The special theory of relativity as applied to the Born–Oppenheimer–Huang approach

In two recent publications (Int. J. Quant. Chem. 114, 1645 (2014) and Mole. Phys. 114, 227 (2016)) it was shown that the Born–Hwang (BH) treatment of a molecular system perturbed by an external field yields a set of decoupled vectorial wave equations, just like in electro-magnetism. This finding led us to declare on the existence of a new type of Fields, which were termed Molecular Fields. The fact that such fields exist implies that at the vicinity of conical intersections exist a mechanism that transforms a passing-by electric beam into a field which differs from the original electric field. This situation is reminiscent of what is encountered in astronomy where Black Holes formed by massive stars may affect the nature of a near-by beam of light. Thus, if the non-adiabatic-coupling-terms (NACT) with their singular points may affect the nature of such a beam (see the above two publications), then it would be interesting to know to what extend NACTs (and consequently also the BH equation) will be affected by the special theory of relativity as introduced by Dirac. Indeed, while applying the Dirac approach we derived the relativistic affected NACTs as well as the corresponding BH equation.     

The phase structure of Einstein–Cartan theory

In the Einstein–Cartan theory of torsion-free gravity coupling to massless fermions, the four-fermion interaction is induced and its strength is a function of the gravitational and gauge couplings, as well as the Immirzi parameter. We study the dynamics of the four-fermion interaction to determine whether effective bilinear terms of massive fermion fields are generated. Calculating one-particle-irreducible two-point functions of fermion fields, we identify three different phases and two critical points for phase transitions characterized by the strength of four-fermion interaction: (1) chiral symmetric phase for massive fermions in strong coupling regime; (2) chiral symmetric broken phase for massive fermions in intermediate coupling regime; (3) chiral symmetric phase for massless fermions in weak coupling regime. We discuss the scaling-invariant region for an effective theory of massive fermions coupled to torsion-free gravity in the low-energy limit.     

The Schrödinger group in molecular quantum mechanics: beyond the Born–Oppenheimer approximation

The Galillei transformation invariant form of molecular quantum mechanics is obtained, which is not restricted by the Born–Oppenheimer approximation. The molecular Hamiltonian takes then the form of a linear combination of operators of the Schrödinger group, which define the new space–time molecular symmetry properties (e.g. helicity).

The puzzling homochirality of the hydrogen bonded biomolecular systems appears then as a simple result of the molecular space-time symmetry.     

Adiabatic Decoupling and Time-Dependent Born–Oppenheimer Theory

We reconsider the time-dependent Born–Oppenheimer theory with the goal to carefully separate between the adiabatic decoupling of a given group of energy bands from their orthogonal subspace and the semiclassics within the energy bands. Band crossings are allowed and our results are local in the sense that they hold up to the first time when a band crossing is encountered. The adiabatic decoupling leads to an effective Schr?dinger equation for the nuclei, including contributions from the Berry connection. Received: 10 July 2000 / Accepted: 30 July 2001     

On the structure of open–closed topological field theory in two dimensions

I discuss the general formalism of two-dimensional topological field theories defined on open–closed oriented Riemann surfaces, starting from an extension of Segal's geometric axioms. Exploiting the topological sewing constraints allows for the identification of the algebraic structure governing such systems. I give a careful treatment of bulk-boundary and boundary-bulk correspondences, which are responsible for the relation between the closed and open sectors. The fact that these correspondences need not be injective nor surjective has interesting implications for the problem of classifying ‘boundary conditions’. In particular, I give a clear geometric derivation of the (topological) boundary state formalism and point out some of its limitations. Finally, I formulate the problem of classifying (on-shell) boundary extensions of a given closed topological field theory in purely algebraic terms and discuss reducibility of boundary extensions.     

Algebraic connectivity analysis in molecular electronic structure theory I: coulomb potential,tensor connectivity, ε-approximation

In this (first) paper we attempt to generalize the notion of tensor connectivity, subsequently studying how this property is affected in different tensorial operations. We show that the often implied corollary of the linked diagram theorem, namely individual size-extensivity of arbitrary connected closed diagrams, can be violated in Coulomb systems. In particular, the assumption of the existence of localized Hartree–Fock orbitals is generally incompatible with the individual size-extensivity of connected closed diagrams when the interaction tensor is generated by the true two-body part of the electronic Hamiltonian. Thus, in general, size-extensivity of a many-body method may originate in specific cancellations of super-extensive quantities, breaking the convenient one-to-one correspondence between the connectivity of arbitrary many-body equations and the size-extensivity of the expectation values evaluated by those equations (for example, when certain diagrams are discarded from the method). Nevertheless, assuming that many-body equations are evaluated for a stable many-particle system, it is possible to introduce a workaround, called the ε-approximation, which restores the individual size-extensivity of an arbitrary connected closed diagram, without qualitatively affecting the asymptotic behavior of the computed expectation values. No assumptions concerning the periodicity of the system and its strict electrical neutrality are made.     

A Time-Dependent Born–Oppenheimer Approximation with Exponentially Small Error Estimates

We present the construction of an exponentially accurate time-dependent Born–Oppenheimer approximation for molecular quantum mechanics. We study molecular systems whose electron masses are held fixed and whose nuclear masses are proportional to ε⁻⁴, where ε is a small expansion parameter. By optimal truncation of an asymptotic expansion, we construct approximate solutions to the time-dependent Schr?dinger equation that agree with exact normalized solutions up to errors whose norms are bounded by , for some C and γ >0. Received: 13 February 2001 / Accepted: 13 July 2001     

Energy–density functional plus quasiparticle–phonon model theory as a powerful tool for nuclear structure and astrophysics

During the last decade, a theoretical method based on the energy–density functional theory and quasiparticle–phonon model, including up to three-phonon configurations was developed. The main advantages of themethod are that it incorporates a self-consistentmean-field and multi-configuration mixing which are found of crucial importance for systematic investigations of nuclear low-energy excitations, pygmy and giant resonances in an unified way. In particular, the theoretical approach has been proven to be very successful in predictions of new modes of excitations, namely pygmy quadrupole resonance which is also lately experimentally observed. Recently, our microscopically obtained dipole strength functions are implemented in predictions of nucleon-capture reaction rates of astrophysical importance. A comparison to available experimental data is discussed.     

A modification of Einstein–Schrödinger theory that contains Einstein–Maxwell–Yang–Mills theory

The Lambda-renormalized Einstein–Schrödinger theory is a modification of the original Einstein–Schrödinger theory in which a cosmological constant term is added to the Lagrangian, and it has been shown to closely approximate Einstein– Maxwell theory. Here we generalize this theory to non-Abelian fields by letting the fields be composed of d × d Hermitian matrices. The resulting theory incorporates the U(1) and SU(d) gauge terms of Einstein–Maxwell–Yang–Mills theory, and is invariant under U(1) and SU(d) gauge transformations. The special case where symmetric fields are multiples of the identity matrix closely approximates Einstein–Maxwell–Yang–Mills theory in that the extra terms in the field equations are < 10^?13 of the usual terms for worst-case fields accessible to measurement. The theory contains a symmetric metric and Hermitian vector potential, and is easily coupled to the additional fields of Weinberg–Salam theory or flipped SU(5) GUT theory. We also consider the case where symmetric fields have small traceless parts, and show how this suggests a possible dark matter candidate.     

Chern–Simons–Rozansky–Witten topological field theory

We construct and study a new topological field theory in three dimensions. It is a hybrid between Chern–Simons and Rozansky–Witten theory and can be regarded as a topologically-twisted version of the N=4$N=4$d=3$d=3$ supersymmetric gauge theory recently discovered by Gaiotto and Witten. The model depends on a gauge group G and a hyper-Kähler manifold X with a tri-holomorphic action of G. In the case when X is an affine space, we show that the model is equivalent to Chern–Simons theory whose gauge group is a supergroup. This explains the role of Lie superalgebras in the construction of Gaiotto and Witten. For general X, our model appears to be new. We describe some of its properties, focusing on the case when G is simple and X is the cotangent bundle of the flag variety of G. In particular, we show that Wilson loops are labeled by objects of a certain category which is a quantum deformation of the equivariant derived category of coherent sheaves on X.     

Time-dependent generalized Kohn–Sham theory

Generalized Kohn–Sham (GKS) theory extends the realm of density functional theory (DFT) by providing a rigorous basis for non-multiplicative potentials, the use of which is outside original Kohn–Sham theory. GKS theory is of increasing importance as it underlies commonly used approximations, notably (conventional or range-separated) hybrid functionals and meta-generalized-gradient-approximation (meta-GGA) functionals. While this approach is often extended in practice to time-dependent DFT (TDDFT), the theoretical foundation for this extension has been lacking, because the Runge–Gross theorem and the van Leeuwen theorem that serve as the basis of TDDFT have not been generalized to non-multiplicative potentials. Here, we provide the necessary generalization. Specifically, we show that with one simple but non-trivial additional caveat – upholding the continuity equation in the GKS electron gas – the Runge–Gross and van Leeuwen theorems apply to time-dependent GKS theory. We also discuss how this is manifested in common GKS-based approximations.     

Picard–Vessiot theory and integrability

We survey some recent applications of the Differential Galois Theory of linear differential equations to the integrability (or solvability) of Dynamical Systems and Spectral Problems.     

Quantum field theory as effective BV theory from Chern–Simons

The general procedure for obtaining explicit expressions for all cohomologies of Berkovits' operator is suggested. It is demonstrated that calculation of BV integral for the classical Chern–Simons-like theory (Witten's OSFT-like theory) reproduces BV version of two-dimensional gauge model at the level of effective action. This model contains gauge field, scalars, fermions and some other fields. We prove that this model is an example of “singular” point from the perspective of the suggested method for cohomology evaluation. For arbitrary “regular” point the same technique results in AKSZ (Alexandrov, Kontsevich, Schwarz, Zaboronsky) version of Chern–Simons theory (BF theory) in accord with [N. Berkovits, Covariant quantization of the superparticle using pure spinors, JHEP 0109 (2001) 016, hep-th/0105050; N. Berkovits, ICTP lectures on covariant quantization of the superstring, hep-th/0209059; M. Movshev, A. Schwarz, On maximally supersymmetric Yang–Mills theories, Nucl. Phys. B 681 (2004) 324, hep-th/0311132; M. Movshev, A. Schwarz, Algebraic structure of Yang–Mills theory, hep-th/0404183].     

Algebraic structure and Poisson＇s theory of mechanico-electrical systems

The algebraic structure and Poisson's integral theory of mechanico-electrical systems are studied. The Hamilton canonical equations and generalized Hamilton canonical equations and their the contravariant algebraic forms for mechanico-electrical systems are obtained. The Lie algebraic structure and the Poisson's integral theory of Lagrange mechanico-electrical systems are derived. The Lie algebraic structure admitted and Poisson's integral theory of the Lagrange--Maxwell mechanico-electrical systems are presented. Two examples are presented to illustrate these results.     

Einstein–Rosen solutions from Kaluza–Klein theory

From a time-dependent boost-rotational symmetric vacuum solution of the Einstein Equations in five dimensions, through the Kaluza–Klein reduction the corresponding Einstein–Maxwell-dilaton solutions are obtained. The four dimensional counterpart turns out to be generalized Einstein–Rosen spacetimes representing unpolarized gravitational waves traveling in an inhomogeneous cosmology. Restricting the parameters we are able to obtain different 4D time-dependent solutions equipped with scalar and electromagnetic fields.