Re: Where is Invariant Mass?

From: George Jones (george_llew_jones_at_yahoo.com)
Date: 10/13/04 

Date: Wed, 13 Oct 2004 08:24:27 -0400

Paul Draper" <pdraper@yahoo.com> wrote in message
news:74768d2d.0410120720.56d3ef7e@posting.google.com...
> regenesis0@aol.com (Derik Smith) wrote in message
news:<20041010232304.08241.00001689@mb-m10.aol.com>...
> > What does invariant mass originate from? What causes it? What
> > reastion/interrelation... etc.
> >
> > Google has been of little help to me, as every discussion that
mentions
> > invariant mass seems to do so just to explain relativistic. (on the
plus side,
> > I found the best explanation of relativistic mass I've ever read in the
> > proscess. It uses snow plows to explain energy density.)
> >
> > -Derik
> > "I'm torn between being pissed off at you and being in complete
> > awe of you." - Zobovor
> > "I'm a sucker for G1 homages." - The Wombat King
> > Luke Skywalker was a terrorist.
>
> This is a little like asking where does momentum conservation come
> from. Or, actually more to your underlying point, where does momentum
> come from?
>
> The simple answer sounds flippant, but it's not: It's what nature
> tells us. That is, we notice that in some experiments we've done, that
> there is a quantity in a system that is the same before and after an
> intra-system interaction, regardless of the nature of the interaction.
> Then we look further and notice that the same quantity is "conserved"
> in *all* experiments we choose to do. So this looks like a useful
> quantity, and we define it and give it a name (momentum) and suggest
> that momentum is conserved in all closed systems. Thus the definition
> springs from utility and the *discovery* of a regularity in nature.
> The same goes for invariant mass. There happens to be a quantity that
> is the same in a closed system regardless of the motion of an inertial
> observer.
>
> A deeper answer comes from a suggestion by Noether that all conserved
> quantities are due to a symmetry of nature. Now, to some this has
> essentially the same "Huh, how about that?" quality that a
> conservation law does and nothing more, but there really is something
> deeper about symmetry. The reason is that conservation laws (and there
> are several) all seem like separate items in a catalog, but symmetries
> can be combined -- with the result of predictions of other conserved
> quantities to look for. I would argue that symmetry is no more
> fundamental than a conservation law, but symmetry is more
> theoretically useful.
>
> I actually don't know (or have forgotten) what the symmetry is that's
> associated with invariant mass in the Noether sense, unless Lorentz
> symmetry accounts for both the spacetime interval and invariant mass.
> I had the impression there was a 1-1 correspondence between a symmetry
> and a conserved quantity in Noether's theorem. Some folks have said
> that Lorentz invariance is coupled to charge conservation; I thought
> charge conservation had to do with the U(1) symmetry of QED.
>
> While I'm on the topic, I recall how the U(1) symmetry of QED demands
> charge conservation, and similarly where lepton number conservation
> comes from, but I don't think I've ever seen a demonstration of the
> symmetry that accounts for boson number conservation. Is there one?

I don't think boson (or fermion) number is conserved, but maybe
I'm missing something subtle. For example, consider

e- + e+ --> 2 gamma,

where 2 photons are created by the annihilation of an electron and
positron. I think that there also might be examples of boson
non-conservation in meson decay, and in the decay of W+- and Z bosons.

The full symmetry group for Minkowski spacetime is the Poincare group,
i.e., the group generated by the Lorentz group together with spacetime
translations. Noether's theorem applied to the spacetime translation
part gives conservation of 4-momentum, and "invariant mass" is the
Lorentz square of this.

As you said above, what is fundamental and what is derived is sometimes
a matter of taste. Rest mass pops out of a very interesting (at least to
me) 1939 analysis of Wigner of elementary systems, which takes as
fundamental:
  1) special relativity;
  2) quantum state are rays in a Hilbert space;
  3) the frame invariance of transition probabilities.

He then defines the state space of an elementary system to be an
irreducible representation space for the Poincare group. This makes a
lot of physical sense (again, at least to me). Supose that I have a
state space that is associated with a reducible representation of the
Poincare group. Because the representation is reducible, it contains a
subspace that is invariant under all Poincare transformations, i.e.,
elements of the subspace don't get tossed out of the subspace by
Poincare transformations. The subspace is more "elementary" than the
whole state space.

Spin and invariant mass for elementary systems fall out of this
analysis as labels for the possible elementary systems! The actual
allowed values for spin also pop out, while the values for mass are
only restricted to the set of non-negative real numbers.

The analysis has as input only the fairly general principles that I
outlined above. The analysis itself is quite technical and is the only
paper in the last 100 years or so that I know to have made fundamental
advances in both physics and pure mathematics by the standards of each.

String theorists might take exception to this last statement.

Regards,
George

Relevant Pages

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