Geometric Structure for Quantum Mechanics

A geometric connection between quantum mechanics and classical mechanics is described and an operator version of the Poisson bracket is developed.     

Classical Mechanics and Quantum Mechanics

The Newton equation of motion is derived from quantum mechanics.     

Zitterbewegung in Quantum Mechanics

The possibility that zitterbewegung opens a window to particle substructure in quantum mechanics is explored by constructing a particle model with structural features inherent in the Dirac equation. This paper develops a self-contained dynamical model of the electron as a lightlike particle with helical zitterbewegung and electromagnetic interactions. The model admits periodic solutions with quantized energy, and the correct magnetic moment is generated by charge circulation. It attributes to the electron an electric dipole moment rotating with ultrahigh frequency, and the possibility of observing this directly as a resonance in electron channeling is analyzed in detail. Correspondence with the Dirac equation is discussed. A modification of the Dirac equation is suggested to incorporate the rotating dipole moment.     

No-Signaling in Quantum Mechanics

We review some concepts and reasonings regarding the notion of no-signaling and its relation to quantum mechanics in bipartite Bell-type scenarios. We recapitulate the no-signaling property of joint conditional probability distributions in geometrical and information theoretic terms. We summarize the reasons why quantum mechanics does not enable instantaneous communication. We make some comments on quantum field theoretic aspects.     

Recurrence in Quantum Mechanics

We first compare the mathematical structure of quantum and classical mechanics when both are formulated in a C^*-algebraic framework. By using finite von Neumann algebras, a quantum mechanical analogue of Liouville's theorem is then proposed. We proceed to study Poincaré recurrence in C^*-algebras by mimicking the measure theoretic setting. The results are interpreted as recurrence in quantum mechanics, similar to Poincaré recurrence in classical mechanics.     

Individuation in Quantum Mechanics

It has been claimed that the Principle of the Identity of Indiscernibles (PII) is incompatible with quantum mechanics, considered as a complete theory. Van Fraassen has argued specifically that a conflict between the two arises due to the requirements of Bose-Einstein statistics when imposed on two-particle quantum states. It is shown here that this apparent contradiction of the PII with quantum mechanics can be removed by the introduction of a natural criterion of individuality.     

Time-Symmetric Quantum Mechanics

A time-symmetric formulation of nonrelativistic quantum mechanics is developed by applying two consecutive boundary conditions onto solutions of a time- symmetrized wave equation. From known probabilities in ordinary quantum mechanics, a time-symmetric parameter P₀ is then derived that properly weights the likelihood of any complete sequence of measurement outcomes on a quantum system. The results appear to match standard quantum mechanics, but do so without requiring a time-asymmetric collapse of the wavefunction upon measurement, thereby realigning quantum mechanics with an important fundamental symmetry.     

Glafka 2004: Generalizing Quantum Mechanics for Quantum Gravity

Familiar quantum mechanics assumes a fixed spacetime geometry. Quantummechanics must therefore be generalized for quantum gravity where spacetime geometry is not fixed but rather a quantum variable. This extended abstract sketches a fully fourdimensional generalized quantum mechnics of cosmological spacetime geometries that is one such generalization.This contribution to the proceedings of the Glafka Conference is an extended abstract of the author's talk there. More details can be found in the references cited at the end of the abstract expecially (Hartle, 1995).     

A Toy Model for Quantum Mechanics

The toy model used by Spekkens (Phys. Rev. A 75, 032110, 2007) to argue in favor of an epistemic view of quantum mechanics is extended by generalizing his definition of pure states (i.e. states of maximal knowledge) and by associating measurements with all pure states. The new toy model does not allow signaling but, in contrast to the Spekkens model, does violate Bell-CHSH inequalities. Negative probabilities are found to arise naturally within the model, and can be used to explain the Bell-CHSH inequality violations.     

A Foundational Principle for Quantum Mechanics

In contrast to the theories of relativity, quantum mechanics is not yet based on a generally accepted conceptual foundation. It is proposed here that the missing principle may be identified through the observation that all knowledge in physics has to be expressed in propositions and that therefore the most elementary system represents the truth value of one proposition, i.e., it carries just one bit of information. Therefore an elementary system can only give a definite result in one specific measurement. The irreducible randomness in other measurements is then a necessary consequence. For composite systems entanglement results if all possible information is exhausted in specifying joint properties of the constituents.     

Generalized Quantum Mechanics for Separated Systems

We give seven necessary physical conditions on a property lattice for to describe two quantum systems when they are separated.     

Simple New Axioms for Quantum Mechanics

The space of pure states of any physical system,classical or quantum, is identified as a Poisson spacewith a transition probability. These two structures areconnected through unitarity. Classical and quantum mechanics are each characterized by asimple axiom on the transition probability p. Unitaritythen determines the Poisson bracket of quantum mechanicsup to a multiplicative constant (identified with Planck's constant).     

Quantum Phase for an Electric Multipole Moment in Noncommutative Quantum Mechanics

We study the noncommutative nonrelativistic quantum dynamics of a neutral particle, which possesses an electric multipole moment, in the presence of an external magnetic field. First, by introducing a shift for the magnetic field we give the Schrödinger equations in the presence of an external magnetic field both on a noncommutative space and a noncommutative phase space, respectively. Then by solving the Schrödinger equations, we obtain quantum phases of the electric multipole moment both on a noncommutative space and a noncommutative phase space. We demonstrate that these phase are geometric and dispersive.     

Quantum Mechanics is About Quantum Information

I argue that quantum mechanics is fundamentally a theory about the representation and manipulation of information, not a theory about the mechanics of nonclassical waves or particles. The notion of quantum information is to be understood as a new physical primitive---just as, following Einsteins special theory of relativity, a field is no longer regarded as the physical manifestation of vibrations in a mechanical medium, but recognized as a new physical entity in its own right.     

Compression Helps Deep Learning in Image Classification

The impact of JPEG compression on deep learning (DL) in image classification is revisited. Given an underlying deep neural network (DNN) pre-trained with pristine ImageNet images, it is demonstrated that, if, for any original image, one can select, among its many JPEG compressed versions including its original version, a suitable version as an input to the underlying DNN, then the classification accuracy of the underlying DNN can be improved significantly while the size in bits of the selected input is, on average, reduced dramatically in comparison with the original image. This is in contrast to the conventional understanding that JPEG compression generally degrades the classification accuracy of DL. Specifically, for each original image, consider its 10 JPEG compressed versions with their quality factor (QF) values from {100,90,80,70,60,50,40,30,20,10}. Under the assumption that the ground truth label of the original image is known at the time of selecting an input, but unknown to the underlying DNN, we present a selector called Highest Rank Selector (HRS). It is shown that HRS is optimal in the sense of achieving the highest Top k accuracy on any set of images for any k among all possible selectors. When the underlying DNN is Inception V3 or ResNet-50 V2, HRS improves, on average, the Top 1 classification accuracy and Top 5 classification accuracy on the whole ImageNet validation dataset by 5.6% and 1.9%, respectively, while reducing the input size in bits dramatically—the compression ratio (CR) between the size of the original images and the size of the selected input images by HRS is 8 for the whole ImageNet validation dataset. When the ground truth label of the original image is unknown at the time of selection, we further propose a new convolutional neural network (CNN) topology which is based on the underlying DNN and takes the original image and its 10 JPEG compressed versions as 11 parallel inputs. It is demonstrated that the proposed new CNN topology, even when partially trained, can consistently improve the Top 1 accuracy of Inception V3 and ResNet-50 V2 by approximately 0.4% and the Top 5 accuracy of Inception V3 and ResNet-50 V2 by 0.32% and 0.2%, respectively. Other selectors without the knowledge of the ground truth label of the original image are also presented. They maintain the Top 1 accuracy, the Top 5 accuracy, or the Top 1 and Top 5 accuracy of the underlying DNN, while achieving CRs of 8.8, 3.3, and 3.1, respectively.     

Non-exponential Decay in Quantum Field Theory and in Quantum Mechanics: The Case of Two (or More) Decay Channels

We study the deviations from the exponential decay law, both in quantum field theory (QFT) and quantum mechanics (QM), for an unstable particle which can decay in (at least) two decay channels. After a review of general properties of non-exponential decay in QFT and QM, we evaluate in both cases the decay probability that the unstable particle decays in a given channel in the time interval between t and t+dt. An important quantity is the ratio of the probability of decay into the first and the second channel: this ratio is constant in the Breit-Wigner limit (in which the decay law is exponential) and equals the quantity Γ ₁/Γ ₂, where Γ ₁ and Γ ₂ are the respective tree-level decay widths. However, in the full treatment (both for QFT and QM) it is an oscillating function around the mean value Γ ₁/Γ ₂ and the deviations from this mean value can be sizable. Technically, we study the decay properties in QFT in the context of a superrenormalizable Lagrangian with scalar particles and in QM in the context of Lee Hamiltonians, which deliver formally analogous expressions to the QFT case.     

Hamiltonian Formalism in Quantum Mechanics

Abstract

Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties are formulated in a series of Conjectures.

To Vladimir Igorevich Arnol’d with admiration, on occasion of his 60^th birthday.