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  Are timelike q-correlations beables in the thermal interpretation?

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In my review of "Foundations of quantum physics II. The thermal interpretation", I uncritically claimed that the collection of all q-expectations and q-correlations would be a state in the thermal interpretation, without noticing that timelike q-correlations could be problematic:

The states of a system in the thermal interpretation are encoded by density operators. However, a state itself is rather the collection of all q-expectations for that state. Section "4.1 Beables and observability in quantum field theory" states this as follows:
  "According to the thermal interpretation, there is nothing in quantum field theory apart from q-expectations of the fields and q-correlations. The quantities accessible to an observer are those q-expectations and q-correlations whose arguments are restricted to the observer’s world tube. More precisely, what we can observe is contained in the least oscillating contributions to these q-expectations and q-correlations. The spatial and temporal high frequency part is unobservable due to the limited resolution of our instruments."

This quote is relevant for a number of reasons. The q-expectations have a spatial and temporal dependence (as parameters). The q-correlations even depend on more than one different spatial and temporal parameter. (This is how I interpret the difference between q-expectations and q-correlations in this quote.) We are given an explicit reason why some q-expectations are not observable (because their high frequency dependence of the spatial and temporal parameters exceed the resolution limits of our instruments).

My claim is problematic for (non-qft) quantum mechanics, because I explicitly claimed that the state is encoded by a density operator, and at the same time claim that q-correlation with two or more temporal parameters would be part of the state. But the density matrix at a given time could only encode those "two times q-correlations" if it additionally used some Hamiltonian in a Schrödinger equation to compute those. But then that Hamiltonian would have to be part of that encoding too.

I think that timelike q-correlations might be problematic (as state) in quantum field theory too, because it is unclear which measurement setup should be able to measure even their lowest frequency components. (But maybe those timelike q-correlations would never have low frequency components anyway, by an effect similar to evanescent modes. But even in this case, it would be nice if the situation could be further clarified.)

asked Mar 31 in Theoretical Physics by Thomas Klimpel (280 points) [ revision history ]

1 Answer

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Time correlations are well-defined only in the Heisenberg picture. There the state is time-independent and the time dependence is encoded in the operators. Time correlations are defined as q-expectations of (suitably ordered) products of field operators. As the field operators themselves, these products and hence their q-expectations, are defined only as distributions. As distribution-valued q-expectations, time correlations are distribution-valued beables. After smearing in space and time, they become complex-valued beables.

answered Apr 2 by Arnold Neumaier (15,488 points) [ no revision ]

Thanks for your interesting answer. (This made me reread many parts of the paper and the book, and pondering many small "answerable" questions about details.) I actually tried to ask a simpler question than the one you answered, but apparently I failed. At least I expected a simpler answer, like that for non-qft quantum mechanics timelike q-correlations are indeed not part of the state (because they are not needed for determining the state), but that the state for quantum field theory cannot avoid those timelike q-correlations, because ... or that it is at least not obvious whether or not the state can avoid those timelike q-correlations.

In relativistic QFT, time correlations are no different in principle than space correlations. The whole formalism needs the Heisenberg picture, where space and time appear on equal footing, even in the nonrelativistic case.

Time-dependent states are not fundamental. They are restricted to single-time formulations, where it is assumed that the restriction to sharp times is well-defined. This requires an approximation step when one starts with an interacting relativistic QFT. (This can be studied in great detail and fully rigorously in 2D, where renormalization issues are essentially absent.)

Of course, time-dependent states  are very important in practice, especially for quantum chemistry and for sequences of measurements.

Even in relativistic QFT, there is one difference between timelike correlations and spacelike correlations, namely the one hinted at by "(suitably ordered) products of field operators" in your answer: For timelike correlations, there is a preferred order, and the order is important, but for spacelike correlations, there is no preferred order, and the order is irrelevant.

Trying to avoid timelike correlations is not the same as requiring sharp times. The point is more that measuring timelike quantum correlations is conceptually more complicated than measuring spacelike quantum correlations: measuring implies interaction with a system, so as soon as a measurement starts, it risks to modify what will happen with the system in the future. Therefore it is unclear whether it is even possible in principle to measure timelike quantum correlations in a similar way as it is possible to measure spacelike quantum correlations.

Field correlations (both timelike and spacelike) are experimentally observable quantities, in practice measured indirectly through linear response theory. 

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