1. A method for finding complete observables in classical mechanics
Vladimir Cuesta † and Jos´ David Vergara † †
e
Instituto de Ciencias Nucleares, Universidad Nacional Aut´ noma de M´ xico, 70-543, Ciudad de M´ xico, M´ xico
o e e e
Abstract. In the present work a new method for finding complete observables is discussed. In first place is presented the algorithm
for systems without constraints, and in second place the method is exemplified for gauge systems. In the case of systems with first class
constraints we begin with a set of clocks (non gauge invariant quantities) that are equal to the number of constraints and another non gauge
invariant quantity, being all partial observables, and we finish with a gauge invariant quantity or complete observable. The starting point
is to consider a partial observable and a clock or clocks being both functions of the phase space variables, that is a function of the phase
space variables P (q, p) and a clock T (q, p) or clocks T1 (q, p), . . . , Tn (q, p), where n is the number of first class constraint. Later, we
take the equations of motion for the system and we found constants of motion and with the help of these at different times, we can find a
gauge invariant phase space function associated with the partial observable P (q, p) and the set of clock or clocks.
Introduction
Loop Quantum Gravity is one of the most known intents to build a quantum theory of gravity, for making this theory successful,
one must solve the set of constraints that appear in the theory, or equivalently one should find functions that can commute with
all gravity constraints and in this way quantizing the theory using the methods developed in Loop Quantum Gravity.
One outstanding result in Loop Quantum Gravity is that the length, area and volume operator have discrete spectra. However
these mathematical entities are not fully gauge invariant. Following with this line of reasoning we can find inside the scientific
literature three crucial concepts: a partial observable (a non fully gauge invariant quantity), a system of clock or clocks (that in
the case of first class systems this number is equal to the number of first class constraints) and using different procedures we can
obtain a complete observable (a gauge invariant quantity) [1-3]. Different studies have shown that these gauge invariant quantities
could have two kinds of spectra, we mean discrete and continuous spectra [4-7]. However, the main reason to study this problem
is the fact that in Loop Quantum Gravity we must find gauge invariant quantities with respect to all Gravity constraints and then
the following question remains open: Do the spectra of these gauge invariant quantities are discrete or continuous?
In the present section we will show the algorithm (the interested reader can compare this method with the presented in
[5-6]), we need two phase space functions that do not commute with the set of constraints or a hamiltonian (a clock and a partial
observable) for finishing with a phase space function that commute with all the constraints or with the hamiltonian.
Now, the idea is to find for the system constants of motion and define these at two different values of the evolution
parameter, we will take one given at the initial value of the parameter and another to arbitrary value. In this expressions the
complete observable will correspond to write the partial observable in terms of initial values. And the parameter of evolution
will correspond to the clock in terms of initial values. In this way the complete observable will commute with all the constraints
or will be a constant of motion for a system without constraints.
We begin with the set of equations of motion q i = E i (q1 , . . . , qn , p1 , . . . , pn , N1 , . . . , Na ) and pi =
˙ ˙
i
F (q1 , . . . , qn , p1 , . . . , pn , N1 , . . . , Na ) where 2n is the dimension of the phase space and a is the number of first class constraints
and Na are Lagrange multipliers, the second step is to combine the previous equations and to eliminate the dependence on time,
we obtain a set of differential relations ρs (dq1 , . . . , dqn , dp1 , . . . , dpn ) = 0 where s = 2n − a, solving these differential
equations it is possible to obtain 2n − a constants of motion (in the case of systems without constraints the number of constants
of motion are 2n − 1 and probably these are not independent). To finish we choose a set of clocks Ta (q1 , . . . , qn , p1 , . . . , pn )
and a partial observable f (q1 , . . . , qn , p1 , . . . , pn ), now if we take τa = Ta (q1 (t = 0), . . . , qn (t = 0), p1 (t = 0), . . . , pn (t = 0))
and as a complete observable the initial condition F (q1 , . . . , qn , p1 , . . . , pn , τ1 , . . . , τa ) = f (q1 (t = 0), . . . , qn (t = 0), p1 (t =
0), . . . , pn (t = 0)) the dependence lies on the parameters τ1 , . . . , τa and the constants of motion C1 , . . . , C2n−a .
† vladimir.cuesta@nucleares.unam.mx
†† vergara@nucleares.unam.mx

2. A method for finding complete observables in classical mechanics 2
Particle on a gravitational field
We begin with one of the simplest models, a massive particle immerse in a constant gravitational field, the importance of this
system lies into the fact that it has a finite number of degrees of freedom and not constraints, the phase space coordinates are
1 p
(xµ ) = (q, p), with hamiltonian H = 2m p2 − mgq, the equations of motion are q = m and p = mg. In this case a constant of
˙ ˙
motion is the hamiltonian, making the selection of the clock as T = p and as a partial observable f = q one can find the complete
observable in the following way: we take the parameter τ = p0 = p(t = 0) and the complete observable as F = q0 = q(t = 0),
p2 p2 τ2
and using the constant of motion at t = 0 and t arbitrary, we write 2m − mgq = 2m − mgq0 = 2m − mgq0 and solving for q0
0
τ 2 −p2
the complete observable is F1 = q + 2m2 g .
Model with two constraints
The present model has the characteristic of being a system with a finite number of degrees of freedom more precisely has 2
degrees of freedom per point of the phase space. In this case the phase is labeled by the coordinates (xµ ) = (q1 , q2 , q3 , p1 , p2 , p3 )
and the set of constraints is
1 1
D1 = [−(p1 )2 + (p2 )2 + (p3 )2 ], D2 = − [q1 p1 + q2 p2 + q3 p3 ], (1)
2 2
with equations of motion
1 1 1 1 1 1
q1 = −N p1 − M q1 , q2 = N p2 − M q2 , q3 = N p3 − M q3 , p1 = M p1 , p2 = M p2 , p3 = M p3 , (2)
˙ ˙ ˙ ˙ ˙ ˙
2 2 2 2 2 2
and in this case we obtain the constants of motion C1 , C2 , C3 and C4
p2 p3 p1 p2 p2
C1 = , C2 = , C3 = q 3 − p 1 q 2 , C 4 = q 1 p1 + 1 q 3 , (3)
p1 p1 p3 p3
if we take as a partial observable f = q1 and as a pair of clocks T1 = q3 and T2 = p1 , the final complete observable is
p 1 q 1 p3 + q 3 p1
F = − τ1 . (4)
p3 τ2
However, taking T1 = q1 − q2 and T2 = p2 + p2 as partial observable we obtain the complete observable
2 2
1 2
2
−C1 C3 p1 + C1 C4 p1 C3 p2 − 2C1 C3 C4 p2 + C1 C4 p2 + p4 τ1 − C1 p4 τ1
2
1 1
2 2
1 1
2
1
F = ± ,
C1 (p2 − 1)
2
1 C1 (p2 − 1)
2
1
this shows that the set of complete observables depend of the clock in the same way as [5].
sl(2, ) model
The model sl(2, ) is an outstanding model [7] quadratic in the phase space coordinates, it resembles the structure of the
constraints for General Relativity, the model has 2 degrees of freedom per point of the space. In this case the phase space
coordinates are (xµ ) = (u1 , u2 , v 1 , v 2 , p1 , p2 , π 1 , π 2 ) with constraints
1 1
H1 = (p 2 − v 2 ), H2 = (π 2 − u 2 ), D = u· p − v· π, (5)
2 2
whose Poisson bracket between them are {H1 , H2 } = D, {H1 , D} = −2H1 , {H2 , D} = 2H2 , with this set of constraints one
can find the following equations of motion
˙ ˙ ˙
u = N p + λu, v = M π − λv, p = M u − λp, π = N v + λπ, ˙ (6)
after a straiightforward calculation we find the constants of motion
O1 = u1 p2 − p1 u2 , O 2 = u 1 v 1 − p1 π 1 , O 3 = u 1 v 2 − p1 π 2 ,
O4 = u2 v 1 − p2 π 1 , O5 = u 2 v 2 − p2 π 2 , O6 = π 1 v 2 − v 1 π 2 . (7)
1 2 1 1
if we take τ1 = u i , τ2 = u i , τ3 = v i as clocks and f1 = π as a partial observable and following the algorithm presented in
the present work we obtain the complete observable
τ2 (u1 v 1 − p1 π 1 ) − τ1 (u2 v 1 − p2 π 1 )
F1 = , (8)
u 1 p2 − p 1 u 2

3. A method for finding complete observables in classical mechanics 3
For finishing this section we can take the 3 clocks T1 = u1 , T2 = u2 and T3 = v 1 , with partial observable
f2 = A sin(u2 + βπ 2 ), and we can deduce that the complete observable is
O1 + βO2 O4
F2 = A sin τ2 − β τ1 . (9)
O1 O1
An example in field theory
As a final example we show a model of field theory, in this case the original model can be thought as a model with 2
degrees of freedom per point. In this model the coordinates (xµ ) = (e, φ, πe , πφ ) label the phase space that we are working, the
hamiltonian for the theory is (see [8] for instance)
2
eπe πe πφ
H=− + + eV = 0, (10)
2D 2 D
dD(φ)
where D = dφ and V = V (φ). Using D(φ) = Qφ, V (φ) = λ we obtain the equations of motion
2
eπe πφ ˙ πe πe
e=
˙ − + N, φ= N, πe =
˙ − λ N, πφ = 0,
˙ (11)
q2 Q Q 2q2
and we find the following constants of motion
2 2
C1 = πφ , C2 = φ − Q ln(πe − 2λq2 ), C3 = (πe − 2q2 λ)e − 2πφ Qπe , (12)
if we take as clock πe and as partial observable f = e the complete observable that we obtain is
2
(πe − 2q2 λ)e − 2πφ Qπe + 2πφ Qτ
F = . (13)
τ 2 − 2q2 λ
And for finishing the present work we can suppose that T = aφ2 + bπe as a a different clock and if we choose as partial
τ1 −a2 C1
observable f2 = e the result is F = a1 .
Conclusions and perspectives
In the present work we have presented a method for finding complete observables we mean functions of the phase space that
commute with the hamiltonian for systems without constraints or that commute with all the constraints for gauge systems. In the
first case we need a clock and a partial observable for finishing with a complete one or a constant of motion, in the second case we
begin with a set of clocks with the number equal to the number of first class constraints and we finish with a complete observable
or a gauge invariant function. In all the cases we put the complete observables in terms of initial conditions. This method can
illustrate the difficulties found in Loop Quantum Gravity to solve the constraints or to obtain gauge invariant quantities with
respect to the full set of General Relativity constraints.
The procedure presented in this work differs of [6], this algorithm lies in to solve the flow (or a formal serie) of the set
of clocks under the set of first class constraints and inserting the solution into the flow of the partial observable and the final
result is the complete observable. However, the physical interpretation in that article is not clear. In fact, in the present work we
have presented a different procedure and we have interpreted the final result or the complete observable as initial conditions or
equivalently as a function of full gauge invariant quantities and a set of parameters τ1 , . . . , τa .
[1] V. Cuesta, M. Montesinos and J. D. Vergara, Phys. Rev. D 76, 025025 (2008),
[2] C. Rovelli, Class. Quant. Grav. 8 (1991) 297-316,
[3] C. Rovelli, Phys. Rev. D, 65, 124013 (2002),
[4] M. Montesinos, Gen. Rel. Grav. 33, 1 (2001),
[5] B. Dittrich, Gen. Rel. Grav. 39 (2007) 1891-1927,
[6] B. Dittrich and T. Thiemann, J. Math. Phys. 50, 012503 (2009),
[7] M. Montesinos, C. Rovelli and T. Thiemann, Phys. Rev. D, 60 044009 (1999),
[8] T. Banks and M. O’loughlin, Nucl. Phys. B, 362, 3 (1991) 649-664.