Mastering the master field

The basic concepts of non-commutative probability theory are reviewed and applied to the large- N limit of matrix models. We argue that this is the appropriate framework for constructing the master field in terms of which large- N theories can be written. We explicitly construct the master field in a number of cases including QCD ₂. There we both give an explicit construction of the master gauge field and construct master loop operators as well. Most important we extend these techniques to deal with the general matrix model, in which the matrices do not have independent distributions and are coupled. We can thus construct the master field for any matrix model, in a well defined Hilbert space, generated by a collection of creation and annihilation operators—one for each matrix variable—satisfying the Cuntz algebra. We also discuss the equations of motion obeyed by the master field.