Conservation of energy and wave function collapse

[KleinMoretti]

TL;DR

Is conservation of energy violated by the collapse of the wave function in standard quantum mechanics?

I was reading an old paper titled “Wavefunction Collapse and Conservation Law” where it is explicitly mentioned that the collapse of the wave function in standard quantum mechanics violates conservation of energy. “It is not generally appreciated that the collapse postulate of standard quan- tum theory (SQT) can violate the geometric conservation laws, e.g., the conser- vation of energy and momentum[1].”
I thought that conservation of energy was respected in QM.


Link to the paper. https://arxiv.org/pdf/quant-ph/0004067

 

[PeterDonis]

I thought that conservation of energy was respected in QM.

Conservation of energy requires a well-defined dynamics that has the requisite properties (basically that the Lagrangian or Hamiltonian is independent of time). Generally speaking that is true for unitary dynamics in QM, but it is not true for collapse because collapse has no well-defined dynamics; it's just declaring by fiat that the wave function changes discontinuously. So one would not expect conservation laws to hold in such a case.

However, it should be noted that, in basic QM (i.e., without adopting any particular interpretation--discussion of how particular QM interpretations deal with collapse belongs in the interpretations subforum), "collapse" is not claimed to be an actual physical process; it's just an update you make in your mathematical model when you know the result of a measurement. Since making a measurement on a quantum system requires the system to be open, i.e., it has to interact with other systems, one would not expect conservation laws to hold for the measured system alone, since conserved quantities can be exchanged through the interaction. For example, the measured system could gain or lose energy from the interaction that takes place during meaurement, so its energy taken in isolation would not be conserved. Only the energy of the whole larger system, including the measuring device and anything else the measuring device interacted with, would be conserved.

The paper you reference (which, btw, is a preprint and it's not clear whether it has actually been published) does not appear to take such things into account. Its view of "collapse" also appears to be interpretation dependent--it treats it as an actual physical process, which, as above, is not how basic QM independent of any intepretation treats it.

 

[KleinMoretti]

The paper you reference (which, btw, is a preprint and it's not clear whether it has actually been published) does not appear to take such things into account. Its view of "collapse" also appears to be interpretation dependent--it treats it as an actual physical process, which, as above, is not how basic QM independent of any intepretation treats it.

I thought when he said that the collapse postulate of standard quantum mechanics violates conservation of energy/momentum he was talking about QM free of interpretation and thus not treating the collapse as a physical process.
He turns says “This violation of the conservation of particle energy has also been the focus of criticism of the CSL model[23, 24]. As has been emphasized in section 1, this criticism also deserves to be applied to SQT plus the collapse postulate.” Which I took it as him saying that even QM independent of any interpretation has this problem

 

[pines-demon]

I thought when he said that the collapse postulate of standard quantum mechanics violates conservation of energy/momentum he was talking about QM free of interpretation and thus not treating the collapse as a physical process.
He turns says “This violation of the conservation of particle energy has also been the focus of criticism of the CSL model[23, 24]. As has been emphasized in section 1, this criticism also deserves to be applied to SQT plus the collapse postulate.” Which I took it as him saying that even QM independent of any interpretation has this problem

I have no idea of what CSL is, but as far as I know objective-collapse models (Ghirardi–Rimini–Weber, Penrose) suffer from the non-conservation of energy, and this feature can be used to test those models. Many-worlds interpretation may have the same issue, at least as argued by Sean Carroll (https://arxiv.org/pdf/2101.11052) but it might be an original take on it. I do not think the problem exists for other interpretations.

 

[KleinMoretti]

I have no idea of what CSL is, but as far as I know objective-collapse models (Ghirardi–Rimini–Weber, Penrose) suffer from the non-conservation of energy, and this feature can be used to test those models. Many-worlds interpretation may have the same issue, at least as argued by Sean Carroll (https://arxiv.org/pdf/2101.11052) but it might be an original take on it. I do not think the problem exists for other interpretations.

Funny that you mention Sean Carroll’s paper because that is where I came across this paper which he uses as a reference. (btw I think Carroll’s paper actually argues that MWI is the only interpretation that does conserve energy)

 

[pines-demon]

Funny that you mention Sean Carroll’s paper because that is where I came across this paper which he uses as a reference. (btw I think Carroll’s paper actually argues that MWI is the only interpretation that does conserve energy)

Kind of. When he says that it conserves energy, it is in the sense of the entirety of many-worlds (or for many measurements), for an observer that sees a single branch after a single measurement, energy is not conserved... Or at least that is what I get out of it.

 

[KleinMoretti]

Conservation of energy requires a well-defined dynamics that has the requisite properties (basically that the Lagrangian or Hamiltonian is independent of time). Generally speaking that is true for unitary dynamics in QM, but it is not true for collapse because collapse has no well-defined dynamics; it's just declaring by fiat that the wave function changes discontinuously. So one would not expect conservation laws to hold in such a case.

However, it should be noted that, in basic QM (i.e., without adopting any particular interpretation--discussion of how particular QM interpretations deal with collapse belongs in the interpretations subforum), "collapse" is not claimed to be an actual physical process; it's just an update you make in your mathematical model when you know the result of a measurement. Since making a measurement on a quantum system requires the system to be open, i.e., it has to interact with other systems, one would not expect conservation laws to hold for the measured system alone, since conserved quantities can be exchanged through the interaction. For example, the measured system could gain or lose energy from the interaction that takes place during meaurement, so its energy taken in isolation would not be conserved. Only the energy of the whole larger system, including the measuring device and anything else the measuring device interacted with, would be conserved.

The paper you reference (which, btw, is a preprint and it's not clear whether it has actually been published) does not appear to take such things into account. Its view of "collapse" also appears to be interpretation dependent--it treats it as an actual physical process, which, as above, is not how basic QM independent of any intepretation treats it.

I guess my question is if what the paper says about conservation of energy being violated by the collapse of the wave function in standard QM is an accepted view nowadays, like I said in another reply I originally came across this paper because Sean Carroll used it as reference for his paper arguing in support of the MWI which came out fairly recently.

 

[KleinMoretti]

Kind of. When he says that it conserves energy, it is in the sense of the entirety of many-worlds (or for many measurements), for an observer that sees a single branch after a single measurement, energy is not conserved... Or at least that is what I get out of it.

Yes you are right, that is what I meant by only interpretation that does conserve energy even if it is across all branches.

 

[pines-demon]

Yes you are right, that is what I meant by only interpretation that does conserve energy even if it is across all branches.

The problem I see with interpretationless quantum mechanics, is that it does not tell anything about the dynamics of your measurement device. Suppose that you prepare a superposition of energy eigenstates $$|\psi\rangle=\frac1{\sqrt2}(|0\rangle+|1\rangle)$$
where 0 is the label for the ground state with energy $E_0$ and 1 is the label for an excited state with energy $E_1>E_0$. You start with a superposition and after a measurement you get either $E_0$ or $E_1$ how do you account for the energy conservation here?

No matter how you prepare the system, after measurement the energy is less or more than expected (the expectation value of the energy before measurement is $(E_0+E_1)/2$ ). Now clearly we ignored what happened with the measurement device (did it gain/loss energy? ). And how to model the device seems to be interpretation dependent.

 

[PeterDonis]

I thought when he said that the collapse postulate of standard quantum mechanics violates conservation of energy/momentum he was talking about QM free of interpretation and thus not treating the collapse as a physical process.

If it's not a physical process it can't violate anything. The only way the claim that conservation of energy is violated in collapse even makes sense is if collapse is a physical process.

 

[PeterDonis]

The problem I see with interpretationless quantum mechanics, is that it does not tell anything about the dynamics of your measurement device.

You can include the measuring device in your dynamics, but that of course makes things much, much more complicated. It is done, however, in some of the analysis in the decoherence literature.

 

[PeterDonis]

how to model the device seems to be interpretation dependent.

No, not interpretation dependent, just very complicated. All QM interpretations use equivalent mathematical models.

 

[PeterDonis]

if what the paper says about conservation of energy being violated by the collapse of the wave function in standard QM is an accepted view nowadays

I don't think so. Since the question of what "collapse of the wave function" actually means is still open (there is no single "accepted view"), any other questions related to collapse must also be open.

 

[KleinMoretti]

If it's not a physical process it can't violate anything. The only way the claim that conservation of energy is violated in collapse even makes sense is if collapse is a physical process.

I believe he is saying that the violation is found in the difference of the initial and final values after measurement, "It is emphasized that the collapse postulate of standard quantum the-ory can violate conservation of energy-momentum and there is no in-dication from where the energy-momentum comes or to where it goes."

 

[PeterDonis]

I believe he is saying that the violation is found in the difference of the initial and final values after measurement

But, as I've already said, the system is not isolated during measurement so just looking at it, without looking at the measuring device and anything else that might interact with either, is not a valid way of assessing any conservation law.

there is no in-dication from where the energy-momentum comes or to where it goes

This is a drastic overstatement. We cannot possibly keep track of things like the energy of a measuring device, let alone the rest of the environment, accurately enough to be able to say that the tiny amount of energy required to balance the books in a measurement cannot come from the measuring device. The fact that we can't detect such an energy exchange is no reason at all to say that it can't possibly happen and claim a violation of conservation of energy. Similar remarks would apply to any conservation law.

 

[PeterDonis]

I believe he is saying that the violation is found in the difference of the initial and final values after measurement

Btw, this claim is actually interpretation dependent; it assumes that the wave functions before and after the measurement describe the actual physical state of the system being measured. Not all interpretations agree with that.

 

[KleinMoretti]

But, as I've already said, the system is not isolated during measurement so just looking at it, without looking at the measuring device and anything else that might interact with either, is not a valid way of assessing any conservation law.


This is a drastic overstatement. We cannot possibly keep track of things like the energy of a measuring device, let alone the rest of the environment, accurately enough to be able to say that the tiny amount of energy required to balance the books in a measurement cannot come from the measuring device. The fact that we can't detect such an energy exchange is no reason at all to say that it can't possibly happen and claim a violation of conservation of energy. Similar remarks would apply to any conservation law.

so you are saying that the paper doesn't take into account the openness of the system and any exchange of energy that this entails?

 

[PeterDonis]

so you are saying that the paper doesn't take into account the openness of the system and any exchange of energy that this entails?

It doesn't appear to, no.

 

[Demystifier]

The observables such as energy, momentum, angular momentum and charge are conserved in the sense that the corresponding operator does not have an explicit time dependence. Let $A$ be any observable conserved in that sense. If you measure A (and hence collapse the state) and then measure A again (and hence collapse the state again), the two observed values of $A$ will be the same. In this sense, the collapse does not violate the conservation of $A$.

 

[Lord Jestocost]

I have no idea of what CSL is ........

CSL: Continuous Spontaneous Localization

 

[pines-demon]

CSL: Continuous Spontaneous Localization

I mean those clearly are words, but what does it mean? Is it an interpretation? A theory?

 

[DrClaude]

The paper you reference (which, btw, is a preprint and it's not clear whether it has actually been published)

Found.Phys. 30 (2000) 1145-1160
https://doi.org/10.1023/A:1003677103804

 

[Lord Jestocost]

I mean those clearly are words, but what does it mean? Is it an interpretation? A theory?

I think the term “CSL” is used in the realm of "objective-collapse theories".

 

[physika]

I mean those clearly are words, but what does it mean? Is it an interpretation? A theory?

Another model.
Akin to GRW.

 

[Vanadium 50]

Am I missing something? To test conservation of energy, you need a well-defined energy in the "before" and "after" states. If your system is not in an energy eigenstate, of course energy is not conserved.

 

[weirdoguy]

If your system is not in an energy eigenstate, of course energy is not conserved.

If it's not in an energy eigenstate, does it even make sense to consider conservation of energy? If not, then you can't say energy is or is not conserved.

 

[Vanadium 50]

Exactly. It makes no sense to talk about the energy of a system not in an energy eigenstate.

 

[pines-demon]

Am I missing something? To test conservation of energy, you need a well-defined energy in the "before" and "after" states. If your system is not in an energy eigenstate, of course energy is not conserved.

Well I do not think that is justifiable. You put some energy to prepare the state, then you measure some energy (plus the energy you gain-lose in order to perform the measurement). The balance can be 0, positive or negative.

However when we say "energy is not conserved in quantum mechanics" it seems that the remark is focused on the single particle states, it kinda ignores how to take into account the measurement and the environment.

 

[Lord Jestocost]

If one accepts that the quantum mechanical state function $\Psi$ is nothing more than a catalog of knowledge following from one observed fact and determining the probabilities for possible future events, issues such as “energy is not conserved in quantum mechanics” become obsolete.

 

[Nugatory]

You put some energy to prepare the state, then you measure some energy (plus the energy you gain-lose in order to perform the measurement). The balance can be 0, positive or negative.

That is, we consider the total system including the preparation apparatus and measuring device? This works as long we treat the total energy of this larger system classically so that the net energy flow is defined. But if we think quantum mechanically, we cannot speak of the energy of that larger system without preparing the system and measuring the energy; we end up with an infinite recursion of ever larger systems under consideration.

The moral of the story: the notion of a classical/quantum split is very convenient even if the interpretational foundations are shaky.