Modern Physics Letters A

The Yang–Mills gauge field theory, which was proposed 60 years ago, is extremely successful in describing the basic interactions of fundamental particles. The Yang–Mills theory in the course of its developments also stimulated many important field theoretical machinery. In this brief review I discuss the path integral techniques, in particular, the fermionic path integrals which were developed together with the successful applications of quantized Yang–Mills field theory. I start with the Faddeev–Popov path integral formula with emphasis on the treatment of fermionic ghosts as an application of Grassmann numbers. I then discuss the ordinary fermionic path integrals and the general treatment of quantum anomalies. The contents of this review are mostly pedagogical except for a recent analysis of path integral bosonization.

In this paper, we use our recently generalized black hole entropy formula to propose a quantum version of the Friedmann equations. In particular, starting from the differential version of the first law of thermodynamics, we are able to find Planckian (noncommutative) corrections to the Friedmann flat equations. The so modified equations are formally similar to the ones present in Gauss–Bonnet gravity, but in the ordinary 3+1 dimensions. As a consequence of these corrections, by considering negative fluctuations in the internal energy that are allowed by quantum field theory, our equations imply a maximum value both for the energy density $\rho$ and for the Hubble flow H, i.e. the big bang is ruled out. Conversely, by considering positive quantum fluctuations, we found no maximum for $\rho$ and H. Nevertheless, by starting with an early time energy density $\rho \sim 1/t^2$, we obtain a value for the scale factor $a(t)\sim e^{\sqrt{t}}$, implying a finite Planckian universe at t = 0, i.e. the point-like big bang singularity is substituted by a universe of Planckian size at t = 0. Finally, we found possible higher order Planckian terms to our equations together with the related corrections of our generalized Bekenstein–Hawking entropy.

We present exact solutions of solitonic profile mass Schrödinger equation with a modified Pöschl–Teller potential. We find that the solutions can be expressed analytically in terms of confluent Heun functions. However, the energy levels are not analytically obtainable except via numerical calculations. The properties of the wave functions, which depend on the values of potential parameter $\nu$ are illustrated graphically. We find that the potential changes from single well to a double well when parameter $\nu$ changes from minus to positive. Initially, the crest of wave function for the ground state diminishes gradually with increasing $\nu$ and then becomes negative. We notice that the parities of the wave functions for $n>1$ also change.

The recently introduced two- and three-parameter $(p,q)$- and $(p,q,\mu )$-deformed extensions of the Heisenberg algebra (HA) were explored under the condition of their connectedness with the respective nonstandard (other than known ones) deformed quantum oscillator algebras. In this paper, we show that such connection dictates certain new $\eta (N)$-pseudo-Hermitian conjugation rule between the creation and annihilation operators with $\eta (N)$ depending on the particle number operator $N$. In turn, that leads to the related $\eta (N)$-pseudo-hermiticity of the position–momentum operators, though the involved Hamiltonian is Hermitian. Different possible cases are studied, and some interesting features implied by the use of such $\eta (N)$-based conjugation rule are emphasized.

We consider the three conjectures stated in a 2003 paper of Wu, concerning the asymptotics of particular sums of products of binomials, powers and logarithms. These sums relate to the form of the regularised integrals used in loop regularisation. We show all three are true, extend them to more general powers and produce their full asymptotic series. We also extend a classical result to produce an exact formula for the sum in the last.

We show that the elementary modes of the planar harmonic oscillator can be quantized in the framework of quantum mechanics based on pseudo-hermitian Hamiltonians. These quantized modes are demonstrated to act as dynamical structures behind a new Jordan–Schwinger realization of the SU(1, 1) algebra. This analysis complements the conventional Jordan–Schwinger construction of the SU(2) algebra based on hermitian Hamiltonians of a doublet of oscillators.

We consider collision of two particles in rotating spacetimes without horizons. If the metric coefficient responsible for rotation of spacetime is big enough, the energy of collisions in the center of mass frame can be as large as one likes. This can happen in the ergoregion only. The results are model-independent and apply both to relativistic stars and wormholes.