Today I will repeat the argument using Rovelli's language of partial and complete variables [gr-qc/0110035], which I found useful to organize my thoughts.
Rovelli distinguishes between partial observables, which can be measured but not predicted, and complete observables, which can be predicted once we know the initial conditions. In a simple mechanical system there are two partial observables:
- \(\phi\): Result, measured by a detector.
- \(t\): Time, measured by a clock.
The partial observables can be measured but not predicted. What can be predicted is the complete observable \(\phi(t)\), the reading of the detector at time \(t\). In the quantum case predictions are statistical, so complete observables are fuzzy.
Now let us turn to field theory. There are now three partial observables:
- \(\phi\): Result, measured by a detector.
- \(t\): Time, measured by a clock.
- \(x\) or \(q\): Position, measured e.g. by a ruler or a GPS receiver.
In QFT we construct the complete observable \(\phi(x,t)\), the reading of the detector at time \(t\) and location \(x\), from these three partial observables.
Here is the problem with QFT. The experiment takes place inside a detector, and we can in fact predict its future location. The detector is part of the physical world and has dynamics of its own. Instead we can construct two different complete observables from the same three partial observables:
- \(q(t)\): The detector's location at time \(t\).
- \(\phi(t)\): The reading of the detector at time \(t\), at the detector's location.
We can think of \(\phi(t)\) as \(\phi(q(t),t)\), the value of the field at the detector's location. Since complete observables are fuzzy, we now have two types of fuzziness: the horizontal fuzziness of \(q(t)\) and the vertical fuzziness of \(\phi(t)\). In QFT we only have vertical fuzziness.
The obvious objection is that all information about the field outside the detector's worldline is lost. However, we can add more partial observables which can also be measured inside the detector:
- \(\phi_m\): The \(m\):th derivative of the field, also measured inside the detector.
From the complete observables \(\phi_m(t)\) we can construct the generating function
\[
\phi(x,t) = \sum_m {1\over m!} \phi_m(t) (x-q(t))^m,
\]
which can be viewed as the Taylor series expansion of the field \(\phi(x,t)\). To the extent an infinite Taylor series is equivalent to the field itself, we recover the field that we started from.
Hence we again arrive at Quantum Jet Theory.
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