Quantum numbers represent quantities that can be measured .i.e OBSERVABLES. Every observable will have an operator associated with that measurable quantity.
ASIDE: To reach a quantum description the observables become operators. The operators, in general, transform the quantum states.
Some observables (Q#s) are continuous such as momentum. Others are discrete such as angular momentum or mass. We are looking to find those quantum numbers that are useful for labeling quantum states.
If things do not change under a transformation we will refer to this as a symmetry and an associate conserved quantity can be found. For example, translational symmetry implies momentum conservation.
ASIDE: Sometimes a subset of operators may be sufficient to establish particle id so not all Q#s are necessary. Baryon number, B, and charge, Q, are sufficient for establishing u‑ness or d‑ness so these are not needed for labeling states.
In the table below important symmetries, observables and related conservation principles are listed.
The following will list observables, symmetries and their roles.
Space time symmetries 

Momentum 
P 
Translation; Conserved ; labels states of a particle 
Energy 
H 
Time evolution: Conserved; labels states of a particle 
Angular momentum 
L 
Rotations: Sometimes, labels composite state rotation 
Spin 
S 
Rotations: Magnitude S for a particle identifies particles; zprojection possible 
Total 
J=L+S 
Rotations: Conserved 
Time reversal 
Sym. 
Change time direction things remain same 
Isotopic space 
Sym. 
Rotation and translation symmetry 
Lorentz Inv. 
Sym. 
Relativity and frames of reference equivalence 
Time evolution 
Sym. 
Physics is same at all times 
U(1) Symmetry 

Charge 
Q 
Electric charge ,Conserved; labels particles 
Phase 

Always introduce an overall phase 
Particle same as an anti particle 

Charge Conjugation 
C 
Good Q# when particle=antiparticle Almost always conservved; labels particles sometimes. 
Particleanti 
Sym. 
Change particle to anti without impact (partially true) 
Flavor approximate 




Flavor 

u,d,s,c,t,b; ; labels particles; good for E&M and QCD, NOT weak since (u,d) and due to CKM mass matrix. Flavor mixing, Partially for the QCD 
strangeness 
S 
Minus # strange quarks; labels particles see above comments 
charm 
C 
Number of charmed labels particles see above comments 
beauty 
B~ 
Minus number of bottom labels particles see above comments 
topness 
T~ 
Number of top labels particles see above comments 
Baryon No 
B 
1/3[N(q)N(qbar)] labels particles see above comments 
Charge 
Q 
Defined as above but =2/3[Nu+Nc+Nt]  1/3[Nd+Ns+Nb] 
Complete flavor 

S,C,T~,B~,Q,B establish flavor uniquely 
Isospin 
I 
Combination of uness and dness 
Lepton number conservation 

Electron no. 

Conserved as far we know; labels particles 
Muon no. 

Conserved as far we know; labels particles 
Tau no. 

Conserved as far we know; labels particles 
CPT 

Any field theory will require that the product of CPT is conserved 
Chirality 
helicity 
Right handed left handed, for m=0 this character is maintained 
e,μ,τ,ν_{e},ν_{μ},ν_{τ}, 

Lepton particles; labels particles 















To try and find the appropriate labels for an elementary particle there are two aspects of the problem to consider.
BULLETIN
(New Series) OF THE
AMERICAN
MATHEMATICAL SOCIETY
Volume 39,
Number 3, Pages 433{439
S
02730979(02)009448
Article
electronically published on
A
particle described by a quantum field theory is characterized by its mass and
spin. The notion of spin is an important one and merits an explanation. These
two characteristics, mass and spin, come naturally from representation theory
in the following way: the single particle states of the quantum field theory
essentially describe a particle moving with momentum P. These
states form an irreducible representation of the Poincare
group. Irreducible representations of Poincare are labelled by two Casimir
invariants. The first Casimir is P_{μ}P^{μ} = M2, where M is the
mass of the particle.
The second
invariant can be described as follows: while P_{μ}P^{μ} is
invariant under
the action of the Lorentz
group, any particular momentum P is only
left invariant by a subgroup of the Lorentz group, Spin(3; 1). This
subgroup is called the little group of Spin(3;1). In
determining the spin of a particle, we need to consider two distinct cases: in
the case of massive particles where M ≠0, it is
not hard to see that the little group is SU(2). The
single particle states with a fixed momentum P transform
irreducibly under the little group. Let the dimension of this SU(2) representation be 2j + 1. The
quantum number j is the spin of the particle, and M^{2}j(j + 1) is
the second Casimir of the Poincare
group. If we boost to a frame where the particle is stationary so only P0 = M is
nonzero, we see that spin indeed describes how the particle rotates in three
space.
For massless particles where P_{μ}P^{μ} = 0, the
situation is different. The little group which leaves any particular P invariant
has three generators which we label J3, B1 and B2. In a
convenient frame where the particle is moving along the third axis, J3 generates
rotations around this axis, while B1 and B2 generate
particular boosts. These generators satisfy the Lie algebra relations
[B1;B2] = 0, [J3,B1] = iB2, [J3,B2] = iB1.
The
representations of this algebra which correspond to physical particles are
quite restricted: B1 and B2 must act
trivially on allowed representations. Further, the eigenvalue
of J3, known as the helicity of the particle, is
quantized. The helicity can be either integral or
halfintegral. There is, however, a standard abuse of notation under which a massless particle is assigned spin, as if it were massive.
For example, a massless photon is a spin 1 particle
even though it consists only of helicity ±1 states.
Likewise, a graviton is a spin 2 particle. With this caveat in mind, we can now
state the spinstatistics theorem which follows from quite general properties
of quantum field theory: bosons must have integral spin, while fermions must
have halfintegral spin.
We conclude that P, M, S are the appropriate labels to describe free particles. M and S will identify specific particles while P will label possible states of this particle.
ASIDE: The momentum operators commute are therefore abelian. When a group is abelian, then all its unitary irreducible representations are one dimensional.
The notion of free particle is not always clear. It implies that the Hamiltonian or Lagrangian that describes the system can be cleanly split into two pieces. One describes the situation when the interactions are off H_{o} and the other is the piece due to the interactions H_{I}. The total Hamilotonian becomes H = H_{o} + H_{I}. This separation is not always unique.
ASIDE: These ideas are complicated by possible selection of large and small parts for a Hamitonian or Lagrangian. In the vicinity of other particles the flavor eigenstates are the best description. So for
_{}
the
neutrino flavor eigenstates are used and are created
in the decay but as they propagate the mass eigenstates
are the better states to use and the states will actually oscillate among the
flavors. The meaning
of particle quantum numbers are blurred by the fact that in different
situations the notion of what a particle is changes from the mass eigenstate to flavor eigenstate. In the above process the electron is paired
with a neutrino as particle doublet for lepton number. So in addition to the
question of the relevance of the mass or flavor states, one can view the (ν_{e}, e) as two states of the same particle
(Weak Isospin).
Further refinement reveals the separation of particle into chiral (L,R) or equivalently when
mass=0 helicity states: (ν_{e},
e)_{L}, e_{R}. Right and left handed
electrons behave differently. There are an assortment
of states that one might choose as the particle states.
Weinberg suggests (see the article posted on Web) that the fields that are incorporated into the Lagrangian are perhaps the elementary particles. In essences these are the lines of the Feynman diagrams. One chooses a set of free fields and interactions between them when building the Lagrangian. In general there is a certain amount of freedom as to how to split up the parts. The standard model does have a prescription for generating the Lagrangian and so the free particles are chosen and the interactions and masses are generated via gauge transformations and symmetry breaking (Higgs). However in all practical problems scientists are forced to choose frameworks to represent the critical physics in the most transparent model. The practical choices for the particle states can be dictated by the problem. As an example consider the quarks that compose the hadrons. Typically, they are assigned mass values of a few hundred MeV. These are the constituent quarks not the current quarks that have a mass of a few MeV. Current quarks pick up an effective mass as they travel through the vacuum and are better represented as more massive objects when trying to model the hadrons.
The difficulty encountered with different interpretations of “basic or elementary” states may be evidence that the true elementary particles have yet to be discovered.
COMPOSITE SYSTEMS:
Are particle labels meaningful as one builds composite systems? Can the combination of two or more quantum entities become more than the sum of the ingredients? The answer is yes. This feature is referred to as entanglement. The composite nature of particles is usually a bit confounding because of the way states combine. A perfect example of this complexity is the definition of the π_{o}, η, η_{ 8}, η’.
_{}
Combining states into composite systems such as the proton or neutron makes these composite systems have an intrinsic uness and dness that remains fixed. The labels for the composite state may involve udd (neutron, uud proton). All linear combinations of states in the composite system will be uud but amplitudes and phases for the combinations can change giving rise to Isospin(u,d) structure which can be extended using SU(3)(u,d,s) to octoplet or decuplet. This character is the same as one finds for combining spin states. The basic labels S_{1} , S_{2} are present in all the combinations although they are often dropped because they do not change._{}
_{ }
ASIDE: Notice that for states that are labeled by some Q#s there is not always a formal operator introduced. This is often the case when the Q# labels a state that doesn’t change so your states are never constructed as a sum of states with this Q# differing.
Time evolution
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