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 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.

Nonlinear Dirac Equations

“…This current paper is a step towards a quantitative study of the suggestions in [18]. We have discussed a framework for systematically constructing nonlinear Dirac equations, at the quantum mechanical level, that satisfy other conventional properties such as Hermiticity, Poincaré invariance and ψ → λψ invariance although, as shown, even those can be relaxed.…”

“…As mentioned in Section 1, one application of such equations is to study neutrino oscillations [43] which would be an ideal probe of quantum nonlinearities, with of without a simultaneous Lorentz violation [18]. Other examples we hope to study with the nonlinear equations are CP violation and dark matter/energy.…”

“…We will consider nonlinear terms F which simultaneously violate Lorentz invariance [18]. The Lorentz violation will be implemented via constant background fields: in the terminology of [37,38] our equations will preserve the observer Lorentz covariance but break the particle Lorentz symmetry which involves boosting the particles and local fields but not background fields [37,38].…”

“…As there are various obstacles to generalising the non-relativistic information theory approach of [18] to the relativistic domain, we proceed in a different manner here. We write the nonlinear equation as…”

Nonlinear Quantum Mechanics at the Planck Scale

“…See Markopoulou and Smolin 2003, Parwani 2004, Singh 2003, and Svetlichny 2004a, 2004b 6 Nonlinear quantum mechanics and causality As valiant as these efforts might be, one still cannot say that we have an explicit and consistent causal relativistic nonlinear quantum mechanical theory, however, if nonlinearities are of Plank scale should one worry? At Planck energies space-time is thought to be ill defined, the causal structure also ill defined, and so it makes little sense to talk of its violation.…”