What gets lost in the classical --> quantum transition

[Pete]

In a discussion I got into off the forum there was an objection about a concept that finds use in relativity. The person (who shall remain nameless) who objected based his objection, in part, on the notion that the concept, supposedely, has no meaning in relativistic quantum mechanics. This is clearly a very poor arguement since there is absolutely no reason to expect that all useful ideas in classical mechanics will still be useful in quantum mechanics. For example, the concept of a classical trajectory/worldline looses its meaning when one transitions to quantum mechanics. But one cannot argue that this is a reason to abandon the very useful idea of worldlines in classical relativity. Infact its a very important concept. I believe that force is another concept that gets lost in the transition?

 

I wanted to ask for help in finding more examples of this, i.e. what concepts get lost in the classical --> quantum transition? I'd greatly appreciate your help folks. Thank you in advance.

 

Pete

[booker]

Pete. I think I'd be hard-pressed to find anything that isn't lost, or redefined, really. That which is kept would include Minkowski space...

[Gilded]

Force as a concept regarding acceleration doesn't magically disappear when discussing quantum mechanics, although it can be expressed differently/explained further. But certain forces and concepts are definitely more relevant in, let's say nuclear physics than classical mechanics concerning a ball rolling down a hill. But as booker said, most of the stuff is redifined, especially since you can't make the classical kind of approximations anymore to describe things accurately.

[Klaynos]

It depends what you mean by lost...

 

If you follow the classical -> quantum route by going

 

Newtonian - Lagrangian - Hamiltonian - hamilton jacobi

 

Then you lose things like force quite early on (Lagrangian mechanics uses just energies, no forces at all)

[Gilded]

(Lagrangian mechanics uses just energies, no forces at all)

 

I thought that wasn't "until" Hamiltonian mechanics?

[Pete]

Pete. I think I'd be hard-pressed to find anything that isn't lost, or redefined, really. That which is kept would include Minkowski space...

and that lost is the notion of a worldline. I know of plenty of things that are unchanged when going to QM. Those things aren't why I created this thread. Its those things which become meaningless that I'm interested in. Also, if something needs to redefined in QM then that in itself a strong indication that it might be meaningless in QM, hence the need for redefinition. The newly defined concept then becomes meaningful. But is not the same definition.

 

Perhaps I need to clarify more. Forget "lost" since it seems that term is either vauge or misleading. Instead consider the example I gave of the fact that classical physics uses trajectories (continuous position of particle with respect to time) is a well definend and meaningful concept in relativity while in its meaningles in QM. These serve as an illustrative examples of what I'm looking for. Seems that just asking the question has led me to think of answers I could't think of before.

 

Determinisim is another example of something that does not hold in QM. Force is another example in that force cannot be defined as the time rate of change in momentum, at least not in the Schrodinger picture. Momentum is an operator and in the Schrodinger picture operators are not functions of time so taking the derivatives of them is meaningless.

 

Another concepts is that of reality, e.g. to talk of the "actual" state of an object without reference to an observer in meaningless, e.g., "Cat is alive/Cat is dead" has no meaning unless an observation is made, unobserved position is also meaningless in quantum theory.

 

Pete

Edited August 4, 2008 by Pete

[booker]

and that lost is the notion of a worldline. I know of plenty of things that are unchanged when going to QM. Those things aren't why I created this thread. Its those things which become meaningless that I'm interested in. Also, if something needs to redefined in QM then that in itself a strong indication that it might be meaningless in QM, hence the need for redefinition. The newly defined concept then becomes meaningful. But is not the same definition.

 

Right, right. Now I get it: "those things which become meaningless". Excellent topic.

 

Even thought force doesn't seem to have an equivalent, potential does as it appears in Schrodinger's equation and [math] F= \int \int \Phi [/math]

[foodchain]

and that lost is the notion of a worldline. I know of plenty of things that are unchanged when going to QM. Those things aren't why I created this thread. Its those things which become meaningless that I'm interested in. Also, if something needs to redefined in QM then that in itself a strong indication that it might be meaningless in QM, hence the need for redefinition. The newly defined concept then becomes meaningful. But is not the same definition.

 

Perhaps I need to clarify more. Forget "lost" since it seems that term is either vauge or misleading. Instead consider the example I gave of the fact that classical physics uses trajectories (continuous position of particle with respect to time) is a well definend and meaningful concept in relativity while in its meaningles in QM. These serve as an illustrative examples of what I'm looking for. Seems that just asking the question has led me to think of answers I could't think of before.

 

Determinisim is another example of something that does not hold in QM. Force is another example in that force cannot be defined as the time rate of change in momentum, at least not in the Schrodinger picture. Momentum is an operator and in the Schrodinger picture operators are not functions of time so taking the derivatives of them is meaningless.

 

Another concepts is that of reality, e.g. to talk of the "actual" state of an object without reference to an observer in meaningless, e.g., "Cat is alive/Cat is dead" has no meaning unless an observation is made, unobserved position is also meaningless in quantum theory.

 

Pete

 

Where do I put the limit on observation though with QM. Such as with the observer effect, do I say just because I am not viewing such in some way in an experiment QM no longer holds? Such as with relativity, I would expect the universe as far as we understand it to still operate the same tomorrow when I wake up, such as the speed of light will still be constant, with QM though, do I think my stepping on the ground is turning into phonons, or do I say such is something else?

[ajb]

Classical worldlines/configurations ([math]\delta S =0[/math]) don't get lost but what does happen is that you need to include off mass-shell (non-minimising configurations) to the path integral*. In fact, the classical solutions contribute the most to the path integral. They are by far "the most important".

 

One thing you have to watch for when going from a classical system to a quantum one is if the classical symmetries are preserved when you quantise the theory. If the symmetries are global then this can be desirable. If they are local then this can spoil the picture completely. Look up anomalies.

 

 

 

* I assume you mean the path integral as we are talking about worldlines.

[Pete]

Classical worldlines/configurations ([math]\delta S =0[/math]) don't get lost but what does happen is that you need to include off mass-shell (non-minimising configurations) to the path integral*. In fact, the classical solutions contribute the most to the path integral. They are by far "the most important".

Note that I chose not to consider anything containing the term "lost" since it has an ill-defined meaning in this context. Correct me if I'm wronge by [math]\delta S =0[/math] refers specifically to geodesics, not worldlines in general. I was speaking about general trajectories of particles. This is, which is different than that used in path intergrals. This is a case of something that was perhaps changed/redefined. I.e. the classical trajectory was replaced with path integrals. Mind you, these two quantities are not the same thing.

I assume you mean the path integral as we are talking about worldlines.

No. I was talking about trajectories (i.e. worldlines in spacetime) in general. For example; a classical charged partilce moving in nn EM field would move on a trajectory/worldline (not simply geodesics). Such a trajectory is defined by the set of points for which X(t) = [t, x(t), y(t), z(t)]. That is quite different than a path integral and has quite a different meaning. In relativity a particle can be considered to be observed at each point on the trajectory (i.e. is located at different point spacetime for all t). That is quite different than the notion of a path integral.

Where do I put the limit on observation though with QM.

A theory is limited by its axioms.

Such as with the observer effect, do I say just because I am not viewing such in some way in an experiment QM no longer holds?

I didn't say that an experiment didn't hold. The two are not the same thing. It is well known that nothing can be said about a quantum mechanical system unless an observation has taken place. That is a fundamental part of QM.

Right, right. Now I get it: "those things which become meaningless". Excellent topic.

 

Even thought force doesn't seem to have an equivalent, potential does as it appears in Schrodinger's equation and [math] F= \int \int \Phi [/math]

Force is not the integral of the potential energy function, its the gradient of that function, i.e. F = -grad [math]\Phi[/math].

 

Pete

Edited August 4, 2008 by Pete
multiple post merged

[ajb]

Note that I chose not to consider anything containing the term "lost" since it has an ill-defined meaning in this context. Correct me if I'm wronge by [math]\delta S =0[/math] refers specifically to geodesics, not worldlines in general. I was speaking about general trajectories of particles. This is, which is different than that used in path intergrals. This is a case of something that was perhaps changed/redefined. I.e. the classical trajectory was replaced with path integrals. Mind you, these two quantities are not the same thing.

 

"[math]\delta S=0[/math]" more generally I just mean the classical equations of motion hold.

 

No. I was talking about trajectories (i.e. worldlines in spacetime) in general. For example; a classical charged partilce moving in nn EM field would move on a trajectory/worldline (not simply geodesics). Such a trajectory is defined by the set of points for which X(t) = [t, x(t), y(t), z(t)]. That is quite different than a path integral and has quite a different meaning. In relativity a particle can be considered to be observed at each point on the trajectory (i.e. is located at different point spacetime for all t). That is quite different than the notion of a path integral.

 

A (classical) charged particle would move along a path that satisfied the classical equations of motion. In a path integral, you include these paths plus many others that don't satisfy the equations of motion.

 

As for things that get "lost" I will have to think about it. Apart from the obvious like absolute position/momentum etc, things like Newton's laws get modified to average values etc. Things like the ordering of x's and p's in classical observables don't matter, but for the (differential) operators it is important. But I can't think of any key concepts that are completely lost.

Edited August 4, 2008 by ajb

[Pete]

A (classical) charged particle would move along a path that satisfied the classical equations of motion. In a path integral, you include these paths plus many others that don't satisfy the equations of motion.

Actually a path integral contains all paths, not just a subset of them hence the term explore all paths was coined/.

 

I understand what your opinion on this ajb, I just disagree with it. A classical trajedctory gives the position (i.e. location of point) as a function of time.

 

Lets drop that example for herein because it will only serve as a distraction. I started this thread seeking examples of those classical concepts which are meaningless in quantum mechanics, not those which have a qm counterpart. Can you think of any?

As for things that get "lost"

I myself am not interested in such things. It was a mistake for me to use that word in the first place.

 

Please note: I'm sure all of you have your own opinions of the corresponding quantum mechanical concept of a given classical concept. Lets stick to those examples of concepts which are well defined classically but not quantum mechanically. I would appreciate it. Thanks.

 

Pete

Edited August 4, 2008 by Pete

[ajb]

Formally, you use all paths/configurations that satisfy the boundary conditions required. In practice using the loop expansion you only consider those "near" the classical one order by order in Planck's constant. And of course more generally you may have gauge symmetries to contend with. But anyway...

 

The only sort of things I can think of that are well defined classically but not quantum mechanically are anomalies. That is classical conservation laws are broken at the one loop level.

 

What also can be interesting is the fact that two different classical systems can have the same quantum system. For example quantum Thirring models models are known to be dual to quantum scalar field theories via bosonisation. But this is not what you were thinking of.

[Pete]

The only sort of things I can think of that are well defined classically but not quantum mechanically are anomalies. That is classical conservation laws are broken at the one loop level.

Thanks. That in itself is helpful.

 

Pete

[booker]

Force is not the integral of the potential energy function, its the gradient of that function, i.e. F = -grad [math]\Phi[/math].

 

OK, but you're missing the point. Replace the potential in Schodinger's equation with the function as a centrally acting force. but so what.

 

Alll this discussion is on one level.

 

On a meta-level, what classical laws and definitions have no equivalent in quantum mechanics?

 

Conversely, what quantum mechanical laws and definitions have no equivalent in classical mechanics.

 

We are talking about the two mechanics, right?

Edited August 5, 2008 by booker

[Pete]

OK, but you're missing the point.

The only point I was making with that comment was F = -grad [math]\Phi[/math]. Nothing more and nothing less. If I don't comment on something else it doesn't mean that I didn't/don't understand it, okay?

 

Pete

[ajb]

My MSc project was on anomalies, you can find a version of the thesis here. There are bound to be some typos and some further corrections that are needed.

 

A great book on the topic is

 

Bertlmann, R. A.

Anomalies in quantum field theory

Oxford, UK: Clarendon (1996) 566 p. (International series of monographs on physics: 91)

 

Most other books on QFT should say something about anomalies.

[pioneer]

One way to look at it is, quantum breaks down at large size. While classical breaks down at small size. There is no quantum of rocks. We can locate a rock with great certainty. It will not disappear, go back or forward into time or have coordinated partners or even move as a wave. But as we go small, these things begin to happen. I am not sure where transition no mans land is. Maybe at bio-molecules, who knows???? Medicine uses sort of an uncertainty approach, not due to nature but due to technical limitations. But soon it will be very certain where to put the medicine. Unless the bacteria has a quantum trick up it sleeve. It all depends where the transition is.

[Pete]

One way to look at it is, quantum breaks down at large size. While classical breaks down at small size. There is no quantum of rocks. We can locate a rock with great certainty.

Something like a rock can be considerd to be a huge quantum mechanical system. Quantum mechanics applies here as well. The properties of quantum mechanical systems hold for rocks in this sense. E.g. the uncertainty in position is so small as to be beyond human senses can detect and the human imagination can picture (i.e. we have no experience in variations in position of rocks which are of atom dimensions).

It will not disappear, go back or forward into time or have coordinated partners or even move as a wave.

In non-relativistic mechanics particles not disappear either.

 

Pete

[booker]

The only point I was making with that comment was F = -grad [math]\Phi[/math]. Nothing more and nothing less. If I don't comment on something else it doesn't mean that I didn't/don't understand it, okay?

 

Pete

 

If you're going to pit-pick, you should at least get it right. The potential in Schodingers equation is voltage, not gravitational potential.

Edited August 7, 2008 by booker
multiple post merged

[Pete]

If you're going to pit-pick, ..

Correcting a formula is not nit-picking, its simply pointing out a mistake.

...you should at least get it right.

I already did.

The potential in Schodingers equation is voltage, not gravitational potential.

Yipes! Where did you get that idea from???

 

Note - I was using your notation. Normally the potential function is labeled with "V" and not [math]\Phi[/math].

 

The potential that appears in Schrodingers equation is not " potential," its potential energy. I.e. the quantity whose -grad gives force is potential energy and has units of energy. Potential is something different. In the case of an electric field it has the value of potential energy per unit charge. In a gravitational field it has units of potential energy per unit mass. This is confused a bit in QM since they use the term "potential" to refer to "potential energy." Its a very unfortunate convention.

 

Notice that Energy is given by E = K + V where K = kinetic energy and V = potential energy. The operator corresponding to this value is the Hamiltonian operator and given by

 

H = p^2/2m + V

 

V must have units of energy in order to be able to add it to kinetic energy to get the value of H which also has units of energy.

 

Pete

[Klaynos]

If you're going to pit-pick, you should at least get it right. The potential in Schodingers equation is voltage, not gravitational potential.

 

AFAIK it can be any potential energy, whether that is caused by EM, strong, weak or gravity...

[Pete]

AFAIK it can be any potential energy, whether that is caused by EM, strong, weak or gravity...

But you're right. It could very well be the Yukawa potential. In fact that is a common example that is used in QM for a potential energy function.

 

Pete

 

ps - I never mentioned gravitational potential in my post so I don't know where booker got that idea from.