From: <[email protected] <mailto:[email protected]>>
On Thursday, June 7, 2018 at 11:32:23 AM UTC, Bruce wrote: From: <[email protected]>On Tuesday, June 5, 2018 at 3:05:40 AM UTC, Bruce wrote: From: <[email protected]>On Tuesday, June 5, 2018 at 1:18:29 AM UTC, Bruce wrote: From: <[email protected]>Remember that the analysis I have given above is schematic, representing the general progression of unitary evolution. It is not specific to any particular case, or any particular number of possible outcomes for the experiment. Bruce*OK. For economy we can write, **(|+>|e+> + |->|e->), where e stands for the entire universe other than theparticle whose spin is being measured. What is the status of the interference between the terms in this superposition? For a quantum superposition to make sense, there must be interference between the terms in the sum. At least that's my understanding of the quantum principle of superposition. But the universe excluding the particle being measured seems to have no definable wave length; hence, I don't see that this superposition makes any sense in how superposition is applied. Would appreciate your input on this issue. TIA, AG*A superposition is just a sum of vectors in Hilbert space. If these vectors are orthogonal there is no interference between them. Your quest for a wavelength in every superposition is the wrong way to look at things. Macroscopic objects have vanishingly small deBroglie wavelengths, but the can still be represented as vectors in a HIlbert space, so can still form superpositions. I think you are looking for absolute classicality in quantum phenomena -- that is impossible, by definition. Bruce *If that's the case, why all the fuss about Schrodinger's cat? AG*Is there a fuss about Schrödinger's cat? Whatever fuss there is, is not about the possibility of a superposition of live and dead cats. It is about choosing the correct basis in which to describe the physical situation. The Schrödinger equation does not specify a basis, and that is its main drawback. In fact, that observation alone is sufficient to sink the naive many-worlds enthusiast -- he doesn't know in which basis the multiplication of worlds occurs. Bruce * Interesting point. Do you mean that if one solved the SE for some standard quantum problem (nothing fancy, no decoherence modeled), one can generally expand the solution in different bases, say p, E, or x, and each expansion would imply a different set of worlds using the MWI? Are there other bases besides these three? I'm thinking there could be an infinite set of basis vectors since, by analogy, IIUC, for the simple 2-dimensional vector space of "little pointy things", I think every pair of non co-linear vectors could form a basis (so most bases are not orthogonal). AG*There are an indefinite number of possible sets of basis vectors in any Hilbert space. Think of the 2-dimensional space for a spin half particle -- one can form a set of orthonormal basis vectors for every direction in the 3-sphere. Different bases are not different observables such as p, E, or x. Each such observable has its own Hilbert space and an infinite set of possible bases. Each set of basis vectors is just a linearly independent set of sums over some other basis. It is easier to visualize this in the case of a simple linear vector space. Think of 3-dimensional Euclidean space. You can choose a set of three axes, but these can be rotated into any direction. Or linear combinations can be formed that are not necessarily orthogonal. For physical situations in QM, some bases are more useful than others, but the choice of basis is by no means unique. Bruce*OK. I understand your comments .But let me rephrase the issues as I conflated some of them above. In the spin half case, were you claiming that each orientation of the SG device implies a different world according to the MWI, and if so, does the MWI make no sense since the SWE does not indicate which orientation is in play?*
The SWE does not give a preferred basis. Basing MWI on the Schrödinger equation runs into the basis problem. Few MWI advocates actually take this seriously. And they should.
*In this situation, what is the role of the SWE since the wf is usually asserted without any reference to it? Now consider a general case where the wf for a system is determined using the SWE. Since the solution can be expanded using difference bases, say E or p, does each possible expansion, each implying a different possible set of measurements, imply a different set of worlds using the SWE? TIA, AG*
The Schrödinger equation merely gives the time evolution of the system. To define the problem you have to specify a wave function. It is in the expansion of this wave function in terms of a set of possible eigenvalues that the preferred basis problem arises. So it is not really down to the SE itself, it is a matter for the wave function. Each expansion basis defines a set of worlds, and all bases give different worlds.
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