Download Maxwell-Chern-Simons

November 19th 2004
A journey inside planar pure QED
By Bruno Bertrand
CP3 lunch meeting
INTRODUCTION
November 19th 2004
A journey inside planar pure QED
CP3 lunch meeting
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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.
November 19th 2004
A journey inside planar pure QED
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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
November 19th 2004
A journey inside planar pure QED
CP3 lunch meeting
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FIRST PART
Maxwell theory in 2+1 and 3+1 dim.
November 19th 2004
A journey inside planar pure QED
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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]
November 19th 2004
A journey inside planar pure QED
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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
November 19th 2004
A journey inside planar pure QED
CP3 lunch meeting
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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-scalar
Pseudo-vector
Invariance of Max. lagrangian under Parity
(x1, x2, t) ! (-x1, x2, t)
November 19th 2004
(x1, x2, x3, t) ! (-x1, -x2, -x3, t)
A journey inside planar pure QED
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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
November 19th 2004
A journey inside planar pure QED
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SECOND PART
Maxwell-Chern-Simons theory
November 19th 2004
A journey inside planar pure QED
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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’’ :
November 19th 2004
A journey inside planar pure QED
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The Chern-Simons theory
Is it a boring, uninteresting and simply trivial theory ?
November 19th 2004
A journey inside planar pure QED
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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
Canonical hamiltonian = 0  Phys. States of zero energy !
√ Sometimes : analytical (non perturbative) solutions exist.
NB : In quantum field theory, a physical state or observable is gauge invariant
November 19th 2004
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NO ! 2) Numerous theoretical applic.
Chern-Simons
Alone
2+1 d. gravity
Pure
quantum
gravity
String theory
Chern-Simons
coupled
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Knot
theory in
math.
Mathematics
Landau
Problem
CHERNSIMONS
String
theory /
AdS/CFT
Solid state physics
Vortices
+SUSY
Yang
-MillsC-S
2+1 d. Field theories
A journey inside planar pure QED
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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 :
November 19th 2004
A journey inside planar pure QED
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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
November 19th 2004
A journey inside planar pure QED
CP3 lunch meeting
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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
November 19th 2004
A journey inside planar pure QED
CP3 lunch meeting
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CONCLUSION
November 19th 2004
A journey inside planar pure QED
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November 19th 2004
A journey inside planar pure QED
By Bruno Bertrand
CP3 lunch meeting