The thermal interpretation of quantum physics

[EPR]

Not all of reality. The appearance of definite macroscopic reality is not understood. Hence my statement

The so called world is a collection of fields that produce, 'create'(okay i will back off slightly here) and substitute that term with the more acceptable term - 'emerge' a classical world at the classical limit.

 

[PeterDonis]

Not all of reality.

Ok, that clarifies it.

 

[ftr]

The description of C60 as a field is appropriate for C60 beams (as prepared in double slit experiments); these are described by a current (constructible for arbitrary bound states according to a method due to Sandhas).

I have googled for W. Sandhas but not sure which paper describes the method. Can you point to a specific paper, thanks.

 

[A. Neumaier]

I have googled for W. Sandhas but not sure which paper describes the method. Can you point to a specific paper, thanks.

• W. Sandhas, Definition and existence of multichannel scattering states, Comm. Math. Phys. 3 (1966), 358-374.

Section 5 constructs for any bound state asymptotic creation and annihilation operators. Section 6 proves that they satisfy standard commutation relations, providing a bosonic field description. As in any bosonic field theory, this gives associated densities and currents.

 

[Demystifier]

@A. Neumaier I have one question for you. The thermal interpretation is an ontological interpretation, so it must be described by a nonlocal theory, as the Bell theorem implies. Is this fundamental nonlocality consistent with fundamental Lorentz covariance? Or does it mean that Lorentz covariance of QFT is emergent, rather than fundamental?

 

[A. Neumaier]

@A. Neumaier I have one question for you. The thermal interpretation is an ontological interpretation, so it must be described by a nonlocal theory, as the Bell theorem implies. Is this fundamental nonlocality consistent with fundamental Lorentz covariance? Or does it mean that Lorentz covariance of QFT is emergent, rather than fundamental?

The nonlocality of the thermal interpretation is a direct consequence of quantum field theory, hence as consistent with fundamental Lorentz covariance as quantum field theory itself. Thus provably in space-time dimensions 2 (quantum wires) and 3 (quantum surfaces), and empirically in space-time dimension 4.

In the thermal interpretation, there are local distribution-valued beables, the q-expectations of fields, and nonlocal distribution-valued beables, the n-point correlation functions, q-expectations of products of fields at different points and their derivatives. These q-expectations are manifestly Lorentz covariant. In a first order formulation (involving with each field also the canonically conjugate field), the time derivative of the local field expectations is given by some (infinite) linear combination of q-expectations of products of fields and their spatial derivatives, hence depends on nonlocal properties. (In the hydrodynamic approximation, these are approximated by products of local terms, giving rise to the equations of fluid mechanics and their relativistic and quantum generalizations. This approximation suffices for the description of the macroscopic regime.)

More precisely, the dynamics is given by the quantum field theory version of the Ehrenfest equations, which express the second time derivative of q-expectations of products of fields with respect to the most time-advanced arguments as a renormalized limit of sums of q-expectations of other products of fields (in the interacting case, some of them have more factors) and their spatial derivatives. However, the conventional treatment is in a Lorentz covariant 4D view, where it is more convenient to express products of fields in terms of products of fewer fields and their spatial, temporal and mixed derivatives. This is encoded in the operator product expansion, which are the modern form of the equation of motion of quantum fields.

 

[vanhees71]

I'm a bit puzzled by what you define as "local" vs. "nonlocal". Why do you consider the (non-observable) n-point functions as "non-local"? They are given by something like (for a scalar self-adjoint field as the most simple example)
$$G^{(n)}(x_1,\ldots,x_2)=\langle \mathcal{T} \hat{\phi}(x_1) \hat{\phi}(x_2) \cdots \hat{\phi}(x_n) \rangle,$$
where $\mathcal{T}$ is some "time-ordering prescription" (like time-ordering for vacuum QFT, contour-ordering for the most general case of real-time many-body QFT), and the expecation is meant as the trace with the stat. op.

The field operators are "local" in the sense that they obey the local transformation laws under Poincare transformations as their classical analogues, and also the interactions are usually taken as local, i.e., the Hamiltonian is a functional of field-operator products at one space-time point and the local observable-operators obey the microcausality constraint. So what's non-local in the thermal interpretation what is considered local in the standard interpretation? The mathematical functions are obviously the same (and also in (1+3)d there are the usual mathematical troubles with the formalism, but this is not under debate here, I guess).

Of course, relativistic QFT is "non-local" in the sense of any type of QT that it admits states where parts of a quantum system that are observable at large spatial distances have correlations described by entanglement, but imho to call this "non-locality" is misleading, because it's rather long-ranged correlations that are stronger than any classical correlation witin a local classical theory can be (which is the content of the violation of Bell's inequality).

It's this subtle balance between "locality of the dynamics" (microcausality condition fulfilled) and "non-locality of correlations" which makes relativistic local QFT consistent with the causality structure of special-relativistic spacetime. In how far this is the case for the more complicated case of QFTs in non-flat "background spacetimes" or even for a possible future quantum-gravity theory, I can't say of course.

 

[A. Neumaier]

I'm a bit puzzled by what you define as "local" vs. "nonlocal". Why do you consider the (non-observable) n-point functions as "non-local"?

Nonlocal = dependent on more than one space-time position (which can be arbitrarily far apart in space).
This is the same notion of nonlocality that is used to decide whether a Lagrangian density is local or nonlocal. It is also the notion of nonlocality that is excluded in assumptions proving Bell inequalities, and is the kind of nonlocality established experimentally in long distance entanglement experiments.

In this sense, quantum fields, their dynamics, and their q-expectations are local, but correlation functions are nonlocal.

They are given by something like (for a scalar self-adjoint field as the most simple example)
$$G^{(n)}(x_1,\ldots,x_2)=\langle \mathcal{T} \hat{\phi}(x_1) \hat{\phi}(x_2) \cdots \hat{\phi}(x_n) \rangle,$$
where $\mathcal{T}$ is some "time-ordering prescription" (like time-ordering for vacuum QFT, contour-ordering for the most general case of real-time many-body QFT), and the expectation is meant as the trace with the stat. op.

Yes, and there are also the unordered correlations corresponding to Wightman n-point functions.

In general, the trace with the statistical operator is a formal q-expectation only, not an expectation in the statistical sense. For a complex scalar field, $\Phi(x)$ and its smeared versions are not selfadjoint operators, hence $\langle\Phi(x)\rangle=Tr\rho\phi(x)$ and their smeared versions have no Born interpretation as statistical expectation values. Let alone the correlation functions.

But the 2-point correlations are in principle observable through linear response theory.

 

[Demystifier]

In the thermal interpretation, there are local distribution-valued beables, the q-expectations of fields, and nonlocal distribution-valued beables, the n-point correlation functions, q-expectations of products of fields at different points and their derivatives. These q-expectations are manifestly Lorentz covariant.

I think I get it. You avoid Bell theorem by something I would call multi-ontology in a single world (as opposed to many-world interpretation, which could be called single-ontology in many worlds). For instance, let $s_A$ and $s_B$ be the ontological spins of two entangled particles, and let their ontological product be $s_A\circ s_B$. In theories covered by the Bell theorem one has
$$s_A\circ s_B=s_As_B$$
while in the thermal interpretation
$$s_A\circ s_B \neq s_As_B$$
The ontology with the inequality above does not make much sense to me, but that's essentially what the thermal interpretation, as far as I understood it, claims to be the case.

 

[PeterDonis]

let their ontological product be

What is an "ontological product"?

 

[Demystifier]

What is an "ontological product"?

An ontological quantity that in the classical limit is given by an ordinary product of two ontological quantities.

 

[PeterDonis]

An ontological quantity that in the classical limit is given by an ordinary product of two ontological quantities.

What is "an ordinary product of two ontological quantities"? I only know how to multiply numbers and other mathematical objects; I don't know how to multiply "ontological quantities".

 

[A. Neumaier]

I think I get it. You avoid Bell theorem by something I would call multi-ontology in a single world (as opposed to many-world interpretation, which could be called single-ontology in many worlds). For instance, let $s_A$ and $s_B$ be the ontological spins of two entangled particles, and let their ontological product be $s_A\circ s_B$. In theories covered by the Bell theorem one has
$$s_A\circ s_B=s_As_B$$
while in the thermal interpretation
$$s_A\circ s_B \neq s_As_B$$
The ontology with the inequality above does not make much sense to me, but that's essentially what the thermal interpretation, as far as I understood it, claims to be the case.

Informally, in the thermal interpretation, the whole is more than its parts, which makes perfect sense to me. Whereas Bell assumed that, as in classical n-particle mechanics, the complete description of the parts furnishes a complete description of the whole.

I don't understand your reformulation in terms of an ontological product which neither figures in Bell's work nor in mine.

 

[vanhees71]

Nonlocal = dependent on more than one space-time position (which can be arbitrarily far apart in space).
This is the same notion of nonlocality that is used to decide whether a Lagrangian density is local or nonlocal. It is also the notion of nonlocality that is excluded in assumptions proving Bell inequalities, and is the kind of nonlocality established experimentally in long distance entanglement experiments.

In this sense, quantum fields and their q-expectations are local, but correlation functions are nonlocal.

Then it's simply "non-local" because I make an experiment with two or more detectors at distant points in space, but that's trivial and has nothing to do with the complicated implications of what's usually meant by "non-locality" in the sense of "spooky actions at a distance", and then it's of course consistent with the standard interpretation of relativistic local QFT.

Of course, the $N$-point functions of various kinds (including the fixed-ordered field-operator products, or Wightman functions) do not correspond directly to expectation values of observables but they are used to calculate them.

E.g., my beloved dilepton and photon production rates as measured in heavy-ion collisions are indeed derivable from the (thermal) retarded two-point correlation function. It's basically the imaginary part or spectral function of the electromagnetic current-current correlation function with some kinematical factors.

It's of course related to the mentioned connection with the linear-response theory, where the retareded two-point correlation functions of appropriate (composite-)field operators are the corresponding response functions. In equilibrium one can use them to evaluate transport coefficients using the famous Kubo formula.

 

[Demystifier]

What is "an ordinary product of two ontological quantities"? I only know how to multiply numbers and other mathematical objects; I don't know how to multiply "ontological quantities".

An ordinary product of two ontological quantities is just an ordinary multiplication of numbers, as in classical physics. The thermal interpretation replaces this with something very non-classical (and in my opinion too weird to make sense ), which, in effect, can be expressed as a weird way of multiplication. It has its roots in the well-known multiplication of operators in QM, which also looks weird if one attempts to interpret operators as ontological. For that reason operators are not interpreted as ontological in any quantum interpretation I am aware of. But the thermal interpretation takes the expectation value of operator products as ontological, which can be expressed as a strange multiplication. This multiplication is mathematically well defined (because the product of operators is mathemicaly well defined), but it is very strange when interpreted ontologically, on which the thermal interpretation insists.

 

[Demystifier]

I don't understand your reformulation in terms of an ontological product which neither figures in Bell's work nor in mine.

I reformulated it in this way to better understand the thermal interpretation in my own terms.

 

[Demystifier]

Informally, in the thermal interpretation, the whole is more than its parts. Whereas Bell assumed that the complete description of the parts furnishes a complete description of the whole.

OK, that's another way to express the fact that the Bell theorem does not apply to the thermal interpretation.

 

[A. Neumaier]

Then it's simply "non-local" because I make an experiment with two or more detectors at distant points in space, but that's trivial and has nothing to do with the complicated implications of what's usually meant by "non-locality" in the sense of "spooky actions at a distance", and then it's of course consistent with the standard interpretation of relativistic local QFT.

"spooky actions at a distance" is not a physical phenomenon but the result of a poor interpretation.

 

[A. Neumaier]

But the thermal interpretation takes the expectation value of operator products as ontological, which can be expressed as a strange multiplication.

No, the expectation value of operator products cannot be expressed as a product of q-expectations, which are the beables of the parts! It gives truly additional beables - whence the whole has more properties than the parts.

OK, that's another way to express the fact that the Bell theorem does not apply to the thermal interpretation.

Yes, and a very intuitive one!

 

[PeterDonis]

An ordinary product of two ontological quantities is just an ordinary multiplication of numbers, as in classical physics.

I think you are confusing the model with reality. Numbers are not "ontological quantities"; they are things in our mathematical model.

 

[ftr]

I think you are confusing the model with reality. Numbers are not "quantities"; they are things in our mathematical model.

I think maybe Demystifier can do a write up on what is meant by "ontological" for various interpretations. He has been always good at summarizing contentious issues. Does "ontology" have a universal agreed upon meaning? or maybe I should open a thread.

 

[A. Neumaier]

Does "ontology" have a universal agreed upon meaning? or maybe I should open a thread.

A discussion of this question surely does not belong to this thread.

 

[ftr]

A discussion of this question surely does not belong to this thread.

yes. This is what I was suggesting.

 

[PeterDonis]

Does "ontology" have a universal agreed upon meaning? or maybe I should open a thread.

You should open a separate thread for this topic. But before doing so, I would advise looking at the literature, since there are already plenty of published papers on the term "ontology" as it pertains to quantum mechanics.

 

[Demystifier]

No, the expectation value of operator products cannot be expressed as a product of q-expectations, which are the beables of the parts! It gives truly additional beables - whence the whole has more properties than the parts.

In ontological theories such as Bohmian mechanics of many worlds, if $A$ and $B$ are beables, then so is $AB$. In thermal interpretation, it is not so. I find it too weird for my taste.

Yes, and a very intuitive one!

Would you say that the thermal interpretation denies reductionism?

 

[Demystifier]

I think maybe Demystifier can do a write up on what is meant by "ontological" for various interpretations. He has been always good at summarizing contentious issues. Does "ontology" have a universal agreed upon meaning? or maybe I should open a thread.

I would categorize all the interpretations into 3 categories:
1) Interpretations without ontology (most variants of Copenhagenish interpretations)
2) Interpretations with ontology but without primitive ontology (consistent histories, thermal interpretation)
3) Interpretations with primitive ontology (Bohmian, many worlds, objective collapse)

Primitive ontology is the fundamental ontological quantity to which all other ontological quantities can be reduced. In Bohmian mechanics it is particle positions of all particles in the Universe. In many worlds it is the wave function of the multiverse.

 

[A. Neumaier]

In ontological theories such as Bohmian mechanics of many worlds, if $A$ and $B$ are beables, then so is $AB$. In thermal interpretation, it is not so. I find it too weird for my taste.Would you say that the thermal interpretation denies reductionism?

In the TI, the product of the q-expectations of A and B is a different beable than the q-expectation of the product AB. Nothing weird is involved.

I don't care about reductionism. What counts is what is explained.

 

[Demystifier]

In the TI, the product of the q-expectations of A and B is a different beable than the q-expectation of the product AB. Nothing weird is involved.

In TI, the q-expectation does not have a statistical interpretation. It is a property of a single system, not of an ensemble of systems. From that perspective, I understand how a q-expectation of the product AB is calculated, but I can't understand what a q-expectation of the product AB is. Is there perhaps some analogy?

 

[ftr]

I would categorize all the interpretations into 3 categories:
1) Interpretations without ontology (most variants of Copenhagenish interpretations)
2) Interpretations with ontology but without primitive ontology (consistent histories, thermal interpretation)
3) Interpretations with primitive ontology (Bohmian, many worlds, objective collapse)

Primitive ontology is the fundamental ontological quantity to which all other ontological quantities can be reduced. In Bohmian mechanics it is particle positions of all particles in the Universe. In many worlds it is the wave function of the multiverse.

Thanks. I hope to open a thread soon.

 

[PeterDonis]

In ontological theories such as Bohmian mechanics of many worlds, if $A$ and $B$ are beables, then so is $AB$.

I still don't understand how you multiply beables. Beables aren't numbers. Maybe a specific example would help me to understand what you are saying here.

 

[A. Neumaier]

In TI, the q-expectation does not have a statistical interpretation. It is a property of a single system, not of an ensemble of systems. From that perspective, I understand how a q-expectation of the product AB is calculated, but I can't understand what a q-expectation of the product AB is. Is there perhaps some analogy?

If A and B are local properties at different location, it is a nonlocal property of the system, a property that figures in the dynamical law. In general it has no interpretation except as a part of the dynamical law - something needed to determine the evolution of the whole system. Asking what it is is like asking in Bohmian mechanics what the wave function is.

However, some nonlocal properties can be given an interpretation. The 2-point correlations $C(x,y):=\langle \Phi(x)\Phi(y)\rangle$ can be Wigner transformed and then produce a (not necessarily positive) phase space density of the kind that figures in the Boltzmann equation (which can be obtained from the field dynamics in a suitable approximation). In general, many 2-point correlations can be observed through linear response theory, hence are true properties of a system.

 

[A. Neumaier]

I still don't understand how you multiply beables. Beables aren't numbers. Maybe a specific example would help me to understand what you are saying here.

Numerical beables are meant here. One can multiply the time traveled (a nonlocal beable) with the speed (another beable) and gets the distance traveled (a third beable).

 

[Demystifier]

I still don't understand how you multiply beables. Beables aren't numbers. Maybe a specific example would help me to understand what you are saying here.

Beables are represented by numbers. Consider, for example, a classical harmonic oscillator. The fundamental beable is the particle position represented by the number $X$. Its potential energy can also be considered a beable, but it's not fundamenatal. It is represented by the number $V=kX^2/2$.

 

[A. Neumaier]

Beables are represented by numbers. Consider, for example, a classical harmonic oscillator. The fundamental beable is the particle position represented by the number $X$. Its potential energy can also be considered a beable, but it's not fundamenatal. It is represented by the number $V=kX^2/2$.

With your distinction, the fundamental beables in the thermal interpretation are [represented by] the (distributional) q-expectations of products of fields (generalized Wightman n-point functions) , and the other beables are what is computable from them, such as smeared field expectations, Wigner functions, etc..

 

[Demystifier]

With your distinction, the fundamental beables in the thermal interpretation are [represented by] the (distributional) q-expectations of products of fields (generalized Wightman n-point functions) , and the other beables are what is computable from them, such as smeared field expectations, Wigner functions, etc..

That's helpful for the sake of comparison with other ontological interpretations. Bell introduced the notion of local beables. Those are not beables with local interactions, but beables defined locally at space points. In this sense Bohmian mechanics is a theory of fundamental local beables (particles have well defined positions in space) with nonlocal interactions. Many-world interpretation, on the other hand, is a theory of nonlocal fundamental beables (the state in the Hilbert space is not defined at a space point). Thermal interpretation is somewhere in between, because it contains both local fundamental beables (e.g. $\langle\phi(x)\rangle$) and nonlocal fundamental beables (e.g. $\langle\phi(x)\phi(y)\rangle$).