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started an entry Planck’s constant with a remark on its meaning from the point of view of geometric quantization (and nothing else, so far).
Added Planck constant – Basic definition, just for fun. (And also since it’s a point rarely made explicit, simple as it is.)
Am a little surprised that nothing (or barely anything) was said about the dimension of $h$ as a physical unit, as measured in $(kg)m^2 s^{-1}$ or whatever. Is that considered too low-level?
It’s not too low level, no. Please add it if you have energy.
Well, I had the time and energy (measured in joule-seconds?) to put in a little something about Planck’s constant as a physical constant, but it might be considered embarrassingly low-level. In which case, please feel free to jazz it up to suit taste.
Mentioning the Planck scale there would give a peg to hang on a funny observation of John L. Bell in some version of ch.1. of his primer of infinitesimal analysis, namely that the infinitesimal neighborhood of 0 is akin to the Planck scale in that there the order-theoretic structure of the smooth line breaks down. I wonder whether more could be made of this observation. Unfortunately, I can’t give a more precise reference here, because I have this version in electronic form only, and it isn’t contained in the first print edition nor can I find a corresponding text on his homepage.
Thanks, Todd.
I have moved your addition up to the top. Then I added an Idea-section which points to the various subsections.
Will have to create an entry physical unit now.
added a brief History-comment just so as to link back to entries such a black body radiation and ultraviolet catastrophe
Here’s a potentially stupid question connected to the Planck constant: do we expect the laws of physics to be invariant under the action of Galois group of real numbers, i.e., the complex conjugation?
For example, consider an electron traveling around a small loop in an electromagnetic field. Then the change of phase will be given by the holonomy, which is a complex number of the form $\exp(s)$, where $s$ is a small purely imaginary number.
Now the purely imaginary line $\{i t\mid t\in \mathbf{R}\}$ splits into two rays, and in mathematics there is no canonical way to distinguish them (i.e., equip the purely imaginary line with an orientation), until one makes a noncanonical choice of a square root of $-1$.
Is there a physical experiment that can distinguish the two rays from each other? That is to say, can we say when the above change in phase $\exp(s)$ has “positive” $s$?
That phase associated to the worldline of a charged particle in an electromagnetic field is locally the exponentiated integral of the 1-form $q A$, where $A$ is the “gauge potential” (the local connection 1-form) and the number $q$ is the particle’s charge. For the electron $q = e$ in usual conventions, while for the positron $q =-e$.
Hence that sign difference reflects the charge of the particle. This is measurable from the Lorentz force that the particle feels, whose expression is obtained by Euler-Lagrange variation of $e A$ and this way reflects the same sign difference.
I see, so perhaps from a mathematical viewpoint we could say that the charges of an electron and positron are the square roots of $-1$, with no canonical way to distinguish between them since the CPT symmetry exchanges electrons and positrons?
I was wondering whether we could distinguish them on fundamental grounds.
In practice (say in the famous cloud chamber track pictures) one does distinguish the positively from the negatively charged particles by whether they turn right or left, due to the Lorentz force, when moving through a transverse magnetic field.
But of course this requires first fixing a sign convention for the magnetic field (the Farady tensor, hence the curvature 2-form of the line bundle on spacetime) which probably always comes down to the converse argument.
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