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PHYS 3446 – Lecture #19

PHYS 3446 – Lecture #19. Wednesday, Nov. 15, 2006 Dr. Jae Yu. Symmetries Local gauge symmetry Gauge fields. Announcements. 2 nd 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

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PHYS 3446 – Lecture #19

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  1. PHYS 3446 – Lecture #19 Wednesday, Nov. 15, 2006 Dr. JaeYu • Symmetries • Local gauge symmetry • Gauge fields PHYS 3446, Fall 2006 Jae Yu

  2. Announcements • 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 minutes PHYS 3446, Fall 2006 Jae Yu

  3. Homework Assignments • 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 are • Prove that if is a solution for the Schrödinger equation , then is also a solution for it. • Due for this is Monday, Nov. 27 PHYS 3446, Fall 2006 Jae Yu

  4. Why symmetry? • 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 theories PHYS 3446, Fall 2006 Jae Yu

  5. Local Symmetries • 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!! PHYS 3446, Fall 2006 Jae Yu

  6. Local Symmetries • Let’s consider time-independent Schrödinger Eq. • If is a solution, should also be a solution for a constant a • 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. PHYS 3446, Fall 2006 Jae Yu

  7. Local Symmetries • 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 obtain PHYS 3446, Fall 2006 Jae Yu

  8. Local Symmetries • 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 operator PHYS 3446, Fall 2006 Jae Yu

  9. Additional Field Local Symmetries • Now requiring the vector potential to change under transformation as • Makes • And the local symmetry of the modified Schrödinger equation is preserved under the transformation PHYS 3446, Fall 2006 Jae Yu

  10. Local Symmetries • 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 group PHYS 3446, Fall 2006 Jae Yu

  11. U(1) Local Gauge Invariance Dirac Lagrangian for free particle of spin ½ and mass m; is invariant under a global phase transformation (global gauge transformation) since . However, if the phase, q, varies as a function of space-time coordinate, xm, is L still invariant under the local gauge transformation, ? No, because it adds an extra term from derivative of q. PHYS 3446, Fall 2006 Jae Yu

  12. U(1) Local Gauge Invariance Requiring the complete Lagrangian be invariant under l(x) local gauge transformation will require additional terms to free Dirac Lagrangian to cancel the extra term Where Am is a new vector gauge field that transforms under local gauge transformation as follows: Addition of this vector field to L keeps L invariant under local gauge transformation, but… PHYS 3446, Fall 2006 Jae Yu

  13. U(1) Local Gauge Invariance The new vector field couples with spinor through the last term. In addition, the full Lagrangian must include a “free” term for the gauge field. Thus, Proca Largangian needs to be added. This Lagrangian is not invariant under the local gauge transformation, , because PHYS 3446, Fall 2006 Jae Yu

  14. Free L for gauge field. Vector field for gauge invariance U(1) Local Gauge Invariance The requirement of local gauge invariance forces the introduction of a massless vector field into the free Dirac Lagrangian. is an electromagnetic potential. And is a gauge transformation of an electromagnetic potential. PHYS 3446, Fall 2006 Jae Yu

  15. Gauge Fields and Local Symmetries • To maintain a local symmetry, additional fields must be introduced • This is in general true even for more complicated symmetries • A crucial information for modern physics theories • A distinct fundamental forces in nature arises from local invariance of physical theories • The associated gauge fields generate these forces • These gauge fields are the mediators of the given force • This is referred as gauge principle, and such theories are gauge theories • Fundamental interactions are understood through this theoretical framework PHYS 3446, Fall 2006 Jae Yu

  16. Gauge Fields and Mediators • To keep local gauge invariance, new particles had to be introduced in gauge theories • U(1) gauge introduced a new field (particle) that mediates the electromagnetic force: Photon • SU(2) gauge introduces three new fields that mediates weak force • Charged current mediator: W+ and W- • Neutral current: Z0 • SU(3) gauge introduces 8 mediators (gluons) for the strong force • Unification of electromagnetic and weak force SU(2)xU(1) gauge introduces a total of four mediators • Neutral current: Photon, Z0 • Charged current: W+ and W- PHYS 3446, Fall 2006 Jae Yu

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