From Geometry to Algebra in Physics, with Heisenberg
This chapter considers the new relationships between physics and mathematics that emerge with Heisenberg’s discovery of matrix mechanics and its development in the work of Born, Jordan, and Heisenberg himself, and in Dirac’s version of the formalism. Taking as its point of departure Einstein’s view of “the Heisenberg method” as “a purely algebraic method of description of nature,” Section 4.1 examines the shift from geometry to algebra in quantum mechanics as a reversal of the philosophy that governed classical mechanics by grounding it mathematically in the geometrical description of the behavior of physical objects in space and time. Heisenberg’s matrix mechanics abandons any attempts to develop this type of description and instead offers essentially algebraic machinery for predicting the outcomes of experiments observed in measuring instruments. By the same token, a new nonrepresentational type of relationship between mathematics and physics is established, compelling Bohr to speak, in the wake of Heisenberg’s discovery, of “a new era of mutual stimulation of mechanics and mathematics.” Section 4.2 addresses these relationships and their implications.