question about noether's theorem in QM
May 20, 2010 1:02 AM Subscribe
Noether's Theorem states that if a system is invariant (degenerate/symmetric ) with respect to a certain variable, it has an associated conserved quantity - the best example is that if you can rotate the axis of a system without changing it, then angular momentum is conserved with respect to time. If you can translate your axes along a line, linear momentum is conserved in that direction.
I'm a mathematician who doesn't know much physics, but...
I think the reverse implication doesn't necessarily hold: symmetry implies an invariant, but having an invariant wouldn't necessarily imply the existence of a symmetry... except...
Invariants come up all over the place, and, mathematically, can be thought of as properties or values that are preserved by an allowed set of transformations. Such an invariant needn't be unique. For example, if you observe a red ball, there are many properties of the ball invariant under the stresses of everyday life, such as redness, ballness, radius, physical composition, and so on.
The Really Mathy Paragraph: You can change the allowed set of transformations and end up with different sets of invaraints. For example, in topology, one set of functions is the collection of all continuous functions from a space X to a space Y. Very, very little is preserved by this set. One example is that if X is a compact space and f a continuous function to Y, then the image f(X) is also compact. Another set of functions is the collection of homeomorphisms, which are continuous functions that are also one-to-one and onto. Shit-tons of interesting properties are preserved by homoemorphism, including some incredibly nuanced things. The homology of a space is an example of such an invariant, and homology is not just a number but an entire algebraic structure.
For QM, my understanding is that the set of allowed transformations is the collection of unitary operators, so I would guess that your invariants would be invariant under the collection of all unitary operators. But IANAP, so take that recommendation with a grain of salt.
posted by kaibutsu at 1:33 AM on May 20, 2010 [1 favorite]
I think the reverse implication doesn't necessarily hold: symmetry implies an invariant, but having an invariant wouldn't necessarily imply the existence of a symmetry... except...
Invariants come up all over the place, and, mathematically, can be thought of as properties or values that are preserved by an allowed set of transformations. Such an invariant needn't be unique. For example, if you observe a red ball, there are many properties of the ball invariant under the stresses of everyday life, such as redness, ballness, radius, physical composition, and so on.
The Really Mathy Paragraph: You can change the allowed set of transformations and end up with different sets of invaraints. For example, in topology, one set of functions is the collection of all continuous functions from a space X to a space Y. Very, very little is preserved by this set. One example is that if X is a compact space and f a continuous function to Y, then the image f(X) is also compact. Another set of functions is the collection of homeomorphisms, which are continuous functions that are also one-to-one and onto. Shit-tons of interesting properties are preserved by homoemorphism, including some incredibly nuanced things. The homology of a space is an example of such an invariant, and homology is not just a number but an entire algebraic structure.
For QM, my understanding is that the set of allowed transformations is the collection of unitary operators, so I would guess that your invariants would be invariant under the collection of all unitary operators. But IANAP, so take that recommendation with a grain of salt.
posted by kaibutsu at 1:33 AM on May 20, 2010 [1 favorite]
Noether's theorem is only concerned with continuous symmetries. Conservation of lepton number (which only changes in discrete quantities) will have symmetry implications in group theory instead.
posted by fatllama at 6:21 AM on May 20, 2010
posted by fatllama at 6:21 AM on May 20, 2010
One answer is that lepton number isn't always conserved; it's violated in neutrino oscillation. Physicists do use the theorem this way (to rule out some lagrangians), but doing so requires you to know what's really conserved.
Some (p351) sources point to the way that gauge invariance (or U(1) ) invariance in scalar fields creates charge conservation in EM and suggest that something similar is happening with other "number" conservations like lepton, baryon, strangeness, etc. but doesn't go through how the relevant force does that. The first link goes through how gauge symmetry creates electric charge.
posted by a robot made out of meat at 7:00 AM on May 20, 2010
Some (p351) sources point to the way that gauge invariance (or U(1) ) invariance in scalar fields creates charge conservation in EM and suggest that something similar is happening with other "number" conservations like lepton, baryon, strangeness, etc. but doesn't go through how the relevant force does that. The first link goes through how gauge symmetry creates electric charge.
posted by a robot made out of meat at 7:00 AM on May 20, 2010
Best answer: Neutrino oscillations break the conservation of lepton flavor, a.r.m.o.o.m. The analogy is the change of strangeness or charm in heavy quark decays, or the strange-antistrange oscillations in neutral K mesons. There's not any known process that changes the number of leptons minus the number of antileptons.
fatllama gave my short answer: Noether's theorem only applies to continuous symmetries, since its proof requires that the functions involved be differentiable. I don't know if there is a similar concise statement that can be made in general about discrete symmetries. Electric charge conservation does come from the U(1) gauge symmetry in electrodynamics, but in that theory charge is a continuous quantity; I think there is no good explanation of why charge is quantized (unless you like Dirac's conjecture about magnetic monopoles and angular momentum). Parity conjugation and time reversal are discrete transformations, more or less analogous to translation or angular momentum, but with no differentiable quantity you can label as a Noether current.
One interesting transformation to think about is particle exchange. Systems of particles that are symmetric or antisymmetric under exchange obey Bose-Einstein or Fermi-Dirac statistics, which changes how they absorb heat at low temperatures. The mathematical model for this involves restrictions on the relative phases of different bits of the wavefunction.
I like to think of the different flavor quantum numbers as defining sets of particles with different exchange symmetry restrictions. So for instance, the transformation associated with strong isospin is the "rotation" between protons and neutrons. In small nuclei where you can ignore electric charge, isospin is a good quantum number and there are sets of excited states with energy depending on the isospin of the state but not on the charge of the nucleus. These are called "mirror states," and the most famous one is the spin-zero state of the deuteron, which is analogous to the di-proton and the di-neutron and therefore falls apart. If you wanted to write down a wavefunction for a nucleus that accounts for isospin symmetry, your state would have to be antisymmetric under exchange of any two neutrons, antisymmetric under exchange of any two protons, and antisymmetric under exchange of any neutron with any proton. The "transformation" that "rotates" between the heavier flavors is the CKM matrix of weak decays, although maybe that's stretching things a bit.
posted by fantabulous timewaster at 10:20 AM on May 20, 2010
fatllama gave my short answer: Noether's theorem only applies to continuous symmetries, since its proof requires that the functions involved be differentiable. I don't know if there is a similar concise statement that can be made in general about discrete symmetries. Electric charge conservation does come from the U(1) gauge symmetry in electrodynamics, but in that theory charge is a continuous quantity; I think there is no good explanation of why charge is quantized (unless you like Dirac's conjecture about magnetic monopoles and angular momentum). Parity conjugation and time reversal are discrete transformations, more or less analogous to translation or angular momentum, but with no differentiable quantity you can label as a Noether current.
One interesting transformation to think about is particle exchange. Systems of particles that are symmetric or antisymmetric under exchange obey Bose-Einstein or Fermi-Dirac statistics, which changes how they absorb heat at low temperatures. The mathematical model for this involves restrictions on the relative phases of different bits of the wavefunction.
I like to think of the different flavor quantum numbers as defining sets of particles with different exchange symmetry restrictions. So for instance, the transformation associated with strong isospin is the "rotation" between protons and neutrons. In small nuclei where you can ignore electric charge, isospin is a good quantum number and there are sets of excited states with energy depending on the isospin of the state but not on the charge of the nucleus. These are called "mirror states," and the most famous one is the spin-zero state of the deuteron, which is analogous to the di-proton and the di-neutron and therefore falls apart. If you wanted to write down a wavefunction for a nucleus that accounts for isospin symmetry, your state would have to be antisymmetric under exchange of any two neutrons, antisymmetric under exchange of any two protons, and antisymmetric under exchange of any neutron with any proton. The "transformation" that "rotates" between the heavier flavors is the CKM matrix of weak decays, although maybe that's stretching things a bit.
posted by fantabulous timewaster at 10:20 AM on May 20, 2010
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posted by Dr Dracator at 1:18 AM on May 20, 2010