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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” )

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Richard Kass When something doesn\'t happen there is normally a reason! Protection Laws Read: M&S Chapters 2, 4, and 5.1, That something is a preservation law ! A moderated amount is identified with a symmetry in the Lagrangian that portrays the association. ( "Noether\'s Theorem" ) A symmetry is connected with a change that leaves the Lagrangian invariant. time invariance prompts vitality protection translation invariance prompts straight energy preservation rotational invariance prompts rakish force preservation 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 top Y Y N found 1995 unusual quality Y Y N found 1947 appeal Y Y N found 1974, Nobel 1976 base Y Y N found 1977 Isospin Y N proton = neutron (m u » m d ) Charge conjugation (C) Y Y N particle Û against molecule 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 N works for pions just Neutrino motions give first proof of lepton # infringement! These analyses were intended to search for baryon # infringement!! Exemplary case of weirdness infringement in a rot : L® p - (S=-1 ® S=0) Very unobtrusive case of CP infringement : expect: K o long ® p + p 0 p - BUT K o long ® p + p - ( » 1 section in 10 3 )

Richard Kass Problem 2.1 (M&S page 43): a) Consider the response v u +p ®m + +n What power is included here? Since neutrinos are included, it must be WEAK communication. Is it true that this is response permitted or illegal? Consider amounts saved by feeble association: lepton #, baryon #, q, E, p , L , and so forth muon lepton number of v u =1, m + =-1 (molecule Vs. hostile to molecule) Reaction not permitted! b) Consider the response v e +p ® e - + p + +p Must be feeble connection since neutrino is included. moderates all powerless collaboration amounts Reaction is permitted c) Consider the response L® e - + p + (hostile to v e ) Must be frail connection since neutrino is included. preserves electron lepton #, however not baryon # (1 ® 0) Reaction is not permitted d) Consider the response K + ®m - + p 0 + (hostile to v m ) Must be powerless collaboration since neutrino is included. moderates all feeble connection (e.g. muon lepton #) amounts Reaction is permitted Some Reaction Examples

Richard Kass Let\'s consider the accompanying responses to check whether they are permitted: a) K + p ®Lp + p + b) K - p ®L n c) K 0 ®p + p - First we ought to make sense of which powers are included in the response. Every one of the three responses include just unequivocally associating particles (no leptons) so it is characteristic to consider the solid connection first. Unrealistic by means of solid communication since weirdness is abused (1 ® - 1) Ok through solid connection (e.g. peculiarity –1 ® - 1) Not conceivable by means of solid association since bizarreness is abused (1 ® 0) If a response is conceivable through the solid drive then it will happen that way! Next, consider if responses an) and c) could happen through the electromagnetic association. Since there are no photons required in this response (beginning or last state) we can disregard EM. Additionally, EM rations weirdness. Next, consider if responses an) and c) could happen through the frail communication. Here we should recognize connections (crashes) as in an) and rots as in c). The likelihood of a communication (e.g. an) including just baryons and mesons happening through the frail connections is small to the point that we disregard it. Response c) is a rot. Numerous particles rot by means of the frail collaboration through unusual quality evolving rots, so this can (and occurs) by means of the powerless communication. To compress: Not conceivable through powerless communication OK by means of frail collaboration Don\'t significantly try to consider Gravity! More Reaction Examples

Richard Kass Conserved Quantities and Symmetries Every protection law compares to an invariance of the Hamiltonian (or Lagrangian) of the framework under some change. We call these invariances symmetries. There are 2 sorts of changes: ceaseless and irregular Continuous ® give added substance protection laws x ® x+dx or q ® q +d q cases of moderated amounts: electric charge momentum baryon # Discontinuous ® give multiplicative preservation laws parity change: x, y, z ® (- x), (- y), (- z) charge conjugation (molecule « antiparticle): e - ® e + cases of monitored amounts: parity (in solid and EM) charge conjugation (in solid and EM) parity and charge conjugation (solid, EM, quite often in frail)

Richard Kass Example of established mechanics and force protection. By and large a framework can be portrayed by the accompanying Hamiltonian : H=H(p i ,q i ,t) with p i =momentum coordinate, q i =space coordinate, t=time Consider the variety of H because of an interpretation q i as it were. Saved Quantities and Symmetries For our case dp i =dt=0 so we have: Using Hamilton\'s standard conditions: We can modify dH as: If H is invariant under an interpretation (dq) then by definition we should have: This must be valid if: Thus every p segment is consistent in time and energy is moderated.

Richard Kass Conserved Quantities and Quantum Mechanics In quantum mechanics amounts whose administrators drive with the Hamiltonian are monitored. Review: the desire estimation of an administrator Q is: How does <Q> change with time? Review Schrodinger\'s condition: H + = H *T = hermitian conjugate of H Substituting the Schrodinger condition into the time subsidiary of Q gives: Since H is hermitian ( H + = H ) we can revise the above as: So if ¶ Q/¶ t=0 and [Q,H]=0 then <Q> is monitored.

Richard Kass Conservation of electric charge: S Q i = S Q f Conservation of electric charge and gage invariance Evidence for preservation of electric charge: Consider response e - ®g v e which damages charge protection however not lepton number or whatever other quantum number. On the off chance that the above move happens in nature then we ought to see x-beams from nuclear moves. The nonattendance of such x-beams prompts the utmost: t e > 2x10 22 years There is an association between charge preservation, gage invariance, and quantum field hypothesis. Review Maxwell\'s Equations are invariant under a gage change: A Lagrangian that is invariant under a change U=e i q is said to be gage invariant. There are two sorts of gage changes: local: q = q (x,t) global: q =constant, free of (x,t) Maxwell\'s EQs are locally gage invariant Conservation of electric charge is the consequence of worldwide gage invariance Photon is massless because of neighborhood gage invariance

Richard Kass Gage invariance, Group Theory, and Stuff Consider a change (U) that follows up on a wavefunction ( y ): y¢= U y Let U be a ceaseless change then U is of the structure: U=e i q is an administrator. On the off chance that q is a hermitian administrator ( q = q *T ) then U is a unitary change: U=e i q U + =(e i q ) *T = e - i q *T = e - i q Þ UU + = e i q e - i q =1 Note: U is not a hermitian administrator since U ¹ U + In the dialect of gathering hypothesis q is said to be the generator of U There are 4 properties that characterize a gathering: 1) conclusion: if An and B are individuals from the gathering then so is AB 2) personality: for all individuals from the set I exists with the end goal that IA=A 3) Inverse: the set must contain a backwards for each component in the set AA - 1 =I 4) Associativity: if A,B,C are individuals from the gathering then A(BC)=(AB)C If q = ( q 1 , q 2 , q 3 ,..) then the change is "Abelian" if: U( q 1 )U( q 2 ) = U( q 2 )U( q 1 ) i.e. the administrators drive If the administrators don\'t drive then the gathering is non-Abelian. The change with one and only q shapes the unitary abelian bunch U(1) The Pauli (turn) networks create the non-Abelian bunch SU(2) S= "special"= unit determinant U=unitary n=dimension (e.g.2)

Richard Kass Global Gage Invariance and Charge Conservation The relativistic Lagrangian for a free electron is: This Lagrangian gives the Dirac condition: y is the electron field (a 4 part spinor) m is the electrons mass g u = "gamma" frameworks, four (u=0,1,2,3) 4x4 grids that fulfill g u g v + g v g u =2g uv ¶ u = ( ¶ 0 , ¶ 1 , ¶ 2 , ¶ 3 )= ( ¶/¶ t, ¶/¶ x, ¶/¶ y, ¶/¶ z) Let\'s apply a worldwide gage change to L By Noether\'s Theorem there must be a rationed amount connected with this symmetry!

Richard Kass The Dirac condition: The Dirac Equation on One Page An answer (one of 4, two with +E, two with - E) to the Dirac condition is : The capacity U is a (two-segment) SPINOR and fulfills the accompanying condition: Spinors are most ordinarily utilized as a part of material science to depict turn 1/2 objects. For instance: Spinors likewise have the property that they change sign under a 360 0 turn!

Richard Kass Global Gage Invariance and Charge Conservation We have to discover the amount that is preserved by our symmetry. All in all if a Lagrangian thickness, L=L( f , ¶f/¶ x u ) with f a field, is invariant under a change we have: Result from field hypothesis For our worldwide gage change we have: Plugging this outcome into the condition above we get (after some variable based math… ) E-L condition in 1D The primary term is zero by the Euler-Lagrange condition. The second term gives us a coherence condition.

Richard Kass Global Gage Inva