Is Energy an Illusion in the Many-Worlds Interpretation?

[lomidrevo]

TL;DR

How conservation of energy works in many-worlds interpretation of QM?

I think I have a rough idea about it, but I am not sure whether it is correct. At least I feel that my understanding is a bit vague. Here it is:

Globally (I mean across all worlds), the energy is conserved because the universal wavefunction evolves strictly according to Schrödinger equation. That is clear. But how is that reflected in observations done by someone living in one of the worlds? Here is my clue:

When the wavefunction branches, the "existence" of new worlds is generally not equal. Instead, each new world (or branch) comes with a weight that is proportional to the amplitude squared (as described by the wavefunction). So if we sum energies of all new branches $E_i$, the result would be equal to the overall energy of the "parent" world $E$ as it was just before branching. The energies of new branches are indeed the overall energy $E$ multiplied by the corresponding weights $W_i$:
$$E = \sum{E_i} = \sum{W_i E}$$
Let's say that Alice is going to do some quantum experiment, and just before she manages to measure overall energy of the world $E$.
After the experiment, the world has branched, and she find herself in a particular new branch where one of the outcomes of the experiment is realized. Now, when she measures the energy of the world it must be again $E$, (and not $E_i$) otherwise she would conclude that energy is not conserved. So how should be this interpreted? According to Alice in particular new branch, the measured total energy is still $E$, but the "real" contribution of this branch to overall energy is only $W_iE = E_i$?

Am I wrong or do I miss something? If you can correct me or provide any relevant texts on this topic, I would be welcome.

 

[PeroK]

There was a long thread about this recently:

https://www.physicsforums.com/threads/conservation-of-energy-in-everetts-mwi.994897/

The simplest answer is that energy conservation in MWI is not fundamentally different from energy conservation in orthodox QM. In MWI, however, the superposition continues after measurement, with the measurement apparatus part of the the system

 

[Demystifier]

According to the many-worlds interpretation (MWI), the only real thing is the wave function. Everything else is not real. Energy is not real, momentum is not real, position is not real, etc. They are just illusions. So when we observe that energy is conserved, it's just an illusion. Where do these illusions come from? They come from measurements. For instance, when the shape of the wave function of letters (on the screen of the measuring apparatus) takes the shape of letters "E=1J", it looks as if the energy is equal to 1 Joule.

Now what happens when the wave function is split into two wave functions? If the split conserves energy, then splitting does not change the shape responsible for the information about energy. In other words, "E=1J" label splits into two "E=1J" labels. It's almost like xeroxing a paper on which "E=1J" is written. In this way, it looks as if each branch has E=1J energy. In reality, it does not have energy at all. It just has an "E=1J" label on it.

More formally, if the state $|E=1J\rangle$ has energy 1 Joule, then what is the energy of the state
$$\frac{1}{\sqrt{2}}|E=1J\rangle?$$
Naively, one might think that its energy is 0.5J, but it isn't. Its energy is still E=1J. The energy is encoded in the state $|E=1J\rangle$, not in its norm.

 

[lomidrevo]

There was a long thread about this recently:

https://www.physicsforums.com/threads/conservation-of-energy-in-everetts-mwi.994897/

I was searching the forum before creating this thread, but surprisingly "energy many worlds" aren't the correct keywords to hit this one . I am not able to add direct quotes from that thread, probably because it is locked, so let me try like this:

@PeterDonis said in post #14:

You left out the part about a weighted average. Each branch has the same energy as before the split according to the observer in that branch. But each branch contributes less energy than before the split to the weighted average, since before the split there was only one branch so the weighted average was just the energy in that branch, but after the split each branch's contribution to the weighted average is its energy "from the inside" multiplied by the appropriate weighting factor, which is less than 1.

and illustrated by an example (spin measurement) in post #23.

I used different words, but I think my understanding is similar. "From the inside", Alice would measure $E$ before and after the experiment, so for her the energy is conserved. From the global point of view (universe described by single universal wavefunction), the total energy must be calculated as weighted average, following the rules of superposition. So, after all, it makes sense to write
$$E = \sum{W_iE}$$
Writing $E=\sum{E_i}$ is not strictly incorrect, but perhaps it is misleading, because $E_i$ individually doesn't have any physical meaning, and cannot be measured. It makes more sense to me now, when I reformulate my understanding like this.

In post #28, you (@PeroK) wrote:

Now you're not so happy. You accepted that a silver atom could be in a superposition of states without violating conservation of energy, but you are not happy that the detector and you and macroscopic things can be in a superposition of states. These things can't be in a superposition without violating conservation of energy.

Sorry for taking your sentence out of the context, but could you please explain what do you mean? If we stick with the weighted average described before, energy is conserved even in the superposition, no?

 

[lomidrevo]

In MWI, however, the superposition continues after measurement, with the measurement apparatus part of the the system

I think I understand this part and I have no problem to accept it. Actually, this is what I like about MWI, how elegantly it gets rid of collapse of the wavefunction and solves the measurement problem. What is more tricky to me, is to map the idea of "all universe being in a superposition described by single wavefunction" to our observational experience and hence my previous confusion about conservation of energy.

 

[lomidrevo]

According to the many-worlds interpretation (MWI), the only real thing is the wave function. Everything else is not real. Energy is not real, momentum is not real, position is not real, etc. They are just illusions. So when we observe that energy is conserved, it's just an illusion. Where do these illusions come from? They come from measurements. For instance, when the shape of the wave function of letters (on the screen of the measuring apparatus) takes the shape of letters "E=1J", it looks as if the energy is equal to 1 Joule.

Now what happens when the wave function is split into two wave functions? If the split conserves energy, then splitting does not change the shape responsible for the information about energy. In other words, "E=1J" label splits into two "E=1J" labels. It's almost like xeroxing a paper on which "E=1J" is written. In this way, it looks as if each branch has E=1J energy. In reality, it does not have energy at all. It just has an "E=1J" label on it.

More formally, if the state $|E=1J\rangle$ has energy 1 Joule, then what is the energy of the state
$$\frac{1}{\sqrt{2}}|E=1J\rangle?$$
Naively, one might think that its energy is 0.5J, but it isn't. Its energy is still E=1J. The energy is encoded in the state $|E=1J\rangle$, not in its norm.

This is interesting point of view. I've never thought about it this way. I will try to consider it in more details. For now, I can just say it severely hurts my intuition, but honestly I am not so surprised anymore. I fully realize that intuition based on our daily experience is not the best advisor when it comes to fundamentals of physics