The inevitable nonlinearity of quantum gravity falsifies the many-worlds interpretation of quantum mechanics

There are fundamental reasons as to why there should exist a reformulation of quantum mechanics which does not refer to a classical spacetime manifold. It follows as a consequence that quantum mechanics as we know it is a limiting case of a more general nonlinear quantum theory, with the nonlinearity becoming significant at the Planck mass/energy scale. This nonlinearity is responsible for a dynamically induced collapse of the wave-function, during a quantum measurement, and it hence falsifies the many-worlds interpretation of quantum mechanics. We illustrate this conclusion using a mathematical model based on a generalized Doebner-Goldin equation. The non-Hermitian part of the Hamiltonian in this normpreserving, nonlinear, Schrödinger equation dominates during a quantum measurement, and leads to a breakdown of linear superposition. This essay received an Honorable Mention in the Gravity Research Foundation Essay Competition, 2007 e-mail address: [email protected] Talk given at the Meeting ‘Himalayan Relativity Dialogue’, Mirik, India, 18-20 April, 2007 1 There are two fundamental unsolved problems in our understanding of quantum mechanics. The first is the famous problem of quantum measurement, for which one of the possible solutions is the mechanism of decoherence, in conjunction with the many-worlds interpretation of quantum mechanics. An alternative explanation of a quantum measurement is a dynamically induced collapse of the wave-function, which requires modification of the Schrödinger equation in the measurement domain. The second unsolved fundamental problem is the need for a reformulation of quantum mechanics, which does not refer to a classical spacetime manifold [1]. In this essay we show that these two unsolved problems have a deep connection, and the resolution of the second problem implies that quantum measurement is explained by dynamically induced collapse of the wave-function. This, in turn, falsifies the many-worlds interpretation of quantum mechanics. The standard formulation of quantum theory depends on an external classical time. The need for a reformulation of quantum mechanics which does not refer to a classical spacetime manifold arises because the geometry (metric and curvature) of the manifold is produced by classical matter fields. One can envisage a Universe in which there are only quantum, and no classical, fields. This will cause the spacetime geometry to undergo quantum fluctuations, which, in accordance with the Einstein hole argument, destroy the underlying classical spacetime manifold. However, one should still be able to describe quantum dynamics; hence the need for the aforementioned reformulation. The new formulation becomes equivalent to standard quantum mechanics as and when an external classical spacetime geometry becomes available. When one tries to construct such a reformulation of quantum mechanics, it follows from very general arguments [1] that quantum gravity is effectively a nonlinear theory. What this means is that the ‘quantum gravitational field’ acts as a source for itself. Such a nonlinearity cannot arise in the standard canonical quantization of general relativity, which is inherently based on linear quantum theory, and which leads to the Wheeler-DeWitt equation. It also follows as a consequence that at the Planck mass/energy scale, quantum theory itself becomes an effectively nonlinear theory [because of self-gravity], and that the Hamiltonian describing a quantum system depends nonlinearly on the quantum state. The standard linear quantum theory is recovered as an approximation at energy scales much smaller than the Planck mass/energy scale. In [1] we have developed a model for the above-mentioned reformulation of quantum mechanics, based on noncommutative differential geometry. One of the outcomes of this model is that the non-relativistic