I'm just curious about why many physical identities build relationship with the same units as angular momentum like the action, Lagrangian$\cdot$time, Hamiltonian$\cdot$time, phase space area etc?
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$\begingroup$ Related: physics.stackexchange.com/q/28957/2451 $\endgroup$– Qmechanic ♦Commented Sep 23, 2015 at 5:45
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2 Answers
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Well, OK, this is a resolutely vague question, but there is something special, actually, about angular momentum dimensions.
In quantum mechanics, the fundamental constant, $\hbar$ has dimensions of angular momentum (and is very small in terms of angular momenta, actions, or phase-space areas of our macroscopic world experience). Classical mechanics results as the "small-$\hbar$ limit" of quantum mechanics, when the action (Lagrangian-times-time) of a specific problem is much larger than $\hbar$, as first observed by Dirac and Wentzel about 80 years ago, and exploited by Feynman in developing path-integral quantization. Before that, Bohr had noticed area rules in phase space that led to an early version of quantum mechanics.
So you might say that nature, mysteriously, by dint of quantum mechanics, has chosen a fundamental constant with units of angular momentum.
answered Sep 22, 2015 at 16:22
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$\begingroup$ Also newton chooses Angular Momentum as the first theorem in Principia, and of course it is Kepler Area law too. Not a peculiarity of quantum mechanics. $\endgroup$– ariveroCommented Sep 23, 2015 at 0:45
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1$\begingroup$ Of course Angular momentum occurs all over the place in classical mechanics, and both Lagrangian and Hamiltonian mechanics run on it--but that is not the OP's question. In fact, the extremal principle in Lagrangian mechanics is traceable to the classical limit of QM, in the path integral, as focused on in the answer. $\endgroup$ Commented Oct 26, 2017 at 13:42
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$\begingroup$ Or you could say that a rigorous way to implement the extremal principle in Lagrangian mechanics is to use QM :-) Perhaps the only rigorous way. Or at a least, the one that Nature chooses. $\endgroup$– ariveroCommented Oct 26, 2017 at 23:42
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1$\begingroup$ Indeed, when, decades ago, I first read p 69 in Dirac's 1933 paper I heard the angels' trumpets in heaven.... $\endgroup$ Commented Oct 27, 2017 at 0:29
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$\begingroup$ I think that Feynman had sometimes commented about that paper :-) I am been always intrigued that the path integral weight looks as a dirac delta, but actually executes a dirac delta prime (it finds the extremal of the langrangian). I have never seen a nice formalization if this point, althought. $\endgroup$– ariveroCommented Oct 30, 2017 at 1:31
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The Lagrangian and Hamiltonian both have units of energy. You can get a long way in classical mechanics by only thinking about energy. In field theories the relevant unit becomes energy density.
The action must have units of energy times time, as it is the time integral of the Lagrangian.
I don't think there's any profound significance to the fact that angular momentum has the same units as the action, any more than to the fact that torque has units of energy, or that pressure has units of energy density. (Actually of those three the pressure one is probably the most interesting.)
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$\begingroup$ I took the liberty of correcting the second paragraph. $\endgroup$ Commented Sep 22, 2015 at 17:38
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$\begingroup$ power? You are mistranslating some word, or am I? Watts? $\endgroup$– ariveroCommented Sep 23, 2015 at 0:24
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