208 THE NEW QUANTUM THEORY 



Where the trained mathematician has the advantage is 

 that he can use it, and in the past year or two it has 

 been used in physics with very great advantage indeed. 

 It leads not only to those phenomena described by the 

 older quantum laws such as the h rule, but to many 

 related phenomena which the older formulation could 

 not treat. 



On the right-hand side, besides h (the atom of action) 

 and the merely numerical factor 2tt, there appears i (the 

 square root of — i) which may seem rather mystical. 

 But this is only a well-known subterfuge; and far back 

 in the last century physicists and engineers were well 

 aware that V — i in their formulae was a kind of sig- 

 nal to look out for waves or oscillations. The right- 

 hand side contains nothing unusual, but the left-hand side 

 baffles imagination. We call q and p co-ordinates and mo- 

 menta, borrowing our vocabulary from the world of 

 space and time and other coarse-grained experience; 

 but that gives no real light on their nature, nor does 

 it explain why qp is so ill-behaved as to be unequal 

 to pq. 



It is here that the three theories differ most essen- 

 tially. Obviously q and p cannot represent simple 

 numerical measures, for then qp — pq would be zero. 

 For Schrodinger p is an operator. His "momentum" 

 is not a quantity but a signal to us to perform a certain 

 mathematical operation on any quantities which may 

 follow. For Born and Jordan p is a matrix — not one 

 quantity, nor several quantities, but an infinite number 

 of quantities arranged in systematic array. For Dirac 

 p is a symbol without any kind of numerical interpreta- 

 tion; he calls it a ^-number, which is a way of saying 

 that it is not a number at all. 



I venture to think that there is an idea implied in 



