## Quantum mechanics and unitarity (part 3 of 4)

In part 2 we have see how to construct the Grothendieck group. Can we do this for the composability monoid in the case of classical or quantum mechanics? The construction works only if we have an equivalence relationship and this naturally exists only for quantum mechanics.

There is no Grothendieck group of the tensor product for classical mechanics, and there is no “ontological collapse” there other than an epistemic update of information in an ignorance interpretation.

In quantum mechanics the situation is different because of unitarity and one can construct an equivalence relationship starting from a property called envariance : http://arxiv.org/abs/quant-ph/0405161 Skipping the boring technical details on how to prove the usual properties of an equivalence relationship, here is the basic idea: whatever I can change using unitarity for the system over here, can be undone by another unitary evolution on the environment over there.

Therefore the correct way to write a wavefunction in quantum mechanics is not |psi>, but as in Grothendieck way: the Cartesian product: (|psi>, null) with the second element representing the “negative elements”, or the environment degrees of freedom which will absorb the “collapsed information” during measurement.

The measurement device should be represented as (null, |measurement apparatus and environment>) and the contact between the system and the measurement device should be represented as the tensor product of the two Grothendieck elements resulting into:

(|system to be measured>, |measurement apparatus and environment>)

By the equivalence relationship this is the same as:

(|collapsed system A>, |measurement apparatus displaying A and environment>)

as well as all other potential experimental outcomes:

(|collapsed system B>, |measurement apparatus displaying B and environment>)

(|collapsed system C>, |measurement apparatus displaying C and environment>)…

But then since only one outcome is recorded, we either need to resort to MWI interpretation, or we need to find another explanation for this.

The explanation is that the measuring apparatus is an unstable system which provides massive information copies (think of the Wilson’s cloud chamber in Mott’s problem). Measurement is not a neat and primitive operation, and the one and only outcome creates an extremely large number of information copies which dwarfs the information about the other potential outcomes which are now hidden in the environmental degrees of freedom.

Sir Neville Mott showed that in a cloud chamber two atoms cannot both be ionized unless they lie in a straight line with the radioactive nucleus. In other words, we only need to understand the very first ionization. Similarly, in the SchrĂ¶dinger’s cat scenario, we only need to understand the first decay, and we do not need to hide “the other cat” in the environment degrees of freedom.

Please stay tuned for the conclusion in part 4.

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