A journey inside planar pure QED

1. November 19th 2004 A journey inside planar pure QED CP3 lunch meeting By Bruno Bertrand

2. INTRODUCTION A journey inside planar pure QED CP3 lunch meeting

3. Why do we work in 2+1 dimensions ? • Theoretical ‘‘test’’ laboratory => Understanding & methods of quant. field theories + Simpler than 3+1 d. models, sometimes with exact solutions Possible generic results, interests of dim. reduction, etc. + Less trivial than 1+1 d. models (often trivial dynamics) • Specific properties of models with even number of spatial dim. => 1+1 d. models closer to 3+1 d. than 2+1 d. models -2+1 d. models are less ‘‘realistic’’ - Problem in the extension of 2+1 d. methods to 3+1 d. case + Great interest : Surprising phenomenon. e- and  beavior differing in many ways. A journey inside planar pure QED CP3 lunch meeting

4. Table of contents • 1st part : Maxwell theory in 2+1 & 3+1 d. => A case of common quantum field theory √ Lagrangian of pure QED √ Differences between 2+1 & 3+1 dim. cases √ Classical hamiltonian analysis • 2nd part : Maxwell Chern Simons theory => Quantum field theory specific to 2+1 d. case √ The Chern-Simons theory √ Interests and theoretical applications √ The Maxwell-Chern-Simons theory √ Hamiltonian analysis A journey inside planar pure QED CP3 lunch meeting

5. FIRST PARTMaxwell theory in 2+1 and 3+1 dim. A journey inside planar pure QED CP3 lunch meeting

6. Lagrangian of pure QED I • Pure gauge QED action and fields (without matter) √ Action & lagrangian in d+1 dim. : √ Minkowski metric  in d+1 dim. / Flat manifold Rd+1 √ Strenth field (Faraday) antisym. tensor (curvature) [L-2] √ Fundamental Gauge vector field A (connection) [L-1] Scalar potential Vector potential √ Gauge group coupling constant ‘‘e’’ [E-1/2 L-2+d/2] A journey inside planar pure QED CP3 lunch meeting

7. Lagrangian of pure QED II •  S = 0 => Euler-Lagrange equations of motion => Maxwell equation in the vacuum : • Lagrangian invariance under U(1) gauge transf. √ U(1) ! Abelian group of phase transf. : √ Action on the gauge field : • At this level planar Maxwell theory quite similar to the familiar 3+1 dim. Maxwell theory A journey inside planar pure QED CP3 lunch meeting

8. What are changing from now ? A journey inside planar pure QED CP3 lunch meeting

9. d+1-dim. class. Hamiltonian analysis • Phase space degrees of freedom (df) √ 2 df coming from the potential vector √ 2 df => conjugate momentum : the electric field √ A0 non-physical (Lagrange multiplier) • Symplectic structure on the phase space => Antisym. Poisson bracket : • Classical can. hamiltonian $ Class. energy density • Constraint : Gauss law A journey inside planar pure QED CP3 lunch meeting

10. SECOND PART Maxwell-Chern-Simons theory A journey inside planar pure QED CP3 lunch meeting

11. The Chern-Simons theory • Pure Chern-Simons lagrangian √ Topologically invariant (thus Lorentz invariant) lagrangian : √ Non invariant under parity & gauge inv. up to surface term => Boundary terms : • Completely  type of gauge theory specific to 2+1 d. √ 1st -order in spacetime deriv. √ Quadratic in A • Source-free eq. of motion √ ‘‘Flat connection’’ : A journey inside planar pure QED CP3 lunch meeting

12. The Chern-Simons theory Is it a boring, uninteresting and simply trivial theory ? A journey inside planar pure QED CP3 lunch meeting

13. NO ! 1) TQFT !!! • Structures in differentiable geometry 1. Topological space (plane, sphere, torus) 2. Manifold with differentiable structure and coordinate system 3. Metric  Notion of distance • Topological quantum field theory (TQFT) √ Phys. Observables topologically invariant √ Phys. states invariant under reparametrisation √ Sometimes : analytical (non perturbative) solutions exist. Canonical hamiltonian = 0  Phys. States of zero energy ! NB : In quantum field theory, a physical state or observable is gauge invariant A journey inside planar pure QED CP3 lunch meeting

14. NO !2) Numerous theoretical applic. String theory A journey inside planar pure QED CP3 lunch meeting

15. Maxwell-Chern-Simons theory I • Lagrangian (only 2+1 d.) √ Coupling Maxwell + Chern-Simons : viable gauge theory √ CS term breaks parity inv. of Maxwell theory • E-L equation of motion => 2+1 d. pseudo-vector dual field : => Proca-type equation of massive field with mass : A journey inside planar pure QED CP3 lunch meeting

16. Maxwell-Chern-Simons theory II • 3+1 d. examples of mass generation √ Proca mass term …BUT breaks gauge invariance √ Higgs mecanism • 2+1 d. mass generation • New surprising mass generation induced by the CS term ! √ Gauge invariant √ No introduction of other field √ Parity breaking A journey inside planar pure QED CP3 lunch meeting

17. MCS Hamiltonian analysis • Phase space degrees of freedom (df) √ Potential vector : ! Conjugate momentum : √ A0 is non-physical • Symplectic structure on the phase space √ Antisym. Poisson bracket : ! Non commutating electric field components • Classical can. hamiltonian • ! Constraint : Gauss law A journey inside planar pure QED CP3 lunch meeting

18. CONCLUSION A journey inside planar pure QED CP3 lunch meeting

19. November 19th 2004 A journey inside planar pure QED CP3 lunch meeting By Bruno Bertrand