This investigation examines the one-dimensional Dirac oscillator in the framework of Doubly Special Relativity (DSR), with a comparative analysis of the Magueijo–Smolin (MS) and Amelino-Camelia (AC) formulations up to order <math altimg='si1.svg'><mrow><mi mathvariant='script'>O</mi><mo>(</mo><msup><mi>E</mi><mn>2</mn></msup><mo linebreak='goodbreak'>/</mo><msup><mi>k</mi><mn>2</mn></msup><mo>)</mo></mrow></math>. Exact closed-form energy spectra and non-relativistic (NR) expansions are derived for both particle and antiparticle sectors, revealing distinctive features inherent to each model: a gradual flattening of the energy-level dispersion in the MS realisation and an <italic>n</italic>-dependent critical deformation scale <italic>k<inf>c</inf></italic>( <italic>n</italic>) in the AC realisation that constrains the accessible excitation states. <italic>Crucially, when k is set to the Planck energy, authentic DSR modifications remain undetectable in laboratory spectroscopic measurements: for energies spanning eV to GeV, relative deviations are bounded by</italic> <math altimg='si2.svg'><mrow><mo>&#8818;</mo><msup><mn>10</mn><mrow><mo>&#8722;</mo><mn>38</mn></mrow></msup></mrow></math> <italic>to</italic> <math altimg='si3.svg'><msup><mn>10</mn><mrow><mo>&#8722;</mo><mn>18</mn></mrow></msup></math> <italic>.</italic> In light of this, two viable experimental strategies are proposed to optimise detectability: (i) <italic>analogue simulation</italic> employing a tunable effective deformation parameter <italic>k</italic> <inf>eff</inf> (e.g., via trapped ions or photonic/microwave lattices) to replicate the anticipated <italic>functional dependence</italic> of level-spacing perturbations; and (ii) <italic>Dirac materials</italic> (such as graphene and moir&#233; heterostructures), wherein engineered departures from linear dispersion mimic DSR-inspired kinematics. Across these platforms, the undeformed regime is seamlessly restored as <italic>k</italic> <inf>eff</inf>&#8239;&#8594;&#8239;&#8734;. The analytically exact solutions presented here offer precise directives, including parameter regimes, the significance of <italic>k<inf>c</inf></italic>( <italic>n</italic>), and NR approximations, for empirical inquiries aimed at probing the <italic>form</italic> of DSR modifications, despite the inaccessibility of genuine Planck-scale effects.

Kazakhstan

Azerbaijan

Nosratollah Jafari

Algeria

Abdelmalek Boumali

Dirac oscillator in DSR: A comparative study of Magueijo–Smolin and Amelino-Camelia models

We examine the thermal and statistical properties of the one-dimensional Klein–Gordon oscillator within two prominent Doubly Special Relativity (DSR) frameworks: Amelino-Camelia and Magueijo–Smolin. Using the modified dispersion relations specific to each formulation, we derive the positive energy spectra, construct the partition function via the Euler–Maclaurin method, and compute key thermodynamic quantities, including the specific heat Cv, as functions of temperature and the deformation scale. Planck-scale corrections produce distinct, theoretically resolvable shifts in both the position and magnitude of the Cv peak in the two models. An accompanying entropy analysis reveals that these peaks correspond to smooth Schottky-type anomalies: the specific heat curves remain analytic and positive across the explored temperature range, and thus do not indicate latent or continuous thermodynamic phase transitions. These comparative results provide a robust diagnostic framework for differentiating DSR prescriptions in relativistic quantum systems and reinforce the transition-free character of their thermal response.

Bekdaulet Shukirgaliyev

Fadila Serdouk

Thermal properties of Klein–Gordon oscillator in the context of Amelino-Camelia and Magueijo–Smolin doubly special relativity (DSR) frameworks

We derive the interesting result that the two asymptotically flat Universes classically linked by the Einstein-Rosen bridge may also be quantum mechanically connected in their far out regions. This would be felt by the Newtonian potential far away from a black/white hole system, and raises the possibility of establishing communication via perturbations. We obtain our results by means of wavepackets with a small variance in the mass, solving the equations derived from a maximally symmetry-reduced canonical quantisation method. Mass and a proxy of the Newtonian potential appear as canonical duals, leading to a Heisenberg uncertainty relation between the two. Coherent states are then built, which become non-semiclassical only in two regions: asymptotic spatial infinity (where unitarity forces the packets to “feel” the other asymptotic spatial infinity), and inside the horizon at <math altimg='si1.svg'><mi>r</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mi>G</mi><mi>m</mi></math> where there is ringing. Whilst the latter has been noted in the literature, the former&#8212;the quantum wormhole at spatial infinity&#8212;seems to have eluded past scrutiny. Even under a coherent state there is a free parameter determining the distance beyond which the states becomes non-semiclassical, given the ambiguity in defining quadratures and squeezing, with departures eventually becoming of order one and cutting off the Newtonian potential. Further studies are required to examine the stability of these conclusions beyond their symmetry-reduced test tube.

United Kingdom

João Magueijo

Ganga Singh Manchanda

Quantum wormholes at spatial infinity

In this article, we investigate the phenomenology of the Magueijo-Smolin (MS) model of Doubly Special Relativity by examining the non-relativistic (NR) limit of the Dirac equation and its implications for spin dynamics. A deformed Dirac equation is derived within the MS model framework and is used to explore the continuity equation and the NR limit, leading to a deformed Schrödinger-Pauli equation. The effectiveness of this NR limit is compared to the Foldy–Wouthuysen transformation. Additionally, we obtain the classical limit of the deformed Schrödinger-Pauli equation and evaluate the gyromagnetic factor of the Dirac particle. The spin motion equation and a deformed magnetization current are also derived within the MS model framework. Furthermore, using the mass difference between the tau and anti-tau particles induced by the MS model, we establish a lower bound on the parameter <italic>&#954;</italic>. The MS model's impact is analyzed in both relativistic and NR systems through the continuity equation, and the anomalous Zeeman effect is extensively studied in this context. This study highlights the influence of the MS model on both relativistic and NR dynamics.

Ilyas Haouam

Phenomenology of the Magueijo-Smolin model of doubly special relativity using the non-relativistic limit of the Dirac equation, spin and Zeeman effect

In this paper we entertain a Machian setting where local physics is non-locally affected by the whole Universe, taking the liberty to identify the local (“Newton's bucket”) with our visible Universe, and the whole Universe (Mach's “fixed stars”) with the global Universe beyond our horizon. Crucially, we allow for the two to have different properties, so that we are beyond the traditional FRW setting. For definiteness we focus on theories where non-locality arises from evolution in the laws of physics in terms of spatially global time variables dual to the constants of Nature. Since non-local theories are foliation-dependent, the local (but not the global) Hamiltonian constraint is lost. This is true not only while non-locality is taking place, but also after it ceases: the local Hamiltonian constraint is only recovered up to a constant in time, keeping a memory of the integrated past non-locality. We show that this integration constant is equivalent to preserving the local Hamiltonian constraint and adding an extra fluid with the same cosmological properties as conventional pressureless dark matter. The equivalence breaks down in terms of clustering properties, with the new component attracting other matter, but not budging from its location. This is the ultimate “painted-on” dark matter, attracting but not being attracted, and nailing down a preferred frame.

Mach's principle and dark matter

We relate physical time with the topology of magnetic field vortices. We base ourselves on a formulation of unimodular gravity where the cosmological constant Λ appears as the canonical dual to a variable which on-shell becomes four-volume time. If the theory is restricted to a topological axionic form (viz. a parity-odd product of an electric and a magnetic field), such a time variable becomes the spatial integral of the Chern-Simons density. The latter equates to helicity, so that unimodular time is transmuted into the linking number of the vortices of the topological magnetic field, times their flux. With the added postulate that this flux is a universal constant, the flow of time can thus be interpreted as the progressive weaving of further links between magnetic field vortices, each link providing a quantum of time with value related to the fixed flux. Non-abelian extensions, and targetting parameters other than Λ are briefly examined, exposing different types of vortices and a possible role for inter-linking leading to new phenomenology.

Altay Etkin

Farbod-Sayyed Rassouli

Vortices, topology and time

We examine local physics in the presence of global variables: variables associated with the whole of the spacelike surfaces of a foliation. These could be the (pseudo)constants of nature and their conjugate times, but our statements are more general. Interactions between the local and the global (for example, dependence of the local action on global times dual to constants) degrades full space-time diffeomorphism invariance down to spatial diffeomorphism invariance, and so an extra degree of freedom appears. When these presumably primordial global interactions switch off, the local action recovers full invariance and so the usual two gravitons, but a legacy matter component is left over, bearing the extra degree of freedom. Under the assumption that the preferred foliation is geodesic, this component behaves like dark matter, except that 3 of its 4 local degrees of freedom are frozen, forcing its rest frame to coincide with the preferred foliation. The nonfrozen degree of freedom (the number density of the effective fluid) is the survivor of the extra “graviton” present in the initial theory and keeps memory of all the past global interactions that took place in a given location in the preferred foliation. Such “painted-on” dark matter is best distinguished from the conventional one in situations where the preferred frame would be preposterous if all 4 degrees of freedom of dark matter were available. We provide one example: an outflowing halo of legacy matter with exact escape speed at each point and a very specific profile, surrounding a condensed structure made of normal matter.

Dark matter and spacetime symmetry restoration

We study the Hamiltonian structure of unimodular-like theories, where the cosmological constant (or other supposed constants of nature) are demoted from fixed parameters to classical constants of motion. No new local degrees of freedom are present as a result of a U(1) gauge invariance of the theory. Hamiltonian analysis of the action reveals that the only possible gauge fixing that can be enforced is setting the spatial components of the four-volume time vector  $$\mathcal{T}^{i}\approx 0$$  <math> <mrow> <msup> <mrow> <mi>T</mi> </mrow> <mi>i</mi> </msup> <mo>≈</mo> <mn>0</mn> </mrow> </math>  . As a consequence of this, the gauge-fixed unimodular path integral is equivalent to the minisuperspace unimodular path integral. However, should we break the U(1) gauge invariance, two things happen: a massless propagating degree of freedom appears, and the (gauge-invariant) zero-mode receives modified dynamics. The implications are investigated, with the phenomenology depending crucially on the target “constant”.

Raymond Isichei

Unimodular proca theory: breaking the U(1) gauge symmetry of unimodular gravity via a mass term

In theories where physics depends on a global foliation of space-time, a black hole’s horizon is surrounded by an “eternity skin”: a pile-up of spacelike leaves that in the far-out region cover all times from the start of collapse to future eternity. Any future foliation-dependent change in the laws of physics would be enacted in this region and affect the last stages of collapse toward black hole formation. We show how in some cases the black hole never forms but, rather, bounces into an explosive event. There is also a nonlocal transfer of energy between the asymptotic Universe and the formed black hole precursor, so that the back hole (if formed) or the exploding star (otherwise) will have a different mass from what was initially thrown in. These last matters are generic to nonlocal theories and can be traced to the breakdown of the local Hamiltonian constraint.

Black holes and foliation-dependent physics

We couple the issue of evolution in the laws of physics with that of violations of energy conservation. Avoiding a time dependence in terms of coordinate time, we define evolution as a function of time variables canonically dual to “constants” (such as <math><mrow><mi>Λ</mi></mrow></math>, the Planck mass, or the gravitational coupling), mimicking a procedure associated with one formulation of unimodular gravity. We then introduce variability via a dependence of other fundamental “constants” on these clocks. Although this is not needed, sharper results are obtained if this procedure violates local Lorentz invariance, which we define in the spirit of Horava-Lifshitz theories (modifying a <math><mn>3</mn><mo>+</mo><mn>1</mn></math> split action, so that a Lorentz invariant 4D reassembly is no longer possible). We find that variability in the “laws of physics” generically leads to violations of energy conservation if either a matter parameter varies as a function of a gravitational clock, or a gravity parameter depends on a matter clock, with the other combinations sterile. In general the matter components associated with the varying parameter or the clock absorb or give off the violated energy. We illustrate this with a variety of clocks (ticking unimodular time, Ricci time, etc.) and parameters (mainly the gravitational and matter speed of light, but also <math><mi>Λ</mi></math>). We can accommodate in this construction (and improve on) several early varying speed of light models, allowing for variability effects related to the spatial curvature and <math><mi>Λ</mi></math> to cause creation of radiation and a hot big bang.