I Does mass conservation correspond to a pseudo-symmetry?

[Feynstein100]

Emmy Nöther proved that mathematically, if a certain quantity is conserved, there must be a corresponding symmetry somewhere. Momentum conservation stems from spatial symmetry, charge conservation stems from complex phase symmetry at the quantum level and energy conservation stems from time symmetry iirc. The last one is a bit controversial because the universe is not time-symmetric on cosmic scales but that's a discussion for another time.
Right so I was wondering, locally (at ordinary scales), we also observe mass conservation. Does this also correspond to some kind of symmetry? It can't be a true symmetry because ultimately mass isn't conserved and can be converted to energy. So perhaps it's a kind of pseudosymmetry? Or does it fall under energy conservation and thus time symmetry? Hmm but if that were true, momentum can be converted to energy too. So it should also fall under energy conservation. Yet, it has its own symmetry

 

[wrobel]

Besides first integrals there are a lot of other tensor fields that the phase flow of a system may conserve.

For example the Euler equations
$$J\boldsymbol{\dot\omega}+\boldsymbol\omega\times J\boldsymbol{\omega}=0,\quad J=\mathrm{diag}\,(A,B,C)$$ conserve the standard volume:
$$d\omega_1\wedge d\omega_2\wedge d\omega_3,\quad \boldsymbol\omega=(\omega_1,\omega_2,\omega_3)^T$$

 

[Feynstein100]

Besides first integrals there are a lot of other tensor fields that the phase flow of a system may conserve.

For example the Euler equations
$$J\boldsymbol{\dot\omega}+\boldsymbol\omega\times J\boldsymbol{\omega}=0,\quad J=\mathrm{diag}\,(A,B,C)$$ conserve the standard volume:
$$d\omega_1\wedge d\omega_2\wedge d\omega_3,\quad \boldsymbol\omega=(\omega_1,\omega_2,\omega_3)^T$$

I don't get it Could you unpack that a bit?

 

[vanhees71]

That's a very interesting question. First of all, mass conservation holds only in Newtonian but not in relativistic physics. E.g., if you heat up a solid body at rest, it's invariant mass gets larger within relativistic physics.

So the question is, where does the additional mass conservation in Newtonian physics come from. I don't have an answer within classical mechanics but only within quantum mechanics, and it's not a "Noether symmetry", i.e., it's not coming from a (global) symmetry in the sense of Noether's first theorem, but it has to do with the properties of the Galilei group and its Lie algebra.

One can define as "Newtonian quantum mechanics" the quantum theory, which obeys Galilei symmetry. In QT a symmetry is described as a unitary ray representation of the corresponding symmetry group or its Lie algebra.

Now you can always lift a unitary ray representation to a unitary representation of a central extension of its covering group. Now the Lie algebra of the Galilei group indeed has a non-trivial central charge, i.e., one which cannot be eliminated from any given ray representation by simply redefining the operators with corresponding phase factors, and as it turns out that's the mass, and thus mass can be introduced as an additional self-adjoint operator in addition to the 10 operators making up the Galilei group (Hamiltonian, 3 momentum components, 3 angular-momentum components, and 3 center-of-mass positions, generating time translations, space translations, spatial rotations, and Galilei boosts, respectively), which commutes with all these other operators.

Now for any irreducible representation of the Galilei group, defining an "elementary quantum object/aka particle" mass simply takes a fixed real eigenvalue. As it turns out, only positive masses lead to a physically useful dynamics. For negative masses the energy wouldn't be bounded from below, and for 0 mass you simply don't get a physically interpretible dynamics. For $m>0$ you get standard non-relativistic quantum mechanics.

In addition you have a mass-superselection rule, i.e., since mass is a central charge of the Galilei algebra, there cannot be superpositions of states belonging to representations of the extended Galilei group with different masses.

All this is not the case in the (special) relativistic case since the Poincare algebra doesn't have any non-trivial central charges, and there is no additional mass-conservation law in relativistic physics.

 

[haushofer]

Excellent question. Mass conservation doesn't follow from a spacetime symmetry, but corresponds to a central extension of the Galilei algebra. This extension is the result of the lagrangian of a particle going to a total derivative under boosts. This changes the corresponding Noether charge, such that the Poisson brackets between boosts and translations are proportional to the particle's mass. Since these brackets are isomorphic to the corresponding Lie algebra, the commutator between boosts and translations is centrally extended. See e.g. pg 48-51 of

https://www.google.com/url?sa=t&sou...0QFnoECA8QAQ&usg=AOvVaw1yTvvq3k78fxXgAXiRG8Ih

From a 5-dim. perspective you can regard mass conservation as the result of translational invariance along the fifth direction. Maybe I can find some references for that.

 

[lightarrow]

... It can't be a true symmetry because ultimately mass isn't conserved and can be converted to energy.

Mass isn't conserved when the system does not move (more precisely its momentum is 0) and it exchanges energy with the environment. But where is this different with a system which momentum changes because of external forces or its energy changes because it exchanges some energy with the environment? The spatial translational symmetry and the time translational symmetry still hold when they can be applied to the system!
More precisely, when you can find a lagrangian for the system (and in classical mechanics it's not so easy to find a system which does not have a lagrangian), then, if the system's lagrangian is invariant under space translations, the system's momentum is conserved, if it's invariant under time translations, energy is conserved, if it's invariant under space rotations, angular momentum is conserved,...
So the above symmetries don't hold always: it depends on the system and on its environment.
About "mass converted into energy", I strongly disapprove this concept, even if you can find it in a lot of books.
I've already discussed it in a previous thread. Just consider this (it's not the only reason of my disapproval): energy is not frame invariant, mass is.
What you can actually convert is just a kind of energy into another kind of energy.

--
Lightarrow

 

[Feynstein100]

From a 5-dim. perspective you can regard mass conservation as the result of translational invariance along the fifth direction

Wait, I thought the universe was only 4-dimensional?