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.

• The uncertainty principle demands that certain observables be treated as pairs which are mutually exclusive. Measure position and momentum must be uncertain. The choice of Q#’s must respect this relationship and only mutually observable quantities can be used to label states.
• Q#s identify fundamental constituents. Examples of these Q#s are: charge spin, mass, lepton number, and quark flavor.
• Q#s also classify possible states. Examples include momentum, position and energy which label different states of basic particle.
• Useful Q#s include those that don’t change in a process and therefore follow a conservation law. In particle physics we find Q#s that are:
• always conserved,
• sometimes conserved,
• almost always conserved but occasionally violated.

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; z-projection 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.
Particle-anti	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 u-ness and d-ness
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,νe,νμ,ντ,		Lepton particles; labels particles

To try and find the appropriate labels for an elementary particle there are two aspects of the problem to consider.

1. Space-Time labels: Need to choose the space time operators that commute and label the states for elementary particles. The goal is to search for states that do not mix. {Remember monopole, dipole…}. The following statement is a more rigorous statement as to how particle states are labeled by mass and spin based on the space-time properties.

BULLETIN (New Series) OF THE

AMERICAN MATHEMATICAL SOCIETY

Volume 39, Number 3, Pages 433{439

S 0273-0979(02)00944-8

Article electronically published on April 12, 2002

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_μP^μ = -M2, where M is the mass of the particle.

The second invariant can be described as follows: while P_μP^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 -M²j(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 non-zero, we see that spin indeed describes how the particle rotates in three space.

For massless particles where P_μP^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 half-integral. 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 spin-statistics theorem which follows from quite general properties of quantum field theory: bosons must have integral spin, while fermions must have half-integral 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 u-ness and d-ness 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 S₁ , S₂ 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