Consistency and linearity in quantum theory

Ariel Caticha

Quantum theory is formulated as the uniquely consistent way to manipulate probability amplitudes. The crucial ingredient is a consistency constraint: if the amplitude of a quantum process can be computed in two different ways, the two answers must agree. The constraint is expressed in the form of functional equations the solution of which leads to the usual sum and product rules for amplitudes. An immediate consequence is that the Schrödinger equation must be linear: non-linear variants of quantum mechanics violate the requirement of consistency. PACS: 03.65.Bz, 03.65.Ca. The formalism of quantum mechanics is very robust; modifications are invariably unsuccessful. A number of non-linear alternative theories have been proposed, [1] experimentally tested, [2] and promptly discarded, but the question of whether new non-linear variants might prove successful is not settled. In fact, interest in the subject remains in spite of, or perhaps, partly because of

D. I. Blokhint︠s︡ev

Ali Shojai, Fatimah Shojai

In this paper we have applied Bohmian quantum theory to the linear field approximation of gravity and present a self-consistent quantum gravity theory in the linear field approximation. The theory is then applied to some specific problems, the Newtonian limit, and the static spherically symmetric solution. Some observable effects of the theory are investigated.

Alfred Landé, E. Richard Cohen

Stefan Scheel, Dirk‐Gunnar Welsch

We present a consistent quantum theory of the electromagnetic field in nonlinearly responding causal media, with special emphasis on Chi(2) media. Starting from QED in linearly responding causal media, we develop a method to construct the cubic Hamiltonian expressed in terms of the complex nonlinear susceptibility in a quantum mechanically consistent way. In particular, we show that the method yields the nonlinear noise polarization, which together with the linear one is responsible for intrinsic quantum decoherence.