If U|Ψ⟩ describes the same physical state as |Ψ⟩ and is normalized, U is a unitary symmetry satisfying [H,U] = 0 and U^†U = UU^† = 1. If F is an operator representing an observable, and we can write U(ε) = exp[±iFε], then F is said to be the generator of the transformation U. For example, the momentum operator p performs space translations, the angular momentum operator L performs space rotations, and the Hamiltonian operator H performs time translations. These are examples of Nöther's Theorem, that generators of space-time symmetries must correspond to conserved quantities. If we discover a conserved quantity experimentally, and it is not connected to a symmetry transformation, we might suspect that the quantity is “not really” conserved, but that we just haven't yet seen an example of its nonconservation!

Eugene Wigner was awarded the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles".

Local gauge transformation: U = exp[iε(r,t)Q]. Here Q is the charge operator. This form of the transformation leads to charge conservation and the gauge invariance that the electromagnetic field must display. We do not know how to handle any fundamental force unless its coupling constant operator generates a local gauge transformation, and this is possible only if the bosons of the field are massless.

Global Gauge Transformation

Local Gauge Transformation

In order that the laws of physics be invariant under a local gauge transformation (bottom line) we must introduce vector fields which display gauge invariance... as in the example of electromagnetism. This gauge freedom can be used to cancel out the unwanted terms from the transformation of ψ and thus recover the original equation, for example the Schrödinger Equation.

Heaviside

Nöther

Other conserved(?) quantities:

• Baryon number A.
• Lepton number L.
• Flavor numbers L_e, L_μ, L_τ.
• Strangeness S.   S = -(n_s - n_anti-s).
• Charm C.   C = (n_c - n_anti-c).
• Topness or Truth.  T = (n_t - n_anti-t).
• Bottomness or Beauty.   B = (n_b - n_anti-b).

Strong interactions change color but nothing else, Electromagnetic interactions change nothing, and weak interactions always change flavor. Therefore weak interactions DO NOT CONSERVE flavor-related quantum numbers, such as the flavor L's or S, C, T and B. The discovery of this for S was the first example seen by physicists of a quantity that was conserved for some fundamental interactions but not others!

Strong process... flavors don't change, color does. About 10-23 sec.

Electromagnetic process, nothing changes.

Weak process, flavors change. (Note strong contribution!) 10-8 sec.

Remember strong process lifetimes are typically 10^-23 sec, electromagnetic processes are typically 10^-17 sec, and weak processes are typically 10^-10 sec, so the decay process involved is instantly evident from the observed lifetime.

• Hypercharge,  Y = A + S + C + T + B = 2⟨(q/e)⟩.  Thus Y is a quantum number equal to twice the average electric charge of a particle multiplet.  It makes more sense to write hypercharge in terms of the types of quarks present in a baryon or meson.  That is, Y = (1/3)n_u + (1/3)n_d + (4/3)n_c − (2/3)n_s + (4/3)n_t − (2/3)n_b.  Hypercharge is conserved in strong processes, but not in weak processes, for obvious reasons.

For more details see text Chs. 7 and 8.


A hypothetical process conserving A - L, coupling to currently unknown gauge fields with X and Y bosons.  If this process existed, protons would be unstable and all matter would ultimately decay to photons.   [What about black holes?  They would slowly evaporate to photons via Hawking radiation.]  Modern research on the origins of matter focuses on possible processes that could generate baryons and leptons.  The problem is that the processes have to generate a slight excess of particles over antiparticles; no known process comes anywhere close to doing this.
 
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