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

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A journey inside planar pure QED CP3 lunch meeting 2 INTRODUCTION

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 3 Why do we work in 2+1 dimensions ? Theoretical ‘‘test’’ laboratory Theoretical ‘‘test’’ laboratory => U nderstanding & methods of quant. field theories + Simpler than 3+1 d. models, sometimes with exact solutions Possible generic results, interests of dim. reduction, etc. + 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) + Less trivial than 1+1 d. models (often trivial dynamics) Specific properties of models with even number of spatial dim. 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’’ - 2+1 d. models are less ‘‘realistic’’ - Problem in the extension of 2+1 d. methods to 3+1 d. case - 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. + Great interest : Surprising phenomenon. e - and beavior differing in many ways.

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 4 Table of contents 1 st part : Maxwell theory in 2+1 & 3+1 d. 1 st part : Maxwell theory in 2+1 & 3+1 d. => A case of common quantum field theory √ Lagrangian of pure QED √ Lagrangian of pure QED √ Differences between 2+1 & 3+1 dim. cases √ Differences between 2+1 & 3+1 dim. cases √ Classical hamiltonian analysis √ Classical hamiltonian analysis 2 nd part : Maxwell Chern Simons theory 2 nd part : Maxwell Chern Simons theory => Quantum field theory specific to 2+1 d. case √ The Chern-Simons theory √ The Chern-Simons theory √ Interests and theoretical applications √ Interests and theoretical applications √ The Maxwell-Chern-Simons theory √ The Maxwell-Chern-Simons theory √ Hamiltonian analysis √ Hamiltonian analysis

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November 19th 2004A journey inside planar pure QED CP3 lunch meeting 5 FIRST PART Maxwell theory in 2+1 and 3+1 dim.

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 6 Lagrangian of pure QED I Pure gauge QED action and fields (without matter) Pure gauge QED action and fields (without matter) √ Action & lagrangian in d+1 dim. : √ Action & lagrangian in d+1 dim. : √ Minkowski metric in d+1 dim. / Flat manifold R d+1 √ Minkowski metric in d+1 dim. / Flat manifold R d+1 √ Strenth field (Faraday) antisym. tensor (curvature) [L -2 ] √ Strenth field (Faraday) antisym. tensor (curvature) [L -2 ] √ Fundamental Gauge vector field A (connection) [L -1 ] √ Fundamental Gauge vector field A (connection) [L -1 ] Scalar potentialVector potential √ Gauge group coupling constant ‘‘e’’ [E -1/2 L -2+d/2 ] √ Gauge group coupling constant ‘‘e’’ [E -1/2 L -2+d/2 ]

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 7 Lagrangian of pure QED II S = 0 => Euler-Lagrange equations of motion S = 0 => Euler-Lagrange equations of motion => Maxwell equation in the vacuum : Lagrangian invariance under U(1) gauge transf. Lagrangian invariance under U(1) gauge transf. √ U(1) ! Abelian group of phase transf. : √ U(1) ! Abelian group of phase transf. : √ Action on the gauge field : √ Action on the gauge field : At this level planar Maxwell theory quite similar to the familiar 3+1 dim. Maxwell theory At this level planar Maxwell theory quite similar to the familiar 3+1 dim. Maxwell theory

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 8 What are changing from now ? 2+1 dim. 3+1 dim. :. Electric field :. 2 dim. Vector [E] 3 dim. Vector [E/L] Magnetic field Pseudo-scalar : Pseudo-vector Spin Pseudo-scalarPseudo-vector Invariance of Max. lagrangian under Parity (x 1, x 2, t) ! (-x 1, x 2, t) (x 1, x 2, x 3, t) ! (-x 1, -x 2, -x 3, t)

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 9 d+1-dim. class. Hamiltonian analysis Phase space degrees of freedom (df) Phase space degrees of freedom (df) √ 2 df coming from the potential vector √ 2 df coming from the potential vector √ 2 df => conjugate momentum : the electric field √ 2 df => conjugate momentum : the electric field √ A 0 non-physical (Lagrange multiplier) √ A 0 non-physical (Lagrange multiplier) Symplectic structure on the phase space Symplectic structure on the phase space => Antisym. Poisson bracket : => Antisym. Poisson bracket : Classical can. hamiltonian $ Class. energy density Classical can. hamiltonian $ Class. energy density Constraint : Gauss law Constraint : Gauss law

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November 19th 2004A journey inside planar pure QED CP3 lunch meeting 10 SECOND PART Maxwell-Chern-Simons theory

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 11 The Chern-Simons theory Pure Chern-Simons lagrangian Pure Chern-Simons lagrangian √ Topologically invariant (thus Lorentz invariant) lagrangian : √ Topologically invariant (thus Lorentz invariant) lagrangian : √ Non invariant under parity & gauge inv. up to surface term √ Non invariant under parity & gauge inv. up to surface term => Boundary terms : Completely type of gauge theory specific to 2+1 d. Completely type of gauge theory specific to 2+1 d. √ 1 st -order in spacetime deriv. √ 1 st -order in spacetime deriv. √ Quadratic in A √ Quadratic in A Source-free eq. of motion Source-free eq. of motion √ ‘‘Flat connection’’ : √ ‘‘Flat connection’’ :

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 12 The Chern-Simons theory Is it a boring, uninteresting and simply trivial theory ?

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 13 NO ! 1) TQFT !!! Structures in differentiable geometry Structures in differentiable geometry 1. Topological space (plane, sphere, torus) 1. Topological space (plane, sphere, torus) 2. Manifold with differentiable structure and coordinate system 2. Manifold with differentiable structure and coordinate system 3. Metric Notion of distance 3. Metric Notion of distance Topological quantum field theory (TQFT) Topological quantum field theory (TQFT) √ Phys. Observables topologically invariant √ Phys. Observables topologically invariant √ Phys. states invariant under reparametrisation √ Phys. states invariant under reparametrisation √ Sometimes : analytical (non perturbative) solutions exist. √ 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

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 14 NO ! 2) Numerous theoretical applic. Mathematics Solid state physics 2+1 d. gravity Chern-SimonsAlone Chern-Simonscoupled 2+1 d. Field theories String theory

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 15 Maxwell-Chern-Simons theory I Lagrangian (only 2+1 d.) Lagrangian (only 2+1 d.) √ Coupling Maxwell + Chern-Simons : viable gauge theory √ Coupling Maxwell + Chern-Simons : viable gauge theory √ CS term breaks parity inv. of Maxwell theory √ CS term breaks parity inv. of Maxwell theory E-L equation of motion E-L equation of motion => 2+1 d. pseudo-vector dual field : => Proca-type equation of massive field with mass :

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 16 Maxwell-Chern-Simons theory II 3+1 d. examples of mass generation 3+1 d. examples of mass generation √ Proca mass term √ Proca mass term …BUT breaks gauge invariance √ Higgs mecanism √ Higgs mecanism 2+1 d. mass generation 2+1 d. mass generation New surprising mass generation induced by the CS term ! √ Gauge invariant √ Gauge invariant √ No introduction of other field √ No introduction of other field √ Parity breaking √ Parity breaking

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November 19th 2004 A journey inside planar pure QED CP3 lunch meeting 17 MCS Hamiltonian analysis Phase space degrees of freedom (df) Phase space degrees of freedom (df) √ Potential vector : √ Potential vector : ! Conjugate momentum : ! Conjugate momentum : √ A 0 is non-physical √ A 0 is non-physical Symplectic structure on the phase space Symplectic structure on the phase space √ Antisym. Poisson bracket : √ Antisym. Poisson bracket : ! Non commutating electric field components ! Non commutating electric field components Classical can. hamiltonian Classical can. hamiltonian ! Constraint : Gauss law ! Constraint : Gauss law

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November 19th 2004A journey inside planar pure QED CP3 lunch meeting 18 CONCLUSION

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

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