This document discusses gauge systems with constraints and complete observables at the quantum level. It examines two cases: a non-relativistic free particle with one constraint and a model with two constraints. For the particle, choosing a clock such that its Poisson bracket with the constraint is one solves problems of the clock not being defined at all times and the operator for a complete observable not being self-adjoint. For the two constraint model, choosing clocks conjugate to the constraints results in self-adjoint operators. The self-adjointness of operators for complete observables in more complex systems like field theory and general relativity requires further study.

1. Gauge systems and functions, hermitian operators and clocks as conjugate
functions for the constraints
Vladimir Cuesta 1,†
, Jos´e David Vergara 1,††
and Merced Montesinos 2,∗
1
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, 70-543, Ciudad de M´exico, M´exico
2
Departamento de F´ısica, Centro de Investigaci´on y de Estudios Avanzados del Instituto Polit´ecnico Nacional, Instituto Polit´ecnico Nacional 2508, San Pedro
Zacatenco, 07360, Gustavo A. Madero, Ciudad de M´exico, M´exico
Abstract
We work with gauge systems and using gauge invariant functions we study its quantum
counterpart and we find if all these operators are self adjoint or not. Our study is divided in two
cases, when we choose clock or clocks that its Poisson brackets with the set of constraints is one
or it is different to one. We show some transition amplitudes.
Introduction
The study of gauge systems or systems with first class constraints is an outstanding branch
of theoretical physics and mathematics, its importance lies in the fact that a major number
of physical systems have first class constraints, including parametric systems, quantum electro-
dynamics, the standard model, general relativity and a lot of systems with a finite number of
degrees of freedom and so on. A special case of these theories are the covariant systems. In this
case the canonical Hamiltonian vanishes and the system is invariant under the reparametrization
of the coordinates. In consequence the time is not a priori defined.
Let us consider a phase space with coordinates (q1, . . . , qn, p1, . . . , pn) and first class
constraints γa, where the set of constraints obeys {γa, γb} = Cab
cγc and a = 1, . . . , m
and m is the number of first class constraints. We take the action principle S[q, p, λa] =
σ2
σ1
pi ˙qi − λaγa dσ, then if we vary the action we can obtain the equations of motion for our
system. Now, we define a gauge invariant function R or a complete observable as a phase space
function such that the Poisson brackets with the full set of constraints is zero, i. e. {R, γa} = 0.
Then, all the phase space functions such that its Poisson bracket with all the constraints is not
zero are called partial observables. If we take m partial observables as clocks T1, . . . , Tm, and
a partial observable f, then we can find a complete observable F or a gauge invariant func-
tion (see [1] and [2] for instance). Now, our aim is to consider what happen to the quantum
level. In that case we study two problems, the first problem is to define our set of clocks all
along the real line and the second is the self adjoint character for the ˆF operator, where F
is a complete observable. If Dim[Ker( ˆF + ı)] = Dim[Ker( ˆF − ı)], then ˆF is self adjoint; if
Dim[Ker( ˆF + ı)] = Dim[Ker( ˆF − ı)], then ˆF is not self adjoint and it has not self adjoint ex-
tensions (see [3] for instance). In the covariant systems we have two problems first for arbitrary
defined clock the
Det




{Ti, Tj } {Ti, Dj }
{Di, Tj } {Di, Dj }



 ,
is zero for some regions at the phase space and then clocks are not globally well defined, the
second problem is that ˆF is not self adjoint, if we find one or two of the previous problems we
propose that the clocks must be selected in such a way that {Ti, Dj } = δij (see [4] for instance)
and we show in this poster that this form of select the time solves the problems for several
systems.
Non relativistic parametric free particle
In this case, the constraint for the one dimensional non-relativistic free particle is
D = p0 +
p2
2m
= 0 and the phase space coordinates are (x0, x, p, p0).
Example 1
For the first example, we can take the T clock like T = x0 − ax and the partial observable
like f = x, if we make that we will obtain the complete observable:
F =
q +
p
m
τ
1 − a
p
m
, q = x −
px0
m
. (1)
However, we find a first problem for the present system, in this case we have
Det
{T, T } {T, D}
{D, T } {D, D}
=
0 1 −
ap
map
m
− 1 0
= 1 −
ap
m
2
,
and our choice for the time is not defined all along the real line.
Now, we consider the pair of equations ( ˆF + ı)ψ+ = 0 and ( ˆF − ı)ψ− = 0 to determine if
ˆF is a self adjoint operator or it is not. In this case the solutions are:
ψ± = r± 1 −
ap
m
exp


ıp2τ
2m

p −
ap2
2m



 , (2)
where r+ and r− are constant, then |ψ±|2 = |r±|2|1 −
ap
m
| exp 2p −
ap2
m
; ψ+ is not
square integrable and ψ− is square integrable, then Dim[Ker( ˆF + ı)] = 0, Dim[Ker( ˆF + ı)] = 1
and we have found a second problem: the ˆF operator is not self adjoint (see [3] for instance).
Example 2
We will study our previous system and we correct the problems of the previous example,
we will use T such that {T, D}=1 (see [4] for instance) and the general solution for T will be:
T = mx
p
+ f p, x0 − mx
p
, p0 , then
Det
{T, T } {T, D}
{D, T } {D, D}
=
0 1
−1 0
= 1,
and our T clock is well defined all along the real line.
We continue, for the clock T = mx
p
+ αp + β x0 − mx
p
+ ρp0 with {T, D} = 1 and the
partial observable f = x we obtain
F =
pτ
m
−
αp2
m
+ βx −
βpx0
m
−
ρp0p
m
. (3)
To determine the self adjointness for ˆF we must find Dim[Ker( ˆF + ı)], Dim[Ker( ˆF − ı)] and
compare our answers, with the previous purpose we consider ( ˆF + ı)ψ+ = 0 and ( ˆF − ı)ψ− = 0.
We take ψ±(p, p0) = δ p0 +
p2
2m
and the solutions for ( ˆF ± ı)ψ± = 0 are:
g± = r± exp


p
β
+ ı
p2τ
2mβ
− ı
αp3
3mβ
+ ı
ρp4
8m2β
− ı
p2ρ(2mp0 + p2)
4m2β

 , (4)
1 ,† vladimir.cuesta@nucleares.unam.mx
1 ,†† vergara@nucleares.unam.mx
2 ,∗ merced@fis.cinvestav.mx
in this case we define ˜g(p) = g p, −
p2
2m
and ψ1|ψ2 F is = ∞
−∞ dp˜g∗
1 (p)˜g2(p), ˆF is self adjoint
and we have not the second problem of the first example.
Now, we take the eigenvalue equation ˆXψ = x1ψ, then we take ψ(p, p0) = δ p0 +
p2
2m
g(p)
and g(p) = r1 exp
ıρp4
8βm2 −
ıαp3
3βm
+
ıp2τ
2mβ
−
ıx1p
β
, with the previous result we obtain:
ψ
x1,τ
|ψx,τ = |r1|
2 2mπβ
ı(τ − τ )
exp


ım(x1 − x1)2
2β(τ − τ )

 . (5)
This is the correct result.
Two constraints
The constraints for our model are 1 D1 = 1
2
[−(p1)2 + (p2)2 + (p3)2], D2 = − 1
2
[q1p1 +
q2p2 +q3p3], and {D1, D2} = D1. We take similar restrictions like our previous example, in this
way we define clocks all along the real line and the quantum operator associated with a complete
observable is self adjoint, with the restrictions {T1, D1} = 1, {T2, D2} = 1 and {T2, D1} = 0,
we have,
Det








{T1, T1} {T1, T2} {T1, D1} {T1, D2}
{T2, T1} {T2, T2} {T2, D1} {T2, D2}
{D1, T1} {D1, T2} {D1, D1} {D1, D2}
{D2, T1} {D2, T2} {D2, D1} {D2, D2}







 = 1,
and the general solutions for T1 and T2 are:
T1 = −
q1
p1
+ f p1, p2, q2 +
q1p2
p1
, p3, q3 +
q1p3
p1
,
T2 = 2 ln(p1) + g
p2
p1
,
p1
p2
(p1q2 + q1p2),
p3
p1
,
p1
p2
(−p3q2 + p2q3) , (6)
and our clocks are defined all along the real line.
Example 1
We can take T1 = −
q1
p1
, T2 = 2 ln(p1) as clocks and f = q2 as partial observable, then the
complete observable will be:
F = (q1p2 + q2p1) exp −
τ2
2
+
p2
p1
τ2 exp
τ2
2
, (7)
we consider ˆF ± ı ψ± = 0 and we obtain the solutions:
ψ± = p
ı exp(τ2)τ1
1 (p1 + p2)
exp
τ2
2 p
2
2 − p
2
1 g±(p3), (8)
then we have |ψ±|2 = |p1 + p2|
2 exp
τ2
2 p2
2 − p2
1
2
|g±(p3)|2 and we can choose g±(p3)
such that ψ+ and ψ− are square integrable; Dim[Ker( ˆF + ı)] = 1, Dim[Ker( ˆF − ı)] = 1 and
ˆF will be self adjoint.
Example 2
We use now T1 = −
q1
p1
and T2 = ln(p2) as clocks and f = q1 like partial observable, then
the complete observable will be:
F = −
p1
p2
τ1 exp
τ2
2
, (9)
we must solve the equations ˆF ± ı ψ± = 0 to determine if ˆF is self adjoint or it is not, we find
the solutions:
ψ± = exp
q1p2
τ1
exp −
τ2
2
g±(p3), (10)
and later we obtain |ψ±|2 = exp
2q1p2
τ1
exp −
τ2
2
|g±(p3)|2. We can choose g±(p3) such
that Dim(Ker( ˆF + ı)) = Dim(Ker( ˆF − ı)) = 0 and ˆF will be self adjoint.
Conclusions and perspectives
In the first example for the one dimensional non-relativistic free particle we found two
problems: our clock was not defined all along the real line and the operator associated with a
complete observable was not self adjoint, we correct the problem when we choose our clock like
a phase space function such that the Poisson bracket with the constraint is one.
For the case of two constraints, we begin with clocks that are conjugate to the set of con-
straints and we did not find problems with the operators associated with the complete observable.
For the one dimensional non-relativistic free particle we studied the partial observable f = x
with different choices for the clocks and we correct some problems for the system. However, if
we choose the partial observable f = x0, we can choose a clock such that {T, D} = 1 and ˆF is
not self adjoint, where F is a complete observable constructed for f = x0, we have made that
study, but is not presented in the present poster.
For the model with two constraints we studied some partial observables with different se-
lections for the clocks in such a way that Det




{Ti, Tj } {Ti, Dj }
{Di, Tj } {Di, Dj }



 = 1 and we
found that ˆF is self adjoint, where F is a complete observable constructed for a partial observ-
able f and the clocks are globally well defined. However, with a different choice for the previous
determinant, we have the problem that our clocks are not defined all along the real line and the
problem to the self adjointness for the operator associated with a complete observable is open.
The self adjoint character for operators associated with complete observables in field theory
is open and it must be studied.
The definition of clocks and self adjoint operators associated with complete observables or
gauge functions is open for complex systems, including general relativity.
References
[1] B. Dittrich, Gen. Rel. Grav. 39 (2007) 1891-1927,
[2] B. Dittrich and T. Thiemann, J. Math. Phys. 50, 012503 (2009),
[3] R. Gambini and R. A. Porto, Phys. Rev D, 63, 105014, (2001).
[4] F. Calogero and F. Leyvraz, J. Phys. A: Math. Theor. 40, (2007), 12931-12944.
1The phase space coordinates are (q1, q2, q3, p1, p2, p3).