
16 Commutation Relations, Normal Ordering, and Stirling Numbers
with Heisenberg , they derived for any function f (q, p), which can be formally expressed as
power series in p and q,therule
pf − f p =(−i)
∂f
∂q
,
as well as
qf − f q =(−i)
∂f
∂p
. (1.33)
Dirac, who independently found (1.31) in [349], considered in his subsequent work [351]
algebraic consequences of (1.31) and also derived (1.33). Furthermore, Dirac showed that
f(q, p)e
iαq
= e
iαq
f(q, p + α),
which one recognizes as (1.12) already discussed by Boole – or, even earlier, by Cauchy
(1.11). Dirac [350] considered many other interesting consequences of (1.31). Coutinho [304]
gave a beautiful account ...