Born rule and thermodynamics

In summary, the Born rule in quantum mechanics states that the probability of finding a particle in a certain state is given by the square of the amplitude of its wave function. In thermodynamics, a similar rule can be seen in the statistical operator, which represents the state of a system. The probability of finding a certain value for an observable is given by the trace of the product of the statistical operator and the projector to the corresponding eigenspace. This can be seen as a classical analogue of the Born rule, where the classical variables take on any value and the laws governing their behavior can be derived from, but differ from, Newton's laws. This suggests that classical systems with regions of higher probability of occurrence may be akin to the Born rule in quantum
  • #71
Demystifier said:
I didn't say that there is no preferred basis at all. There isn't in MWI, but there is in Bohmian interpretation.

About Bohmian, I have difficulty integrating consciousness to Bohmian mechanics. Can you write a paper relating Consciousness to Bohmian Mechanics to explore its conceptual plausibility? Why didn't you write such paper..

Say. Do you have any access to say a Raman Spectroscope? Neumaier too?

Anywhere you are in the planet, I (and a lot many others) can influence any molecular system on Earth that one can test with instruments. Other people I know can do this and extensively tested by the Chinese Academy of Science. I have spent 10 years trying to understand the physics of it.

I'm still undecided whether it's Bohmian, a MWI thing or Copenhagen or even a new force of nature. It's such a weary search.

Such difficulty is natural for every new stuff, it is hard in the beginning really.
 
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  • #73
Demystifier said:
I can't write such a paper, but I have written this:
https://arxiv.org/abs/1112.2034Because it would contradict what I wrote here:
http://philsci-archive.pitt.edu/12325/

I just read this. All I can say is that there are more things in the world than are dreamt of by most physicists.. and I can't continue describing it because it's against forum rule here. But you can read it in quality journals. Thanks anyway for stuff you shared in this thread which make many concepts more clear.
 
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  • #74
Blue Scallop said:
there are more things in the world than are dreamt of by most physicists.. and I can't continue describing it because it's against forum rule here

Exactly. So please don't post about it. If you do so again, you will receive a warning.
 
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  • #75
Demystifier said:
The Born rule is about the probability ##p##, not the average value. So the general Born rule is actually
$$p = \mathrm{Tr} (\rho \pi),$$
where ##\pi=\pi^2## is a projector.

In physics, probabilities and averages give the same information, so I accept vanhees71's definition of the Born rule.
 
  • #76
atyy said:
In physics, probabilities and averages give the same information, so I accept vanhees71's definition of the Born rule.

I was about to say that, but I wasn't sure that it's true. From probabilities you can compute averages, but I'm not sure about the other way around. For example, suppose that I tell you that [itex]\langle S_z \rangle = 0[/itex]. That doesn't give me much information about probabilities of various values of [itex]S_z[/itex].

But maybe if for some observable [itex]A[/itex], I know [itex]\langle A^n \rangle[/itex] for every [itex]n[/itex], does that uniquely determine the probability?
 
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  • #77
stevendaryl said:
But maybe if for some observable [itex]A[/itex], I know [itex]\langle A^n \rangle[/itex] for every [itex]n[/itex], does that uniquely determine the probability?

Yes, it's like Taylor series. For example, the cumulant expansion specifies a generating function or characteristic function, which is equivalent to specifying the probability distribution. The generating or characteristic function is analogous to the partition function of statistical physics and QFT.

This can fail, but I think such cases are not physical.
 
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  • #78
stevendaryl said:
But maybe if for some observable [itex]A[/itex], I know [itex]\langle A^n \rangle[/itex] for every [itex]n[/itex], does that uniquely determine the probability?
This is called the moment problem. Although correct under suitable additional conditions on the probability measure, it has a negative answer in general, i.e., the expectations of all powers of ##A## do not determine its probability distribution. For counterexamples see, e.g.,

J. Stoyanov, Inverse Gaussian distribution and the moment problem, J. Appl. Statist. Science 9 (1999), 61-71.
https://www.researchgate.net/publication/246535073

These counterexamples are not exotic as they include, for example, certain inverse Gaussian distributions.
 
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  • #79
Demystifier said:
I didn't say that there is no preferred basis at all. There isn't in MWI, but there is in Bohmian interpretation.

Demystifier, please clarify something.

Zurek summarized it:

Let us summarize our results for the environment-induced selection of preferred states and discuss the implications for the general preferred-basis-problem outlined in Sect. 2.5.2 and for our observation of only particular phyiscal quantities in the world around us.

System-environment interaction Hamiltonians frequency described a scattering process of surrounding particles (photons, air molecules, etc.) interacting with the system under study. Since the force laws describing such processes typically depend on some power of distance (such as @ r ^-2 in Newton's or Coulomb's force law), the interaction Hamiltonian will usually commute with the position operator. According to the commutativity requirement (2.89), the pointer states will therefore be approximate eigenstates of position. The fact that position is typically the determinate property of our experience can thus be explained by referring to the dependency of most interactions on distance.

Bill Hobba kept mentioning the above. For example Bill stated that:

The basis is chosen in theory ie the position basis is implicit in the normal form of the Schrodinger equation. When expressed in that basis virtually all interactions turn out to be radial ie the V(x) in the Schrodinger equation. Its purely arbitrary what basis you chose to write your equations in - writing them in the position basis is what's usually done and you can easily see their radial nature.

Now let's take your nutshell about MWI:

"In a nutshell, the argument is this:
To define separate worlds of MWI, one needs a preferred basis, which is an old well-known problem of MWI. In modern literature, one often finds the claim that the basis problem is solved by decoherence. What J-M Schwindt points out is that decoherence is not enough. Namely, decoherence solves the basis problem only if it is already known how to split the system into subsystems (typically, the measured system and the environment). But if the state in the Hilbert space is all what exists, then such a split is not unique. Therefore, MWI claiming that state in the Hilbert space is all what exists cannot resolve the basis problem, and thus cannot define separate worlds. Period! One needs some additional structure not present in the states of the Hilbert space themselves.

As reasonable possibilities for the additional structure, he mentions observers of the Copenhagen interpretation, particles of the Bohmian interpretation, and the possibility that quantum mechanics is not fundamental at all. "

My question:

Does the above mean that without preferred basis or additional structure, Bill stuff about the radial nature of interactions choosing position as the basis won't even occur?

Or is this radial thing (or interaction Hamiltonian within the pure MWI without preferred basis) a separate thing from the concept of additional preferred basis put in by hand?
 
  • #80
Blue Scallop said:
Does the above mean that without preferred basis or additional structure, Bill stuff about the radial nature of interactions choosing position as the basis won't even occur?
Yes.
 
  • #81
Demystifier said:
Yes.

But then according to Peterdonis, he suggested the interaction Hamiltonian (which can be where the Hobba radial argument was based) was independent of the preferred basis as when he wrote "(One point that I have not raised is that the wave function is not the only "structure" present in QM; there is also the Hamiltonian, or Lagrangian if you are doing QFT. So one possibility that we have not discussed is that the "additional structure" is in the Hamiltonian, not the wave function; that the Hamiltonian of the cat, or the cat/environment system, is what picks out the alive/dead basis as the one that gets decohered.)"

What is your say about this Hamiltonian itself able to create subsystems without additional structure (prepared basis) put in by hand? Any counterarguments for his statements?
 
  • #82
atyy said:
In physics, probabilities and averages give the same information, so I accept vanhees71's definition of the Born rule.
Consider a physical system in which the average value of position is ##<x>=0##. What is the probability that the position is ##x=1## nm?
 
  • #83
Blue Scallop said:
What is your say about this Hamiltonian itself able to create subsystems without additional structure (prepared basis) put in by hand? Any counterarguments for his statements?
For the sake of conceptual simplicity, consider a classical system (e.g. a planet) with a measured trajectory ##x(t)##. And suppose that this trajectory can be explained theoretically by two different Hamiltonians. Without considering any other experiments/measurements, is it possible to determine which Hamiltonian is the correct one? It seems obvious to me that the answer is - no. Therefore, the Hamiltonian itself is not a physical object. Therefore it cannot define physical objects, including physical subsystems.
 
  • #84
To reconstruct the probability distribution, you need all moments of it, the average is not enough. This has nothing to do with the foundations of QT. So please for get this irrelevant discussion of standard theorems of probability theory and take Born's rule to mean the probabilities for the outcome of measurements given the operator algebra of observables and the state in terms of the statistical operator.
 
  • #85
Demystifier said:
For the sake of conceptual simplicity, consider a classical system (e.g. a planet) with a measured trajectory ##x(t)##. And suppose that this trajectory can be explained theoretically by two different Hamiltonians. Without considering any other experiments/measurements, is it possible to determine which Hamiltonian is the correct one? It seems obvious to me that the answer is - no. Therefore, the Hamiltonian itself is not a physical object. Therefore it cannot define physical objects, including physical subsystems.

Hope Peterdonis can address this as it's his belief the state vector may be sufficient in itself to create the subsystems.

Anyway. What would happen if you use the Schroedinger equations without any implicit basis in the pure MWI (without any basis). Can the Schroedinger Equation still produce output with all nonorthogonal results? And what good would be this output?

And can you propose experiments more complex than the Stern-Gerlach setup to tell whether the state vector in MWI can or really can't produce a basis without additional structure. What efforts are being done in this department?
 
  • #86
Blue Scallop said:
Anyway. What would happen if you use the Schroedinger equations without any implicit basis in the pure MWI (without any basis). Can the Schroedinger Equation still produce output with all nonorthogonal results? And what good would be this output?
One could get the spectrum of Hamiltonian.

Blue Scallop said:
And can you propose experiments more complex than the Stern-Gerlach setup to tell whether the state vector in MWI can or really can't produce a basis without additional structure.
I cannot.
 
  • #87
Demystifier said:
One could get the spectrum of Hamiltonian.I cannot.

Theoretically if say you suddenly removed the position basis of an object let's say a post office postcard.. what would happen to the atoms and molecules inside.. would they become damaged like being exposed to fire where all atoms/molecules are rearrange permanently or destroyed??

And can you think of a theoretical way to save the information of position somewhere so that you can dematerialize the post office postcard.. and rematerialize it later with all positions intact?
 
  • #88
Blue Scallop said:
Theoretically if say you suddenly removed the position basis of an object let's say a post office postcard.. what would happen to the atoms and molecules inside.. would they become damaged like being exposed to fire where all atoms/molecules are rearrange permanently or destroyed??
The question does not make much sense to me. What does it mean to remove the position basis of an object?
 
  • #89
Demystifier said:
The question does not make much sense to me. What does it mean to remove the position basis of an object?

Simple.. to remove the position basis so the object would no longer have position.. or in terms of Bohmian Mechanics.. to remove position as preferred.. so it's like transforming the object into pure MWI vector or Hilbert space. Now if this occurs.. are the information of the arrangements of the molecules of the object still intact? Only they become pure Hilbert space vectors? and if you enable the position basis again.. would the information return or would the object become unrecognizable and become a mere blob because the nonorthogonal conversion to orthogonal is random hence the position information can't be stored and reread?
 
  • #90
Blue Scallop said:
Simple.. to remove the position basis so the object would no longer have position.. or in terms of Bohmian Mechanics.. to remove position as preferred.. so it's like transforming the object into pure MWI vector or Hilbert space. Now if this occurs.. are the information of the arrangements of the molecules of the object still intact? Only they become pure Hilbert space vectors? and if you enable the position basis again.. would the information return or would the object become unrecognizable and become a mere blob because the nonorthogonal conversion to orthogonal is random hence the position information can't be stored and reread?
I think such a question can be meaningfully asked only by using mathematical equations.
 
  • #91
Demystifier said:
Consider a physical system in which the average value of position is ##<x>=0##. What is the probability that the position is ##x=1## nm?

But if you also know [itex]\langle x^2 \rangle, \langle x^3 \rangle, ...[/itex]?
 
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  • #92
Demystifier said:
I think such a question can be meaningfully asked only by using mathematical equations.

Ok, mathematically.. if we remove the position basis.. then there would be no decoherence and no subsystems if there are no other basis to define it. And it's back to pure Hilbert space vectors with unit trace 1.

So the question is.. what kind of information can be stored in the Hilbert space nonorthogonal states? How complex the information it can hold? Can they store a Barbie doll information? or nothing? just want to have idea...
 
  • #93
Demystifier said:
suppose that this trajectory can be explained theoretically by two different Hamiltonians

Is this possible? Can you give an actual example?
 
  • #94
Demystifier said:
Consider a physical system in which the average value of position is ##<x>=0##. What is the probability that the position is ##x=1## nm?

To be more conventional, one can use the term "expectation" in place of "average". Conceptually, there is no difference. You can see A. Neumaier's post #78 for the mathematical fine print.
 
  • #95
Blue Scallop said:
Ok, mathematically.. if we remove the position basis.. then there would be no decoherence and no subsystems if there are no other basis to define it. And it's back to pure Hilbert space vectors with unit trace 1.
Yes.
 
  • #96
PeterDonis said:
Is this possible?
Yes.

PeterDonis said:
Can you give an actual example?
Solution ##x(t)=0##.
$$H_1=\frac{p^2}{2m}$$
$$H_2=\frac{p^2}{2m}+\frac{k x^2}{2}$$
 
<h2>1. What is the Born rule in thermodynamics?</h2><p>The Born rule is a fundamental principle in quantum mechanics that describes the probability of obtaining a particular measurement outcome for a quantum system. In thermodynamics, the Born rule is used to calculate the probability of a system being in a particular thermodynamic state.</p><h2>2. How does the Born rule relate to entropy in thermodynamics?</h2><p>The Born rule is closely related to entropy, which is a measure of the disorder or randomness in a system. In thermodynamics, the Born rule is used to calculate the probability of a system being in a particular microstate, which is a specific arrangement of particles within the system. The more microstates a system can occupy, the higher its entropy.</p><h2>3. Can the Born rule be applied to macroscopic systems in thermodynamics?</h2><p>Yes, the Born rule can be applied to macroscopic systems in thermodynamics. While the rule is often used to describe the behavior of individual particles, it can also be applied to larger systems by considering the probabilities of different macroscopic states.</p><h2>4. How does the Born rule account for irreversible processes in thermodynamics?</h2><p>The Born rule does not explicitly account for irreversible processes in thermodynamics. However, it can be used to calculate the probabilities of different states in a system, which can then be used to analyze the likelihood of irreversible processes occurring.</p><h2>5. Is the Born rule a fundamental law of thermodynamics?</h2><p>No, the Born rule is not a fundamental law of thermodynamics. It is a principle in quantum mechanics that is often used in thermodynamics to calculate probabilities and analyze systems. The fundamental laws of thermodynamics include the laws of conservation of energy and the increase of entropy in isolated systems.</p>

1. What is the Born rule in thermodynamics?

The Born rule is a fundamental principle in quantum mechanics that describes the probability of obtaining a particular measurement outcome for a quantum system. In thermodynamics, the Born rule is used to calculate the probability of a system being in a particular thermodynamic state.

2. How does the Born rule relate to entropy in thermodynamics?

The Born rule is closely related to entropy, which is a measure of the disorder or randomness in a system. In thermodynamics, the Born rule is used to calculate the probability of a system being in a particular microstate, which is a specific arrangement of particles within the system. The more microstates a system can occupy, the higher its entropy.

3. Can the Born rule be applied to macroscopic systems in thermodynamics?

Yes, the Born rule can be applied to macroscopic systems in thermodynamics. While the rule is often used to describe the behavior of individual particles, it can also be applied to larger systems by considering the probabilities of different macroscopic states.

4. How does the Born rule account for irreversible processes in thermodynamics?

The Born rule does not explicitly account for irreversible processes in thermodynamics. However, it can be used to calculate the probabilities of different states in a system, which can then be used to analyze the likelihood of irreversible processes occurring.

5. Is the Born rule a fundamental law of thermodynamics?

No, the Born rule is not a fundamental law of thermodynamics. It is a principle in quantum mechanics that is often used in thermodynamics to calculate probabilities and analyze systems. The fundamental laws of thermodynamics include the laws of conservation of energy and the increase of entropy in isolated systems.

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