Rajesh R. Parwani

An approximate expression for the large angle period of a simple pendulum

A heuristic but pedagogical derivation is given of an explicit formula which accurately reproduces the period of a simple pendulum even for large amplitudes. The formula is compared with others in the literature.

An information-theoretic link between spacetime symmetries and quantum linearity

Abstract A nonlinear generalisation of Schrodinger’s equation is obtained using information-theoretic arguments. The nonlinearities are controlled by an intrinsic length scale and involve derivatives to all orders thus making the equation mildly nonlocal. The nonlinear equation is homogeneous, separable, conserves probability, but is not invariant under spacetime symmetries. Spacetime symmetries are recovered when a dimensionless parameter is tuned to vanish, whereby linearity is simultaneously established and the length scale becomes hidden. It is thus suggested that if, in the search for a more basic foundation for Nature’s Laws, an inference principle is given precedence over symmetry requirements, then the symmetries of spacetime and the linearity of quantum theory might both be emergent properties that are intrinsically linked. Supporting arguments are provided for this point of view and some testable phenomenological consequences are highlighted. The generalised Klien–Gordon and Dirac equations are also studied, leading to the suggestion that nonlinear quantum dynamics with intrinsically broken spacetime symmetries might be relevant to understanding the problem of neutrino mass (lessness) and oscillations: among other observations, this approach hints at the existence of a hidden discrete family symmetry in the Standard Model of particle physics.

The Constraints and Spectra of a Deformed Quantum Mechanics

We examine a deformed quantum mechanics in which the commutator between coordinates and momenta is a function of momenta. The Jacobi identity constraint on a two-parameter class of such modified commutation relations (MCR’s) shows that they encode an intrinsic maximum momentum; a sub-class of which also imply a minimum position uncertainty. Maximum momentum causes the bound state spectrum of the one-dimensional harmonic oscillator to terminate at finite energy, whereby classical characteristics are observed for the studied cases. We then use a semi-classical analysis to discuss general concave potentials in one dimension and isotropic power-law potentials in higher dimensions. Among other conclusions, we find that in a subset of the studied MCR’s, the leading order energy shifts of bound states are of opposite sign compared to those obtained using stringtheory motivated MCR’s, and thus these two cases are more easily distinguishable in potential experiments.

Information measures for inferring quantum mechanics

Starting from the Hamilton–Jacobi equation describing a classical ensemble, one may infer a quantum dynamics using the principle of maximum uncertainty. That procedure requires an appropriate measure of uncertainty. Such a measure is constructed here from physically motivated constraints. It leads to a unique single parameter extension of the classical dynamics that is equivalent to the usual linear quantum mechanics.

Pressure of hot g2 phi4 theory at order g5.

The order

Can degenerate bound states occur in one-dimensional quantum mechanics?

g^5

Universality in an information-theoretic motivated nonlinear Schrodinger equation

contribution to the pressure of massless

A Physical Axiomatic Approach to Schrodinger's Equation

g^2 \phi^4

Why is Schrödinger's equation linear?

theory at nonzero temperature is obtained explicitly. Lower order contributions are reconsidered and two issues leading to the optimal choice of rearranged Lagrangian for such calculations are clarified.

Nonlinear Dirac Equations

We point out that bound states, degenerate in energy but differing in parity, may form in one-dimensional quantum systems even if the potential is non-singular in any finite domain. Such potentials are necessarily unbounded from below at infinity and occur in several different contexts, such as in the study of localised states in brane-world scenarios. We describe how to construct large classes of such potentials and give explicit analytic expressions for the degenerate bound states. Some of these bound states occur above the potential maximum while some are below. Various unusual features of the bound states are described and after highlighting those that are ansatz independent, we suggest that it might be possible to observe such parity-paired degenerate bound states in specific mesoscopic systems.