Why global conservation of certain physical quantities is not enough?

[arroy_0205]

What is the implication of the fact that quantities like electric charge, color etc are conserved not just globally but in local region of space too? I understand the later is more restrictive but what advantage does it offer?

 

[blechman]

Actually, I've recently done a lot of thinking about this, and now is an opportunity for me to correct some things I've said before:

The GLOBAL symmetry that always appears for free when there is a local symmetry (just take the local parameter as being a constant - why not?!) is a SYMMETRY - that means by Noether's Theorem that there is a conservation law. This is important for obvious reasons (charge cannot appear out of nowhere or vanish into nothingness).

The LOCAL symmetry is not in fact a symmetry, but is a statement that there are LONG RANGE FORCES (gauge fields) that couple to this conserved charge. That is the implication of the local part of the symmetry.

 

[genneth]

Actually, I've recently done a lot of thinking about this, and now is an opportunity for me to correct some things I've said before:

The GLOBAL symmetry that always appears for free when there is a local symmetry (just take the local parameter as being a constant - why not?!) is a SYMMETRY - that means by Noether's Theorem that there is a conservation law. This is important for obvious reasons (charge cannot appear out of nowhere or vanish into nothingness).

The LOCAL symmetry is not in fact a symmetry, but is a statement that there are LONG RANGE FORCES (gauge fields) that couple to this conserved charge. That is the implication of the local part of the symmetry.

Also, from a quantum mechanical point of view, the so-called local or gauge symmetries are not in fact symmetries of anything. Symmetry in quantum mechanics are operators which act on the Hilbert space of states, and which commute with the Hamiltonian; thus they are good quantum numbers to label states with. Local or gauge symmetry (which can be treated as a real symmetry in classical mechanics (kinda...)) are not operators at all; they are over-labelling of the *same* physical state, which is why when "quantising" things like electromagnetism it is often necessary to fix a gauge, which then reveal the correct number of degrees of freedom.

 

[tom.stoer]

Also, from a quantum mechanical point of view, the so-called local or gauge symmetries are not in fact symmetries of anything. Symmetry in quantum mechanics are operators which act on the Hilbert space of states, and which commute with the Hamiltonian; thus they are good quantum numbers to label states with. Local or gauge symmetry (which can be treated as a real symmetry in classical mechanics (...) are not operators at all; they are over-labelling of the *same* physical state, which is why when "quantising" things like electromagnetism it is often necessary to fix a gauge, which then reveal the correct number of degrees of freedom.

Yes and no.

Of course local gauge symmetries are over-labelling using unphysical degrees of freedom which must be fixed (eliminated) for quantization. But one CAN contrsuct quantum mechanical operators acting in an (enlarged) unphysical Hilbert space. In quantum electro dynamics the Gauss law constraint (in A°=0 gauge) acts as a qm generator of residual, time-independent gauge transformations. The Gauss law constraint is an operator commuting with the Hamiltonian and therefore describes a conserved charge (electric charge). The kernel of the Gauss law G|phys>=0 is just the physical Hilbert space.

Of course - when carrying out the Dirac constraint quantization procedure - this is an intermediate step which eventually leads to the physical sector of the Hilbert space, physical degrees of freedom, gauge fixing etc. but nevertheless in can be treated entirely quantum mechanically. In that sense it is a kind of symmetry.