# QM and a new way to calculate probabilities

1. Apr 4, 2014

### DaTario

Hi All,

Would it be reasonable to state that, together with the quantum mechanics appearance, there appeared also a new way to calculate probabilities?

I have never heard any teacher pointing out that QM has brought a new probability method. They speak of amplitudes of probability but never say that the quantum revolution is happening in the probability theory as well.

Any Comment?

Best wishes,

DaTario

2. Apr 4, 2014

### micromass

How exactly is it a new way of calculating probabilities? To my knowledge, every probability calculations in QM still conforms to classical probability theory and Kolmogorov's axioms.

3. Apr 4, 2014

### DaTario

The rule that for a composite event which is formed by two elementary events, in QM we are to sum up the two amplitudes and only after this step we square it to get the probability itself.

4. Apr 4, 2014

### DaTario

when it takes to the probability itself it is interpreted in an usual way. But the method used to produce its value seems to have suffered some modification. I may be wrong of course, but in fact it is a subject that has long annoyed me.

Last edited: Apr 4, 2014
5. Apr 5, 2014

### atyy

There isn't necessarily a probability revolution. In the Copenhagen interpretation, the Born rule can be interpreted using classical probability, but there is a measurement problem. In the Bohmian interpretation, classical probability is all that's needed, and there is no measurement problem. Whether the Bohmian interpretation can be satisfactorily extended to relativistic quantum mechanics is still being researched.

However, in some approaches, quantum mechanics can be seen as a generalization of classical probability.

http://arxiv.org/abs/quant-ph/0101012
Quantum Theory From Five Reasonable Axioms
Lucien Hardy

http://arxiv.org/abs/1107.5849
Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
M. S. Leifer, R. W. Spekkens

6. Apr 5, 2014

### strangerep

Then you haven't read Ballentine's textbook. See below.

Not quite. If you have a copy Ballentine, check p30. Although he uses the Cox axioms rather than Kolmogorov, the crucial point is that he defers discussion of the 4th axiom (involving notions like "A and B" which are problematic in QM with noncommuting observables). He eventually returns to this in section 9.6 on "Joint and Conditional Probabilities", and shows how one must introduce time-ordering to cope with this difficulty.

7. Apr 5, 2014

### DaTario

In fact I haven't.

Is it the "Quantum Mechanics: A Modern Development" ?

8. Apr 5, 2014

### Staff: Mentor

Indeed it is.

Best book on QM I know.

Thanks
Bill

9. Apr 5, 2014

### Staff: Mentor

QM does nothing to modify probability theory.

In fact it builds on it. An area in math these days is so called generalised probability models that build on standard probability theory which is seen as the simplest generalised probability model. The modern view is its simply the most reasonable generalised probability model that has features suited to modelling physical systems. In particular standard probability theory does not allow continuous transformations between its so called pure states.

The argument goes something like this. Suppose we have a system in 2 states represented by the vectors [0,1] and [1,0]. These states are called pure. These can be randomly presented for observation and you get the vector [p1, p2] where p1 and p2 give the probabilities of observing the pure state. Such states are called mixed. Now consider the matrix A that say after 1 second transforms one pure state to another with rows [0, 1] and [1, 0]. But what happens when A is applied for half a second. Well that would be a matrix U^2 = A. You can work this out and low and behold U is complex. Apply it to a pure state and you get a complex vector. This is something new. Its not a mixed state - but you are forced to it if you want continuous transformations between pure states.

QM is basically the theory that makes sense out of pure states that are complex numbers. There is really only one reasonable way to do it - by the Born rule (you make the assumption of non contextuality - ie the probability is not basis dependant, plus a few other things no need to go into here) - as shown by Gleason's theorem. But it can also be done without such high powered mathematical machinery. Check out (the link which Atty has also given):
http://arxiv.org/pdf/quant-ph/0101012.pdf

Thanks
Bill

Last edited: Apr 6, 2014