As this is the 50th anniversary of Everett’s Many-World’s Interpretation (MWI) of quantum mechanics, thanks to a post on Matt Leifer’s blog, Quantum Quandaries, I dug out some two-year old notes I had on MWI and permutation invariance and am throwing them up here, unedited (except for typos) but with comments, to see what sort of response I might get (I am hoping Matt takes a serious look at them, for one, since I have started to find his thoughts ever more insightful). As a warning, this gets a tad philosophical at times. So, without further ado, here they are:
Assume that states and (Sab) represent different ‘universes’ in the multiverse. If Sab is unitary it is its own inverse leading to P3 (van Fraassen) where Sij represents the permutation of the ith and jth universes in the multiverse.
P3 refers to the third description of permutation invariance given in van Fraassen’s Quantum Mechanics: An Empiricist’s View.
Since subspace as a whole is invariant under permutations the universes must at the very least share the same facets (underlying laws, etc.) though since the vectors in the subspace are not invariant there could be differences in actual form and outcome.
However, Huggett’s work seems to imply that the universes must be nearly identical if they consist of the same parts (i.e. have same type of origin such that matter and anti-matter as we know it are likewise produced). Or, we could say if the ‘universes’ within the multiverse are all possible outcomes then they are simply permutations (a selection between given choices) which seems to imply all observables should have the same expectation values in all universes (though not necessarily the same results).
Note that by ‘all possible outcomes’ I mean ‘all possible outcomes of all possible probabilistically determined measurement or action.’
In the case of Hermitean operators (self-adjoint) the laws of physics must either be identical or entirely different (i.e. no resemblance at all). This would correspond to a ‘bosonic’-type case in the former and a ‘fermionic’-type case in the latter. It would prohibit ‘para’-like statistics but not necessarily mixtures of states (e.g. one might imagine entangled universes). As French and Redhead point out, however, states with the wrong symmetry are not accessible but this does not mean they do not exist! So there could conceivably be some universes accessible to others and some that are not. This seems to mesh with Messiah and Greenburg’s notion that it really isn’t the states we’re limiting but the number of observables. So the number of possible universes is infinite but the number of universes accessible to us is finite.
One potentially sticky question: in Leibniz’ Principle of the Identity of Indiscernibles (PII) there are two versions – weak and strong – having to do with the treatment of spatial location. But, in a multiverse how does one deal with spatial location?
Again, as French and Redhead point out, in discussing PII, we are not restricted to attributes that are actually observable. This restricts the discussion to symmetric combinations. Even so, in the case of fermions French and Redhead have pointed out that two fermions in an entangled state can have the same monadic and relational properties which violates the weakest form of PII.
Again, F&R point out that the Exclusion Principle has no bearing on PII “if we adhere to the orthodox view that actualizations do not correspond to antecedently existing possessed values” (F&R 1988, p.242) unless, of course, we’re dealing with hidden variable theories in which case the actualizations are determined by the values of our mysterious hidden variables.
Finally, F&R again support the idea that there should be at least some differences since even in the case of bosons there can be some discernibility. Parastatistics are ruled out, though. However, there are issues here in relation to spatio-temporal continuity (STC) and transcendental individuality (TI). First, the former: in MWI, STC depends on the way the multiverse is defined. If it is true that each universe contains space and time within itself and space and time are meaningless concepts outside or between the various universes then STC is not applicable here. However, if there is some link between the various ‘fabrics’ of space and time for each universe, then STC would need to be considered. This poses a problem with TI which is the latter of the two issues I mentioned. Since the universes are being treated as quantal particles and they are being treated as individuals then this individuality must be conferred by TI. STC requires well-defined trajectories which seems meaningless if the individual universes are spatio-temporally self-contained. But STC is generally used to label macroscopic bodies – and not much is more macroscopic than an entire universe! – in which case STC conflicts with the TI individuation of the universes composing the multiverse. In the case of particles, F&R point out that interference is never entirely absent. But what about the case of universes? It seems there’s a whole new ballgame here. Perhaps quantum field theory could be useful, but even then it’s not clear how to deal with the inter-universe portions of the multiverse where such interference might be transmitted.
OK, here’s something I might rethink (and I’ve got to find these references again – I know French & Redhead as well as Huggett are in either the British Journal for the Philosophy of Science or Studies in the History and Philosophy of Modern Physics). Hmmm…
The underlying question, then, is whether or not quantum mechanics can even be applied to entire universes. Of course, the classic case of this is the Hartle-Hawking paper of 1983 simply entitled “Wave function of the Universe” (H&H 1983). H&H do recognize that space and time are meaningless concepts outside of the universe itself, supporting the notion that MWI has a problem with inter-universe transitions via permutation invariance. They are careful to maintain Hermiticity which I already pointed out leads to an ‘all or nothing’ situation – no parastatistics here for MWI. H&H certainly is accurate in its predictions and has been elaborated upon numerous times in the past 22 years. This seems to indicate the fact that QM can indeed be applied to universes leaving only the conclusion that MWI is inconsistent with, at the very least, a modal interpretation of QM.
And there you have it. My somewhat dusty thoughts on the subject, distilled in my memory to ‘MWI is inconsistent with permutation invariance (PI) since PI would seem to require no interference between the various “universes.”‘