In 1925 Dirac published a paper building on Heisenberg’s sketchy outline of a mathematical structure for quantum theory. He followed this up in 1926 with a paper demonstrating how the Schrödinger and Heisenberg approaches were di erent ways of looking at the same basic mathematical picture. In essence, Dirac showed how basis states could be set up, how a state can be expanded over this basis, and how one basis can be converted into another equally valid description using a di erent basis.
In 1925 Dirac published a paper building on Heisenberg’s sketchy outline of a mathematical structure for quantum theory. He followed this up in 1926 with a paper demonstrating how the Schrödinger and Heisenberg approaches were di erent ways of looking at the same basic mathematical picture. In essence, Dirac showed how basis states could be set up, how a state can be expanded over this basis, and how one basis can be converted into another equally valid description using a di erent basis.
Between 1925 and 1933, Paul Adrian Maurice Dirac made a series of crucial contributions to the development of quantum mechanics and set much of the agenda for fundamental research through the following decades. Any one of his insights would have been sufficient to ensure his lasting recognition in physics, but together they amount to the work of a genius. In 1925 Dirac published a paper building on Heisenberg’s outline of a mathematical structure for quantum theory. Dirac’s next two papers came out in 1927 and developed the theory to describe the interaction between electromagnetic waves and atoms. This work set the foundations of quantum field theory and developed the technique called ‘second quantization’. Between 1932 and 1933, Dirac wrote three more papers on quantum field theory, managing to cast it in a fashion that made its consistency with relativity apparent.
The relationship between physical notions and the structures expressing them is an issue of primary importance the history and philosophy of pre-modern natural philosophy and of modern science. Historical evidence suggests that, when investigating the relationship between physics and mathematics, one should pay attention to the material, sensually perceivable forms mediating it: spoken or written words, drawings, artefacts, descriptions of experimental set-ups or, as the case discussed the present paper, symbolic notation. The reason for this is that, historically, physical notions and their expressions have very often been developed as one, and only distinguished from each other at a later stage. This was for example the case with what we today regard as mathematical and classical mechanics, which were developed the course of the eighteenth century as the discipline of the analysis of motion. When considered from a later point of view, both the physical notions and the concepts involved historical developments appear as being in flux, vague, and, as far as the side is concerned, non-rigorous. In such cases, verbal description, mechanical artefacts or symbols do not simply mediate between physical notions and structures, as might be the case at a later stage, but represent both of them one, allowing the historical actors to construct and manipulate physical-mathematical meaning by following, at times, their physical reasoning and, at times, their criteria of rigour. The history of quantum mechanics offers many examples of how new physics and new mathematics can emerge as one, with symbolic notation playing a central role shaping both. Among the many creators of quantum theory, Paul Dirac was particularly sensitive to questions of notation:
In an attempt to give an exact mathematical formulation of Bohr*s Correspondence Principle, Heisenberg (June 1925) discovered the rules governing the behaviour of quantum-theoretical magnitudes.In fall 1925 Born, Heisenberg and Jordan and, independently, Dirac, formulated consistent algebraic schemes of quantum mechanics.Early in 1926 Schrodinger developed wave mechanics.In quick succession were discovered : Bora's probability interpretation of the wave function, the transformation theory of Dirac, Jordan and E. London, Heisenberg's uncertainty relations and Bohr's principle of complementarity.By September 1927 the basis of a complete theory of atomic phenomena had been established.Aspects of this development, in which Heisenberg played a central role, will be presented as a tribute to his memory.
This essay offers an epistemological reinterpretation of the foundational divide between matrix mechanics and wave mechanics. Though formally equivalent, the two theories embody distinct modes of knowing: procedural construction and recognitional verification. These epistemic architectures anticipate, in philosophical form, the logical asymmetry expressed by the P versus NP problem in computational complexity. Here, the contrast between efficient generation and efficient recognition is treated not as a mathematical taxonomy but as a framework for understanding how knowledge is produced and validated across physics, computation, and cognition. The essay reconstructs the mathematical history of quantum mechanics through the original derivations of Werner Heisenberg, Max Born, Pascual Jordan, Paul Dirac, Erwin Schrödinger, Paul Ehrenfest, and Wolfgang Pauli, culminating in John von Neumann's unification of both approaches within the formalism of Hilbert space. By juxtaposing Heisenberg's algorithmic formalism with Schrödinger's representational one, it argues that their divergence reveals a structural feature of scientific reasoning itself-the enduring tension between what can be procedurally constructed and what can only be recognized.
This chapter presents the origins of the transformation theory and related concepts. It shows how, in 1925, a procedure initiated by Werner Heisenberg was developed by himself, Max Born, Pascual Jordan, and a little later by Paul Dirac, into a new system of quantum theory—the first complete system of quantum theory which physics has possessed. A little later Erwin Schrödinger developed the “wave mechanics” from an entirely different starting point. This accomplished the same ends, and soon proved to be equivalent to the Heisenberg, Born, Jordan, and Dirac system. On the basis of the Born statistical interpretation of the quantum theoretical description of nature, it was possible for Dirac and Jordan to join the two theories into one, the “transformation theory,” in which they make possible a grasp of physical problems which is especially simple mathematically.