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Invariance Principles

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Invariance Principles Invariance Principles • Invariance of states under operations implies conservation laws – Invariance of energy under space translation → momentum conservation – Invariance of energy under space rotation → angular momentum conservation • Continuous Space-Time Transformations – Translations – Rotations – Extension of Poincare group to include fermionic anticommuting spinors (SUSY) • Discrete Transformations – Space Time Inversion (Parity=P) – Particle-Antiparticle Interchange (Charge Conjugation=C) – Time Reversal (T) – Combinations of these: CP, CPT • Continuous Transformations of Internal Symmetries – Isospin – SU(3)flavor – SU(3)color – Weak Isospin – These are all gauge symmetries Symmetries and Conservation Laws SU(2) SU(3) 8 generators; 2 can be diagonalized at the same time: rd I3 … 3 component of isospin Y … hypercharge SU(3) Raising and lowering operators Y = +1/2√3 Y = +1/√3 SU(3) SU(3) Adding 2 quarks SU(3) Adding 3 quarks Quantum Numbers • Electric Charge • Baryon Number • Lepton Number • Strangeness • Spin • Isospin • Parity • Charge Conjugation Electric Charge Quantum Numbers are quantised properties of particles that are subject to constraints. They are often related to symmetries Electric Charge Q is conserved in all interactions Strong ✔ Interaction Weak ✔ Interaction Baryon Number Baryon number is the net number of baryons or the net number of quarks ÷ 3 Baryons have B = +1 Quarks have B = +⅓ Antibaryons have B = -1 or Antiquarks have B = -⅓ Everything else has B = 0 Everything else has B = 0 Baryons = qqq = ⅓ + ⅓ + ⅓ = 1 Mesons = qq = ⅓ + (-⅓) = 0 Baryon Number B is conserved in Strong, EM and Weak interactions Total (quarks – antiquarks) is constant Baryon Number Strong ✔ Interaction Weak ✔ Interaction Since the proton is the lightest baryon it cannot decay if B is conserved e.g: ✗ Lepton Number Leptons have L = +1 Antileptons have L = -1 Everything else has L = 0 Lepton Number L is conserved in Strong, EM and Weak interactions but is also separately conserved within lepton families: – + e and ѵe have Le = 1 e and ѵe have Le= -1 μ– and ѵ have L = 1 + μ μ μ and ѵμ have Lμ = -1 �– and ѵ have L = 1 + � � � and ѵ� have L� = -1 Le, L μ and L � are separately conserved Lepton Number Pair ✔ Production Pion Decay ✔ Muon Decay ✔ Lepton Number Radiative Forbidden ✗ Decay ✗ OK ✔ L is conserved but neither Le or Lμ separately The decay has not been observed and has a Branching Ratio < 10-9 Spin Spin is an intrinsic property of all particles: 0ħ, 1ħ, 2ħ, 3ħ, . Bosons ½ħ, 3/2ħ, 5/2ħ, . Fermions Spin is like angular momentum but a Quantum Mechanical effect. For spin S there are 2S+1 states of different Sz (like 2J+1 in Angular Momentum) For Spin S = ½, Sz can be +½ or -½ (2 states) For Spin S = 1, Sz can be +1, 0, -1 (3 states) For a process a + b → c + d the cross section is This can be used to determine the spin of unknown particles Helicity • All particles have spin and three momentum p • For fermions S=½, two possible arrangements p! p! Left Right Handed Handed f f Spin vector Spin vector opposite same as momentum momentum vector vector Photon Helicity • The photon has spin 1. Therefore jZ =-1, +1 • Massless photon cannot have 0 helicity p! p! p! jZ= −1 jZ=0 jZ= 1 γ γ γ Isospin Used mostly in Nuclear Physics from charge independence of nuclear force p ↔ p = n ↔ n = p ↔ n sometimes called Isobaric Spin/Isotopic Spin T (or t!) Isospin is represented by a 'spin' vector I with component I3 along some axis I = ½ : p has I3 = +½ (↑), n has I3 = -½ (↓) – 3 0 3 + 3 I = 1 : π has I = -1, π has I = 0, π has I = +1 u has I3 = +½ u has I3 = -½ d has I3 = -½ d has I3 = +½ I3 really only counts the number of u and d quarks - p = uud = ½ + ½ + (-½) = ½ π = ud = -½ + (-½) = -1 Isospin Symmetry Proton and neutron are different states of the nucleon, with the quantum number of Isospin (I = 1/2) Heisenberg Complete analogy with the quantum number of spin. Strong interactions conserve isospin (I) and do not depend on I3: do not distinguish p from n • Observed equivalence of n-p, n-n, p-p forces once EM effects are subtracted (the proton is heavier than the neutron thanks to the energy needed to “bring” electric charge on the proton) • The isospin symmetry is transferred from the p-n equality to the quark level u-d → their nearly mass equality is the source of the isospin symmetry • Therefore, the symmetry extend to all baryons and mesons that are associated by u-d quark exchange π + = ud 1 + - π 0 = (dd − uu) Mass of π and π (equal for C-symmetry): 140 MeV 2 Mass of π0: 135 MeV € π − = ud € € Isospin I3 can be related to charge Q and baryon number B: For a proton B = 1, I3 = +½ and hence Q = 1 Since the Strong Interaction doesn't distinguish p from n or u from d, I and I3 are conserved in Strong Interactions This is equivalent to saying that the number of (u – u) – (d – d) = constant In Weak Interactions where u ⇌ d, I and I3 are NOT conserved In EM Interactions u and d are not changed but because of the different charges, u and d can be distinguished. Hence I3 is conserved but I is NOT conserved Isospin EM and weak interactions do not conserve I Strangeness Associated production of 'strange' particles π– + p → K0 + Λ0 K0 and Λ0 'Strange' – decay weakly not strongly Strangeness Assume pair production of new quark s and antiquark s by Strong Interaction but once produced s and s can only decay weakly Strong Production Weak Decay Strange particles are produced in pairs (ΔS = 0) via strong interaction but they decay weakly (ΔS = ±1) Strangeness 0 0 0 Λ is uds Λ0 is uds The K has an antiparticle the K 0 0 although it is neutral, unlike the K is ds K is ds π0 which is its own antiparticle K+ is us K– is us Strangeness can be combined with Isospin if Gell Mann – Nishijima relation The s quark has strangeness S = -1 Strangeness is conserved in Strong and EM Interactions but NOT in Weak Interactions Likewise charm, bottom, top quantum numbers Strong and EM Interactions do not change quark flavours. Number of (u – u), (d – d), (s – s), (c – c), (b – b) , (t – t) constant Weak Interaction changes one quark type to another Isospin, Strangeness and Hypercharge There is a compact way to express the relation between electric charge, third component of the isospin and baryon number: Q B = I + e 3 2 N(q) − N(q) (its conservation implies the stability of Baryon number is B = 3 matter: no proton decay) € If we also include the other quantum number associated to the s quark (strangeness S): € Q B + S Y = I + = I + With Y = B + S defined as hypercharge | e | 3 2 3 2 € Strangeness Strangeness is conserved in strong and EM interactions, not in weak processes strong EM weak weak Associated production of a ss quark pair, weak decay of the s (s) quark Parity Parity is a Quantum Mechanical concept For a wavefunction ѱ(r) and Parity If an operator Ô acts on a operator P, the Parity Operator wavefunction ѱ such that ѱ reverses the coordinates r to –r is unchanged Pψ(r) =ψ(−r) ѱ is an Eigenfunction of Ô and λ is the Eigenvalue € Hence the eigenvalues of Parity are +1 (even) and -1 (odd) Global and discrete operations: e.g. translation r → r+δr is global, reflection through the origin of coordinates: x, y, z → -x, -y, -z is discrete. Parity of Particles • Intrinsic parity – Fermions – Consider electrons and positrons represented by a wave function Ψ. Pˆ x!,t P x!,t Ψ( ) = e± Ψ(− ) – Dirac equation is satisfied by a wave function representing both electrons and positrons ⇒ related and it can be shown P P 1 e+ e− = − – Strong and EM reactions always produce e+e- pairs. – Arbitrarily have to set one =1 and the other = -1. • P2 = 1 (P is unitary operator) and its eigenvalues (if any) are ±1 (the parity of the system) • Parity is a multiplicative quantum number: if ψ = φ + η then Pψ = Pφ x Pη Parity The Parity Operator reverses the coordinates r to –r Equivalent to a reflection in the x-y plane followed by a rotation about the z axis Reflection in x-y plane Rotation about z axis ψ = cos x ⇒ Pψ = cos(−x) = cos x = +ψ ⇒ ψ is even (P = +1) ψ = sin x ⇒ Pψ = sin(−x) = −sin(x) = −ψ ⇒ ψ is odd (P = −1) ψ = cos x + sin x ⇒ Pψ = cos x − sin x ≠ ±ψ ⇒ ψ has no defined P Parity The hydrogen atom with a potential V(r) = V(-r) must have a well defined parity Its wave functions are the product of radial and angular functions (spherical harmonics): m ψ(r,θ,φ) = χ(r) Yl (θ,φ) r → −r ⇒ θ → π −θ, φ → π + θ m l and P Yl = (−1) Electric dipole transitions with photon emission have Δl= ± 1. In order to conserve the parity of the global system (atom + photon) the latter must have NEGATIVE PARITY. € Parity IS FOUND to be conserved in EM and strong interactions, but not in weak € interactions • The “intrinsic” parity of the proton and of the neutron are assumed by convention +1 (baryons are conserved) • The “intrinsic” parity of the charged pion is -1: from an experiment on π- + d → n + n • While pions can be created singly, particles carrying “strange quarks” are created in pairs, whose parity is -1 Link between the total angular momentum of a particle: J = L + S and the parity: J P =0+ scalar particle J P =0- pseudoscalar particle J P =1- vector particle J P =1+ axial-vector particle Parity Parity is a multiplicative quantum number.
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