  The elements of a deterministic quantum theory are developed, which
reformulates and extends standard quantum theory. The proposed theory is
`realistic' in the sense that in it, a general M-level quantum state is
represented by a single real number r. Surprising as it may seem, this real
number is shown to contain the same probabilistic information as the standard
Hilbert-space state, plus additional information from which measurement outcome
is determined. A crucial concept in achieving this is that of Borelian
(number-theoretic) normality. The essential role of complex numbers in standard
quantum theory is subsumed by the action of a set of self-similar permutation
operators on the digits and places of the base-M expansion of a base-M
Borelian-normal r; these permutation operators are shown to have complex
structure and leave invariant the normality of the underlying real number. The
set of real numbers generated by these permutations defines not only the
Hilbert space of standard quantum theory, but also, in addition, the sample
space from which quantum measurement outcomes can be objectively determined.
Dynamical real-number state reduction is precisely described by deterministic
number-theoretic operators that reduce the degree of normality of r; from the
degree of normality one can infer the standard quantum-theoretic trace rule for
measurement probability. All the foundational difficulties of standard quantum
theory are described in terms of the proposed theory. It is shown that the
real-number states of the proposed theory are precisely its beables.
