I've been directed to a few articles, and I am sure there is a related post, but can someone explain the procedure by which we can view classic electromagnetism through quantum mechanics? Indeed we need to be able to look at any field as an ensemble of particles (photons), but how can we develop classic field theory assuming quantum mechanics?
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2$\begingroup$ Classical electrodynamics is Lorentz invariant, therefore any sensible underlying theory needs to be as well. Quantum mechanics alone is therefore not enough. QM + SRT, which pretty much means QFT, would be needed. $\endgroup$– DanuCommented Sep 3, 2014 at 5:07
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$\begingroup$ Okay, but even so how do we go from speaking of particles to speaking of fields? $\endgroup$– user24082Commented Sep 3, 2014 at 5:34
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2$\begingroup$ We go from quantum fields to classical fields. Roughly speaking, from creation/annihilation operators on the Fock space to functions belonging to a suitable functional space. See also this recent post. $\endgroup$– yuggibCommented Sep 3, 2014 at 7:05
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2$\begingroup$ @Anthony via SRT. The crucial point is that you cannot discuss particles in the sense that QM does it when you talk of relativistic interactions; particle number is simply not conserved because of mass-energy equivalence! $\endgroup$– DanuCommented Sep 3, 2014 at 7:17
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$\begingroup$ There exists a blog post that treats this subject motls.blogspot.com/2011/11/… $\endgroup$– anna vCommented Sep 3, 2014 at 16:28
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1 Answer
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I think the fundamental problem that many people have with quantum mechanics is, that is seems to be about particles, when, in reality, it is about quanta. A quantum is not the same thing as a particle!
A quantum is the exchanged amount of physical quantities between two parts of a physical system. That can be a quantum of energy, a quantum of momentum, a quantum of spin etc.. Quanta do NOT have to be discrete amounts (e.g. integer multiples of a unit). The numerical amount of the quantum of energy exchanged between an atom and a field can, for instance, be from the continuum of the atom's spectrum, in which case it is not discretized.
What is "discrete" about the exchange of quanta is the "before and after" picture. In case of quantum systems, one can't divide the events "before" and "after" into arbitrarily small fractions or time slices. It's an all or nothing kind of deal. Either the interaction has happened, or it has not, but one can't "watch" it happen halfway trough, as one can in classical mechanics.
As a result, we are not looking at fields as an ensemble of particles. Even quantum fields are, as the name implies, "smooth" objects. What differentiates them from classical fields is, that they can only interact by exchanging their physical states in form of quanta and that we can only measure the differences between initial and final states in terms of quanta, which, in many important cases, can be interpreted as particles. What determines the physical dynamics, however, is not the individual particle that shows up in our measurement devices, but the totality of quanta that can be exchanged in the process of interest.
answered Sep 3, 2014 at 14:52
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$\begingroup$ Just curious... what's not agreeable about my answer? $\endgroup$ Commented Sep 3, 2014 at 15:07
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$\begingroup$ I did not downvote but it seems to me you are not addressing the question of going from quanta to maxwell's equations. $\endgroup$– anna vCommented Sep 3, 2014 at 16:27
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$\begingroup$ But I am. The OP wrote "Indeed we need to be able to look at any field as an ensemble of particles (photons)", which is not a correct description of QFT. The OP needs to develop a better understanding of what QM is. Right now he is stuck in the naive phase where he believes that quantum fields are collections of particles. While I have seen seen some high energy physicists portray it that way, it's simply not true. We aren't "just" dealing with complicated gases made up of point particles. That's a suggestion that comes from an unreflected interpretation of perturbation theory. $\endgroup$ Commented Sep 3, 2014 at 19:27