Dirac analysis of all constraints

Let us turn to whether the equations of motion preserve the constraints. We do this in the Hamiltonian picture. With the Lagrangian (3.46), linear in the time derivatives, we can immediately find the Hamiltonian, which is itself constrained to vanish. If we introduce theprimary Hamiltonian

H^′[π, ω](t) =z(t)F_n(t)[π, ω] +λ(t)

2 D[π, ω] + cc., (3.47) we can write the evolution equations in the most covariant way possible:

D∂tω^A={H^′, ω^A}, D∂tπ^A={H^′, π^A}. (3.48) The canonical commutation relations are{π^A, ω^B}=ǫ^AB, andD∂t is again the covari-antsl(2,C)derivativeD∂tπ^A=∂_tπ^A+Φ^A_Bπ^B, withΦ^A_Bbeing the self-dual connection contracted with the tangent vector of the edge—just as defined in (3.43).

To prove that the Hamiltonian vector field preserves the constraints we discuss each of them separately.

(i) stability of the area-matching constraintE˙ = 0, and ofD= 0 The area-matching constraintE˙ = 0guarantees the area of a triangle is the same seen from all tetrahedra it belongs to. The Hamiltonian vector field of E=πAω^Aacts as follows:

{E, π^A}=−π^A, {E, ω^A}=ω^A, {E,π¯^A¯}= 0 ={E,ω¯^A¯}. (3.49) We thus easily get

E˙ =D∂tE={H^′, E}=z(t)F −z(t) ¯¯ F ∝0, (3.50) where∝means equality up to constraints. SinceD= iE/(β+ i) + cc., hence linear in E, equation (3.50) also implies that the reality condition D= 0 holds for all times:

D˙ ={H^′, D} ∝0. (3.51)

Therefore, the Hamiltonian time evolution along a spinfoam edge preserves both the area-matching constraint (3.14, 3.38) and the Lorentz invariant part D = 0 of the simplicity constraintsK_i+βL_i = 0.

(ii) stability ofFⁿ= 0 Before we explore under which conditions our primary Hamil-tonian (3.47) is compatible with the constraintF_n= 0, let us first recall^* all solutions of the reality conditionsF_n= 0 =D. They are parametrised by a real numberJ 6= 0, and tell us the momentum π is proportional to ω. We find, in fact,¯

πA=−i√

2(β+ i)Jn_AA¯ω¯^A¯ kωk²n

, (3.52)

with the SU(2) norm kωk²n = −i√

2n_AA¯ω^Aω¯^A¯. Notice that we can always assume J > 0. We have mentioned, in the lines shortly above equation (3.16), that there is a discrete symmetry simultaneously exchanging all π and ω spinors. Since

J = πAω^A

β+ i, (3.53)

a transformation exchanging π and ω, maps J into −J, hence J > 0 without loss of generality.

The quantityJ parametrising the solutions of the reality conditions also has a clean geometrical interpretation. It measures the area A[τ]of the triangleτ under consider-ation. A short calculation gives the precise relation:

A²[τ] = Σⁱ[τ]Σi[τ] =−2Σ^ABΣAB =βℓ²_PJ

~ 2

. (3.54)

We are now ready to come back to our original problem, and show how the Hamilto-nian can preserve the reality conditions. Since the normal is covariantly constant, we get for the time evolution ofF_n= 0 the equation:

F˙_n= d

dt n^AA¯π_Aω¯A¯

=D∂t n^AA¯π_Aω¯A¯

=

H^′, F_n ∝z(t)¯ F¯_n, F_n . (3.55)

*We will derive these solutions explicitly in chapter 3.5, further details can be found in references [152] and [151].

We calculate the missing Poisson bracket in the gauge wheren^α=n^α_o and find:

F¯no, Fno = 1

2δ_AA¯δ_BB¯

π¯^A¯ω^A, π^Bω¯^B¯ = 1

2δ_AA¯δ_BB¯

h

¯

ǫ^A¯B¯ω^Aπ^B+ǫ^AB¯π^A¯ω¯^B¯i

=

=−1

2 π_Aω^A−cc.

=−iJ. (3.56)

The result being manifestlySL(2,C)invariant we can conclude that

F˙n∝ −i¯z(t)J. (3.57)

We have assumed the area of the triangle does not vanish, hence J 6= 0. This implies the Hamiltonian flow preserves the constraint Fn = 0 only if we put the Lagrange multiplierz(t)to zero. Reinserting this restriction on the Lagrange multiplier into the primary Hamiltonian (3.47) we get the secondary Hamiltonian

H^′′=λ(t)D[π, ω]. (3.58)

(iii) stability of the Gauß law The secondary Hamiltonian (3.58) generates the edge-evolution compatible with the simplicity constraints for one pair of spinors. There are, however, four of these pairs per edge—one twistor Z = (¯πA¯, ω^A) for each adjacent triangle. The Gauß law^* is an example of an observable depending on all of them.

Its time evolution is governed by the physical Hamiltonian, which is the sum over the secondary Hamiltonians (3.58) of all four triangles:

H^phys = X4

I=1

λ_(I)(t)D[π_(I), ω_(I)]. (3.59) The Hamiltonian has this simple form, just because the action for an edge (3.46) splits into a sum over adjacent triangles, without any “interaction-terms” appearing. Since the Hamiltonian vector field of the constraintD= 0 acts as

X_D[ω^A] ={D, ω^A}= i

β+ iω^A, X_D[π^A] ={D, π^A}=− i

β+ iπ^A, (3.60) we immediately get for any choice of λ, that the Gauß constraint is covariantly pre-served:

D∂tG_AB =

H^phys, G_AB = 0. (3.61)

The partial derivative, on the other hand, vanishes weakly:

d

dtG_AB ∝0. (3.62)

which follows from the commutation relations of the Lorentz algebra:

{L_i, L_j}=−ǫ_ij^lL_l, {L_i, K_j}=−ǫ_ij^lK_l, {K_i, K_j}= +ǫ_ij^lL_l. (3.63)

*The Gauß law follows from the stationarity of the action (3.46) under variations ofΦ^AB (3.43).

Since the time normals depend on Φ^AB through (3.40), and linearly appear in the Fn-term of the action (3.46), this adds a term to the Gauß law (3.44) which is linear in the multipliersz(I). We will later prove that allz(I)must vanish, henceGAB= 0as in (3.44).

First- and second-class constraints We got the constraint equations on an edge by varying the Lagrange multipliers Φ ∈ sl(2,C), z_(I) ∈ C and λ_(I) ∈ R in the action (3.46). If we want to quantise the theory we have to compute the constraint algebra and identify first-class and second-class constraints therein. The set of constraints consists of both the rotational and boost part of the Gauß law, together with the simplicity constraints on the triangles. Only some of these constraints are independent: If we impose the simplicity constraints K_i+βL_i = 0 on three triangles only, the Gauß law P4

I=1K_i^(I) = P4

I=1L^(I_i ⁾ = 0 implies them on the fourth. These constraints can be rearranged to treat all triangles equally. We can then impose just the rotational part of the Gauß law, and require the simplicity constraints (3.19) on all four triangles. This rearrangement leads us to the following system of constraints:

G^rot_i = X4

I=1

L_i[π_(I), ω_(I)]= 0,^! (3.64a) D_(I)≡D[π_(I), ω_(I)]= 0,^! ∀I = 1, . . . ,4, (3.64b) F_(I)≡F_no[π_(I), ω_(I)]= 0,^! ∀I = 1, . . . ,4. (3.64c) Notice, that in a general gauge, where the time-normal does not assume the canonical formn^α =n^α_o, we must boost the constraints into the direction of n^α. We would then work with the constraints

G^rot_i(n):= exp X

G^boost_i ηⁱ

G^rot_i , and F_n= exp X

G^boost_i ηⁱ

F_no (3.65) instead. Here, X

G^boost_i = {G^boost_i ,·} denotes the Hamiltonian vector field of the boost part of the Gauß law and the generic normal nhas been parametrised as:

(n⁰, nⁱ) =

ch(|η|), sh(|η|)ηⁱ

|η|

, where: |η|=q

δ_ijηⁱη^j. (3.66) All of the constraints (3.64) are preserved by the physical Hamiltonian generating the time evolution along an edge, e.g. D∂tG^rot_i(n) = {H^phys, G^rot_i(n)} ∝ 0. To identify first-and second-class constraints within this set, we have to study their mutual Poisson brackets. We find:

{G^rot_i , G^rot_j }=−ǫijlG^rot_l(n), {G^rot_i , D_(I)}= 0, {G^rot_i , F_(I)}= 0 ={G^rot_i ,F¯_(I)}, (3.67a) {D_(I), F_(J)}=− 2iβ

β²+ 1δ_IJF_(I), {D_(I),F¯_(J)}= 2iβ

β²+ 1δ_IJF¯_(I), (3.67b) {F_(I),F¯_(J)}= iδ_IJIm(π_A^(I)ω_(I)^A ) = iδ_IJIm(E_(I)). (3.67c) The set of first-class constraints consists of the rotational component of the Gauß law, attached to each edge, together with the Lorentz invariant simplicity constraint D= 0, attached to each triangle. The constraint F_n= 0 is second class and generates an additional su(2) algebra. This becomes more explicit once we define the ladder operatorsJ_± together with the generator J₃:

J₋=J₁−iJ₂ =:−√

2 ¯F_no, J₊=J₁+ iJ₂:=−√

2F_no, J₃ :=Im(E), (3.68)

with the Poisson bracket of the rotation group:

{Ji, J_k}=−ǫijkJ_k. (3.69) In our case J₋ and J₊ are constrained to vanish, while J₃ 6= 0, reflects the fact that the constraints form a second-class system. A last comment on the time gauge (3.39):

if we want to relax this condition, little will happen, the constraints get boosted as in (3.65), but the structure constants appearing in the constraint algebra (3.67) remain the same.

Im Dokument The Chiral Structure of Loop Quantum Gravity (Seite 101-105)