  The algebras of interacting "Lie random fields" that were introduced in J.
Math. Phys. 48, 122302 (2007) are developed further. The conjecture that the
vacuum vector defines a state over a Lie random field algebra is proved. The
difference between Lie random field algebras and quantum field algebras is the
triviality of the field commutator at time-like separation, the field
commutator being trivial at space-like separation in both cases. Many
properties that are usually taken to be specific to quantum theory, such as the
superposition of states, entanglement, quantum fluctuations, and the violation
of Bell inequalities, are also properties of Lie random fields.
