On the equivalence principle in quantum theory

The role of the equivalence principle in the context of non-relativistic quantum mechanics and matter wave interferometry, especially atom beam interferometry, will be discussed. A generalised form of the weak equivalence principle which is capable of covering quantum phenomena too, will be proposed. It is shown that this generalised equivalence principle is valid for matter wave interferometry and for the dynamics of expectation values. In addition, the use of this equivalence principle makes it possible to determine the structure of the interaction of quantum systems with gravitational and inertial fields. It is also shown that the path of the mean value of the position operator in the case of gravitational interaction does fulfill this generalised equivalence principle.     

Topological orbifold models and quantum cohomology rings

We discuss the topological sigma model on an orbifold target space. We describe the moduli space of classical minima for computing correlation functions involving twisted operators, and show, through a detailed computation of an orbifold ofCP ¹ by the dihedral groupD ₄, how to compute the complete ring of observables. Through this procedure, we compute all the rings of dihedralCP ¹ orbifolds. We then considerCP ²/D ₄, and show how the techniques of topologicalanti-topological fusion might be used to compute twist field correlation functions for nonabelian orbifolds.Supported in part by Fannie and John Hertz Foundation     

On the general covariance and strong equivalence principles in quantum general relativity

The various physical aspects of the general relativistic principles of covariance and strong equivalence are discussed, and their mathematical formulations are analyzed. All these aspects are shown to be present in classical general relativity, although no contemporary formulation of canonical or covariant quantum gravity has succeeded to incorporate them all. This has, in part, motivated the recent introduction of a geometro-stochastic framework for quantum general relativity, in which the classical frame bundles that underlie the formulation of parallel transport in classical general relativity are replaced by quantum frame bundles. It is shown that quantum frames can take over the role played by complete sets of observables in conventional quantum theory, so that they can mediate the natural transference of the general covariance and the strong equivalence principles from the classical to the quantum general relativistic regime. This results in a geometrostochastic mode of quantum propagation in general relativistic quantum bundles, which is mathematically implemented by path integration methods based on parallel transport along horizontal lifts of geodesics for the vacuum expectation values of a quantum gravitational field in a quantum spacetime supermanifold. The covariance features of this field are embedded in a quantum gravitational supergroup, which incorporates Poincaré as well as diffeomorphism invariance, and resolves the issue of time in quantum gravity.     

Tomographic quantum cryptography: equivalence of quantum and classical key distillation

The security of a cryptographic key that is generated by communication through a noisy quantum channel relies on the ability to distill a shorter secure key sequence from a longer insecure one. For an important class of protocols, which exploit tomographically complete measurements on entangled pairs of any dimension, we show that the noise threshold for classical advantage distillation is identical with the threshold for quantum entanglement distillation. As a consequence, the two distillation procedures are equivalent: neither offers a security advantage over the other.     

On the equivalence of KMS and Gibbs conditions for states of quantum lattice systems

The equivalence of a Gibbsian equilibrium condition and the KMS condition is proven for one-dimensional quantum lattice systems with a finite range interaction at arbitrary temperature, and for quantum lattice systems of arbitrary dimension, with a finite body interaction, at high temperature.     

The quantum orbifold cohomology of toric stack bundles

We study Givental’s Lagrangian cone for the quantum orbifold cohomology of toric stack bundles. Using Gromov–Witten invariants of the base and combinatorics of the toric stack fibers, we construct an explicit slice of the Lagrangian cone defined by the genus 0 Gromov–Witten theory of a toric stack bundle.     

On the equivalence of theKMS condition and the variational principle for quantum lattice systems

For quantum spin systems on a lattice of an arbitrary dimension, theKMS condition and the variational principle are shown to be equivalent at an arbitrary temperature for translationally invariant states.     

Gromov-Witten classes,quantum cohomology,and enumerative geometry

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov-Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given.     

On the equivalence between magnetic-field-induced phase transitions in the integer quantum Hall effect

Magnetic-field-induced phase transitions in the two-dimensional electron system in a AlGaAs/InGaAs/GaAs heterostructure are studied. Two kinds of magnetic-field-induced phase transitions, plateau-plateau (P-P) and insulator-quantum Hall conductor (I-QH) transitions, are observed in the integer quantum Hall effect regime at high magnetic fields. In the P-P transition, both the semicircle law and the universality of critical conductivities are broken and we do not observe the universal scaling. However, the P-P transition can still be mapped to the I-QH transition by the Landau-level addition transformation, and as the temperature decreases the critical points of these two transitions appear at the same temperature. Our observations indicate that the equivalence between P-P and I-QH transitions can be found by the suitable analysis even when some expected universal properties are invalid.     

The cohomology of the classical and quantum Weyl CCR in curved spaces

An analysis is presented of the cohomological underpinnings for the Weyl group of the canonical commutation relations on manifolds of constant negative curvature. Several uniqueness results are obtained leading from purely classical considerations to the group associated with the systems of imprimitivity of the orthodox approach to quantum mechanics.     

Regularity of gauge equivalence in quantum mechanics and the Aharonov-Bohm effect

A general new proof of the Aharonov-Bohm effect is given. The proof is essentially based on the regularity theorem for elliptic operators and does not depend on the geometry or symmetry of the system. We also establish the inequivalence of the different canonical quantum realizations of classical systems whose configuration spaces are multiply connected.     

Simple proof of equivalence between adiabatic quantum computation and the circuit model

We prove the equivalence between adiabatic quantum computation and quantum computation in the circuit model. An explicit adiabatic computation procedure is given that generates a ground state from which the answer can be extracted. The amount of time needed is evaluated by computing the gap. We show that the procedure is computationally efficient.     

On the equivalence of boundary conditions

We show that ifb andb are two boundary conditions (b.c.) for general spin systems on ^d such that the difference in the energies of a spin configuration in ^d is uniformly bounded, |H _,b ()–H _,b()|C < , then any infinite-volume Gibbs states and obtained with these b.c. have the same measure-zero sets. This implies that the decompositions of and into extremal Gibbs states are equivalent (mutually absolutely continuous). In particular, if is extremal,=. Application of this observation yields in an easy way (among other things) (a) the uniqueness of the Gibbs states for one-dimensional systems with forces that are not too long-range; (b) the fact that various b.c. that are natural candidates for producing non-translation-invariant Gibbs states cannot lead to such an extremal Gibbs state in two dimensions.Supported in part by NSF Grant PHY 78–15920 and by the Swiss National Foundation For Scientific Research.     

On the equivalence of additive and analytic renormalization

The equivalence of additive and analytic renormalization is proved for any choice of finite renormalizations and any fixed generalized evaluator.     

On the equivalence of methods of Darwin and Ewald-Laue

The fundamental system of equations of the dynamical theory of scattering by an ideal crystal is given by Darwin's method for any number of strong waves. The equivalence of this system and the fundamental system of equations of Ewald-Laue is demonstrated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 108–113, April, 1972.     

On the equivalence of adiabatic invariance and the KMS-condition

The paper outlines the reasons why for infinite quantum systems the KMS-states are exactly the ones which are invariant under local adiabatic perturbations.