# Was Dirac really trying to take the square root of the Klein-Gordon operator?

As a student of physics one will, on several occasions, indubitably hear professors or other physicists (here is an example, from Physics.SE's highest-rep user John Rennie) tell the famous story that Paul Dirac came upon his relativistic wave equation for the electron, now called the Dirac equation, by "trying to take the square root of the Klein-Gordon operator".

The story comes in many different versions, some dressing it up in a funny anecdote about Dirac, when asked what he was working on, casually replying with "Oh, just trying to take the square root of something". Others simply claim that this was the train of thought he followed in his derivation. Wikipedia has another version, which features Dirac pondering by the fireplace...

A quick look at the first few sections of the paper where Dirac introduced his wave equation, however, shows no trace of "taking square roots" of any kind. Of course, it is clear that Dirac was looking for an operator that was linear---as opposed to the quadratic Klein-Gordon equation---but that doesn't really justify saying that he was looking to take a square root.

Nonetheless, this doesn't really debunk the entire story, since it seems quite reasonable to assume that Dirac may have streamlined his derivation considerably before publishing. Thus, the intuitive idea (maybe he thought it would sound a little silly to publish something so vague-sounding) may have been carefully hidden behind a more rigorous derivation and justification.

My question is: What is this story about "taking the square root of the Klein-Gordon operator" based on? Is there any evidence that Dirac was indeed thinking along these lines? If not, can the story be traced back to any other source?

• You seem not easily convinced! – Francois Ziegler Apr 14 '18 at 0:41
• I guess I somehow forgot to accept your answer... corrected that now! – Danu Apr 14 '18 at 11:40

One routinely speaks of the Dirac operator as a square root of the Laplacian (or Dalembertian as the case may be), and Dirac himself rather supports this heuristic in his Recollections of an exciting era, History of twentieth century physics, Academic Press 1977, pp. 109-146:

I was playing around with the three components $\sigma_1, \sigma_2,\sigma_3,$ which I had used to describe the spin of the electron, and I noticed that if you formed the expression $\sigma_1p_1+\sigma_2p_2+\sigma_3p_3$ and squared it, $p_1$, $p_2$ and $p_3$ being the three components of momentum, you got just $\smash{p_1^2+p_2^2+p_3^2}$, the square of the momentum (1). This was a pretty mathematical result. I was quite excited over it. It seemed that it must be of some importance. (...)

I suddenly realized that there was no need to stick to the quantities $\sigma$, which can be represented by matrices with just two rows and columns. Why not go over to four rows and columns? Mathematically there was no objection to this at all. Replacing the $\sigma$-matrices by four-row and column matrices, one could easily take the square root of the sum of four squares, or even five squares if one wanted to.

Accordingly, as I am sure is well known, in The Principles of Quantum Mechanics (4th ed., §67; it would be interesting to compare the first of 1930, which I can’t find online) he starts with “the relativistic Hamiltonian [that] leads to the wave equation $$\{p_0 - (m^2c^2+p_1^2+p_2^2+p_3^2)^{\frac12}\}\psi=0, \tag5$$ where the $p$'s are interpreted as operators in accordance with [$p_\mu=i\hbar\partial/\partial x^\mu$]”, then multiplies by the conjugate to obtain $$\{p_0^2 - m^2c^2-p_1^2-p_2^2-p_3^2\}\psi=0, \tag6$$ and then seeks “a wave equation that is linear in $p_0$ and that is roughly equivalent to (6)” in the form $$\{p_0 - \alpha_1p_1-\alpha_2p_2-\alpha_3p_3 -\beta\}\psi=0, \tag7$$ which precisely leads him to the “square root of the sum of four squares” problem.

(1) An identity found explicitly, with $p_r=-i\partial/\partial x_r$, in the 1928 paper you quote (bottom of p. 618, case $\mathbf A=0$).

Edit: Dirac’s first edition (1930) is now online, and its §74 already has the exact same development as above (but for minimal differences in notation).

Dirac was a mathematician above all who liked the beauty of mathematics. This is one of the reasons why he objected the ugliness of renormalization theories. You should read Pretty Mathematics International Journal of Theoretical Physics, Vol. 21, Nos. 8//9, 1982 , where he essentially explains that he was playing with 2X2 matrices, quoting him

The resulting wave equation for the electron turned out to be very successful. It led to correct values for the spin and the magnetic moment. This was quite unexpected. The work all followed from a study of pretty mathematics, without any thought being given to these physical properties of the electron.

Such papers are largely ignored by physicists in the field, though.