Dirac's method for time-dependent Hamiltonian systems in the extended phase space

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by  Angel Garcia-Chung

Dirac’s method for time-dependent Hamiltonian systems in the extended phase space
Angel Garcia-Chung,∗ Daniel Guti´errez Ruiz,† and J. David Vergara‡
Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de M´exico,
Ciudad Universitaria, Ciudad de M´exico 04510, Mexico
(Dated: January 26, 2017)
The Dirac’s method for constrained systems is applied to the analysis of time-dependent Hamil-
toniansintheextendedphasespace. Ouranalysisprovidesaconceptuallycompletedescriptionand
offersadifferentpointofviewofearlierworks. WeshowthattheLewisinvariantisaDirac’sobserv-
able and in consequence, it is invariant under time-reparametrizations. We compute the Feynman
propagator using the extended phase space description and show that the quantum phase of the
Feynmanpropagatorisgivenbytheboundarytermofthecanonicaltransformationoftheextended
7
phasespace. Wealso give a new light tothe physical character of this phase.
1
0
2

n
a I. INTRODUCTION
J

5 Time-dependent Hamiltonian systems are broadly used in physics in both classical and quantum mechanics. A
2
particular feature of these systems is that its Hamiltonians are not conserved quantities and hence it makes difficult

the analytical study of the system, particularly its quantum description. However, this difficulty can be addressed
]
h usingvariousmethodsamongwhichstandsthemethodofexactinvariants[1–3]. Aparticularclassofexactinvariants
p is the well known Lewis invariant given in [4–6] and subsequently applied to the quantum time-dependent harmonic
-
oscillator and a charged particle in a time-dependent electromagnetic field [3].
h
t For a Hamiltonian of the form
a
m p2 ω(t)q2
H(q,p,t)= + , (1)
[ 2 2

1 wherepandqareconjugatecanonicalcoordinatesandω(t)isasmoothtime-dependentfrequency,Lewis[4,5]showed
v that
0
2 1 q2
1 I(q,p,t)= (ρp−ρ˙q)2+ , (2)
2 2ρ2
7
0
is an exact invariant, dI(q,p,t)=0, when the auxiliary function ρ(t) satisfies
. dt
1
0 1
ρ¨+ω2(t)ρ= . (3)
7 ρ3
1
:
v This result can be extended to more general potentials V(q,t), although only certain potentials admit such exact
i invariants [7]. The method used by Leach in [7] is an ad hoc derivation which finds explicitly all invariants that
X
are either linear or quadratic in the momentum p. Moreover, additional methods have been applied to attribute an
r invariant to time-dependent non-autonomous systems, e.g. [8–12].
a
A noteworthy example, is given by Leach [8], where it was shown that an exact invariant for the time-dependent
harmonic oscillator can be found by a two-step time-dependent linear transformation. This method was later gen-
eralized by Struckmeier and Riedel [11, 12] including three-dimensional time-dependent Hamiltonians. The two-step
transformation is given by a time-dependent canonical transformation first and secondly, by a specific time-scaling
transformation. The first transformation contains an arbitrary function, which is further related to the auxiliary
function ρ and gives rise to the auxiliary equation (3). The time-scaling, on the other hand, removes entirely the
explicit time-dependence of the Hamiltonian and leads to the Lewis invariant. In this way, time-dependent canonical
transformations,aswellastime-scalingtransformationsemergeinthe analysisofexactinvariantsfortime-dependent
Hamiltonian systems. Nonetheless, these transformations suffer restrictions in the context of time-dependent Hamil-
tonian mechanics [13] and some caveats are in order.
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
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Thefirstrestrictionisthattime-dependentcanonicaltransformationsmustpreservetime,thatistosay,theoriginal
andthedestinationsystemarealwayscorrelatedatthesameinstantoftheirrespectivetimescales[13,14]. Thesecond
restrictionis thattime-scaling transformationsarenotcanonicaltransformations[13,15]. Struckmeier[16,17]solved
this issue showing that the appropriate frame to study the exact invariants for a time-dependent Hamiltonian using
time-dependent canonical transformations and time-scaling transformations, is the extended phase space formalism.
The extended phase space is just an enlargement of the standard phase space where the time parameter t and
its conjugate momentum p are promoted as additional canonical variables of the system. As a consequence, the
t
symplectic group is also enlarged, admitting in this way the time-dependent transformations on the standard phase
space and time-scaling transformations as particular cases of symplectic maps.
Nevertheless, the analysis given in [16, 17] lacks of some results which might be useful for the quantization of such
systems. Forexample,itisnotclearwhichisthecommutationrelationtobeemployedinordertocanonicallyquantize
the system or whether there is a more general transformation to that used in [16] leading to the exact invariant of
the system. Thesequestions,amongothers,canbe naturallyansweredusingDirac’smethod forconstrainedsystems.
As we mentioned, the enlarging of the standard phase space to the extended phase space is achieved by adding two
more variables: t and p . These additional variables are not really physical ones. Dirac’s method is a procedure to
t
consistently remove these spurious degrees of freedom. It is broadly used in particle physics as well as in field theory
physics, inasmuch as it provides the tools to canonically quantize constrained systems [18, 19].
Another advantage of using Dirac’s method for time-dependent Hamiltonian systems in the extended phase space
is that it paves the way to the path integral analysis of these systems. This is an alternative route to the known
methods to determine the Feynmanpropagatorfortime-dependent Hamiltoniansystems [20,21]. These methods use
the two-step transformation to modify the measure DqDp in the Feynman propagator. In this case, it is possible
to consider non-canonical transformations because the variables q and p on each infinitesimal interval are not
j j
canonically conjugated variables [21]. The net effect of the first transformation is a factor depending exclusively in
the auxiliary function ρ. On the other hand, the time-scaling transformation does not modify the measure DqDp
and contributes to the amplitude with a phase. In this derivation, it is unclear why the time-scaling transformation
does not affect the measure DqDp. To attend this question, the Dirac’s method can be used in the path integral
description. In this case, the measure will be of the form DqDtDpDp , thus allowing modifications of time-scaling
t
transformations.
In this work we show that, in addition to the analysis of the exact invariants for time-dependent Hamiltonians in
the extended phase space, the Dirac’s method for constrained systems [18, 19] has also to be applied, in order to fill
the gaps of the analysis given by Struckmeier [16, 17]. We also obtain the Feynman propagator using the Dirac’s
method in the extended phase space analysis and study the change in the measure of the path integral amplitude.
This paper is organized as follows. In Section II, we briefly summarize the main aspects of the extended phase
spaceanalysisusing Dirac’smethod. SectionIII studies the canonicaltransformationandthe emergenceofthe Lewis
invariant as a Dirac observable. The Feynman propagator using the path integral analysis in the extended phase
space within the Dirac’s method is provided in Section IV. Finally, we discuss our results in Section V.
II. DIRAC’S METHOD IN THE EXTENDED PHASE SPACE
In this Section, we are going to derive the main relations of time-dependent systems using Dirac’s method within
the extended phase space. To begin with, consider the following action
dq(t) t2 m(t) dq 2
S q(t), = −V(q,t) dt, (4)
dt 2 dt
(cid:20) (cid:21) Zt1 " (cid:18) (cid:19) #
where the mass m(t) andthe time-dependent potential V(q,t) aresmoothfunctions of the time parametert. Despite
this time-dependency, the standard Hamiltonian analysis leads to the two-dimensional phase space with coordinates
(q,p) and Hamiltonian function
p2
H(q,p,t):= +V(q,t). (5)
2m(t)
Clearly, the Hamilton equations are explicitly time-dependent
dq p dp ∂V(q,t)
= , =− , (6)
dt m(t) dt ∂q
3
and the Hamiltonian is not a conserved quantity
dH ∂H m˙ p2 ∂V(q,t)
= =− + 6=0. (7)
dt ∂t m2 2 ∂t
(cid:18) (cid:19)
Our aim is to consider the time parameter t as an additional degree of freedom for the system described by (4) -
(7). To do so, we consider the arbitrary time-scaling transformation t = t˜(τ), where the parameter τ plays the role
of the new time parameter. The function t˜(τ) is so chosen that it gives a smooth one-to-one correspondence of the
domains of τ and t (see for instance [22]). This transformation changes the dependency of the coordinate function
q(t) thus the following definition q(t˜(τ))=:q˜(τ) is required. In this scheme, t is no longer the time but an additional
dynamical variable of the system and consequently, a new functional expression for the action S in (4), denoted by
S˜, emerges
dq˜ dt˜ τ2 m(t˜) q˜′ 2
S˜ q˜,t˜, , = −V(q˜,t˜) t˜′dτ, (8)
(cid:20) dτ dτ(cid:21) Zτ1 " 2 (cid:18)t˜′(cid:19) #
where the prime means derivative in τ. Notice that both, q˜and t˜, constitute the generalized configuration variables
onthis extended spacewhereasq˜′ andt˜′ correspondto their velocities. This implies thatinadditionto the boundary
conditionsfortheactionin(4)givenbyq˜(t )=q andq˜(t )=q ,twomoreboundaryconditionshavetobeconsidered
1 1 2 2
t˜(τ )=t and t˜(τ )=t .
1 1 2 2
Notice that if we consider a new reparametrizationτ =τ˜(σ), it gives rise to the following relation
S˜ q˜,t˜, dq˜, dt˜ =S˜ q˜˜,t˜˜, dq˜˜, dt˜˜ , (9)
dτ dτ dσ dσ
(cid:20) (cid:21) " #
whereq˜˜(σ):=q˜(τ˜(σ))andt˜˜(σ):=t˜(τ˜(σ)). ThisshowsthattheactionS˜isinvariantunderreparametrizations[22,23].
Naturally, the emergence of this symmetry (9) is a direct consequence of the enlarging of the dynamical degrees of
freedom and it will affect the Hamiltonian analysis as we will see further.
Consider now the Hessian matrix for (8) given as
H= ∂∂q2˜′S˜2 ∂∂q˜′2∂S˜t˜′ = m(t˜) t˜′2 −q˜′t˜′ , (10)
∂∂t˜′2∂S˜q˜′ ∂∂t2˜′S˜2 ! t˜′3 (cid:18)−q˜′t˜′ q˜′2 (cid:19)
and whose determinant is equal to zero. This Hessian matrix (10) is not invertible and consequently, the Euler-
Lagrange equation for t˜is not independent of the Euler-Lagrange equation for q˜. This implies that the extended
system in the variables t˜and q˜is a constrained system [18, 19]. Let us go over to the analysis of such system by first
considering the canonical conjugate momenta
δS m(t˜)q˜′
p = = , (11)
δq˜′ t˜′
δS m(t˜)q˜′2
p = =− +V(q˜,t˜) . (12)
t˜ δt˜′ 2 t˜′2
(cid:20) (cid:21)
Using Eq (11) we can express the derivative q˜′ in terms of the momentum p as
t˜′
q˜′ = p, (13)
m(t˜)
and inserting the former relation in the expression for the time momentum p given in (12) we obtain the primary
t˜
constraint
φ=p +H(q˜,p,t˜)≈0, (14)
t˜
where H(q˜,p,t˜) is the Hamiltonian givenin (5). The weak equality symbol ≈ stands for the following reason: we can
notice that the Poisson bracket {B ,B } of two arbitrary functions B (q˜,t˜,p,p ) and B (q˜,t˜,p,p ) on the extended
1 2 1 t˜ 2 t˜
phase spaceevaluatedonthe constraintsurface φ=0is differentto thatofthe Poissonbracketofthe same functions
but first evaluated on the constraint B (q˜,t˜,p,−H) and B (q˜,t˜,p,−H), that is to say
1 2
{B ,B } 6={B ,B }. (15)
1 2 φ=0 1φ=0 2φ=0
4
To avoid this contradiction, we take as a rule to first work out the Poisson brackets before we make use of the
constraint equation φ = 0. Therefore, the weak equality sign stands in order to remind us this rule [18, 19]. Notice
that the relation (14), gives rise to the Schr¨odinger equation once the coordinates t˜, q˜ and p, p are promoted to
t˜
operators in a Hilbert space as a result of the quantum canonical formalism.
The canonical Hamiltonian takes the form
m(t˜) q˜′ 2
H =pq˜′+p t˜′−L=t˜′φ, where L:= −V(q˜,t˜) t˜′, (16)
c t˜ " 2 (cid:18)t˜′(cid:19) #
hencethetotalHamiltonianH isproportionaltotheconstraintφandthecoefficientofproportionalityisaLagrange
T
multiplier denoted by λ
H =λφ. (17)
T
The Lagrange multiplier λ is a function of the parameter τ and it is independent of the phase space points. This
kind of Lagrange multipliers is referred to as non-canonical gauge [19]. This particular case of constrained system in
whichthe totalHamiltonianisnullwhenthe constraintisstronglyzeroareusuallycalledreparametrizationinvariant
systems. The free relativistic particle or the canonical formulation of the General Relativity are examples of such
systems [22].
ThePoissonbracketfortwoarbitrarysmoothfunctions B andB inthis extendedphasespacetakesthefollowing
1 2
form
∂B ∂B ∂B ∂B ∂B ∂B ∂B ∂B
1 2 1 2 1 2 1 2
{B ,B } = + − − . (18)
1 2 PB ∂q˜ ∂p ∂t˜ ∂pt˜ ∂p ∂q˜ ∂pt˜ ∂t˜
The Hamilton equations, derived with this Poisson bracket and Hamiltonian H , are given as
T
p ∂V ∂H m˙ p2 ∂V
q˜′ =λ , p′ =−λ , t˜′ =λ, p′ =−λ =λ − . (19)
m(t˜) ∂q˜ t˜ ∂t˜ m2 2 ∂t˜
(cid:20) (cid:21)
As it is expected, there is no equation for the Lagrange multiplier λ and in consequence, two different Lagrange
multipliers induce different evolution of the same initial point on the extended phase space.
The process in which a value for the Lagrange multiplier λ is fixed is usually called ‘fixing the gauge’ [24, 25]. For
instance,the mostcommongaugefixingisthe caseinwhichλ= (t2−t1). This gaugesolvesthe thirdequationin(19)
(τ2−τ1)
whose solution is
(t −t )(τ −τ )
t˜= 2 1 1 +t , (20)
1
(τ −τ )
2 1
and notice that the conditions t˜(τ )=t with i=1,2 hold. The first two equations in (19) give rise to the Hamilton
i i
equations (6) and turns the fourth equation into (7), if we make the following identification p′ = −∂H. As a result,
t ∂t
we recover the system given by (6) with Hamiltonian (5) and energy evolution (7).
However, this procedure, although quite direct and simple, does not shows the Poisson bracket structure for the
resulting physical degrees of freedom that might appear, for instance, in more complicated systems. In order to do
so, the gauge fixing condition is promoted as an additional constraint surface η ≈0 and such that {φ,η}6≈0. Under
this premise, the Poisson bracket (18) is replaced by the so called Dirac bracket which, in this case, is given by
1
{B ,B } ={B ,B } + [{B ,φ} {η,B } −{B ,η} {φ,B } ]. (21)
1 2 DB 1 2 PB 1 PB 2 PB 1 PB 2 PB
{φ,η}
PB
In the present case, the constraint
(t −t )(τ −τ )
η =t˜− 2 1 1 −t ≈0, (22)
1
(τ −τ )
2 1
is tantamount as to fixing the gauge λ= (t2−t1) and it gives the following non-null Dirac brackets
(τ2−τ1)
∂φ p ∂φ ∂V
{q˜,p} =1, {q˜,p } =− =− , {p,p } = = . (23)
DB t˜ DB ∂p m t˜ DB ∂q˜ ∂q˜
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It can be seen that the last two relations differ from their similar in terms of the Poisson brackets {q˜,p } =
t˜ PB
{p,p } = 0. Notably, the Dirac bracket {t˜,p } = 0 differs from its Poisson bracket pair {t˜,p } = 1. In this
t˜ PB t˜ DB t˜ PB
way,to addanother constraintη ≈0, allowsto define areducedphase spacewhichis the subspace ofthe phase space
(q˜,t˜,p,p ) such that the constraints are fulfilled, i.e., φ = 0 and η = 0. It is in this reduced phase space where the
t˜
dynamic of the system takes place and it is generated by the Hamiltonian H together with the Dirac brackets (23).
We can see this by first selecting the physical degrees of freedom, which in this case, are trivially chosen due to the
constraints yield t˜= (t2−t2)(τ−τ1) +t and p = −H(τ,q˜,p). Hence the coordinates q˜and p can be selected as the
(τ2−τ1) 1 t˜
physical degrees of freedom using τ as the time parameter or instead we can use q and p with t as time parameter in
accordance with the initial description.
To conclude this brief analysis, consider the Hamiltonian form of the action (8)
τ2
S˜= pq˜′+p t˜′−λφ dτ, (24)
t˜
Zτ1
(cid:2) (cid:3)
and let us evaluate (24) on the constraints φ=0 and η =0. As a result, we will obtain the action
t2
S˜= [pq˙−H(q,p,t)] dt, (25)
Zt1
which is the standard Hamiltonian form of the action (4).
Summarizing,wehaveshownthattheDiracformalismforthesystemgivenbytheaction(4)intheextendedphase
space, gives rise to the standard Hamiltonian analysis with Hamiltonian H and coordinates (q,p). In this formalism
the time variable t is promoted as an additional dynamical variable t˜. Its conjugate momentum results from the
consideration of the extended action given in (8). The converse procedure, namely, removing these extra degrees of
freedom, is accomplished by the gauge fixing process or by the introduction of an additional constraint η ≈0. These
steps are a result of the implementation of the Dirac’s method. It is worth mentioning that, the gauge fixing process
is mainly used within the path integral formalism, while the additional constraint procedure, in order to derive the
Dirac brackets, is commonly used in the canonical quantization scheme.
As we mentioned, a direct implication of the aforementioned enlarging of the phase space is that it also enlarges
the symplectic group of the system. Having this in mind, in the next section we are going to study a canonical
transformation of the extended phase space such that the final dynamical description of the reduced phase space is
no longer time-dependent. To achieve this, we will use the new energy conservation law as an auxiliary equation for
the arbitrary coefficient in the canonical map. This canonical transformation is a generalization of the Struckmeier
transformation given in [12]. Moreover, we will show that this transformation gives rise to a boundary term which
can be related to the Lewis phase in the path integral quantization.
III. CANONICAL TRANSFORMATION IN THE EXTENDED PHASE SPACE
Recallthatacanonicaltransformationinthe2n−dimensionalsymplecticspace(Γ(2n),ω)isamapM:Γ(2n) →Γ(2n)
such that the symplectic two form ω is preserved [14]. This definition is mathematically expressed as
MT JM=J, (26)
where J is the complex structure map J2 =−1 which in matrix form reads as
0 1
J = n×n . (27)
−1 0
n×n
(cid:18) (cid:19)
The Leach-Struckmeier transformation [8, 12] consists of two transformations in the standard phase space with
coordinates(q,p)oneofwhichisacanonicaltransformationandthesecondisatime scaling,whichisnotacanonical
transformation[15]. However,as a resultof the enlargingof the symplectic group,the time scaling canbe considered
as a canonical transformation in the extended phase space.
Our goal is to consider the more general canonical transformation in the extended phase space containing the
Struckmeier transformation. Of course, in this case, time scaling can be considered as a part of the canonical
transformation together with a given variation for the momentum of the time variable p .
t˜
Let us consider a coordinate transformation of the form
q˜ A(Q,T)
t˜ B(Q,T)
= , (28)
 p  C(Q,T)P +D(Q,T)
p G(Q,T,P,P )
 t˜  T 
   
6
where the coefficients A(Q,T), B(Q,T), C(Q,T), D(Q,T) together with G(Q,T,P,P ) are smooth functions to be
T
determined in order to make (28) canonical. We take the variables q˜ and t˜to be momenta independent to avoid
more sophisticated dependencies of the new function K(Q,P,T) := H(q˜(Q,T),˜t(Q,T),p(Q,T,P)) in terms of the
new momentum P. For the same reason, we select a linear relation between the momenta p and P to preserve the
square order of the kinetic term on K. We remark that this transformation (28) is ‘time independent’ since τ is the
new ‘time’ variable. Furthermore, notice that the momentum p is also P -independent to avoid a term of the form
T
P2 which, after the gauge fixing procedure, gives rise to negative energy solutions.
T
The canonical transformation matrix resulting from (28) is given as
∂∂Qq˜ ∂∂Tq˜ ∂∂Pq˜ ∂∂Pq˜T A′ A˙ 0 0
∂t˜ ∂t˜ ∂t˜ ∂t˜ B′ B˙ 0 0
M= ∂∂Qp ∂∂Tp ∂∂Pp ∂∂PpT = C′P +D′ C˙P +D˙ C 0 . (29)
 ∂∂∂pQQt ∂∂∂pTTt ∂∂∂pPPt ∂∂∂PPpTTt   ∂∂GQ ∂∂GT ∂∂PG ∂∂PGT 
Henceforth, the prime and the dot symbols denote the operators ∂ and ∂ respectively. In order to make (28)
∂Q ∂T
canonical, M must satisfies the condition (26). When inserting (29) into (26) we obtain the following system of
partial differential equations
A˙(C′P +D′)+B˙G′ = A′(C˙P +D˙)+B′G˙, (30)
∂G
A˙C+B˙ = 0, (31)
∂P
∂G
B′ +A′C = 1, (32)
∂P
∂G
B˙ = 1, (33)
∂P
T
∂G
B′ = 0, (34)
∂P
T
whose general solution is
1 P A˙P 1
t˜=B =B(T), C = , p =G= T − + A′D˙ −A˙D′ dQ. (35)
A′ t˜ B˙ A′B˙ B˙
Z (cid:16) (cid:17)
Consequently, the old coordinates (q˜,t˜,p,p ) can be written in terms of the new coordinates as
t
A(Q,T)
q˜
t˜ B(T)
 p = A1′P +D(Q,T) . (36)
pt˜  PB˙T − AA˙′PB˙ + B1˙ A′D˙ −A˙D′ dQ 
 
 R (cid:16) (cid:17) 
Notably, the momentum p results to be a linear function in terms of the momenta P and P, i.e., a contact trans-
t˜ T
formation. On the other hand, notice that the functions B(T), A(Q,T) as well as D(Q,T) are arbitrary so far. We
will see further that additional conditions to the new Hamiltonian are necessary to remove its time-dependency and
that these conditions can be used to fix these coefficients via some differential equations.
The constraint φ can be written in terms of the new variables after inserting (36) in (14) as
1 B˙ P2 A˙ B˙ D B˙ D2
φ= P − − + − P − A′D˙ −A˙D′ dQ− −B˙ V , (37)
B˙ ( T " 2mA′2! A′ mA′! 2m #)
Z (cid:16) (cid:17)
and notice that the constraint is naturally split into two parts. The first is the linear term P and the second one is
T
the expression inside the square brackets. The linear term is connected with the fact that p is linear in P and that
t˜ T
p is P -independent. Furthermore, due to this linear relation, we select the new momentum P as a non-physical
T T
degree of freedom, once we solve the constraint. The second term in (37), the one inside the square brackets, will
contribute to the new Hamiltonian function and to a boundary term associated with this canonical transformation.
7
The equations of motion in these variables take the form
dQ λ B˙ B˙ D A˙
={Q,λφ}= P + − , (38)
dτ B˙ " mA′2! mA′ A′!#
dP λ ∂ B˙ ∂ B˙ D A˙ ∂ B˙D2
={P,λφ}=− P2+ − P + A′D˙ −A˙D′ + +B˙V , (39)
dτ B˙ "∂Q 2mA′2! ∂Q mA′ A′! ∂Q 2m !#
(cid:16) (cid:17)
dT λ
={T,λφ}= , (40)
dτ B˙
dP λ ∂ B˙ ∂ B˙ D A˙ ∂ B˙D2
T ={P ,λφ}=− P2+ − P + dQ(A′D˙ −A˙D′)+ +B˙V .
dτ T B˙ (∂T 2mA′2! ∂T mA′ A′! ∂T " 2m #)
Z
(41)
In the last equation we made the constraint strongly equal to zero in order to remove from the final expression the
term proportionalto the constraint−λ ∂ 1 φ. We now look upon the gauge fixing process and solvethe equation
∂T B˙
(40). In this case, we have to consider the(cid:16)con(cid:17)ditions T(τ1)=T1 and T(τ2)=T2 and one simple solution is given by
(T −T )(τ −τ )
2 1 1
T = +T , (42)
1
(τ −τ )
2 1
which implies that λ= (T2−T1)B˙ in which case the previous equations read as
(τ2−τ1)
dQ B˙ B˙ D A˙
= P + − , (43)
dT mA′2 mA′ A′
! !
dP ∂ B˙ ∂ B˙ D A˙ ∂ B˙D2
=− P2+ − P + A′D˙ −A˙D′ + +B˙V , (44)
dT ∂Q 2mA′2 ∂Q mA′ A′ ∂Q 2m
" ! ! !#
(cid:16) (cid:17)
dP ∂ B˙ ∂ B˙ D A˙ ∂ B˙D2
T =− P2+ − P + dQ(A′D˙ −A˙D′)+ +B˙V . (45)
dT ∂T 2mA′2 ∂T mA′ A′ ∂T 2m
( ! ! " #)
Z
The first two equations are the Hamilton equations once Q and P are fixed as the physical degrees of freedom. In
consequence, the last equation provides the energy conservation law for this system in the new coordinates (Q,P).
As we already know, the system in the reduced phase space is not conservative due to H(q,p,t) is explicitly time-
dependent (see Eqs (19)). Nevertheless, using this last equation(45), the system will be a conservativesystem if and
only if
dP
T
=0. (46)
dT
This condition must hold on the full reduced phase space (Q,P) hence, the right hand side of (45) results in the
following equations
∂ B˙ ∂ B˙ D A˙ ∂ B˙D2
=0, − =0, dQ(A′D˙ −A˙D′)+ +B˙V =0. (47)
∂T 2mA′2 ∂T mA′ A′ ∂T 2m
! ! " #
Z
The first equation can be easily solved for B˙ as
∂B m(B(T)) ∂A 2
B˙(T)= = , (48)
∂T m ∂Q
0 (cid:18) (cid:19)
where 1/m is the integration constant. Due to B is Q-independent then A(Q,T) must be of the form
0
A(Q,T)=A˜(T)Q, (49)
8
whereA˜(T)isasmoothfunctionofT tobefixedandweassumethatm isQ-independent. Ofcourse,amoregeneral
0
solutionimplies toconsiderm =m (Q) butthis results ina Q-dependentmasstermforthe new Hamiltonianwhich
0 0
exceeds the purpose of the present work. We now combine (48) with (49) and obtain
1 dt
t=B(T):=B˜−1 A˜2(T)dT , where B˜(t):= . (50)
m m(t)
(cid:18) 0 Z (cid:19) Z
Inserting (48), (49) and (50) in the second equation of (47) we obtain the expressionfor D(Q,T) in terms of A˜(T)
as
m A˜˙
0
D(Q,T)= κ(Q)+ Q , (51)
A˜ " A˜ #
where κ(Q) is the integration function (T-independent) of the second equation in (47). In order to fix A˜(T), we use
the third equation in (47) which, after inserting the former definitions, takes the form
∂ m Q2 A¨˜ 2A˜˙2 m(T)A˜2
0 − + V(q˜(Q,T),˜t(T)) =0. (52)
∂T " 2 A˜ A˜2 ! m0 #
This is the equation for A˜(T) in terms of m(T) and V(q˜(Q,T),˜t(T)). As can be seen, for a general potential
V(A(Q,T),B(T)) this equation is not necessarily Q-independent. This is an inconvenient result implying that only
potentials of the form V(q˜,t˜) ∼ q˜2 give rise to an equation (52) independent of the coordinate Q [12]. The term
proportionalto Q2 is the contribution of the canonical transformation in addition to the term given by the potential
V(q˜(Q,T),˜t(T)). To solve equation (52) for an arbitrary potential V(q˜,t˜) the equations (43) and (44) have to be
considered. An alternative analysis implies to consider m as a function of Q and such that the Q-dependency is
0
removed from (52) but again, this only works for particular cases of the potential V.
Once we obtain the expression for A˜(T), we integrate (52) and use (50) and (51) to obtain
m κ(Q)2 m Q2 A¨˜ 2A˜˙2 m(T)A˜2
0 + 0 − + V(q˜(Q,T),˜t(T))=:V(Q), (53)
2 2 A˜ A˜2 ! m0
where the function V(Q) is the time-independent potential bringing about the dynamics of the physical degrees of
freedom. This new potential leads to the new Hamiltonequations as we cansee by inserting (49), (50), (51) and (53)
into (43) and (44)
dQ 1 dP ∂κ(Q) ∂V(Q)
= P +κ(Q), =− P − , (54)
dT m dT ∂Q ∂Q
0
which is, clearly, an autonomous Hamiltonian system. To derive the Hamiltonian function giving rise to these equa-
tions, let us consider the Hamiltonian form of the action (24) in the new variables
τ2 dQ dT
S = (P +A′D) + P +A˙D+ A′D˙ −A˙D′ dQ −λφ dτ, (55)
T
dτ dτ
Zτ1 (cid:26) (cid:20) Z (cid:16) (cid:17) (cid:21) (cid:27)
where φ is given in (37) and recall that the Lagrange multiplier given by Eq (40) is λ = (T2−T1)B˙. Solving the
(τ2−τ1)
constraint (37) for the momentum P and inserting the result in (55), gives
T
T2 dQ d T2 dQ Q(T2) T2
S = P − I(Q,P)− dQA′D dT = P −I(Q,P) dT − dQA′D , (56)
dT dT dT
ZT1 (cid:26) (cid:20) (cid:18)Z (cid:19)(cid:21)(cid:27) ZT1 (cid:26) (cid:27) "ZQ(T1) #T1
where
P2
I(Q,P):= +κ(Q)P +V(Q), (57)
2m
0
is the new Hamiltonian and the last term in (56) is the boundary term associatedwith the canonicaltransformation.
9
The boundary term given in (56) will play a major role in the quantum description of the system, particularly in
the path integral analysis. The Hamiltonian (57), on the other hand, is the Lewis invariant given by Struckmeier et
al[12] when it is written in terms of the old coordinates(q,p,t), of course, I(Q,P)is time-independent whenever the
A˜ is a solution of (52) and B(T) and D(Q,T) are given by (50) and (51) respectively. Moreover, I(Q,P) is a gauge
invariant observable, that is to say, it commutes with the constraint {I,φ} = 0. The significance of this result
PB
is that I(Q,P) is not only time-independent but gauge independent. Due to the gauge symmetry in this system is
a time-reparametrization symmetry, the Lewis invariant is nothing but a reparametrization invariant: we can use a
different gauge for λ and the result will be the same I(Q,P).
On the other hand, there is another way to derive the Hamiltonian system in the reduced phase space using (55).
Inserting the expressions for A′D and A˙D+ (A′D˙ −A˙D′)dQ in (55) we obtain
τ2 RQA˜˙ dQ m Q2 ∂ A˜˙ dT
0
S = P +m κ(Q)+ + P + −λφ dτ, (58)
Zτ1 (" 0 A˜ !# dτ " T 2 ∂T A˜!# dτ )
where as can be notice, the boundary term is now modifying our further definition of the momenta in the extended
phase space. This can be seen by considering
δS QA˜˙ ∂
P˜ := =P +m κ(Q)+ =P + dQA′D , (59)
δddQτ 0 A˜ ! ∂Q(cid:18)Z (cid:19)
δS m Q2 ∂ A˜˙ ∂
P˜ := =P + 0 =P + dQA′D . (60)
T δddTτ T 2 ∂T A˜! T ∂T (cid:18)Z (cid:19)
Let us replace P and P in the action (58) by P˜ and P˜ using the previous expressions
T T
τ2 dQ dT ∂F(Q,T) ∂F(Q,T)
S = P˜ +P˜ −λ P˜ − +I Q,P˜− dτ, (61)
T T
dτ dτ ∂T ∂Q
Zτ1 (cid:26) (cid:20) (cid:18) (cid:19)(cid:21)(cid:27)
where the function F(Q,T) is the boundary term given by
A˜˙
F(Q,T)= dQA′D =m κ(Q)+ Q dQ. (62)
0 " A˜! #
Z Z
The explicit expression for the action (61) is different to that given in the expression (55). Let us now solve the
constraint in (61) using the same gauge as before
T2 dQ ∂F(Q,T) ∂F(Q,T)
S = P˜ −I˜(Q,P˜,T) dT, where, I˜(Q,P˜,T)=I Q,P˜− − (63)
dT ∂Q ∂T
ZT1 (cid:26) (cid:27) (cid:18) (cid:19)
isanewHamiltonianfunction. Ascanbeseen,theHamiltonianI˜(Q,P˜,T)isatime-dependentHamiltonianduetothe
boundary term F(Q,T) can be a time-dependent function. Nevertheless, it is worth to mention that both systems,
the one described by (Q,P) together with the Lewis invariant I(Q,P) and the new system described by (Q,P˜)
together with the Hamiltonian I˜(Q,P˜,T) are both canonically related. The relations (59) and (60) are canonical
transformations in the extended phase space. Moreover, the transformation (59) is also a canonical transformation
between the reduced phase spaces given by the degrees of freedom (Q,P) and (Q,P˜). Of course, in this case, (59) is
a time-dependent canonical transformation. Due to I˜is a time-dependent Hamiltonian, we are going to work with
I(Q,P) instead of I˜(Q,P˜,T).
As aremarkrelatedwiththe boundaryterm(62), notice thatthe infinitesimalactionofthe constraintonF(Q,T),
combined with the equations of motion, gives rise to a total derivative in the extended phase space
∂F(Q,T) ∂φ ∂F(Q,T) ∂φ ∂F(Q,T) ∂F(Q,T) dF(Q,T)
δ F(Q,T)={F(Q,T),λφ}=λ +λ = Q˙ + T˙ = , (64)
φ
∂Q ∂P ∂T ∂P ∂Q ∂T dτ
T
where the equations (38) and (40) were used. This total derivative is null only in case that F(Q,T) is constant in τ.
Let us now calculate the Dirac brackets in the new variables Q, T, P, P which can be obtained using the general
T
expression
1
{B ,B } :={B ,B } + ({B ,φ} {η,B } −{B ,η} {φ,B } ), (65)
1 2 DB 1 2 PB 1 PB 2 PB 1 PB 2 PB
{φ,η}
PB
10
where η :=T − (T2−T1)(τ−τ1) −T ≈0 and φ is givenin (37). Notice that {φ,η} =−1/B˙ and as a result, the only
(τ2−τ1) 1 PB
non-null Dirac brackets are
∂φ ∂φ
{Q,P} =1, {Q,P } =−B˙ , {P,P } =B˙ . (66)
DB T DB T DB
∂P ∂Q
(cid:18) (cid:19) (cid:18) (cid:19)
From these relations we can conclude that, although the transformation (28) is canonical, it does not preserve the
Dirac brackets. Therefore, the Hamilton equations for the physical degrees of freedom q and p are different to the
Hamilton equations for the new degrees of freedom Q and P. In others words, the canonical transformation (28) in
the extended phase space is not canonical when it is restricted to the reduced phase space formed with the physical
degrees of freedom.
Let us conclude this sectionby calculating the Generating function F =q˜p+t˜p +F (p,p ,Q,T)relatedwith this
t˜ 3 t˜
canonical transformation. In this case, F is now given by
3
F (p,p ,Q,T)=−B(T)p −A(Q,T)p+ A′(Q,T)D(Q,T)dQ, (67)
3 t˜ t˜
Z
where B(T), D(Q,T) and A(Q,T) are given in (50), (51) and (49) and (52) respectively. This Generating function
will be used in the path integral analysis on the next section.
Let us summarize up to this point. We begin with the standard description of the system in the phase space
with coordinates (q,p). Due to the Hamiltonian H(q,p,t) is time-dependent, we enlarge the phase space by adding
two additional degrees of freedom, t˜and p . This enlarging of the phase space allows us to consider a canonical
t˜
transformation which is not only time-dependent but also p -dependent. The aim is to transfer the time-dependency
t˜
oftheHamiltonequationsintothecoefficientsofthiscanonicaltransformationrenderingthefinalHamiltonequations
time-independent. As a result, we obtain three auxiliary equations given in (47). The first two of them can be easily
solvedwhile the thirdgivesriseto the equation(52),whichisageneralizationofthe auxiliaryequation(3). Thefinal
outcomes of this procedure are the Hamilton equations given by (54) with a time-independent Hamiltonian given by
theLewisinvariant(57). Ofcourse,thisisthemainresultofthissectiontogetherwiththederivationoftheboundary
term in (56). Notably, the Lewis invariant I(Q,P) is also the gauge invariant observable. A different election for
the gauge (another reparametrization) gives the same invariant. The boundary term is absent in previous classical
analysis given by Struckmeier et al. [12] and we argue that it plays an important role in the quantization of such
systems as we will see in the next section. Additionally, we obtained a Hamiltonian system described by I˜(Q,P˜,T)
which, although is a time-dependent Hamiltonian, is canonically related with the Lewis invariant system.
IV. PATH INTEGRAL ANALYSIS
InthepreviousSection,weattendedtheconstrainedHamiltoniananalysisofthetime-dependentHamiltoniangiven
in (5) using Dirac method for constrained systems. In this section, we will study the path integral formulation of
this system using the action given in (4). In this case, the extended phase space formalism, allows us to consider
the transformation (36) as a canonical transformation on each of the infinitesimal intervals in which the quantum
extended phase space is split. It will be explained, in this way, that the boundary term given in (56) is the result of
this transformation and that it is a generalizationof the term reported in [21].
Consider the amplitude given by
hqf,tf|qi,tii= DqDpe~i Rttifdt[pq˙−H(q,p,t)], (68)
Z
which is the formalexpressionof the path integralform ofthe Feynman propagatorand the Hamiltonian H(q,p,t)is
given in (5). The infinitesimal expression of this amplitude takes the form
hqf,tf|qi,tii=Nl→im∞ N−1dqj N 2dπp~j e~i PNj=1ǫhpj(cid:16)qj−qǫj−1(cid:17)−H(cid:16)qj+q2j−1,pj,tj+t2j−1(cid:17)i. (69)
Z j=1 j=1
Y Y
The expression (69) can be obtained from the path integral formulation in the extended phase space using the
amplitude given by
hq˜f,t˜f,τf|q˜i,t˜i,τii(0) = Dq˜DpDt˜Dpt˜δ(t˜−g(τ))δ(pt˜+H)e~i Rττif dτ[pq˜′+pt˜t˜′−λ(pt˜+H(q˜,p,t˜))], (70)
Z