Quantum of Action and the Uncertainty Principle

This is a revision to my June post (subsequently revised to match this post) about action being fundamental and how it's being quantized in integral multiples of Planck's constant, h, or integral multiples of h/2pi implies the Uncertainty Principle.

Action, S, is a fundamental quantity. Let's examine a single particle. Its action is given by E_p t_p, where the subscript p means in the frame of the particle and E_p =m_pc².

Action, S = m _pt_p c², is invariant. If an observer sees a particle moving with velocity v then they measure action as 

E_obst_obs - p_obsds = m_obsc² - m_obs v ds = m_obsc² t_obs- m_obs v vt_obs = m_obsc² t_obs- m_obs v²t_obs

but m_obs = gamma m_p and t_obs  = gamma t_p, where gamma = 1/sqrt(1-v² /c² ) 

Thus, m_obsc² t_obs- m_obs v²t_obs = 

1/sqrt(1-v² /c² )m _pc²1/sqrt(1-v² /c² )t_p - 1/sqrt(1-v² /c² )m _p v²1/sqrt(1-v² /c² )t_p

=[1/sqrt(1-v² /c² )]²[m _pc²t_p - m _pv²t_p ]

=[1/sqrt(1-v² /c² )]²m _pt_p [c² - v² ] multiplying by c² /c² yields

= m _pt_p c²

While invariance of action does not necessarily imply fundamental importance, it is reassuring.

Do all measurements of E require some finite time and all measurements of p require some finite distance? If so, can they be thought of as really being measurements of action? Let us assume this is true.

Energy is the time rate of change in action. A particle's intrinsic energy is m_pc². All changes in action are quantized in the quantity h/2pi, which we'll call hbar where h is Planck's Constant. Particles have a period, tau, associated with their change in action , given by tau =h/m _pc². One can establish a related quantity, tau bar, the time for a change in action of h/2pi or hbar. Tau bar = 1/omega

Discussions which follow remain essentially the same if the 

The Uncertainty Principal follows naturally from the quantization of action. If one measures the change in action over duration t_meas , one gets an energy measurement, E_meas = change in S /t_meas . 

If one performs the measurement so that it begins just after a change in S of magnitude hbar occurs and it ends exactly when the following change in S by hbar occurs, i.e. if t_meas= tau bar, one gets E_meas = E_p . However, if the duration of the measurement is less than tau bar but the measurement still encompasses the minimum change in S of hbar, then E_meas > E_p . E_meas = change in S /t_meas so that E_meas = hbar/ t_meas or E_meas t_meas= hbar , and the smaller t_meas is the greater E_meas  .

If the duration of the measurement is over several periods, tau (say N tau) the uncertainty in energy is reduced.

Assume the change in S during the measurement is N(h/2pi). To capture the N changes in S, the measurement duration could be less than or more than N tau bar by as much as tau bar less an instant. For example, t_meas could begin just before the first change in S and end just after the last change. It would record the full change in S, but would be almost one tau bar shorter than N tau bar. Then E_meas = Nhbar /[(N-1)tau bar]. However, if the measurement began just after a change in S to the intial value and ended after the final change to action of Nhbar , the duration of the measurement would be N tau bar and the energy measured would be Nhbar /(N tau bar). The uncertainty in energy for the the measurements of the same change of Nhbar in action S , would be Nhbar /{tau bar [1/(N-1)-1/(N)] } which for moderately large N comes to about hbar /(N tau hbar) . Thus, by extending the duration of the measurement by a factor of N one reduces the uncertainty in energy by the same factor

We can also explain uncertainty using the DeBroglie relationship lambda = h/p:

Just as energy is related to the quantum of action hbar ,by E = hbar/tau bar, momentum, p, is related to the quantum of action by p = hbar/lambda bar; and p lambda bar = hbar indicates that a particle of momentum p must travel a distance lambda bar to accumulate one hbar of increase in total action, S.

If you measure the particle’s action over a length r shorter than lambda bar, you may detect a change in action of hbar. Then you can conclude that the momentum may be as high as hbar/r. However, most of the accumulation of momentum times distance since the last change in total action may have occurred in the particle path before your measurement, so p could be much lower so your uncertainty in p is hbar/r. If you measure change in action over r’ = 7 lambda bar, you will measure between 6 hbar and 7 hbar change in total action, and the uncertainty becomes 1 hbar/r’ or 1/7 of hbar/r.

Here is a slightly different approach assuming the quanta of action is h not hbar yielding a similar result if you substitute hbar and taubar  appropriately:

Suppose one measures the change in total action over a time interval, t,  less than tau. Say sometime during that interval, t,  the total action changes by one h.  The only things you know are that sometime during t total action changed by h. From this you conclude that E could be as high as h/t. On the other hand if you think about it a bit you realize that your interval t could have started after a large fraction of tau had passed since the last increase in total action by h, so that most of the accumulation of energy time leading to an addition of h in total action occurred prior to your beginning your measurement. Therefore the energy could be much lower( near zero?). Thus you are uncertain about E by h/t.

Now suppose your measurement takes place over a period, t’,  several times tau, lets say  t’ = 7 tau for example. Depending on exactly when you begin your measurement vis a vis  when a period begins [ i.e. the time when the last increase in total action by h occurred prior to the measurement]  you will measure either 6 h or 7h as the change in total action. So you know the energy is between 6h/t’ and 7h/t’  but since t’ is 7 x t , you have reduced your uncertainty by a factor of 7.