Dirac’s Hidden Geometry

There’s an interesting article by Graham Farmelo in last week’s Nature, entitled Dirac’s Hidden Geometry. Most people think of Dirac as a brilliant algebraist, but he himself claimed that his motivations and way of thinking were much more geometrical than algebraic. Farmelo’s article contains an amusing account of how Roger Penrose tried to get Dirac to explain how projective geometry had influenced his work in quantum mechanics. Dirac gave a talk about this at Boston University in 1972, but, after giving a presentation about projective geometry, stopped before explaining the relation to quantum mechanics. Penrose, the moderator, asked Dirac about the relation to quantum mechanics, and in answer “Dirac gave his trademark shake of the head, and declined to speak.”

Several historians of science have tried to figure out what Dirac’s geometrical motivations were. This question is dealt with in Olivier Darrigol’s very interesting book (which is now available on-line) From c-numbers to q-numbers: The Classical Analogy in the History of Quantum Theory. The material about Dirac and projective geometry is in chapter XI. On the same topic, there’s also an article by Peter Galison published in 2000 in the journal Representations, entitled The Suppressed Drawing: Paul Dirac’s Hidden Geometry.

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7 Responses to Dirac’s Hidden Geometry

  1. icecube says:

    Wow..that’s really fascinating…never knew about that before.

  2. Unfortunately I don’t have access to the first and third link, but nevermind, the Darrigol’s book seems very interesting, thanks a lot for this.

  3. Alejandro Rivero says:

    Hmm, at least in one paper Dirac was explicit about the geometric character of his non-commutative p, q.

  4. Nigel says:

    ‘Most people think of Dirac as a brilliant algebraist, but he himself claimed that his motivations and way of thinking were much more geometrical than algebraic.’

    What strikes me in reading chapter XI of Darrigol is Dirac’s love of relativity. This must make it strange for many that Dirac followed Einstein into ‘embarrassing’ ideas about a 3-D fabric of spacetime:

    ‘… with the new theory of electrodynamics we are rather forced to have an aether.’ – P.A.M. Dirac, ‘Is There an Aether?,’ Nature, v.168, 1951, p.906. See also Dirac’s paper in Proc. Roy. Soc. v.A209, 1951, p.291.

    ‘Looking back at the development of physics, we see that the ether, soon after its birth, became the enfant terrible of the family of physical substances. First, the construction of a simple mechanical picture of the ether proved to be impossible and was discarded. This caused to a great extent the breakdown of the mechanical point of view. Second, we have to give up the hope that through the presence of the ether sea, one co-ordinate system will be distinguished and lead to the recognition of absolute and not only relative motion. … After such bad experiences, this is the moment to forget the ether completely and to try never to mention its name. We shall say our space has the physical property of transmitting waves and so omit the use of a word we have decided to avoid. The omission of a word from our vocabulary is of course no remedy; the troubles are indeed much too profound to be solved in this way. Let us now write down the facts which have been sufficiently confirmed by experiment without bothering any more about the ‘e—r’ problem.’ – Albert Einstein and Leopold Infeld, Evolution of Physics, 1938, pp. 184-5

    ‘The idealised physical reference object, which is implied in current quantum theory, is a fluid permeating all space like an aether.’ – Sir Arthur Eddington, MA, DSc, LLD, FRS, Relativity Theory of Protons and Electrons, Cambridge University Press, Cambridge, 1936, p. 180.

    ‘It has been supposed that empty space has no physical properties but only geometrical properties. No such empty space without physical properties has ever been observed, and the assumption that it can exist is without justification. It is convenient to ignore the physical properties of space when discussing its geometrical properties, but this ought not to have resulted in the belief in the possibility of the existence of empty space having only geometrical properties… It has specific inductive capacity and magnetic permeability.’ – Professor H.A. Wilson, FRS, Modern Physics, Blackie & Son Ltd, London, 4th ed., 1959, p. 361.

    ‘To deny the ether is ultimately to assume that empty space has no physical qualities whatever… Recapitulating, we may say that according to the general theory of relativity, space is endowed with physical qualities… therefore there exists an ether. According to the general theory of relativity space without ether is unthinkable.’ – Albert Einstein, Leyden University, 1920. (Einstein, A., Sidelights on Relativity, Dover, New York, 1952, pp. 15, 16, and 23.)

    ‘But if, meanwhile, someone explains gravity along with all its laws by the action of some subtle matter, and shows that the motion of planets and comets will not be disturbed by this matter, I shall be far from objecting.’ – Isaac Newton, Letter to Leibniz, 1693.

    ‘The Michelson-Morley experiment has thus failed to detect our motion through the aether, because the effect looked for – the delay of one of the light waves – is exactly compensated by an automatic contraction of the matter forming the apparatus.’ – A.S. Eddington, Space Time and Gravitation, Cambridge, 1921, p. 20.


  5. andy.s says:

    I can’t see those links, but the Geometric Algebra folks have been going on about that for a while. See:


  6. D R Lunsford says:

    Dirac’s book on GR is one of his miraculous works. It’s only 69 pages long and reads like one of his papers. I remember it used to sit next to MTW on my shelf – here was this burlesque routine filled with hot air and dubious leaps of faith, and there was Dirac.

    This is good:


    The papers about deSitter space are fascinating and should be read by anyone interested in extra dimensions.


  7. plato says:


    These pictures were not for pedagogical purposes: Dirac kept them hidden. They were not for popularization—even when speaking to the wider public, Dirac never used the diagrams to explain anything. Astonishing: across the great divide of visualization and formalism that has, for generations, split both physics and mathematics, we read here that Dirac published on one side and worked on the other.