Preferred Citation: Darrigol, Olivier. From c-Numbers to q-Numbers: The Classical Analogy in the History of Quantum Theory. Berkeley:  University of California Press,  1992.

Chapter XII Queer Numbers


In the fall of 1925, Dirac scrutinized Heisenberg's fundamental papers and perceived three essential elements: the new quantum product, the endeavor to maintain as many classical relations as possible, and the direct connection between the quantum amplitudes and the properties of the emitted radiation. Misled by the latter point, Dirac originally interpreted Heisenberg's theory as a modification of the BKS theory and tried to draw

[71] Dirac 1926f, 668.


from it a new conception of virtual oscillators. But he quickly abandoned this line of thought and addressed a more fruitful question: Where did Heisenberg's new quantum rule come from? In his paper, Heisenberg pointed to the possibility of deducing his quantum rule from the high-frequency limit of Kramers's dispersion formula. Consequently, Dirac went back to the Kramers-Heisenberg paper (or to Fowler's account of it) and observed that the dispersion formula was the symbolic translation of a Poisson bracket (Poisson brackets are differential expressions involving two dynamic quantities: they appear when considering infinitesimal transformations in Hamiltonian mechanics, and they enjoy remarkably simple algebraic properties). Together with Heisenberg's remark, this led him to postulate that quantum mechanics was obtained by expressing the fundamental equations of mechanics in terms of Poisson brackets, and by replacing the brackets by purely algebraic expressions, the commutators (divided by ih /2p ).

This conception implied a deep structural analogy between classical and quantum mechanics, from which Dirac drew maximum profit. First of all, his "fundamental equations" were expressed in a very homogeneous form, one involving only algebraic operations (except for time differentiation), whereas the mechanics developed in Göttingen awkwardly mixed algebraic and differential operations (with respect to matrix coordinates!). On the practical side, Dirac imagined a quantum-mechanical analogue of the canonical methods for solving mechanical problems, particularly an analogue of the powerful technique of action-angle variables. This led him, within a semester, to results comparable, and sometimes superior, to those obtained in Göttingen. At the end of 1925 (a little after Pauli) he solved the hydrogen atom, and, soon after, he derived the algebra of angular momenta and made a relativistic calculation of Compton scattering probabilities.

The superiority of Dirac's method lay in his personal appraisal of the classical analogy in the new mechanics. While Göttingen's physicists judged that this analogy had been integrated, once for all, into the foundation of the theory, Dirac believed that it could still be used profitably in the development of the theory. Accordingly, Dirac exaggerated the analogy between classical and quantum mechanics. He initially underplayed the revolutionary character of quantum mechanics and asserted that only the physical interpretation, but not the equations of classical mechanics, was at fault. Heisenberg corrected him: the revolution affected the very concept of motion (kinematics); furthermore, the formal analogy between the two mechanics was not quite as close as Dirac first thought. One could not,


without contradiction, replace all Poisson brackets of the classical theory with commutators, and the energy expression in terms of action variables was not the same in the classical and the quantum cases, contrary to what Dirac suggested in his first paper on quantum mechanics. But to Dirac these were only points of rigor, which did not affect his general view or strategy.

The success of Dirac's adaptation of classical methods depended on another unique aspect of his approach, namely his notion of quantum algebra. In an Eddingtonian manner, Dirac formulated the fundamental equations of quantum mechanics in a purely abstract way, without having formerly interpreted the symbols entering these equations. The symbols, or "q -numbers," were defined only by their mutual relations, which for him constituted a "quantum algebra." The physical interpretation of these symbols occurred in two steps. The introduction of a quantum analogue of action-angle variables first suggested a matrix representation of the symbols; then the matrices were identified with collections of transition amplitudes, as suggested by some formal properties and a touch of "correspondence." This strategy was reminiscent of Whitehead's extensive abstraction, insofar as the relations defining the symbols were abstracted from ordinary mechanics; and it was akin to Eddington's principle of identification insofar as it purported to deduce the interpretation of the symbols from their formal properties. Dirac's symbols, however, in contrast with Whitehead's geometric objects, were not interpreted on the basis of their empirical origin; and contrary to Eddington's symbols of the world, they could not be interpreted without comparing the theory with an already interpreted theory of the same phenomena, Bohr's old quantum theory.

Dirac did some purely mathematical work on the quantum algebra, in the course of which he ventured to subject q -numbers to axioms similar to those found in Baker's Principles and Kelland and Tait's Quaternions . Some of these axioms did not suit quantum mechanics, as quickly noticed by Jordan and Brillouin. But Dirac already knew that the guilty axioms were not necessary to his practical calculations. In general, he wished to maintain a certain flexibility in his notion of q -number: the algebra had to be adapted to the needs of the developing theory. Also he did not require rigorous mathematical definitions of the objects he was manipulating; it was sufficient for him that the symbolic operations performed on these objects would not lead to contradiction.

By the spring of 1926 Dirac's progress had amazed all his colleagues; yet it seemed to have reached a peak. The method of transposing classical methods indeed had a defect: essentially, it could only solve problems that


had a solvable counterpart in the old quantum theory based on classical orbits. To make it worse, through a very ingenious argument Dirac proved that action-angle variables could not exist for quantum-mechanical systems containing two or more indistinguishable particles (this case includes all atoms beyond hydrogen!). A new method had to be found to solve the fundamental equations of quantum mechanics.


Chapter XII Queer Numbers

Preferred Citation: Darrigol, Olivier. From c-Numbers to q-Numbers: The Classical Analogy in the History of Quantum Theory. Berkeley:  University of California Press,  1992.