I The thermal interpretation of quantum physics

[akhmeteli]

@A. Neumaier: "Points 4 and 5 also show that at finite times (i.e., outside its use to interpret asymptotic S-matrix elements), Born’s rule cannot be strictly true in relativistic quantum field theory, and hence not in nature."

So it seems that you fault the Born's rule for what is actually the Schrödinger equation's fault. The Born's rule is no relative of mine, but this doesn't look like a strong point of your critique of the Born's rule. Maybe you should use the results of Allahverdyan et al., who show that the Born's rule is only approximately correct?

 

[Demystifier]

@A. Neumaier , I have finished the papers and I am now beginning my closer second read through.

Good, I hope you will present us an unbiased summary with advantages and disadvantges in comparison with other interpretations.

 

[A. Neumaier]

Good, I hope you will present us an unbiased summary with advantages and disadvantges in comparison with other interpretations.

My obviously biased comparison is in Section 5 of Part III.

 

[Demystifier]

My obviously biased comparison is in Section 5 of Part III.

A typo: At page 51 (part III) you have the item
"•has no split between classical and quantum mechanics – the former emerges naturally as the macroscopic limit of the latter;"
twice.

And by the way, your thermal interpretation has nothing to do with temperature, am I right?

 

[vanhees71]

Yes, but the expectation can still be defined via the trace. That's why the probability interpretation is secondary. In the thermal interpretation it is not assumed but (under additional assumptions) derived.

But expectation values are defined via the probabilities and vice versa. The probablities are also special expectation values like, e.g., for the position of a particle
$$P(\vec{x})=\langle \delta(\vec{x}-\vec{X}) \rangle.$$
To evaluate this expectation value from the statistical operator within the standard mathematical QT formalism you need the spectral theorem and the "generalized eigenvectors"
$$P(\vec{x})=\mathrm{Tr} (\hat{\rho} \delta(\vec{x} -\hat{\vec{x}}))=\rho(\vec{x},\vec{x}) \quad \text{with} \quad \rho(\vec{x},\vec{x}')=\langle \vec{x} |\hat{\rho}|\vec{x}' \rangle.$$

It is needed in rigorous interactive relativistic QFT precisely because of Haag's theorem. One cannot use a Fock space (which is separable) but needs a nonseparable Hilbert space with superselection rules for unitarily inequivalent representations of the field algebra. I recommend studying the massless free scalar field in 1+1 dimensions, which is an exactly solvable toy example where the problem of superselection rules appears already in the free case.

So far, I was very pragmatic with regard to Haag's theorem. I just introduced a quantization volume to regularize the mathematically meaningless abusive treatment of distributions and then take the limit at the appropriate point in the calculation. Physically there's anyway never an infinite space available since so far even the LHC is a device occupying lab space of finite volume ;-)).

The formal treatment of exact toy models of interacting fields in lower space-time dimensions is very interesting. I know the Schwinger model (massless scalar QED in (1+1) dimensions), but only in the usual nonrigorous physicists' treatment and without the use of superselection rules. Do you have a reference for some interacting model like that, where this is done rigorously?

The definition of q-expectation is given by eq. (14) of Part II, and involves neither probabilities nor eigenvalues nor the spectral theorem. It works even for nonhermitian operators such as creation and annihilation operators, and is meaningful there. An operational meaning has to be given only for those q-expectations which actually correspond in a simple way to measurements (which is a small number of all possible ones). The operational meaning is therefore dependent on what you measure and how you do it. It is based on the general uncertainty principle (GUP) from page 15 of Part II, and made more concrete by the measurement principle (MP) on p.6 of Part III. For equilibrium thermodynamics, we may use the principle introduced by H.B. Callen in his famous textbook ''Thermodynamics and an introduction to thermostatistics'',

But (II.14) is the standard definition with the usual trace. I still don't see, how you even evaluate this for concrete observables without the spectral theorem, let alone how you understand its application to real measurements in a lab. As I said, I cannot understand the interpretation, before this is clearly stated. It's not necessary to state it in general, concrete (preferrably the most simple) examples are enough. I'd try to mathematically and operationally to define how you interpret Stern-Gerlach experiments (as I summarized in my previous posting) as an example for non-relativistic QT for the most simple case of an (even idealized) von Neumann filter measurement

This implies that q-expectations of 1-particle operators and energy in QFT are measured by the established techniques of equilibrium thermodynamics. This is fully operational, using single measurements only, without any need to check whether the operator is self-adjoint (which is difficult to verify), and without ever having mentioned probabilities. Callen formulated the principle for thermodynamics, but its analogue is valid everywhere in scientific modeling:

• Callen's principle: Operationally, a system is in a given state if its properties are consistently described by the theory for this state.

This together with (GUP) and (MP) is enough to find out in each single case how to measure a q-expectation.

I think, the key of my trouble is that I don't understand, how to decide Callen's principle without the usual operational definitions of "properties". For a thermal system (i.e., a system in equilibrium) it's clear how to define the thermodynamical properties operationally, i.e., how to measure temperature, chemical potential(s), pressure etc. observables, using the adequate devices to do so. However, you don't give an operational definition. You give the mathematics of expectation values in terms of the state-of-the-art version of Born's rule (using the trace formula quoted above), but at the same time claim that you have a deterministic interpretation of this procedure by introducing some Lie algebra acting on an algebra built by functions on these expectation values (in analogy to the symplectic structure of classical phase space with the Poisson bracket defining Lie derivatives). This doesn't provide an operational definition of the averages though. In the standard interpretation, applied to thermal equilibrated systems, there's a clear definition: You use a thermometer in thermal contact with the measured system in equilibrium with the system, providing a reading of the temperature like the height of a Hg column in an old-fashioned thermometer. How is this simple example to be understood within your thermal interpretation?

q-probabilities are special cases of q-expectations, namely those of self-adjoint Hermitian operators with spectrum in $[0.1]$. The weak law of large numbers, discussed in Subsection 3.3 of Part II applies to probabilities and gives operational recipes for measuring them in certain cases, see Subsection 3.5 of Part II; in particular, Born's probabilistic rule follows for ideal binary measurements; see Subsection 3.4 of Part II.
Thus, in the thermal interpretation, Born's rule is not assumed but derived (where appropriate)!

As I said above, this is the very point I misunderstood in reading your papers. For me you use Born's rule to define the entire formalism and then claim you rederive it from these definitions. For me that's circular. Otherwise your thermal interpretation, is completely in accordance with how the QT formalism is used in physicists' lab practice.

See Subsection 2.5 of Part III for the ease with which POVMs arise in the thermal interpretation. Instead, in an approach based on statistical mechanics lecture notes (where POVM do not even appear, though they account for many more measurements than the ideal ones you formalize with your version of Born's rule!) one would need for the justification of POVMs as measurements a very unnatural exptension of the physical Hilbert space by adding a fictitious ancilla degrees of freedom.

Maybe I'll discuss this in Part IV.

The POVM is not in contradiction with the standard fundamentals of Born's rule, but just an extension of the description of measurement protocols generalizing the special case of idealized von Neumann measurements which are indeed only rare cases, applicable to very simple few-particle systems. It's of course of some merit to give a more natural derivation of this extension than the usual one (as e.g., given in Asher Peres's nice textbook).

For how the thermal interpretation views coarse-grained descriptions see Section 4.2 of Part III. One takes a subspace of q-expectations - in case of the Kadanoff-Baym equations the subspace of field expectations and pair correlation functions - and derives an approximate closed dynamics for these.
The approximation is in the equations of motion, not in the meaning of the expectations! Nowhere in the derivation of the Kadanoff-Baym equations is a need to interpret the q-expectations as ensemble averages. [Over which ensemble?? People successfully apply the equations to the early universe, of which there is only one observable case, not an ensemble; the ensembles are pure imaginations without any operational content, as discussed in Subsection 2.4 of Part II!]

Your notion of "ensemble" is too narrow. Particularly for this case, it's not an abstract ensemble a la Gibbs. The Kadanoff-Baym equations are exact equations derived from the 2PI formalism. There are no approximations so far. Of course, it's impossible to solve them (even to state them) without further approximations. One is to use the gradient expansion, which is a coarse-graining procedure leading to semiclassical transport equations (in the most simple case to the standard BUU equations). The "averaging" is here done quite implicitly using this coarse-graining in terms of the gradient expansion, but nowhere is the idea of a Gibbs ensemble envoked. You just interpret the coarse-grained Wigner function as a classical phase-space distribution function, which is consistent as long as the coarse graining is coarse enough to guarantee the positive semidefiniteness of the so defined phase-space distribution functions.

Neither do the derivations (see, e.g., the slides of your 2002 lecture) invoke any observations or any average over fluctuations. Instead just a gradient expansion! Everything you do in your slides is shut-up-and-calculate - no probabilities, no observations, no measurements, no operational recipes for observing the bilocal field correlations. You just worked out approximate formulas for the dynamics of some q-expectations, without ever entering the interpretation of these q-expectations! And you end up with something where the q-expectations have a semiclassical, nonstatistical interpretation - as the thermal interpretation requires, without invoking any of the ghosts of the past that you invoked!

In this talk, there's no gradient expansion. It's a formal thing how to renormalized self-consistent 2PI approximations in thermal equilibrium. Of course the observable quantities are given as the usual QFT expectation values of the corresponding correlation functions (e.g., the em. current-current correlation functions to calculate dilepton-production rates in heavy-ion collisions assuming (local) thermal equilibrium of the source (QGP, hadron-resonance gas) or the electric conductivity via Green-Kubo). Of course, I use the clearly defined notion of expectation values, which is the trace formula you simply don't call Born's rule. As I said, I still did not understand what the averages are within your thermal interpretation if there is no alternative definition to "the ghosts of the past".

I describe indeed usual q-expectations, except that I don't interpret them probabilistically as averages, since this is unnecessary baggage that caused nearly a century of unsettled dispute. This makes all the difference:

Probability is not an irreducible input to quantum mechanics but a consequence of not being able to make observations at arbitrary scales: one needs to introduce approximations, and in many (but not all) cases, these approximation introduce stochastic aspects. Just as the classical dynamics of a bistable system coupled to a bath becomes in some approximation a binary stochastic hopping process. That this analogy is not far-fetched is discussed in Section 5.1 of Part III.

You always say what the q-expectations are not. You have to say what they are, if it's forbidden to interpret them probabilistically. If there's a deterministic theory, I don't see, where I need expectation values to begin with. Of course, in practice, it's impossible to describe even in a classical picture the entire microscopic details of a real-world condensed-matter system, but within classical physics there's no need for expectation values at all in principle. That's not the case in the standard interpretation of QT, where the whole meaning of the formalism is based on probability theory.

It's of course clear that the famous Feynman-Vernon treatment of an open quantum system is also a good example, but also there they use standard probabilistic interpretations of the quantum state.

 

[A. Neumaier]

But expectation values are defined via the probabilities and vice versa.

Only in the minimal statistical interpretation. In the thermal interpretation, q-expectations are defined by $\langle A\rangle := Tr~\rho A$ for arbitrary operators $A$. Nowhere a claim (or even a hint) that this should be an average, except in the unfortunate name tradition gives to this object. The equivalence you talk about is not present, since $A$ may not even have a spectral resolution. As long as you do not realize this distinction, there is no hope to understand the thermal interpretation.

To evaluate this expectation value from the statistical operator within the standard mathematical QT formalism you need the spectral theorem and the "generalized eigenvectors"

This is not needed in the thermal interpretation. For example, if $\rho=\psi\psi^*$ is a pure state and $A=p^2-q^6$ we can calculate the q-expectation $\langle A\rangle = Tr~\rho A=\psi^* (p^2-q^6)\psi=\| p\psi\|^2-\|q^3\psi\|^2$ and evaluate this explicitly for every continuously differentiable wave function $\psi(x)$ decaying fast enough at infinity. The spectral theorem nowhere figures; in fact, you'd have a hard time to figure out whether $A$ is self-adjoint (probably it isn't).

You always say what the q-expectations are not. You have to say what they are, if it's forbidden to interpret them probabilistically. If there's a deterministic theory, I don't see, where I need expectation values to begin with.

I defined explicitly what a q-expectation is: a formal concept in the mathematical part of quantum mechanics, just like a phase space function is a formal concept in the mathematical part of classical mechanics. In both cases, these are claimed to be beables of the theory. In classical mechanics you don't give an operational definition of the concept of a phase space function - only a mathematical one. Why do you demand more in the quantum case?

If there's a deterministic theory, I don't see, where I need expectation values to begin with.

You need q-expectations (not statistical expectation values) since they are the beables of quantum physics, in the same way as you need phase space functions, since they are the beables of classical physics. In both cases, there is a deterministic dynamics for them, given in terms of a Lie algebra on the beables.

But (II.14) is the standard definition with the usual trace. I still don't see, how you even evaluate this for concrete observables without the spectral theorem, let alone how you understand its application to real measurements in a lab

I just demonstrated how to evaluate some without the spectral theorem. For some particular cases, I demonstrated how to understand their application in the lab in my previous posting #21: If $\rho$ is a grand canonical state, the expectation of any conserved quantity $A$ is an extensive thermodynamic variable and can be determined deterministically by the standard thermodynamic methods (which I don't have to repeat here); if you want to see details, read the chapter on equilibrium thermodynamics in my online book. For microscopic projection states, I referred to the discussion in Subsection 2.5 of Part III, and for probabilities I referred to the discussion in Subsection 3.5 of Part II. I find this very clear and don't know how to spell things out more clearly.

For me you use Born's rule to define the entire formalism and then claim you rederive it from these definitions. For me that's circular. Otherwise your thermal interpretation, is completely in accordance with how the QT formalism is used in physicists' lab practice.

Can we agree to call the definition $\langle A\rangle := Tr~\rho A$ the formal Born rule? Clearly, the formal Born rule is a purely mathematical statement, only introducing a symbolic abbreviation for the right hand side. Thus - unlike Born's rule, which refers to measurement - it has no interpretational content at all. That's why I cannot call it Born's rule - the latter establishes some idealized (but claimed to be universal) relation between the mathematical formalism and reality (aka measurement practice), while the formal Born rule doesn't.

What I claim is that I derived - in the contexts where it is valid - the statistical interpretation of the formal Born rule (and hence the actual Born rule) from the (GUP) of Section 2.5 of Part II and the (MP) of Section 2.1 of Part II, which are the interpretational assumptions made by the thermal interpretation.

On the other hand, you assume the statistical interpretation of the formal Born rule as a general property of measurements, and later have to correct for the idealization by admitting that there are other measurements governed instead by POVMs.

The formal treatment of exact toy models of interacting fields in lower space-time dimensions is very interesting. I know the Schwinger model (massless scalar QED in (1+1) dimensions), but only in the usual nonrigorous physicists' treatment and without the use of superselection rules. Do you have a reference for some interacting model like that, where this is done rigorously?

Please first understand the free massless field in 1+1 dimensions; e.g.,

• Dereziński, Jan, and Krzysztof A. Meissner. "Quantum massless field in 1+ 1 dimensions." Mathematical physics of quantum mechanics. Springer, Berlin, Heidelberg, 2006. 107-127.

which has a simple and fairly elegant exact description. Interacting exactly solvable fields are much more technical, for the Schwinger model, see, e.g.,

• Morchio, Giovanni, Dario Pierotti, and Franco Strocchi. "http://preprints.sissa.it/xmlui/bitstream/handle/1963/626/30_88.pdf?sequence=1&isAllowed=y." Annals of Physics 188, no. 2 (1988): 217-238.

and for more general exactly solvable models in 1+1D, the book

• Abdalla, Elcio, M. Cristina B. Abdalla, and Klaus D. Rothe. Non-perturbative methods in 2 dimensional quantum field theory. 1991.

I think, the key of my trouble is that I don't understand, how to decide Callen's principle without the usual operational definitions of "properties". For a thermal system (i.e., a system in equilibrium) it's clear how to define the thermodynamical properties operationally, i.e., how to measure temperature, chemical potential(s), pressure etc. observables, using the adequate devices to do so.

Callen's point (if you read the context of his book) is that these operational definitions are all based on the theory - for example, to measure temperature with a gas thermometer you already need to know what an ideal gas is and that the gas you use behaves like this. Thus once a theory is fully mature (as thermodynamics is) you can start with only the theory and some guesses about how to relate it to practice, and you can check whether your guesses are correct by checking whether the predicted consequences of the theory actually hold. See this toy situation for how this can work without being circular.

However, you don't give an operational definition.

I gave operational definitions for several special cases; why do you require more? In your statistical physics course you also give only very simple operational examples. For example, you claim that self-adjoint operators are observable, but you don't give an operational definition of how to measure the operator $qpqpq$ or $p+q$.

The Kadanoff-Baym equations are exact equations derived from the 2PI formalism. There are no approximations so far.

The approximations arise once you approximate the infinite series of contributions from all loop orders by the first few. But my point was that you are always using the formal Born rule; you never check that the arguments are actually self-adjoint; in fact they are not even Hermitian! Thus you only use the formal Born rule, which has no a priori relation at all to observation...

... unless you give it an interpretation. You give it the statistical interpretation, I give it the thermal interpretation, which are different in character.

But your statistical interpretation is inappropriate as you use q-expectations of nonhermitian operators and as you never consider any observation that would justify the statistical interpretation!

The "averaging" is here done quite implicitly using this coarse-graining in terms of the gradient expansion, but nowhere is the idea of a Gibbs ensemble envoked.

The Gibbs ensemble is directly encoded in your postulates, which say that the expectation of any operator (and hence in particular every two-point function $W(x,y)=\langle\phi(x)\phi(y)\rangle$ is to be interpreted as an average over repeated preparations of the system under consideration, i.e., a significant space-time region containing the collision center of the CERN accelerator, say. Thus:

• You need a large ensemble of many CERN accelerators, of which only one is realized!
• This average is undefined in terms of your postulates! (Indeed, $\phi(x)\phi(y)$ is not self-adjoint, not even in a finite lattice regularization.)
• The operational meaning of $W(x,y)$ is quite different and very indirect only! (This can be seen by considering the use made of 2-point functions in interpreting actual experiments.)

Thus the fairy story you tell fails on three different accounts to be even a meaningful proxy to what really happens.

 

[A. Neumaier]

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.

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.

@A. Neumaier:
So it seems that you fault the Born's rule for what is actually the Schrödinger equation's fault. [...] this doesn't look like a strong point of your critique of the Born's rule.

The Schrödinger equation just says that the Hamiltonian is the infinitesimal generator of time translations, and hence is always valid - even for a system described by the Dirac equation, where $H=cp_0$.

If someone finds fault with the Schrödinger equation then the whole of quantum mechanics breaks down completely!

 

[vanhees71]

Only in the minimal statistical interpretation. In the thermal interpretation, q-expectations are defined by $\langle A\rangle := Tr~\rho A$ for arbitrary operators $A$. Nowhere a claim (or even a hint) that this should be an average, except in the unfortunate name tradition gives to this object. The equivalence you talk about is not present, since $A$ may not even have a spectral resolution. As long as you do not realize this distinction, there is no hope to understand the thermal interpretation.


This is not needed in the thermal interpretation. For example, if $\rho=\psi\psi^*$ is a pure state and $A=p^2-q^6$ we can calculate the q-expectation $\langle A\rangle = Tr~\rho A=\psi^* (p^2-q^6)\psi=\| p\psi\|^2-\|q^3\psi\|^2$ and evaluate this explicitly for every continuously differentiable wave function $\psi(x)$ decaying fast enough at infinity. The spectral theorem nowhere figures; in fact, you'd have a hard time to figure out whether $A$ is self-adjoint (probably it isn't).

Ok, that's the problem. You do not tell me, how to calculate this expectation value. If I'm not allowed to use the spectral theorem, I cannot even write down what the abstract definitions mean. E.g., take a Gaussian wave packet of the traditional formalism. How do you define that without using the position or momentum basis. Then it's easy to calculate this expectation value (even without caring, whether it's self-adjoint or not). You define the state ket via your favorite representation, e.g., in the position representation
$$\psi(x)=N \exp(-x^2/(4 \Delta^2)+\mathrm{i} p x)$$
and then calculate your trace with the corresponding stat. op., leading to
$$\langle f(x,p) \rangle=\int_{\mathbb{R}} \mathrm{d} x \psi^*(x) f(x,-\mathrm{i} \partial_x) \psi(x).$$
Then you have a concrete way to calculate this expectation value. Now you forbid to use the spectral decomposition of the involved operators. So how is within the thermal representation this expectation value evaluated?

 

[A. Neumaier]

You do not tell me, how to calculate this expectation value.

I did it for the case $f(q,p)=p^2-q^6$, but you didn't take it up. So let me give a few more details: By the definition of $p$ and $q$ in the position representation and the rules for multiplying operators, you get $p\psi(x)$ by differentiation and $q^3\psi(x)$ by multiplication with $x^3$. Given a Gaussian wave package $\psi(x)=N \exp(-x^2/(4 \Delta^2)+\mathrm{i} p x)$ as you suggest, you can calculate the two norm squares in my previous post using integration by parts, and you get an explicit, exact value for the q-expectation!

This is an elementary exercise that already good high school pupils can do.

On the other hand, the spectral theorem for unbounded self-adjoint operators is heavy machinery (in Vienna 3rd year math bachelor) that takes a lot of math to prove. Thus it should not figure in a first introduction to quantum mechanics.

Essentially the same works for any expression in $p,q$ involving only sums of products of powers of $p,q$ and more generally for any expression $A$ that is polynomial in $p$ and often enough differentiable in $q$. In the latter case, to evaluate $A\psi$, one uses the product rule to move all $p$s to the right of all $q$s, and ends up with a sum of terms of the form $f_i(q)p^n\psi$ which can be evaluated directly in the position representation and then integrated numerically after multiplication with $\psi^*$. If $A$ is nonpolynomial in $p$ one can use limits, power series, etc...
Only the definition of the operators $p$ and $q$ is needed for all this, no spectral theorem!

 

[vanhees71]

I gave operational definitions for several special cases; why do you require more? In your statistical physics course you also give only very simple operational examples. For example, you claim that self-adjoint operators are observable, but you don't give an operational definition of how to measure the operator $qpqpq$ or $p+q$.

My notes describe the established theory in the shutup-and-calculate interpretation. I make no claim, I'd have a new interpretation which solves all (apparent philosophical) problems of this standard interpretation. I also assume that the reader is familiar with QM at the level of the QM 1 lecture. Perhaps at this time, I was not as careful to formulate it, but I never claim that an observable is a self-adjoint operator on Hilbert space. Observables are defined operationally by giving a measurement procedure. The position observable is defined by a reference frame implying rulers to measure distances from a given point along three perpendicular directions in space.

In QM these observables are represented by self-adjoint operators on Hilbert space, and the states have a probabilistic meaning, relating the formalism to the operationally defined observables in the lab.

Of course, you always need a theory to define the observables, and particularly Callen's book is very clear about these foundations.

As I understood only now after the discussion here, in your thermal interpretation it's not even allowed to use the spectral theorem. Then for me the formulae have no more relation to anything observable in the sense of physics. This is also formally clear since the statistical operator is even picture dependent. For itself it cannot have a physical meaning. You also need the (also picture dependent) observable operators (forming the observable algebra) and the spectral theorem to make physical sense to the formalism.

If you want an alternative physical interpretation of QT, you have to make this connection between observables and the formalism representing these observables in the mathematical constructions clear. As it stands $\mathrm{Tr} \hat \rho \hat{A}$ is not even defined in a way that I can calculate anything with it. Of course, it's my mistake to think that your text associates the usual physical (!) meaning of these symbols, but how else can I understand the text, if I don't make this assumption, if the text doesn't give a clear definition of what's meant by the symbols? This should at least be done for the most simple examples.

I also don't need to think about POVMs if I haven't even understood the most simple case of von Neumann measurements first! If I haven't even understood the most simple textbook case within the new interpretation, I don't dare to hope to understand an even more general and even more abstract concept!

The approximations arise once you approximate the infinite series of contributions from all loop orders by the first few. But my point was that you are always using the formal Born rule; you never check that the arguments are actually self-adjoint; in fact they are not even Hermitian! Thus you only use the formal Born rule, which has no a priori relation at all to observation...

Since when are position and momentum operators within non-relatvistic first-quantized QT not self-adjoint?

... unless you give it an interpretation. You give it the statistical interpretation, I give it the thermal interpretation, which are different in character.

But your statistical interpretation is inappropriate as you use q-expectations of nonhermitian operators and as you never consider any observation that would justify the statistical interpretation!


The Gibbs ensemble is directly encoded in your postulates, which say that the expectation of any operator (and hence in particular every two-point function $W(x,y)=\langle\phi(x)\phi(y)\rangle$ is to be interpreted as an average over repeated preparations of the system under consideration, i.e., a significant space-time region containing the collision center of the CERN accelerator, say. Thus:

• You need a large ensemble of many CERN accelerators, of which only one is realized!
• This average is undefined in terms of your postulates! (Indeed, $\phi(x)\phi(y)$ is not self-adjoint, not even in a finite lattice regularization.)
• The operational meaning of $W(x,y)$ is quite different and very indirect only! (This can be seen by considering the use made of 2-point functions in interpreting actual experiments.)

Thus the fairy story you tell fails on three different accounts to be even a meaningful approximation to what really happens.

We argue in circles. So far you haven't given any interpretation but a mathematical formalism which doesn't even allow me to calculate the most simple things, which are not problem in the standard interpretation. That's not a very attractive alternative.

Of course, I don't need a large ensemble of the LHC. One LHC is enough to generate zillions of pp, pA, and AA collision events to collect "enough statistics".

Averages are very well defined with the same formula you give. The difference is that it's also clearly said, how to really calculate it, namely by using the observable algebra to construct the probabilities or equivalently the expectation values.

Just take the most simple thinkable QM 1 system:

There's a separable Hilbert space, on which the Heisenberg algebra with the fundamental self-adjoint operators $\hat{x}$ and $\hat{p}$ is realized. All you need is the general Hilbert-space structure and a notion of self-adjointness. The physics is defined operationally via Noether's theorem (or even more simply and heuristically motivated as "canonical quantization", i.e.,
$$[x,p]=\mathrm{i}.$$
From this it follows
$$\langle x|p \rangle=\frac{1}{\sqrt{2 \pi}} \exp(\mathrm{i} p x),$$
and from this everything else follows in a not too complicated way. Of course, the mathematician has to specify this much more carefully, defining the domains of the self-adjoint operators, the nuclear space and its dual of the rigged Hilbert space etc. But that's not the point of the interpretational issues.

The interpretation comes in, when I interpret pure states in the usual probabilistic way, i.e., interpreting the wave functions $\psi(x)=\langle x|\psi \rangle$ (with $|\psi \rangle$ in the Hilbert space and thus $\psi(x)$ in $L^2$) via Born's rule as giving the position probability distribution $P(x)=|\psi(x)|^2$. Then expectation values of properly defined operator functions $f(x,p)$ can be evaluated since the construction of the rigged Hilbert space with the observable algebra implies that in the position representation
$$f(\hat{x},\hat{p}) \mapsto f(x,-\mathrm{i} \partial_x)$$
and in this sense
$$\langle f(x,p) \rangle = \int_{\mathbb{R}} \mathrm{d} x \psi^*(x) f(x,-\mathrm{i} \partial_x) \psi(x).$$
The generalization to general ("mixed" states) is also straight forward, as long as you allow the use of the spectral decomposition with respect to (a complete set of compatible) observable operators.

If you now claim, you have a new interpretation of expectation values underlying a new interpretation, you have to give a concrete description of how to calculate these expectation values, if it's "forbidden" to use the standard meaning of the symbols!

 

[vanhees71]

I did it for the case $f(q,p)=p^2-q^6$, but you didn't take it up. So let me give a few more details: By the definition of $p$ and $q$ in the position representation and the rules for multiplying operators, you get $p\psi(x)$ by differentiation and $q^3\psi(x)$ by multiplication with $x^3$. Given a Gaussian wave package $\psi(x)=N \exp(-x^2/(4 \Delta^2)+\mathrm{i} p x)$ as you suggest, you can calculate the two norm squares in my previous post using integration by parts, and you get an explicit, exact value for the q-expectation!

This is an elementary exercise that already good high school pupils can do.

On the other hand, the spectral theorem for unbounded self-adjoint operators is heavy machinery (in Vienna 3rd year math bachelor) that takes a lot of math to prove. Thus it should not figure in a first introduction to quantum mechanics.

Essentially the same works for any expression in $p,q$ involving only sums of products of powers of $p,q$ and more generally for any expression $A$ that is polynomial in $p$ and often enough differentiable in $q$. In the latter case, to evaluate $A\psi$, one uses the product rule to move all $p$s to the right of all $q$s, and ends up with a sum of terms of the form $f_i(q)p^n\psi$ which can be evaluated directly in the position representation and then integrated numerically after multiplication with $\psi^*$. If $A$ is nonpolynomial in $p$ one can use limits, power series, etc...
Only the definition of the operators $p$ and $q$ is needed for all this, no spectral theorem!

You told me that precisely this is not allowed to do within your thermal interpretation since it's forbidden to use the spectral theorem. Of course, I know how to calculate the said expectation values. Even a machine can do this nowadays ;-)).
I don't talk about the complicated rigorous definition in a functional-analysis course at all. For me the usual sloppy physicists approach is enough. I want to concentrate on the interpretational aspect, but if it is not allowed to use even this in principle strictly definable rules of the rigged-Hilbert-space formalism, I cannot make sense of the symbols at all. I must have something that let's me boil it down "to the numbers"! As I said in my previous posting, in standard QT there's no problem with the construction of these observables down to the level of making it computable down to the numbers, but I don't see, how this is done without the use of a concrete representation (naturally the position and/or momentum representation in this case).

 

[A. Neumaier]

You told me that precisely this is not allowed to do within your thermal interpretation since it's forbidden to use the spectral theorem.

I was not using the spectral theorem. I was only using the definition of $p$ and $q$ in the Hilbert space $L^2(R)$. This has nothing to do with the spectral theorem. Schrödinger didn't know the spectral theorem when he invented the position representation! (It was first formulated by von Neumann a few years later.)

Moreover, I do not forbid the spectral theorem as a mathematical tool to work with operators that are self-adjoint. Whenever this is the case one can of course work with the spectral theorem on the shut-up-and-calculate level (but one still need not impose a statistical interpretation). But the self-adjoint operators are not general enough - they are neither closed under addition nor under multiplication with a scalar, while these operations are ubiquitous in calculations with q-expectations!

And they are not needed for much of quantum mechanics, including your cited lecture on the Kadanoff-Baym equations.

I don't see, how this is done without the use of a concrete representation (naturally the position and/or momentum representation in this case).

Neither do I. Without giving some Hilbert space and a definition of $p,q$ satisfying the CCR one cannot do any quantum mechanics. But once this is given (as a space of sequences or functions, or in still other ways), one has a representation and can do most of the calculations without the spectral theorem. The latter may be needed when one wants to change to a different representation (even this can often be done without it), but not when working in a fixed representation.

Since when are position and momentum operators within non-relatvistic first-quantized QT not self-adjoint?

They are, but in your derivation in this lecture, you are not taking q-expectations of position and momentum operators but of (regularized) products of fields such as $\phi(x)\phi(y)$. According to you, it is an ordinary expectation, hence the argument of the expectation should (according to your definition of expectation) be self-adjoint. Thus this product should be self-adjoint - but it is not even Hermitian!

 

[vanhees71]

Well, if you always redefine standard language, it's really hard to discuss :-(. For me the position representation is an application of the spectral theorem for the position operator. For me QT is formulated representation independently by an observable algebra and its representation as self-adjoint operators on a Hilbert space. The representations are constructed from this via the spectral theorem. If you start in the position representation that's fine with me, but then what's the difference between the thermal and the standard probabilistic interpretation? It becomes more and more enigmatic to me rather than clear!

It also doesn't make sense to discuss about non-Hermitean operators which may have some applications in QT. All I want is a clear definition of the thermal interpretation, particularly how to understand the expression $\mathrm{Tr}(\hat{\rho} \hat{A})$ if not probabilistically as an expectation value of a random variable as in standard QT.

Finally, it's of course clear that you use calculational expressions which are not related to observables like propagators, i.e., something $\propto \langle \hat{\phi}(x) \hat{\phi}^{\dagger}(y)$. It's not claimed that this directly refers to observational (probabilistic) quantities. As usual in QFT the N-point functions (and building blocks as the connected and the proper vertex functions) are used as calculational tools towards directly observable quantities (if you want you can use the modern word "beables" for this too) like cross sections or expectation values of observable quantities etc.

To understand better the meaning of your formalism, it would really be helpful to stick to the most simple examples like the Stern-Gerlach experiment of non-relativistic QT first. This is so close to classical mechanics because of the linearity of the forces involved that this must be also simple in your thermal representation.

 

[A. Neumaier]

Well, if you always redefine standard language, it's really hard to discuss :-(. For me the position representation is an application of the spectral theorem for the position operator.

My language is at least as standard as yours: Before you can apply the spectral theorem in some Hilbert space to some operator, you need definitions of both! I define an inner product on $L^2(R)$ and then the operators $p$ and $q$, to get the necessary Hilbert space and two particular operators on it. Having these definitions, I don't need the spectral theorem at all - except when I need to define transcendental functions of some operator.

If you start in the position representation that's fine with me, but then what's the difference between the thermal and the standard probabilistic interpretation?

The difference, given the position representation (or any other representation), is as follows:

What you call the minimal statistical or standard probabilistic interpretations uses this representation for defining irreducible probabilities of measurement in an ensemble of repeated observations, and thus introduces an ill-defined notion of measurement (and hence the measurement problem - though you close your eyes to it) into the very basis of quantum mechanics. It is no longer clear when something counts as a measurement (so that the unitary evolution is modified) and when the Schrödinger equation applies exactly; neither does it tell you why the unitary evolution of the big system consisting of the measured objects and the detector produces definite events. All this leads to the muddy reasoning visible in the literature on the measurement problem.

The thermal interpretation uses this representation instead to define the formal q-expectation of an arbitrary operator $A$ for which the trace in the formal Born rule can be evaluated. (There are many of these, including many nonhermitian ones and many Hermitian, non-selfadjoint ones.) This is the way q-expectations are used in all of statistical mechanics - including your slides. All this is on the formal side of the quantum formalism, with no interpretation implied, and no relation to observations. This eliminates the concept of probability from the foundations and hence allows progress to be made in the interpretation questions.

Then I note that the collection of all these q-expectations has a deterministic dynamics given by a Lie algebra structure, just as the collection of phase space functions in classical mechanics. In the thermal interpretation, the elements of both collections are considered to be beables.

Then I note that in statistical thermodynamics of local equilibrium, the q-expectations of the fields are actual observables, as they are the classical observables of fluid mechanics, whose dynamics is derived from the 1PI formalism - in complete analogy to your 2PI derivation of the Kadanoff-Baym equations. In practice one truncates to a deterministic dissipative theory approximating the deterministic dynamics of all q-expectations. This gives a link to observable deterministic physics - all of fluid mechanics, and thus provides an approximate operational meaning for the field expectations. This is not worse than the operational meaning of classical fields, which is also only approximate since one cannot measure fields at a point with zero diameter.

Then I prove that under certain other circumstances and especially for ideal binary measurements (rather than assume that always, or at least under unstated conditions), Born's interpretation of the formal Born rule as a statistical ensemble mean is valid. Thus I recover the probabilistic interpretation in the cases where it is essential, and only there, without having assumed it anywhere.

calculational expressions which are not related to observables like propagators, i.e., something $\propto \langle \hat{\phi}(x) \hat{\phi}^{\dagger}(y)\rangle$. It's not claimed that this directly refers to observational (probabilistic) quantities.

What then is the meaning of the expectation in this case? It is just a formal q-expectation defined via the trace. Thus you should not complain about my notion!

Born's rule only enters when you interpret S-matrix elements or numerical simulation results in terms of cross sections.

 

[DarMM]

It also doesn't make sense to discuss about non-Hermitean operators which may have some applications in QT. All I want is a clear definition of the thermal interpretation, particularly how to understand the expression $\mathrm{Tr}(\hat{\rho} \hat{A})$ if not probabilistically as an expectation value of a random variable as in standard QT.

It's a "beable" in Bell's terminology, that is a property of the system in question no different from properties in classical mechanics. Or at least thus is my understanding so far.

 

[DarMM]

I'm currently thinking a bit about the Bell inequalities for this interpretation.

For now as a side question have you considered the SIC-POVM conjecture, i.e. that specifying the $d^2$ collection of SIC-POVMs is enough to characterize the state $\rho$ completely. If true could this be taken into the Thermal Interpretation as the SIC-POVMs being the fundamental beables/quantities?

 

[A. Neumaier]

have you considered the SIC-POVM conjecture, i.e. that specifying the $d^2$ collection of SIC-POVMs is enough to characterize the state $\rho$ completely. If true could this be taken into the Thermal Interpretation as the SIC-POVMs being the fundamental beables/quantities?

They are interesting from a combinatorial point of view but nothing fundamental. Zauner wrote his thesis on these under my supervision.

 

[Fra]

I defined explicitly what a q-expectation is: a formal concept in the mathematical part of quantum mechanics, just like a phase space function is a formal concept in the mathematical part of classical mechanics. In both cases, these are claimed to be beables of the theory. In classical mechanics you don't give an operational definition of the concept of a phase space function - only a mathematical one. Why do you demand more in the quantum case?.

I personally demand much more in the quantum case because in my vision, I understand and expect future foundations of QM to be a theory of rational expectations of observers possessing incomplete information and having limited processing resources to process the information at hand. IMO this lies at another level of scientific standard than does classical physics. Ie. introducing non-inferrable concepts are against what i thinks are constructing principles. In a sense i an holding an evolved form of logical positivism here, but motivated by contraints of the physical observing system, rather than "human empirical observation".

But I agree completely what is the key problem with QM as it stands, and that is how to interpret or attach expectations of the the P-spaces without relating to fictive infinite ensembles. Statistical ensembles from repetitive experiments of identically prepared setup are fine for the typical HEP accelerator experiments however, but the problem is the cases (QG and unification) where this breaks down.

As I understand it the thermal interpretation aims to be an effective somewhat pragmatic interpretation, is that correct? In that case i figure it may not be worse than others but i do not see what advantage it offers for attaching QG and unification? Is it supposed to? If so i probably don't understand something.

/Fredrik

 

[DarMM]

They are interesting from a combinatorial point of view but nothing fundamental.

Thanks.

Zauner wrote his thesis on these under my supervision.

I saw that under footnote 27 in the first paper, I must read the thesis.

 

[akhmeteli]

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.

Could you please give a reference?

The Schrödinger equation just says that the Hamiltonian is the infinitesimal generator of time translations, and hence is always valid - even for a system described by the Dirac equation, where $H=cp_0$.

If someone finds fault with the Schrödinger equation then the whole of quantum mechanics breaks down completely!

I only had in mind the original nonrelativistic Schrödinger equation.

 

[A. Neumaier]

I'm currently thinking a bit about the Bell inequalities for this interpretation.

For now as a side question have you considered the SIC-POVM conjecture, i.e. that specifying the $d^2$ collection of SIC-POVMs is enough to characterize the state $\rho$ completely. If true could this be taken into the Thermal Interpretation as the SIC-POVMs being the fundamental beables/quantities?

They are interesting from a combinatorial point of view but nothing fundamental.

The point is that the construction principles for them is irregular, hence has not enough mathematical structure for something that could be considered fundamental.

More importantly, every physical system that can move or vibrate is represented in an infinite-dimensional Hilbert space. Hence anything dependent on finitely many dimensions cannot be fundamental. Despite their recent popularity, foundations of quantum mechanics just based on quantum information theory are highly defective since they do not even have a way to represent the canonical commutation relations, which are fundamental for all of spectroscopy, scattering theory, and quantum chemistry.

 

[A. Neumaier]

I only had in mind the original nonrelativistic Schrödinger equation.

The original nonrelativistic Schrödinger equation is only for a collection of spinless particles. It is far from what is today considered as the Schrödinger equation: the equation $i\hbar \dot \psi = H \psi$ for an arbitrary Hamiltonain $H$. One needs other forms of $H$ almost everywhere - in spectroscopy, in quantum chemistry, in quantum optics, in quantum information theory.

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

Could you please give a reference?

I don't have a reference; this seems to have not been considered before. When you work out the solution in terms of the Fourier transform you get for $\psi(x,t+x_0)$ a convolution of $\psi(x,t)$ (assumed to have compact support) with a function that does not have causal support.

 

[AlexCaledin]

. . . any interpretation as inadequate that cannot account for the meaning of quantum physics at a time before any life existed...

James B. Hartle shows that this problem is solved in the "post-Everett" CH-generalization of the QM , in his lectures "Spacetime QM and the QM of spacetime" (2014)
https://arxiv.org/abs/gr-qc/9304006

Page 21 of the PDF:

"There is nothing incorrect about Copenhagen quantum mechanics. Neither is it, in any sense, opposite to the post-Everett formulation"

 

[A. Neumaier]

Page 21 of the PDF

Please link to the pdf.

 

[AlexCaledin]

- sorry! here you are:
https://arxiv.org/pdf/gr-qc/9304006.pdf
" Spacetime Quantum Mechanics and the Quantum Mechanics of Spacetime
James B. Hartle
(Submitted on 5 Apr 1993 (v1), last revised 14 Jan 2014 (this version, v3))
These are the author's lectures at the 1992 Les Houches Summer School, "Gravitation and Quantizations". They develop a generalized sum-over-histories quantum mechanics for quantum cosmology that does not require either a preferred notion of time or a definition of measurement. The "post-Everett" quantum mechanics of closed systems is reviewed. Generalized quantum theories are defined by three elements (1) the set of fine-grained histories of the closed system which are its most refined possible description, (2) the allowed coarse grainings which are partitions of the fine-grained histories into classes, and (3) a decoherence functional which measures interference between coarse grained histories. Probabilities are assigned to sets of alternative coarse-grained histories that decohere as a consequence of the closed system's dynamics and initial condition. Generalized sum-over histories quantum theories are constructed for non-relativistic quantum mechanics, abelian gauge theories, a single relativistic world line, and for general relativity. For relativity the fine-grained histories are four-metrics and matter fields. Coarse grainings are four-dimensional diffeomorphism invariant partitions of these. The decoherence function is expressed in sum-over-histories form. The quantum mechanics of spacetime is thus expressed in fully spacetime form."
https://arxiv.org/abs/gr-qc/9304006
(it's the size of a good book though)

 

[vanhees71]

My language is at least as standard as yours: Before you can apply the spectral theorem in some Hilbert space to some operator, you need definitions of both! I define an inner product on $L^2(R)$ and then the operators $p$ and $q$, to get the necessary Hilbert space and two particular operators on it. Having these definitions, I don't need the spectral theorem at all - except when I need to define transcendental functions of some operator.The difference, given the position representation (or any other representation), is as follows:

What you call the minimal statistical or standard probabilistic interpretations uses this representation for defining irreducible probabilities of measurement in an ensemble of repeated observations, and thus introduces an ill-defined notion of measurement (and hence the measurement problem - though you close your eyes to it) into the very basis of quantum mechanics. It is no longer clear when something counts as a measurement (so that the unitary evolution is modified) and when the Schrödinger equation applies exactly; neither does it tell you why the unitary evolution of the big system consisting of the measured objects and the detector produces definite events. All this leads to the muddy reasoning visible in the literature on the measurement problem.

The thermal interpretation uses this representation instead to define the formal q-expectation of an arbitrary operator $A$ for which the trace in the formal Born rule can be evaluated. (There are many of these, including many nonhermitian ones and many Hermitian, non-selfadjoint ones.) This is the way q-expectations are used in all of statistical mechanics - including your slides. All this is on the formal side of the quantum formalism, with no interpretation implied, and no relation to observations. This eliminates the concept of probability from the foundations and hence allows progress to be made in the interpretation questions.

That's a cultural difference between physicists and mathematicians. While the mathematician can live with a set of rules (called axioms) without any reference to the "real world". Of course you can just start in the position representation and define a bunch of symbols calling them q-expectation and then work out the mathematical properties of this notion. The physicist however needs a relation of the symbols and mathematical notions to observations in the lab. That's what's called interpretation. As with theory and experiment (theory is needed to construct measurement devices for experiments, which might lead to observations that contradict the very theory; then the theory has to be adapted, and new experiments can be invented to test its consequences and consistency etc. etc.) also the interpretation is needed already for model building.

Now, I don't understand why I cannot interpret your q-expectations as usually as probabilistic expectation values. So the first very natural connection to experiments, which always need statistical arguments to make objective sense. For each measurement to be of credibility you need to repeat the experiment under the same circumstances (in q-language preparations of ensembles) and analyze the results both statistically as well as for systematical errors. The true art of experimenalists is not just to measure something but have a good handle about the errors, and statistics, based on mathematical probability theory is one of the basic tools of every physicist. This you get already in the first lesson of the introductory physics lab (to the dismay of most students, particularly the theoretically inclined, but it's indeed of vital importance particularly for them ;-)).

Concerning QT another pillar to make sense of the formalism, which is also already part of the interpretation, is to find the operators that describe the observables. The most convincing argument is to use the symmetries known from classical physics, defining associated conserved quantities via Noether's theorem. The minimal example for the first lessons of the QM1 lecture is the one-dimensional motion of a non-relativistic particle. There you have time-translation invariance leading to the time-evolution operator (in q-language called Hamiltonian) by finding the corresponding symmetry transformations (unitary for continuous smooth representations of Lie groups thanks to Wigner's theorem) and the generators defining the observable operators. I guess here comes the first place, where the observable operators should be represented by essentially self-adjoint operators, leading to the unitary representations of the (one-parameter) Lie symmetry groups. Then of course you also have momentum from translation invariance along the one direction the particle is moving and Galileo boosts to get also a position operator from the corresponding center-of-mass observable (I leave out the somewhat cumbersome discussion of mass in non-relativistic physics, which can fortunately postponed to the QM 2 lecture if you want to teach it at all ;-)).

Then you may argue to work in the position representation to begin with, and then the above considerations indeed lead to the operators of the "fundamental observables" position and momentum:
$$\hat{p} \psi(t,x) =-\mathrm{i} \partial_x \psi(t,x),$$
and the time-evolution equation (aka Schrödinger equation)
$$\mathrm{i} \partial_t \psi(t,x)=\hat{H} \psi(t,x).$$
Ok, but now if not having the Born interpretation (for the special case of pure states and precise measurements) at hand, I don't know, how to get the connection with real-world experiments.

It's an empirical fact that we can measure positions and momenta with correspondingly constructed macroscopic measurement devices. So we don't need to discuss the complicated technicalities of a particle detector which measures positions or a cloud chamber with a magnetic field to measure momenta and via the energy loss (also based on theory by Bethe and Bloch by the way) to have particle ID etc. etc.

However, I don't see how you make contact with these clearly existing macroscopic "traces" of the microworld, enabling to get quantitative knowledge about these microscopic entities we call, e.g., electrons, $\alpha$ particles etc. Having the statistical interpretation at hand, it's well known how the heuristics procedes, and as long as you don't insist that there is a "measurement problem" there is indeed none, because all I can hope from a theory, together with some consistent theoretical interpretation about its connection to these real-world observations, is to be consistent with these observations. You cannot expect it to satisfy your intuition from your macroscopic everyday experience which appears to be well-described by deterministic classical theories. The point is that this is also true for coarse-grained macroscopic observables, and this is in accordance with quantum statistics too. To coarse grain of course you need a description of the coarse grained observables, for which you need again statistics.

So the big for me still unanswered question is indeed this interpretive part of the "thermal interpretation". It's an enigma to me, how to make contact between the formalism (which includes also Ehrenfest's theorem which seems to be another corner stone of your interpretation too, but I don't see how it helps to make contact with the above described observations).

Then I note that the collection of all these q-expectations has a deterministic dynamics given by a Lie algebra structure, just as the collection of phase space functions in classical mechanics. In the thermal interpretation, the elements of both collections are considered to be beables.

Then I note that in statistical thermodynamics of local equilibrium, the q-expectations of the fields are actual observables, as they are the classical observables of fluid mechanics, whose dynamics is derived from the 1PI formalism - in complete analogy to your 2PI derivation of the Kadanoff-Baym equations. In practice one truncates to a deterministic dissipative theory approximating the deterministic dynamics of all q-expectations. This gives a link to observable deterministic physics - all of fluid mechanics, and thus provides an approximate operational meaning for the field expectations. This is not worse than the operational meaning of classical fields, which is also only approximate since one cannot measure fields at a point with zero diameter.

Yes, this is all very clear, as soon as I have the statistical interpretation and have extended it to "incomplete knowledge" and thus statstical operators to define non-pure states (i.e., states of non-zero entropy and thus implying incomplete knowledge). If I have just an abstract word like "q-expectations" there's no connection with classical (ideal or viscous) hydro. If I'm allowed to interpret "field expecations" in the usual way probabilistically, this is all well established. BTW. it's not a principle problem to use QFT instead of using the "first-quantization" formalism.

Then I prove that under certain other circumstances and especially for ideal binary measurements (rather than assume that always, or at least under unstated conditions), Born's interpretation of the formal Born rule as a statistical ensemble mean is valid. Thus I recover the probabilistic interpretation in the cases where it is essential, and only there, without having assumed it anywhere.

Well, but you need this probabilistic interpretation before you can derive hydro from the formalism. If not, I've obviously not realized, where and how this crucial step is done within your thermal interpretation.

What then is the meaning of the expectation in this case? It is just a formal q-expectation defined via the trace. Thus you should not complain about my notion!

Born's rule only enters when you interpret S-matrix elements or numerical simulation results in terms of cross sections.

It was about the Green's function in QFT or field correlators like $$\mathrm{i} G^{>}(x,y)=\mathrm{Tr} \hat{\rho} \hat{\phi}(x) \hat{\phi}(y)$$. Of course, that's not an expectation value of anyting observable. It's not forbidden to use such auxiliary functions in math to evaluate the observable quantities. Why should it be? As already Heisenberg learned from Einstein, the strictly positivistic approach (i.e., to work only with observable quantities) is neither necessary nor possible in theoretical physics. Also in classical electrodynamics you quite often work with the clearly unobservable potentials to derive the observable quantities (electromagnetic fields, or to be more precise the observable facts we understand as caused by the interaction of the charged matter building the detectors (e.g., our eyes) with the field in the standard interpretation of classical electromagnetism).

 

[DarMM]

The point is that the construction principles for them is irregular, hence has not enough mathematical structure for something that could be considered fundamental.

More importantly, every physical system that can move or vibrate is represented in an infinite-dimensional Hilbert space. Hence anything dependent on finitely many dimensions cannot be fundamental. Despite their recent popularity, foundations of quantum mechanics just based on quantum information theory are highly defective since they do not even have a way to represent the canonical commutation relations, which are fundamental for all of spectroscopy, scattering theory, and quantum chemistry.

I appreciate the construction point, but since your interpretation uses insights from AQFT (quite rightly, the reference to Yngvason is quite refreshing, I have often wondered how Many Worlds would deal with that result) would the "compactness criterion" of Haag, Sweica, Wichmann and Buchohlz be of any relevance?

In attempting to characterize those local algebras which admit asymptotic particle states Haag & Sweica proposed that the space of states on a local algebra $\mathcal{A}\left(\mathcal{O}\right)$ with energy below a threshold $E$ should be finite dimensional. Buchholz and Wichmann replaced this by a stronger property called the "Nuclearity condition" see:
Buchholz, Detley and Eyvind H. Wichmann. 1986. Causal independence and the energy-level density of states in local quantum field theory. Comm. Math. Phys.106: 321-344

With this condition you can demonstrate both decent thermodynamics and a particle interpretation.

So there is a chance that for QFT infinite-dimensional Hilbert spaces are just unphysical idealizations like pure states.

 

[vanhees71]

It's a "beable" in Bell's terminology, that is a property of the system in question no different from properties in classical mechanics. Or at least thus is my understanding so far.

How is it related to the outcomes of measurements in the lab, if I'm not allowed to interpret as an average in the probabilistic/statistical sense? That's my question. It's no question within the standard interpretation, where macroscopic measurement outcomes are derivable from the very notion of expectation values in probability theory.

 

[DarMM]

How is it related to the outcomes of measurements in the lab, if I'm not allowed to interpret as an average in the probabilistic/statistical sense? That's my question. It's no question within the standard interpretation, where macroscopic measurement outcomes are derivable from the very notion of expectation values in probability theory.

My understanding is that lack of knowledge of the unmodelled environment in which the measuring device is embedded will ensure that the measured value $A_m$ will deviate from the true value $\langle A\rangle$.

In a sense we invert the typical conclusion. Rather than $\langle A\rangle$ predicting the average value of our "precise" measurements, our imprecise noisy measurements prevent us from directly measuring the value $\langle A\rangle$ and we use the statistics of multiple such measurements to compute our measured value of $\langle A\rangle$.

Ultimately it is no different from measuring a Classical quantity. There are measurement errors which one controls by building a large sample.

 

[DarMM]

@A. Neumaier a few questions:

1. Do you have a physical picture for $\mathbb{L}^{*}$ the dual of the Lie algebra of q-expectations? I mean simply what is it/how do you imagine it physically. Just to get a better sense of the Hamiltonian dynamics.
2. What is the significance of $\mathbb{L}^{*}$ not being symplectic? Note for both these questions I know the mathematical theory, it's easy to show $\mathfrak{g}^{*}$ is a Poisson manifold for a Lia algebra $\mathfrak{g}$. I'm more looking for the physical significance in the Thermal interpretation
3. Should I understand $\mathbb{L}$ formally, i.e. the algebra of expectation "symbols" as such, not the algebra of expectations of a specific state $\rho$? In other words it isn't truly $\mathbb{L}_{\rho}$

Forgive the naivety of these, the interpretation has yet to solidify in my head