# Which publication did Dirac introduce the “Associative axiom of multiplication” in?

I want to know which publication to cite when I reference the "Associative axiom of multiplication" in the Bra-Ket notation of Quantum mechanics. Sakurai only attributes it to Dirac, but doesn't name the source. As far as I can tell, it was not introduced in "A new notation for Quantum mechanics".

• I don't actually think the associativity of operator multiplication needs to be postulated as an axiom; it seems to follow from the definition. Operator "multiplication" really refers to functional composition, which is associative by definition. – tparker Apr 26 '20 at 15:55
• ok. Do you know where he first or most prominently coined that phrase? To quote Sakurai, Modern QM, page 16: "Dirac calls this the associative axiom of multiplication." – Markus Gratis Apr 26 '20 at 16:00
• HSMPE specializes in these issues. – Cosmas Zachos Apr 26 '20 at 21:08

## 2 Answers

I don't know if Dirac used that phrase in that paper, as the paper is behind a paywall, but the purpose of the paper was to define rules for bra-ket notation, including the use of bras and kets as operators, and it is clear that he defined the way in which bras and kets obey associativity such that for expressions involving inner products, outer products, and/or linear operators, written in bra–ket notation, the parenthetical groupings do not matter. For example:

\begin{aligned}\langle \psi |{\bigl (}A|\phi \rangle {\bigr )}={\bigl (}\langle \psi |A{\bigr )}|\phi \rangle \,&{\stackrel {\text{def}}{=}}\,\langle \psi |A|\phi \rangle \\{\bigl (}A|\psi \rangle {\bigr )}\langle \phi |=A{\bigl (}|\psi \rangle \langle \phi |{\bigr )}\,&{\stackrel {\text{def}}{=}}\,A|\psi \rangle \langle \phi |\end{aligned}

and so forth. Now we would usually do it a little differently, by defining kets as vectors, bras as vectors in the dual space, the braket as the inner product, and simply using the axioms of vector space.

Dirac did use the phrase "associative axiom of multiplication for the triple product of $$\alpha$$ $$\beta$$, and $$|A\rangle$$" to refer to the definition of operator composition $$\{\alpha \beta \}|A\rangle = \alpha\{\beta|A\rangle\}$$ on p24 of The Principles of Quantum Mechanics. An axiom is really just a definition of a mathematical structure; it was not unusual for Dirac to use idiosyncratic terminologies.

I doubt that anyone ever meaningfully coined the phrase "Associative axiom of [operator] multiplication", because the associativity of operator multiplication does not need to be given as an axiom - it follows directly from the definition of operator "multiplication" as functional composition. So either Sakurai misquoted Dirac (my guess), or else Dirac made a mistake (and there's probably not much value in tracking down the exact location of such a mistake).