Volume 5: Pages 97-114, 1992

How Nonlinearity Determines the Laws of Nature

R. Mirman ¹

¹155 East 34 Street, New York, New York 10016 U.S.A.

Objects interact; they obey nonlinear equations. This is sufficient to give the dimension of space and the need for quantum mechanics and many (all?) of its properties. Quantum mechanics is analyzed by viewing it as being, necessarily, a representation theory of perhaps the fundamental law of physics; the transformations (which need not be symmetries) of the objects in space are given by the Poincaré group. While ordinary quantum mechanics is linear, for good reasons, interactions, which are nonlinearities, restrict the results of linear quantum mechanics. Gravitation is nonlinear, so a quantum theory of gravitation leads to consequences that differ substantially from those of other, more usual, theories. And besides being nonlinear, it also belongs to a massless representation of the Poincaré group — these are quite unlike massive ones, having solvable little groups. These representations, describing electromagnetism, gravitation, and their characteristics arising from masslessness and nonlinearities, are considered, these underlying the next step, a quantum theory of gravitation.

Keywords: dimension of space, reason for quantum mechanics, properties of quantum mechanics, Poincaré mass‐zero representations, gravitation, reason for gauge transformations, solvable little groups

Received: October 4, 1990; Published Online: December 15, 2008