International Journal of Geometric Methods in Modern Physics

In this paper we discuss various geometric aspects related to the Schrödinger and the Pauli equations. First we resume the Madelung–Bohm hydrodynamical approach to quantum mechanics and recall the Hamiltonian structure of the Schrödinger equation. The probability current provides an equivariant moment map for the group $G=\mathrm{\text{sDiff}}(R^3)$ of volume-preserving diffeomorphisms of $R^3$ (rapidly approaching the identity at infinity) and leads to a current algebra of Rasetti–Regge type. The moment map picture is then extended, mutatis mutandis, to the Pauli equation and to generalized Schrödinger equations of the Pauli–Thomas type. A gauge theoretical reinterpretation of all equations is obtained via the introduction of suitable Maurer–Cartan gauge fields and it is then related to Weyl geometric and pilot wave ideas. A general framework accommodating Aharonov–Bohm and Aharonov–Casher effects is presented within the gauge approach. Furthermore, a kind of holomorphic geometric quantization can be performed and yields natural “coherent state” representations of $G$. The relationship with the covariant phase space and density manifold approaches is then outlined. Comments on possible extensions to nonlinear Schrödinger equations, on Fisher-information theoretic aspects and on stochastic mechanics are finally made.

In the framework of the Joos–Weinberg $2(2S+1)$-theory for massless particles, the dynamical invariants have been derived from the Lagrangian density which is considered to be a 4-vector. A la Majorana interpretation of the 6-component “spinors”, the field operators of $S=1$ particles, as the left- and right-circularly polarized radiation, leads us to the conserved quantities which are analogous to those obtained by Lipkin and Sudbery. The scalar Lagrangian of the Joos–Weinberg theory is shown to be equivalent to the Lagrangian of a free massless field, introduced by Hayashi. As a consequence of a new “gauge” invariance this skew-symmetric field describes physical particles with the longitudinal components only. The interaction of the spinor field with the Weinberg’s $2(2S+1)$-component massless field is considered. New interpretation of the Weinberg field function is proposed.

In this paper the dynamical equation of the scale factor of the universe is investigated to describe the effect of bulk viscosity on the early evolution of the universe. We assume the inhomogeneous equation of state of the form $p=(\gamma -1)\rho +\mathrm{\Lambda }(t)$, where the adiabatic parameter $\gamma$ varies with the scale factor $R$ proposed by Carvalho [Unified description of early universe, Int. J. Theor. Phys. 35 (1996) 2019] and $\mathrm{\Lambda }$ is a time-dependent parameter in the framework of the flat FRW model. From this modified equation of state the exact solution of the field equations is obtained by considering the bulk viscosity is a linear combination of two terms of the form: $\zeta ={\zeta }_0+{\zeta }_{1}H$ and cosmological constant $\mathrm{\Lambda }={\mathrm{\Lambda }}_{0}H+{\mathrm{\Lambda }}_1{H}^2$, where ${\mathrm{\Lambda }}_0,{\mathrm{\Lambda }}_1,{\zeta }_0$ and ${\zeta }_1$ are constants, in which an inflationary phase is followed by the radiation dominated phase. For a special choice of the parameter we can explain the dark energy dominant universe and Friedmann equations are solved for two different phases of the universe and obtain the $H$–$z$ relation.

The problem of minimality for constrained variational calculus is analyzed within the class of piecewise differentiable extremaloids. A fully covariant representation of the second variation of the action functional based on a family of local gauge transformations of the original Lagrangian is proposed. The necessity of pursuing a local adaptation process, rather than the global one described in [1] is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both the necessary and the sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the prolongability of the Jacobi fields across the corners. Eventually, in the appendix, an alternative approach to the concept of strength of a corner, more closely related to Pontryagin’s maximum principle, is presented.

In this paper, we consider invariant $(\alpha ,\beta )$-metrics which are induced by invariant Riemannian metrics $ã$ and invariant vector fields $\tilde{X}$ on homogeneous spaces. We study the flag curvatures of invariant $(\alpha ,\beta )$-metrics. We first give an explicit formula for the flag curvature of invariant $(\alpha ,\beta )$-metrics arising from invariant Riemannian metrics on homogeneous spaces and Lie groups. We then give some explicit formula for the flag curvature of invariant Matsumoto metrics, invariant Kropina metrics and invariant Randers metrics.

We introduce the notion of a standard static Finsler spacetime $ℝ\times M$ where the base $M$ is a Finsler manifold. We prove some results which connect causality with the Finslerian geometry of the base extending analogous ones for static and stationary Lorentzian spacetimes.

It is shown that, when there is a magnetic field present, in the framework of classical or quantum mechanics, the active translations differ from the passive ones and that the canonical momentum is not the generator of them. It is also shown that an infinitesimal generator of passive translations or rotations exists only if the magnetic field is invariant under these transformations.

Bianchi-IX four metrics are SU(2) invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux–Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux–Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang–Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux–Halphen system and occurring in the study of Bianchi-IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.

In a slight different way from the previous one, we propose a modified non-Euclidean transformation on the $\frac{\mathrm{\text{SO}}(2N+2)}{U(N+1)}$ Grassmannian which gives the projected $\mathrm{\text{SO}}(2N+1)$ Tamm–Dancoff equation. We derive a classical time-dependent (TD) $\mathrm{\text{SO}}(2N+1)$ Lagrangian which, through the Euler–Lagrange equation of motion for $\frac{\mathrm{\text{SO}}(2N+2)}{U(N+1)}$ coset variables, brings another form of the previous extended-TD Hartree–Bogoliubov (HB) equation. The $\mathrm{\text{SO}}(2N+1)$ random phase approximation (RPA) is derived using Dyson representation for paired and unpaired operators. In the $\mathrm{\text{SO}}(2N)$ HB case, one boson and two boson excited states are realized. We, however, stress non-existence of a higher RPA vacuum. An integrable system is given by a geometrical concept of zero-curvature, i.e. integrability condition of connection on the corresponding Lie group. From the group theoretical viewpoint, we show the existence of a symplectic two-form $\omega$.

The existence of a Killing symmetry in a gauge theory is equivalent to the addition of extra Hamiltonian constraints in its phase space formulation, which imply restrictions both on the Dirac observables (the gauge invariant physical degrees of freedom) and on the gauge freedom. When there is a time-like Killing vector field only pure gauge electromagnetic fields survive in Maxwell theory in Minkowski space-time, while in ADM canonical gravity in asymptotically Minkowskian space-times only inertial effects without gravitational waves survive.

Geometric structures of dynamical systems are investigated based on a differential geometric method (Jacobi stability of KCC-theory). This study focuses on differences of Jacobi stability of two-dimensional second-order differential equation from that of one-dimensional second-order differential equation. One of different properties from a one-dimensional case is the Jacobi unstable condition given by eigenvalues of deviation curvature with different signs. Then, this geometric theory is applied to an overhead crane system as a two-dimensional dynamical system. It is shown a relationship between the Hopf bifurcation of linearized overhead crane and the Jacobi stability. Especially, the Jacobi stable trajectory is found for stable and unstable spirals of the two-dimensional linearized system. In case of the linearized overhead crane system, the Jacobi stable spiral approaches to the equilibrium point faster than the Jacobi unstable spiral. This means that the Jacobi stability is related to the resilience of deviated trajectory in the transient state. Moreover, for the nonlinear overhead crane system, the Jacobi stability for limit cycle changes stable and unstable over time.

In this paper, we explore teleparallel conformal vector fields in non-static plane symmetric space-times in the teleparallel theory of gravitation using the direct integration technique and diagonal tetrads. This study will also cover the static plane symmetric space-times as well. In the teleparallel theory curvature of the non-static plane symmetric space-times is zero and the presence of torsion allows more symmetries. In this study after solving the integrabilty conditions it turns out that the dimension of teleparallel conformal vector fields are 5, 6, 7 or 8.

In this paper, the extremals of curvature energy actions on non-null regular curves in 2-dimensional de Sitter space are studied. We completely solve the Euler–Lagrange equation by quadratures. By using the Killing field, we construct three special coordinate systems and express the generalized elastica in 2-dimensional de Sitter space $S_1^2$ by integral explicitly.

In this paper, we construct a discrete simulation of an expanding homogeneous and isotropic space-time that expands via expansion of its basic elements to figure out properties and characteristics of such a space-time and derive conclusions. We prove that in such an expanding space-time, the geodesics are curved and more precisely, they fluctuate on the boundaries of the expanding basic elements. The non-existence of privileged expansion direction leads to the existence of an infinity of fluctuating geodesics between any two locations in this space-time, that provides a prediction of polarization in geometric optics, and a prediction of an earlier acceleration of the expansion as for the cosmic inflation model. This simulation is a case study and an example of space-time with variable topology using the principle of variation of topology via a transformation that creates holes.

We compute, both explicitly, at least, up to next-to-leading order and in a proof by induction for all loop levels, the critical exponents for thermal Lorentz-violating O($N$) self-interacting scalar field theory. They are evaluated in a massless theory renormalized at arbitrary external momenta, where a reduced number of Feynman diagrams is needed. The results are presented and shown to be identical to that found previously in distinct theories renormalized at different renormalization schemes. Finally, we give both mathematical explanation and physical interpretation for them based on coordinates redefinition techniques and symmetry ideas, respectively.

In this paper, we consider a truncated harmonic oscillator without reflection symmetry and a truncated harmonic oscillator with reflection symmetry. For both algebra, we investigate the representation, correct Hamiltonian and uncertainty relation. Especially, for a truncated harmonic oscillator with reflection symmetry, we discuss thermodynamics of a particle obeying this algebra.

Sobouti proposes an action-based $f(R)$ modification of Einstein’s gravity, which admits a similar Schwarzschild metric. A test star moving in such a space–time acquires a constant asymptotic speed at large distances. As we are concerned with two classical tests of Einstein’s theory which are gravitational red shift of spectral lines and time delay of radar echo passing the sun, we shall calculate them in the $f(R)$-gravity and show that the results are consistent with the experimental observation data.