## Wednesday, 26 December 2012

### Quantum Probability

Have being having yet another debate (or rather a continuation of one) about the meaning or otherwise of quantum mechanics on the physics fora. Those who have been reading this blog, know, that I think the simplest way to understand quantum mechanics is to see it as an extension of classical  probability, to embrace the concept of a probability amplitude which is a complex number, the modulus squared of which gives rise to a probability density function.  This is to take the Born Interpretation to it's logical conclusion, the standard view point is a curious mix of probability and the implications of this to a single particle or system. It is this switch between an essentially statistical idea and the claim that we can predict what will happen to one particle that I believe to leads what Karl Popper calls the great quantum muddle.

In a couple of previous posts, I have indicated how such a perspective clears up the interpretation of the two slit experiment. To recap the so called wave properties are essentially statistical applying to a population or ensemble. Thus questions which are essentially statistical tend to be confused with what we can say about individual particles. The claim being for example that a particle splits into two as it passes through the slits that magically recombines to form a single particle at the screen. The screen being alleged to 'collapse the wavefunction' so that it appears as a single point on the screen. Of course empirically what happens is that it is only after a significantly number of events have occurred that anything like a wave pattern can be discerned. This would imply that the 'wave properties' are related to the probabilistic aspects associated with an ensemble of photons and not to an individual particle. I show how this can be explained by the concept of complex probability amplitudes and the same applies to beam splitters etc in these posts

http://chrisfmathsphysicsmusic.blogspot.co.uk/search?q=Two+state

Anyway I am gratified to learn that there is a whole branch of mathematics called quantum probability which includes classical probability. On this view, the so called collapse of the wave packet is a mapping from the complex number space C^n to the real numbers. There is nothing physical about it all. Indeed those who did M338 will be familiar with the concept of a projection from a higher dimensional space to the real numbers.

Anyway a good introduction to the whole subject is given here

http://igitur-archive.library.uu.nl/math/2001-0713-122250/km.pdf

What is apparent is that it is clear that classical probability theory cannot account for many quantum phenomenon. The paper also makes it clear, that in so called Aspect experiments, where a correlation has been measured between the spins of two photons emitted from the same source and sent in opposite directions, The attempt to understand it's implications which has caused so much fuss over the past 20 -30 years, measurement of the spin of one particle does not affect the spin of the other. This is counter to so much of the prevailing wisdom (or at least that which dominates the debate in the media and the popular literature) that I really think this approach deserves further attention. What is gratifying about the quantum probabilistic approach is the way in which the correlation between two particles emitted from the same source just pops out of the formalism. I will expand on this in later posts.

The Open University physics project course includes one on quantum entanglement and I'm seriously contemplating doing this within the next year or so based on the above paper.  I'm swithering between doing the OU course SM358 in October, which I'm not sure I would need, or just diving in straight away.