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A few answers on here seem to suggest to something like the following:

In the Many Worlds Interpretation, the amplitude of a branch exponentially decreases over time and so the energy of that branch also decreases exponentially. This seems to suggest the branch amplitude and energy conserved within that branch have a very simple linear relationship.

However, do the amplitudes not just give probabilities for transitions, some of which can be interpreted as measurements of energy? To say that we know the energy has decreased in direct proportion to the branch amplitude is a simplification?

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    $\begingroup$ Why do you think energy decreases with amplitude? Amplitudes are like probabilities of states, and you can have a constant energy situation where some states are becoming more or less likely. $\endgroup$ – Anders Sandberg Feb 14 at 11:54
  • $\begingroup$ Can you elaborate a bit more on what you mean by "a constant energy situation"? I just read knzhou's comment on this thread which confused me: physics.stackexchange.com/questions/420610/… $\endgroup$ – User11 Feb 14 at 12:06
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    $\begingroup$ Sorry, in retrospect that was phrased badly. The contribution to the expectation value of the energy from a branch falls with the amplitude. But not the energy itself. $\endgroup$ – knzhou Feb 14 at 12:18
  • $\begingroup$ ahh ok that makes sense. So is there a conserved amount of energy in each branch (and thus the multiverse itself), it's just the likelihood of inhabiting that branch that decreases (rather than inhabiting a branch with "less" energy)? Apologies if this is rather basic! $\endgroup$ – User11 Feb 14 at 13:10
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In the Many Worlds Interpretation, the amplitude of a branch exponentially decreases over time

This is not true. Possibly you've gotten confused by a discussion of MWI as an approximation to decoherence, i.e., decoherence as MWI done right. In decoherence, what falls off exponentially with time is the off-diagonal elements of the density matrix.

MWI involves only unitary evolution of the wavefunction (unlike Copenhagen, which postulates an additional collapse process). This unitary evolution is given by the Schrodinger equation, which is basically a statement of conservation of energy.

By the way, these ideas like splitting and branching aren't really mandatory as part of MWI. The most austere versions of MWI are simply standard quantum mechanics: unitary evolution of the wavefunction, and that's all. Different people mean different things by MWI.

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  • $\begingroup$ Hi Ben, thanks for answering this. So is it in a way flawed to ask a question about the energy associated with a particular branch? The energy is conserved over the entire multiverse, and crudely speaking, a certain amount of energy is associated with each branch? I guess a more intuitive way to think of it is either 'overlapping' or the 'diverging' interpretations... $\endgroup$ – User11 Feb 14 at 15:35
  • $\begingroup$ @User11: So is it in a way flawed to ask a question about the energy associated with a particular branch? No, I think that's legit to talk about. The energy is conserved over the entire multiverse, and crudely speaking, a certain amount of energy is associated with each branch? No, not if what you have in mind is sharing of energy between branches. If the initial state is a state of definite energy $E$, then each of the branches is also a state with the same definite energy $E$. $\endgroup$ – Ben Crowell Feb 15 at 4:23

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