I would like to re-start the discussion regarding conservation of energy in the many worlds interpretation. Kith did not agree with my example, so hopefully it becomes clearer now where I see problems regarding the interpretation.(adsbygoogle = window.adsbygoogle || []).push({});

Let's prepare a particle in a state like

##|\psi\rangle = a_h|h\rangle + a_v|v\rangle##

##a_h^2 + a_v^2 = 1##

Both states with horizontal (h) and vertical (v) polarization are energy eigenstates

##H|h,v\rangle = E|h,v\rangle##

In an experiment with an appropriate polarizator we observe either horizontal or vertical polarization. It seems that the other component has vanished, collapsed, became invisible or something like that. The energy - which was distributed over both component is now contained in the remaining or visible one.

So for the initial preparation we find

##\langle H \rangle = a_h^2 E + a_v^2 E = E##

After the measurement due to decoherence and after tracing out environmental d.o.f. the reduced density matrix can be approximated (!) by

##\rho \simeq a_h^2 |h\rangle\langle h| + a_v^2 |v\rangle\langle v| + \ldots##

where ... represents effectively suppressed interferences terms.

Then again we find for the energy

##\langle H \rangle = a_h^2 E + a_v^2 E = E##

Mathematically this is fine, but the interpretation is rather weird.

1) In the collapse interpretation (which I don't like b/c of unphysical ingredients) the wave function collapses, e.g.

##\rho \to |h\rangle\langle h| ##

so the energy which was "contained" the vertical component is contained in the surviving horizontal component after the collapse.

2) in the many worlds interpretation both components remain real, but one component becomes "effectively invisible", so for example I observe the horizontal component only.

In order to calculate the probability whether I will observe branch "h" in future I use

##p_h = \text{tr} (\rho \, P_h) = a_h^2##

After the observation I can't use the same formula to find what I observe (namely "h" with certainty).

So the logical problem is this

1) I have a formula to calculate what I will observe in future (the probability)

2) There is no formula to calculate what I did observe

In order to calculate the energy I always have to use this formula, so before and after the observation I use

##\langle H \rangle = a_h^2 E + a_v^2 E = E##

So the logical problem I see is the following:

1) the polarization "v" became invisible for me due to decoherence and branching

2) the energy contained in branch "v" is still visible to me (even so the polarization "v" is invisible)

3) the formulas to calculate the energy E before and after the observation are identical

So I can't calculate what did observe (!) and I have to use a formula to calculate the energy I measure where the energy which is contained in the invisible branch is still visible.

Of course all this is mathematically consistent, but there's - at least to me - still some weird stuff in the interpretation.

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# Many Worlds and conservation of energy

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