PHYS 3446 – Lecture #19Slide 2Homework AssignmentsSlide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10U(1) Local Gauge InvarianceU(1) Local Gauge InvarianceSlide 13Slide 14Slide 15Slide 16Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu1PHYS 3446 – Lecture #19Wednesday, Nov. 15, 2006Dr. Jae Yu1. Symmetries•Local gauge symmetry•Gauge fields2. Parity•Properties of Parity•Determination of Parity•Conservation and violation of parityWednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu2•2nd term exam–Next Wednesday, Nov. 22–Covers: Ch 4 – whatever we finish on Nov. 20•Workshop on Saturday, Dec. 2–I heard from Liquid supply system construction–But still don’t have requests for•Kerosene pump + Liquid Nitrogen for cooling•Blue filter films•Mid-tem grade discussion in the last 20 minutesAnnouncementsWednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu3Homework Assignments1. Construct the Lagrangian for an isolated, two particle system under the potential that depends only on the relative distance between the particles and show that the equations of motion from are2. Prove that if is a solution for the Schrödinger equation , then is also a solution for it.3. Due for this is Monday, Nov. 27( )ryr( ) ( ) ( ) ( )222H r V r r E rmy y y� �= - � + =� �� �rr h r r r( )ie rayr2 2m r =r&&1 1m r =r&&( )1 1 2V r r- � - =rr r( )1 21V r rr�- -�r rr( )2 1 2V r r- � - =rr r( )1 22V r rr�- -�r rr0i iL Lddt rr� �- =��rr&Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu4•When does a quantum number conserved?–When there is an underlying symmetry in the system–When the quantum number is not affected (or is conserved) under changes in the physical system•Noether’s theorem: If there is a conserved quantity associated with a physical system, there exists an underlying invariance or symmetry principle responsible for this conservation.•Symmetries provide critical restrictions in formulating theoriesWhy symmetry?Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu5•All continuous symmetries can be classified as–Global symmetry: Parameters of transformation are constant•Transformation is the same throughout the entire space-time points•All continuous transformations we discussed so far are global symmetries–Local symmetry: Parameters of transformation depend on space-time coordinates•The magnitude of transformation is different from point to point•How do we preserve a symmetry in this situation?–Real forces must be introduced!!Local SymmetriesWednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu6•Let’s consider time-independent Schrödinger Eq.•If is a solution, should also be a solution for a constant –Any quantum mechanical wave functions can be defined up to a constant phase–A transformation involving a constant phase is a symmetry of any quantum mechanical system–Conserves probability density Conservation of electrical charge is associated w/ this kind of global transformation.Local Symmetries( )H ry =r( )ryr( )ie rayr( ) ( )222V r rmy� �- � + =� �� �rh r r( )E ryrWednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu7•Let’s consider a local phase transformation–How can we make this transformation local?•Multiplying a phase parameter with an explicit dependence on the position vector•This does not mean that we are transforming positions but just that the phase is dependent on the position•Thus under local transformation, we obtainLocal Symmetries( )ry �r( )( )i re ray� �� =� �rrr( )i rear( )ryr( )( )( )( ) ( )i re i r r raa y y� �� +�� �rr rr r r( )( )i re ray� �rrrWednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu8•Thus, Schrödinger equation•is not invariant (or a symmetry) under local phase transformation–What does this mean?–The energy conservation is no longer valid.•What can we do to conserve the energy?–Consider an arbitrary modification of a gradient operatorLocal Symmetries( ) ( ) ( ) ( )222H r V r r E rmy y y� �= - � + =� �� �rr h r r rѮr( )iA r�-rrrWednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu9•Now requiring the vector potential to change under transformation as•Makes•And the local symmetry of the modified Schrödinger equation is preserved under the transformationLocal Symmetries( ) ( )( )( ) ( ) ( )22' ' '2H r iA r V r r E rmy y y� �= - �- + =� �� �rrr r r r rh( )A r �rr( )( )( )iA r ry�- �rrr r( ) ( )A r ra+�rrr r( ) ( )( )( )( )( )i riA r i r e raa y� ��- - � =� �rrr rr r r( )( )( )( )i re iA r ray�-rrrr r( )A rrrAdditional FieldWednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu10•The invariance under a local phase transformation requires the introduction of additional fields–These fields are called gauge fields–Leads to the introduction of a definite physical force•The potential can be interpreted as the EM vector potential•The symmetry group associated with the single parameter phase transformation in the previous slides is called Abelian or commuting symmetry and is called U(1) gauge group Electromagnetic force groupLocal Symmetries( )A rrrWednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu11U(1) Local Gauge Invariance Dirac Lagrangian for free particle of spin ½ and mass m;( )( )2i c mcmmy g y yy= � -hLis invariant under a global phase transformation (global gauge transformation) since .ieqy y�ieqy y-�However, if the phase, , varies as a function of space-time coordinate, x, is L still invariant under the local gauge transformation, ?( )i xeqy y�No, because it adds an extra term from derivative of .Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu12U(1) Local Gauge InvarianceRequiring the complete Lagrangian be invariant under (x) local gauge transformation will require additional terms to free Dirac Lagrangian to cancel the extra termWhere A is a new vector gauge field that transforms under local gauge transformation as follows:Am�( )( )2i c mcmmy g y yy� �= � -� �hLAddition of this vector field to L keeps L invariant under local gauge transformation, but… ( )q Ammy g y-Am ml+�Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006Jae Yu13This Lagrangian is not invariant under the local gauge transformation, , because U(1) Local Gauge InvarianceThe new
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