Energy Non-Conservation in Quantum Mechanics

[Coelum]

TL;DR Summary

Is energy conserved (not in a statistical sense) during a measurement in QM?

Dear PFer's,
this is a problem I have been struggling with for years. Is energy conserved (not in a statistical sense) in QM? The so-called collapse of the wavefunction, occurring during a measurement process, is incompatible with energy conservation - at least in the general case. A starting point for the discussion is Sean Carrol's paper with the same title as this post (https://arxiv.org/abs/2101.11052). In Carroll's classification, I favour the idea that our description is not complete, though not in the sense of some "hidden variable" theory but rather as an incomplete description of the measurement process itself.
As a side note, I understand that, on a cosmological level, energy is not conserved. Both the non-zero value of the cosmological constant and the time asymmetry in the history of our universe show it is not. I just wonder if we can somehow recover the energy conservation at a more "local" level, like we do in Classical Mechanics thanks to Noether's theorem.

I'd really appreciate your point of view.

 

[PeterDonis]

Is energy conserved (not in a statistical sense) in QM? The so-called collapse of the wavefunction, occurring during a measurement process, is incompatible with energy conservation - at least in the general case.

The system is not a closed system during measurement, so we would not expect conservation laws to hold during measurement for the system taken by itself. Carroll's paper shows explicitly that energy conservation will not in general hold during a measurement, but that should not be surprising.

 

[PeterDonis]

Sean Carrol's paper

Note that this paper adopts a particular interpretation, the MWI. Other interpretations might approach the question differently. Discussion of QM interpretations belongs in the interpretations subforum.

 

[Coelum]

The system is not a closed system during measurement, so we would not expect conservation laws to hold during measurement for the system taken by itself. Carroll's paper shows explicitly that energy conservation will not in general hold during a measurement, but that should not be surprising.

I agree, my point is: is energy conserved for the closed system made up of the quantum system and the observer? Quoting Sean Carroll: "One purpose of this paper is to show that this is not the case, and that the total energy of the system plus apparatus plus environment can change."

 

[Coelum]

Note that this paper adopts a particular interpretation, the MWI. Other interpretations might approach the question differently. Discussion of QM interpretations belongs in the interpretations subforum.

OK, how can I move it?

 

[PeterDonis]

is energy conserved for the closed system made up of the quantum system and the observer?

This is not a closed system either, since it is interacting with the surrounding environment (such interaction is part of decoherence). So again Carroll's conclusion should not be surprising. Even when Carroll includes "the enviroment" in the analysis, that still is not the entire universe, which would be the only truly closed system we could analyze.

 

[PeterDonis]

OK, how can I move it?

I can move it, but that might not be necessary. Carroll notes at the end of the paper that, although he has used the MWI in his analysis, the conclusions should not change for other interpretations. So it depends on whether you want to discuss this question purely from the standpoint of the MWI, or whether you want to discuss it generally, as a feature of QM independent of any interpretation.

 

[Coelum]

I can move it, but that might not be necessary. Carroll notes at the end of the paper that, although he has used the MWI in his analysis, the conclusions should not change for other interpretations. So it depends on whether you want to discuss this question purely from the standpoint of the MWI, or whether you want to discuss it generally, as a feature of QM independent of any interpretation.

I have no issue with the interpretation. I suspect that the measurement process is not correctly modelled. After all, the measurement process is irreversible, while QM is not. Are you aware of any analysis along this line of thought?

 

[PeterDonis]

the measurement process is irreversible, while QM is not

More precisely, the measurement process is irreversible, while unitary evolution is not. But the bare math of QM, without adopting any interpretation, says that unitary evolution (i.e., evolution according to the Schrodinger equation or equivalent) only applies to an isolated quantum system in the absence of measurement. See, for example, Rule 3 in PF's 7 Basic Rules of QM:

https://www.physicsforums.com/threads/the-7-basic-rules-of-quantum-mechanics.971724/

Are you aware of any analysis along this line of thought?

The observations made above are ubiquitous in the QM literature.

 

[Coelum]

I am having a hard time presenting my concerns about the measurement process. So, let me try a different point of view. How can it be that the Universe behaves differently when a human being observes (i.e., measures)? We can try and save the measurement postulate (von Neumann projection postulate) by assuming that, instead of "measurement" we have "irreversibility". That leaves human beings out of the loop. But irreversibility is incompatible with the evolution equation of QM, just like in CM. The situation reminds me of Boltzmann's "molecular chaos" hypothesis. Maybe we are missing a similar, additional hypothesis, possibly of thermodynamic nature?

 

[PeroK]

How can it be that the Universe behaves differently when a human being observes (i.e., measures)?

Human beings have no special place in QM.

But irreversibility is incompatible with the evolution equation of QM, just like in CM.

It isn't. Statistical irreversibility is compatible with fully reversible laws of nature.

Maybe we are missing a similar, additional hypothesis, possibly of thermodynamic nature?

Or, maybe you don't understand QM and Quantum Statistical mechanics?

 

[Coelum]

Human beings have no special place in QM.

It isn't. Statistical irreversibility is compatible with fully reversible laws of nature.

Or, maybe you don't understand QM and Quantum Statistical mechanics?

1- what is the meaning of measurement? how is it different from any other interaction?
2- how can we derive irreversible laws from reversible laws without additional assumptions?
3- maybe - I'm here to learn.

 

[PeroK]

1- what is the meaning of measurement? how is it different from any other interaction?

The key concept you may be missing is quantum decoherence.

https://en.wikipedia.org/wiki/Quantum_decoherence

Whether decoherence fully explains the measurement problem is an open question. Here's something I found on this.

https://arxiv.org/pdf/2301.01207.pdf

2- how can we derive irreversible laws from reversible laws without additional assumptions?

The point is certain processes are statistically very likely to happen (to the point where they are essentially inevitable) and the reverse process is statistically very unlikely to happen (to the point where they are essentially impossible).

One simple example is to consider a large number of lights (let's say 1,000). Each light can be on or off. The law is that each second one of the lights is chosen at random and changes: if it is on, it switches off; and vice versa. The next second another light is chosen. This law is symmetric and if you observe an equilibrium state, where approx half the lights are on and half are off, then you cannot tell whether time is running forwards or backwards. Every individual change is equally likely and the time evolution looks generally the same forwards and backwards in time. I.e. this simple law is fully reversible.

However, if we start with a state where all 1000 lights are on, then we quickly transition to an equilibrium state. And, of course, even with only 1000 lights, we will statistically never return to the original state, where all light are on. This process is, therefore, irreversible. And, if you watch this process in reverse, with gradually all the lights ending up back on again, then you know statistically that you are watching the process in reverse. Although, slightly paradoxically, you cannot identify any particular change as being out of the ordinary.

This is the same idea if you stir a chocolate sauce into something. Technically, if you precisely reversed your stirring, the chocolate sauce would unmix. But, of course, statistically that is never going to happen - even though the mixing process is fully reversible on a microscopic scale.

3- maybe - I'm here to learn.

Sean Carroll is a great physicist, but IMO he has taken his preference for MWI too far. I agree that the MWI is the simplest interpretation of QM, but he shouldn't promote that interpretation to the exclusion of all else. And, in fact, there is an argument that MWI simply shifts the measurement problem, rather than solves it. In the sense that the MWI has no real explanation for the Born rule.

Personally, I remain agnostic when it comes to interpretations.

 

[Vanadium 50]

An atom is in an excited state. It falls to its ground state, emitting a photon (which may or may not be detected). Now it has less energy.

Does this bother you? We can put more bells and whistles on the problem, but that adds to the confusion, not the clarity.

 

[anuttarasammyak]

On this topic a blog by Dr. Hotta, Tohoku University "Where did the missing energy come from when you find a particle in the tunnel region? " in Japanese, https://mhotta.hatenablog.com/entry/2014/04/28/194922 was informative to me. We can read it by web translation service to our mother tongues.　

 

[PeterDonis]

How can it be that the Universe behaves differently when a human being observes (i.e., measures)?

QM does not say this. A measurement does not require human intervention.

irreversibility is incompatible with the evolution equation of QM

As already noted, the evolution equation you refer to only applies to an isolated quantum system in the absence of measurement. And under those circumstances, there is no irreversibility.

 

[PeterDonis]

1- what is the meaning of measurement? how is it different from any other interaction?

A measurement occurs when an irreversible record is made. The modern view is that this happens when decoherence occurs.

2- how can we derive irreversible laws from reversible laws without additional assumptions?

We don't. The irreversibility of measurement is not derived from the reversible unitary evolution law.

 

[romsofia]

Seems like your question is digging moreso at measurement than conservation of energy, and so this paper may answer your question, or help you phrase your question better: https://arxiv.org/abs/quant-ph/0312059

 

[vanhees71]

A measurement occurs when an irreversible record is made. The modern view is that this happens when decoherence occurs.We don't. The irreversibility of measurement is not derived from the reversible unitary evolution law.

Well, decoherence can be derives from unitary time evolution, looking at an open quantum system, i.e., a subsystem of a larger closed system. The most clear approach is via the Schwinger-Keldysh real-time formalism and the 2PI approach to derive the Kadanoff-Baym equations, guaranteeing thermodynamical consistency in the equilibrium limit as well as the evolution to the proper thermal equilibrium state for the sub system when it is coupled to an "environment" in thermal equilibrium ("heat bath").

 

[PeterDonis]

decoherence can be derives from unitary time evolution

Yes, but by itself, decoherence in principle could be reversible. It's only irreversible for all practical purposes (FAPP, as Bell liked to say). For true in principle irreversibility, something else has to be added.

Of course, whether true in principle irreversibility is actually needed depends on which QM interpretation you adopt.

 

[Coelum]

First of all, I want to thank all the kind people who took the time to try and help me understand better.

I'm aware I am touching a critical point. To make things worst, I was not able to present my thoughts in a clean, ordered way...
As next step, I will read the paper quoted by @romsofia. After that, I hope I'll be able to pose a better defined question.

 

[LittleSchwinger]

Just to add decoherence isn't really the main factor responsible for the classical limit. Even before the investigation of decoherence in the 1970s there were detailed models of classicality being caused by ergodic effects or kinematic effects reducing the algebra of observables, such as in the WAY theorem.

In fact if one thinks about it, diagonalisation of the density matrix alone can't really give you classicality since for a qubit even the purely mixed state with a diagonal density matrix can be easily shown to display quantum effects.

A good, and very long, guide to all this is the well known paper of Allahverdyan et al:
https://arxiv.org/abs/1107.2138

 

[Coelum]

Just to add decoherence isn't really the main factor responsible for the classical limit. Even before the investigation of decoherence in the 1970s there were detailed models of classicality being caused by ergodic effects or kinematic effects reducing the algebra of observables, such as in the WAY theorem.

In fact if one thinks about it, diagonalisation of the density matrix alone can't really give you classicality since for a qubit even the purely mixed state with a diagonal density matrix can be easily shown to display quantum effects.

A good, and very long, guide to all this is the well known paper of Allahverdyan et al:
https://arxiv.org/abs/1107.2138

Great reference, thanks!