# Why mass, charge and spin are good labels for elementary particles?

Why mass and charge are good quantities to labels elementary particles?

[I know that what qualifies to be a good label must be a quantity that is invariant under some kind of transformation, but why this criteria? I just do not understand what good does using such invariant quantities buy us]

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Do you have any other suggestions? There is space and there are things in space. As far as the Lagrangian formalism goes, these numbers are the ones which specify your energy functional (The remaining number is the dimension of the space you're working in). Roughly speaking, the group representation of the symmetry group says which fields are used, the mass value is the "weight" of its energy terms and the charges are the coupling among fields. I don't know if it gets deeper within that formalism. But interesting thought. –  NikolajK Jan 13 '12 at 14:32
@NickKidman Please see my reply to annav below –  Revo Jan 13 '12 at 15:31
Well, in physics you observe how specific things behave differently, build a model (a mathematical theory explaining the dynamics) and then characterize the systems behaving differently in real live by associating a priori free values in the theoriy with them. In fluid dynamics, you generalize particle momentum conservation to media (Navier-Stokes equations), incidentally introduce the stress tensor $T$ and the new parameters, which are left open by the theory (viscosities $\mu$ and $\kappa$), specify a certain fluid. Viscosities are like masses and a fluid is always alone with 3 components. –  NikolajK Jan 13 '12 at 16:27
The updated question(v2) basically seems to answer its own question. –  Qmechanic Jan 15 '12 at 11:33

The states of a system are in a big Hilbert space $\mathcal{H}$. The states seen by different inertial observers are related by a representation $U(g)$ of the Poincare group. So, if Alice prepares a state $|\psi\rangle$, then Bob sees this as state $|\psi'\rangle=U(g)|\psi\rangle$; a general state $|\psi\rangle$ will be sent to some state $|\psi'\rangle$ with no simple relation to $|\psi\rangle$. However, by using the structure of the group, the big Hilbert space $\mathcal{H}$ decomposes into a direct sum of smaller irreducible subspaces as, $$\mathcal{H}=\oplus \mathcal{L}_{au}$$ where the subspaces are the $\mathcal{L}_{au}$ and $a$ labels the inequivalent subspaces and $u$ labels the copies of subspace $a$. The states which span the subspace $\mathcal{L}_{au}$ are $|a,i,u\rangle$ for $i=1,\ldots,n_{a}$. Now, if Alice prepares a system in state $|a,i,u\rangle$, then Bob sees this system in state $U(g)|a,i,u\rangle \in \mathcal{L}_{au}$; Bob's state remains in the small subspace $\mathcal{L}_{au}$ because $U(g)|a,i,u\rangle$ is just a linear combination of the $|a,j,u\rangle$ for different values of $j$. So, if a state belongs to an irreducible subspace $\mathcal{L}_{au}$, it is the most elementary kind of system that can exist. Such a system is called a "particle" because the states in the subspace $\mathcal{L}_{au}$ constitute the smallest set of states which Alice and Bob and all other inertial observers can agree to call different views of the same thing. (This argument applies to any space which carries a group representation and not just the states seen by inertial observers connected by the Poincare group; this notation reflects the general case.)

What we call the "properties" of the particle or an elementary system are just the labels $a$ of the inequivalent irreducible subspaces $\mathcal{L}_{au}$. If one finds the irreducible subspaces that carry a unitary representation of the Poincare group, the states are $|M,s;p,m\rangle$ where $M$ is mass, $s$ is spin, $p$ is 4-momentum and $m=s,s-1,\ldots,-s$ is the z-component of spin. The pair $(M,s)$ form the label $a$ of the inequivalent irreducible subspaces; they are the "properties" of the particle, the pair $(p,m)$ form the label $i$ that selects the states spanning the subspace $\mathcal{L}_{au}$ and there are no copies $u$ in this case. The momentum labels $p$ are coords of points on an orbit $p^{2}=M^{2}$ and this is the clue that the label $M$ is what we call mass.

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This might be above the level of the question, but I found it rather enlightening. Good answer! –  David Z Feb 14 '12 at 21:08

Well one could label them as they were being found, as was the case when the first cosmic ray and bubble chamber studies of charged tracks in cloud chambers and emulsions and later electronic detectors were detected. That is what happened at the beginning, they took the greek alphabet and named alpha particles and beta particles and gamma particles and later we get delta( a nucleon resonance) and eta and pi and rho and psi and xi and ...

Is that a better way of labeling them?

Noticing that there exist invariant quantities in the relativistic and quantum mechanical frameworks allowed to organize these over a hundred particles into a structure, analogous to a crystal structure in solids, called the standard model. This, in addition to charge mass and spin organized the particles into group structures that appear naturally or imply naturally theories needed to study elementary particle interactions, and thus extend our knowledge of nature.