Richard Kass When something doesn’t happen there is usually a reason! Conservation Laws Read: M&S Chapters 2, 4, and 5.1, That something is a conservation law ! A conserved quantity is related to a symmetry in the Lagrangian that describes the interaction. (“Noether’s Theorem”) A symmetry is associated with a transformation that leaves the Lagrangian invariant. time invariance leads to energy conservation translation invariance leads to linear momentum conservation rotational invariance leads to angular momentum conservation Familiar Conserved Quantities Quantity Strong EM Weak Comments energy Y Y Y sacred linear momentum Y Y Y sacred ang. momentum Y Y Y sacred electric charge Y Y Y sacred
Richard Kass Other Conserved Quantities Quantity Strong EM Weak Comments Baryon number Y Y Y no p®p+p 0 Lepton number(s) Y Y Y no m-®e-g Nobel 88, 94, 02 topY YNdiscovered 1995 strangenessY YNdiscovered 1947 charmY YNdiscovered 1974, Nobel 1976 bottomY YNdiscovered 1977 Isospin Y NN proton = neutron (mu»md) Charge conjugation (C) Y Y N particle Û anti-particle Parity (P) Y Y N Nobel prize 1957 CP or Time (T) Y Y y/n small No, Nobel prize 1980 CPT Y Y Y sacred G Parity Y NN works for pions only Neutrino oscillations give first evidence of lepton # violation! These experiments were designed to look for baryon # violation!! Classic example of strangeness violation in a decay:L®pp- (S=-1® S=0) Very subtle example of CP violation: expect: Kolong® p+p0p- BUT Kolong® p+p-(»1 part in 103)
Richard Kass Problem 2.1 (M&S page 43): a) Consider the reaction vu+p®m++n What force is involved here? Since neutrinos are involved, it must be WEAK interaction. Is this reaction allowed or forbidden? Consider quantities conserved by weak interaction: lepton #, baryon #, q, E, p, L, etc. muon lepton number of vu=1, m+=-1 (particle Vs. anti-particle) Reaction not allowed! b) Consider the reaction ve+p®e-+p++p Must be weak interaction since neutrino is involved. conserves all weak interaction quantities Reaction is allowed c) Consider the reaction L®e-+p++(anti-ve) Must be weak interaction since neutrino is involved. conserves electron lepton #, but not baryon # (1 ® 0) Reaction is not allowed d) Consider the reaction K+®m-+p0+(anti-vm) Must be weak interaction since neutrino is involved. conserves all weak interaction (e.g. muon lepton #) quantities Reaction is allowed Some Reaction Examples
Richard Kass • Let’s consider the following reactions to see if they are allowed: • a) K+p ®Lp+p+ b) K-p ®Ln c) K0®p+p- • First we should figure out which forces are involved in the reaction. • All three reactions involve only strongly interacting particles (no leptons) • so it is natural to consider the strong interaction first. • Not possible via strong interaction since strangeness is violated (1®-1) • Ok via strong interaction (e.g. strangeness –1®-1) • Not possible via strong interaction since strangeness is violated (1® 0) • If a reaction is possible through the strong force then it will happen that way! • Next, consider if reactions a) and c) could occur through the electromagnetic interaction. • Since there are no photons involved in this reaction (initial or final state) we can neglect EM. • Also, EM conserves strangeness. • Next, consider if reactions a) and c) could occur through the weak interaction. • Here we must distinguish between interactions (collisions) as in a) and decays as in c). • The probability of an interaction (e.g. a) involving only baryons and mesons occurring through • the weak interactions is so small that we neglect it. • Reaction c) is a decay. Many particles decay via the weak interaction through strangeness • changing decays, so this can (and does) occur via the weak interaction. • To summarize: • Not possible via weak interaction • OK via weak interaction • Don’t even bother to consider Gravity! More Reaction Examples
Richard Kass Conserved Quantities and Symmetries Every conservation law corresponds to an invariance of the Hamiltonian (or Lagrangian) of the system under some transformation. We call these invariances symmetries. There are 2 types of transformations: continuous and discontinuous Continuous® give additive conservation laws x ® x+dx or q ®q +d q examples of conserved quantities: electric charge momentum baryon # Discontinuous ® give multiplicative conservation laws parity transformation: x, y, z ® (-x), (-y), (-z) charge conjugation (particle«antiparticle): e-® e+ examples of conserved quantities: parity (in strong and EM) charge conjugation (in strong and EM) parity and charge conjugation (strong, EM, almost always in weak)
Richard Kass Example of classical mechanics and momentum conservation. In general a system can be described by the following Hamiltonian: H=H(pi,qi,t) with pi=momentum coordinate, qi=space coordinate, t=time Consider the variation of H due to a translation qi only. Conserved Quantities and Symmetries For our example dpi=dt=0 so we have: Using Hamilton’s canonical equations: We can rewrite dH as: If H is invariant under a translation (dq) then by definition we must have: This can only be true if: Thus each p component is constant in time and momentum is conserved.
Richard Kass Conserved Quantities and Quantum Mechanics In quantum mechanics quantities whose operators commute with the Hamiltonian are conserved. Recall: the expectation value of an operator Q is: How does <Q> change with time? Recall Schrodinger’s equation: H+= H*T= hermitian conjugate of H Substituting the Schrodinger equation into the time derivative of Q gives: Since H is hermitian (H+= H) we can rewrite the above as: So if ¶Q/¶t=0 and [Q,H]=0 then <Q> is conserved.
Richard Kass Conservation of electric charge: SQi=SQf Conservation of electric charge and gauge invariance Evidence for conservation of electric charge: Consider reaction e-®gve which violates charge conservation but not lepton number or any other quantum number. If the above transition occurs in nature then we should see x-rays from atomic transitions. The absence of such x-rays leads to the limit: te > 2x1022 years There is a connection between charge conservation, gauge invariance, and quantum field theory. Recall Maxwell’s Equations are invariant under a gauge transformation: A Lagrangian that is invariant under a transformation U=eiq is said to be gauge invariant. There are two types of gauge transformations: local: q=q(x,t) global: q=constant, independent of (x,t) Maxwell’s EQs are locally gauge invariant Conservation of electric charge is the result of global gauge invariance Photon is massless due to local gauge invariance
Richard Kass Gauge invariance, Group Theory, and Stuff Consider a transformation (U) that acts on a wavefunction (y): y¢=Uy Let U be a continuous transformation then U is of the form: U=eiq q is an operator. If q is a hermitian operator (q=q*T) then U is a unitary transformation: U=eiq U+=(eiq)*T= e-iq*T = e-iq Þ UU+= eiq e-iq =1 Note: U is not a hermitian operator since U¹U+ In the language of group theory q is said to be the generator of U There are 4 properties that define a group: 1) closure: if A and B are members of the group then so is AB 2) identity: for all members of the set I exists such that IA=A 3) Inverse: the set must contain an inverse for every element in the set AA-1=I 4) Associativity: if A,B,C are members of the group then A(BC)=(AB)C If q = (q1, q2, q3,..) then the transformation is “Abelian” if: U(q1)U(q2) = U(q2)U(q1) i.e. the operators commute If the operators do not commute then the group is non-Abelian. The transformation with only one q forms the unitary abelian group U(1) The Pauli (spin) matrices generate the non-Abelian group SU(2) S= “special”= unit determinant U=unitary n=dimension (e.g.2)
Richard Kass Global Gauge Invariance and Charge Conservation The relativistic Lagrangian for a free electron is: This Lagrangian gives the Dirac equation: y is the electron field (a 4 component spinor) m is the electrons mass gu= “gamma” matrices, four (u=0,1,2,3) 4x4 matrices that satisfy gugv+ gvgu =2guv ¶u= (¶0, ¶1, ¶2, ¶3)= (¶/¶t, ¶/¶x, ¶/¶y, ¶/¶z) Let’s apply a global gauge transformation to L By Noether’s Theorem there must be a conserved quantity associated with this symmetry!
Richard Kass The Dirac equation: The Dirac Equation on One Page A solution (one of 4, two with +E, two with -E) to the Dirac equation is: The function U is a (two-component) SPINOR and satisfies the following equation: Spinors are most commonly used in physics to describe spin 1/2 objects. For example: Spinors also have the property that they change sign under a 3600 rotation!
Richard Kass Global Gauge Invariance and Charge Conservation We need to find the quantity that is conserved by our symmetry. In general if a Lagrangian density, L=L(f, ¶f/¶xu) with f a field, is invariant under a transformation we have: Result from field theory For our global gauge transformation we have: Plugging this result into the equation above we get (after some algebra…) E-L equation in 1D The first term is zero by the Euler-Lagrange equation. The second term gives us a continuity equation.
Richard Kass Global Gauge Invariance and Charge Conservation The continuity equation is: Result from quantum field theory Recall that in classical E&M the (charge/current) continuity equation has the form: (J0, J1, J2, J3) =(r, Jx, Jy, Jz) Also, recall that the Schrodinger equation give a conserved (probability) current: If we use the Dirac Lagrangian in the above equation for L we find: Conserved quantity This is just the relativistic electromagnetic current density for an electron. The electric charge is just the zeroth component of the 4-vector: Therefore, if there are no current sources or sinks (Ñ·J=0) charge is conserved as:
Richard Kass Local Gauge Invariance and Physics Some consequences of local gauge invariance: a) For QED local gauge invariance implies that the photon is massless. b) In theories with local gauge invariance a conserved quantum number implies a long range field. e.g. electric and magnetic field However, there are other quantum numbers that are similar to electric charge (e.g. lepton number, baryon number) that don’t seem to have a long range force associated with them! Perhaps these are not exact symmetries! ® evidence for neutrino oscillation implies lepton number violation. c) Theories with local gauge invariance can be renormalizable, i.e. can use perturbation theory to calculate decay rates, cross sections, etc. Strong, Weak and EM theories are described by local gauge theories. U(1) local gauge invariance first discussed by Weyl in 1919 SU(2) local gauge invariance discussed by Yang&Mills in 1954 (electro-weak) y¢=e it·l(x,t)yt is represented by the 2x2 Pauli matrices (non-Abelian) SU(3) local gauge invariance used to describe strong interaction (QCD) in 1970’s y¢=e it·l(x,t)yt is represented by the 3x3 matrices of SU(3) (non-Abelian)
Richard Kass Local Gauge Invariance and QED Consider the case of local gauge invariance, l=l(x,t) with transformation: The relativistic Lagrangian for a free electron is NOT invariant under this transformation. The derivative in the Lagrangian introduces an extra term: We can MAKE a Lagrangian that is locally gauge invariant by adding an extra piece to the free electron Lagrangian that will cancel the derivative term. We need to add a vector field Au which transforms under a gauge transformation as: Au®Au+¶uL(x,t) with l(x,t)=-qL(x,t) (for electron q=-|e|) The new, locally gauge invariant Lagrangian is:
Richard Kass The Locally Gauge Invariance QED Lagrangian Several important things to note about the above Lagrangian: 1) Au is the field associated with the photon. 2) The mass of the photon must be zero or else there would be a term in the Lagrangian of the form: mgAuAu However, AuAu is not gauge invariant! 3) Fuv=¶uAv-¶vAu and represents the kinetic energy term of the photon. 4) The photon and electron interact via the last term in the Lagrangian. This is sometimes called a current interaction since: Au g e- Ju e- In order to do QED calculations we apply perturbation theory (via Feynman diagrams) to JuAu term. 5) The symmetry group involved here is unitary and has one parameter Þ U(1)