## Matrices do not commute

Heisenberg had not gone very far with his idea when he noticed something in his scheme which worried him. Physical phenomena had up to now been represented by an algebra in which a number A multiplied by B gave the same result as B multiplied by A. This is called the commutative law of multiplication, but matrices do not necessarily obey this law. If we use matrices to determine the values of two physical observables, we will in general get a different result if we change the order in which we make these determinations. Thus if matrix [A] represents observable A, and matrix [B] represents observable B, it is natural to represent a measurement of A followed by a measurement of B by the product [B][A] ('B after A'). If [B][A] * [A][B] we conclude that measuring B after A is not the same as measuring A after B.

Heisenberg felt that the non-commutative nature of matrix algebra gave rise to a fundamental error in his theory. However, Paul Dirac (1902-1984), who had received an early copy of the paper later published by Bohr, Heisenberg and Jordan, quickly realised that non-commutation might in fact be exactly the dominant feature governing basic natural phenomena. The theory was not wrong — quite the opposite, the matrix equations were trying to tell us something!

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