# Is QM necessary for single observable parameters?

1. Jul 7, 2014

### QuestionMarks

Forgive the title if improper, the language of QM is not my native tongue. :)
All scenarios where QM is invoked that I am used to involve parameters that do not commute, but I suppose I've never truly asked myself if QM is necessary to describe scenarios wherein you are not concerned with this. If we were only ever concerned with commuting variables, would we ever need step outside the bounds of the classical? Or even more simply, if we were only ever concerned with one observable (i.e. spin on one axis, or charge) in entangled systems, even in some unusual setup, would we ever need QM to predict the results? Under this (admittedly silly) form of ignorance, would we still always expect a local realist world?

I understand, of course, we are concerned with these measurements contrary to the above. I pose this hypothetical simply as a means to check my own understanding.

Thanks

Last edited: Jul 7, 2014
2. Jul 8, 2014

### Simon Bridge

Why not look at QM of commuting variables and see what happens?

Note: QM is needed, not because some observables are covered by non-commuting operators but because classical mechanics is insuffient for describing Nature. The link between classical and quantum worlds is not the commutation of operators but the average of large numbers of measurements.

3. Jul 8, 2014

### QuestionMarks

Looking for me is a bit problematic without the appropriate educational background and knowing where to begin (perhaps emphasis on that last part). I'm Chemistry by background, and unfortunately direct treatment of QM for us (me) was limited, the rest hand-waving, and personally predominately through what I can scrounge off the internet. Now what you're saying then I agree with intuitively, but I can't conceive of the scenario. Now that you've answered the general, maybe we can run with the specific considerations (i.e. looking at only one parameter) I mentioned to support this?

4. Jul 8, 2014

### WannabeNewton

How would you calculate the thermodynamic properties of ferromagnets using classical statistical mechanics? To a good approximation the only operator involved here is that of spin, which doesn't even exist in classical mechanics.

5. Jul 8, 2014

### QuestionMarks

Fair enough, but you would want to consider multiple axis still. To be franker, what I'm chewing on is that it seems QM comes about or is necessary due to considerations of orientation or reference frame, but I can't think of a scenario where it would be needed otherwise. Maybe charge is the better example for me to run on?

In a sense, QM is almost seeming to me like a relativistic theory for quanta in an odd categorical way.

Last edited: Jul 8, 2014
6. Jul 8, 2014

### bhobba

The rock bottom essense of QM is a generalisation of probability that allows the continuous transformations between pure states.

I have posted it before but will do it again because I believe its an important point.

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. Probability theory is basically the theory of such mixed states. 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 of such weird complex pure states (which are required to have continuous transformations between pure states) - it does so by means of the so called Born rule.

That's an outline of it. In order of increasing sophistication here are more detailed accounts:
http://www.scottaaronson.com/democritus/lec9.html
http://arxiv.org/pdf/quantph/0101012.pdf

An interesting by-product of this approach is it turns out the assumption of continuity is logically equivalent to entanglement in determining quantum mechanics:
http://arxiv.org/pdf/0911.0695v1.pdf

The logical situation is this. A few reasonable assumptions leads to either standard probability theory or QM. QM is uniquely determined if you require continuous transformations between pure states OR entanglement. It would seem entanglement is one of the key, or maybe even the key, ingredient that makes for QM.

Thanks
Bill

7. Jul 9, 2014

### QuestionMarks

Bhobba, your comment and that first link were excellent reads, thanks. I'm still digesting the last two but have a taste of the general ideas. Seeing QM approached from a non-historical/experimental fashion was a huge conceptual difference. Everything seems agree-able though I haven't yet shook my initial intuition that multiple parameters/references/(whatever-the-best-term-is) holds a high QM significance regarding it's importance. Let me list some roadblocks more explicitly:

1) The probability-based formulation, particularly in that first link, arose from the motivation of having a 2-norm system. I didn't quite see the motivation for this? I grant one motivation as simply to create a probability theory consistent with experimental observations, but is there more to this? I would find that question particularly relevant if one wanted to argue QM as accurate but incomplete.

2) If we found the universe not fundamentally continuous, would entanglement be our remaining motivation for QM?

3) If, as per question 2, entanglement was our remaining motivation, I would sensibly want to attach a higher significance to entanglement phenomena. In doing so, I would come to note (as my original quandary), that I have only seen the "spookyness" of QM when regarding categorically similar scenarios as measuring the spin of entity A on one axis while measuring the spin of entity B on another axis. Meanwhile, I have never heard of any such "spookyness" when concerned with only one parameter state. If I'm leaving any ambiguity for what I mean categorically here, I might envision an entangled positron & electron pair. In such, there would be no spookyness (or need for QM in this scenario explicitly) if just concerned with their charge alone (or in the least I am naive to this). The charge would, it would seem, never be uncertain beyond the classical sense. As such, the categorical difference between such scenarios (one requiring QM, the other not so much) would seem to warrant some significance.

I guess in a sense my line of questioning is a matter of significance and motivation, and I'm particularly trying to dispel the intuitive significance (regarding the need for QM vs classical) hinted at in my 3rd questions, but finding it difficult to.

8. Jul 9, 2014

### bhobba

Its basically a slightly different take on my continuity argument leading to complex numbers. You have to start somewhere; he started by saying lets see what we get if we have negative probabilities and want to make sense out of it. I started by saying what happens if you demand continuity for transformations between pure states. The correct axiomatic treatment is in the second link that derives it from stated axioms.

In that approach they are logically equivalent. If one goes so does the other. Either that or the other assumptions change.

Thanks
Bill

9. Jul 10, 2014

### QuestionMarks

Thanks bhobba, I see that now from the last paper, and that my third question wouldn't be needed. I'm still battling my intuition here hah, but I feel that if I used the Math commonly it would be less unsettling, as I agree with it logically in this reading. I think the first take on building QM more as a probability theory rather than a strictly physical theory was most useful.