Energy in Everett's Many-worlds Interpretation

[jxcs]

Hi,

Is anybody able to explain how energy is "distributed" in the many-worlds interpretation. I'm using scare quotes as I think this may be the wrong line of thought. It's tempting to imagine energy being distributed amongst subsequent branches as the wave function evolves but I'm not certain that is a true picture? In this sense, energy would be proportional to the number of branches, at which point it all just starts to sound very fuzzy!

 

[stevendaryl]

Hi,

Is anybody able to explain how energy is "distributed" in the many-worlds interpretation. I'm using scare quotes as I think this may be the wrong line of thought. It's tempting to imagine energy being distributed amongst subsequent branches as the wave function evolves but I'm not certain that is a true picture? In this sense, energy would be proportional to the number of branches, at which point it all just starts to sound very fuzzy!

The various "worlds" in Many-Worlds are different branches of the universal wave function. The branching is just a way of writing the universal wave function as a superposition of wave functions. So in the simplest nontrivial case (two branches), the wave function $\psi(t)$ can be written as a superposition $\alpha \psi_1(t) + \beta \psi_2(t)$. 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.

 

[jxcs]

Hi
Thanks so much for getting back to me. Ok, so one might naively conclude that the energy in a superposition might somehow sum to the energy of its constituents but I guess the "energy" of a state that is not an energy eigenstate is not *defined* and we can only consider energy in a state once it's in an eigenstate?
Could you also point me in the direction of an explanation of how the conservation of energy expectation value is 0?

Thanks!

 

[Cloudwood]

The various "worlds" in Many-Worlds are different branches of the universal wave function. The branching is just a way of writing the universal wave function as a superposition of wave functions. So in the simplest nontrivial case (two branches), the wave function $\psi(t)$ can be written as a superposition $\alpha \psi_1(t) + \beta \psi_2(t)$. 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.

Tempted to break down the logic of how branching implies the current branch will either have too much energy and/or subsequent branches will eventually run to zero energy at the limit. But you seem to be suggesting (intentionally?) a kind of "Many Blocks Interpretation" of MWI—the MWI version of a block universe. In this view, energy is conserved across all branches because the system is already complete and determined, meaning there's no need for branching to create new energy or subdivide existing energy, as all branches already exist in a timeless block (with each slice describing all possible branches to that point). I haven't heard that one before. Interesting. Doesn't fully convince me, but as a Borges-ian subject for a story, it's intriguing.

 

[Nugatory]

@Cloudwood you may be interested in https://arxiv.org/abs/2101.11052 (or this more layman-friendly blog summary). Note that neither had been available back when this thread was last active.

 

[Jonomyster]

Here's a way of thinking about it that I find helpful.

A key intuition behind Many-Worlds is that the laws of quantum mechanics are universal. That is to say, the descriptions we apply to electrons, buckyballs, cats, observers, and the Earth are all fundamentally the same. They are "merely" quantum systems of increasing size.

Therefore, we can ask ourselves simpler versions of the energy question. A question like "how is the energy different when Earth goes into a superposition?" certainly seems confusing. But the answer to that is fundamentally the same as the answer to "how is the energy distributed when Schrödinger's cat goes into a superposition?" And if that's confusing, we may even ask "how is the energy distributed when an electron goes into a superposition?"

I'll answer the question from the perspective of Schrödinger's cat. Suppose we put a cat and a radioactive atom into a perfectly isolated box. After some time, the atom will decay, triggering a detector which releases sleeping gas. (I like to keep the cat alive in both branches.)

Write $\ket{C}$ for the various states of the cat, write $\ket{A}$ as the state of the atom before it has decayed, and write $\ket{D}$ as the state of the atom after it has decayed. I'll also write $\ket{\psi}$ as the state of the entire system. The time evolution then looks something like
$$
\begin{aligned}
\ket{\psi} =&\, \ket{A} \ket{C_\text{awake}} \\
\rightarrow&\, \frac{1}{\sqrt 2} (\ket A + \ket D) \ket{C_\text{awake}} \quad \text{(atom decays)} \\
= &\, \frac{1}{\sqrt 2} \ket A \ket{C_\text{awake}} + \frac{1}{\sqrt 2} \ket D \ket{C_\text{awake}} \\
\rightarrow &\, \frac{1}{\sqrt 2} \ket A \ket{C_\text{awake}} + \frac{1}{\sqrt 2} \ket D \ket{C_\text{asleep}} \quad \text{(cat interacts with atom)}
\end{aligned}
$$ So then, after the time evolution we have two cats, one asleep and one awake. This is analogous to the situation where an experimenter measures, say, a qubit, causing the wavefunction to branch into two worlds.

We can then ask, "how is the energy distributed between the awake-branch and the asleep-branch?" To begin, note that the individual states $\ket A \ket{C_\text{awake}}$ and $\ket D \ket{C_\text{asleep}}$ contain (at least to a very good approximation) the same energy. (The main contribution to this is the rest mass of the cat, which is of course the same in both cases). We can express this using the Hamiltonian $\hat H$ for the entire system:
$$
\begin{aligned}
\hat H \ket{A} \ket{C_\text{awake}} \simeq E \ket{A} \ket{C_\text{awake}} \\[0.8em]
\hat H \ket{D} \ket{C_\text{asleep}} \simeq E \ket{D} \ket{C_\text{asleep}} \\
\end{aligned}
$$ Then, notice what happens when we have a superposition of the asleep-branch and awake-branch. Applying the Hamiltonian operator yields
$$
\begin{aligned}
&\hat H \left( \frac{1}{\sqrt 2} \ket{A} \ket{C_\text{awake}} + \frac{1}{\sqrt 2} \ket{D} \ket{C_\text{asleep}} \right) \\
&= \frac{1}{\sqrt 2} \big( \hat H \ket{A} \ket{C_\text{awake}} \big) + \frac{1}{\sqrt 2} \big( \hat H \ket{D} \ket{C_\text{asleep}} \big) \quad \text{(by linearity)} \\
&\simeq \frac{1}{\sqrt 2} \big( E \ket{A} \ket{C_\text{awake}} \big) + \frac{1}{\sqrt 2} \big( E \ket{D} \ket{C_\text{asleep}} \big) \\
&= E \left( \frac{1}{\sqrt 2} \ket{A} \ket{C_\text{awake}} + \frac{1}{\sqrt 2} \ket{D} \ket{C_\text{asleep}} \right)
\end{aligned}
$$ As we can see, the eigenvalue of the superposition under $\hat H$ is also $E$. That is to say, the energy of the superposition is the same as the energy of the individual parts. (Note that this has nothing to do with the factors of $1/\sqrt 2$; those are just there for normalization.)

How can we interpret this? Think about how this must look from the cat's perspective. It doesn't suddenly feel its mass half in two, it either just feels some sleeping gas getting released, or it keeps doing whatever it was doing before. In the language of quantum mechanics, then, I would say that the energy is shared between the asleep-branch and awake-branch of the wavefunction.

And because there's no fundamental difference between superpositions of cats and superpositions of Earths, the same argument applies whenever an observer measures, say, a qubit.

So that's my answer to the question. I think the confusion arises from thinking of energy as something that exists inside each world - this is the classical intuition that we are used to. But in Many-Worlds, the universe fundamentally lives in Hilbert space, not Euclidean space. And in Hilbert space, multiple worlds can share the same energy.

 

[bhobba]

This has touched on a key difference between Many Worlds and Decoherent Histories and a possible experimental way to tell the difference:


It would be instant Nobel material if proven one way or the other.

Just as an aside, I am in the potentially real camp. I even go further and believe QM is just a limiting case of QFT - but that limit is a bit strange:
https://arxiv.org/abs/1712.06605

Are quantum issues like those alluded to here best looked at using QFT?

Thanks
Bill

 

[Demystifier]

Roughly speaking, the energy$E$ in quantum mechanics is described by the oscillatory factor $e^{-iEt/\hbar}$ of the wave function. The energy is related to frequency of the wave function oscillation. The splitting of the wave function in MWI does not affect the frequency of oscillation. Two branches with the same frequency have the same energy. The total frequency of two branches together is not the sum of their individual frequencies, hence the energies of the two branches are not to be added.