Is There a Definitive Answer to the Existence of Action in Quantum Mechanics?

[olgranpappy]

Of course there is always conservation of energy. I wasn't talking about conservation of energy. I was talking about the fact that fundamentally, there is no such thing as a nonconservative force, and forces seem nonconservative only when degrees of freedom are neglected.
As it says in the Feynman Lectures on Physics:
"We have spent a considerable time discussing conservative forces; what about nonconservative forces? We shall take a deeper view of this than is usual, and state that there are no nonconservative forces! As a matter of fact, all the fundamental forces in nature appear to be conservative. This is not a consequence of Newton's laws. In fact, so far as Newton himself knew, the forces could be nonconservative, as friction apparently is. When we say friction apparently is, we are taking a modern view, in which it has been discovered that all the deep forces, the forces between particles at the most fundamental level, are conservative."

...

This is why I am so surprised that the electromagnetic force is nonconservative, as it seems to contradict these quotes.

Apparently when Feynman uses the phrase "conservative" in reference to forces he does *not* mean "are the gradient of a scalar potential." If that had been what he meant then, as we have seen already in great detail, he would be wrong. But that is not what he means. Of course, he probably did not have electromagnetism in mind when he was giving these lecture... hence the misunderstanding.

What he means (what he must mean) is just what we have already discussed in regards to friction--that energy is conserved. And not that the force is derived as the gradient of a scalar.

 

[lugita15]

Apparently when Feynman uses the phrase "conservative" in reference to forces he does *not* mean "are the gradient of a scalar potential." If that had been what he meant then, as we have seen already in great detail, he would be wrong. But that is not what he means. Of course, he probably did not have electromagnetism in mind when he was giving these lecture... hence the misunderstanding.

What he means (what he must mean) is just what we have already discussed in regards to friction--that energy is conserved. And not that the force is derived as the gradient of a scalar.

He did mean conservative force in the sense of a gradient of a scalar field. This is because he says:
"If we calculate how much work is done by a force in moving an object from one point to another along some curved path, in general the work depends upon the curve, but in special cases it does not. If it does not depend upon the curve, we say that the force is a conservative force." So he did mean "conservative force" in the usual sense of the term.

 

[olgranpappy]

okay, then he was wrong.

 

[smallphi]

I am reading "The variational principles of mechanics" by Cornelius Lanczos (4th ed.) and I bumped on something that may clarify the polemics about system of charges in external (possibly time dependent) electromagnetic field.

page 114:

Hamilton's principle holds for arbitrary mechanical systems which are characterized by monogenic forces and holonomic auxiliary conditions.

The definition of 'monogenic' forces acting on the system is explained on pages 30/31 and is less restrictive than 'conservative' forces. The monogenic forces must be derivable from a work function U that depends on generalized coordinates, velocities and time:

$U = U(q_1, ..., q_n; \dot{q_1}, ..., \dot{q_n}; t)$

$F_i = \frac{\partial U}{\partial q_i} - \frac{d}{dt} \frac{\partial U}{\partial \dot{q_i}}$

Conservative forces are particular case of monogenic forces when the work function U depends only on coordinates but not on velocities or time. For conservative forces, the work function U is the usual potential energy with minus sign.

The Lorentz forces acting on the charges of the system from the external EM field must be derivable from a work function. In this case the external Lorentz forces are monogenic but not conservative and according to Lanczos, the system can be treated with action methods.

Another case is a closed system consisting of charges AND the field of those charges. The external forces acting on that system are zero i. e. trivially conservative.

 

[olgranpappy]

I am reading "The variational principles of mechanics" by Cornelius Lanczos (4th ed.) and I bumped on something that may clarify the polemics about system of charges in external (possibly time dependent) electromagnetic field.

page 114:


The definition of 'monogenic' forces acting on the system is explained on pages 30/31 and is less restrictive than 'conservative' forces. The monogenic forces must be derivable from a work function U that depends on generalized coordinates, velocities and time:



Conservative forces are particular case of monogenic forces when the work function U depends only on coordinates but not on velocities or time. For conservative forces, the work function U is the usual potential energy with minus sign.

The Lorentz forces acting on the charges of the system from the external EM field must be derivable from a work function. In this case the external Lorentz forces are monogenic but not conservative and according to Lanczos, the system can be treated with action methods.

interesting

 

[smallphi]

Another quote from "The variational principles of mechanics" by Cornelius Lanczos relevant to the original question:

page 351:

... the basic feature of the differential equations of wave-mechanics [quantum mechanics of Schroedinger] is their self-andjoint character, which means that they are derivable from a variational principle.