Time-dependent conformal transformations and the propagator for quadratic systems

The method proposed by Inomata and his collaborators allows us to transform a damped Caldiroli-Kanai oscillator with time-dependent frequency to one with constant frequency and no friction by redefining the time variable, obtained by solving a Ermakov-Milne-Pinney equation. Their mapping ``Eisenhart-Duval'' lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.

The rigourous definition and calculation of (I.2) is beyond our scope here. However the semiclassical approximation leads to the van Vleck-Pauli formula [2][3][4], whereĀ(x , t |x , t ) = t t L(γ(t),γ(t), t)dt is the classical action calculated along the (supposedly unique 1 ) classical pathγ(τ ) from (x , t ) and (x , t ). This expression involves data of the classical motion only. We note here also the van Vleck determinant ∂ 2Ā ∂x ∂x in the prefactor [4].
For ω ≡ 0, i.e., for a free non-relativistic particle of unit mass in 1+1 dimensions with coordinates X and T , the result is [1][2][3], An harmonic oscillator with dissipation is in turn described by the Caldirola-Kanai (CK) Lagrangian and equation of motion, respectively [5]. For constant damping and harmonic frequency we have, with λ 0 = const. > 0 and ω 0 = const.. A lengthy calculation then yields the exact 1 This condition is satisfied away from caustics [2,3,6]. Morevover (I. 5) and (I.7a) are valid only for 0 < T − T and for 0 < t − t < π, respectively as it will be discussed in sec.IV.
Inomata and his collaborators [10][11][12] generalized (I.7) to time-dependent frequency by redefining time, t → τ , which allowed them to transform the time-dependent problem to one with constant frequency (see sec.II). Then they follow by what they call a "time-dependent conformal transformation" (x, t) → (X, T ) such that which allows them to derive the propagator from the free expression (I. 5). When spelled out, (I.8) boils down to a generalized version, (II.11), of the correspondence found by Niederer [13].
It is legitimate to wonder : in what sense are these transformations "conformal" ? In sec.III we explain that in fact both mappings can be interpreted in the Eisenhart-Duval (E-D) framework as conformal transformations between two appropriate Bargmann spaces [14][15][16][17][18].
Moreover, the change of variables x, t → X, T is a special case of the one put forward by Arnold [19], and will be shown to be convenient to study explicitly time-dependent systems.
In sec.V B we illustrate our theory by the time-dependent Mathieu profile ω 2 (t) = a − 2q cos 2t , a, b const. whose direct analytic treatment is complicated.

II. THE JUNKER-INOMATA DERIVATION OF THE PROPAGATOR
Starting with a general quadratic Lagrangian in 1+1 spacetime dimensions with coordinatesx and t, Junker and Inomata derive the equation of motion [10] x which describes a non-relativistic particle of unit mass with dissipation λ(t). The driving force F (t) can be eliminated by subtracting a particular solution h(t) of (II.1), x(t) = x(t) − h(t), in terms of which (II.1) becomes homogeneous, This equation can be obtained from the time-dependent generalization of (I.6a), The friction can be eliminated by setting x(t) = y(t) e −λ(t)/2 which yields an harmonic oscillator with no friction but with shifted frequency [25], For λ(t) = λ 0 t and ω = ω 0 = const., for example, we get a usual harmonic oscillator with constant shifted frequency, Ω 2 = ω 2 0 − λ 2 0 /4 = const. The frequency is in general time-dependent, though, Ω = Ω(t), therefore (II.4) is a Sturm-Liouville equation that can be solved analytically only in exceptional cases.
Junker and Inomata [10] follow another, more subtle path. Eqn. (II.2) is a linear equation with time-dependent coefficients whose solution can be searched for within the Ansatz 2 where A, B andω are constants and ρ(t) and τ (t) functions to be found. Inserting (II.5) into (II.2), putting the coefficients of the exponentials to zero, separating real and imaginary parts and absorbing a new integration constant into A, B provides us with the coupled system for Manifestlyτ > 0. Insertingτ into (II.6a) then yields the Ermakov-Milne-Pinney (EMP) equation [26] with time-dependent coefficients, 2 A similar transcription was proposed, independently, also by Rezende [24].
Conversely, the constancy of the r.h.s. here can be verified using the eqns (II.6). Equivalently, starting with the Junker-Inomata condition (I. 8), To sum up, the strategy we follow is [10,27] Then the trajectory is given by (II.5).
Junker and Inomata show, moreover, that substituting into (II.3) the new coordinates allows us to present the Caldirola-Kanai action as 3 , where we recognize the action of a free particle of unit mass. One checks also directly that X, T satisfy the free equation as they should. The conditions (I.8) are readily verified.
The coordinates X and T describe a free particle, therefore the propagator is (I.5) (as anticipated by our notation). The clue of Junker and Inomata [10] is that, conversely, trading X and T in (I.5) for x and t allows to derive the propagator for the CK oscillator (see also [9], sec.5.1), 4 , 3 Surface terms do not change the classical equations of motion and multiply the propagator by an unobservable phase factor, and will therefore dropped. 4 The extension of (II.13) from 0 <ω(τ − τ ) < π to all t [2, 3, 6, 9], will be discussed in sec.IV.
This remarkable formula says that in terms of "redefined time", τ , the problem is essentially one with constant frequency. Eqn. (II.13) is still implicit, though, as it requires to solve first the coupled system (II.6) that we can do only in particular cases.
• When the oscillator is turned off, ω 0 = 0 but λ 0 > 0, we have motion in a dissipative medium. The coordinate transformation propagator (II.11) and (II.13) become (II. 19) respectively. A driving force F 0 (e.g. terrestrial gravitation) could be added and then Further examples can be found in [11,12]. An explicitly time-dependent example will be presented in sec. V B.

III. THE EISENHART-DUVAL LIFT
Further insight can be gained by "Eisenhart-Duval (E-D) lifting" the system to one higher dimension to what is called a "Bargmann space" [14][15][16][17][18]. The latter is a d+1+1 dimensional manifold endowed with a Lorentz metric whose general form is which carries a covariantly constant null Killing vector ∂ s . Then : Theorem 1 [15,17] : Factoring out the foliation generated by ∂ s yields a non-relativistic space-time in d + 1 dimensions. Moreover, the null geodesics of the Bargmann metric g µν project to ordinary space-time consistently with Newton's equations.
is a solution of the non-relativistic equations of motion, then its null lifts to Bargmann space where s 0 is an arbitrary initial value.
Let us consider, for example, a particle of unit mass with the Lagrangian of where g ij (x k )dx i dx j is a positive metric on a curved configuration space Q with local coor- The coefficients α(t) and β(t) may depend on time t and V (x i , t) is some (possibly time-dependent) scalar potential. The associated equations of motion are where the Γ i jk are the Christoffel symbols of the metric g ij . For d = 1, g ij = δ ij and V = 1 2 ω 2 (t)x 2 for α = β = 1 resp. for α = β −1 = e −λ(t) we get a (possible time-dependent) 1d oscillator without resp. with friction, eqn. (I.6), [5,7,25]. Equation (III.4) can also be obtained by projecting a null-geodesic of d+1+1 dimensional Bargmann spacetime with coordinates (x µ ) = (x i , t, s), whose metric is Choosing λ(t) = ln m(t) would describe motion with a time-dependent mass m(t). The friction can be removed by the conformal rescaling x → y = √ m x and the null geodesics of the rescaled metric describe, consistently with (II.4), an oscillator with no friction but with time-dependent frequency, Ω 2 = ω 2 −m 2m + ṁ 2m ) 2 [28]. The friction term −(α/α)ẋ i in (III.4) can be removed also by introducing a new timeparametert, defined by dt = α dt [18]. For λ(t) = λ 0 t, for example, puttingt = −e −λ 0 t /λ 0 and eliminates the friction -but it does it at the price of getting manifestly time-dependent frequency [29,30] A. The Junker-Inomata Ansatz as a conformal transformation The approach outlined in sec.II admits a Bargmannian interpretation. For simplicity we only consider the frictionless case λ = 0.
Theorem 2 : The Junker-Inomata method of converting the time-dependent system into one with constant frequency by switching from "real" to "fake time", induces a conformal transformation between the Bargmann metrics Proof : Putting µ = lnτ allows us to present the constant-frequencyω (II.8) as (III.10) Then with the notation • ξ= dξ/dτ we find, Let us now recall that the null lift to Bargmann space of a space-time curve is obtained by subtracting the classical action as vertical coordinate, Setting here ξ =τ 1/2 x and dropping surface terms yields, using the same procedure for the time-dependent-frequency case, up to surface terms. Then inserting all our formulae into (III.8a) and (III.8b) yields (III.9), as stated. In Junker-Inomata language (I.8), f (t) =τ 1/2 sec(ωτ ), g(t) = (ω) −1 tan(ωτ ) .
Our investigation have so far concerned classical aspects. Now we consider what happens quantum mechanically. Restricting our attention at d = 1 space dimensions as before 5 we posit that the E-D lift ψ of a wave function ψ be equivariant, Then the massless Klein-Gordon equation for ψ associated with the 1+1+1 = 3 d Barmannmetric implies the Schrödinger equation in 1+1 d, where ∆ g is the Laplace-Beltrami operator associated with the metric. In d = 1 it is of It is implemented on a wave function lifted to Bargmann space as In secs.IV B these formulae will be applied to the Niederer map (IV.12). 5 In d > 2 conformal-invariance requires to add a scalar curvature term to the Laplacian.

B. The Arnold map
The general damped harmonic oscillator with time-dependent driving force can be solved by an Arnold transformation [19] which "straightens the trajectories" [18,25,31]. To this end one introduces new coordinates, where Then in the new coordinates the motion becomes free [19], Eqn (III.16) can be obtained by projecting a null geodesic of the Bargmann metric Completing (III.17) by lifts the Arnold map to Bargmann spaces, (x, t, s) → (X, T, S) 6 , The oscillator metric (III.20) is thus carried conformally to the free one, generalizing earlier results [15,16,32]. For the damped harmonic oscillator with λ(t) = λ 0 t and F (t) ≡ 0, u p ≡ 0 is a particular solution. When ω = ω 0 = const., for example, are two independent solutions of the homogeneous equation with initial conditions (III. 18) and provide us with In the undamped case λ 0 = 0 thus Ω 0 = ω 0 , and (III.24) reduces to that of Niederer [13] lifted to Bargmann space [16,17], The Junker-Inomata construction in sec.II can be viewed as a particular case of the Arnold transformation. We choose u p ≡ 0 and the two independent solutions The initial conditions (III.18) at t 0 = 0 imply τ (0) =ρ(0) = 0, ρ(0) =τ (0) = 1. Then spelling out (III.21), completes the lift of (II.11) to Bargmann spaces. In conclusion, the one-dimensional damped harmonic oscillator is described by the conformally flat Bargmann metric, The metric (III.28) is manifestly conformally flat, therefore its geodesics are those of the free metric, X(T ) = aT + b. Then using (III.17) with (III.26) yields The bracketed quantity here describes a constant-frequency oscillator with "time" τ (t). The original position, x, gets a time-dependent "conformal" scale factor.

IV. THE MASLOV CORRECTION
As mentioned before, the semiclassical formula (I.7) is correct only in the first oscillator half-period, 0 < t − t < π/Ω 0 . Its extension for all t involves the Maslov correction. In the constant-frequency case with no friction, for example, assuming that Ω 0 (t − t )/π is not an integer, we have [2,3,6], is called as the Maslov index (where Ent[x] is the integer part of x). counts the completed half-periods, and is related also to the Morse index which counts the negative modes of Now we generalize (IV.1) to time-dependent frequency : Theorem 4 : In terms ofω and τ introduced in sec.II, • Outside caustics, i.e., forω(τ − τ ) = π , the propagator for the harmonic oscillator with time-dependent frequency and friction is • At caustics, i.e., forω we have instead [3,6], cf. (III.7) and using the notation Thus the problem is reduced to one with time independent frequency,ω in (II.8) 7 .
Let us now recall the formula #(19) of Junker and Inomata in [10] which tells us how propagators behave under the coordinate transformation (ξ, τ ) ←→ (x, t) : Here K 2 = K ext is the propagator of an oscillator with time-dependent frequency and friction, ω(t) and λ(t), respectively -the one we are trying to find. Notice that (IV.3) is regular at the points r k ∈ J k where sin = ±1. However at caustics, τ − τ = (π/ω) , K ext diverges and we have instead (IV.5).
Henceforth we limit our investigations to λ = 0.

A. Properties of the Niederer map
More insight is gained from the perspective of the generalized Niederer map (II.11). We first study their properties in some detail. For simplicity we choose, in the rest of this section, x = t = 0 and x ≡ x and t ≡ t. is mapped by (II.11) onto the full range −∞ < T < ∞. Therefore the inverse mapping is multivalued, labeled by integers k, where arctan k ( · ) = arctan 0 ( · ) + kπ with arctan 0 ( · ) the principal determination i.e. in (−π/2, π/2). Then lim t→r k − tan t = ∞ and lim t→r k + tan t = −∞ imply that Therefore the intervals I k and I k+1 are joined at r k+1 and the I k form a partition the time axis, − ∞ < t < ∞ = ∪ k I k .
By (III.29) the classical trajectories are regular at t = r k . Moreover, for arbitrary initial velocities, (IV.17) Note that t depends on k, t = N k (T ), but x and s do not.
Proof : These formulae follow at once by inverting (III.25) at once with the cast ω 0 ⇒ ω, t ⇒ τ . Alternatively, it could also be proven as for of Theorem 2.
For each integer k (IV.17) maps the real line −∞ < T < ∞ into the "open strip" [16] r k , r k+1 × R 2 ≡ I k × R 2 with r k defined in (IV.10). Their union covers the entire Bargmann manifold of the oscillator. consistently with s(t) = s 0 −Ā osc , as it can be checked directly. Note that the s coordinate oscillates with doubled frequency.
• At t = r k = ( 1 2 + k)π (where the Niederer maps are joined), we have, lim t→r k x(t) = (−1) k+1 a , lim t→r k s(t) = S 0 . Thus the pull-backs of the Bargmann-lifts of free motions are glued to smooth curves.
• Similarly at t caustics t = t = π we infer from (IV. 19) that for all initial velocity a and for all lim t→t x(t) = 0, lim t→t s(t) = S 0 . Thus the lifts are again smooth at t and after each half-period all motions are focused above the initial position (x(0) = 0, s(0) = S 0 ).

B. The propagator by the Niederer map
Now we turn at the quantum dynamics. Our starting point is the free propagator (I.5) which (as mentioned before) is valid only for 0 < T − T . Its extension to all T involves the sign of (T − T ) [16]. (IV.20) In conclusion, the formula valid for all T is, is the free action calculated along the classical trajectory. Let us underline that (IV.21) already involves a "Maslov jump" e −iπ/2 -which, for a free particle, happens at T = 0. For Accordingly, the wave function Ψ ≡ Ψ f ree of a free particle is, by (I.1), Now we pull back the free dynamics using the multivalued inverse Niederer map. It is sufficient to consider the constant-frequency caseω = const. and denote time by t. Let t belong to the range of N k in (IV.12), t ∈ I k = [r k , r k+1 ] = N k {−∞ < T < ∞} . Then applying the general formulae in sec.III A yields [16], However the second exponential in the middle line combines with the integrand in the braces in the last line to yield the action calculated along the classical oscillator trajectory, Thus using the equivariance we end up with, Now we recover the Maslov jump which comes from the first line here. For simplicity we consider again t = 0, x = 0 and denote t = t, x = x.
Firstly, we observe that the conformal factor cosωt has constant sign in the domain I k and changes sign at the end points. In fact, The cosine enters into the van Vleck factor while the phase combines with exp − iπ 4 sign( tanωt ω ) . Recall now that t k+1 = N k T = 0 divides I k into two pieces, fig.1. But t k+1 is precisely where the tangent changes sign : this term contributes to the phase in [r k , t k+1 ] −π/4, and +π/4 in [t k+1 , r k+1 ]. Combining the two shifts, we end up with the phase which is the Maslov jump at t .
Intuitively, that the multivalued N k "exports" to the oscillator at t +1 the phase jump of the free propagator at T = 0. Crossing from J to J +1 shifts the index by one.

A. For constant frequency
We assume first that the frequency is constant. We split the propagator K(x, t) ≡ K(x, t|0, 0) in (IV.1) as, The probability density, viewed as a surface above the x − t plane, diverges at t = t = π , = 0, ±1, . . . .
Representing the phase of the propagator would require 4 dimensions, though. However, recall that that the dominant contribution to the path integral should come from where the phase is stationary [1], i.e., from the neighborhood of classical pathsx(t), distinguished by the vanishing of the first variation, δAx = 0. Therefore we shall study the evolution of the phase along classical pathsx(t) for which (III.29) yields, for =ω = 1 and a ∈ R, b = 0, x a (t) = a sin t and P a (t) = exp − iπ depicted in Fig.2. An intuitive understanding comes by noting that when t = π = t , then different initial velocities a yield classical pathsx a (t)s with different end points, and thus contribute to different propagators. However approaching from the left -times a half period, t → (π )− , all classical paths get focused at the same end-point (x = 0 for our choice) and for all a, which is precisely the Maslov phase. Thus all classical paths contribute equally, by P , and to the same propagator. Comparing with the right-limit, P a (t → π +) = e −i π 4 (1+2( +1) = P +1 = e − iπ 2 P .
the Maslov jump is recovered. Choosing instead y = 0 there will be no classical path from (0, 0) to (y, π ) and thus no contribution to the path integral.
To conclude this section we just mention with that the extended Feynman method [6] with the castω = constant frequency and τ = "fake time" would lead also to (IV. 3  Alternatively, we can use the Junker-Inomata -Arnold transformation (III.17) [19,31].
Eqn (II.6) is solved by following the strategy outlined in sec.II. Carrying out those steps numerically provides us with Fig.3.
From the general formula (II.13) we deduce, for our choice x = x, t = t, x = t = 0, the probability density 8 happens not depend on the position, and can therefore be plotted as in Fig.4. The propagator K and hence the probability density (V.7) diverge at t , which are roughly t 1 ≈ 1.92, t 2 ≈ 4.80, t 3 ≈ 7.83 . The classical motions are regular at the caustics, 8 The wave function is multiplied by the square root of the conformal factor, cf. (III.9).

VI. CONCLUSION
The Junker-Inomata -Arnold approach yields (in principle) the exact propagator for any quadratic system by switching from time-dependent to constant frequency and redefined time, ω(t) →ω = const. and t → "fake time" τ . (VI.1) The propagator (IV.3)-(IV.5) is then derived from the result known for constant frequency.
A straightforward consequence is the Maslov jump for arbitrary time-dependent frequency ω(t) : everything depends only on the productω τ .
By switching from t to τ the Sturm-Liouville-type difficulty is not eliminated, though, only transferred to that of finding τ = τ (t) following the prcedure outlined in sec.II. We have to solve first solve EMP equation (II.7) for ρ(t) (which is non-linear and has time-dependent coefficients), and then integrate ρ −2 , see (II.10). Although this is as difficult to solve as solving the Sturm-Liouville equation, however it provides us with theoretical insights.
When no analytic solution is available, we can resort to numerical calculations.
The Junker-Inomata approach of sec.II is interpreted as a Bargmann-conformal transformation between time-dependent and constant frequency metrics, see eqn (III.9).
Alternatively, the damped oscillator can be converted to a free system by the generalized While the "Maslov phase jump" at caustics is well established when the frequency is constant, ω = ω 0 = const., its extension to the time-dependent case ω = ω(t) is more subtle.
In fact, the proofs we are aware of [21][22][23][24] use sophisticated mathematics, or a lengthy direct calculation of the propgagator [35]. A bonus from the Junker-Inomata transcription (I. 8) we follow here is to provide us with a straightforward extension valid to an arbitrary ω(t).
Caustics arise when (IV.4) holds, and then the phase jump is given by (IV.27).
The subtle point mentioned above comes from the standard (but somewhat sloppy) expression (I.5) which requires to choose a branch of the double-valued square root function.
Once this is done, the sign change of T − T induces a phase jump π/2. Our "innocentlylooking" factor is in fact the Maslov jump for a free particle at T = 0 (obscured when one considers the propagator for T > 0 only). Moreover, it then becomes the key tool for the ocillator : intuitively, the multivalued inverse Niederer map repeates, all over again and again, the same jump. Details are discussed in sec.IV.
The transformation (I.8) is related to the non-relativistic "Schrödinger" conformal symmetries of a free non-relativistic particle [36][37][38] later extended to the oscillator [13] and an inverse-square potential [39]. These results can in fact be derived using a time-dependent conformal transformation of the type (I.8) [16,33].
The above results are readily generalized to higher dimensions. For example, the oscillator frequency can be time-dependent, uniform electric and magnetic fields and a curl-free "Aharonov-Bohm" potential (a vortex line [40]) can also be added [32]. Further generalization involves a Dirac monopole [41].