Saturday, June 15, 2013

Quantum mechanics and unitarity (part 2 of 4)

When talking about measurement, one talks about the collapse postulate. Let us take a look of what happens with the underlying Hilbert space. During collapse, the dimensionality of the Hilbert space is reduced to the dimensionality of the subspace where the wavefunction is projected to. A key point is that the dimensionality of a Hilbert space is its sole characteristic.

Measurement is initiated by first doing the tensor product of the Hilbert space of system wavefunction with the Hilbert space of the measurement apparatus. This operation increases the dimensionality of the original Hilbert space. Then the collapse decreases the dimensionality.

As an abstract operation, the tensor product respects the properties of a commutative monoid. Short of the existence of an inverse element, this is almost a mathematical group

To model the collapse in a fully unitary way (and free of interpretations) we would like to construct the tensor product group from the tensor product commutative monoid. Is such a construction possible? Indeed it is and it is called the Grothendieck group construction Let us explain this using a simple challenge: let’s construct the group of integers Z starting from the abelian monoid of natural numbers N. We would need to introduce negative integers using only positive numbers!!! At first sight this seems impossible. How can such a thing be even possible? N and by itself is not enough, but with the addition of an equivalence relationship it can be done.

So consider a Cartesian product NxN and we would call the first element a positive number, and the second element a negative number: p = (p,0)  n = (0, n) We would like to do something like this (p,0)+(0,n) = (p, n) = p-n

Also : (0,-q) = (q, 0) All this works in general, but the definition of a Z number is no longer unique. For example: 7=(7,0) =(8,1)=(9,2)=… and -3=(0,3)=(1,4)=(2,5)=…

Therefore we need an equivalence relationship such that two pairs (a,b) and (p,q) are considered equivalent if a+q=b+p Notice that in the equivalence relationship we used only the “+” operation of the original monoid N. The formal definition of the equivalence relationship is slightly more complex due to the need to prove the transitivity property of an equivalence relationship. We call two pairs equivalent: (a,b)~(p,q) if there is a number t such that a+q+t =b+p+t

Now since Grothendieck construction is categorical (universal), it can be applied to the tensor product commutative monoid and this will explain the collapse postulate in a pure unitary way. Please stay tuned for part 3.

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