The thermal interpretation of quantum physics

[akhmeteli]

(concerning: "an analogous statement about a free relativistic particle somehow prepared at time t in a small region of spacetime suffers the same problem."
I have cited Hegerfeldt's https://arxiv.org/abs/quant-ph/9806036, which is general enough to include both relativistic and nonrelativistic cases because it depends only on positive energy. Reeh-Schlieder can be construed as essentially the same property for QFT. Also significant, in my view, is "Anti-Locality of Certain Lorentz-Invariant Operators", I. E. SEGAL and R. W. GOODMAN, Journal of Mathematics and Mechanics, Vol. 14, No. 4 (1965), pp. 629-638.
In another comment (also two weeks old), the locality of the retarded propagator was mentioned, but quantum field theory is mostly concerned with the propagator $\int 2\pi\mathrm{e}^{\mathrm{i}k{\cdot}x}\delta(k{\cdot}k-m^2)\theta(k_0)\frac{\mathrm{d}k}{(2\pi)^4}$ (noting the restriction to positive frequency, $\theta(k_0)$, which puts it in the frame for Hegerfeldt's result), and its time-ordered variant, the Feynman propagator, both of which are nonlocal in that they are nonzero at space-like separation.

But Hegerfeldt requires positivity of energy, and there is no positivity of energy, say, for the Dirac equation.

 

[A. Neumaier]

But Hegerfeldt requires positivity of energy, and there is no positivity of energy, say, for the Dirac equation.

That's the main reason why the Dirac equation is poorly suited for relativistic quantum physics. Physical energy must be bounded below; so one has to discard the solutions below some energy level, and thus change the Hilbert space of the Dirac equation to a space dependent on the external field.

 

[A. Neumaier]

From what I can gather as a lay person after a quick glance and mostly guessing what it is about, superficially, the thermal interpretation appears like a redressing of the ensemble interpretation. [...] terms such as deterministic, realistic, objective and local are no longer describing the same thing as in most other interpretations.

It changes the ensemble interpretation to a foundation that is independent of ensembles, probabilities, and measurement. The latter become secondary concepts that fit certain (but by far not all) situations where quantum mechanics is used.

I would say at least, it is not objective or deterministic, and that it is silent on locality vs objectively unique measurement results, and that it may be non-complete by not being able to explain single measurement results fully.

However, as I said at the beginning, I haven’t read the source material closely so could be missing a lot

Yes, you are missing a lot. In relativistic quantum field theory, q-expectations of fields refer to single measurements, all fields are local and objective, and single measurements are fully explained through the effects of coarse-graining and the resulting dissipation.

 

[Peter Morgan]

But Hegerfeldt requires positivity of energy, and there is no positivity of energy, say, for the Dirac equation.

The anticommutative structure ensures that in the vacuum sector the Hamiltonian operator of the free Dirac quantum field has a spectrum that is bounded below by the vacuum energy.

 

[akhmeteli]

But Hegerfeldt requires positivity of energy, and there is no positivity of energy, say, for the Dirac equation.

The anticommutative structure ensures that in the vacuum sector the Hamiltonian operator of the free Dirac quantum field has a spectrum that is bounded below by the vacuum energy.

That's the main reason why the Dirac equation is poorly suited for relativistic quantum physics. Physical energy must be bounded below; so one has to discard the solutions below some energy level, and thus change the Hilbert space of the Dirac equation to a space dependent on the external field.

Let me remind you the context of the discussion. Previously I wrote:

So it looks like the statement I quoted is indeed true only for nonrelativistic quantum mechanics, if the atom in a trap cannot be modeled by relativistic quantum mechanics, without using quantum field theory.

A. Neumaier replied:

The statement about the ion trap yes, but an analogous statement about a free relativistic particle somehow prepared at time $t$ in a small region of spacetime suffers the same problem.

At this point I asked for a reference.

So Peter Morgan explains to me that if we use the anticommutative structure we get a problem with the Dirac equation, but I spoke about "relativistic quantum mechanics, without using quantum field theory." If you use the anticommutative structure and the spectrum is bounded below by the vacuum energy, you actually use quantum field theory.

A. Neumaier explains to me that the Dirac equation is not good enough. But that was exactly my point: if the original nonrelativistic Schrödinger equation and the Dirac equation have problems, how can one fault the Born's rule for that? And again, I have no investment in the Born's rule, I am just trying to say that the specific critique of the Born's rule does not look well-founded.​

 

[Lord Jestocost]

But maybe consciousness is not preprogrammed and allows for some user decisions...

Decisions with respect to what in a pre-programmed world?

 

[Peter Morgan]

Let me remind you the context of the discussion. Previously I wrote:

A. Neumaier replied:

At this point I asked for a reference.

So Peter Morgan explains to me that if we use the anticommutative structure we get a problem with the Dirac equation, but I spoke about "relativistic quantum mechanics, without using quantum field theory." If you use the anticommutative structure and the spectrum is bounded below by the vacuum energy, you actually use quantum field theory.

A. Neumaier explains to me that the Dirac equation is not good enough. But that was exactly my point: if the original nonrelativistic Schrödinger equation and the Dirac equation have problems, how can one fault the Born's rule for that? And again, I have no investment in the Born's rule, I am just trying to say that the specific critique of the Born's rule does not look well-founded.​

@akhmeteli, I read through the whole thread in one sitting after a few weeks of all of you discussing away, so I've no doubt I missed some of the nuances of what each of you hope to get from it. The lack of threading on PF makes it difficult enough to get into a long conversation that I don't usually bother once there are many posts, but I thought that you might find the Hegerfeldt reference of some interest, given that no-one else responded to your request. The connection of positive frequency with analyticity and locality/nonlocality has always seemed to me less emphasized than it might be, particularly given the analogy it suggests with the perhaps simpler connection of the Hilbert transform with analyticity in signal analysis.
In any case, I didn't notice that you wanted to discuss only "relativistic quantum mechanics, without using quantum field theory", about which I know too little to contribute much, but I have always taken it that problems such as you mention with the Dirac equation as relativistic QM are what pushed physicists towards the Dirac equation as the basis of a QFT.

 

[vanhees71]

No. Like you did in the classical case, I nowhere assume anything probabilistic in the quantum case. Randomness appears as in classical physics by breaking metastability through effects of the deterministic noise neglected in coarse-graining. Of course it will be in agreement with the standard interpretation where the latter is based on actual measurement.

That's what you claim! Your entire formalism is just equivalent to the standard formalism, you just rename "probability" as "q-expectation" and forbid people to translate it back to the standard language which is standard language, because it makes the physical meaning (i.e., the interpretation!) of the formalism clear instead of hiding it for some ununderstandable philosophical quibbles unrelated with physics.

 

[vanhees71]

It doesn't matter how one calls it. Tradition calls it expectation value and denotes it by pointed brackets, even in situations like:

where it is very clear (and where you agree) that it cannot have this meaning:

Therefore the right way is to regard all these as calculational tools, as you emphasized in this particular case. I call them q-expectations to emphasize that they are always calculational tools and not expectations values in the statistical sense, but using any other name (e.g., ''reference values'', as I did very early in the development of the thermal interpretation) would not change anything.

In some instances (often at the end of long calculations) q-expectations may refer to sample means of actual measurements, but in as many other instances, they refer to single actual measurement only. No matter which interpretation of quantum mechanics is used, the interpretation is done only at this very last stage, namely where actual measurement is involved. There, depending on the situation, one must give them a deterministic (currents, densities) or a statistical (event counts) interpretation.

Thus to account for the actual practice of quantum mechanics in all applications, one needs both deterministic measurements and statistical measurements. The thermal interpretation accounts for both, without having to start with fundamental probability.

As I stressed many times before, the problem is not the formalism, which is anyway leading to the correct results, i.e., the QT results. The problem is that you just rename everything and claim, there's no probability involved just to solve some phyilosophical quibble which are no problems for physics anyway. Then the meaning of the formalism as a physical theory (not an empty mathematical game!) gets lost although that's the very purpose of any interpretation.

 

[ftr]

The average taken is not over microscopic degrees of freedom but over a small spacetime region where the measurement is performed

sample means of actual measurements, but in as many other instances, they refer to single actual measurement only.

The problem is that you just rename everything and claim, there's no probability involved just to solve some phyilosophical quibble

I am wondering that the TI sort of takes this view based on some idea that I have thought about before. I will explain but Arnold may correct me of course. There is a class of math branch which is called geometric probability( eventually linked to many transforms), while not exactly Schrodinger EQ. type solution (including the density matrix) but a very close cousin that I think illuminates what TI intends . The the class of problems like "line circle picking" or ball triangle picking, and many others that you can check in the links give the probability density function for these picking processes.

Now the density of the line lengths or triangle areas at different values are similar to QM probability density. So what I think TI is trying to say is That these underlying microstates are of no importance in themselves but the mean average is what is important which we actually measure. In this sense Ti denies the physicality of those "random" events and just attributes them to a field like values, and since fields are controversial physical thing, hence the confusion.

My hopeful solution is that these are indeed have physical interpretation(maybe energy since they represent lengths, areas..I don't know) that keeps the interpretations "probabilistic" nature only in the sense that how many line lengths s1 as compared to s2 and so on can fit the area. AND have the TI interpretation as the expectation(average value) translates to the particle "measured" property. I am summering but I hope you get the drift. Of course, I am far from being any certain about these ideas, but at least I hope I am right about my understanding of TI, however general.

P.S. curiously enough many of the pick problems linked to have density functions which look surprisingly similar to what we see in QM

 

[akhmeteli]

@akhmeteli, I read through the whole thread in one sitting after a few weeks of all of you discussing away, so I've no doubt I missed some of the nuances of what each of you hope to get from it. The lack of threading on PF makes it difficult enough to get into a long conversation that I don't usually bother once there are many posts, but I thought that you might find the Hegerfeldt reference of some interest, given that no-one else responded to your request. The connection of positive frequency with analyticity and locality/nonlocality has always seemed to me less emphasized than it might be, particularly given the analogy it suggests with the perhaps simpler connection of the Hilbert transform with analyticity in signal analysis.
In any case, I didn't notice that you wanted to discuss only "relativistic quantum mechanics, without using quantum field theory", about which I know too little to contribute much, but I have always taken it that problems such as you mention with the Dirac equation as relativistic QM are what pushed physicists towards the Dirac equation as the basis of a QFT.

Of course, nobody can demand that you thoroughly read an entire long thread before you post a reply. And I agree that the Hegerfeldt reference is interesting, but I don't have much to say about it. Hegerfeldt himself offers some plausible interpretations of his results.

 

[vanhees71]

It is the trace of the product of the q-observable $j_0(h):=\int_\Omega h(x)j_0(x)dx$ with the density operator, $\langle j(h)\rangle:=Tr~\rho j(h)$, where $\Omega$ is the region in which the coarse-grained current is observed and $h(x)$ is an appropriate smearing function determined by the sensitivity of the measuring instrument or coarse-graining. This is mathematically well-defined; no mystery and no circularity is present (unless you impose your interpretation in addition to mine). The result can be compared with experiment in a single reading, without any statistics involved, giving an operational definition of the meaning of this q-expectation. The average taken is not over microscopic degrees of freedom but over a small spacetime region where the measurement is performed (needed to turn the distribution-valued operator current into a well-defined q-observable,

To the contrary! The average you describe is precisely over "many microscopic degrees of freedom".

Let's first get the notation straight. If I understand it right, your integral is operator valued, i.e., you define a "smeared charge-density operator":
$$\hat{j}_{\Omega}^0(t,\vec{x}_0)=\int_{\Omega} \mathrm{d}^3 x \hat{j}^0(t,\vec{x}).$$
If you want to use this to evaluate a macroscopic charge density, the $\Omega$ must be a "macroscopically small but microscopically large" finite volume around $\vec{x}_0$.

As important is that the state, $\hat{\rho}$, allows for such a "separation of scales" (usually a save assumption is that $\hat{\rho}$ is close to thermal equilibrium). Then
$$\rho(t,\vec{x}_0)=\mathrm{Tr} [\hat{\rho} \hat{j}_{\Omega}^0(t,\vec{x}_0)]$$
leads to a physically (!) interpretible "macroscopic, (semi-)classical" charge-distribution observable.

It's important for the physicist to have such an intuitive/heuristic picture to derive usefull quantities. It's not enough to relabel the standard notation with highly abstract formalized language, loosing the contact with the experimental physicist in the lab. Interpretation is not about formal reformulations but precisely about this connection between the formalism and real-world experiments.

Take the other example about the redefinition of several base units of the SI. It may look trivial: You just take a bunch of fundamental natural constants like $f_{\text{Cs}}$, $c$, $h$, $N_A$, $k_{\text{B}}$, and define values for them. Formally that's it. With choosing these values and naming the units everything is formally well defined, but that's useless for the physicist, and you have to "realize" the units, based on these definitions in terms of real-world experimental setups. That's what took decades of hard work by the metrologists around the world, not the definition, which is nearly what Planck already has worked out (for technical reasons it's today impossible to do the final step and base everything on Planck units because of the trouble with the determination of Newtons gravitational constant $\gamma$ which could be used to eliminate $t_{\text{Cs}}$ an base everything on fixing fundamental constants rather than to a property of a a certain substance, in this case the atomar finestructure of Cs). It is the realization of the units with the best available technology today to reach the precision necessary to keep the changes to the actual meaning of the units minimal. FAPP there shouldn't be a big change in what 1 kg of mass means. That's why for decades the two approaches to realizing the kg, based on fixing the value of $h$ as in the formal definition, have been pursued by various national metrology institutes (e.g, the PTB with the Avogadro project and NIST with the Kibble balance). The advantage of the redefinition simply is that the fundamental definitions do not change while the ever more precise realization of the units may change with the advance of technology.

 

[A. Neumaier]

To the contrary! The average you describe is precisely over "many microscopic degrees of freedom".

Let's first get the notation straight. If I understand it right, your integral is operator valued, i.e., you define a "smeared charge-density operator":
$$\hat{j}_{\Omega}^0(t_0,\vec{x}_0)=\frac{1}{|\Omega|}\int_{\Omega} dtd^3\vec{x}\,\hat{j}^0(t,\vec{x}).$$
If you want to use this to evaluate a macroscopic charge density, the $\Omega$ must be a "macroscopically small but microscopically large" finite space-time volume around $(t_0,\vec{x}_0)$.

As important is that the state, $\hat{\rho}$, allows for such a "separation of scales" (usually a safe assumption is that $\hat{\rho}$ is close to thermal equilibrium). Then
$$\rho(t,\vec{x}_0)=\mathrm{Tr} [\hat{\rho} \hat{j}_{\Omega}^0(t,\vec{x}_0)]$$
leads to a physically (!) interpretible "macroscopic, (semi-)classical" charge-distribution observable.

I added a time integration to account for the smearing in time necessary for a realistic measurement and the factor $1/|\Omega|$ to make the density correctly normalized. Then this is exactly what I wrote, for the special case of a constant weight function $h(x)$ centered at $(t_0,\vec{x}_0).$

It involves nothing but an average over a macroscopically small but microscopically large region $\Omega$ of space-time, because that's due to the macroscopic nature of any charge measurement device.

There is no average over microscopic degrees of freedom - these microscopic degrees of freedom appear nowhere in the formula. Nevertheless, one has the correct intuitive/heuristic picture!

 

[vanhees71]

Obviously we have a completely different understanding about the meaning of the symbols. Of course, your formal additions, i.e., using averages over a space-time region rather than only space instead of the sum, are not the point. The point is that the said integral in fact averages and the following tracing with $\hat{\rho}$ averages indeed over many microscopic degrees of freedom.

The classical analogue a la Lorentz is to start with a "microscopic" current
$$j_{\text{mic}}^{\mu}(t,\vec{x})= \sum_N \int_{\mathbb{R}} \mathrm{d} \tau_N q_N \frac{\mathrm{d} y_N^{\mu}}{\mathrm{d} \tau_N} \delta^{(4)}(x-y_N(\tau_N))$$
and then average in the very same sense as above (I'm not sure whether Lorentz averaged over small time intervals too, but that doesn't change the principle of the argument. It's "coarse-graining" in the sense of "blurring out" many microscopic details and fluctuations, introducing a scale of resolution via the extent of the spatial (and temporal) volume (four-volume) elements, $\Omega$.

 

[A. Neumaier]

tracing with $\hat{\rho}$ averages indeed over many microscopic degrees of freedom.

No. The trace is a calculational tool only, a purely mathematical operation, namely the sum of the diagonal elements in an arbitrary basis.

Instead of starting with the density operator and the trace, I could as well have defined (as standard in algebraic quantum field theory) $\langle\cdot\rangle$ to be a normalized monotone linear functional - where no trace of a trace exists and the connotation of an average is completely absent. Then one can reconstruct (if desired) the representation in terms of a density operator and the trace by the general machinery of Hilbert space operator theory. But this shows that there is no fundamental averaging going on in the foundations!

You defined Green's functions by $W(x,y):=Tr~ \rho \phi(x)^*\phi(y)$ and though calling them expectation values (which they are not, at least not in the sense of Born's rule), you insisted on that they are only calculational tools. In the same spirit you should allow me to use the trace everywhere in this purely formal meaning as a tool. This does not affect at all the relation to experiment. It just provides everything with a different intuition - a different interpretation of quantum mechanics! In general, to establish a valid relation between the formal tools and reality, only the interpretation of the final predictions in terms of actual measurements counts, not how one calls or thinks of the calculational tools one uses.

The trace is therefore not an average over anything - it is the latter only in your interpretation, and only when the operator is self-adjoint! Your arguments are just a rant against any other interpretation than what you are used to.

The classical analogue a la Lorentz is

... irrelevant for a quantum foundation, since classical physics should be derived from quantum physics and not the other way round.

 

[Mentz114]

No. The trace is a calculational tool only, a purely mathematical operation, namely the sum of the diagonal elements in an arbitrary basis.

Instead of starting with the density operator and the trace, I could as well have defined (as standard in algebraic quantum field theory) $\langle\cdot\rangle$ to be a normalized monotone linear functional - where no trace of a trace exists and the connotation of an average is completely absent. Then one can reconstruct (if desired) the representation in terms of a density operator and the trace by the general machinery of Hilbert space operator theory. But this shows that there is no fundamental averaging going on in the foundations!

You defined Green's functions by $W(x,y):=Tr~ \rho \phi(x)^*\phi(y)$ and though calling them expectation values (which they are not, at least not in the sense of Born's rule), you insisted on that they are only calculational tools. In the same spirit you should allow me to use the trace everywhere in this purely formal meaning as a tool. This does not affect at all the relation to experiment. It just provides everything with a different intuition - a different interpretation of quantum mechanics! In general, to establish a valid relation between the formal tools and reality, only the interpretation of the final predictions in terms of actual measurements counts, not how one calls or thinks of the calculational tools one uses.

The trace is therefore not an average over anything - it is the latter only in your interpretation, and only when the operator is self-adjoint! Your arguments are just a rant against any other interpretation than what you are used to.

... irrelevant for a quantum foundation, since classical physics should be derived from quantum physics and not the other way round.

Averaging is difficult to grasp at the foundational level because it seems that if occupation probabilities are averaged this increases entropy (like replacing numbers $n_i$ with $\sum n_i/N$) but averaging over dynamical variables decreases entropy ( disorder, uncertainty) which throws a spanner in the search for homologies between probability evolution and thermodynamic evolution.

 

[vanhees71]

I think we discuss in circles, because you don't want to understand my point.

On the one level you have a formalism. There you can use any calculational tool you like to get the physical numbers out. That's a technical point.

However, from a paper claiming to have a new interpretation I expect an interpretation not just the mathematical formalism. The interpretation is precisely what's beyond the mathematical formalism making it from a pure game of human thought as, say, "group theory", to a physical theory. The Green's function is defined as a trace, and that's why I write it in terms of the expecation value $\langle \hat{\phi}(x) \hat{\phi}^{\dagger}(y) \rangle$. This doesn't imply at all that it is an observable quantity. It's just a useful tool to calculate an observable quantity, e.g., the single-particle phase-space distribution function after precisely using the coarse-graining formalism we discuss right now to its Wigner transform, and it is very clear that the coarse graining is over "many microscopic degrees of freedom", which motivates the entire formalism to begin with. There are different aims for mathematicians and physicists. The mathematician will be satisfied with clarifying the formal sense of the calculational operations physicists use to get their numbers out, but that's not what you expect if you read a paper about interpretation.

I brought the classical example since usually it's easier to understand, in which sense a formal averaging/coarse-graining procedure is about summing/averaging over a huge number of microscopic degrees of freedom. Obviously this was another failed attempt to bring the argument across :-((.

 

[A. Neumaier]

I think we discuss in circles, because you don't want to understand my point.

No; we discuss in nonintersecting lines, because neither gets his point across. This is a thread about the thermal interpetation, so the rules of the thermal interpretation should be applied, and you should want to understand my point rather than try to make a point based on assuming the statistical interpretation.

from a paper claiming to have a new interpretation I expect an interpretation not just the mathematical formalism. The interpretation is precisely what's beyond the mathematical formalism making it from a pure game of human thought as, say, "group theory", to a physical theory.

I fully agree. But my interpretation is opposite from yours, except at those places where theoretical predictions are compared with event statistics, where they agree.

To interpret a spacetime integral of the theoretical q-expectation $\langle j_0(x)\rangle$ of a density as a single measured value of an instrument with a limited spacetime resolution tells directly and in a very intuitive manner what the theoretical q-expectation means in terms of observability. There is no averaging of actual measurements. And there is no need to talk about any postulated unobservable microscopic averaging, which is nowhere in the mathematical formalism, and nowhere in the calculations.

Similarly, to interpret a field correlation, i.e., a theoretical q-expectation $\langle \phi(x)^*\phi(y)\rangle$ in terms of the integral kernel of an instrument measuring the linear response to an external field perturbation requires a fluctuation-dissipation theorem (a theoretical result on the level of the mathematical formalism) and again tells directly and in a very intuitive manner what the theoretical correlation function means in terms of observability. There is no averaging of actual measurements. And there is no need to talk about any postulated unobservable microscopic averaging, which is nowhere in the mathematical formalism, and nowhere in the calculations.

Similarly, to interpret a spectral density function, i.e., the Fourier transform of another theoretical q-expectation in terms of an instrument measuring the positions and width of spectral lines again tells directly and in a very intuitive manner what the theoretical spectral density function means in terms of observability. There is no averaging of actual measurements. And there is no need to talk about any postulated unobservable microscopic averaging, which is nowhere in the mathematical formalism, and nowhere in the calculations.

The same holds for all other connections to experiment, including those where some statistics is done to get the observations! In the latter case, only the same kind of statistics done in classical measurement theory to go from many highly uncertain actual observations to a much more predictable statistical average. But there is never a need to talk about any postulated unobservable microscopic averaging, which is nowhere in the mathematical formalism, and nowhere in the calculations.

The talk about averaging unobservable stuff has in the thermal interpretation the same dispensable superficial (and often misleading) meaning as the talk about virtual particles explaining how particles mediate interactions between fields. They are an artifact of simplified imagery applied to formal developments in a dubious attempt to make them less abstract and more intuitive. The expert knows in which sense and with how many grains of salt it has to be taken to guide calculations without being led astray, but whoever tries to take the talk at face value is lead into the morass that 93 years of quantum interpretations had not been able to clear up.

 

[A. Neumaier]

it is very clear that the coarse graining is over "many microscopic degrees of freedom"

In the thermal interpretation, the coarse-graining is not done by averaging over many microscopic degrees of freedom (which nowhere figure in my description of coarse-graining in Subsection 4.2 of Part III).

Coarse-graining is done by picking a vector space of relevant q-expectations and restricting the exact dynamics of all q-expectations to an exact or approximate dynamics of the relevant q-expectations, using one of a number of theoretical techniques described purely on the mathematical level. Examples are the Dirac-Frenkel variational principle, the projection operator formalism, or the closed-time-path (CTP) formalism. Where is the averaging over many microscopic degrees of freedom in the CTP formalism as described in the book by Calzetta and Hu? They talk informally about looking at slow variables and ignoring obervationally inaccessible observables (p.20), just as I do in Subsection 4.2 of Part III. When they come to the CTP formalism in Chapter 6, I see q-expectations and integrals over spacetime and frequency modes, but not over microscopic degrees of freedom.

 

[vanhees71]

I have given my arguments why the very formalism of integrating over spatial (or temporal-spatial) volumes (four-volumes) is averaging/summing over many microscopic degrees of freedom. That's the view since the very beginning of statistical physics. In the field-theoretical context it's, as far as I know, first applied by Lorentz within his classical "electron theory". This is also what a macroscopic measurement device does, providing a single "pointer reading" in an experiment. I think that's very clear, and I don't know, how I can reformulate it to bring this point across better.

Obviously it's also hard to give an explanation, what your thermal interpretation is about in physical terms. The math is obviously the same as in standard quantum theory, based on the statistical interpretation. For me the only difference is that you forbid to call the formal procedure of taking the trace over observable operators times the statistical operator "averaging". What's lacking for me to understand this obviously crucial point is, where there is (a) the interpretation (i.e., where is the connection between the formalism and the physical meaning) and (b) in which sense I can understand it as "thermal" (which hinges for me also essentially on the statistical interpretation of the quantum state and the interpretation of the trace formula as averaging over many microscopic degrees of freedom).

 

[A. Neumaier]

I have given my arguments why the very formalism of integrating over spatial (or temporal-spatial) volumes (four-volumes) is averaging/summing over many microscopic degrees of freedom.

Yes, this is the well-known statistical interpretation, and I understand your arguments, so there is no point in repeating these.

But this interpretation is replaced in the thermal interpretation by alternative, philosophically less problematic imagery rooted in the formal mathematical core alone rather than in a reference to microscopic degrees of freedom.

In purely mathematical terms, spatiotemporal smearing is a filtering procedure that suppresses high frequency contributions to the field. This has an immediate and intuitive physical meaning since high frequency contributions in space or time cannot be resolved and thus must be suppressed somehow. One way of doing so is by regularization using spatial and temporal frequency cutoffs. Almost nobody introduces these as averaging, since the direct interpretation is much more intuitive.

General filtering is just a more adaptive linear regularization operation. The filtering procedure relevant for a localized measuring device (e.g., the human ear for measuring pressure oscillations) is a convolution with a function with negligible support outside the active region of the device, the Fourier transform of a function weighting each frequency according to its ability to cause a measurable response. The precise weight function to be used in the convolution is measurement dependent, and can be determined by calibration from measurements of prepared harmonic high frequency signals - audiometry in case of the ear, but nothing is special to the ear.

All this applies either on the operator level or on the level of q-expectations. The thermal interpretation treats the q-expectation $\rho(x)=\langle j_0(x)\rangle$ as an unregularized observable distribution-valued density field, and determines the regularized response of a particular measurement device by calibration, as in the case of the ear. All physical contents can be extracted in this way.

Nothing at all depends on the internal structure of the q-expectations. Their statistical interpretation as means over unmeasured microscopic degrees of freedom can simply be dispensed with, without any loss in understanding. Mathematically (i.e., with more than metaphorical invocation), this statistical interpretation works well anyway only in very simple situations such as very low density gases. Thus, like the particle picture, it is best regarded as an elementary precursor simplifying the real thing.

Obviously it's also hard to give an explanation, what your thermal interpretation is about in physical terms. The math is obviously the same as in standard quantum theory, based on the statistical interpretation.

My interpretation is not at all hard. The math of standard quantum theory is completely independent of the interpretation, not anything special to the statistical interpretation. (It is shared by the Copenhagen interpretation, the many-histories interpretation, and all other fancy variations.)

For me the only difference is that you forbid to call the formal procedure of taking the trace over observable operators times the statistical operator "averaging". What's lacking for me to understand this obviously crucial point is, where there is (a) the interpretation (i.e., where is the connection between the formalism and the physical meaning)

I gave in the last few mails very precise connections between the formalism and the physical meaning, sufficient to interpret everything done in the textbooks and in actual experiments in intuitive terms. I have no idea what you could possibly find missing.

Unless you insist on the identity ''physical = averaging over microscopic degrees of freedom''. But this is just your statistical interpretation of ''physical'', not the God-given meaning of the term!

(b) in which sense I can understand it as "thermal" (which hinges for me also essentially on the statistical interpretation of the quantum state and the interpretation of the trace formula as averaging over many microscopic degrees of freedom).

I had explained ''thermal'' in Section 1 of Part II as follows:

Essential use is made of the fact that everything physicists measure is measured in a thermal environment for which statistical thermodynamics is relevant. This is reflected in the characterizing adjective ''thermal'' for the interpretation.

Specifically, the local values of an extensive field in local equilibrium thermal statistical mechanics (which describes all measurement devices) are q-expectation of the corresponding field operator, in a state that depends on the local temperature. Suitably filtered values (in the sense above) are directly measurable.
Invoking statistical concepts (which unfortunately still figure in the historical names of the concepts) do not add anything that would help in motivating or structuring the mathematical tools or their application to real experiments.

Thus the thermal setting provides a well-grounded point of departure for an interpretation, the thermal interpretation, in the same way as the double-slit experiment and the Stern-Gerlach experiment for few-particle experiments provided in the past a well-grounded point of departure for Born's statistical interpretation.

 

[vanhees71]

So you say yourself what I say from the very beginning:

"Essential use is made of the fact that everything physicists measure is measured in a thermal environment for which statistical thermodynamics is relevant. This is reflected in the characterizing adjective ''thermal'' for the interpretation."

Also renormalization, which you quote above, interpreted in the Wilsonian sense, and that's the sense of the abstract mathematical procedure in physical terms, is precisely about coarse graining through averaging over microscopic degrees of freedom.

It's not as easy to get rid of statistics in the natural sciences. At least you cannot just use the very arguments of statistical thermodynamics and then forbid to use the usual probabilistic language just to hide the statistical foundations. It's a contradictio in adjecto!

 

[DarMM]

Thermal Interpretation Summary Redux

1. Q-Expectations and Q-correlators are physical properties of quantum systems, not predicted averages. This makes these objects highly "rich" in terms of properties for $\langle\phi(t_1)\phi(t_2)\rangle$ is not merely a statistic for the field value, but actually a property itself and so on for higher correlators.
2. To emphasize the point above remember that Probability theory is merely a formal construct whose elements are interpreted based on the context in which they are used. Thus we are not required to view things like $Tr(\rho A)$ as statistical statements fundamentally. Here in the Thermal Interpretation we view them as property assignment. They are merely propeties whose values have the same relation to each other as the formal relations of Probability theory.
3. This richness of physical properties is not compatible with the notion of a system being purely decomposable into its subsystems in all cases. There are often properties such as a correlators like $\langle\phi(t_1)\phi(t_2)\rangle$ that are properties of the total system that don't arise from properties of subsystems.
4. Since some properties are assigned to the system as a whole, which can be quite extended, they provide the nonlocal beables required by Bell's theorem. This is a combination of points above. Consider an extended two photon system. This has correlator properties like $\langle AB\rangle$ that are assigned to the whole system, no matter how extended it is and by the above these properties are not merely a property or combination of properties of any of the subsystems.
   
   Note in the Thermal Interpretation, once we have these nonlocal properties, violation of the CHSH inequalities for example has a simple explanation. There are simply four nonlocal total system properties:
   $$\langle AB\rangle, \langle BC\rangle, \langle CD\rangle, \langle AD\rangle$$
   for whom the following sum:
   $$\langle AB\rangle + \langle BC\rangle + \langle CD\rangle - \langle AD\rangle$$
   is greater than $2$.
5. Properties in the Thermal Interpretation are intrinsically "Fuzzy". For example for a particle its position property $\langle q\rangle$ has an associated property $\Delta q$, in orthodox terminology called the uncertainty, that indicates the delocalised nature of the particle's position. Rather than the particle being located at points along a world line $q(t)$ it in fact constitutes a world tube centered on $\langle q\rangle_t$ and with width $\Delta q_t$. (Question for @A. Neumaier , do higher statistics like skew and kurtosis give one further structure of the world tube?)
   
   All properties are blurred/fuzzy like this, not just position. Note in particular that spin for example is not discrete at the fundamental ontological level since $\langle S_z\rangle$ may take a continuous range of values. And thus we have a "spin tube" of width $\Delta S_z$.
   
   For this reason there is a more basic notion of uncertainty to quantum properties akin to asking "What is the position of Paris". Since Paris is an extended object there is no precise answer to this question.
6. Nonetheless as per standard scientific practice even this basic notion of uncertainty may be treated statistically.
7. In measurements our devices (for reasons given in the next point) unfortunately only become correlated with a single point within the world tube or blurred range of a property. This gives measurements on quantum systems discrete results that don't faithfully represent the "tubes". Thus we must reconstruct them from multiple observations.
8. The origin of this discreteness is in the metastability of the system-device-environment interaction. The device as a physical system can be partitioned into slow large scale and fast small scale modes. The space of slow large scale modes has the form of a set of disconnected manifolds. After interaction with the system the total state of the device initially faithfully records the uncertain quantity $\langle A\rangle$ however it is metastable. Noise from the environment causes it to quickly decay into one of the slow mode manifolds giving a discrete outcome not fully reflective of $\langle A\rangle$.
   From our perspective, ignorant of the environmental dynamics, this constitutes a stochastically driven discrete outcome of a measurement.

 

[A. Neumaier]

So you say yourself what I say from the very beginning:

"Essential use is made of the fact that everything physicists measure is measured in a thermal environment for which statistical thermodynamics is relevant. This is reflected in the characterizing adjective ''thermal'' for the interpretation."

Also renormalization, which you quote above, interpreted in the Wilsonian sense, and that's the sense of the abstract mathematical procedure in physical terms, is precisely about coarse graining through averaging over microscopic degrees of freedom.

It's not as easy to get rid of statistics in the natural sciences. At least you cannot just use the very arguments of statistical thermodynamics and then forbid to use the usual probabilistic language just to hide the statistical foundations. It's a contradictio in adjecto!

We talk of a vector space even if its elements are not vectors in the sense of little arrows or n-tuples. Similarly I refer to statistical thermodynamics for the results of the traditional theory without interpreting the items used there as mean - i.e., statistical mechanics as treated in Part II of my online book. You are allowed to use the old interpretation but then you are in the statistical interpretation and not in the thermal interpretation.

 

[vanhees71]

Well, you always tell me how you don't interpret the "symbolism of atomic measurements" (Schwinger). Paradoxically, although claiming to provide a new interpretation, you never tell us its content. It's not enough to write down the formalism and forbid to use the 93 years old accepted probabilistic interpretation without providing a new one. Is @DarmMM 's posting #268 provides the correct interpretation of your interpretation, it's the more confirmed that it is still the usual probabilistic one too.

 

[A. Neumaier]

Paradoxically, although claiming to provide a new interpretation, you never tell us its content. It's not enough to write down the formalism and forbid to use the 93 years old accepted probabilistic interpretation without providing a new one. If @DarMM 's posting #268 provides the correct interpretation of your interpretation, it's the more confirmed that it is still the usual probabilistic one too.

You never listened to the content descriptions I gave (last in #263 and #266).

Of course I don't forbid anything; so you can always add probabilistic interpretations. DarMM's summary is essentially correct; I'll point to small corrections later. It never assumes anything probabilistic; thus one doesn't need it.

Unlike you, I allow people to dismiss statistical connotations in all situations where no actual measurement results are averaged over - without loss of physics but with a resulting improvement of the foundations. No irreducible probability, no dependence of the foundations on measurement, the same clarity in the association of predictions and experimental results. But the traditional foundational problems are gone!

You, of course, never saw any foundational problems in the statistical interpretation. This is because you are far more liberal in the use of intuition than strict adherence to the axioms in any statistical foundations would allow. At this level of liberality in the discussion we agree. But many others (e.g., @Stevendaryl, or Steven Weinberg) were never satisfied with such liberal (and hence poorly defined) foundations. The thermal interpretation is for those.

Our whole discussion shows that I didn't miss anything in the thermal interpretation, and really covered the actual physical usage of the quantum formalism.

 

[A. Neumaier]

Thermal Interpretation Summary Redux

Excellent summary, thanks!

7. In measurements our devices (for reasons given in the next point) unfortunately only become correlated with a single point within the world tube or blurred range of a property. This gives measurements on quantum systems discrete results that don't faithfully represent the "tubes". Thus we must reconstruct them from multiple observations.

The ''single point'' here is of course also a blurred point only, as a detector always responds to a small space-time region and not to a point only. The measurement result will often be discrete, however, due to the reasons given in point 8.

It is like measuring the rate of water flow into a basin by the number of buckets per unit time needed to keep the water at a roughly fixed level of height. As long as there is enough flow the bucket is very busy and the flow is measured fairly accurately. But at very low rates it is enough to occasionally take out one bucket full of water and the bucket number is a poor approximation of the flow rate unless one takes very long times.

Photocells and Geiger counters act as quantum buckets, by the same principles.

 

[A. Neumaier]

Question for @A. Neumaier , do higher statistics like skew and kurtosis give one further structure of the world tube?

The world tube is just a visualization, even a fuzzy one since it cannot be said how many sigmas the world tube is wide. I don't think skewness and kurtosis are valuable on this fuzzy visualization level.

Rather, the next level of naturally visualizable detail would be to think of the particle's world tube as being actually a moving cloud of mass (or energy, or charge, etc.), with mass density $\rho(z)=\langle m\delta(x_k-z)\rangle$ for a distinguishable particle $k$ inside a multiparticle system. For a single particle state, this would be $\rho(z)=m|\psi(z)|^2$, recovering Schrödinger's intuition about the meaning of the wave function. Of course, this is already the field picture, just in first quantized form.

Mathematically, of course, there is no world tube at all but just a host of loosely related beables. One family of them is the reference path $\langle x(t)\rangle$. It has a clear intuitive meaning (as the path one sees at low resolution), just as the reference path of a classical donut traced out by the center of mass, though in both cases nothing is precisely at the reference path.

 

[ftr]

For a single particle state, this would be ρ(z)=m|ψ(z)|2ρ(z)=m|ψ(z)|2\rho(z)=m|\psi(z)|^2, recovering Schrödinger's intuition about the meaning of the wave function. Of course, this is already the field picture, just in first quantized form

I most definitely agree with this picture based on the example I gave in post #255. The only thing is that your system does not tell exactly what is going on in detail in that small region, my idea supplements and give the interpretation of the "line" picking distances as energy which upon calculation adds to mass according to good old Einstein's formula.

P.S. I think you were right Arnold. The wavefunction is nothing, only density has a physical reality.

 

[A. Neumaier]

does not tell exactly what is going on in detail in that small region

Nothing exact can be said. Differences of the order of the uncertainty are physically meaningless.

 

[ftr]

Nothing exact can be said. Differences of the order of the uncertainty are physically meaningless.

So what do you think of this
https://www.newscientist.com/articl...en-atom-youre-on-quantum-camera/#.UeSbfW2i18E

 

[A. Neumaier]

So what do you think of this
https://www.newscientist.com/articl...en-atom-youre-on-quantum-camera/#.UeSbfW2i18E

They say, ''Rather than taking an image of a single atom, they sampled a bunch of atoms. '' Thus they are explicitly averaging over many atoms. It is like creating an average picture of a human face...

 

[vanhees71]

Unlike you, I allow people to dismiss statistical connotations in all situations where no actual measurement results are averaged over - without loss of physics but with a resulting improvement of the foundations. No irreducible probability, no dependence of the foundations on measurement, the same clarity in the association of predictions and experimental results. But the traditional foundational problems are gone!

Again, just to understand, because I think this is the main source of our mutual misunderstanding:

Let's take my example for the measurement of a current with a galvanometer. From a quantum-theoretical point of view electric current consists of the motion of many conduction electrons in a wire at room temperature.

My point of view is that at a microscopic level you have a many-electron system, and the current density is described by the operator
$$\hat{j}^{\mu}(x)=-e \bar{\psi}(x) \gamma^{\mu} \psi(x).$$
What's measured by the galvanometer is the expectation value
$$J^{\mu}(x) = \mathrm{Tr} (\hat{\rho} \hat{j}(x)),$$
where $\hat{\rho}$ is a statistical operator at (or close) to thermal equilibrium, i.e., something like $\propto \exp[-(\hat{H}-\mu \hat{Q})/T]$.
Now the galvanometer provides the current,
$$I=\int \mathrm{d} \vec{f} \cdot \vec{J}$$
right away as a pointer reading, because it is providing the above formally noted expectation value due to precisely resolving the measured quantity at this level of resolution.

This is within the standard statistical interpretation of the formalism. Now, how does your thermal interpretation explain this situation, if it denies the validity of the statistical arguments.

 

[DarMM]

Again, just to understand, because I think this is the main source of our mutual misunderstanding:

Let's take my example for the measurement of a current with a galvanometer. From a quantum-theoretical point of view electric current consists of the motion of many conduction electrons in a wire at room temperature.

My point of view is that at a microscopic level you have a many-electron system, and the current density is described by the operator
$$\hat{j}^{\mu}(x)=-e \bar{\psi}(x) \gamma^{\mu} \psi(x).$$
What's measured by the galvanometer is the expectation value
$$J^{\mu}(x) = \mathrm{Tr} (\hat{\rho} \hat{j}(x)),$$
where $\hat{\rho}$ is a statistical operator at (or close) to thermal equilibrium, i.e., something like $\propto \exp[-(\hat{H}-\mu \hat{Q})/T]$.
Now the galvanometer provides the current,
$$I=\int \mathrm{d} \vec{f} \cdot \vec{J}$$
right away as a pointer reading, because it is providing the above formally noted expectation value due to precisely resolving the measured quantity at this level of resolution.

This is within the standard statistical interpretation of the formalism. Now, how does your thermal interpretation explain this situation, if it denies the validity of the statistical arguments.

In the thermal interpretation $J^{\mu}(x)$ (or a smeared version thereof) is a quantity itself not the average outcome for an indeterminate quantity.

 

[vanhees71]

Hm, well, that's just the usual sloppy language of some textbooks on QM to identify the expectation values with the actually observed quantities. I think there's much more behind Arnold's interpretation, which however I still miss to understand :-(.