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In the original 1928 paper (pdf) the Dirac equation appears on page 615 in equation (9) as

$$ [p_0+\rho_1\left(\boldsymbol{\sigma},\boldsymbol{p}\right)+\rho_3mc]\psi=0\qquad(1) $$

Using the definitions

$$ p_0=i\hbar\frac{\partial}{c\partial t};\quad\boldsymbol{p}=-i\hbar\boldsymbol{\nabla} $$

from pages 613 and 611 as well expressing the matrices given at the bottom of page 614 in modern notation

$$ \rho_1\rightarrow\gamma_5;\quad\boldsymbol{\sigma}\rightarrow\boldsymbol{\Sigma};\quad\rho_3\rightarrow\beta $$

and finally multiplying through with the speed of light $c$, we obtain the equation

$$ [i\hbar\frac{\partial}{\partial t}+c(\boldsymbol{\alpha}\cdot\boldsymbol{p})+\beta mc^2]\psi=0 $$

However, this is not how the Dirac equation is written today; the term with the time derivative has the opposite sign !

The form (1) is also found in the third edition (1947) of Dirac's book The Principles of Quantum Mechanics (see eq (8) on page 256).

In the fourth edition (1958), however, in equation (10) on page 257, we find

$$ [p_0-\rho_1\left(\boldsymbol{\sigma},\boldsymbol{p}\right)-\rho_3mc]\psi=0\qquad(2) $$

that translates into the form we know today. Basically, eq (2) is obtained from eq (1) by changing the sign of time.

Do we know why Dirac modified his equation in this way ?

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  • $\begingroup$ Just a comment instead of an answer because I can't check right now: it looks like the more recent expression is simply a covariant rewrite. In this context, it's common to have a relative minus sign between temporal and spatial terms. $\endgroup$
    – Miyase
    May 10, 2022 at 16:40
  • $\begingroup$ It is true that contrary to the third edition of PQM, in section 66 of the fourth edition, Dirac makes a distinction between covariant and contravariant vectors (although his convention in terms of sub/superscript is opposite to current convention). In (4) he writes the 4-momentum operator on covariant form, but to me the resulting operator is identical to the definitions given in (1) and (2) of the third edition. $\endgroup$
    – Trond Saue
    May 13, 2022 at 12:51
  • $\begingroup$ The crucial difference occurs in section 67, between eq (5) in the third edition and eq (7) of the fourth edition. Here the sign difference that I have pointed out occurs. As already stated, it appears to me that the momentum operator given in these equations are strictly identical, and by comparing (7) of the third edition and (9) of the fourth edition, the matrices are also identical. So this is still a mystery to me ! $\endgroup$
    – Trond Saue
    May 13, 2022 at 12:51

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