# Meaning of the "Thermal Interpretation" of quantum mechanics (A. Neumaier)

I would like to understand the follow statements in the context of the thermal interpretation:

https://arnold-neumaier.at/physfaq/therm/topics/s24.html

If one knows the state of the universe completely, one can, according to the thermal interpretation, predict the results of all single measurements. In particular, one should therefore be able to predict the impact positions of particles in a Stern-Gerlach experiment. Since we do not, in fact, know the state, this is of course an idle speculation, but it is the principle that is important. We work in the Schrödinger picture, and denote the state of the universe at time $$t$$ by $$ρ(t)$$. This is a density matrix, an operator on the Hilbert space of the universe

What can be measured is therefore clearly determined by the state of the universe. If God

1. knows this state at the time $$t=0$$,
2. knows the Hamiltonian of the universe, and
3. can solve the von Neumann equation, then he can calculate $$ρ(t)$$ and thereby predict the distribution of silver at any given time.”

“According to the thermal interpretation, if we make a single run of a Stern-Gerlach experiment, there is indeed only one result. Which result it is (i.e., whether a silver atom is deposited at the left or the right spot) depends on in practice unpredictable details in the deterministic environment.”

https://arxiv.org/abs/1904.12721

“In particular, if a beam contains only a single particle, the quantum field representing the beam is in a state with sharp particle number $$N = 1$$, but otherwise nothing changes. Conservation of mass, together with the instability of macroscopic superpositions and randomly broken symmetry forces that only one of the two spots gets marked by a silver atom, just as a classical bar under vertical pressure will bend into only one direction. It is not clear how Nature achieves the former, but this lack of explanation is common to all interpretations of quantum physics.”

Apart from the technical aspects of thermal interpretation, these claims seem quite clear:

In principle

1. ρ(t) is subject to a deterministic time evolution;
2. ρ(t) determines a single and unique measurement result.

As I understand it, this is in contradiction with decoherence:

1. ρ(t) is subject to deterministic time evolution;
2. ρ(t) determines a set of possible measurement outcomes, each of which corresponds to a decoherent/stable branch (on top of decoherence the many-worlds-interpretation claims that all these branches do exist in reality)

So to me it seems that the thermal interpretation

• does not provide a different interpretation for the same ρ(t), but that it claims that ρ(t) itself is different;
• claims that nature is fully deterministic (similar to classical deterministic chaos).

To phrase it in mathematical terms i.e. q-expectation values: for a single run and therefore a position measurement of a single particle, the quantity

$$a(x,t) = \text{tr}\,A \, U(t) \, \rho_0 \, U^\dagger(t)$$

has always one single peak at some $$x = x_0$$ indicating the unique result of the position measured. Probability is just due to our lack of knowledge of $$\rho_0$$ and $$H$$, and the fact that in practice we are not able to calculate $$x = x_0$$.

I would like to know whether my understanding is correct and whether this has been discussed in more detail for standard experiments like

• Stern-Gerlach
• Bell-type experiments, e.g. two Spin-entangled particles with two Stern-Gerlach devices and two position measurements with 2*2 possible outcomes
• double-slit
• cloud chamber tracks i.e. a “sequence of measurements” = droplets (Mott‘s problem)

Your 1. and 2. for decoherence talk about different $$\rho(t)$$.

In 1. it is (as in your 1. and 2. for the thermal interpretation) the density operator of the universe, never explicitly considered in decoherence, whereas in 2. it is the density operator of a small and hence open system that can be measured.

This explains the discrepancy; the stochasticity in 2. (and in decoherence) comes from the neglect of the environment.

Compared to https://arxiv.org/abs/1904.12721 , an improved exposition of experiments is given in my book 'Coherent quantum physics' https://www.degruyter.com/document/doi/10.1515/9783110667387/html

A derivation of the thermal interpretation from quantum tomography is given in my recent paper 'Quantum mechanics via quantum tomography' https://arxiv.org/abs/2110.05294

• This is a misunderstanding; I always mean the same state $\rho$. Also acc. to the proponents of the many-worlds interpretation, decoherence is a process at the level of the state of the universe (split into relevant and environmental d.o.f. for technical reasons). The apparent stochastic collapse is due to decoherence, the superposition in the state of the universe remains. Your thermal interpretation seems to make a distinctly different claim: there is no such superposition mathematically (so the superposition in the context of decoherence must be an artifact of tracing out the environment).
– TomS
Aug 28, 2022 at 9:50
• Regardless, thank you very much, I will read the new paper. However, after your reply, I assume that I have correctly understood the key statements quoted. Correct?
– TomS
Aug 28, 2022 at 9:53
• not quite. Your a(x,t) is not a function of x, but is the q-expectation value of the observable A. This is what would be observed in an ideal experiment for measuring A at time t. But it is known only approximately (with error bars compatible with the uncertainty relation) since we know precisely neither the density operator of the universe nor its Hamiltonian. Aug 28, 2022 at 11:50
• The state of the universe cannot be sensibly be described in terms of superpositions, since the notion of a superposition requires a preferred basis of the space of wave functions of the universe. What decoherence does is consider a vastly reduced space only, where the reduced density operator acts on. Therefore all talk about superpositions is completely spurious and can be dropped without loss of physical information. Aug 28, 2022 at 11:53
• Ok, understood - perhaps besides some technical details. But these are really bold claims: 1) there are - at least in principle - unique measurement results encoded mathematically, i.e. no need for any interpretation at all. 2) all interpretations along the line of decoherence are totally irrelevant.
– TomS
Aug 28, 2022 at 12:05