sci.physics.research

On 2007-08-10, JohnMS wrote:
> On 10 Aug., 00:58, Igor Khavkine wrote:

>> If by the word "action" you mean "integral of the Lagrangian over some
>> path", what I called the "action functional" in my previous post, then I
>> don't know of any experiments that measure it, in whatsoever continuous
>> or discrete units. If you think that such experiments exist, then I
>> invite you to give an example.
>>
>> If you can provide such an example, then we can discuss whether the
>> quantum mechanical operator corresponding to the same measurement has
>> discrete or continuous spectrum.

> Well, if action could not be measured, it would not be a
> pysical quantity and would be part of theology, not
> of physics. So I hope everybody agrees that action can be
> measured (yes, the integral of the Lagrangian).

More like theology, would be blankly agreeing to the above statement
without demanding a clarification of what you mean by "action".

> I am confused somewhat; you said that J is
> sometimes given by an expansion in h; so why do you
> say now that it cannot be measured? (Or do I misunderstand)?
> The expansion formula must have been checked with
> experiment?

I hope you paied sufficient attention to my previous posts to realize
that that there are (at least) two possible meanings to the word
"action" and that you are confusing them.

There is the "action functional", S[x(t)], which is an integral
expression involging the Lagrangian, which associates a number to any
motion of the system, x(t). The variation of S[x(t)] with respect to
x(t) gives the dynamical equations of motion for ths system. I don't
know of any way to determine, empirically, the action functional for any
given system. In fact, there are often different but equivalent action
functionals that can correspond to the same physical system! So, the
"action functional" is not a likely candidate for a physical observable
(this statement can be made more technical and more forceful). Whether
this fact brings it into the realm of theology is for you to decide.

There is also something called an "action variable", which I called J
previously. It is a specific choice of variables that is convenient in
the Hamiltonian treatment of periodic motion, alternative to the usual q
and p. This topic is usually titled "action-angle variables" and can be
found in many books on analytical mechanics (some of which I've already
referenced for the definition). Part of the confusion comes from the
fact that S[x(t)] is seen much more often than J in the usual
undergraduate physics curriculum, while unqualified references to
"action" in the sense of "action variables" still remain in common
terminology.

Action variables, like any other physical observables, can be measured.
Different systems possess different action variables. For instance, the
simple harmonic oscillator's action variable is proportional to its
energy. More specifically, for a simple harmonic oscillator,

J = E/nu or equivalently E = nu*J,

where J is the action variable, E is the total energy, and w is the
oscillation frequency (the radial frequency is w = 2*pi*nu). If you
squint, you'll notice that replacing J by n*h gives, n an integer,

E = nu*n*h,

the allowed energy levels of a quantized harmonic oscillator (up to the
zero point energy nu*h/2). If you further replace n by 1, you get

Delta E = nu*h,

which is the energy of a single oscillation quantum. Now, if, like
Planck, you treat a single mode of light of frequency nu as a simple
harmonic oscillator, you arrive at the conclusion that the energy of a
quantum of oscillation of this mode of light is nu*h, and you call this
quantum the photon.

Notice the close relationship between the action variable J and the
Planck constant h. I strongly suspect that the name of "quantum of
action" often associated to h is a consequence of this relation and is
not directly related to the "action functional". Unfortunately, I don't
have a definitive reference to back this guess up.

Now, let me just restate that the generalization of the above
quantization rule is known as the Wilson-Sommerfeld rule and is now
known to be an approximation to the discrete spectrum of an action
variable operator in modern quantum theory. These action variables are
perfectly measurable and, where applicable, the Wilson-Sommerfeld rule
has been tested experimentally, and so have been the corrections to it
that are provided by modern quantum theory.

Hope this helps.

Igor