A Classical-Quantum Correspondence and Backreaction

We work in the Heisenberg picture to demonstrate the classical-quantum correspondence (CQC) in which the dynamics of a quantum variable is equivalent to that of a complexified classical variable. The correspondence provides a tool for analyzing quantum backreaction problems which we illustrate by a toy model in which a rolling particle slows down due to quantum radiation. The dynamics found using the CQC is in excellent agreement with that found using the much more laborious full quantum analysis.

A large class of physical systems involve classical dynamics that is coupled to quantum degrees of freedom that get excited as the classical system evolves. Examples of such systems include particle production during cosmological evolution, Hawking radiation during gravitational collapse and Schwinger pair production in an electric field. The key question we address in this paper is: how do we account for the backreaction of the quantum excitations on the classical background? The question is of fundamental interest as its solution may hold the key to many problems of current interest including the black hole information paradox.
Past work on the backreaction question is usually framed as a perturbative-iterative process; the radiation is calculated in perturbation theory, the backreaction is then calculated semiclassically, which then leads to modified radiation, and so on. In the present work, we instead develop a classical-quantum correspondence (CQC) using which we can transform the quantum radiation problem into a classical radiation problem. Then the entire problem, including backreaction, can be cast as a set of classical equations with definite initial conditions [1,2]. These equations can then be solved numerically. (Other work on classical-quantum connections includes [3][4][5][6][7]. ) The system we have in mind consists of a classical background variable that couples to a free quantum field. Expanding the field in modes, the mode coefficients behave like an infinite set of simple harmonic oscillators with time-dependent mass and frequency. By redefining variables, it is possible to further reduce the problem to that of decoupled simple harmonic oscillators with time-dependent frequencies. The time-dependence is prescribed by the background variable. Thus the field theory problem can be mapped into a quantum mechanics problem consisting of an infinite set of simple harmonic oscillators with time-dependent frequencies.
The quantum simple harmonic oscillator (qSHO) with a time-dependent frequency has been solved in terms of a two-dimensional classical SHO (cSHO) in early work [8][9][10] and more recently [11]. In Ref. [2], we have used this connection in the Schrodinger picture to show the equivalence of the quantum and classical systems, and we have discussed the excitations produced due to the time-dependence of the frequency. This result can be applied to a mode by mode analysis of a free quantum field to show that the quantum field dynamics of a real scalar field in a time-dependent background is equivalent to the classical dynamics of a complex scalar field with prescribed initial conditions. However, particle production is usually discussed in the Heisenberg picture using the method of Bogoliubov transformations (e.g. [12]) while functional Schrodinger derivations are less familiar. In this paper we first close this gap by demonstrating the classical-quantum correspondence (CQC) in the Heisenberg picture.
The production of particles due to the time varying background will cause dissipation in the time variation. The CQC provides a simple tool to study this backreaction because quantum dynamics can be replaced by classical dynamics. To assess the validity of the CQC approach to backreaction, we construct a toy model in which we can find the backreaction using the CQC and also in the full quantum theory. The results are in excellent agreement and the accuracy of the CQC approach increases as the background becomes more classical. This paper is organized as follows. We first show the CQC between a qSHO and two cSHOs in the Heisenberg picture when the frequency of the SHOs is an arbitrarily varying function of time. This is done in three steps. First, in Sec. I A we derive the Heisenberg equations of motion for the ladder operators with a time-dependent frequency. Then in Sec. I B we find the energy radiated in particles by the method of Bogoliubov transformations. In Sec. I C we show that the dynamics of the radiation and, in particular, the energy in quantum radiation, can be found by a purely classical calculation that involves doubling the radiative degrees of freedom, or equivalently complexifying these degrees of freedom. Having thus established the CQC, we turn to the quantum radiation backreaction on the dynamics of the classical variable. We find the results obtained using the CQC and compare them to the full quantum dynamics that are found by using novel, though laborious, numerical methods described in the appendix. Our conclusions are given in Sec. III. The Hamiltionian for a simple harmonic oscillator with time-dependent frequency is where ω = ω(t) is an unspecified function. We define ladder operators in the usual way It is straight-forward to check that [a, a † ] = 1 even for a time-dependent ω. Then, and We now go to the Heisenberg picture. Then the equation of motion for a is and similarly

B. Bogoliubov transformation
To obtain the excitation of the simple harmonic oscillator due to the time-dependence of ω, we write where a 0 and a † 0 are the annihilation and creation operators in Eq. (2) at the initial time, t = 0. The commutation relation [a, a † ] = 1 leads to the constraint and Eqs. (5), (6) lead tȯ These equations also lead to the constraint The expectation value of the energy in the vacuum state is C. The CQC Here we show that the quantum dynamics of the timedependent simple harmonic oscillator is given by the classical dynamics of two classical simple harmonic oscillators if we impose certain initial conditions. In contrast to the earlier derivation in the functional Schrodinger picture [2,13,14], here we show this correpondence in the Heisenberg picture. Particle production is then given by the Bogoliubov transformation method, and the energy in quantum excitations, E q , is also identical to the energy of two classical simple harmonic oscillators.
We solve Eqs. (9) and (10) by writing where is a complex variable, ξ and χ are real variables, and we have introduced an explicit (constant) phase factor to emphasize that the overall phase is arbitrary. However, such a phase can only be relevant when there are several radiative modes (such as in field theory) and we shall take δ = 0 for the rest of the paper. The overall factor of √ m in Eqs. (13) and (14) ensures that α and β have the correct mass dimensions equal to zero when z has dimensions of length.
The expressions for α and β are identical to the definition of the annihilation operator a in Eq. (2) if we think of z and mż as representing the complexified position and momentum operators for one dynamical variable and similarly z * and mż * for a second dynamical variable. Similarly the complex conjugates α * and β * then correspond to the expression for the creation operator a † in Eq. (2).
Inserting Eqs. (13) and (14) in Eqs. (9) and (10) we find the equation of motion satisfied by z, Hence ξ and χ satisfy the classical equations of motion The initial condition, a(0) = a 0 , corresponds to: α(0) = 1, β(0) = 0, which imply and are equivalent to . (The expressions differ from those in [2] in the factors of √ 2 and m because of different conventions.) The initial conditions and Eq. (17) imply that the Wronskian is conserved, which can also be written as where p z ≡ mż.
The initial conditions have the simple interpretation that the two-dimensional cSHO initially has the same energy as the qSHO in its ground state. This is easy to see because β(0) = 0 in Eq. (12) gives E q (0) = ω 0 /2. What is more novel is that the initial conditions are such that the cSHO also has angular momentum 1/2. Furthermore, the angular momentum, equivalently the Wronskian, is conserved during the evolution. If we think of the qSHO as the mode coefficient of a free scalar quantum field, the initial conditions imply that each mode of the corresponding classical complex scalar field must carry a conserved non-zero global charge.
The quantum dynamical problem has thus transformed into a classical evolution problem for any time-dependent frequency ω(t). To emphasize this point we write the full time-dependent annihilation operator in terms of classical solutions, where z denotes a classical solution with the initial conditions given above. Thus we have a mapping between the quantum solution and the classical solution.
Finally we re-express the quantum energy in Eq. (12) in terms of the ξ and χ variables, To summarize: to find the energy in quantum excitations, we simply have to solve the classical problem in Eq. (16) with the initial conditions in (18) and then calculate E q using (23). This is the CQC, earlier derived in the functional Schrodinger picture [2], but derived here in the Heisenberg picture via Bogoliubov transformations.

II. BACKREACTION
In the quantum problem, the time-dependent frequency produces quantum excitations and must backreact on the source responsible for the time dependence. In many situations, especially in gravitational settings, the quantum backreaction is difficult to calculate. However, the backreaction in the corresponding classical problem is in principle straightforward to evaluate because the classical equations of motion are known. If the classical equations are difficult to solve analytically, we can always, in principle, solve them numerically. We will now illustrate such a backreaction calculation for a toy problem that can be solved completely. This will tell us if the solution using the CQC is a good approximation to the full quantum solution.
Our toy model consists of two quantum degrees of freedom, x and z, where x represents a particle rolling down a linear potential and z represents a simple harmonic oscillator that couples to the rolling particle. (The model has similarities to field theories used in inflationary cosmology and to the "bottomless" potentials considered in Ref. [15].) The Hamiltonian for the system is which we shall rescale and write with re-defined a, ω 0 and λ as Here a corresponds to the constant classical acceleration while rolling, ω 0 is the simple harmonic oscillator frequency in the absence of any coupling to the rolling particle, and λ is the coupling. We are mainly interested in the dynamics of the rolling particle and how the presence of the simple harmonic oscillator backreacts on the dynamics. So we will first solve the classical rolling problem, then find the simple harmonic oscillator solution in the "fixed background" approximation. Next we will solve for the full dynamics using the CQC described above. Finally we will solve the full quantum problem and compare with the result obtained using the CQC. That is, the rolling particle continues to roll with constant acceleration while the simple harmonic oscillator degree of freedom is not excited.

B. Fixed background analysis
In the fixed background of the rolling particle, the CQC is exact and the Hamiltonian for the simple harmonic oscillator is where z = ξ + iχ and Then the energy of the simple harmonic oscillator can be found from Eq. (23) where we need to solve the classical equations of motion in Eq. (17) with the initial conditions in Eq. (19). With ω 0 = 0, Eq. (17) can be solved in terms of Bessel functions but for ω 0 = 0 we have to resort to a numerical computation. The result for E q (t) with ω 0 = 1, a = 1, λ = 1 is shown in Fig. 1. Note that total energy is not conserved in the fixed background analysis: initially the energy is ω 0 /2 = 0.5 while at t = 20 we see that it has grown to ∼ 25.

C. Backreaction with CQC
To obtain dynamics with backreaction with the CQC, we need to solve the classical equations with initial conditions This system of equations is solved numerically.
In Fig. 2 we show how the energy in the simple harmonic oscillator grows with time and that in the rolling particle decreases with time. The total energy is conserved. We will show the solution for x(t) using the CQC below, after we have discussed the solution of the full quantum problem.

D. Full quantum treatment
To solve for the full quantum dynamics, we have to solve the time-dependent Schrodinger equation with H given in Eq. (25). The initial wavefunction is taken to consist of Gaussian wavepackets in both the x and z variables, The parameter σ x is a free parameter in the full quantum problem and we shall study the dynamics for several values of σ x . With the initial condition in Eq. (38), we have x = 0 at t = 0 for all σ x . Ehrenfest's theorem in the absence of backreaction (λ = 0) gives the classical result for the evolution of the expectation value of x, We are interested in determining the effect of backreaction on this evolution. We solve the Schrodinger equation numerically using the technique described in the Appendix. The solution yields the wavefunction at all times. We then calculate the expectation value of the position of the rolling particle, x . (Symmetry under z → −z gives z = 0 at all times.) In Fig. 3 we show the dynamics of the rolling particle in all the different treatments: first the evolution ignoring backreaction, then the full quantum calculation for several values of σ x where backreaction is automat- ically included, and finally the evolution with backreaction evaluated using the CQC. We zoom into the late time evolution in a log-log plot in Fig. 4. This plot shows that x ∝ t 1.76 for the quantum evolution as well as the CQC evolution. To quantify how the quantum evolution tends to the CQC evolution for large σ x , we plot the difference of quantum and CQC evolutions at a fixed time (t = 5) for different values of σ x . The fit to the line gives Therefore the full quantum result goes to the CQC result in the limit of large σ x . To understand why quantum evolution tends to CQC evolution at late times and for large σ x , we consider the time-dependent wavepacket solution for a free particle, At late times the width of the wavepacket grows as So the rate of wavepacket spreading is (2σ x m) −1 . This wavepacket spreading is a completely quantum effect. For a rolling particle to behave classically, the rate of spreading should be much less than the rate at which it rolls, where a is the constant acceleration of the particle. Thus a rolling particle behaves more classically at late times, and the time at which it starts behaving classically occurs earlier if the initial width of the initial wavepacket is larger. As the rolling becomes more classical, the CQC becomes more exact, and at late times the CQC matches the quantum evolution. This explains our results in Figs. 4 and 5.

III. CONCLUSIONS
We have derived the CQC in the Heisenberg picture. This shows that the dynamics of a quantum simple harmonic oscillator with a time-dependent frequency is given by the dynamics of two classical simple harmonic oscillators with the same time-dependent frequency and prescribed initial conditions. Equivalently, the quantum dynamics can be recovered by complexifying the phase space variables of the classical simple harmonic oscillator. Since the modes of a free quantum field in a background can be treated as an infinite set of simple harmonic oscillators with time-dependent frequencies, the CQC can be extended to field theory. Then the dynamics of a quantum real scalar field is given by the dynamics of a classical complex scalar field, again with prescribed initial conditions.
The CQC provides a tool to study the backreaction of quantum radiation on classical dynamics. We have investigated the backreaction in a toy model that involves a particle rolling down a linear potential and coupled to other simple harmonic oscillator degrees of freedom. We solved this toy problem using the CQC and compared it to the full quantum solution. The dynamics in the two approaches agree remarkably well, especially as the initial quantum state in the full treatment is taken to be more classical (i.e. larger σ x ). Furthermore, the analysis using the CQC is trivial to implement numerically, whereas the full quantum treatment is a non-trivial numerical task.
We expect the CQC to have wide applicability since quantum excitations on classical backgrounds occur in many physical systems. The approach could prove invaluable in the gravitational context where one considers quantum fields in curved spacetime. Then the common approach is to work with "semiclassical gravity" [12], i.e. with the Einstein equation modified to where G µν is the Einstein tensor. The right-hand side is the expectation value of the renormalized energymomentum tensor of the radiation fields in a suitable quantum state -this is the energy-momentum tensor after the vacuum energy has been subtracted out and other bare couplings have been adjusted to reduce the equation to the above form. In principle semiclassical gravity and extensions may provide an iterative scheme for calculating the backreaction of quantum fields on the spacetime. The CQC approach however is to solve the classical equations where the prime on the right-hand side denotes that it is the classical energy-momentum tensor for the complexified fields minus the vacuum energy contribution. (Depending on the physical situation of interest, we could include a cosmological constant term.) This modified Einstein equation would then be solved together with the classical field equations ∇ ν T µν = 0 (46) with suitable initial conditions as discussed in this paper. The solution would provide the complete time dependence of the fields as well as the spacetime. A successful analysis in the case of gravitational collapse promises to shed light on black hole formation and the information paradox as already indicated in Ref. [2].