Symmetries and Conservation in Quantum Mechanics

1. PHYS 3446 – Lecture #22 Monday, Apr 23, 2012 Dr. Andrew Brandt • Local Symmetry • Gauge Fields/Invariance • Standard Model PHYS 3446, Andrew Brandt

2. Symmetry • When is a quantum number conserved? • When there is an underlying symmetry in the system • When the quantum number is not affected by 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, Andrew Brandt

3. Symmetries in Lagrangian Formalism? • Consider an isolated non-relativistic physical system of two particles interacting through a potential that only depends on the relative distance between them • EM and gravitational force • The total kinetic and potential energies of the system are: and • The equations of motion are then PHYS 3446, Andrew Brandt

4. Symmetries in Lagrangian Formalism • If we perform a linear translation of the origin of coordinate system by a constant vector • The position vectors of the two particles become • But the equations of motion do not change since is a constant vector • This is due to the invariance of the potential V under the translation PHYS 3446, Andrew Brandt

5. Symmetries in Lagrangian Formalism • A symmetry of a system is defined by any set of transformations that keep the equation of motion unchanged or invariant • Equations of motion can be obtained through • Lagrangian formalism: L=T-V where the Equation of motion is what minimizes the Lagrangian L under changes of coordinates • Hamiltonian formalism: H=T+V with an equation of motion that minimizes the Hamiltonian under changes of coordinates • Either of these notations can be used to discuss symmetries in classical cases or relativistic cases and quantum mechanical systems PHYS 3446, Andrew Brandt

6. Symmetries in Lagrangian Formalism? • The translation of the coordinate system for an isolated two particle system defines a symmetry of the system (recall Noether’s theorem) • This particular physical system is invariant under spatial translation • What is the consequence of this invariance? • From the form of the potential, the total force is • Since PHYS 3446, Andrew Brandt

7. Symmetries in Lagrangian Formalism? • What does this mean? • Total momentum of the system is invariant under spatial translation • In other words, translational symmetry results in linear momentum conservation • This holds for multi-particle system as well PHYS 3446, Andrew Brandt

8. Symmetries in Lagrangian Formalism • For multi-particle system, using Lagrangian L=T-V, the equations of motion can be generalized • By construction, • As previously discussed, for the system with a potential that depends on the relative distance between particles, The Lagrangian is independent of particulars of the individual coordinate and thus PHYS 3446, Andrew Brandt

9. Symmetries & Conserved Quantities • Symmetry under linear translation • Linear momentum conservation • Symmetry under spatial rotation • Angular momentum conservation • Symmetry under time translation • Energy conservation • Symmetry under isospin space rotation • Isospin conservation PHYS 3446, Andrew Brandt

10. Symmetries in Quantum Mechanics • In quantum mechanics, an observable physical quantity corresponds to the expectation value of the corresponding operator on a given quantum state • The expectation value is given as a product of wave function vectors about the physical quantity (operator) • The wave function ( )is the probability distribution function of a quantum state at any given space-time coordinates • The observable is invariant or conserved if the operator Q commutes with Hamiltonian PHYS 3446, Andrew Brandt

11. Types of Symmetry • All symmetry transformations of the theory can be divided in two categories • Continuous symmetry: Symmetry under continuous transformation • Spatial translation • Time translation • Rotation • Discrete symmetry: Symmetry under discrete transformation • Transformation in discrete quantum mechanical system PHYS 3446, Andrew Brandt

12. Local Symmetries • Continuous symmetries can be classified as either • Global symmetry: Parameters of transformation are constant • Transformation is the same throughout the entire space-time points • All continuous transformations we have 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, Andrew Brandt

13. Local Symmetries • Let’s consider the 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, Andrew Brandt

14. 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, Andrew Brandt

15. Local Symmetries • Thus, Schrödinger equation • is not invariant (or a symmetry) under local phase transformation • What does this mean? • Energy conservation is no longer valid. • What can we do to re-establish conservation of energy? • Consider an arbitrary modification of a gradient operator PHYS 3446, Andrew Brandt

16. 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, Andrew Brandt

17. 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, Andrew Brandt

18. 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, Andrew Brandt

19. U(1) Local Gauge Invariance Requiring the complete Lagrangian to be invariant under l(x) local gauge transformation will require additional terms in the 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 PHYS 3446, Andrew Brandt

20. 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 • 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, Andrew Brandt

21. 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, Andrew Brandt