  QJT (Quantum Jet Theory) is the quantum theory of jets, which can be
canonically identified with truncated Taylor series. Ultralocality requires a
novel quantization scheme, where dynamics is treated as a constraint in the
history phase space. QJT differs from QFT since it involves a new datum: the
expansion point. This difference is substantial because it leads to new gauge
and diff anomalies, which are necessary to combine background independence with
locality. Physically, the new ingredient is that the observer's trajectory is
explicitly introduced and quantized together with the fields. In this paper the
harmonic oscillator and free fields are treated within QJT, correcting previous
flaws. The standard Hilbert space is recovered for the harmonic oscillator, but
there are interesting modifications already for the free scalar field, due to
quantization of the observer's trajectory. Only free fields are treated in
detail, but the complications when interactions are introduced are briefly
discussed. We also explain why QJT is necessary to resolve the conceptual
problems of quantum gravity.
