Lectures on Quantum Mechanics by Paul A. M. Dirac

Author:Paul A. M. Dirac
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2001-04-19T16:00:00+00:00

DR. DIRAC

Lecture No. 3

QUANTIZATION ON CURVED SURFACES

We started off with a classical action principle. We took our action integral to be Lorentz-invariant. This action gives us a Lagrangian. We then passed from the Lagrangian to the Hamiltonian, and then to the quantum theory by following through certain rules. The result is that, starting with a classical field theory, described by an action principle, we end up with a quantum field theory. Now you might think that that finishes our work, but there is one important problem still to be considered: whether our quantum field theory obtained in this way is a relativistic theory. For the purposes of discussion, we may confine ourselves to special relativity. We have then to consider whether our quantum theory is in agreement with special relativity.

We started from an action principle and we required that our action should be Lorentz-invariant. That is sufficient to ensure that our classical theory shall be relativistic. The equations of motion that follow from a Lorentz invariant action principle must be relativistic equations. It is true that when we put these equations of motion into the Hamiltonian form, we are disturbing the four-dimensional symmetry. We are expressing our equations in the form

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