arXiv subjects

F. G. Scholtz

Publications and source records attributed to F. G. Scholtz.

At least 19 recordsLinked to original sources

arXiv2025

A formalism of Gravitation based on a Physical Field Strength

We propose a reformulation of gravitation in which the gravitational interaction is treated as a genuine force rather than an inertial effect arising from spacetime geometry. Within this framework, the difference between the affine connection and a flat reference connection defines a tensor $\mathrm{K}^μ_{αβ}$, identified as the gravitational field strength. This object cannot be eliminated by coordinate transformations, demonstrating that gravity possesses true physical degrees of freedom. The formalism introduces vector fields $ξ_a{}^μ$ that extend the notion of infinitesimal translations to curved spacetime and naturally yield a gauge-invariant field strength $\mathfrak{F}^{ξa}{}_{μν}$. The dynamics of the gravitational field are governed by a Lagrangian of Yang--Mills type with an additional scalar degree of freedom $ϕ^{2}$, corresponding to the Newtonian potential. In the limit of vanishing gravitational coupling $\mathfrak{g}\to0$, the theory reduces to General Relativity, while for nonzero $\mathfrak{g}$ it constitutes an $\mathrm{SU(2)\times U(1)}$ gauge theory of gravity. The framework provides a unified description in which dark energy emerges as the self-interaction energy of the $ϕ$ field, and dark-matter-like effects arise from the extended gravitational degrees of freedom. This formulation offers a consistent bridge between classical and quantum descriptions of gravity and clarifies the conceptual foundations of the gravitational interaction.

physics.gen-ph↗

arXiv2020

Emergent entropy of exotic oscillators and squeezing in three-wave mixing process

We demonstrate the existence of entanglement between the spatial degrees of freedom of a system of harmonic oscillators placed in the noncommutative Moyal plane ("exotic oscillators") by computing the entanglement entropy as measured by the von Neumann entropy of the reduced density matrix. It is explicitly verified that the entanglement arises from the noncommutativity, which controls the coupling strength between the spatial modes. This can easily be generalised to the case where the momentum components also satisfy noncommutative relations, so that the entire phase space becomes noncommutative. In the former case, i.e. when only the spatial noncommutativity is present, the underlying mathematical structure is reminiscent of the Unruh effect, as observed by a Rindler observer whose acceleration now gets related to the noncommutative parameter. It is shown that the Landau problem in the presence of a harmonic interaction gives a concrete physical realisation of this effect. Finally, we show that phase-space noncommutativity can give rise to a the non-classical effect of squeezing, which results from the non-linearity of a medium in a three-wave mixing process.

hep-th↗

arXiv2020

The spin-polarized ferromagnetic state of a cold Fermi gas

The spin-polarized ferromagnetic state of a cold Fermi gas is investigated for interacting and non-interacting charge-neutral and $β$-equilibrated gases. The standard minimal couplings between the magnetic field and the fermions' charges and magnetic dipole moments define the fermions' interaction with the magnetic field. Assuming a variable coupling strength between the magnetic field and the fermion (baryon) dipole moments, it is shown that a ferromagnetized state can be achieved that corresponds to a lower energy spin-polarized state with a magnetic field entirely due to the gas's magnetic response. We find that, depending on the density, a very large increase in the baryon dipole moments is needed to achieve this ferromagnetized state. While the required increase seems unlikely, the induced magnetic field is of the order $\sim10^{17}$ gauss. Furthermore, while externally magnetized Fermi gases have an anisotropic pressure, the pressure of the ferromagnetized gas is completely isotropic and the thermodynamically preferred magnetized state.

nucl-th↗

arXiv2020

Ferromagnetism in neutron and charge neutral beta-equilibrated nuclear matter

Ferromagnetism in infinite neutron matter as well as beta equilibrated, charge neutral, dense, and infinite nuclear matter is investigated using a model of interacting baryons and mesons. The standard minimal couplings between the magnetic field and the particle charges as well as the baryon dipole moments are included in the Lagrangian density. Minimizing the energy density with respect to the magnetic field yields a self-consistent expression for the ferromagnetic field. We calculate the phase boundary at a given density by increasing the strength of the baryon dipole moments till the energy density of magnetized matter is lower than that of unmagnetized matter. We find that, depending on the density, it is crossed when the baryon dipole moments are increased by a factor of 35. It is also sensitive to the details of the nuclear matter parameterizations and crossing it induces a magnetic field of $\sim 10^{17}$ gauss.

nucl-th↗

arXiv2019

On the role of the Vacuum Energy in the Thermodynamics of Neutron Matter

The only way neutron matter can couple to the electromagnetic field is through an anomalous coupling, which plays an important role in the thermodynamics of pure neutron matter. Such theories are, however, perturbatively non-renormalisable, which presents a difficulty in terms of the unambiguous treatment of the divergencies. Here we show that despite this, an unambiguous expression can be obtained for the vacuum energy contribution to the grand canonical potential in the case of a constant magnetic field. We find that this contribution is quite small, which justifies the no-sea approximation usually made. We also discuss the density and temperature dependence of the full grand canonical potential.

nucl-th↗

arXiv2019

Fuzzy Classical Dynamics as a Paradigm for Emerging Lorentz Geometries

We show that the classical equations of motion for a particle on three dimensional fuzzy space and on the fuzzy sphere are underpinned by a natural Lorentz geometry. From this geometric perspective, the equations of motion generally correspond to forced geodesic motion, but for an appropriate choice of noncommutative dynamics, the force is purely noncommutative in origin and the underpinning Lorentz geometry some standard space-time with, in general, non-commutatuve corrections to the metric. For these choices of the noncommutative dynamics the commutative limit therefore corresponds to geodesic motion on this standard space-time. We identify these Lorentz geometries to be a Minkowski metric on $\mathbb{R}^4$ and $\mathbb{R} \times S ^2$ in the cases of a free particle on three dimensional fuzzy space ($\mathbb{R}^3_\star$) and the fuzzy sphere ($S^2_\star$), respectively. We also demonstrate the equivalence of the on-shell dynamics of $S^2_\star$ and a relativistic charged particle on the commutative sphere coupled to the background magnetic field of a Dirac monopole.

hep-th↗

arXiv2016

Scattering in three-dimensional fuzzy space

We develop scattering theory in a non-commutative space defined by a $su(2)$ coordinate algebra. By introducing a positive operator valued measure as a replacement for strong position measurements, we are able to derive explicit expressions for the probability current, differential and total cross-sections. We show that at low incident energies the kinematics of these expressions is identical to that of commutative scattering theory. The consequences of spacial non-commutativity are found to be more pronounced at the dynamical level where, even at low incident energies, the phase shifts of the partial waves can deviate strongly from commutative results. This is demonstrated for scattering from a spherical well. The impact of non-commutativity on the well's spectrum and on the properties of its bound and scattering states are considered in detail. It is found that for sufficiently large well-depths the potential effectively becomes repulsive and that the cross-section tends towards that of hard sphere scattering. This can occur even at low incident energies when the particle's wave-length inside the well becomes comparable to the non-commutative length-scale.

hep-th↗

arXiv2016

A manifestly Lorentz covariant, interacting and non-commutative Dirac equation

We propose a manifestly Lorentz covariant, non-commutative Dirac equation for charged particles interacting with an electromagnetic field. The equation is formulated on the operator level, but operators are not composed through the normal operator product, but a modified product that restores the Lorentz covariance. This equation is solved for the free particle and a particle moving in a constant magnetic field. An abstract action, constructed on the operator level, that yields this equation as equation of motion is also derived. To relate this formalism to current formulations of non-commutative quantum field theories, this action is written in a coherent state basis, leading to an action in 4-dimensional Minkowski space-time. The resulting action differs from existing non-commutative actions, but still exhibits non-commutative effects through non-locality.

hep-th↗

arXiv2015

Duality constructions from quantum state manifolds

The formalism of quantum state space geometry on manifolds of generalised coherent states is proposed as a natural setting for the construction of geometric dual descriptions of non-relativistic quantum systems. These state manifolds are equipped with natural Riemannian and symplectic structures derived from the Hilbert space inner product. This approach allows for the systematic construction of geometries which reflect the dynamical symmetries of the quantum system under consideration. We analyse here in detail the two dimensional case and demonstrate how existing results in the AdS_2/CFT_1 context can be understood within this framework. We show how the radial/bulk coordinate emerges as an energy scale associated with a regularisation procedure and find that, under quite general conditions, these state manifolds are asymptotically anti-de Sitter solutions of a class of classical dilaton gravity models. For the model of conformal quantum mechanics proposed by de Alfaro et. al. the corresponding state manifold is seen to be exactly AdS_2 with a scalar curvature determined by the representation of the symmetry algebra. It is also shown that the dilaton field itself is given by the quantum mechanical expectation values of the dynamical symmetry generators and as a result exhibits dynamics equivalent to that of a conformal mechanical system.

hep-th↗

arXiv2015

The thermodynamics of Fermi gases in three dimensional fuzzy space

We use the recently derived density of states for a particle confined to a spherical well in three dimensional fuzzy space to compute the thermodynamics of a gas of non-interacting fermions confined to such a well. Special emphasis is placed on non-commutative effects and in particular non-commutative corrections to the thermodynamics at low densities and temperatures are computed where the non-relativistic approximation used here is valid. Non-commutative effects at high densities are also identified, the most prominent being the existence of a minimal volume at which the gas becomes incompressible. The latter is closely related to a low/high density duality exhibited by these systems, which in turn is a manifestation of an infra-red/ultra violet duality in the single particle spectrum. Both non-rotating and slowly rotating gasses are studied. Approximations are benchmarked against exact numerical computations for the non-rotating case and several other properties of the gas are demonstrated with numerical computations. Finally, a non-commutative gas confined by gravity is studied and several novel features regarding the mass-radius relation, density and entropy are highlighted.

hep-th↗

arXiv2013

Saturated symmetric nuclear matter in strong magnetic fields

Strongly magnetized symmetric nuclear matter is investigated within the context of effective baryon-meson exchange models. The magnetic field is coupled to the charge as well as the dipole moment of the baryons by including the appropriate terms in the Lagrangian density. The saturation density of magnetized, symmetric nuclear matter was calculated for magnetic fields of the order of 10^17 gauss. For the calculated range of saturation densities the binding energy, symmetry energy coefficient and compressibility of nuclear matter were also calculated. It is found that with an increasing magnetic field the saturation density increases, while the system becomes less bound. Furthermore, the depopulation of proton Landau levels leaves a distinct fluctuating imprint on the symmetry energy coefficient and compressibility. The calculations were also performed for increased values of the baryon magnetic dipole moment. By increasing the dipole moment strength the saturation density is found to decrease, but the system becomes more tightly bound while the fluctuations in the symmetry energy coefficient and compressibility persist.

nucl-th↗

arXiv2012

Spectral triplets, statistical mechanics and emergent geometry in non-commutative quantum mechanics

We show that when non-commutative quantum mechanics is formulated on the Hilbert space of Hilbert-Schmidt operators (referred to as quantum Hilbert space) acting on a classical configuration space, spectral triplets as introduced by Connes in the context of non-commutative geometry arise naturally. A distance function as defined by Connes can therefore also be introduced. We proceed to give a simple and general algorithm to compute this function. Using this we compute the distance between pure and mixed states on quantum Hilbert space and demonstrate a tantalizing link between statistics and geometry.

hep-th↗

arXiv2012

Eigenvalue distributions from a star product approach

We use the well-known isomorphism between operator algebras and function spaces equipped with a star product to study the asymptotic properties of certain matrix sequences in which the matrix dimension $D$ tends to infinity. Our approach is based on the $su(2)$ coherent states which allow for a systematic 1/D expansion of the star product. This produces a trace formula for functions of the matrix sequence elements in the large-$D$ limit which includes higher order (finite-$D$) corrections. From this a variety of analytic results pertaining to the asymptotic properties of the density of states, eigenstates and expectation values associated with the matrix sequence follows. It is shown how new and existing results in the settings of collective spin systems and orthogonal polynomial sequences can be readily obtained as special cases. In particular, this approach allows for the calculation of higher order corrections to the zero distributions of a large class of orthogonal polynomials.

math-ph↗

arXiv2010

Bound state energies and phase shifts of a non-commutative well

Non-commutative quantum mechanics can be viewed as a quantum system represented in the space of Hilbert-Schmidt operators acting on non-commutative configuration space. Within this framework an unambiguous definition can be given for the non-commutative well. Using this approach we compute the bound state energies, phase shifts and scattering cross sections of the non- commutative well. As expected the results are very close to the commutative results when the well is large or the non-commutative parameter is small. However, the convergence is not uniform and phase shifts at certain energies exhibit a much stronger then expected dependence on the non-commutative parameter even at small values.

quant-ph↗

arXiv2010

Ferromagnetic phase diagram of neutron matter

The magnetic properties of matter under extreme conditions are of particular importance to understanding the neutron star interior. One contributing factor to the magnetic field of a neutron star could be the ferromagnetic phase of nuclear matter. In this letter we present a self-consistent, relativistic description of ferromagnetism in dense matter, within which the ferromagnetic phase diagram for neutron matter is calculated.

nucl-th↗

arXiv2007

Non-Hermitian Hamiltonians, Metric, Other Observables and Physical Implications

The metric associated with a quasi-Hermitian Hamiltonian and its physical implications are scrutinised. Consequences of the non-uniqueness such as the question of the probability interpretation and the possible and forbidden choices of additional observables are investigated and exemplified by specific illustrative examples. In particular it is argued that the improper identification of observables lies at the origin of the claimed violation of the brachistchrone transition time between orthogonal states. The need for further physical input to remove ambiguities is pointed out.

quant-ph↗

arXiv2007

Spectrum of the non-commutative spherical well

We give precise meaning to piecewise constant potentials in non-commutative quantum mechanics. In particular we discuss the infinite and finite non-commutative spherical well in two dimensions. Using this, bound-states and scattering can be discussed unambiguously. Here we focus on the infinite well and solve for the eigenvalues and eigenfunctions. We find that time reversal symmetry is broken by the non-commutativity. We show that in the commutative and thermodynamic limits the eigenstates and eigenfunctions of the commutative spherical well are recovered and time reversal symmetry is restored.

hep-th↗

arXiv2007

On the role of twisted statistics in the noncommutative degenerate electron gas

We consider the problem of a degenerate electron gas in the background of a uniformly distributed positive charge, ensuring overall neutrality of the system, in the presence of non-commutativity. In contrast to previous calculations that did not include twisted statistics, we find corrections to the ground state energy already at first order in perturbation theory when the twisted statistics is taken into account. These corrections arise since the interaction energy is sensitive to two particle correlations, which are modified for twisted anti-commutation relations.

hep-th↗