E−En
(1.2.28)
due to the completeness of the eigenstates. In this form, the Green’s function can now be related to the density of states. Observing that a Lorentzian of width
f(x) = π
1
[x2+2] (1.2.29)
approximates a Dirac delta function in the limit astends to 0, and by writing the Lorentzian in a different way, we get the relation
δ(x) = lim
→0−1 πIm 1
x+ i (1.2.30)
This allows us to express the density of states in terms of the trace of the Green’s function
d(E) = lim
→0−1
πIm TrG(E+ i) (1.2.31)
1.3 Semiclassical approximations
In general, the full quantum mechanics of a system cannot be solved analytically, but we can gain insight into the behaviour by looking at the semiclassical regime.
In the quantum mechanical equations we take the limit ¯h→0, and we can see that the differential evolution equations have a non-analytic singularity. However, due to the correspondence principle, we should arrive at classical mechanics on the other side of the singularity. In the semiclassical regime, as we shall see, the essence of the quantum mechanics is described by entirely classical quantities. Of course, ¯h is a fixed physical constant, therefore in practice when taking the limit ¯h → 0, we are studying the regime in which physical quantities, like the action, become large compared to ¯h.
1.3.1 Propagator and Green’s function
To obtain the semiclassical approximation to the propagator, we start (see, for example, Gutzwiller, 1990) from Feynman’s path integral form. The propagator can be obtained by combining propagators of shorter time as in equation (1.2.25), and Feynman (1948) considered splitting the time interval intoN short steps, which we will assume to be equally long and denoted byt (witht=N t). This gives the following relation for the propagator
K(q0,q, t) = Z
dq1. . .dqN−1K(q0,qN−1, t). . . K(q1q, t) (1.3.1)
where we integrate over the N −1 intermediate positions. In the limit where N becomes large, and the time of each step small, we can make a short time approx-imation for each propagator. For a particle of mass m moving under a potential V(q), in Cartesian coordinates, the Hamiltonian is
Hˆ =−¯h2
2m∆ +V(q) (1.3.2)
where ∆ is the Laplacian. The short time propagator is then given by
K(q0,q, t)≈
where we have made approximations up to linear order in the small time step. These approximations include that the particle remains near the pointq during the short time step, so the potential can be considered as constant over the small region, that the velocity is approximated by q0t−q
, and that we can commute the Laplacian and the potential.
When we substitute the short time propagator into equation (1.3.1) we obtain
K(q0,q, t)≈
whereq0 =q and qN =q0. The term in the complex exponential includes the ap-proximation to Hamilton’s principle function (full action) evaluated along a polyg-onal path through all the intermediate points qi at their respective times. The principle function is given by the integral
R(q0,q, t) = Z t
0
dt0L( ˙q,q), L( ˙q,q) = ˙qp−H(q,p) (1.3.5)
where L( ˙q,q) is the Lagrangian. Taking the limit N → ∞, the integrals of equa-tion (1.3.4) can be evaluated using a staequa-tionary phase approximaequa-tion, and the sta-tionary points of the principle function are trajectories travelling fromqtoq0in time t. Evaluating the integrals (Gutzwiller, 1967), we obtain the Van Vleck propagator
K(q0,q, t)≈ 1 (2πi¯h)f2
X
ζ
|DR|12e¯hiRζ(q0,q,t)−iπ2ν˜ζ (1.3.6)
which is a sum over all classical trajectories ζ connecting q and q0 in time t. DR
is the determinant of the matrix formed by the second derivative of the principle function with respect to the final and initial positions, or equivalently
DR= det∂p0
∂q (1.3.7)
A geometrical interpretation of this determinant can be given as follows. Consider our classical trajectories starting at the point q, with unfixed momentum due to the uncertainty principle. When a region of trajectories spanning a small volume δpin the momentum space is transported by the classical dynamics for a time tthe trajectories end up spread over the volumeδqin the position space around the point q0. The ratio of the initial and final volume is the determinant in equation (1.3.7).
The approximation for the semiclassical propagator fails, however, if DR becomes too large along the trajectoryζ, or equivalently if its inverse becomes 0. The inverse depends on the Jacobean
J = ∂q00
∂p0 (1.3.8)
which we evaluate for the moment up to an intermediate pointq00. Every time we reach a conjugate point, where more than one trajectory (from the initial momentum neighbourhood) has the same position coordinateq00, at the same time, the determi-nant of this matrix becomes zero and the semiclassical approximation diverges. The divergence can be avoided by making a change of coordinates (eg changing some position coordinates to momentum ones) before the conjugate point, and changing back afterwards. This results in multiplying the propagator by the phase factor e−iπ2 a number of times equal to the loss in rank of J (namely once for each coordinate change necessary). We add this phase for every conjugate point along the trajectory from q toq0 and record it via a topological index ˜νζ.
To obtain the semiclassical approximation to the Green’s function, we now take the Laplace transform of the semiclassical approximation to the propagator. For long times we can again approximate the integral using the stationary phase ap-proximation. The result is that we can write the semiclassical Green’s function in terms of all the classical trajectories linking the two end points as follows
G(q0,q, E)≈ 1 i¯h(2πi¯h)f−12
X
ζ
|DS|12e¯hiSζ(q0,q,E)−iπ2νζ (1.3.9)
where the sum is over all trajectoriesζ linkingqandq0at the energyE. Sζ(q0,q, E) is the classical action of the trajectoryζand is given by the integral of the momentum along the path
Sζ(q0,q, E) = Z q0
q
dq00pζ(q00,q, E) (1.3.10) The topological index νζ in equation (1.3.9) counts the number of conjugate points along the trajectoryζ (at the energy E).DS is the determinant of the matrix formed by the second derivative of the action with respect to the final and initial positions, and the energy
DS = det
∂2S
∂q∂q0
∂2S
∂q∂E
∂2S
∂q0∂E
∂2S
∂E2
(1.3.11)
This determinant can be simplified by using a coordinate system where one axis
points along the trajectory and the others are orthogonal. The semiclassical ap-proximation for the Green’s function given by equation (1.3.9) is only valid as long as q and q0 remain separate in position or time, and is referred to as a ‘long’ tra-jectory approximation. However, the propagator diverges for short times (as a delta function) and the Laplace transform of the short time form of the propagator (equa-tion (1.3.3)) can be expressed as (see Cvitanovi´c et al., 2005)
G0(q0,q, E) =−im 2¯h2
p 2π¯h|q0−q|
f−22 H+f−2
2
p|q0−q|
¯ h
(1.3.12)
where H+ is a Hankel function of the first kind and p = p
2m(E−V(q)). This semiclassical approximation for the Green’s function is only valid as long asqandq0 remain close in position and time and is known as a ‘short’ trajectory approximation.
Both approximations are important in what follows.
1.3.2 Trace formula
For the semiclassical approximation to the density of states, we need to take the trace of the semiclassical Green’s function
TrG(E) = Z
dqG(q,q, E) (1.3.13)
by integrating over all position space. Inside the integral, the two positions in the Green’s function are identical, so we use the short trajectory form of the semiclassical approximation of equation (1.3.12) (alternatively one can proceed directly from the propagator in equation (1.3.3)). Performing this integral, using the asymptotics of the Hankel function of the first kind for small argument, we obtain Weyl’s formula and recover the average density of states
d(E)¯ ≈ 1 (2π¯h)f
Z
dxδ(E−H(x)) = Ω(E)
(2π¯h)f (1.3.14)
which is now expressed in terms of the classical volume of the energy shell. This result has the following interpretation. Because of the Heisenberg uncertainty prin-ciple, the smallest resolvable space is a (Planck) cell of size (2π¯h)f in phase space.
If each cell can support a quantum state, then the average number of states ¯N(E) can be estimated by simply counting the number of cells that will fit in the phase space volume below the energyE. This estimate gives the above result for the mean density of states.
The idea of Planck cells allows us to provide a better picture of the separation between local and global properties of a quantum chaotic system. For the purely classical motion, once a small region around the trajectory is stretched (and com-pressed) so as to become as large as the typical size of the system, the evolution starts to become ergodic (and mixing) and, depending on the application, can be treated as a random variable. The time this takes depends on the size of the region and can be made longer by shrinking the initial region. Quantum mechanically, however, there is a shortest possible length scale given by the sides of a Planck cell.
The time this takes to grow to the typical size of the system is called the Ehrenfest timeTE. Because the stretching is dominated by the largest Lyapunov exponent λ, the Ehrenfest time is such that
(2π¯h)eλTE ≈1, TE≈ 1 λln
1 2π¯h
(1.3.15)
Semiclassically then, below this time scale we can concentrate on the local hyperbolic motion, while above it we can focus on the global ergodic behaviour.
The ‘zero-length’ trajectories in the trace considered above are not the only ones that can connect the point q to itself. Any trajectory that passes through the same point twice will count, and, as long as they are separated in time, we use the long trajectory form of the Green’s function. The integral over position space is performed using a stationary phase approximation. The condition that the trajectory is a stationary point turns out to be equivalent to the requirement that the momentum is the same each time it passes through q, meaning that the
trajectory must be periodic. The result of the integral then gives the oscillating part of the density of states as a sum over the periodic orbits of the system (Gutzwiller, 1971)
dosc(E)≈Re 1 π¯h
X
γ,r
Aγ,reh¯irSγ(E) (1.3.16) whereγ labels the primitive periodic orbits, andr their repetitions. The orbits have classical action Sγ and the amplitude is given by
Aγ,r = Tγe−iπ2rµγ
q|det(Mγr−1)| (1.3.17)
which incorporates the periodTγ, the stability matrixMγ and the Maslov indexµγ. The latter counts the number of conjugate point along the periodic orbit, but also has a geometrical interpretation due to Creagh et al. (1990) and Robbins (1991). The stable and unstable manifold of the periodic orbit can rotate as they are transported along the periodic orbit, and the Maslov index is the number of times the manifolds rotate by half a turn along the orbit. After each loop along the periodic orbit, the manifolds must be back where they started, so the Maslov index will be an integer.
If the system involves reflections on hard walls (with Dirichlet boundary conditions), then we also need to add twice the number of reflections to the Maslov index.
1.3.3 Form factor
From the trace formula for the density of states we can obtain the semiclassical approximation for the form factor by substituting the expression for the oscillating part of the density of states in terms of periodic orbits (equation (1.3.16)) into the form factor equation (1.2.19). When we make this substitution, we also make some simplifications and assumptions. We ignore any differences in the slowly varying prefactor and we expand the action as a Taylor series up to first order as
Sγ
E±η
2
≈Sγ(E)±η 2
dSγ(E)
dE =Sγ(E)±η
2Tγ(E) (1.3.18)
Because the sum over orbits in the density of states includes the complex conjugate, we obtain terms with action sums and action differences in the form factor. The terms with action sums will add destructively, and should average to zero, so we only retain terms with an action difference. Now when we substitute into the form factor we obtain
K(τ) = 1 2π¯hTH
*Z
dη X
γ,r γ0,r0
h
Aγ,rA∗γ0,r0eh¯i(rSγ−r0Sγ0)e2¯iηh(rTγ+r0Tγ0)+ c.c.i
e−iηh¯ τ TH +
(1.3.19) This is the quantity we wish to consider in the semiclassical limit ¯h→0. When we perform the integral the result is
K(τ) = 1 TH
* X
γ,r γ0,r0
Aγ,rA∗γ0,r0eh¯i(rSγ−r0Sγ0)δ
τ TH−rTγ+r0Tγ0
2
+
(1.3.20)
Since the orbits have positive periods Tγ, and because τ is positive, we retain only the delta function that contributes. We will examine the semiclassical evaluation of this quantity for closed systems, in the regime τ <1, in Chapter 2.
1.3.4 Equidistribution
In section 1.1.3 we considered the ergodic property of almost all trajectories in a classical chaotic system. So far, however, we have refrained from exploring similar properties of long periodic orbits, which are crucial for the semiclassical evaluation of the form factor in terms of periodic orbits. As we have seen, ergodicity does not hold for the periodic orbits, but as a group, the long orbits should still spread evenly over the energy shell. In fact, the ensemble of (weighted) long periodic orbits is uniformly distributed over the energy shell, and sums over the ensemble can be replaced with an energy shell average.
This uniformity of the long periodic orbits is reflected in the sum rule of Hannay and Ozorio de Almeida (1984). They considered the time average of a (smoothed) delta function for a typical trajectory. Due to ergodicity (see equation (1.1.11)), in
the limit of long times, the time average is equivalent to an energy shell average.
Integrating both sides of this equivalence over the energy shell (see also Ozorio de Almeida, 1988), the delta function picks out the periodic orbits and gives them a weight that depends on their stability. This led to a sum rule over the periodic orbits
Tlim→∞
1 T
X
γ,r rTγ<T
Tγ
|det(Mγr−1)| = 1 (1.3.21) The fraction in this sum is very similar to the stability amplitudes (see equa-tion (1.3.17)), and in fact coincides with |ATγ,r|2
γ . We now rearrange this result to get the form of the Hannay–Ozorio de Almeida sum rule that we will use later. Looking at the large time asymptotics of the sum, the contribution of the repetitions of the periodic orbits can be neglected as their number is exponentially smaller than that of the primitive orbits. We can ‘differentiate’ to obtain
X
γ
|Aγ|2δ(T−Tγ)∼T, T → ∞ (1.3.22)
where the delta function is smoothed by the width so that we average over a small range of time. In this formula, the exponential growth in the number of orbits essentially balances the exponential decay of their amplitudes.
The Hannay–Ozorio de Almeida sum rule is in fact an example of a more general equidistribution theorem (Bowen, 1972; Parry and Pollicott, 1990). Imagine that we measure some functionF(x) along a periodic orbit γ
Fγ= 1 Tγ
Z Tγ
0
dt F(x(t)) (1.3.23)
where the pointx(0) is on the periodic orbitγ. If we sum over all primitive periodic orbits weighted as before, we can replace the sum with a phase space average
X
γ
|Aγ|2Fγδ(T −Tγ)∼ T Ω
Z
dyF(y) =ThFi, T → ∞ (1.3.24)
This result is similar to the property of ergodicity in equation (1.1.11), and will allow us to simplify periodic orbit sums later. The upshot of this result is that we can effectively replace the function Fγ for each orbit with the phase space average value hFi (from the right hand side). The equidistribution theorem will therefore be important in the evaluation of the semiclassical form factor that we consider in Chapter 2.