The distributive law is well known in algebra and number theory:
a(b + c) = ab + ac
The product of one number with a sum of other numbers is the same as the
sum of the individual products.
This is widely used in algebra and number theory, but the question is,
'Does this work for quantum theory?'
It is assumed in things like many-worlds theory, where we have
(|1> + |0>)|e> = |1>|e> + |0>|e>
(ignoring normalization factors) where |1> and |0> are two eigenvalues in a
superposition, and |e> is the environment, including the apparatus and the
observer.
Under unitary evolution, the equation above proceeds to
(|1> + |0>)|e> = |1>|e> + |0>|e> = |1>|e_1> + |0>|e_0>
where |e_1> represents the observer and environment recording outcome 1,
and similarly for |e_0>.
But when we ask what this means physically we are faced with a problem. The
final states in which we have environments recording 1 and 0 outcomes,
respectively, are produced by decoherence from the |1> and |0>
eigenfunctions respectively. But for this to happen, each component of the
superposition must interact with an original environment. In other words,
the distributive law applied to the tensor product of the initial
superposition and the environment means that the original environment must
be duplicated *before any interaction takes place*. It is very hard to make
physical sense of this. What causes the environment to duplicate? How can
this even be possible? If the environment is not duplicated before the
interaction, then the only possible interpretation is that environmental
decoherence records no more than the original superposition, and we know
that this is not what happens physically.
The conclusion I draw is that the simple distributive law cannot apply in
quantum mechanics and, consequently, Everettian many-worlds theory cannot
get off the ground.
Bruce
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