Quantum Jet Theory

In the previous posts I argued that what is missing from Quantum Field Theory (QFT) is horizonal fuzziness. Today it is finally time to discuss how this can be implemented. Obviously, horizontal fuzziness requires that the theory has horizontal observables. QFT only has vertical observables, the values of the field at given points in space and time, and hence we can not formulate horizontal fuzziness within the framework of QFT. But there is a simple way to introduce a horizontal observable: expand the field in a Taylor series. A Taylor series has both vertical observables (the Taylor coefficients) and a horizontal observable (the base point a.k.a. the detector's position), and hence it can express both vertical and horizontal fuzziness. This is the idea behind Quantum Jet Theory (QJT). The name comes from the notion of jets in mathematics; a \(p\)-jet is locally the same as a Taylor series truncated at order \(p\). There might be other ways to include the detector's trajectory in a theory, but QJT is what naturally arises from the representation theory of the multi-dimensional Virasoro algebra that I will discuss next week.

QFT is quantum physics applied to fields, i.e. systems with infinitely many degrees of freedom. A field \(\phi(x,t)\) represents the result of an experiment \(\phi\) done at the spacetime point \((x,t)\). The result \(\phi\) is fuzzy, but the location \(x\) is known precisely at all times. Hence QFT has vertical fuzziness but no horizontal fuzziness. In QJT we expand the field in a Taylor series,
\[
\phi(x,t) = \sum_m {1\over m!} \phi_m(t) (x-q(t))^m.
\]
This formulation may seem to be limited to one dimension, but with multi-index notation it makes sense in higher dimensions as well.

The Taylor coefficients \(\phi_m\) are the vertical observables. They describe the result of the experiment inside the detector. The corresponding momenta \(\pi^m\) describe the experiment's rate of change. Here we put the index upstairs, because the momenta belong to the dual space. The product of the uncertainties obey
\[
\Delta \phi_m \cdot \Delta \pi^n \geq {\hbar\over2} \delta^n_m
\]
where \(\delta^n_m\) is nonzero only when \(m = n\). This is vertical fuzziness.

The expansion point \(q\) is the horizontal observable. It represents the location of the detector. If \(p\) is the detector's momentum, the uncertainty relation becomes
\[
\Delta q \cdot \Delta p \geq \hbar/2.
\]
This is horizontal fuzziness.

Hence by replacing the fields with the corresponding Taylor series we get a theory in which both the result of the experiment and its location are fuzzy; QJT implements both vertical and horizontal fuzziness. 

To replace the fields by their Taylor series might sound trivial, and it would be if the expansion point were a fixed point in space. But the last equation means that we expand the fields around a fuzzy point.

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