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- The thermal interpretation includes a non-ensemble interpretation of q-expectations which also applies to single systems, not just to ensembles. Callen's criterion is used to provide operational meaning to the state of a system. Trying instead to use Callen's criterion to provide an "operational falsification" interpretation of expectations (and hence probability) might be instructive (and easier than trying to understand the whole thermal interpretation and its way to use quantum field theory).
In the thermal interpretation, the collection of all q-expectations (and q-correlations) is the state of a system. The interpretation of q-expectations is used only to provide an ontology, the apparent randomness is analysed and explained separately. This may be non-intuitive. Callen's criterion is evoked to provide operational meaning. As an answer to how specific q-expectations can be determined operationally, it occasionally felt unsatisfactory to me.
However, if we limit our goal to "falsification" instead of "measurement"/"determination"/"estimation", then Callen's criterion (section "2.5 Formal definition of the thermal interpretation" [TI], section "6.6 Quantities, states, uncertainties" [CQP]) becomes quite concrete:
Expectations allow us to have partial information about the state. In the thermal interpretation, ⟨f(A)⟩x would be a q-expectation depending on a space-time point x, and ⟨f(A,B)⟩(x,y) would be a q-correlation depending on two space-time points x and y. It makes sense to assume that the space-time points x and y for which we want to assert values for q-expectations and q-correlations are close to our current position in space and time. (So thinking of the state as the initial state is dangerous, but ignoring the inital state and only thinking about the current state is dangerous too.) And it also makes sense that we only want to assert values for q-expectations up to a certain reasonable precision, especially in cases where additional precision would make no difference in terms of how available (and future) observations might falsify our assertions about the state. More important for falsification is that often the theory also gives a standard deviation (to be interpreted as an uncertainty, see below).
But what is a q-correlation like ⟨f(A,B)⟩(x,y) operationally, and how can it be observed? This is one of those cases where evoking Callen's criterium to provide operational meaning felt unsatisfactory to me. Most q-expectations (and especially q-correlations) are not directly observable. It is rather the opposite: we can do a number of different observations of the real system, and it is part of our idealized mathematical model of that system that those observations corresponds to certain q-expectations of our mathematical model in its given state (or more generally to functions of q-expectations and q-correlations). And those observations themselves should not be interpreted probabilistically: for example when we observe which photosensitive pixels changed their color, then this itself is the observation. And if the change of color is only gradually, then again this is an observation all by itself, there is no probabilistic averaging involved over atomic constituents of the pixel. And the probabilities (that we try to interpret) and expectations too are just properties of the relation between our simplified mathematical model and the real system, not properties of the real system itself. (Edit: Oh no, talking of "the" real system might be a mistake, since probabilistic models are also used to describe a class of similarly prepared systems.)Despite all that has been said, and all the limitations that have been pointed out, even the modest goal of "operational falsification" is still far from trivial: The observed deviations of a system from expectations (derived from the theory of a given state) can be more or less significant. If a frequentist (virtual) ensemble interpretation of probability is used to quantify and interpret this significance, then it still remains unclear whether a true operational interpretation of probability has been achieved. The non-ensemble interpretation of q-expectations overcomes this difficulty (section "2.3 Uncertainty" [TI], section "3.1 Uncertainty" [CQP]):
[TI] A. Neumaier, Foundations of quantum physics, II. The thermal interpretation (2018), https://arxiv.org/abs/1902.10779
[CQP] A. Neumaier, Coherent Quantum Physics (2019), De Gruyter, Texts and Monographs in Theoretical Physics
The intention of the above was to describe an "operational falsification" interpretation of probability in the hope that it would be easier to understand than the whole thermal interpretation. But even an interpretation of probability is hard to describe and involves many subtle issues, so in the end I often had to reference the thermal interpretation, because I wanted the description to stay reasonably short and avoid "my own original ideas". But the description is different from the thermal interpretation, because it prefers to explicitly say things and give examples, even if they might be slighly wrong, or against the intentions of the thermal interpretation.
However, if we limit our goal to "falsification" instead of "measurement"/"determination"/"estimation", then Callen's criterion (section "2.5 Formal definition of the thermal interpretation" [TI], section "6.6 Quantities, states, uncertainties" [CQP]) becomes quite concrete:
So if the available observations of a system seem inconsistent with the system being in a given state, then operationally the system is not in that state. This risks being a bit tautological, especially if we don't properly distinguish between the real system that provided the available observations, our idealized mathematical model of that system, and the given state. The state belongs to the mathematical model, because otherwise there would be no theory for the state. Even so an inconsistency between observations and the theory for the state might also be the fault of the model, I want to limit the "falsification" here explicitly to the state. Operationally this could mean to exclude observations which seem to be caused by effects not included in the model, like cosmic rays, radioactive decay, or human errors of the operator making the observations.Callen’s criterion: Operationally, a system is in a given state if its properties are consistently described by the theory for this state.
Expectations allow us to have partial information about the state. In the thermal interpretation, ⟨f(A)⟩x would be a q-expectation depending on a space-time point x, and ⟨f(A,B)⟩(x,y) would be a q-correlation depending on two space-time points x and y. It makes sense to assume that the space-time points x and y for which we want to assert values for q-expectations and q-correlations are close to our current position in space and time. (So thinking of the state as the initial state is dangerous, but ignoring the inital state and only thinking about the current state is dangerous too.) And it also makes sense that we only want to assert values for q-expectations up to a certain reasonable precision, especially in cases where additional precision would make no difference in terms of how available (and future) observations might falsify our assertions about the state. More important for falsification is that often the theory also gives a standard deviation (to be interpreted as an uncertainty, see below).
But what is a q-correlation like ⟨f(A,B)⟩(x,y) operationally, and how can it be observed? This is one of those cases where evoking Callen's criterium to provide operational meaning felt unsatisfactory to me. Most q-expectations (and especially q-correlations) are not directly observable. It is rather the opposite: we can do a number of different observations of the real system, and it is part of our idealized mathematical model of that system that those observations corresponds to certain q-expectations of our mathematical model in its given state (or more generally to functions of q-expectations and q-correlations). And those observations themselves should not be interpreted probabilistically: for example when we observe which photosensitive pixels changed their color, then this itself is the observation. And if the change of color is only gradually, then again this is an observation all by itself, there is no probabilistic averaging involved over atomic constituents of the pixel. And the probabilities (that we try to interpret) and expectations too are just properties of the relation between our simplified mathematical model and the real system, not properties of the real system itself. (Edit: Oh no, talking of "the" real system might be a mistake, since probabilistic models are also used to describe a class of similarly prepared systems.)Despite all that has been said, and all the limitations that have been pointed out, even the modest goal of "operational falsification" is still far from trivial: The observed deviations of a system from expectations (derived from the theory of a given state) can be more or less significant. If a frequentist (virtual) ensemble interpretation of probability is used to quantify and interpret this significance, then it still remains unclear whether a true operational interpretation of probability has been achieved. The non-ensemble interpretation of q-expectations overcomes this difficulty (section "2.3 Uncertainty" [TI], section "3.1 Uncertainty" [CQP]):
to eliminate any trace of a priori statistics from the terminology, we frequently use the terminology uncertain value instead of q-expectation value, and uncertainty instead of q-standard deviation
[TI] A. Neumaier, Foundations of quantum physics, II. The thermal interpretation (2018), https://arxiv.org/abs/1902.10779
[CQP] A. Neumaier, Coherent Quantum Physics (2019), De Gruyter, Texts and Monographs in Theoretical Physics
The intention of the above was to describe an "operational falsification" interpretation of probability in the hope that it would be easier to understand than the whole thermal interpretation. But even an interpretation of probability is hard to describe and involves many subtle issues, so in the end I often had to reference the thermal interpretation, because I wanted the description to stay reasonably short and avoid "my own original ideas". But the description is different from the thermal interpretation, because it prefers to explicitly say things and give examples, even if they might be slighly wrong, or against the intentions of the thermal interpretation.
Even so I know that my explanations are less polished than A. Neumaier's papers and book, I hope that they help to see vanhees71 how the thermal interpretation might be different from the minimal statistical interpretation, and what it could mean that it also applies to single observations of macroscopic systems and not just to ensembles of similarly prepared systems. And I hope A. Neumaier can live with the intentional explicit differences to the "thermal interpretation". After all this is intended as an interpretation of probability, not as a faithful description of the thermal interpretation.gentzen said:Based on the current discussion, it occurred to me that the non-ensemble interpretation of q-expectations of the thermal interpretation could be combined with Callen's criterion to arrive at an "operational falsification" interpretation of expectations (probability). That interpretation would be closely related to the frequentist interpretation, but fix its problem related to the assumption/requirement of "virtual" ensembles that allow to arbitrarily often repeat identical experiments (which makes the frequentist interpretation non-operational and non-applicable to many practically relevant scenarios).vanhees71 said:I always thought that's also the explanation of your "thermal interpretation" until you told me that your expectation values must not be intepreted in the usual statistical sense but as something abstractly defined in the mathematical formalism without relation to an operational realization by a measurement device.
In order not to hijack this thread, I will open a separate thread with more explanations when I find the time.
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