Ivancevic_Applied-Diff-Geom

34 Applied Differential Geometry: A Modern IntroductionFigure 1.2).Recall that the Standard Model is a theory which describes the strong,weak, and electromagnetic fundamental forces, as well as the fundamentalparticles that make up all matter. Developed between 1970 and 1973, itis a quantum field theory, and consistent with both quantum mechanicsand special relativity. The Standard Model contains both fermionic andbosonic fundamental particles. Fermions are particles which possess half–integer spin, obey the Fermi–Dirac statistics and also the Pauli exclusionprinciple, which states that no fermions can share the same quantum state.On the other hand, bosons possess integer spin, obey the Bose–Einsteinstatistics, and do not obey the Pauli exclusion principle. In the StandardModel, the theory of the electro–weak interaction (which describes the weakand electromagnetic interactions) is combined with the theory of quantumchromodynamics. All of these theories are gauge theories, 48 meaning thatthey model the forces between fermions by coupling them to bosons which48 Recall that the familiar Maxwell gauge field theory(or, in the non–Abelian case,Yang–Mills gauge field theory) is defined in terms of the fundamental gauge field (whichgeometrically represents a connection) A µ = (A 0 , ⃗ A), that is µ = 0, 3. Here A 0 is thescalar potential and ⃗ A is the vector potential. The Maxwell LagrangianL M = − 1 4 FµνF µν − A µJ µ (1.1)is expressed in terms of the field strength tensor (curvature) F µν = ∂ µA ν − ∂ νA µ, and amatter current J µ that is conserved: ∂ µJ µ = 0. This Maxwell Lagrangian is manifestlyinvariant under the gauge transformation A µ → A µ + ∂ µΛ; and, correspondingly, theclassical Euler-Lagrange equations of motion∂ µF µν = J ν (1.2)are gauge invariant. Observe that current conservation ∂ νJ ν = 0 follows from theantisymmetry of F µν.Note that this Maxwell theory could easily be defined in any space–time dimension dsimply by taking the range of the space–time index µ on the gauge field A µ to be µ =0, 1, 2, . . . , (d − 1) in dD space–time. The field strength tensor is still the antisymmetrictensor F µν = ∂ µA ν − ∂ νA µ, and the Maxwell Lagrangian (1.1) and the field equationsof motion (1.2) do not change their form. The only real difference is that the numberof independent fields contained in the field strength tensor F µν is different in differentdimensions. (Since F µν can be regarded as a d × d antisymmetric matrix, the numberof fields is equal to 1 d(d − 1).) So at this level, planar (2 + 1)D Maxwell theory is quite2similar to the familiar (3 + 1)D Maxwell theory. The main difference is simply that themagnetic field is a (pseudo–) scalar B = ɛ ij ∂ i A j in (2 + 1)D, rather than a (pseudo–)vector B ⃗ = ∇ ⃗ × A ⃗ in (3 + 1)D. This is just because in (2 + 1)D the vector potential⃗A is a 2D vector, and the curl in 2D produces a scalar. On the other hand, the electricfield E ⃗ = −∇A ⃗ 0 − ˙⃗ A is a 2D vector. So the antisymmetric 3 × 3 field–strength tensorhas three nonzero field components: two for the electric field E ⃗ and one for the magneticfield B.