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

The Dirac's method for constrained systems is applied to the analysis of time-dependent Hamiltonians in the extended phase space. Our analysis provides a conceptually complete description and offers a different point of view of earlier works. We show that the Lewis invariant is a Dirac's observable 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 Feynman propagator is given by the boundary term of the canonical transformation of the extended phase space. We also give a new light to the physical character of this phase.


I. INTRODUCTION
Time-dependent Hamiltonian systems are broadly used in physics in both classical and quantum mechanics. A 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 using various methods among which stands the method of exact invariants [1][2][3]. A particular class of exact invariants is the well known Lewis invariant given in [4][5][6] and subsequently applied to the quantum time-dependent harmonic oscillator and a charged particle in a time-dependent electromagnetic field [3].
For a Hamiltonian of the form where p and q are conjugate canonical coordinates and ω(t) is a smooth time-dependent frequency, Lewis [4,5] showed that is an exact invariant, d dt I(q, p, t) = 0, when the auxiliary function ρ(t) satisfies This result can be extended to more general potentials V (q, t), although only certain potentials admit such exact invariants [7]. The method used by Leach in [7] is an ad hoc derivation which finds explicitly all invariants that are either linear or quadratic in the momentum p. Moreover, additional methods have been applied to attribute an invariant to time-dependent non-autonomous systems, e.g. [8][9][10][11][12].
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 generalized 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, as well as time-scaling transformations emerge in the analysis of exact invariants for time-dependent Hamiltonian systems. Nonetheless, these transformations suffer restrictions in the context of time-dependent Hamiltonian mechanics [13] and some caveats are in order.
The first restriction is that time-dependent canonical transformations must preserve time, that is to say, the original and the destination system are always correlated at the same instant of their respective time scales [13,14]. The second restriction is that time-scaling transformations are not canonical transformations [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 t are promoted as additional canonical variables of the system. As a consequence, the 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. For example, it is not clear which is the commutation relation to be employed in order to canonically quantize the system or whether there is a more general transformation to that used in [16] leading to the exact invariant of the system. These questions, among others, can be naturally answered using Dirac's method for constrained systems. 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 t . These additional variables are not really physical ones. Dirac's method is a procedure to 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 Feynman propagator for time-dependent Hamiltonian systems [20,21]. These methods use the two-step transformation to modify the measure Dq Dp in the Feynman propagator. In this case, it is possible to consider non-canonical transformations because the variables q j and p j on each infinitesimal interval are not 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 Dq Dp and contributes to the amplitude with a phase. In this derivation, it is unclear why the time-scaling transformation does not affect the measure Dq Dp. 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 Dq Dt Dp Dp t , thus allowing modifications of time-scaling 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 space analysis using Dirac's method. Section III studies the canonical transformation and the emergence of the 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 where the mass m(t) and the time-dependent potential V (q, t) are smooth functions of the time parameter t. Despite this time-dependency, the standard Hamiltonian analysis leads to the two-dimensional phase space with coordinates (q, p) and Hamiltonian function Clearly, the Hamilton equations are explicitly time-dependent and the Hamiltonian is not a conserved quantity 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 functiont(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 bỹ S, emergesS where the prime means derivative in τ . Notice that both,q andt, constitute the generalized configuration variables on this extended space whereasq ′ andt ′ correspond to their velocities. This implies that in addition to the boundary conditions for the action in (4) given byq(t 1 ) = q 1 andq(t 2 ) = q 2 , two more boundary conditions have to be considered Notice that if we consider a new reparametrization τ =τ (σ), it gives rise to the following relatioñ whereq(σ) :=q(τ (σ)) andt(σ) :=t(τ (σ)). This shows that the actionS is invariant under reparametrizations [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 and whose determinant is equal to zero. This Hessian matrix (10) is not invertible and consequently, the Euler-Lagrange equation fort is not independent of the Euler-Lagrange equation forq. This implies that the extended system in the variablest andq is a constrained system [18,19]. Let us go over to the analysis of such system by first considering the canonical conjugate momenta Using Eq (11) we can express the derivativeq ′ in terms of the momentum p as and inserting the former relation in the expression for the time momentum pt given in (12) we obtain the primary constraint where H(q, p,t) is the Hamiltonian given in (5). The weak equality symbol ≈ stands for the following reason: we can notice that the Poisson bracket {B 1 , B 2 } of two arbitrary functions B 1 (q,t, p, pt) and B 2 (q,t, p, pt) on the extended phase space evaluated on the constraint surface φ = 0 is different to that of the Poisson bracket of the same functions but first evaluated on the constraint B 1 (q,t, p, −H) and B 2 (q,t, p, −H), that is to say 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 Schrödinger equation once the coordinatest,q and p, pt are promoted to operators in a Hilbert space as a result of the quantum canonical formalism. The canonical Hamiltonian takes the form hence the total Hamiltonian H T is proportional to the constraint φ and the coefficient of proportionality is a Lagrange multiplier denoted by λ 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 which the total Hamiltonian is null when the constraint is strongly zero are usually called reparametrization invariant systems. The free relativistic particle or the canonical formulation of the General Relativity are examples of such systems [22]. The Poisson bracket for two arbitrary smooth functions B 1 and B 2 in this extended phase space takes the following form The Hamilton equations, derived with this Poisson bracket and Hamiltonian H T , are given as 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 most common gauge fixing is the case in which λ = (t2−t1) (τ2−τ1) . This gauge solves the third equation in (19) whose solution ist and notice that the conditionst(τ i ) = t i with i = 1, 2 hold. The first two equations in (19) give rise to the Hamilton equations (6) and turns the fourth equation into (7), if we make the following identification p ′ t = − ∂H ∂t . As a result, 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} ≈ 0. Under this premise, the Poisson bracket (18) is replaced by the so called Dirac bracket which, in this case, is given by In the present case, the constraint is tantamount as to fixing the gauge λ = (t2−t1) (τ2−τ1) and it gives the following non-null Dirac brackets It can be seen that the last two relations differ from their similar in terms of the Poisson brackets {q, pt} P B = {p, pt} P B = 0. Notably, the Dirac bracket {t, pt} DB = 0 differs from its Poisson bracket pair {t, pt} P B = 1. In this way, to add another constraint η ≈ 0, allows to define a reduced phase space which is the subspace of the phase space (q,t, p, pt) such that the constraints are fulfilled, i.e., φ = 0 and η = 0. It is in this reduced phase space where the 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 yieldt = (t2−t2)(τ −τ1) (τ2−τ1) + t 1 and pt = −H(τ,q, p). Hence the coordinatesq and p can be selected as the 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) and let us evaluate (24) on the constraints φ = 0 and η = 0. As a result, we will obtain the actioñ which is the standard Hamiltonian form of the action (4). Summarizing, we have shown that the Dirac formalism for the system given by the action (4) in the extended phase 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 variablet. 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
Recall that a canonical transformation in the 2n−dimensional symplectic space (Γ (2n) , ω) is a map M : Γ (2n) → Γ (2n) such that the symplectic two form ω is preserved [14]. This definition is mathematically expressed as where J is the complex structure map J 2 = −1 which in matrix form reads as The Leach-Struckmeier transformation [8,12] consists of two transformations in the standard phase space with coordinates (q, p) one of which is a canonical transformation and the second is a time scaling, which is not a canonical transformation [15]. However, as a result of the enlarging of the symplectic group, the time scaling can be 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 pt.
Let us consider a coordinate transformation of the form  where the coefficients A(Q, T ), B(Q, T ), C(Q, T ), D(Q, T ) together with G(Q, T, P, P T ) are smooth functions to be determined in order to make (28) canonical. We take the variablesq andt 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 T -independent to avoid a term of the form P 2 T which, after the gauge fixing procedure, gives rise to negative energy solutions. The canonical transformation matrix resulting from (28) is given as Henceforth, the prime and the dot symbols denote the operators ∂ ∂Q and ∂ ∂T respectively. In order to make (28) canonical, M must satisfies the condition (26). When inserting (29) into (26) we obtain the following system of partial differential equationsȦ whose general solution is Consequently, the old coordinates (q,t, p, p t ) can be written in terms of the new coordinates as Notably, the momentum pt results to be a linear function in terms of the momenta P T and P , i.e., a contact transformation. 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 and notice that the constraint is naturally split into two parts. The first is the linear term P T and the second one is the expression inside the square brackets. The linear term is connected with the fact that pt is linear in P T and that p is P T -independent. Furthermore, due to this linear relation, we select the new momentum P T as a non-physical 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.
The equations of motion in these variables take the form In the last equation we made the constraint strongly equal to zero in order to remove from the final expression the term proportional to the constraint −λ ∂ ∂T 1 B φ. We now look upon the gauge fixing process and solve the equation (40). In this case, we have to consider the conditions T (τ 1 ) = T 1 and T (τ 2 ) = T 2 and one simple solution is given by which implies that λ = (T2−T1) (τ2−τ1)Ḃ in which case the previous equations read as 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 timedependent (see Eqs (19)). Nevertheless, using this last equation (45), the system will be a conservative system if and only if This condition must hold on the full reduced phase space (Q, P ) hence, the right hand side of (45) results in the following equations The first equation can be easily solved forḂ aṡ where 1/m 0 is the integration constant. Due to B is Q-independent then A(Q, T ) must be of the form whereÃ(T ) is a smooth function of T to be fixed and we assume that m 0 is Q-independent. Of course, a more general solution implies to consider m 0 = m 0 (Q) but this results in a Q-dependent mass term for the new Hamiltonian which exceeds the purpose of the present work. We now combine (48) with (49) and obtain Inserting (48), (49) and (50) in the second equation of (47) we obtain the expression for D(Q, T ) in terms ofÃ(T ) as where κ(Q) is the integration function (T -independent) of the second equation in (47). In order to fixÃ(T ), we use the third equation in (47) which, after inserting the former definitions, takes the form This is the equation forÃ(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 proportional to Q 2 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 0 as a function of Q and such that the Q-dependency is removed from (52) but again, this only works for particular cases of the potential V .
Once we obtain the expression forÃ(T ), we integrate (52) and use (50) and (51) to obtain which is, clearly, an autonomous Hamiltonian system. To derive the Hamiltonian function giving rise to these equations, let us consider the Hamiltonian form of the action (24) in the new variables where φ is given in (37) and recall that the Lagrange multiplier given by Eq (40) is λ = (T2−T1) (τ2−τ1)Ḃ . Solving the constraint (37) for the momentum P T and inserting the result in (55), gives where I(Q, P ) := is the new Hamiltonian and the last term in (56) is the boundary term associated with the canonical transformation.
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 thẽ 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} P B = 0. The significance of this result 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 andȦ D + (A ′Ḋ −ȦD ′ )dQ in (55) we obtain 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 Let us replace P and P T in the action (58) byP andP T using the previous expressions where the function F (Q, T ) is the boundary term given by 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 is a new Hamiltonian function. As can be seen, the HamiltonianĨ(Q,P , T ) is a time-dependent Hamiltonian due to the 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 HamiltonianĨ(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 toĨ is a time-dependent Hamiltonian, we are going to work with I(Q, P ) instead ofĨ(Q,P , T ). As a remark related with the boundary term (62), notice that the infinitesimal action of the constraint on F (Q, T ), combined with the equations of motion, gives rise to a total derivative in the extended phase space 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 T which can be obtained using the general expression where η := T − (T2−T1)(τ −τ1) (τ2−τ1) − T 1 ≈ 0 and φ is given in (37). Notice that {φ, η} P B = −1/Ḃ and as a result, the only non-null Dirac brackets are 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 section by calculating the Generating function F =q p +t pt + F 3 (p, pt, Q, T ) related with this canonical transformation. In this case, F 3 is now given by 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 pt. This enlarging of the phase space allows us to consider a canonical transformation which is not only time-dependent but also pt-dependent. The aim is to transfer the time-dependency of the Hamilton equations into the coefficients of this canonical transformation rendering the final Hamilton equations time-independent. As a result, we obtain three auxiliary equations given in (47). The first two of them can be easily solved while the third gives rise to the equation (52), which is a generalization of the auxiliary equation (3). The final outcomes of this procedure are the Hamilton equations given by (54) with a time-independent Hamiltonian given by the Lewis invariant (57). Of course, this is the main result of this section together with the derivation of the boundary 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 byĨ(Q,P , T ) which, although is a time-dependent Hamiltonian, is canonically related with the Lewis invariant system.

IV. PATH INTEGRAL ANALYSIS
In the previous Section, we attended the constrained Hamiltonian analysis of the time-dependent Hamiltonian given 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 generalization of the term reported in [21].
Consider the amplitude given by which is the formal expression of the path integral form of the Feynman propagator and the Hamiltonian H(q, p, t) is given in (5). The infinitesimal expression of this amplitude takes the form The expression (69) can be obtained from the path integral formulation in the extended phase space using the amplitude given by where its infinitesimal expression is given by where the symbol (0) means that the Dirac delta functions are inserted in the measure of the path integral and as a result, this amplitude is not equivalent to that of a motion with an arbitrary Hamiltonian function H(q, p,t, pt) in the quantum extended phase space. The reason for this is that the quantum-physics description only takes place in the Hilbert space related with the reduced phase space [19]. Here, we used the following definitions H j := H q j , p j ,t j , q j :=q(τ j ),t j :=t(τ j ) and a j := a(τj)+a(τj−1) 2 for any function a(τ ). Notice that the function g(τ ) in the Dirac delta gives the time gauge in this system. In section II, we use the gauge (20), which in this case, is equivalent to consider the function g(τ ) = (t2−t1)(τ −τ1) (τ2−τ1) + τ 1 . In this section, we are going to use a more general gauge due to, as we proved in section III, the Lewis invariant is also a gauge invariant observable in the extended phase space formalism. In this way, our former derivation should be consistent with the result of the current section.
The infinitesimal expression of the amplitude (70) coincides with (69) whenever the Dirac delta functions are evaluated. This asseveration plays an important role in our derivation due to (70) is the quantum version of the aforementioned phase space enlarging. For this reason, in the appendix VI we show that whenever g(τ ) satisfiesq Of course, the gauge (20) used in section II fulfills this condition (73) as can be easily checked.
Similarly to the goal of section III, our aim in this section is to show that under a quantum canonical transformation on the extended phase space variables in the path integral (70), the amplitude of the time-dependent Hamiltonian system can be written as the product of two factors. One of the factors is the amplitude of a time-independent Hamiltonian system (related to the invariant I of the previous section) and the other factor is related to the quantum canonical transformation and the boundary term given in (56). In order to do so, let us consider the discrete version of the canonical transformation (36).
In the case of the time scaling, the new time coordinate is related to the old one byt = B(T ) as is given in (36), hence on each interval in (71) we have the relationt j → T j . Accordingly, on each of the intervals [t j ,t j+1 ] we obtaiñ Using (74) we obtain the following useful properties fort j and ∆t j where ∆T j := T j − T j−1 and T j := (T j + T j−1 )/2. With these relations, an arbitrary function of timet, let us say G(t), satisfies the following infinitesimal properties In a similar way, the transformations of the coordinate variablesq j → Q j on each interval in (71) are given bỹ and as a result,q j and ∆q j take the form where ∆Q j := Q j − Q j−1 and Q j := (Q j + Q j−1 )/2. To derive the transformations for the momentum variables appearing in the measure in (70), we have to consider the Generating function F 3 given in (67). First, notice that in the classical analysis, F 3 (p, pt, Q, T ) provides the following relation between the new and the old momenta In the quantum description, the discrete version of these relations provides the relations between both, the new and old discrete momenta [21] as follows (80) A careful calculation using (67) results in the explicit expression for the discrete momenta as where the conditions (76) were used. We now consider the relations in (81) to express p j and p t(j) in terms of the new momenta P j and P T (j) as follows With these results, we are now ready to analyze the transformation of the measure and the Dirac delta product given in (70) or in its infinitesimal version (71). Let us, for simplicity, study the transformation of the measure and the transformation of the Dirac deltas separately.
In the case of the measure, a lengthly calculation gives Notice that in contrast to the analysis given in Chetouani [21], both dt j and dpt (j) contribute to the change of the measure, the first with a factorḂ(T j ) and the second with a factor 1/Ḃ(T j ). In the case of the product of the Dirac deltas, we obtain that it transforms as where T (0) j and K j are given by Notably, both Dirac deltas contribute with factors 1/Ḃ(T (0) j ) andḂ(T j ) respectively. These contributions will cancel the factors coming from the differentials dt j and dpt (j) within the measure as we will see further below.
Finally, let us consider the transformation of the argument on the exponential in (71) under the discrete version of the canonical transformation. Inserting (74), (75), (77), (78), (82) and (83) on the argument of exponential and considering up to first order in ∆Q j and ∆T j we obtain the following expression Φ := p j ∆q j + p t(j) ∆t j , Of course, due to the classical canonical transformation is a particular type of contact transformation, its quantum analog is also a discrete contact transformation. This is the reason for the linearity of the momenta in this expression.
We are now ready to calculate the amplitude (71). By plugging the expressions (84), (85) and (87) together in (71) we obtain Integrating in the momenta P T (j) first and second in the time variables T j and making use of the Dirac deltas, we obtain that (89) takes the form whereΦ It can be noticed that the terms inḂ are no longer present in the expression (90). As we mentioned, the contributions of the differentials dt j and dpt (j) as well as those coming from the Dirac deltas cancel each other. This explains why time-scaling cannot modify the measure in the reduced phase space analysis given by Chetouani [21].
To conclude our calculation, we now use the relation which implies that, at lowest order in ∆T (0) j , we have the approximatioñ We now insert the relation (93) in (90) and obtain that the amplitude takes the form where It can be seen that the continuum formulation, i.e., the limiting process N → +∞, gives This result is a generalization to that obtained in [21]. The difference comes from the fact that we used a generalized canonical transformation given by (36). In this case, we show that time-scaling transformation does not alter the measure of the Feynman propagator: the factors introduced by Dirac deltas transformations cancel the factors introduced by the time-scaling and the time momentum transformation. In addition to this result, the phase in (97) depends exclusively on the boundary term given in (56). This aspect is of course absent in [21].

V. DISCUSSION
Plenty of physical systems are modeled with time-dependent Hamiltonians and this time-dependency can turn very difficult the study of such systems. In this realm, the Lewis invariant can be used to solve the equations at both, the classical and the quantum description. Of particular importance is the analysis of the Lewis invariant in the extended phase space where it is obtained via a canonical transformation. In this work, we applied Dirac's method for constrained time-dependent Hamiltonian systems and showed that the Lewis invariant is a gauge invariant in the extended phase space.
We also showed that the quantum phase relating the Feynman propagator of the time-dependent Hamiltonian H with the Feynman propagator obtained with the Lewis invariant I, is given by the boundary term resulting from the canonical transformation in the extended phase space.
To do so, in Section II we summarized the main aspects of the Dirac's method within the extended phase space formalism with coordinates (q,t, p, pt). We obtained the first class constraint φ given by (14) which results from the reparametrization invariance of the action (8). The Dirac brackets are the canonical relations to be used in the canonical quantization of this system and emerge after fixing the gauge λ = (t2−t1) (τ2−τ1) . In Section III, we studied the canonical transformation (36), which is a generalization of the Struckmeier transformation [16]. We showed that imposing the energy conservation law in the new variables results in the equations given in (47). These equations admit solutions in which the mass of the system in the new variables can be considered as a function of the coordinate Q, that it to say, m 0 (Q). We restrict this work only to the case in which m 0 is constant. This leads to a generalization of the Lewis invariant (57) obtained by [16]. In addition, this proves that more general invariants can be attributed to the Hamiltonian H(q, p, t) when the situation m 0 (Q) is considered.
In this section we also derived the Dirac brackets in the new variables Q, T, P, P T . These brackets are not canonically related with those in the old variables. This is a direct consequence of the fact that the transformation (36) is canonical in the extended phase space and not in the reduced phase space. Additionally, we obtained the boundary term in the relation (56) which emerges as a quantum phase in the Feynman propagator in the last Section.
Using the previous results we calculated the Feynman propagator for the time-dependent Hamiltonian H. We first enlarged the quantum space by considering a measure of the form Dq Dt Dp Dpt δ(t − g(τ )) δ(pt + H). With this measure the transformation can be implemented as a canonical map in the quantum extended phase space. We obtained that the time-scaling modifies the measure but the Dirac deltas within the measure absorbe its contributions as a result of the gauge invariance of the measure. In this case, gauge invariance is referred to a time-reparametrization invariant.
Finally, we obtained that the Feynman propagator (97) can be written in terms of the product of two factors. One factor is the usual Feynman propagator in terms of the new degrees of freedom Q, P and the Lewis invariant I while the other factor is given by where the phase is clearly the boundary term resulting from the canonical transformation in the extended phase space (56). The denominator is also a generalization of the Chetouani's result given in [21]. We showed in (64) that F (Q, T ) does not commute with the constraint φ, therefore this phase is not a gauge invariant quantum phase and a time-reparametrization induces an additional term given by the total time derivative in (64). Usually, the quantum phases are gauge invariant phases as a result from their evaluation in closed trajectories. In the present case, this is no longer possible because closed time-lines are meaningless physical trajectories.
As an additional result, let us point out the relation between the quantum canonical description of the systems given by the Hamiltonian I(Q, P ) and the HamiltonianĨ(Q,P , T ). In the first case, Dirac's method requires that the physical states are those Ψ(Q, T ) such that the constraint is strongly satisfied, that is to say, ΦΨ = ( P T + I( Q, P ))Ψ = 0, where the representation for the operators Q, P , T and P T is the standard representation QΨ = QΨ, P Ψ = −i ∂ Q Ψ, T Ψ = T Ψ, P T Ψ = −i ∂ T Ψ.
Of course, here we are omitting the problems related with the time T and time-momentum P T operators interpretation. Notice that as a result, the constraint equation (99) Recall now that both systems are canonically related in the classical description due to the relation between the momenta P andP is a time-dependent canonical transformation. This implies that the physical states of (101) can be obtained from the physical states of (99) using a unitary transformation and this transformation is the phase factor obtained previouslyΨ (Q, T ) = e i F (Q,T ) Ψ(Q, T ).
Therefore, these systems are equivalent not only at classical level but in their quantum description as well, although one is time-independent I( Q, P ) and the other is time-dependent Ĩ .
To conclude, let us stress that this work indicates that the Dirac's method improves the analysis of time-dependent systems at both, classical and quantum level. It serves as a natural bridge between the two levels. In the classical regime, the Dirac's method offers a natural understanding of the Lewis invariant in the extended phase space as a Dirac observable whereas in the path integral description, it allows the implementation of time-dependent transformations and time-scaling transformations as canonical maps in the quantum extended phase space. In this sense, Dirac's method serves as an unifying scheme to study time-dependent systems within the extended phase space formalism. Naturally, the present results can be expanded, as future work, to other systems in which invariants are often ignored and may yield important physical results, e.g., cosmology, quantum field theory, etc.
The right hand side of this expression is τ -independent if we fix as initial and final times t f ≡t f = g(τ f ) and t i ≡t i = g(τ i ). Moreover, let us assume that the following relationq(τ ) =q(g −1 (t)) = q(t) holds on each of the intervals and that any infinitesimal amount δτ j induces in the function g(τ ) an infinitesimal amount δt j such that q(g −1 (t j )) − q(g −1 (t j−1 )) ≈ dq(t j ) dt ǫ, ǫ := t j − t j−1 .