In discussions of the many-worlds interpretation (MWI) and the process of wavefunction branching, folks sometimes ask whether the branching process conflicts with conservations laws like the conservation of energy.Here are some related questions from around the web, not addressing branching or MWI. None of them get answered particularly well.a There are actually two completely different objections that people sometimes make, which have to be addressed separately.
First possible objection: “If the universe splits into two branches, doesn’t the total amount of energy have to double?” This is the question Frank Wilczek appears to be addressing at the end of these notes.
I think this question can only be asked by someone who believes that many worlds is an interpretation that is just like Copenhagen (including, in particular, the idea that measurement events are different than normal unitary evolution) except that it simply declares that new worlds are created following measurements. But this is a misunderstanding of many worlds. MWI dispenses with collapse or any sort of departure from unitary evolution. The wavefunction just evolves along, maintaining its energy distributions, and energy doesn’t double when you mathematically identify a decomposition of the wavefunction into two orthogonal components.
Second possible objection: “If the universe starts out with some finite spread in energy, what happens if it then ‘branches’ into multiple worlds, some of which overlap with energy eigenstates outside that energy spread?” Or, another phrasing: “What happens if the basis in which the universe decoheres doesn’t commute with energy basis? Is it then possible to create energy, at least in some branches?” The answer is “no”, but it’s not obvious.
The argument is as follows: We describe a sequence of historical events in a quantum universe using a set of consistent histories, i.e. time-ordered strings of Heisenberg-picture projectors . For a pure state of the universe, the condition of consistency is equivalent to the orthogonality of the branches, which are defined by . Because each branch must be orthogonal to all the other ones, they define a basis (on some subspace, at least). They all sum up to the global wavefunction , and the norm of each branch is given by projecting onto the relevant basis vector. Now, if we end up, after many branching events, with a branch with an exact amount of energy (i.e. it’s an energy eigenstate, which might be lying in a degenerate subspace of a given energy), then we can see that the norm of this vector (and hence the probability associated with the branch) must be zero unless the vectors lies in the subspace spanned by the energy eigenstates overlapping with the original global state .
Of course, it might be that we have branches at a given time that aren’t energy eigenstates. In this case it’s hard to even say what you mean by energy conservation. The branch isn’t an eigenstate, so it’s energy is undefined. But if it later decoheres into branches with specific energy, then this energy must lie in the support of the energy spectrum of .
As it turns out, Hartle et al. have a paper that discusses this in pretty good detail:
James B. Hartle, Raymond Laflamme, and Donald Marolf.
“Conservation laws in the quantum mechanics of closed systems.”
Phys. Rev. D 51, 7007 (1995).
Their argument in Sec. 2 (that they attribute to Griffiths) is equivalent to the one given aboveEverything after later in the paper about gauge charges isn’t strictly necessary for this question, but may be of interest.b .
[I thank Elliot Nelson and Luciano Combi for discussion leading to this post.]