REVIEW. Quantum electrodynamics (QED) Quantum electrodynamics is a theory of photons interacting with the electrons and positrons of a Dirac field:
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1 Quantum electrodynamics (QED) based on S-58 Quantum electrodynamics is a theory of photons interacting with the electrons and positrons of a Dirac field: Noether current of the lagrangian for a free Dirac field we want the current to be conserved and so we need to enlarge the gauge transformation also to the Dirac field: global symmetry is promoted into local symmetry of the lagrangian and so the current is conserved no matter if equations of motion are satisfied 229
2 We can write the QED lagrangian as: Proof: covariant derivative (the covariant derivative of a field transforms as the field itself) and so the lagrangian is manifestly gauge invariant! 230
3 We can also define the transformation rule for D: then Now we can express the field strength in terms of D s: as required. 231
4 Then we simply see: no derivatives act on exponentials the field strength is gauge invariant as we already knew 232
5 Nonabelian symmetries Let s generalize the theory of two real scalar fields: to the case of N real scalar fields: based on S-24 the lagrangian is clearly invariant under the SO(N) transformation: orthogonal matrix with det = 1 lagrangian has also the symmetry,, that enlarges SO(N) to O(N) 233
6 infinitesimal SO(N) transformation: there are linearly independent real antisymmetric matrices, and we can write: or R = e iθa T a. The commutator of two generators is a lin. comb. of generators: we choose normalization: antisymmetric real R T ij = δ ij + θ ji R 1 ij Im(R 1 R) ij = Im k = δ ij θ ij R ki R kj = 0 (N^2 linear combinations of Im parts = 0) hermitian, antisymmetric, NxN generator matrices of SO(N) structure constants of the SO(N) group 234
7 e.g. SO(3): Levi-Civita symbol 235
8 consider now a theory of N complex scalar fields: the lagrangian is clearly invariant under the U(N) transformation: we can always write so that. actually, the lagrangian has larger symmetry, SO(2N): group of unitary NxN matrices SU(N) - group of special unitary NxN matrices U(N) = U(1) x SU(N) 236
9 infinitesimal SU(N) transformation: or Ũ = e iθa T a. there are e.g. SU(2) - 3 Pauli matrices hermitian traceless linearly independent traceless hermitian matrices: SU(3) - 8 Gell-Mann matrices the structure coefficients are, the same as for SO(3) 237
10 Nonabelian gauge theory based on S-69 Consider a theory of N scalar or spinor fields that is invariant under: for SU(N): a special unitary NxN matrix for SO(N): a special orthogonal NxN matrix In the case of U(1) we could promote the symmetry to local symmetry but we had to include a gauge field and promote ordinary derivative to covariant derivative: then the kinetic terms and mass terms:,, and, are gauge invariant. The transformation of covariant derivative in general implies that the gauge field transforms as: for U(1): 238
11 Now we can easily generalize this construction for SU(N) or SO(N): an infinitesimal SU(N) transformation: from from to to gauge coupling constant generator matrices (hermitian and traceless): structure constants (completely antisymmetric) the SU(N) gauge field is a traceless hermitian NxN matrix transforming as: 239
12 the covariant derivative is: or acting on a field: NxN identity matrix using covariant derivative we get a gauge invariant lagrangian We define the field strength (kinetic term for the gauge field) as: a new term it transforms as: not gauge invariant separately and so the gauge invariant kinetic term can be written as: 240
13 we can expand the gauge field in terms of the generator matrices: that can be inverted: similarly: thus we have: 241
14 the kinetic term can be also written as: Example, quantum chromodynamics - QCD: flavor index: up, down, strange, charm, top, bottom 1,..., 8 gluons (massles spin 1 particles) color index: 1,2, 3 in general, scalar and spinor fields can be in different representations of the group, ; gauge invariance requires that the gauge fields transform independently of the representation. 242
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