arXiv subjects

Otto C. W. Kong

Publications and source records attributed to Otto C. W. Kong.

At least 19 recordsLinked to original sources

arXiv2025

Quantum Mechanics in Curved Space(time) with a Noncommutative Geometric Perspective

We have previously presented a version of the Weak Equivalence Principle for a quantum particle as an exact analog of the classical case, based on the Heisenberg picture analysis of free particle motion. Here, we take that to a full formalism of quantum mechanics in a generic curved space(time). Our basic perspective is to take seriously the noncommutative symplectic geometry corresponding to the quantum observable algebra. Particle position coordinate transformations and a nontrivial metric assigning an invariant inner product to vectors, and covectors, are implemented accordingly. That allows an analog to the classical picture of the phase space as the cotangent bundle. The mass-independent quantum geodesic equations as equations of free particle motion under a generic metric as a quantum observable are obtained from an invariant Hamiltonian. Hermiticity of momentum observables is to be taken as reference frame dependent. Our results have a big contrast to the alternative obtained based on the Schrödinger wavefunction representation which we argue to be less appealing. Hence, the work points to a very different approach to quantum gravity, plausibly with a quantum Einstein equation suggested.

gr-qc↗

arXiv2024

On Locality of Quantum Information in the Heisenberg Picture for Arbitrary States

The locality issue of quantum mechanics is a key issue to a proper understanding of quantum physics and beyond. What has been commonly emphasized as quantum nonlocality has received an inspiring examination through the notion of Heisenberg picture of quantum information. Deutsch and Hayden established a local description of quantum information in a setting of quantum information flow in a system of qubits. With the introduction of a slightly modified version of what we call the Deutsch-Hayden matrix values of observables, together with our recently introduced parallel notion of the noncommutative values from a more fundamental perspective, we clarify all the locality issues based on such values as quantum information carried by local observables in any given arbitrary state of a generic composite system. Quantum information as the {\em `quantum' values} of observables gives a transparent conceptual picture of all the. Spatial locality for a projective measurement is also discussed. The pressing question is if and how such information for an entangled system can be retrieved through local processes which can only be addressed with new experimental thinking.

quant-ph↗

arXiv2024

Noncommutative Number Systems for Quantum Information

Dirac talked about q-numbers versus c-numbers. Quantum observables are q-number variables that generally do not commute among themselves. He was proposing to have a generalized form of numbers as elements of a noncommutative algebra. That was Dirac's appreciation of the mathematical properties of the physical quantities as presented in Heisenberg's new quantum theory. After all, the familiar real, or complex, number system only came into existence through the history of mathematics. Values of physical quantities having a commutative product is an assumption that is not compatible with quantum physics. The revolutionary idea of Heisenberg and Dirac was pulled back to a much more conservative setting by the work of Schrödinger, followed by Born and Bohr. What Bohr missed is that the real number values we obtained from our measurements are only a consequence of the design of the kind of experiments and our using real numbers to calibrate the output scales of our apparatus. It is only our modeling of the information obtained about the physical quantities rather than what Nature dictates. We have proposed an explicit notion of definite noncommutative values of observables that gives a picture of quantum mechanics as realistic as the classical theory. In this article, we illustrate how matrices can be taken as noncommutative (q-)numbers serving as the values of physical quantities, each to be seen as a piece of quantum information. Our main task is to clarify the subtle issues involved in setting up a conventional scheme assigning matrices as values to the physical quantities.

physics.gen-ph↗

arXiv2024

Equivalence Principle for Quantum Mechanics in the Heisenberg Picture

We present an exact quantum observable analog of the weak equivalence principle for a `relativistic' quantum particle. The quantum geodesic equations are obtained from Heisenberg equations of motion as an exact analog of a fully covariant classical Hamiltonian evolution picture, with the proper identification of the canonical momentum variables as $p_μ$, rather than $p^μ$. We discuss the meaning of the equations in relation to projective measurements as well as equations with solution curves as ones in the noncommutative geometric picture of spacetime, and a plausible approach to quantum gravity as a theory about quantum observables as physical quantities including the notion of quantum coordinate transformation.

gr-qc↗

arXiv2023

Quantum Origin of (Newtonian) Mass and Galilean Relativity Symmetry

The Galilei group has been taken as the fundamental symmetry for 'nonrelativistic' physics, quantum or classical. Our fully group theoretical formulation approach to the quantum theory asks for some adjustments. We present a sketch of the full picture here, emphasizing aspects that are different from the more familiar picture. The analysis involves a more careful treatment of the relation between the exact mathematics and its physical application in the dynamical theories, and a more serious full implementation of the mathematical logic than what is usually available in the physics literature. The article summarizes our earlier presented formulation while focusing on the part beyond, with an adjusted, or corrected, identification of the basic representations having the (Newtonian) mass as a Casimir invariant and the notion of center of mass as dictated by the symmetry. Another result is the necessary exclusion of the time translational symmetry, that otherwise bans interactions between particles.

quant-ph↗

arXiv2023

$E=mc^2$ versus Symmetry for Lorentz Covariant Physics

The famous equation $E=mc^2$ is a version of particle mass being essentially the magnitude of the (energy-)momentum four-vector in the setting of `relativistic' dynamics, which can be seen as dictated by the Poincaré symmetry adopted as the relativity symmetry. However, as Einstein himself suggested, the naive notion of momentum as mass times velocity may not be right. The Hamiltonian formulation perspective gives exactly such a setting which in the case of motion of a charged particle under an electromagnetic field actually has the right, canonical, momentum four-vector with an evolving magnitude. The important simple result seems to have missed proper appreciation. In relation to that, we present clear arguments against taking the Poincaré symmetry as the fundamental symmetry behind `relativistic' quantum dynamics, and discuss the proper symmetry theoretical formulation and the necessary picture of the covariant Hamiltonian dynamics with an evolution parameter that is, in general, not a particle proper time. In fact, it is obvious that the action of any position operator of a quantum state violates the on-shell mass condition. The phenomenologically quite successful quantum field theories are `second quantized' versions of `relativistic' quantum mechanics. We present a way for some reconciliation of that with our symmetry picture and discuss implications.

quant-ph↗

arXiv2022

Quantum Origin of (Newtonian) Mass and Symmetry for Lorentz Covariant Physics

The Galilean symmetry and the Poincare symmetry are usually taken as the fundamental (relativity) symmetries for `nonrelativistic' and `relativistic' physics, respectively, quantum or classical. Our fully group theoretical formulation approach to the theories, together with its natural companion of mechanics from symplectic geometry, ask for different perspectives. We present a sketch of the full picture here, emphasizing aspects which are different from the more familiar picture. The letter summarizes our earlier presented formulation while focusing on the part beyond, with an adjusted, or corrected, identification of the basic representations having the (Newtonian) mass as a Casimir invariant. Discussion on the limitations of the Poincare symmetry for the purpose is particularly elaborated.

quant-ph↗

arXiv2022

Mixed State Parametrization and Two-qubit Entanglement

A generic scheme for the parametrization of mixed state systems is introduced, which is then adapted to bipartite systems, especially to a 2-qubit system. Various features of 2-qubit entanglement are analyzed based on the scheme. Our approach exploit much the interplay between is marked by pure states as Hilbert space vectors and mixed states as density matrices, both for the formulation of the parametrization and the analysis of entanglement properties. Explicit entanglement results, in terms of negativity and concurrence, for all 2-qubit mixed states with one single parameter/coordinate among the full set of fifteen being zero and a few other interesting cases are presented.

quant-ph↗

arXiv2022

Towards Noncommutative Quantum Reality

The implications of the physical theory of quantum mechanics on the question of realism is much a subject of sustaining interest, while the background questions among physicists on how to think about all the theoretical notion and `interpretation' of the theory remains controversial. Through a careful analysis of the theoretical notions with the help of modern mathematical perspectives, we give here a picture of quantum mechanics, as the basic theory for `nonrelativistic' particle dynamics, that can be seen as being as much about the physical reality as classical mechanics itself. The key is to fully embrace the noncommutativity of the theory and see it as a notion about the reality of physical quantities. Quantum reality is then just a noncommutative version of the classical reality.

quant-ph↗

arXiv2022

Noncommutative Coordinate Picture of the Quantum Phase Space

We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schrodinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.

quant-ph↗

arXiv2021

Special Relativity and its Newtonian Limit from a Group Theoretical Perspective

In this pedagogical article, we explore a powerful language for describing the notion of spacetime and particle dynamics intrinsic to a given fundamental physical theory, focusing on special relativity and its Newtonian limit. The starting point of the formulation is the representations of the relativity symmetries. Moreover, that seriously furnishes -- via the notion of symmetry contractions -- a natural way in which one can understand how the Newtonian theory arises as an approximation to Einstein's theory. We begin with the Poincaré symmetry underlying special relativity and the nature of Minkowski spacetime as a coset representation space of the algebra and the group. Then, we proceed to the parallel for the phase space of a spin zero particle, in relation to which we present the full scheme for its dynamics under the Hamiltonian formulation, illustrating that as essentially the symmetry feature of the phase space geometry. Lastly, the reduction of all that to the Newtonian theory as an approximation with its space-time, phase space, and dynamics under the appropriate relativity symmetry contraction is presented. While all notions involved are well established, the systematic presentation of that story as one coherent picture fills a gap in the literature on the subject matter.

physics.class-ph↗

arXiv2021

Quantum Frames of Reference and the Noncommutative Values of Observables

Based on a recent relational formulation of quantum reference frame transformations, especially with a case of quantum spatial translations in particular, we analyzed how the `value' of an observable for a fixed state change. That is the exact analog of the classical description, for example, of the value of the $x$-coordinate for a particle decrease by 2 units when we perform a translation of the reference frame putting the new origin at $x=2$. The essence of the quantum reference frame transformations is to have the quantum fluctuations, and even entanglement, of the physical object which serves as the (new) reference frame, taken into account. We illustrate how the recently introduced notion of the noncommutative values of quantum observables gives such a definite description successfully. Formulations, and an analysis of a case example in qubit systems, of analog transformations for observables with a discrete or finite spectrum is also presented. Issues about the evolving picture of the symmetry system of all quantum reference frame transformations discussed.

quant-ph↗

arXiv2021

The First Physics Picture of Contractions from a Fundamental Quantum Relativity Symmetry Including all Known Relativity Symmetries, Classical and Quantum

In this article, we utilize the insights gleaned from our recent formulation of space(-time), as well as dynamical picture of quantum mechanics and its classical approximation, from the relativity symmetry perspective in order to push further into the realm of the proposed fundamental relativity symmetry $SO(2,4)$ of our quantum relativity project. We explicitly trace how the diverse actors in this story change through various contraction limits, paying careful attention to the relevant physical units, in order to place all known relativity theories -- quantum and classical -- within a single framework. More specifically, we explore both of the possible contractions of $SO(2,4)$ and its coset spaces in order to determine how best to recover the lower-level theories. These include both new models and all familiar theories, as well as quantum and classical dynamics with and without Einsteinian special relativity. Along the way, we also find connections with covariant quantum mechanics. The emphasis of this article rests on the ability of this language to not only encompass all known physical theories, but to also provide a path for extensions. It will serve as the basic background for more detailed formulations of the dynamical theories at each level, as well as the exact connections amongst them.

hep-th↗

arXiv2021

The Noncommutative Values of Quantum Observables

We discuss the notion about physical quantities as having values represented by real numbers, and its limiting to describe nature to be understood in relation to our appreciation that the quantum theory is a better theory of natural phenomena than its classical analog. Getting from the algebra of physical observables to their values on a fixed state is, at least for classical physics, really a homomorphic map from the algebra into the real number algebra. The limitation of the latter to represent the values of quantum observables with noncommutating algebraic relation is obvious. We introduce and discuss the idea of the noncommutative values of quantum observables and its feasibility, arguing that at least in terms of the representation of such a value as an infinite set of complex number, the idea makes reasonable sense theoretically as well as practically.

quant-ph↗

arXiv2021

Observables and Dynamics, Quantum to Classical, from a Relativity Symmetry and Noncommutative Geometric Perspective

With approaching quantum/noncommutative models for the deep microscopic spacetime in mind, and inspired by our recent picture of the (projective) Hilbert space as the model of physical space behind basic quantum mechanics, we reformulate here the WWGM formalism starting from the canonical coherent states and taking wavefunctions as expansion coefficients in terms of this basis. This provides us with a transparent and coherent story of simple quantum dynamics where both the wavefunctions for the pure states and operators acting on them arise from the single space/algebra, which exactly includes the WWGM observable algebra. Altogether, putting the emphasis on building our theory out of the underlying relativity symmetry -- the centrally extended Galilean symmetry in the case at hand -- allows one to naturally derive both a kinematical and a dynamical description of a quantum particle, which moreover recovers the corresponding classical picture (understood in terms of the Koopman-von Neumann formalism) in the appropriate (relativity symmetry contraction) limit. Our formulation here is the most natural framework directly connecting all of the relevant mathematical notions and we hope it may help a general physicist better visualize and appreciate the noncommutative-geometric perspective behind quantum physics. It also helps to inspire and illustrate our perspective on looking at quantum mechanics and quantum physics in general in direct connection to the notion of quantum (deformed) relativity symmetries and the corresponding quantum/noncommutative models of spacetime as various levels of approximations all the way down to the Newtonian.

quant-ph↗

arXiv2020

Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and $c \rightarrow \infty $ Limit

We present in the article the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg-Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors under the Lorentz symmetry. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group $C^*$-algebra giving the algebra of observables. The key feature of the formulation is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators of any `dimension'. Galilean limit of the Lorentz symmetry and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.

quant-ph↗

arXiv2020

Analysis on Complete Set of Fock States with Explicit Wavefunctions for the Covariant Harmonic Oscillator Problem

The earlier treatments of Lorentz covariant harmonic oscillator have brought to light various difficulties, such as reconciling Lorentz symmetry with the full Fock space, and divergence issues with their functional representations. We present here a full solution avoiding those problems. The complete set of Fock states is obtained, together with the corresponding explicit wavefunction and their inner product integrals free from any divergence problem and the Lorentz symmetry fully maintained without additional constraints imposed. By a simple choice of the pseudo-unitary representation of the underlying symmetry group, motivated from the perspective of the Minkowski spacetime as a representation for the Lorentz group, we obtain the natural non-unitary Fock space picture commonly considered though not formulated and presented in the careful details given here. From a direct derivation of the appropriate basis state wavefunctions of the finite-dimensional irreducible representations of the Lorentz symmetry, the relation between the latter and the Fock state wavefunctions is also explicitly shown. Moreover, the full picture including the states with non-positive norm may give consistent physics picture as a version of Lorentz covariant quantum mechanics. Probability interpretation for the usual von Neumann measurements is not a problem as all wavefunctions restricted to a definite value for the `time' variable are just like those of the usual time independent quantum mechanics. A further understanding from a perspective of the dynamics from the symplectic geometry of the phase space is shortly discussed.

quant-ph↗

arXiv2020

Covariant Quantum Mechanics and Quantum Spacetime

We present in the article the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg-Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors under the Lorentz symmetry. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group $C^*$-algebra giving the algebra of observables. The key feature of the formulation is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators of any `dimension'. Galilean limit of the Lorentz symmetry and the classical limit of the Lorentz covariant framework are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space, given both in terms of real/complex number coordinates and noncommutative operator coordinates. The latter gives an explicit picture of the (projective) Hilbert space as a quantum/noncommutative spacetime.

physics.gen-ph↗