Mathematical Principles of Theoretical Physics

6.3. SOLAR NEUTRINO PROBLEM 347 To understand clearly this problem, we begin with a brief introduction to the Standard Solar Mod ...

348 CHAPTER 6. QUANTUM PHYSICS In thep−pchain there are five reactions to yield neutrinos: (6.3.1) p+p−→^2 H+e++νe, (6.3.2) p+p+ ...

6.3. SOLAR NEUTRINO PROBLEM 349 wheresnustands for solar neutrino unit: 1 snu= 10 −^36 reactions/(^37 Cl atom·s). In 1968, R. Da ...

350 CHAPTER 6. QUANTUM PHYSICS It is known that the following reaction (6.3.14) νμ+n−→μ−+p may occur if the energy ofνμsatisfies ...

6.3. SOLAR NEUTRINO PROBLEM 351 There are three discrete eigenvaluesλjof (6.3.17) with eigenstates: (6.3.18) Hˆνj=λjνj for 1≤j ...

352 CHAPTER 6. QUANTUM PHYSICS Inserting (6.3.23) into (6.3.20) we deduce that νμ(t) =cosθ ν 1 (t)+sinθ ν 2 (t) =sinθcosθ(−e−iλ^ ...

6.3. SOLAR NEUTRINO PROBLEM 353 Neutrino masses As masses are much less than kinetic energyc|p|, by the Einstein triangular rela ...

354 CHAPTER 6. QUANTUM PHYSICS It is very difficult to computeEe,Eμ,Eτby (6.3.31). However, sinceA∈SU( 3 )is norm- preserving: E ...

6.3. SOLAR NEUTRINO PROBLEM 355 whereHˆ=−ihc ̄(~α·∇)+mc^2 α 0 , andVis as in (6.3.35). The equations in (6.3.34) are also in the ...

356 CHAPTER 6. QUANTUM PHYSICS In addition, all experiments measuring neutrino velocity had found no violation to the speed of l ...

6.3. SOLAR NEUTRINO PROBLEM 357 whereνk( 1 ≤k≤ 3 )are the two-component Weyl spinors, and~σ= (σ 1 ,σ 2 ,σ 3 ) Based on the massl ...

358 CHAPTER 6. QUANTUM PHYSICS which are generated by the weak interaction attracting force, as demonstrated in the weak charge ...

6.4. ENERGY LEVELS OF SUBATOMIC PARTICLES 359 1) each subatomic particle consisting of two or three weaktons or quarks bound by ...

360 CHAPTER 6. QUANTUM PHYSICS 3) Quark constituents of baryons: Baron=qqq. 4) Quark constituents of mesons: Meson=qq. 5) Weakto ...

6.4. ENERGY LEVELS OF SUBATOMIC PARTICLES 361 Remark 6.17.We need to explain that although each quark has three weak charges, du ...

362 CHAPTER 6. QUANTUM PHYSICS 6.4.2 Spectral equations of bound states In the last subsection we see that the subatomic particl ...

6.4. ENERGY LEVELS OF SUBATOMIC PARTICLES 363 where~σ= (σ^1 ,σ^2 ,σ^3 )is the Pauli matrix operator,~Dis as in (6.4.13). We now ...

364 CHAPTER 6. QUANTUM PHYSICS with~D=∇+i ̄hcg~A, we derive that ~D×~D=ig hc ̄ [ ∇×~A+~A×∇ ] . Note that as an operator we have ...

6.4. ENERGY LEVELS OF SUBATOMIC PARTICLES 365 with (6.4.26) Eˆ=i ̄h ∂ ∂t , pˆ=i ̄h(~σ·∇). As consider the massless bound states ...

366 CHAPTER 6. QUANTUM PHYSICS whereΩ⊂Rnis a bounded domain, and{A,B}=AB+BAis the anti-commutator. Now, we derive the spectral e ...