From: scerir (scerir@libero.it)
Date: Thu Nov 28 2002 - 01:33:43 MST
> Why don't worlds fuse, as well as split?
In the macroscopic superposition of states (exp. by Leggett
et al.) there is a 'sort' of 'fusion'.
> Do splitting worlds imply irreversible physics?
In the quantum theory of measurement the 'collapse' seems
to imply (one kind of) irreversibility. Because the observer
performing measurements gets some information. How his gain
in information is related to the reduction of superpositions,
sometimes connected with huge entropies, is not completely
understood.
There are different lines of thought: irreversibility comes
out from quantum measurements (macroscopic apparata); irreversibility
comes out from finiteness/infinity of the system; irreversibility
comes out from non-equilibrium in the system (interaction with
other systems, Zurek's triangle, etc.).
By means of information theory it is possible to rephrase
the maximun entropy principle in other terms. Suppose that
for some system you know only a few macroscopic quantities,
and you have no further knowledge of it. Then the system is
expected to be in the state with maximal entropy, beause if
it were in a state with a lower entropy it would also contain
more information than previously specified (Jaynes' principle,
1957). However this argument does not *prove* anything.
The entropy of a state describing a physical system is a quantity
expressing the randomness of that system. A gain in potential
information about a system implies a decrease of entropy
(uncertainty) of that system. If rho is the density matrix describing
a quantum system H(rho) = - Tr (rho ln(rho)) is the von Neumann entropy
of that system. The major property of H(rho) is non-negativity
(H(rho) = 0 if rho is a pure state). Actually the density matrix
can be seen as an operator which carries the maximal information
about (the state of) the quantum system. Holevo's theorem gives
the upper bound of the information which can be obtained from a quantum
system. It is also possible to express the *uncertainty* principle in
terms of the von Neumann entropy, but this it is technical stuff.
Now let us have a quantum system S, an apparatus A, an observer O,
uncorrelated. The total initial entropy is then equal to the sum
of the 3 entropies. During the measurement S, A and O become correlated.
The sum of the final (after the measurement) entropies is greater than
the sum of the initial entropies. Essentially that is because we must
take into account the correlations between S, A and O. The final state
of quantum system S, after the measurement, is an eigenstate of the
measured observable, that is to say it is a 'pure' state, which has a lower
entropy than a 'mixture'. Simultaneously the entropies of A and O, of
course,
increase.
The effective gain of information obtained by a measurement
process equals the difference the initial entropy (lower) and
the final entropy (higher). In other words each gain in information
happens at the expense of the entropy of the system. If there is
also a degradation of information is another story ...
Now the MWI is about measurements, so yes ... But we must define
entropy and information in a different way.
Note that the branching structure as such presupposes some
time asymmetry, between past and future, which does *not*
pertain to QM (essentially time-symmetrical).
On the other side, without time asymmetry MWI can not
prevent communication between different 'worlds', as pointed
out by Asher Peres.
Thus, while in QM there is a superposition of states,
in MWI there is a superposition of 'worlds'. Not a very
big gain, imo.
> [H. Everett III]: Thus with each succeeding observation (or
> interaction), the observer state "branches" into a
> number of different states. Each branch represents a
> different outcome of the measurement and the
> corresponding eigenstate for the object-system state.
> All branches exist simultaneously in the superposition
> after any given sequence of observations. The
> "trajectory" of the memory configuration of an
> observer performing a sequence of measurements is thus
> not a linear sequence of memory configurations, but a
> branching tree, with all possible outcomes existing
> simultaneously in a final superposition with various
> coefficients in the mathematical model. [...]
Yes a good quotation from the famous paper on Rev.Modern Phys.,
29, 3, (1957), page 460.
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