*The Mathematical and Philosophical Foundations of Quantum Field Theory*, University of London, 1989.

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In this thesis I wrote a detailed history of relativistic quantum theory from its inception to the mid-thirties. I also wrote a self-contained presentation of the mathematical foundations of QFT, including Von Neumann's classification of operator algebras, Wigner's classification of representations of the Lorentz group, Mackey's theory of imprimitivity, the algebraic approach to QFT, the spin-statistics theorem, Segal quantization, and the theory of Fock and non-Fock representations. This work also included some historical background. I also set out and solved -- at least to my own satisfaction -- a number of philosophical and conceptual questions about quantum field theory, and particularly the relationship of non-relativsitic to relativistics field theory

My main puzzle was this: why is it, when the Galilean group is replaced by the Lorentz group, otherwise retaining the entire structure of quantum theory unchanged, the physical and mathematical content of the theory is so drastically transformed? Thus: within a year or two of the inception of non-relativistic quantum mechanics, several examples of dynamical effects could be modeled with rigour and precision (so by 1928); now, eighty years on, there is still not a single rigourous and precise model of a non-trivial dynamical processes in relativistic quantum field theory in 3+1 dimensions. From a mathematical point of view the theory has fallen off a cliff.

The conclusion I drew is that there are two kinds of complex numbers present in the relativistic case which coincide in non-relativistic spacetime, and that the one relevant to the Hilbert-space theory is non-local; the local complex structure concerns the fields, rather than the state-space. This explains why there is no covariant position operator in RQFT, why local interactions cannot preserve particle number, and hence why such interactions always involve, in principle, all the degrees of freedom of a quantum field and not some finite number of them.

My second puzzle was to see whether the existence of unitarily inequivalent representations of the canonical (anti)commutation relations, for systems of infinitely-many degrees of freedom, could be exploited to give a solution to the problem of measurement (roughly along the lines indicated by Hepp). For a time I believed that it could (see my 1988), but I despaired of it by 1991. This work did, however, provide a point of entry into decoherence theory, important to my work on the Everett interpretation.

Michael Redhead was my supervisor. The thesis was examined by Ray Streater and Roger Penrose.