arXiv subjects

Jan Naudts

Publications and source records attributed to Jan Naudts.

At least 19 recordsLinked to original sources

arXiv2024

Pairs of Subspaces, Split Quaternions and the Modular Operator

We revisit the work of Rieffel and van Daele on pairs of subspaces of a real Hilbert space, while relaxing as much as possible the assumption that all the relevant subspaces are in general positions with respect to each other. We work out, in detail, how two real projection operators lead to the construction of a complex Hilbert space where the theory of the modular operator is applicable, with emphasis on the relevance of a central extension of the group of split quaternions. Two examples are given for which the subspaces have unequal dimension and therefore are not in generic position.

math-ph↗

arXiv2024

Duality of quantum geometries

Quantum connections are defined by parallel transport operators acting on a Hilbert space. They transport tangent operators along paths in parameter space. The metric tensor of a Riemannian manifold is replaced by an inner product of pairs of operator fields, similar to the inner product of the Kubo-Mori formalism of Linear Response Theory. The metric is used to define the dual of a quantum connection.The gradient of the parallel transport operators is the quantum vector potential. It defines the covariant derivative of operator fields. The covariant derivatives are used to quantify the holonomy of the quantum connection. It is shown that a quantum connection is holonomic if and only if its dual is holonomic. If the parallel transport operators are unitary then an alpha-family of quantum connections can be defined in a way similar to Amari's alpha family of connections in Information Geometry. The minus alpha connection is the dual of the alpha connection. In particular, the alpha equal zero connection is self-dual. An operator field can be combined with a path in parameter space to produce a path in operator space. A definition is given for such a path in operator space to be autoparallel. The path in parameter space is then a geodesic for the induced connection.

math-ph↗

arXiv2022

Exponential arcs in manifolds of quantum states

The manifold under consideration consists of the faithful normal states on a sigma-finite von Neumann algebra in standard form. Tangent planes and approximate tangent planes are discussed. A relative entropy/divergence function is assumed to be given. It is used to generalize the notion of an exponential arc connecting one state to another. The generator of the exponential arc is shown to be unique up to an additive constant. In the case of Araki's relative entropy every selfadjoint element of the von Neumann algebra generates an exponential arc. The generators of composed exponential arcs are shown to add up. The metric derived from Araki's relative entropy is shown to reproduce the Kubo-Mori metric. The latter is the metric used in Linear Response Theory. The e- and m-connections describe a dual pair of geometries. Any finite number of linearly independent generators determines a submanifold of states connected to a given reference state. Such a submanifold is a quantum generalization of a dually flat statistical manifold.

math-ph↗

arXiv2021

Update of prior probabilities by minimal divergence

The present paper investigates the update of an empirical probability distribution with the results of a new set of observations. The optimal update is obtained by minimizing either the Hellinger distance or the quadratic Bregman divergence. The results obtained by the two methods differ. Updates with information about conditional probabilities are considered as well.

math.ST↗

arXiv2021

Exponential arcs in the manifold of vector states on a sigma-finite von Neumann algebra

This paper introduces the notion of exponential arcs in Hilbert space and of exponential arcs connecting vector states on a sigma-finite von Neumann algebra in its standard representation. Results from Tomita-Takesaki theory form an essential ingredient. Starting point is a non-commutative Radon-Nikodym theorem that involves positive operators affiliated with the commutant algebra. It is shown that exponential arcs are differentiable and that parts of an exponential arc are again exponential arcs. Special cases of probability theory and of quantum probability are used to illustrate the approach.

math.OA↗

arXiv2020

Parameter-free description of the manifold of non-degenerate density matrices

The paper gives a definition of exponential arcs in the manifold of non-degenerate density matrices and uses it as a starting point to develop a parameter-free version of non-commutative Information Geometry in the finite-dimensional case. Given the Bogoliubov metric the m- and e-connections are each other dual. Convex potentials are introduced. They allow to introduce dual charts. Affine coordinates are introduced at the end to make the connection with the more usual approach.

math-ph↗

arXiv2020

Gauge transformations of a relativistic field of quantum harmonic oscillators

A set of gauge transformations of a relativistic field of quantum harmonic oscillators is studied in a mathematically rigorous manner. Each wave function in the domain of the number operator of a single oscillator generates a Frechet-differentiable field of wave functions. Starting from a coherent wave function one obtains a two-dimensional differentiable manifold of coherent vector states. As an illustration it is shown that the gauge transformation can be chosen in such a way that the resulting fields describe a freely-propagating wave.

hep-th↗

arXiv2019

Log-affine geodesics in the manifold of vector states on a von Neumann algebra

This paper introduces the notion of a log-affine geodesic connecting two vector states on a von Neumann algebra. The definition is linked to the standard notion of Boltzmann-Gibbs states in statistical physics and the related notion of quantum statistical manifolds. In the abelian case it is linked to the notion of exponential tangent spaces.

math-ph↗

arXiv2019

Quantum statistical manifolds: the linear growth case

A class of vector states on a von Neumann algebra is constructed. These states belong to a deformed exponential family. One specific deformation is considered. It makes the exponential function asymptotically linear. Difficulties arising due to non-commutativity are highlighted.

math.FA↗

arXiv2019

Emergent Coulomb forces in reducible Quantum Electrodynamics

This paper discusses an attempt to develop a mathematically rigorous theory of Quantum Electrodynamics (QED). It deviates from the standard version of QED mainly in two aspects: it is assumed that the Coulomb forces are carried by transversely polarized photons, and a reducible representation of the canonical commutation and anti-commutation relations is used. Both interventions together should suffice to eliminate the mathematical inconsistencies of standard QED.

quant-ph↗

arXiv2018

Non-classical correlations in reducible Quantum Electrodynamics

The question is discussed whether the momentum of a photon has a quantum uncertainty or whether it is a classical quantity. The latter assumption is the main characteristic of reducible Quantum Electrodynamics (rQED). Recent experiments in Quantum Optics may resolve the question. The non-classical correlation of quantum noise in color-entangled beams cannot be explained by rQED without modification of the standing explanation. On the other hand, rQED explains uncertainty of the momentum of a single photon when it is entangled with a quantum spin residing in its environment. The explanation of the historical experiment with equally-polarized pairs of photons, showing violation of the Bell inequalities, invokes the argument of collapse of the wave function, also in rQED.

quant-ph↗

arXiv2018

Quantum Statistical Manifolds

Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose, the emphasis is shifted from a manifold of strictly positive density matrices to a manifold of faithful quantum states on the C*-algebra of bounded linear operators. In addition, ideas from the parameter-free approach to information geometry are adopted. The underlying Hilbert space is assumed to be finite-dimensional. In this way technicalities are avoided so that strong results are obtained, which one can hope to prove later on in a more general context. Two different atlases are introduced, one in which it is straightforward to show that the quantum states form a Banach manifold, the other which is compatible with the inner product of Bogoliubov and which yields affine coordinates for the exponential connection.

math-ph↗

arXiv2017

Reducible Quantum Electrodynamics. I. The Quantum Dimension of the Electromagnetic Field

In absence of currents and charges the quantized electromagnetic field can be described by wave functions which for each individual wave vector are normalized to one. The resulting formalism involves reducible representations of the Canonical Commutation Relations. The corresponding paradigm is a space-time filled with two-dimensional quantum harmonic oscillators. Mathematically, this is equivalent with two additional dimensions penetrated by the electromagnetic waves.

quant-ph↗

arXiv2017

On the Emergence of the Coulomb Forces in Quantum Electrodynamics

A simple transformation of field variables eliminates Coulomb forces from the theory of quantum electrodynamics. This suggests that Coulomb forces may be an emergent phenomenon rather than being fundamental. This possibility is investigated in the context of reducible quantum electrodynamics. It is shown that states exist which bind free photon and free electron fields. The binding energy peaks in the long-wavelength limit. This makes it plausible that Coulomb forces result from the interaction of the electron/positron field with long-wavelength transversely polarized photons.

hep-th↗

arXiv2017

Reducible Quantum Electrodynamics. III. The emergence of the Coulomb forces

The assumption is made that only transversely polarized photons are needed for a correct description of Quantum Electrodynamics. A simple mathematical transformation is used to introduce new field operators which satisfy the full Maxwell equations. In particular, they reproduce Coulomb forces between different regions of the charge field. The analogy with the polaron problem can give some insight in the physics underlying the transformation. In this context it is shown that the interaction of the electron field with a transversely polarized photon field can form bound states. The binding energy peaks for long wavelength photons.

math-ph↗

arXiv2016

Reducible Quantum Electrodynamics. II. The charged states of the vacuum

An explicit construction is given of field operators satisfying the free Dirac equation. The quantum expectation of these field operators forms a spinor which satisfies the original Dirac equation. The current operators are defined as pair correlation functions. Explicit expressions in terms of creation and annihilation operators are obtained. A small example shows that all relevant quantities are mathematically well-defined.

math-ph↗

arXiv2015

Extension of Information Geometry to Non-statistical Systems: Some Examples

Our goal is to extend information geometry to situations where statistical modeling is not obvious. The setting is that of modeling experimental data. Quite often the data are not of a statistical nature. Sometimes also the model is not a statistical manifold. An example of the former is the description of the Bose gas in the grand canonical ensemble. An example of the latter is the modeling of quantum systems with density matrices. Conditional expectations in the quantum context are reviewed. The border problem is discussed: through conditioning the model point shifts to the border of the differentiable manifold.

quant-ph↗

arXiv2015

Large deviation estimates involving deformed exponential functions

We study large deviation properties of probability distributions with either a compact support or a fat tail by comparing them with q-deformed exponential distributions. Our main result is a large deviation property for probability distributions with a fat tail.

math-ph↗