1. PHYS 3446 – Lecture #23 Symmetries Why do we care about the symmetry?
Monday, Nov. 28, 2016
Dr. Jaehoon Yu
Symmetries
Why do we care about the symmetry?
Symmetry in Lagrangian formalism
Symmetries in quantum mechanical system
Types of Symmetry
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

2. Announcements Term Exam #2 Quiz 3 results No class this Wednesday!
Monday, Dec. 5
Comprehensive: CH1.1 through what we cover Nov. 28
BYOF
Quiz 3 results
Class Average: 41.2/90
Equivalent to 45.8/100
Previous quizzes: 47.1/100 & 43.5/100
Class top score: 69/90
No class this Wednesday!
Happy and safe Thanksgiving!
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

3. 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!!
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

4. 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.
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

5. 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
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

6. 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The energy conservation is no longer valid.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
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

7. 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
Additional Field
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

8. 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
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

9. 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.
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

10. 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…
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

11. 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
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

12. 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.
Vector field for gauge invariance
is an electromagnetic potential.
And is a gauge transformation of an electromagnetic potential.
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

13. 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
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016

14. 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-
Monday, Nov. 28, 2016
PHYS 3446, Fall 2016