PHYS 3446 Lecture 19 Wednesday Nov 15 2006

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

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 – But still don’t have requests for • • • Kerosene pump + Liquid Nitrogen for cooling Blue filter films Mid-tem grade discussion in the last 20 Wednesday, Nov. 15, PHYS 3446, Fall 2006 2 2006 minutes Jae Yu

Homework Assignments 1. 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 2. Prove that if is a solution for the Schrödinger equation , then is also a solution for it. Wednesday, Nov. 15, PHYS 3446, Fall 2006 3 Jae Yu 3. 2006 Due for this is Monday, Nov. 27

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 4 Wednesday, Nov. 15, PHYS 3446, Fall 2006 formulating theories Jae Yu

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. Fall symmetry in this situation? 5 Wednesday, Nov. 15, PHYS 3446, 2006 Yu – Real forces must be. Jaeintroduced!!

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 Wednesday, Nov. 15, PHYS 3446, Fall 2006 6 of electrical charge is associated w/ this kind of 2006 Jae Yu

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 Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006 Jae Yu 7

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 Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006 Jae Yu 8

Local Symmetries • Now requiring the vector potential to change under transformation as Addition al Field • Makes • And the local symmetry of the modified Schrödinger equation is preserved under the transformation Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006 Jae Yu 9

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) Wednesday, Nov. 15, PHYS 3446, Fall 2006 10 gauge group Electromagnetic force 2006 Jae Yu

U(1) Local Gauge Invariance Dirac Lagrangian for free particle of spin ½ and 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 derivativ Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006 Jae Yu 11

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… Wednesday, Nov. 15, PHYS 3446, Fall 2006 12 2006 Jae Yu

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 Wednesday, Nov. 15, 2006 PHYS 3446, Fall 2006 Jae Yu 13

U(1) Local Gauge Invariance The requirement of local gauge invariance forces the introduction of a massless vector field into the free Dirac Lagrangian. Free L for gauge field. is an electromagnetic Andpotential. is a gauge transformation of an electromagnetic Wednesday, Nov. 15, PHYS 3446, Fall 2006 potential. 2006 Jae Yu Vector field for gauge invariance 14

• 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 PHYS 3446, Fall 2006 15 – Fundamental interactions are understood through this Jae Yu Wednesday, Nov. 15, 2006

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: Z 0 – SU(3) gauge introduces 8 mediators (gluons) for the strong force • Unification of electromagnetic and weak force SU(2)x. U(1) gauge introduces a total of four mediators 0 2006 PHYS 3446, ZFall – Neutral current: Photon, Jae Yu Wednesday, Nov. 15, 2006 16