## Rules of the game

The first part of the problem was to define rules for 'applying' one matrix to another; the second was to construct the relevant matrices for the systems such as 'hydrogen' and for observables such as 'frequency', which would lead to results in agreement with experiment.

Heisenberg returned to Gottingen, where he was fortunate to have as his professor Max Born (1882-1970), who was familiar with the subject of matrix algebra, which had been devised in the 19th century by the English mathematician Arthur Cayley (1821-1895). Cayley had defined a set of self-consistent rules for matrix addition, multiplication and division purely as a mathematical formalism, with no particular relevance to physics. Born drew Heisenberg's attention to the work and enlisted another gifted young post-doctoral student, Pascual Jordan (1902-1980), to help with the project.

It is interesting to note that at that time physicists either did not know what matrices were, or were reluctant to apply them to theoretical models. An illustration of the situation in 1925 is a comment by Heisenberg in a letter to Jordan: 'Now the learned Gottingen mathematicians talk so much about Hermetian matrices, but I do not even know what a matrix is!'

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