1. Lagrange Formalism & Gauge Theories
Roger Wolf
28. April 2015
Prof. Dr. Max Mustermann | Name of Faculty

2. Schedule for today
How do I know that the gauge field should be a boson?
What is the defining characteristic of a Lie group? 3
Lie-groups & (Non-)Abelian transformations 2
Local gauge transformations 1
Lagrange formalism
Prof. Dr. Max Mustermann | Name of Faculty

3. Lagrange formalism & gauge transformations
Joseph-Louis Lagrange
(*25. January 1736, † 10. April 1813)
Prof. Dr. Max Mustermann | Name of Faculty

4. Lagrange formalism (classical field theories)
All information of a physical system is contained in the action integral:
Field:
(Generalization of canonical coordinates)
Action:
Lagrange Density:
( )
Equations of motion can be derived from the Euler-Lagrange formalism:
(From variation of action)
What is the dimension of ?
Prof. Dr. Max Mustermann | Name of Faculty

5. Lagrange formalism (classical field theories)
All information of a physical system is contained in the action integral:
Field:
(Generalization of canonical coordinates)
Action:
Lagrange Density:
( )
Equations of motion can be derived from the Euler-Lagrange formalism:
(From variation of action)
What is the dimension of ?
Prof. Dr. Max Mustermann | Name of Faculty

6. Lagrange density for (free) bosons & fermions
For Bosons:
For Fermions:
Proof by applying Euler-Lagrange formalism (shown only for Bosons here):
NB:
There is a distinction between and
Most trivial is variation by , least trivial is variation by .
Prof. Dr. Max Mustermann | Name of Faculty

7. Global phase transformations
The Lagrangian density is covariant under global phase transformations (shown here for the fermion case only):
(Global phase transformation)
Here the phase is fixed at each point in space at any time .
What happens if we allow different phases at each point in ?
Prof. Dr. Max Mustermann | Name of Faculty

8. Local phase transformations
This is not true for local phase transformations:
(Local phase transformation)
Connects neighboring points in
Breaks invariance due to in .
Prof. Dr. Max Mustermann | Name of Faculty

9. The covariant derivative
Covariance can be enforced by the introduction of the covariant derivative: with the corresponding transformation behavior
(Local phase transformation)
(Arbitrary gauge field)
NB: What is the transformation behavior of the gauge field ?
Prof. Dr. Max Mustermann | Name of Faculty

10. The covariant derivative
Covariance can be enforced by the introduction of the covariant derivative: with the corresponding transformation behavior
(Local phase transformation)
(Arbitrary gauge field)
NB: What is the transformation behavior of the gauge field ?
known from electro-dynamics!
Prof. Dr. Max Mustermann | Name of Faculty

11. The gauge field
Possible to allow arbitrary phase of at each individual point in
Requires introduction of a mediating field , which transports this information
from point to point.
The gauge field couples to a quantity of the external field , which can
be identified as the electric charge.
The gauge field can be identified with the photon field.
Prof. Dr. Max Mustermann | Name of Faculty

12. The interacting fermion
The introduction of the covariant derivative leads to the Lagrangian density of an interacting fermion with electric charge :
free fermion field
IA term
Description still misses dynamic term for a free gauge boson field (=photon).
Prof. Dr. Max Mustermann | Name of Faculty

13. Gauge field dynamics Ansatz: (Field-Strength tensor)
(Free photon field)
Motivation:
Variation of the action integral
is manifest Lorentz invariant.
in classical field theory, leads to
appears quadratically → linear appearance in variation that leads to equations of motion (→ superpo-sition of fields).
Can also be obtained from:
Check that is gauge invariant.
Prof. Dr. Max Mustermann | Name of Faculty

14. Complete Lagrangian density
Application of gauge symmetry leads to Largangian density of QED:
Free Fermion Field
IA Term
Gauge
(Interacting Fermion)
Variation of :
Derive equations of motion for an interacting boson.
Prof. Dr. Max Mustermann | Name of Faculty

15. Complete Lagrangian density
Application of gauge symmetry leads to Largangian density of QED:
Free Fermion Field
IA Term
Gauge
(Interacting Fermion)
Variation of :
(Lorentz Gauge)
(Klein-Gordon equation for a massless particle)
Prof. Dr. Max Mustermann | Name of Faculty

16. Summary (Abelian) gauge field theories
(Local gauge invariance)
(Covariant derivative)
(Field strength tensor)
(Lagrange density)
Prof. Dr. Max Mustermann | Name of Faculty

17. (*17. December 1842, † 18. February 1899)
Review of Lie-Groups
Marius Sophus Lie
(*17. December 1842, † 18. February 1899)
Prof. Dr. Max Mustermann | Name of Faculty

18. Unitary transformations
phase transformation.
is a group of unitary transformations in with the following properties:
Splitting an additional phase from one can reach that :
(Unitary transformations)
(Special unitary transformations)
Prof. Dr. Max Mustermann | Name of Faculty

19. Infinitesimal → finite transformations
The can be composed from infinitesimal transformations with a continuous parameter :
( , )
generators of .
define structure of .
The set of forms a Lie-Group.
The set of forms the tangential-space or Lie-Algebra.
Prof. Dr. Max Mustermann | Name of Faculty

20. Properties of Hermitian: Traceless (example ): !
Dimension of tangential space:
real entries in diagonal.
has generators.
complex entries in off-diagonal.
has generators.
for for det req.
Prof. Dr. Max Mustermann | Name of Faculty

21. Examples that appear in the SM ( )
transformations (equivalent to ):
Number of generators:
NB: what is the Generator?
Prof. Dr. Max Mustermann | Name of Faculty

22. Examples that appear in the SM ( )
transformations (equivalent to ):
Number of generators:
NB: what is the Generator? The generator is 1.
Prof. Dr. Max Mustermann | Name of Faculty

23. Examples that appear in the SM ( )
transformations (equivalent to ):
Number of generators:
i.e. there are 3 matrices , which form a basis of traceless hermitian matrices, for which the following relation holds:
Explicit representation:
(3 Pauli matrices)
algebra closes.
structure constants of
Prof. Dr. Max Mustermann | Name of Faculty

24. Examples that appear in the SM ( )
transformations (equivalent to ):
Number of generators:
i.e. there are 8 matrices , which form a basis of traceless hermitian matrices, for which the following relation holds:
Explicit representation:
(8 Gell-Mann matrices)
algebra closes.
structure constants of
Prof. Dr. Max Mustermann | Name of Faculty

25. (Non-)Abelian symmetry transformations
Example (90º rotations in ): z x y z x y x z y
switch z and y: 1 2 3 4
Prof. Dr. Max Mustermann | Name of Faculty

26. (Non-)Abelian symmetry transformations
Example (90º rotations in ): z x y z x y 2 x z y
switch z and y: 1 2 3 4
Prof. Dr. Max Mustermann | Name of Faculty

27. (Non-)Abelian symmetry transformations
Example (90º rotations in ): z x y z x y 2 x z y
switch z and y: 1 2
cyclic permutation: 3 z x y 4
Prof. Dr. Max Mustermann | Name of Faculty

28. (Non-)Abelian symmetry transformations
Example (90º rotations in ): z x y z x y 2 x z y
switch z and y: 1 3 2
cyclic permutation: 3 z x y 4
Prof. Dr. Max Mustermann | Name of Faculty

29. Abelian vs. Non-Abelian gauge theories
Prof. Dr. Max Mustermann | Name of Faculty

30. Concluding remarks Reprise of Lagrange formalism.
Requirement of local gauge symmetry leads to coupling structure of QED.
Extension to more complex symmetry operations will reveal non-trivial and unique coupling structure of the SM and thus describe all known fundamental interactions.
Next lecture on layout of the electroweak sector of the SM, from the non-trivial phenomenology to the theory.
Prepare “The Higgs Boson Discovery at the Large Hadron Collider” Section 2.2.
Prof. Dr. Max Mustermann | Name of Faculty

31. Backup
Prof. Dr. Max Mustermann | Name of Faculty