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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(*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

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

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

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

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

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

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

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

(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

(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

(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

(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

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

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

Backup Prof. Dr. Max Mustermann | Name of Faculty