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In continuation of another question about Noether's theorem I wonder whether there exists some kind of relationship between this theorem and the Heisenberg uncertainty principle.

Because both the principle and the theorem relate energy with time, momentum with space, direction with angular momentum. When this is a general fact then e.g. electrical charge and electrostatic potential(*) should be partners in an uncertainty relationship too. Are they?

I feel that these results look so basic and general that I hope that a pure physical reasoning (without math or only with a minimal amout of math) exists.

Also compare this question where again momentum and space are connected, this time through a Fourier transform.

(*) i.e. electric potential and magnetic vector potential combined.

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There is just an obvious relation that both Noether's theorem and HUP are derived from the Poisson structure. But Noether requires vanishing of the Poisson bracket (with the Hamiltonian) whereas HUP requires non-vanishing of that bracket (when promoted to commutator). Also, charge and potential are not dual in any sense; I am not sure how you have arrived at that hypothesis. –  Marek Feb 12 '11 at 16:50
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Heisenberg principle is a result of non-commutativeness of two operators, so it can be easily extended over tons of pairs. –  mbq Feb 12 '11 at 16:52
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Actually the uncertainty principle for energy-time and momentum-coordinates have different nature. –  MBN Feb 12 '11 at 17:41
    
@Marek: Just put it into an answer. –  Kostya Feb 12 '11 at 18:31
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@wsc, @Gerard, the conservation of charge comes from the $U(1)$ symmetry. But this is a global symmetry. You need to introduce gauge field only if you promote that to a local symmetry. The gauge field itself is unphysical (it's redundant in the description), it's just a convenient mathematical formalism. So physically there's no sense in which charge would be dual to gauge field. –  Marek Feb 13 '11 at 2:47

2 Answers 2

up vote 6 down vote accepted

Noether theorem is as valid in CM(*) as in QM(**). It deals with conservation laws and symmetries. In CM the variables are certain, in QM they may be uncertain.

HUP belongs to QM and gives a limitation on canonically conjugated variable uncertainties in a given state.

If some variable in QM is uncertain, it does not mean its expectation value is not conserved. A superposition of free motions states $e^{ipr}$ is also a free motion state although the momentum, for example, may be uncertain. The dynamics of the momentum expectation value is determined with an external force, like in CM (see the Ehrenfest's equations). No external force, no variation of the expectation value <p(t)>.

So I do not see any relationship between HUP and Noether.

(*) Classical mechanics (**) Quantum mechanics

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Expanding on Marek's comment, they are related, but not in a deep way. They are related by the notion from Hamiltonian mechanics that every dynamical variable can be interpreted as an infinitesimal generator of some canonical transformation, or the quantum mechnical notion that every Hermitian operator generates a unitary transformation.

The Heisenberg principle is true of any variable with a continuous spectrum and the infinitesimal generator of translations in that variable, just because these variables always have a nonzero commutator in every possible state. Position and momentum, angle and angular momentum, charge and phase, these are all conjugates in classical mechanics. The charge operator generates infinitesimal rotations in the phase of charged-particle wavefunctions, not changes in potential (you were probably thinking of the effect of a gauge transformation on a potential, but a global gauge transformation, the kind that gives you Noether's theorem for charge, does absolutely nothing to the potential).

The Noether theorem states that when translations of a certain variable are a symmetry, the infinitesimal generator of those translations is conserved. So translations in x, translations in angle, and translations in phase give conservation of momentum, angular momentum, and charge. But these generators obey the HUP with their conjugate variables.

The relationship is that both HUP and Noether talk about canonically conjugate pairs.

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