Dirac’s debt to Hamilton

Question

According to Tobias Hurter’s popular exposition Too Big for a Single Mind (narrated in the present tense):

Dirac makes use of an elegant mathematical tool developed by the Irish mathematician William Hamilton in the nineteenth century.

This is from a passage discussing Dirac in Cambridge in the summer of 1925; see pages 127-128.

What “elegant mathematical tool” is Hurter talking about? Was it quaternions? Unfortunately the notes at the back of the book do not say anything more about this episode.

Schrödinger already used Hamiltonian dynamics before Dirac, see Did wave optics anticipate quantum mechanics?, as did Heisenberg in 1926. Heisenberg's way of representing quantum mechanical energy operators as "quantizations" of classical Hamiltonians is now called canonical quantization. Dirac's innovation was to extend the idea to quantum fields in 1927, what is now called second quantization. — Conifold, Nov 22 '23 at 00:43
@Conifold think it is referring to Clifford algebras and quaternions and not to Hamiltonians. — Mauricio, Nov 22 '23 at 15:54
I don’t know if Hurter is correct, but I stress that he is writing about Dirac in 1925, not 1927. — James Propp, Nov 22 '23 at 22:15
Judging by "stumbles upon differential operator that, like matrices, satisfies Heisenberg's multiplication rule" and "no more matrices" in the surrounding passage, Hurter is talking about Dirac's paper The fundamental equations of quantum mechanics submitted in 1925. There he draws parallels between algebras of quantum commutators and Poisson brackets and writes matrix-free Hamilton's equations in the anti-symmetric form for both cases. — Conifold, Nov 24 '23 at 06:31
@Mauricio You probably mean gamma matrices, but those are from 1928. — Conifold, Nov 24 '23 at 06:33

score 3 · Answer 1 · answered Nov 23 '23 at 17:03

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I'd wager that the ambiguous reference is to the appearance of Hamiltonian quaternions (as an instance of a Clifford algebra) in Dirac's construction of a square root (of a certain sort) of a/the Laplacian, which is nowadays called a/the "Dirac operator". (Not to be confused with Dirac's $\delta$.) It is a linear differential operator with coefficients in a Clifford algebra related to a quadratic form attached to the geometry of the situation.

This is quite different from square roots of positive unbounded self-adjoint operators on Hilbert spaces... For example, the Dirac operator still has a robust distributional sense, while the unbounded-op-on-Hilbert-space version is not distributional.

The Hamiltonian quaternions arise as a Clifford algebra attached to a low-dimensional quadratic form.

answered Nov 23 '23 at 17:03

paul garrett

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With the OP specification of the dating to 1925, the Dirac operator is unlikely, it is from 1928. The surrounding passage also talks about a paper sent to Göttingen in 1925, which I think was a draft of this one, where he draws analogies between commutators and Poisson brackets in Hamiltonian dynamics. – Conifold Nov 24 '23 at 06:38
@Conifold, ah, yes, with the dates, you're surely right. But then, as in your earlier comment, it is mildly curious that Schroedinger didn't/doesn't get more credit... – paul garrett Nov 24 '23 at 17:34
@paulgarrett : "This is quite different from square roots of positive unbounded self-adjoint operators on Hilbert spaces... For example, the Dirac operator still has a robust distributional sense, while the unbounded-op-on-Hilbert-space version is not distributional." Could you please explain what you mean or give a reference? This is not a critique, just trying to understand. – akhmeteli Dec 17 '23 at 06:45
@akhmeteli, the functional calculus for unbounded but positive operators such as (a/the self-adjoint extension of...) $-\Delta$ (... restricted to test functions) gives a (positive) square root, yes, but that square root is no longer a differential operator. In contrast, Dirac operators are differential operators, although with coefficients in non-commuting scalars (Hamiltonian quaternions, or, in general, Clifford algebras). – paul garrett Dec 17 '23 at 17:16
@paulgarrett : Thank you very much. – akhmeteli Dec 17 '23 at 18:32

score 1 · Answer 2 · answered Nov 24 '23 at 01:54

The OP noted in a comment:

I don’t know if Hurter is correct, but I stress that he is writing about Dirac in 1925, not 1927.

Therefore, the text is probably about Dirac's 1925 article , where Dirac uses Hamiltonian function. The equations of motion then feature the Poisson bracket. The Dirac's general approach to quantization involves the replacement of the Poisson brackets by commutators.

Dirac’s debt to Hamilton

2 Answers2