Four-vectors in relativity

“Four-vectors in relativity” gives a “soft” introduction to four-vectors by first setting up corresponding properties of three-vectors. These include the triangle rule for vector addition, and rotation of axes by a matrix multiplication. The physics of a three-dimensional system is unchanged by a rotation of the axes within which it is observed. Likewise the physics of a relativistic system is unchanged (“invariant”) under application of a Lorentz transformation.

Thermodynamic Quantities in the High Temperature Limit in the GUP Framework

We study the values of thermodynamic quantities of physical systems at high temperatures in the framework of the generalized uncertainty principle. First, we obtain the semi-classical partition function, internal energy and heat capacity for physical systems in a general case of deformed algebra. We show that at high temperatures limit the heat capacity vanishes and the internal energy is maximized. We calculate this temperature for the ultra-relativistic system in approximate form. Finally, the maximum temperature and maximum energy for the ultra-relativistic system are obtained in the framework of two different GUPs.

Trajectory-state theory of the Klein-Gordon field

We develop a trajectory construction of solutions to the massless wave equation in n + 1 dimensions and hence show that the quantum state of a massive relativistic system in 3 + 1 dimensions may be represented by a stand-alone four-dimensional congruence comprising a continuum of 3-trajectories coupled to an internal scalar time coordinate. A real Klein-Gordon amplitude is the current density generated by the temporal gradient of the internal time. Complex amplitudes are generated by a two-phase flow. The Lorentz covariance of the trajectory model is established.

Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincaré symmetry

As earlier conjectured by several authors and much later established by Solèr (relying on partial results by Piron, Maeda–Maeda and other authors), from the lattice theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. Stückelberg provided some physical, but not mathematically rigorous, reasons for ruling out the real Hilbert space formulation, assuming that any formulation should encompass a statement of Heisenberg principle. Focusing on this issue from another — in our opinion, deeper — viewpoint, we argue that there is a general fundamental reason why elementary quantum systems are not described in real Hilbert spaces. It is their basic symmetry group. In the first part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a real Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the real one, all selfadjoint operators represent observables in accordance with Solèr’s thesis, and the standard quantum version of Noether theorem may be formulated. In the second part of this work, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. This complex structure reveals a nice interplay of Poincaré symmetry and the classification of the commutant of irreducible real von Neumann algebras.

Debye length and plasma skin depth: two length scales of interest in the creation and diagnosis of laboratory pair plasmas

In traditional electron/ion laboratory plasmas, the system size $L$ is much larger than both the plasma skin depth $l_{s}$ and the Debye length $\unicode[STIX]{x1D706}_{D}$. In current and planned efforts to create electron/positron plasmas in the laboratory, this is not necessarily the case. A low-temperature, low-density system may have $\unicode[STIX]{x1D706}_{D}<L<l_{s}$; a high-density, thermally relativistic system may have $l_{s}<L<\unicode[STIX]{x1D706}_{D}$. Here we consider the question of what plasma physics phenomena are accessible (and/or diagnostically exploitable) in these different regimes and how this depends on magnetization. While particularly relevant to ongoing pair plasma creation experiments, the transition from single-particle behaviour to collective, ‘plasma’ effects – and how the criterion for that threshold is different for different phenomena – is an important but often neglected topic in electron/ion systems as well.

On the spin excitation energy of the nucleon in the Skyrme model

In the Skyrme model of nucleons and nuclei, the spin excitation energy of the nucleon is traditionally calculated by a fit of the rigid rotor quantization of spin/isospin of the fundamental Skyrmion (the hedgehog) to the masses of the nucleon and the Delta resonance. The resulting, quite large spin excitation energy of the nucleon of about [Formula: see text] is, however, rather difficult to reconcile with the small binding energies of physical nuclei, among other problems. Here, we argue that a more reliable interval of values for the spin excitation energy of the nucleon, compatible with many physical constraints is between [Formula: see text] and [Formula: see text]. The fit of the rigid rotor to the Delta, on the other hand, is problematic in any case, because it implies the use of a nonrelativistic method for a highly relativistic system.