Download Maxwell-Chern-Simons Theory

Maxwell-Chern-Simons Theory
Marina von Steinkirch, [email protected]
State University of New York at Stony Brook
November 12, 2010
Contents
Gauge Invariance and Equation of Motions
1 Maxwell Gauge Theory
1
2 Chern-Simons Theory
1
3 Topologically Massive Gauge Theory
2
4
2
Mass Spectrum (Excitations)
The Maxwell lagrangian is invariant under gauge
transformation
Aµ → Aµ + ∂µ f,
and the derived equations of motion
∂µ F µν = J ν ,
Introduction
are gauge invariants. With source free, it has planewave solutions.
Chern-Simons theory is a (2+1)-dimensional gauge
theory differently of the (2+1)-Maxwell theory. Together they can be represented by the action
Z
i
h
1
SM CS = d3 x − 2 Fµν F νµ + κµνρ Aµ ∂ν Aρ . (1)
4e
In this paper I discuss basic aspects of the MaxwellChern-Simons theory and then find the equations
of motion and the spectrum of excitations from the
analogy to the Landau levels.
In (2+1) Dimensions
In (2+1) dimensions the magnetic field is a pseudo~ =
scalar B = ij ∂i Aj rather than a pseudovector B
~ ×A
~ in (3+1) dimensions. The electric field E
~ =
∇
~
~
−∇A0 − A is a two dimensional vector.
2
Chern-Simons Theory
The Chern-Simons lagrangian is
1
Maxwell Gauge Theory
LCS =
κ µνρ
e Aµ ∂ ν Aρ − Aµ J µ .
2
(3)
The Maxwell gauge theory is defined in terms of the
~ and its la- Gauge Invariance and Equations of Motion
fundamental gauge field Aµ = (A0 , A),
grangian is
Let us vary the equation (3) by a space-time deriva1
tive
µν
µ
(2)
LM = − Fµν F − Aµ J ,
δLCM ∼ ∂µ (eµνρ ∂ν Aρ ),
4
where the field strength tensor is Fµν = ∂µ Aν −∂ν Aµ , clearly if we do not consider boundary terms, the
and the matter current J µ is conserved (∂µ J µ ).
action will be gauge invariant. Straightforward from
1
the Euler-Lagrange equation, we find the equations
of motion, also gauge invariant,
κ µνρ
Fνρ = J µ .
(4)
2
The source free solution reduces to Fµν = 0, flat
connections. We can show explicitly the current conservation:
κ
∂µ ( µνρ Fνρ ) = ∂µ J µ .
2
µνρ ∂µ Fνρ = 0.
Equations of Motion
Calculating the Euler-Lagrange equation of motion
gives
∂µ F µν +
κe2 νβα
Fβα = 0.
2
(6)
Massive Gauge Theory
From dimensional analysis we see that [e2 ] = [m]
and [κ] = [m0 ] in (2+1) dimensions. Therefore these
equations will describing the propagation of a degree
Chern-Simons + Matter = Anyons
of freedom with mass m = κe2 . We see the origin of
The matter current in equation (4) can be seen better this mass explicitly rewriting equation (6) in terms
~ in terms of components, reveling of a dual gauge invariant field F̃ µ = 1 µνρ Fνρ , and
writing J µ = (ρ, J)
2
a tying of flux to charge and the nature of anyons.
them rewriting the equations of motion
ρ = κB
Charge density is locally
∂ ν ∂ν + (κe2 )2 F̃ µ = 0.
proportional to Mag. Field.
Magnetic flux is proportional
to electric charge.
4
Mass Spectrum (ExcitaJ i = κij Ej
3
tions)
Charge-Flux relation is
preserved under time evolution.
Proof:
ρ̇ = κḂ = κij ∂i Ȧj
∂µ J µ → J i
ij
= −κ Ȧj + ij χ = κij Ej
We shall use the analogy to the classic Landau problem of charge moving in the plane in the presence of
~ perpendicular to the plan to
an external uniform B
find the spectrum of our theory. The quantization of
the Landau problem is well understood, consists of
equally spaced energy levels (Landau levels) by ~ωc ,
B
where ωc = m
is the cyclotron frequency. Each Landau level is infinitely degenerated in the open plane,
but for a finite area A the degeneracy is related to
the net magnetic flux, φ = BA
2π .
Topologically Massive Gauge
Theory
Canonical Chern-Simon
Topological electrodynamics (Chern-Simons charged
particle system) is a theory describing an interaction
of a U(1) gauge field A(x, t), a vector-valued function
on the three-dimensional space, with a charged matter field, characterized by a current J(x, t). When
we put together the two theories we get a surprising
new form of gauge field mass generation, different of
the Higgs mechanism. The Maxwell-Chern-Simons
lagrangian is
κ
1
(5)
LM CS = − 2 F µν Fµν + µνρ Aµ ∂ν Aρ .
4e
2
Let us quantize the lagrangian. We rewrite equation
(5) in a canonical structure
LM CS =
1 2
1
κ
Ei − 2 B 2 + ij Ȧi Aj + κA0 B. (7)
2
2e
2e
2
In the A0 = 0 gauge, Ai are the ’coordinates’ and
we have the momentum fields
Πi =
2
∂LCSM
1
κ
= 2 Ȧi + ij Aj .
e
2
∂ Ȧi
The hamiltonian is
Explicitly, a planar quantum mechanical harmonic
1
system
with a hamiltonian of the kind H = 2m
(pi +
HM CS = Πi Ȧi − L
1
b ij j 2
2 2
c can be separated into two distinct
2 x ) + 2 mω ~
2
harmonic
oscillators
of frequency, let us say for our
2
κ
1
e
Πi − ij Aj + 2 B 2 +A0 ∂i Πi +κB
HM CS =
problem,
2
2
2e
s
i
Then Ai (~x, t), Π (~x, t) satisfy the canonical equal
4ω 2
ωc time Poisson brackets, which becomes the equal-time
1+ 2 ±1 ,
ω± =
2
ωc
canonical commutation,
[Ai (~x), Πj (~y )] = iδ ij δ(~x − ~y ).
(8) where ω is the harmonic well√frequency.
Taking ωc = κe2 and ω = 2ev, the characteristic
frequencies are exactly mass poles m± . In the limit
Analogy to the Landau Problem
which the cyclotron frequency dominates,
We consider the long wavelength limit of 7, in which
2v 2
ω2
we drop all spatial derivatives,
= m−
=
ω− →
ωc
κ
1 2 κ ij
L = 2 Ȧi + Ȧi Aj .
2e
2
and
ω+ → ∞.
Now, thinking about the non-relativistic charged
particle moving in the plane, we have
L=
Therefore, we have the analogy for our the (2+1)
dimensions Maxwell-Chern-Simons, where the gauge
field has two massive modes.
1
B
mẋ2i + ij ẋi xj .
2
2
The momenta is pi =
hamiltonian is
H = pi ẋi − L =
∂L
∂ ẋi
= mẋi +
B ij
2 xj
and the
References
1
B
m
(pi − ij xj )2 = vi2 .
2m
2
2
[1] Tsvelik, Quantum Field Theory in Condensed
Matter.
In the quantum level, [xi , pj ] = iδij implies that [2] Nakahara, Geometry, Topology and Physics.
the velocities do not commute [vi , vj ] = −i mB2 ij . We
now can compare both equations/problems,
[3] Zee, Quantum Field Theory in a Nutshell.
Maxwell-Chern-Simons
e2
κ
mM CS = κe2
Landau
1
m
B
ωc =
B
m
Physical masses of the theory appear as physical
frequencies of the corresponding quantum mechanic
system. The inclusion of the Chern-Simons term in a
gauge theory lagrangian is analogous to the inclusion
of a Lorentz force in a mechanical system. The Landau system shows how to obtain characteristic frequency without introducing a harmonic binding term
(such as in the Higgs mechanics).
3