On the quantum dynamics of a point particle in conical space

A quantum neutral particle, constrained to move on a conical surface, is used as a toy model to explore bound states due to both a inverse squared distance potential and a δ-function potential, which appear naturally in the model. These pathological potentials are treated with the self-adjoint extension method which yields the correct boundary condition (not necessarily a null wavefunction) at the origin. We show that the usual boundary condition requiring that the wavefunction vanishes at the origin is arbitrary and drastically reduces the number of bound states if used. The situation studied here is closely related to the problem of a dipole moving in conical space.     

Self-adjoint extensions of Coulomb systems in 1, 2 and 3 dimensions

We study the nonrelativistic quantum Coulomb hamiltonian (i.e., inverse of distance potential) in Rⁿ, n=1,2,3. We characterize their self-adjoint extensions and, in the unidimensional case, present a discussion of controversies in the literature, particularly the question of the permeability of the origin. Potentials given by fundamental solutions of Laplace equation are also briefly considered.     

Self-adjoint Operators as Functions I

Observables of a quantum system, described by self-adjoint operators in a von Neumann algebra or affiliated with it in the unbounded case, form a conditionally complete lattice when equipped with the spectral order. Using this order-theoretic structure, we develop a new perspective on quantum observables. In this first paper (of two), we show that self-adjoint operators affiliated with a von Neumann algebra ${\mathcal{N}}$ can equivalently be described as certain real-valued functions on the projection lattice ${\mathcal{P}(\mathcal{N}})$ of the algebra, which we call q-observable functions. Bounded self-adjoint operators correspond to q-observable functions with compact image on non-zero projections. These functions, originally defined in a similar form by de Groote (Observables II: quantum observables, 2005), are most naturally seen as adjoints (in the categorical sense) of spectral families. We show how they relate to the daseinisation mapping from the topos approach to quantum theory (Döring and Isham , New Structures for Physics, Springer, Heidelberg, 2011). Moreover, the q-observable functions form a conditionally complete lattice which is shown to be order-isomorphic to the lattice of self-adjoint operators with respect to the spectral order. In a subsequent paper (Döring and Dewitt, 2012, preprint), we will give an interpretation of q-observable functions in terms of quantum probability theory, and using results from the topos approach to quantum theory, we will provide a joint sample space for all quantum observables.     

Classical and quantum ergodicity on orbifolds

We extend classical results on quantum ergodicity due to Shnirelman, Colin de Verdière, and Zelditch to orbifolds, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, the ergodicity of the Hamiltonian flow of p implies the quantum ergodicity for the operator P. We also prove the ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.     

Entropy increase for a class of dynamical maps

The dynamical evolution of a quantum system is described by a one parameter family of linear transformations of the space of self-adjoint trace class operators (on the Hilbert space of the system) into itself, which map statistical operators to statistical operators. We call such transformations dynamical maps. We give a sufficient condition for a dynamical map A not to decrease the entropy of a statistical operator. In the special case of an N-level system, this condition is also necessary and it is equivalent to the property that A preserves the central state.     

Operateurs conjugués et propriétés de propagation

We use an algebraic criteria (based on local positivity of a commutator) which asserts the existence of a direction of propagation for the flowe ^?iHt associated to a self-adjoint operatorH. This criteria is applied to the Hamiltonian of three body quantum systems interacting through long range two body potentials. We found the singular spectral support of the Green functions or equivalently the phase space support of the propagation in the one channel or the coulomb interaction cases. Elementary applications to asymptotic completeness of general three body systems is given in [11b].     

Electron capture and scaling anomaly in polar molecules

We present a new analysis of the electron capture mechanism in polar molecules, based on von Neumann's theory of self-adjoint extensions. Our analysis suggests that it is theoretically possible for polar molecules to form bound states with electrons, even with dipole moments smaller than the critical value D₀=1.63×¹⁰⁻¹⁸ esu cm. We argue that the quantum mechanical scaling anomaly is responsible for the formation of these bound states.     

Theorem of Levinson via the Spectral Density

We deduce Levinson’s theorem in non-relativistic quantum mechanics in one dimension as a sum rule for the spectral density constructed from asymptotic data. We assume a self-adjoint Hamiltonian which guarantees completeness; the potential needs not to be isotropic and a zero-energy resonance is automatically taken into account. Peculiarities of this one-dimension case are explained because of the “critical” character of the free case u(x) = 0, in the sense that any attractive potential forms at least a bound state. We believe this method is more general and direct than the usual one in which one proves the theorem first for single wave modes and performs analytical continuation.     

Multichannel framework for singular quantum mechanics

A multichannel S-matrix framework for singular quantum mechanics (SQM) subsumes the renormalization and self-adjoint extension methods and resolves its boundary-condition ambiguities. In addition to the standard channel accessible to a distant (“asymptotic”) observer, one supplementary channel opens up at each coordinate singularity, where local outgoing and ingoing singularity waves coexist. The channels are linked by a fully unitary S-matrix, which governs all possible scenarios, including cases with an apparent nonunitary behavior as viewed from asymptotic distances.     

New Criteria for Self-Adjointness and its Application to Dirac–Maxwell Hamiltonian

We present a new theorem concerning a sufficient condition for a symmetric operator acting in a complex Hilbert space to be essentially self-adjoint. By applying the theorem, we prove that the Dirac–Maxwell Hamiltonian, which describes a quantum system of a Dirac particle and a radiation field minimally interacting with each other, is essentially self-adjoint. Our theorem covers the case where the Dirac particle is in the Coulomb-type potential.     

A higher order GUP with minimal length uncertainty and maximal momentum II: Applications

In a recent paper, we presented a nonperturbative higher order Generalized Uncertainty Principle (GUP) that is consistent with various proposals of quantum gravity such as string theory, loop quantum gravity, doubly special relativity, and predicts both a minimal length uncertainty and a maximal observable momentum. In this Letter, we find exact maximally localized states and present a formally self-adjoint and naturally perturbative representation of this modified algebra. Then we extend this GUP to D dimensions that will be shown it is noncommutative and find invariant density of states. We show that the presence of the maximal momentum results in upper bounds on the energy spectrum of the free particle and the particle in box. Moreover, this form of GUP modifies blackbody radiation spectrum at high frequencies and predicts a finite cosmological constant. Although it does not solve the cosmological constant problem, it gives a better estimation with respect to the presence of just the minimal length.     

Discrete quantum square well of the first kind

A new exactly solvable cryptohermitian quantum chain model is proposed and analyzed. Its discrete-square-well-like Hamiltonian with the real spectrum possesses a manifestly non-Hermitian form. It is only made self-adjoint by the constructive transition to an ad hoc Hilbert space. Such a space (i.e., the closed form of its inner product, i.e., the “metric” Θ) varies with an N-plet of optional parameters. The simplicity of our model enables one to obtain the complete family of these physics-determining metrics Θ in a user-friendly band-matrix closed form.     

The Hamiltonian Operator Associated with Some Quantum Stochastic Evolutions

We consider the Hamiltonian operator associated to the quantum stochastic differential equation introduced by Hudson and Parthasarathy to describe a quantum mechanical evolution in the presence of a “quantum noise”. We characterize such a Hamiltonian in the case of arbitrary multiplicity and bounded coefficients: we find an essentially self-adjoint restriction of the operator and, in particular, we provide an explicit construction of a dense set of vectors belonging to its domain. An erratum to this article is available at .     

Integrable minisuperspace models with Liouville field: energy density self-adjointness and semiclassical wave packets

The homogeneous cosmological models with a Liouville scalar field are investigated in classical and quantum contexts of Wheeler–DeWitt geometrodynamics. In the quantum case of quintessence field with potential unbounded from below and phantom field, the energy density operators are not essentially self-adjoint, and self-adjoint extensions contain ambiguities. Therefore the same classical actions correspond to a family of distinct quantum models. For the phantom field the energy spectrum happens to be discrete. The probability conservation and appropriate classical limit can be achieved with a certain restriction of the functional class. The appropriately localized wave packets are studied numerically using the Schrödinger’s norm and a conserved Mostafazadeh’s norm introduced from techniques of pseudo-Hermitian quantum mechanics. These norms give a similar packet evolution that is confronted with analytical classical solutions.     

Classical topology and quantum states

Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac’s ‘quantum mechanics’, then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving observables with smooth time evolution: they contain commutative subalgebras from which the spatial slice of spacetime with its topology (and with further refinements of the axiom, its C ^K - and C ^--structures) can be reconstructed using Gel’fand-Naimark theory and its extensions. Classical topology is an attribute of only certain quantum observables for these axioms, the spatial slice emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed.     

Essential Self-Adjointness of¶Translation-Invariant Quantum Field Models¶for Arbitrary Coupling Constants

The Hamiltonian of a system of quantum particles minimally coupled to a quantum field is considered for arbitrary coupling constants. The Hamiltonian has a translation invariant part. By means of functional integral representations the existence of an invariant domain under the action of the heat semigroup generated by a self-adjoint extension of the translation invariant part is shown. With a non-perturbative approach it is proved that the Hamiltonian is essentially self-adjoint on a domain. A typical example is the Pauli–Fierz model with spin 1/2 in nonrelativistic quantum electrodynamics for arbitrary coupling constants. Received: 26 May 1999 / Accepted: 9 November 1999     

Translation-invariant quantizations and algebraic structures on phase space

A class of quantizations, including that of Weyl, called translation-invariant is defined and the phase space formulations of quantum mechanics arising from such quantizations are described. It is then shown that these formulations are co-extensive with non-commutative translation-invariant involutive associative algebraic structures in the linear space of complex polynomials on phase space, in which polynomials with real coefficients are self-adjoint.     

Dynamics on quantum graphs as constrained systems

We study the possibility of regarding the dynamics on a quantum graph as limit, as a small parameter ∈ → O, of a dynamics with a strong confining potential. We define a projection operator along the first eigenfunction of a transversal operator and, under suitable assumptions, we prove that the projection of the solution strongly converges along subsequences to a function that satisfies the Schrödinger equation on each open edge of the graph. Moreover the limit dynamics is unitary. If the limit is independent of the subsequence, one has a limit one-parameter group, generated by one of the self-adjoint extensions of a symmetric operator defined on the open graph (with the vertices deleted). The crucial role of the shape of the confining potential at the vertices is pointed out.     

On Realization of Partially Ordered Abelian Groups

The paper is devoted to algebraic structures connected with the logic of quantum mechanics. Since every (generalized) effect algebra with an order determining set of (generalized) states can be represented by means of an abelian partially ordered group and events in quantum mechanics can be described by positive operators in a suitable Hilbert space, we are focused in a representation of partially ordered abelian groups by means of sets of suitable linear operators. We show that there is a set of points separating ?-maps on a given partially ordered abelian group G if and only if there is an injective non-trivial homomorphism of G to the symmetric operators on a dense set in a complex Hilbert space $\mathcal{H}$ which is equivalent to an existence of an injective non-trivial homomorphism of G into a certain power of ?. A similar characterization is derived for an order determining set of ?-maps and symmetric operators on a dense set in a complex Hilbert space $\mathcal{H}$ . We also characterize effect algebras with an order determining set of states as interval operator effect algebras in groups of self-adjoint bounded linear operators.     

New Dynamics in the Anti-de Sitter Universe AdS 5

This paper deals with the propagation of the gravitational waves in the Poincaré patch of the 5-dimensional Anti-de Sitter universe. We construct a large family of unitary dynamics with respect to some high order energies that are conserved and positive. These dynamics are associated with asymptotic conditions on the conformal time-like boundary of the universe. This result does not contradict the statement of Breitenlohner-Freedman that the hamiltonian is essentially self-adjoint in L ² and thus accordingly the dynamics is uniquely determined. The key point is the introduction of a new Hilbert functional framework that contains the massless graviton which is not normalizable in L ². Then the hamiltonian is not essentially self-adjoint in this new space and possesses a lot of different positive self-adjoint extensions. These dynamics satisfy a holographic principle: there exists a renormalized boundary value which completely characterizes the whole field in the bulk.