The physical interpretation of the Lanczos tensor

Summary

The field equations of general relativity can be written as first-order differential equations in the Weyl tensor; the Weyl tensor, in turn, can be written as a first-order differential equation in a three-index tensor called the Lanczos tensor. Similarly, in the electromagnetic theory Maxwell’s equations can be written as first-order differential equations in the field tensorF _ab and this in turn can be written as a first-order differential equation in the vector potentialA _a; thus the Lanczos tensor plays a similar role in general relativity to that of the vector potential in the electromagnetic theory. The Aharonov-Bohm effect shows that when quantum mechanics is applied to electromagnetic theory, the vector potential is dynamically significant, and this leads to an attempt to quantize the gravitational field by pursuing the analogy between the vector field and the Lanczos tensor.