The thermal interpretation of quantum physics

[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

 

[akhmeteli]

@A. Neumaier: A quote from your work: " When a particle has been prepared in an ion trap (and hence is there with certainty), Born’s rule implies a tiny but positive probability that at an arbitrarily short time afterwards it is detected a light year away"

I guess this is only true if one assumes nonrelativistic equation of motion?

It is true in quantum mechanics, not in quantum field theory. Note that quantum mechanics has no consistent relativistic particle picture, except in the free case. Thus an atom in an ion trap cannot be consistently modeled by (fully) relativistic quantum mechanics.

But for a free particle, if one would know the position at one time to be located in a small compact region of space, it could be the next moment almost everywhere with a nonzero probability.

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'>tt 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.

Your reasoning is not convincing at all (at least not until you provide more details). So far I cannot accept your statement for a free relativistic particle, and the reasoning is as follows. As far as I know, the retarded Green's function for the Klein-Gordon operator has support within the future light cone (including its boundaries) (see, e.g., https://books.google.com/books?id=t...on retarded green function light cone&f=false). It satisfies the Klein-Gordon equation outside the source, for example, for t>0. So the function has a compact support at t=1, evolves in accordance with the Klein-Gordon equation between t=1 and t=2, and has a compact support at t=2. ​

 

[A. Neumaier]

So far I cannot accept your statement for a free relativistic particle, and the reasoning is as follows. As far as I know, the retarded Green's function for the Klein-Gordon operator has support within the future light cone (including its boundaries) (see, e.g., https://books.google.com/books?id=ttuO8-_D_oUC&pg=PA173&lpg=PA173&dq=klein+gordon+retarded+green+function+light+cone&source=bl&ots=24Z2Z4hYeD&sig=ACfU3U1ajzmVFBVlS53NpibBGXJVDovgHA&hl=en&sa=X&ved=2ahUKEwjN3_Szv-zgAhVPZawKHdEaBe04ChDoATAFegQICRAB#v=onepage&q=klein gordon retarded green function light cone&f=false). It satisfies the Klein-Gordon equation outside the source, for example, for t>0. So the function has a compact support at t=1, evolves in accordance with the Klein-Gordon equation between t=1 and t=2, and has a compact support at t=2.

I agree that the retarded Greens functions and their linear combinations.are causal. They form a representation of the physical Hilbert space of the electron.

However, in this representation (for fixed time $t$) , $|\psi(x,t)|^2$ does not have the interpretation of a position probability interpretation! The reason is that multiplication by $x$ is not an operator on a dense subspace of this Hilbert space. It introduces negative energy frequencies! Therefore having compact support $C$ in this representation cannot be interpreted as being localized in $C$.

To get a probability interpretation you need a valid 3D position operator with commuting components. This is the Newton-Wigner operator. See the discussion in the item ''Particle positions and the position operator'' from my Theoretical Physics FAQ, and the remarks following https://www.physicsforums.com/posts/6136475/
If you transform to a representation in which the Newton-Wigner operator is diagonal you get a transformed wave function with a probability interpretation. But in this representation, relativistic causality is lost - since the Newton-Wigner operator is observer dependent.

 

[A. Neumaier]

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 $\mathbb{L}_{\rho}$

Consider first the Lie *-algebra $\mathbb{L}^{*}$ of smooth functions $f(p,q)$ on classical phase space with the negative Poisson bracket as Lie product and * as complex conjugation . The Lie *-algebra can be partially ordered by defining $f\ge 0$ iff $f$ takes values in the nonnegative reals. A state is a (nice enough) monotone *-linear functional on $\mathbb{L}$, hence an element of $\mathbb{L}^{*}$. A general element of $\mathbb{L}^{*}$ may therefore be considered as a ''complex state'' in the same sense as one can generalize measures to complex measures.

Essentially the same holds in the quantum case for the Lie *-algebra of q-expectation symbols (as you observed). In abstract terms it is by definition isomorphic to the Lie *-algebra of linear operators $A$ on a nuclear space in QM with the quantum Lie product and taking adjoints as *, in QFT a more complicated Lie *-algebra (the traditional $C^*$-algebraic setting by Haag is not quite appropriate at is doesn't contain the most relevant physical observables, which are unbounded), with the partial order induced by defining $A\ge 0$ iff $A$ is Hermitian and positive semidefinite. States are again (nice enough) monotone linear functionals. They turn the q-expectation symbols into actual q-expectations (i.e., complex numbers). Thus states are again the most well-behaved elements of $\mathbb{L}^{*}$.

This should answer 1. and 3.. As to 2., a nonsymplectic Poisson manifold can (in finite dimensions) be foliated into symplectic leaves, often characterized by specific values of Casimir operators (i.e., elements in the Lie-Poisson algebra whose Lie product with everything vanishes). The actual Hamiltonian dynamics happens on one of these symplectic leaves since all Casimirs are conserved. In infinite dimensions (needed already for a single thermal oscillator), this too holds in a less rigorous sense,

A simple example is $R^3$ with the cross product as Lie product. It is isomorphic to $so(3)$ and describes in this representation a rigid
rotator. $\mathbb{L}^{*}$ is spanned by the three components of $J$, and the functions of $J^2$ are the Casimir operators. Assigning to $J$ a particular 3-dimensional vector gives the classical angular momentum in a particular state. The Lie-* algebra is the corresponding complexification, hence strictly speaking it is $C^3$.

The same Lie algebra is also isomorphic to $su(2)$, the Lie algebra of traceless Hermitian $2\times 2$ matrices, and then describes (in complexified form) the thermal setting of a single qubit. In this case, we think of $\mathbb{L}^{*}$ as mapping the three Pauli matrices $\sigma_j$ to three numbers $S_j$, and extending the map linearly to the whole Lie algebra. Augmented by $S_0=1$ to account for the identity matrix, which extends the Lie algebra to that of all Hermitian matrices, this leads to the classical description of the qubit discussed in Subsection 3.5 of Part III. (Note: misprints there: all $SS$ should be bold $\mathbf{S}$; there must be a macro problem in the arXiv version!)

 

[A. Neumaier]

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?

I don't know. Much of algebraic QFT is for my taste far too abstract, and I cannot easily read papers on the subject. I just borrowed the simplest aspects, in as far as I found them useful.

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

No. For a satisfactory interpretation, one needs all energies, not only those below some threshold. The contributions of the arbitrarily high energies (with their associated arbitrarily high frequencies) are precisely what makes thermal physics dissipative and hence realistic, and what gives rise to the stochastic aspects of quantum physics!

 

[A. Neumaier]

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.

Well, I told you how to interpret $\langle A\rangle$ for macroscopic q-observables $A$ in terms of a single measurement of a piece of matter in equilibrium, but this didn't reach your understanding. I also told you that Subsections 3.3-3.4 of Part II spell out conditions under which $\langle A\rangle$ can be viewed as a sample average, but you apparently didn't even read it. You simply don't care about how I want things to be interpreted!

Now, I don't understand why I cannot interpret your q-expectations as usually as probabilistic expectation values.

Because then you get your minimal interpretation and not the thermal interpretation. You cannot interpret one interpretation in terms of another nonequivalent one! That you try to do this rather than trying to understand the thermal interpretation in its own terms is the reason why in this thread we practically always talk past each other.

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.

I get it in the same informal way as in the classical case, where there is no a Born interpretation but we still know how to measure the approximate position and momentum of a particle. In both the classical case and the quantum case we measure the position and the momentum (knowing how this is done from experience with lab experiments) and get an approximation for its value. That's it! Your minimal interpretation is that you get in this way an approximation of an eigenvalue; my thermal interpretation is instead that you get an approximation of the q-expectation. Both are compatible with experiment, although quite different in their theoretical implications!

Most interpretations even claim that one gets an exact eigenvalue. But this contradicts experiment: The energy levels of atoms and molecules are only approximately known though they are given exactly by the eigenvalues of the Hamiltonian H, supposedly the only possible results of measurements of the - suitably normalized - energy. And H is the most important ''observable'' in statistical mechanics!

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

The thermal interpretation says that particles are fiction (which may be under special circumstances appropriate). In reality you have beams (states of the electron field, an effective alpha particle field, etc., concentrated along a small neighborhood of a mathematical curve) with approximately known properties (charge densities, spin densities, energy densities, etc.) If you place a detector into the path of a beam you measure these densities - accurately if the densities are high, erratically and inaccurately when they are very low. This is very close to experimental practice, how could it be closer?

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.

No. You only need the 1PI formalism, which nowhere talks about probabilities. It uses q-expectations throughout, nothing else!

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 anything observable.

Thus you use expectation terminology and notation (i.e., q-expectations) for something that is not an expectation value of anything, and you get useful results that you can later interpret in the right context in terms of experimental cross sections, etc. The thermal interpretation just does this consistently, observing that in almost everything done in quantum mechanics and quantum field theory, only q-expectations are computed and worked with, and the experimental interpretation comes only at the very end!

Sometimes, the experiment involves stochastic data (counts of events of certain kinds, many low accuracy measurements) and the theoretical result is interpreted as a probability or sample mean. In many other cases, the experiment involves just a few measurements - for example, of temperature, pressure, and mass, or of spectral lines and spectral widths -, and the theoretical result is interpreted without invoking any probability or statistics.

Therefore there is no need at all to put the statistical/probabilistic stuff into the foundations of quantum physics. As it always was before the advent of quantum mechanics, statistics and probability are experimental techniques for producing reproducible information from nonreproducible (and thus noisy) measurements; nothing more!

 

[A. Neumaier]

Most interpretations even claim that one gets an exact eigenvalue. But this contradicts experiment: The energy levels of atoms and molecules are only approximately known though they are given exactly by the eigenvalues of the Hamiltonian H, supposedly the only possible results of measurements of the - suitably normalized - energy. And H is the most important ''observable'' in statistical mechanics!

Your yesterday revised lecture notes on statistical mechanics (p.20 in the version of 5th March, 2019) is a little more cautious in formulating the traditional Born rule:

A possible result of a precise measurement of the observable O is necessarily an eigenvalue of the corresponding operator O

With this formulation, my argument only shows that there are no ''precise measurements'' of energy.

But then with your foundations, the whole of statistical mechanics hangs in the air because these foundations are too imprecise!

You seem to interpret the total energy in statistical thermodynamics as a mean of somehow measured energies of the zillions of atoms in the macroscopic body.

This is the formal description of an "ensemble average" in the sense that one averages over the microscopic fluctuations by just "blurring" the observation to the accuracy/resolution of typical macroscopic time and space scales, and thus "averaging" over all fluctuations at the microscopic space-time scales.

But your postulates in the lecture notes apply (as stated) only to measurements, not to unmeasured averages over unobserved fluctuations. Thus it seems that you assume that a body in equilibrium silently and miraculously performs $10^{23}$ measurements and averages these. But how are these measured? how often? how long does it take? Where are the recorded measurement results? What is the underlying notion of measurement? And how do these surely very inaccurate and tiny measurements result in a highly accurate q-expectation value? Where is an associated error analysis guaranteeing the observed accuracy of the total energy measured by the thermal engineer?

You cannot seriously assume these zillions of measurements. But then you cannot conclude anything from your postulates, which are explicitly about measured stuff.

Or are they about unmeasured stuff? But then it is not a bridge to the observed world, and the word 'measurement' is just pretense that it were so.

The thermal interpretation has no such problems! It only claims that the q-expectation is approximately measured when it is known to be measured and a measurement result is obtained by the standard measurement protocols.

 

[DarMM]

No. For a satisfactory interpretation, one needs all energies, not only those below some threshold. The contributions of the arbitrarily high energies (with their associated arbitrarily high frequencies) are precisely what makes thermal physics dissipative and hence realistic, and what gives rise to the stochastic aspects of quantum physics!

Could you explain this a bit more? Surely a finite subregion of spacetime contains a maximum energy level and the compactness criterion is known to be valid for free fields (as is the Nuclearity condition), generally in AQFT it is considered that the Hilbert space of states in a finite subregion is finite dimensional as this condition implies a sensible thermodynamics and asymptotic particle interpretation.

I appreciate how dissipation allows a realist account of the stochastic nature of QM in your interpretation (based on the lucid account in section 5.2 of Paper III), so no argument there. I'm simply wondering about the need for infinite-dimensional Hilbert spaces in finite spacetime volumes.

 

[DarMM]

Consider first ...which extends the Lie algebra to that of all Hermitian matrices, this leads to the classical description of the qubit discussed in Subsection 3.5 of Part III. (Note: misprints there: all $SS$ should be bold $\mathbf{S}$; there must be a macro problem in the arXiv version!)

Thank you for this very clear!

So a separate question for let's say a two system state $\rho_{AB}$ with reduced density matrices $\rho_A$ and $\rho_B$ where we have two observables, $\mathcal{O}_A$ and $\mathcal{O}_B$ we can obviously have:
$$\rho_{AB}\left(\mathcal{O}_A\mathcal{O}_B\right) \neq \rho_A\left(\mathcal{O}_A\right)\rho_B\left(\mathcal{O}_B\right)$$
(Obvious abuse of notation here where on the left hand side what is labelled $\mathcal{O}_A$ is really $\mathcal{O}_A \otimes \mathbb{I}_{B}$)

In most "probabilistic interpretations" this is simply correlation. However if $\langle \mathcal{O}_A\mathcal{O}_B\rangle_{\rho_{AB}}$ is an ontic property of the total system what does it mean for it not to simply be the product of the single system ontic properties $\langle \mathcal{O}_A \rangle_{\rho_A}$ and $\langle \mathcal{O}_B \rangle_{\rho_B}$?

 

[akhmeteli]

I agree that the retarded Greens functions and their linear combinations.are causal. They form a representation of the physical Hilbert space of the electron.

However, in this representation (for fixed time $t$) , $|\psi(x,t)|^2$ does not have the interpretation of a position probability interpretation! The reason is that multiplication by $x$ is not an operator on a dense subspace of this Hilbert space. It introduces negative energy frequencies!

Well, this value $|\psi(x,t)|^2$ cannot be a probability density for Klein-Gordon for a different reason - it is not the temporal component of the current. However, $\bar{\psi}\gamma^0\psi$ can be a probability density for the Dirac equation. Your argument against that is about negative energy, therefore, it is based on the fact that there is no consistent one-particle interpretation of the Dirac equation, either free or not (in one of your previous posts you seemed to suggest that using holes is OK for free Dirac, but as soon as you mention holes you don't have a one-particle theory). Therefore, the free Dirac equation also has a serious problem. As I said, you cannot fault the Born's rule for having a problem with a problematic equation.

 

[A. Neumaier]

t's say a two system state $\rho_{AB}$ with reduced density matrices $\rho_A$ and $\rho_B$ where we have two observables, $\mathcal{O}_A$ and $\mathcal{O}_B$ we can obviously have:
$$\rho_{AB}\left(\mathcal{O}_A\mathcal{O}_B\right) \neq \rho_A\left(\mathcal{O}_A\right)\rho_B\left(\mathcal{O}_B\right)$$
(Obvious abuse of notation here where on the left hand side what is labelled $\mathcal{O}_A$ is really $\mathcal{O}_A \otimes \mathbb{I}_{B}$)

In most "probabilistic interpretations" this is simply correlation. However if $\langle \mathcal{O}_A\mathcal{O}_B\rangle_{\rho_{AB}}$ is an ontic property of the total system what does it mean for it not to simply be the product of the single system ontic properties $\langle \mathcal{O}_A \rangle_{\rho_A}$ and $\langle \mathcal{O}_B \rangle_{\rho_B}$?

It means that there are additional correlation degrees of freedom:

Take your observables to be fields you get pair correlations of the fluctuations. Locally via a Wigner transformation this gives kinetic contributions, but if A and B refer to casually disjoint regions, say, you get nonlocal correlations, the beables needed to violate the assumptions of Bell's theorem.