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The problem of infinite divisibility and how QE sheds some light 
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#19
Jun2412, 03:48 PM

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Some of what you write represents some misconceptions about quantum mechanics that are common in popularized discussions.
In the primary usage of the term, where for instance we learn that energy is quantized in units which are called "quanta," we must be more specific and explain that this is most often the case for systems which are bound states. So for example, the electrons in an atom exist in orbitals which have a discrete amount of energy relative to the unbound system. Photons which are emitted when an electron in an excited state decays to a lower energy state will be found to have a discrete spectrum (with a caveat that I will address below). However, the spectrum of free particles in quantum mechanics is not quantized: their energy and momentum lie in the continuous spectrum of values contrained by ##E = \sqrt{(pc)^2 + (m c^2)^2}##. Furthermore, even higherorder transitions between bound states involve a continuous spectrum of photons. For example, the 2s to 1s transition in hydrogen occurs at lowest order by the emission of two photons. Only the sum of their energies is quantized, ##E_1+E_2 = E_{2s}  E_{1s}##. There is a kinematic distribution of energies ##E_1,E_2## that is peaked around ##E_1=E_2##. The second use of the term quanta arises in quantum field theory, which is often itself called "second quantization." This notion of the term refers to particles states being discrete excitations of a classical field. For instance the photon in QFT is the "quantum" of the electromagnetic field. The excitations are quantized in the sense that a classical EM wave corresponds to a finite (though large) number of photons. However, in the freeparticle case, the energymomentum of these individual quanta are not themselves quantized and form a continuum. None of the above addresses any notion of minimal length. That is the realm of gravity and QM alone doesn't address the issue. The uncertainty principle allows us to probe infinitesimally small scales at the expense of having an infinite uncertainty in energymomentum. 


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