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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. –  NiftyKitty95 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. –  NiftyKitty95 Jan 13 '12 at 16:27
    
The updated question(v2) basically seems to answer its own question. –  Qmechanic Jan 15 '12 at 11:33
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2 Answers

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
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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.

To answer the comment:

Is this idea, namely looking for invariants to lable states or particles, natural? It sounds deep to me, I mean is it really an idea that anybody could have "naturally" thought of?

Invariants are used for labeling in all human labeling systems and the answer is self evident in the word. Fingerprints, retinas come easily to mind, DNA of course. Why are they used? Because they do not change through the transformations that a person may undergo. It is a natural organizing principle to look for invariants in landscapes, in biological systems, in evolution, etc. In a similar way, "invariant" for a particle means a quantity that is immutable to transformations that it may undergo and they serve to pin its identity. For particle the transformations are mathematically defined.

Particularly for physics, if one studies classical physics, invariants of motion were very much to the front, so yes, it is natural to look for invariants and utilize them as I illustrated above.

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You totally misunderstood me. What I meant is the following: having discovered such so many different particles, now I am thinking how to organise them and classify them. So far so good. What I do not understand is that how would I think:"I should look for some kind of invariant quantity to lable them". Is this idea, namely looking for invariants to lable states or particles, natural? It sounds deep to me, I mean is it really an idea that anybody could have "naturally" thought of? –  Revo Jan 13 '12 at 15:29
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It sounds natural to me (but maybe that's just because I grew up after the time when the application of group theory in physics had become widespread). If your theory has symmetries (Poincare or internal symmetries), then it seems natural to look for something which emerges naturally from the symmetry transformations - e.g. a Casimir like mass from Poincare transformations, or like a charge which is a generator of SU(N) transformations. But I admit that this is all with the benefit of hindsight ! I have lots of respect for the people who worked it out in the first place. –  twistor59 Jan 13 '12 at 16:29
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