2+1 Dimensional Bulk Spacetime

ρ = ρ

Fig. 3.1.: Pictorial representation of the manifoldM.

The Chern-Simons gauge fieldAis a Lie algebra valued 1-form that can be written as

A=A^a_µdx^µT_a, (3.1)

1Please refer to e.g. [79, 83] for details on how to quantize gauge systems in a Hamiltonian formulation.

withTabeing a basis of some Lie algebragwith commutation relations

[Ta, T_b] =f^c_abTc. (3.2) In addition, I will assume that the Lie algebra admits a non-degenerate invariant bilinear form which will be denoted by

h_ab =hT_a, T_bi. (3.3)

Lie algebra indices(a, b, . . .)are raised and lowered withhab and spacetime indices (µ, ν, . . .)with the spacetime metricg_µν.

One can write (2.3) explicitly in components as S_CS[A] = k

In order to proceed with the canonical analysis it is convenient to use a 2+1 decom-position [84] of the action (2.3), which explicitly separates the manifoldMinto the diskΣand the real lineR, i.e. an explicit split into timetand the coordinates which parametrize the discx^µ¯ ≡x= (ρ, ϕ),µ¯= 1,2. This has the advantage that ultimately one only has to deal with integrals over the two-dimensional (spatial) diskΣand its boundary manifold∂Σ. The decomposition of (3.4) is given by

S_CS[A] = k terms appearing in (3.5) one can seeA^a₀as a Lagrange multiplier, which in turn also means that on-shell the Chern-Simons gauge field has to be gauge flat, i.e. F^a_µν = 0.

Thus, the dynamical fields of the Chern-Simons action areA^a_µ¯.

The starting point of the canonical analysis is a Hamiltonian density. Hence one first has to determine the canonical momentaπaµcorresponding to the canonical variablesA^a_µfrom the Lagrangian density²Lviaπaµ≡ ^∂L

∂A˙^aµ.

The canonical momenta are in general not independent quantities. In case of the Chern-Simons action (3.5) one obtains the following relations³

φa0 :=πa0 ≈0 φaµ¯ :=πaµ¯− k

4π^µ¯¯νh_abA^b_¯ν ≈0, (3.6) which are called primary constraints to emphasize that the equation of motions are not used to obtain these relations. Having identified the canonical variables and

2Where by Lagrangian density I meanSCS=R

Mdx³L.

3From this point onward one has to distinguish between weak (≈) and strong (=) equalities. Two functions in the phase space,fandg, are weakly equalf≈gif restricted to a constraint surface, but not throughout the whole phase space. Iffandgare equal independently of the constraints being satisfiedf=g, they are called strongly equal.

their corresponding canonical momenta one can also define their Poisson brackets as

{A^a_µ(x), πbν(y)}=δ^abδµνδ²(x−y). (3.7) The next step in this analysis is to calculate the canonical Hamiltonian density using a Legendre transformation

H=π_a^µA˙^a_µ− L=− k

4π^µ¯¯νh_abA^a₀F^b_µ¯¯ν+∂_ν¯A^a_µ¯A^b₀. (3.8) As the system under consideration is a constrained Hamiltonian system one also has to include the constraints (3.6) in a Hamlitonian description of the system. This can be done by adding the constraints to (3.8) and thus obtaining a total Hamiltonian H_T using some arbitrary multipliersu^a_µas

H_T =H+u^a_µφ_a^µ. (3.9)

Then, by construction, the equations of motion obtained fromH_T viaF˙ ≈ {F,H_T} for some functionF, which is function of the canonical variables, are the same as the ones obtained from the Lagrangian description.

Since the primary constraints should be conserved after a time evolution, which is generated by the total HamiltonianH_T, one has to require

φ˙_a^µ={φ_a^µ,H_T} ≈0, (3.10) which leads to the following secondary constraints

K_a≡ − k

4π^µ¯¯νh_abF^b_µ¯¯ν ≈0 (3.11a) Dµ¯A^a₀−u^aµ¯ ≈0, (3.11b) whereDµ¯X^a=∂µ¯X^a+f^a_bcA^b_µ¯X^cis the gauge covariant derivative.

The Lagrange multipliers u^a_µ¯ can be determined via the Hamilton equations of motion as

A˙^a_µ¯ = ∂H_T

∂π_a^µ¯ =u^aµ_¯. (3.12)

This allows one to rewrite (3.11b) and in addition yields the following weak equality Dµ¯A^a₀−u^aµ¯ =Dµ¯A^a₀−∂0A^a_µ¯ =F^aµ0¯ ≈0. (3.13) The total Hamiltonian can now be written in the following form

H_T =A^a₀K¯_a+u^a0φa0+∂µ¯(A^a₀πaµ¯), (3.14) with

K¯_a=K_a−D_µ¯φ_a^µ¯. (3.15)

The canonical commutation relations (3.7) can now be used to determine the following Poisson bracket algebra of constraints

{φ_a^µ¯(x), φ_b^ν¯(y)}=− k

2π^µ¯¯νh_abδ²(x−y), (3.16a) {φ_a^µ¯(x),K¯_b(y)}=−f_ab^cφ_c^µ¯δ²(x−y), (3.16b) {K¯_a(x),K¯_b(y)}=−f_ab^cK¯_cδ²(x−y), (3.16c) which are the only non-vanishing Poisson brackets of the constraintsφ_a^µ andK¯_a. This allows one to directly determine which of the constraints are first class and which are second class. First class constraints are constraints whose Poisson brackets vanish weakly with every other constraint. If that is not the case, then the constraint is called second class. This is a crucial distinction of constraints since first class constraints generate gauge transformations whereas second class constraints are used to restrict the phase space and thus promote the Poisson brackets to Dirac brackets, which can be used to consistently quantize the system.

Since only the Poisson brackets ofφa0andK¯_avanish weakly with all other constraints these are first class constraints. On the other handφaµ¯ are second class constraints since they have non-weakly vanishing Poisson brackets with other constraints. Thus, one can use the second class constraintsφaµ¯ in order to promote the Poisson brackets to Dirac brackets by eliminating physically irrelevant degrees of freedom. For the case at hand setting the second class constraintsφaµ¯ strongly to zeroφaµ¯ = 0one obtains the following Dirac brackets for the dynamical variables

{A^a_µ¯(x),A^b_ν¯(y)}_D.B= 2π

k h^abµ¯_¯νδ²(x−y), (3.17) whereµ¯¯ν is obtained using^µ¯¯αα¯¯ν =δ^µ¯ν¯.

Having determined all constraints one can also check that the number of local degrees of freedom of the physical system described by the Chern-Simons action (2.3) is indeed equal to zero. The degrees of freedom of a constrained Hamiltonian system are characterized by the dimension of the phase spaceN, the number of first class constraintsM and the number of second class constraintsS as

№of local physical D.O.F= 1

2(N−2M−S). (3.18) The phase space for a Chern-Simons theory in three dimensions is determined by the gauge fieldsA^a_µ. Denoting the dimension of the Lie algebragasD, then the dimension of the phase space is N = 6D. Accordingly, the number of first class constraints is M = 2D and the number of second class constraints is S = 2D.

Combining N, M and S as in (3.18) one finds that indeed the number of local physical degrees of freedom is zero as expected.