Conservation of energy in Everett's MWI

[name123]

TL;DR

How does conservation of energy work in Everett's MWI

The question seems similar to the one asked here,
https://www.physicsforums.com/threads/energy-in-everetts-many-worlds-interpretation.966266/
but since there didn't seem to be an answer I am asking it again in a slightly different form.

I was watching a youtube video where Sean Carroll explained that the energy of the waveform is conserved, and that the branches can be thought of subdivisions of the waveform, and that the subdivisions contribute less energy than the original. That seems to be a slightly different answer to one given in the thread I referenced, and I assume that is because there are different versions of how to interpret it. But what I am not clear on is how the energy of the waveform can be calculated and how the calculated value can be considered to be conserved. To highlight my point what if I consider an example of a person holding a battery connected to a lightbulb. Sean Carroll explained that our bodies contain radio active nuclei that decay 5000 times a second. Whether the decay happens or not, and presumably the direction of the decay will each lead to a branch within the waveform which would be entangled with the battery + lightbulb system (as long as there would have been a branch where the radiation did interact with that system). I am assuming the emission of a photon from the light bulb would require a certain amount of energy, and that for any branch the amount of photons emitted during the life of the battery would give an indication of the minimum energy the battery released (some being lost elsewhere). With the multiple branches, would the total amount of photons emitted be the total amount of photons over the branches? If so then would the minimum amount of energy that the battery had before the considered branches occurred (the energy that was conserved) be the energy of the sum of the branches?

Sorry if I have been imprecise in my phrasing of the question(s), but hopefully the thrust of what I am asking can be understood and a enlightening answer provided. Thanks.

 

[PeterDonis]

Moderator's note: Thread moved to QM interpretations forum.

 

[PeterDonis]

presumably the direction of the decay will each lead to a branch within the waveform which would be entangled with the battery + lightbulb system (as long as there would have been a branch where the radiation did interact with that system)

This is not a good example. Even if there was some interaction between radioactive decay radiation from a human body and the battery + lightbulb system (which is in itself very unlikely), there is no feasible way of detecting effects of such an interaction. You seem to be thinking that such an interaction would affect the number of photons emitted by the lightbulb; to the extent that is even a good description (which is limited), we have no feasible way of measuring that number. We can measure the light intensity from the lightbulb, but that is not a measurement of photon number and won't tell us anything useful about whether or not some particular radioactive decay radiation has interacted with the battery + lightbulb system.

 

[PeterDonis]

I was watching a youtube video where Sean Carroll explained that the energy of the waveform is conserved, and that the branches can be thought of subdivisions of the waveform, and that the subdivisions contribute less energy than the original. That seems to be a slightly different answer to one given in the thread

No, it isn't. The answer given in the thread was:

The energy of the universe is a weighted average of the energy of the branches. So splitting into two branches doesn't mean each branch has half the energy; each branch will have (approximately) the same energy as before the split.

That is the same answer as Carroll gives, just in different words.

 

[PeterDonis]

With the multiple branches, would the total amount of photons emitted be the total amount of photons over the branches?

No. It would be a weighted average over the branches.

 

[PeterDonis]

would the minimum amount of energy that the battery had before the considered branches occurred (the energy that was conserved) be the energy of the sum of the branches?

No. It would be a weighted average of the energy in each branch.

 

[name123]

This is not a good example. Even if there was some interaction between radioactive decay radiation from a human body and the battery + lightbulb system (which is in itself very unlikely), there is no feasible way of detecting effects of such an interaction. You seem to be thinking that such an interaction would affect the number of photons emitted by the lightbulb; to the extent that is even a good description (which is limited), we have no feasible way of measuring that number. We can measure the light intensity from the lightbulb, but that is not a measurement of photon number and won't tell us anything useful about whether or not some particular radioactive decay radiation has interacted with the battery + lightbulb system.

No I was not thinking that the interaction would affect the number of photons emitted. Just that there would be a different branch depending on whether the battery was hit, or where it was hit or not. I was not thinking that human knowledge of whether the battery was hit or not was relevant. I could have supposed the battery could have a surface that acted as a Geiger counter, and there could be some device recording the hits, so that there would be a difference realisable by humans, but as I said, I wasn't thinking it mattered whether humans could measure it or not, I thought that was a key difference between it and the Copenhagen interpretation.

 

[name123]

No. It would be a weighted average over the branches.

If I considered a given branch, would the number of photons emitted be fixed along with the energy required for that amount of emissions?

 

[PeterDonis]

I was not thinking that the interaction would affect the number of photons emitted. Just that there would be a different branch depending on whether the battery was hit, or where it was hit or not

If there is no effect on the number of photons emitted, that means there's no effect on the battery + lightbulb system, which means there is no interaction when the battery is hit, which means there is no branching based on whether the battery is hit or not. An "interaction" that makes no difference is no interaction at all.

 

[PeterDonis]

If I considered a given branch, would the number of photons emitted be fixed along with the energy required for that amount of emissions?

A given branch based on what? So far you have given nothing on which branching could be based.

If I discard your specific example and just answer your question in general, then, if the branching is based on different results of a measurement of energy, then in each branch, the result of that measurement will be some fixed value. That's what branching means in the MWI.

If the branching is based on some other measurement, not a measurement of energy, then, unless that other measurement commutes with a measurement of energy, each branch will not, in general, have a definite value of energy.

(Technically, there doesn't have to be a "measurement", just some interaction that leads to multiple possible outcomes with decoherence.)

 

[name123]

That is the same answer as Carroll gives, just in different words.

I find it difficult to understand how " each branch will have (approximately) the same energy as before the split." is the same answer as each branch will contribute less energy than before the split (which is what Carroll stated).

To quote Carroll: "How is energy conserved is completely clear in the math, er the energy of the whole wave function is a 100% super duper conserved, but there is a difference between the energy of the whole wave function and the energy that people in each branch perceive. So what you should think of is not duplicating the whole universe but taking a certain amount of the universe and sort of subdividing it. Slicing it into two pieces. The pieces look identical from the inside, except that one has spin up and one has spin down or something like that. But they are really contributing less than the original to the total energy of everything."

 

[name123]

If there is no effect on the number of photons emitted, that means there's no effect on the battery + lightbulb system, which means there is no interaction when the battery is hit, which means there is no branching based on whether the battery is hit or not. An "interaction" that makes no difference is no interaction at all.

So if radiation hits the battery that doesn't count as interacting with the battery? Would there be no energy transferal on impact?

 

[PeroK]

I find it difficult to understand how " each branch will have (approximately) the same energy as before the split." is the same answer as each branch will contribute less energy than before the split (which is what Carroll stated).

MWI is not that much diferent in this respect from orthodox QM before a measurement. A system may be in an evolving superposition of energy eigenstates, for example. Each eigenstate represents a different configuration of the system (in a sense). If you have a superposition of two energy eigenstates, you don't have twice the energy; and, if you have a superposition of one hundred energy eigenstates, you don't have one hundred times the energy.

MWI after a measurement is theoretically the same as orthodox QM before a measurement.

The main difference is that in orthodox QM these superpositions are continually collapsing into definite states; whereas, in MWI the system keeps evolving in a full superposition of all possible states.

 

[PeterDonis]

I find it difficult to understand how " each branch will have (approximately) the same energy as before the split." is the same answer as each branch will contribute less energy than before the split (which is what Carroll stated).

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.

 

[PeterDonis]

So if radiation hits the battery that doesn't count as interacting with the battery? Would there be no energy transferal on impact?

If you're just treating the battery as an absorber of impacts, what's the point of having it be a battery and a light bulb? Why not just a piece of lead or something?

I assumed that your reason for having a battery + lightbulb was to use the light emitted by the bulb as some sort of measurement of the energy being delivered by the radioactive decay radiation. That's what won't work. If you just want something that will absorb impacts from the radioactive decay radiation and heat up, thereby showing that energy is being delivered, that will work fine, although the measurement of the energy delivered will be pretty coarse.

 

[name123]

MWI is not that much diferent in this respect from orthodox QM before a measurement. A system may be in an evolving superposition of energy eigenstates, for example. Each eigenstate represents a different configuration of the system (in a sense). If you have a superposition of two energy eigenstates, you don't have twice the energy; and, if you have a superposition of one hundred energy eigenstates, you don't have one hundred times the energy.

MWI after a measurement is theoretically the same as orthodox QM before a measurement.

The main difference is that in orthodox QM these superpositions are continually collapsing into definite states; whereas, in MWI the system keeps evolving in a full superposition of all possible states.

There is a difference though, because in orthodox, there is never more than the "collapsed" eigenstate for example. But with MWI you have both. And there is the conservation of energies issue. And with Carroll's explanation a given branch energy value seems to tend to infinite as there can tend to infinite branches from there. What cap can you put on it?

 

[name123]

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.

I don't think I left it out, I was just pointing out that the average it was "each branch will have (approximately) the same energy as before the split" (and by the way how did any branch have an energy different to than before the split). With the sum it was each branch was "contributing less than the original to the total energy of everything". You seemed to me to claim that each branch energy equalling the energy before the branch was the same (but in different words) as each branch energy being less than before the branch.

 

[PeroK]

There is a difference though, because in orthodox, there is never more than the "collapsed" eigenstate for example. But with MWI you have both.

Orthodox QM has the same process of superposition upon superposition, almost ad infinitum. With MWI it is ad infinitum.

In MWI a measurement of energy is simply the measuring appartus becoming entangled with the system it's measuring. In fact, if you treat measuring apparatus in this way and apply orthodox QM you get MWI - up to a point, of course.

In MWI you can look at the universe as simply one big system eternally evolving unitarily. You get conservation of energy by default because nothing non-unitary ever happens.

PS It's actually orthodox QM that has to break out of this unitary evolution - and only then is there really any issue with conservation of energy.

 

[PeterDonis]

You seemed to me to claim that each branch energy equalling the energy before the branch was the same (but in different words) as each branch energy being less than before the branch.

No, that's not what I claimed. Go back and read my posts again.

 

[name123]

Orthodox QM has the same process of superposition upon superposition, almost ad infinitum. With MWI it is ad infinitum.

In MWI a measurement of energy is simply the measuring appartus becoming entangled with the system it's measuring. In fact, if you treat measuring apparatus in this way and apply orthodox QM you get MWI - up to a point, of course.

In MWI you can look at the universe as simply one big system eternally evolving under the SDE equation. You get conservation of energy by default because nothing non-unitary ever happens.

Ok, so suppose in one system there is only one universe, and it has an energy of blah, how do you explain later multiple universes each with an energy of blah while conserving energy?

And which were you agreeing with

1) each branch will have (approximately) the same energy as before the split.
2) they are really contributing less than the original to the total energy of everything
3) (1) and (2) were wrong.

 

[PeterDonis]

In MWI you can look at the universe as simply one big system eternally evolving unitarily. You get conservation of energy by default because nothing non-unitary ever happens.

Strictly speaking, this is only true if the universal wave function is an eigenstate of the universal Hamiltonian. Otherwise there is no well-defined notion of "total energy" at all, so there's no "energy" that can be conserved.

 

[PeroK]

Ok, so suppose in one system there is only one universe, and it has an energy of blah, how do you explain later multiple universes each with an energy of blah while conserving energy?

In MWI there is only ever one universe, eternally evolving unitarily. All your measurements and worlds and branches are just part of this. Your mistake, perhaps, is to imagine separating them all out into physically different systems. They are only really logically separate.

There are no measurements of the system, as such. Measurements are simply unitary superpositions within the system.

 

[PeterDonis]

so suppose in one system there is only one universe, and it has an energy of blah, how do you explain later multiple universes each with an energy of blah while conserving energy?

There aren't later "multiple universes". There is only one universe. Assuming its state is an eigenstate of the universal Hamiltonian, its total energy is constant since unitary evolution can't change it.

1) each branch will have (approximately) the same energy as before the split.
2) they are really contributing less than the original to the total energy of everything

Both are true when appropriately understood.

Let's unpack this, using the simplest possible case of a measurement that commutes with the Hamiltonian, so that the measurement does not change the energy of anything. (This will be true, for example, of an idealized spin measurement on a particle.) Let's suppose the measurement has two possible outcomes, whose probabilities are equal.

Before the measurement, there is one branch, and it has energy $E$. So the total energy of the universe is $E$.

After the measurement, there are two branches. As measured by an observer within each branch, each branch has energy $E$. The total energy of the universe is a weighted average of the energies of the branches, with the weighting of each branch being its probability, so it is $\frac{1}{2} E + \frac{1}{2} E = E$.

 

[name123]

Both are true when appropriately understood.

Let's unpack this, using the simplest possible case of a measurement that commutes with the Hamiltonian, so that the measurement does not change the energy of anything. (This will be true, for example, of an idealized spin measurement on a particle.) Let's suppose the measurement has two possible outcomes, whose probabilities are equal.

Before the measurement, there is one branch, and it has energy $E$. So the total energy of the universe is $E$.

After the measurement, there are two branches. As measured by an observer within each branch, each branch has energy $E$. The total energy of the universe is a weighted average of the energies of the branches, with the weighting of each branch being its probability, so it is $\frac{1}{2} E + \frac{1}{2} E = E$.

So an observer within each branch measures each branch to have an energy of $E$.

I am guessing that the Copenhagen interpretation and the wave function collapse, there would only be one actualised branch and the actualised branch would have an energy of $E$ and the other branches that were potential branches before the collapse wouldn't have any energy as they were never actualised.

But with the MWI if each branch had the energy that it would have had if it had been actualised under the Copenhagen interpretation there would seem to be a conservation of energy problem, as if there were two branches with an energy $E$ then 2 * $E$ would equal 2$E$. But with the MWI each branch is said to only have an energy value each to the probability of its branch times $E$ even though an observer within each branch would measure that branch to have an energy of $E$.

A few things I am not clear about. One is the probability of the branch, since each branch occurs the probability would seem to be 1. The other is whether the branch is said to actually have an energy value equal to its probability times $E$ and not $E$ as measured by an observer within it. Or whether each branch has an energy value of $E$ but when adding up the energy values of both branches for example the probability is an ad hoc addition peculiar to the MWI introduced to maintain the conservation of energy.

 

[name123]

In MWI there is only ever one universe, eternally evolving unitarily. All your measurements and worlds and branches are just part of this. Your mistake, perhaps, is to imagine separating them all out into physically different systems. They are only really logically separate.

There are no measurements of the system, as such. Measurements are simply unitary superpositions within the system.

I can understand that as just a mathematical framework, I can consider them as just logically separate, but as I understand it the MWI is a metaphysical suggestion. The only one which avoids concluding "spooky action at a distance" that I am aware of. The suggestion being that there is another dimension to spacetime. That what physically exists at that dimensional spacetime can be different to what physically exists at the same spacetime location but different with regards to that dimension. So in one, the cat could be alive, but in the other it was killed by a poison gas triggered by a radioactive decay event.

 

[PeterDonis]

So an observer within each branch measures each branch to have an energy of $E$.

Yes.

I am guessing that the Copenhagen interpretation and the wave function collapse, there would only be one actualised branch and the actualised branch would have an energy of $E$ and the other branches that were potential branches before the collapse wouldn't have any energy as they were never actualised.

On any interpretation where wave function collapse is taken to be a real physical process, yes. And so the total energy is the same as the energy of the only branch that is actualized.

But with the MWI if each branch had the energy that it would have had if it had been actualised under the Copenhagen interpretation there would seem to be a conservation of energy problem

This has already been explained to you. I even explicitly showed you how the weighted average works in post #23.

as if there were two branches with an energy $E$ then 2 * $E$ would equal 2 of $E$.

Wrong. We are talking about the MWI, and as you say in your very next sentence:

But with the MWI each branch is said to only have an energy value each to the probability of its branch times $E$

Wrong. The fact that you multiply $E$ times the probability of the branch in the weighted average does not mean the MWI says the "energy value" of that branch is only the probability times $E$. The MWI only says that you have to do the weighted average if you want the "total energy of the universe". It does not say that each branch's contribution to that weighted average is the "energy value" of that branch. The "energy value" of each branch is what the observer in that branch measures.

One is the probability of the branch, since each branch occurs the probability would seem to be 1.

No. The MWI does not use the term "probability" the way other interpretations do. In the MWI, the "probability" of each branch is the squared modulus of its amplitude (its complex coefficient in the wave function of the whole universe), even though each branch is actualized so the usual interpretation of the word "probability" does not work.

The other is whether the branch is said to actually have an energy value equal to its probability times $E$ and not $E$ as measured by an observer within it.

No. See above.

Or whether each branch has an energy value of $E$

Yes. See above.

but when adding up the energy values of both branches for example the probability is an ad hoc addition peculiar to the MWI introduced to maintain the conservation of energy.

No. Doing the weighted average is not "ad hoc". It's what the straightforward math of the wave function of the whole universe tells you to do. There is nothing ad hoc about it.

If you have a problem with calling the coefficient of each branch in the weighted average the "probability" of the branch, that's fine; lots of people are not comfortable with the way the MWI uses the term "probability". But that's a matter of words, not physics. The physics, if we assume the MWI is correct, is the same no matter what word you use to describe the coefficients.

 

[name123]

No. Doing the weighted average is not "ad hoc". It's what the straightforward math of the wave function of the whole universe tells you to do. There is nothing ad hoc about it. If you have a problem with calling the coefficient of each branch in the weighted average the "probability" of the branch, that's fine; lots of people are not comfortable with the way the MWI uses the term "probability". But that's a matter of words, not physics. The physics, if we assume the MWI is correct, is the same no matter what word you use to describe the coefficients.

I am assuming with the Copenhagen interpretation the coefficients are the probability of each branch being actualised. That the energy of the actualised branch is $E$, and the energy it contributes to the universe is $E$. The energy the actualised branch contributes to the energy of the universe isn't multiplied by its coefficient. So with the MWI could you explain to me what the coefficient is supposed to represent, and explain why the energy of each actualised branch is multiplied by it (other than as an "ad hoc" attempt to conserve energy) when working out how much that branch contributes to the energy of the universe?

 

[PeroK]

I am assuming with the Copenhagen interpretation the coefficients are the probability of each branch being actualised. That the energy of the actualised branch is $E$, and the energy it contributes to the universe is $E$. The energy the actualised branch contributes to the energy of the universe isn't multiplied by its coefficient. So with the MWI could you explain to me what the coefficient is supposed to represent, and explain why the energy of each actualised branch is multiplied by it (other than as an "ad hoc" attempt to conserve energy) when working out how much that branch contributes to the energy of the universe?

Let's say we have an experiment involving a particle that can - due to quantum randomness - go two ways: to detector A or B. Let's take good old Stern-Gerlach.

Before the magnet we have one silver atom. After the magnet, we have one silver atom in a superposition of two spatial states. This is orthodox QM. Someone asks you where the extra silver atom and extra energy came from? You say: that's okay, that's just a superposition of states: there are not two atoms, there is not twice the energy.

Now, the atom goes to detector A. Which you observe.

An advocate of MWI comes along and says: yes, but the silver atom is still in a superposition of states, but now so are the detectors and so are you.

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.

The MWI advocate says: you and the screen are made of atoms: if atoms can be in as superposition of states, why can't you?

And you say ... QM only applies to small things?

 

[PeterDonis]

I am assuming with the Copenhagen interpretation the coefficients are the probability of each branch being actualised.

Yes, that will be true for any interpretation that treats wave function collapse as a real physical process.

That the energy of the actualised branch is $E$, and the energy it contributes to the universe is $E$.

Yes.

with the MWI could you explain to me what the coefficient is supposed to represent

It represents the "weight" of the branch in the wave function of the whole universe. In other words, the same thing as the coefficient of any term in a wave function with multiple terms.

and explain why the energy of each actualised branch is multiplied by it

Because that's what you do with a wave function that has multiple terms. In a collapse interpretation, the wave function after the measurement only has one term (one branch), because of collapse, so the question of how to treat multiple terms doesn't come into play. That doesn't mean the MWI has to make up what to do with multiple terms ad hoc; what to do with multiple terms is already specified by the basic math of QM. The only difference between the MWI and a collapse interpretation is that the MWI says the multiple terms are still there after a measurement has been made.

 

[name123]

It represents the "weight" of the branch in the wave function of the whole universe. In other words, the same thing as the coefficient of any term in a wave function with multiple terms.

But as you confirmed with the Copenhagen interpretation the coefficient is the probability of a certain outcome. With MWI all outcomes have a probability of 1 so how can the coefficient in the MWI have the same meaning as with the Copenhagen interpretation?

Because that's what you do with a wave function that has multiple terms. In a collapse interpretation, the wave function after the measurement only has one term (one branch), because of collapse, so the question of how to treat multiple terms doesn't come into play. That doesn't mean the MWI has to make up what to do with multiple terms ad hoc; what to do with multiple terms is already specified by the basic math of QM. The only difference between the MWI and a collapse interpretation is that the MWI says the multiple terms are still there after a measurement has been made.

I was assuming that the coefficients we are talking about in with the Copenhagen interpretation are the coefficients given by the Born Rule (the probability of the outcome). Have I misunderstood?