit thermodynamic properties of materials, or
explicit cross sections for high energy experiments can be found in
many textbooks and is not repeated here.

P000.
**
A. Neumaier,
Renormalization without infinities - an elementary tutorial**,
Manuscript (2011).

pdf file (362K)

Renormalization is an indispensable tool for modern theoretical
physics. At the same time, it is one of the least appealing techniques,
especially in cases where naive formulations result in divergences
that must be cured - a step that is often done in a mathematically
dubious way.

In this paper, it is shown how the renormalization procedure works
both in singular cases where it removes naive divergences and in
regular cases where a naive approach is possible but renormalization
improves the quality of perturbation theory. In fact, one can see
immediately that the singular situation is simply a limiting case of the
regular situation.

After discussing generalities, the paper introduces a large family of
toy examples, defined by special perturbations of an arbitrary
Hamiltonian with a discrete spectrum.
The examples show explicitly many of the renormalization effects
arising in realistic quantum field theories such as quantum
chromodynamics: logarithmic divergences, running couplings,
asymptotic freedom, dimensional transmutation, the renormalization
group, and renormalization scheme dependent results at any order of
perturbation theory.

Unlike in more realistic theories, everything is derived rigorously
and nonperturbatively in terms of simple explicit formulas. Thus one
can understand in detail how the infinities arise (if they arise) -
namely as an unphysical infinitely sensitive dependence on the bare
coupling constants. One also sees that all spurious infinities are
cured automatically by the same renormalization process that gives
robust physical results in the case where no infinities occur.

P000.

This lecture (the second of three) discusses work towards a new, classical view of quantum mechanics. It is based on an analysis of polarized light, of the meaning of quantum ensembles in a field theory, of classical simulations of quantum computing algorithms, and resulting optical models for the simulation of quantum mechanics.

In particular, it is shown that classical second-order stochastic optics is precisely the quantum mechanics of a single photon, with all its phenomenological bells and whistles.

P000.

This lecture (the first of three) discusses foundational problems on the nature of light revealed by 1. attempts to define a probability concept for photons, 2. quantum models for photons on demands (and their realization through laser-induced emission by a calcium ion in a cavity), 3. models explaining the photo effect, and 4. Bell-type experiments for single photon nonlocality.

P000.

An experiment is described which proves, using single photons only, that the standard hidden variables assumptions (commonly used to derive Bell inequalities) are inconsistent with quantum mechanics. The analysis is very simple and transparent. In particular, it demonstrates that a classical wave model for quantum mechanics is not ruled out by experiments demonstrating the violation of the traditional hidden variable assumptions.

P000.

On the basis of new, concise foundations, this paper establishes the four laws of thermodynamics, the Maxwell relations, and the stability requirements for response functions, in a form applicable to global (homogeneous), local (hydrodynamic) and microlocal (kinetic) equilibrium.

The present, self-contained treatment needs very little formal machinery and stays very close to the formulas as they are applied by the practicing physicist, chemist, or engineer. From a few basic assumptions, the full structure of phenomenological thermodynamics and of classical and quantum statistical mechanics is recovered.

Care has been taken to keep the foundations free of subjective aspects (which traditionally creep in through information or probability). One might describe the paper as a uniform treatment of the nondynamical part of classical and quantum statistical mechanics ``without statistics'' (i.e., suitable for the definite descriptions of single objects) and ``without mechanics'' (i.e., independent of microscopic assumptions). When enriched by the traditional examples and applications, this paper may serve as the basis for a course on thermal physics.

P000.

The collapse challenge for interpretations of quantum mechanics is to build from first principles and your preferred interpretation a complete, observer-free quantum model of the described experiment (involving a photon and two screens), together with a formal analysis that completely explains the experimental result. The challenge is explained in detail, and discussed in the light of the Copenhagen interpretation and the decoherence setting.

P208.

pdf file (116K)

P117.

Linear optical networks are devices that turn classical incident modes by a linear transformation into outgoing ones. In general, the quantum version of such transformations may mix annihilation and creation operators. We derive a simple formula for the effective Hamiltonian of a general linear quantum network, if such a Hamiltonian exists. Otherwise we show how the scattering matrix of the network is decomposed into a product of three matrices that can be generated by Hamiltonians.

P000.

A mathematically well-defined, manifestly covariant theory of classical and quantum field is given, based on Euclidean Poisson algebras and a generalization of the Ehrenfest equation, which implies the stationary action principle. The theory opens a constructive spectral approach to finding physical states both in relativistic quantum field theories and for flexible phenomenological few-particle approximations.

In particular, we obtain a Lorentz-covariant phenomenological multiparticle quantum dynamics for electromagnetic and gravitational interaction which provides a representation of the Poincaré group without negative energy states. The dynamics reduces in the nonrelativistic limit to the traditional Hamiltonian multiparticle description with standard Newton and Coulomb forces.

The key that allows us to overcome the traditional problems in canonical quantization is the fact that we use the algebra of linear operators on a space of wave functions slightly bigger than traditional Fock spaces.

PN113.

A Gaussian resolution method for the computation of equilibrium density matrices rho(T) for a general multidimensional quantum problem is presented. The variational principle applied to the ``imaginary time'' Schroedinger equation provides the equations of motion for Gaussians in a resolution of rho(T) described by their width matrix, center and scale factor, all treated as dynamical variables.

The method is computationally very inexpensive, has favorable scaling with the system size and is surprisingly accurate in a wide temperature range, even for cases involving quantum tunneling. Incorporation of symmetry constraints, such as reflection or particle statistics, is also discussed.

PS112.

A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization.

Extending the `probability via expectation' approach of Whittle to noncommuting quantities, this paper defines quantities, ensembles, and experiments as mathematical concepts and shows how to model complementarity, uncertainty, probability, nonlocality and dynamics in these terms. The approach carries no connotations of unlimited repeatability; hence it can be applied to unique systems such as the universe.

Consistent experiments provide an elegant solution to the reality problem, confirming the insistence of the orthodox Copenhagen interpretation on that there is nothing but ensembles, while avoiding its elusive reality picture. The weak law of large numbers explains the emergence of classical properties for macroscopic systems.

P000.

The projection formalism for calculating effective Hamiltonians and resonances is generalized to the nonlocal and/or nonhermitian case, so that it is applicable to the reduction of relativistic systems (Bethe-Salpeter equations), and to dissipative systems modeled by an optical potential.

It is also shown how to recover

For practical calculations, it is important that the resulting formulas can be used without computing any projection operators. This leads to a modified coupled reaction channel/resonating group method framework for the calculation of multichannel scattering information.

P103.

We review and further develop the recently introduced numerical approach for scattering calculations based on a so called pseudo-time Schrödinger equation, which is in turn a modification of the damped Chebyshev polynomial expansion scheme.

The method utilizes a special energy-dependent form for the absorbing potential in the time-independent Schrödinger equation, in which the complex energy spectrum E_k is mapped to u_k inside the unit disk, where u_k are the eigenvalues of some explicitly known sparse matrix U.

Most importantly for the numerical implementation, all the physical eigenvalues u_k are extreme eigenvalues of U, which allows one to extract these eigenvalues very efficiently by harmonic inversion of a pseudo-time autocorrelation function using the filter diagonalization method. The computation of 2T steps of the autocorrelation function requires only T sparse real matrix-vector multiplications.

We describe and compare different schemes, effectively corresponding to different choices of the energy-dependent absorbing potential, and test them numerically by calculating resonances of the HCO molecule. Our numerical tests suggest an optimal scheme that provide accurate estimates for most resonance states.

PN97.

The Schrödinger equation with an energy-dependent complex absorbing potential, associated with a scattering system, can be reduced for a special choice of the energy-dependence to a harmonic inversion problem of a discrete pseudo-time correlation function. An efficient formula for Green's function matrix elements is also derived. Since the exact propagation up to time 2t can be done with only t real matrix-vector products, this gives an unprecedently efficient scheme for accurate calculations of quantum spectra for possibly very large systems.

P000.

It is shown that, for a harmonic oscillator in the ground state, Bohmian mechanics and quantum mechanics predict values of opposite sign for certain time correlations.

The discrepancy can be explained by the fact that Bohmian mechanics has no natural way to accomodate the Heisenberg picture, since the local expectation values that define the beables of the theory depend on the Heisenberg time being used to define the operators.

Relations to measurement are discussed, too, and are shown to leave no loophole for claiming that Bohmian mechanics reproduces

CP000.

(in German; English title: Quantum designs - foundations of a non-commutative theory of designs)

Quantum designs are sets of subspaces, or equivalent sets of orthogonal projection matrices, in complex finite dimensional vector spaces with certain properties. These structures are generalizations of classical t-designs (the special case of pairwise commuting matrices), spherical designs, complex t-designs and equi-isoclinic subspaces. All elements of quantum design theory have a natural interpretation in terms of quantum theory.

Apart from general theory (e.g., absolute and special bounds), constructions are given for two classes of quantum designs which generalize the classical balanced incomplete block designs and affine designs. One of them gives rise to the first known class of infinitely many complex 2-designs. Also new tight complex 2-designs are constructed. The constructions have a close analogy to formalisms of quantum theory in infinite-dimensional vector spaces.

P000.

The best mathematical arguments against a realistic interpretation of quantum mechanics - that gives definite but partially unknown values to all observables - are analysed and shown to be based on reasoning that is not compelling.

This opens the door for an interpretation that, while respecting the indeterministic nature of quantum mechanics, allows to speak of definite values for all observables at any time that are, however, only partially measurable.

The analysis also suggests new areas where the foundations of quantum theory need to be tested.

P000.

These comments intend to show that quantum paradoxes are not resolved by the "many-worlds" interpretation or metatheory of quantum mechanics; instead, the latter is full of home-made puzzles and ambiguities.

PS000.

Based on a principal component analysis of 47 published attempts to quantify hydrophobicity in terms of a single scale, we define a representation of the 20 amino acids as points in a 3-dimensional hydrophobicity space and display it by means of a minimal spanning tree. The dominant scale is found to be close to two scales derived from contact potentials.

OPS90.

A smooth empirical potential is constructed for use in off-lattice protein folding studies. Our potential is a function of the amino acid labels and of the distances between the C(alpha) atoms of a protein. The potential is a sum of smooth surface potential terms that model solvent interactions and of pair potentials that are functions of a distance, with a smooth cutoff at 12 Å. Techniques include the use of a fully automatic and reliable estimator for smooth densities, of cluster analysis to group together amino acid pairs with similar distance distributions, and of quadratic programming to find appropriate weights with which the various terms enter the total potential. For nine small test proteins, the new potential has minima within 1.3-4.7Å of the PDB geometry, with one exception that has an error of 8.5Å.

Moreover, a nonuniqueness theorem is given that shows that no set of equilibrium geometries can determine the true effective potential energy function.

NOP85.

This paper discusses the mathematical formulation of and solution attempts for the so-called protein folding problem. The static aspect is concerned with how to predict the folded (native, tertiary) structure of a protein, given its sequence of amino acids. The dynamic aspect asks about the possible pathways to folding and unfolding, including the stability of the folded protein.

From a mathematical point of view, there are several main sides to the static problem:

- the selection of an appropriate potential energy function;

- the parameter identification by fitting to experimental data; and

- the global optimization of the potential.

The dynamic problem entails, in addition, the solution of (because of multiple time scales very stiff) ordinary or stochastic differential equations (molecular dynamics simulation), or (in case of constrained molecular dynamics) of differential-algebraic equations. A theme connecting the static and dynamic aspect is the determination and formation of secondary structure motifs.

The present paper gives a self-contained introduction to the necessary background from physics and chemistry and surveys some of the literature. It also discusses the various mathematical problems arising, some deficiencies of the current models and algorithms, and possible (past and future) attacks to arrive at solutions to the protein folding problem.

P000.

Measurements can be adequately described without reference to ``the collapse of the wave function'' (or to wave functions at all). The collapse, as far as it occurs (i.e., the convergence of the density matrix to one that commutes with the Hamiltonian of the system), is a natural consequence of the reduced description of macroscopic systems in the thermodynamic limit since that leads to a dissipative dynamics. However, in the presence of spin, there is no complete collapse: macroscopic polarization phenomena remain that need 2-state quantum physics, a fact that seems to have escaped notice before. Since polarization is well-understood as a macroscopic phenomenon (no one ever talked about philosophical problems related to macroscopic polarization!), there is no reason to consider the microscopic world as essentially different from the macroscopic world.

P000.

In this paper, an elementary and self-contained axiomatic treatment is given of equilibrium thermodynamics including fluctuations. Among other things, this leads to a natural explanation of the Hilbert space underlying quantum physics, using only a simple quantization condition related to the third law of thermodynamics.

NP73.

The Thiele-Wilson system, a simple model of a linear, triatomic molecule, has been studied extensively in the past. The system exhibits complex molecular dynamics including dissociation, periodic trajectories and bifurcations. In addition, it has for a long time been suspected to be chaotic, but this has never been proved with mathematical rigor.

In this paper, we present numerical results that, using interval methods, rigorously verify the existence of transversal homoclinic points in a Poincarè map of this system. By a theorem of Smale, the existence of transversal homoclinic points in a map rigorously proves its mixing property, i.e., the chaoticity of the system.

NP71.

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Global Optimization

Protein Folding

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my home page (http://www.mat.univie.ac.at/~neum)

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)