Abstract
For quantum theories with a classical limit (which includes the large N limits of typical field theories), we derive a hierarchy of evolution equations for equal time correlators which systematically incorporate corrections to the limiting classical evolution. Explicit expressions are given for next-to-leading order, and next-tonext-to-leading order time evolution. The large N limit of N -component vector models, and the usual semiclassical limit of point particle quantum mechanics are used as concrete examples. Our formulation directly exploits the appropriate group structure which underlies the construction of suitable coherent states and generates the classical phase space. We discuss the growth of truncation error with time, and argue that truncations of the large-N evolution equations are generically expected to be useful only for times short compared to a "decoherence" time which scales like N 1/2 .
I. INTRODUCTION
The time evolution of quantum systems away from equilibrium is of interest in many applications including, but certainly not limited to, phase transition dynamics, inflationary reheating, and heavy ion collisions. Large N expansions have provided a widely used technique for studying equilibrium properties in statistical physics and field theory [1] [2] [3] , and it is natural to apply a similar strategy for studying non-equilibrium problems. The large N limit (as typically formulated) is actually a special type of classical limit [4] . Suitable observables behave classically and the quantum dynamics reduces to classical dynamics on an appropriate phase space.
I. INTRODUCTION
Considerable work has been done examining the dynamics of far from equilibrium states in a variety of applications using leading large-N time-evolution [5] [6] [7] [8] [9] [10] . A major virtue of large N techniques (compared to alternative wholly uncontrolled approximation schemes) is that one should be able to improve the approximation by systematically including sub-leading effects suppressed by powers of 1/N. For a variety of equilibrium problems (such as critical phenomena), this approach can work quite well [11] [12] [13] .
I. INTRODUCTION
For initial value problems, in which one would like to choose a non-equilibrium initial state and then examine the subsequent time evolution, traditional formulations of large-N expansions using graphical or functional integral techniques [3] are very awkward. A major difficulty with these approaches is that they generate integral equations which are non-local in time when sub-leading 1/N corrections are retained. For practical (numerical) applications, one would vastly prefer a formulation in which locality in time is always preserved.
I. INTRODUCTION
In this paper, we describe a formulation of large N (or semi-classical) dynamics which leads to a coupled hierarchy of time-local evolution equations for equal time correlation functions.
I. INTRODUCTION
Our approach directly exploits the appropriate group structure underlying the construction of suitable coherent states and the existence of the classical limit [4] . We specifically focus on the time evolution of initial states chosen to equal one of these coherent states. We will give explicit next-to-leading order (NLO), and next-to-next-to-leading order (NNLO), expressions for the required evolution equations. Somewhat related hierarchies of evolution equations have been discussed in several recent papers [14, 15] . Because of our exploitation of the underlying group structure, the formulation we derive is more efficient, in the sense that it requires integration of fewer coupled equations at a given order in 1/N.
I. INTRODUCTION
A major question which we discuss, but do not fully resolve, is the propagation of errors induced by truncating the exact (infinite) hierarchy at a given order in 1/N. It is known that the N → ∞ limit is not uniform in time. For example, in typical large N field theories the characteristic time scales for scattering or thermalization are known to scale as N to some positive power.
I. INTRODUCTION
1 For a fixed time interval t, results obtained by integrating evolution equations truncated at, for example, next-to-leading order, will have only order 1/N 2 errors.
I. INTRODUCTION
For sufficiently large N, and fixed t, including successively higher orders in the 1/N hierarchy will yield more accurate results. But for fixed N and some given truncation of the 1/N hierarchy, it should be expected that the truncation error will grow with increasing time and eventually become order unity. A key question is how this "breakdown" time scales with N and the order of the truncation. One might hope that a next-to-leading order approximation would be useful
I. INTRODUCTION
[that is, have at most O(1/N) global errors] for times of order N, while a next-to-next-toleading order scheme would be useful out to times of order N 2 , etc. But it is quite conceivable that errors in an order-k truncation will grow with time like (t α /N) k for some positive α, which would imply that all truncations break down after a time of order N 1/α . This behavior, which we consider likely, may well depend on the specific theory and choice of initial state. Available numerical work, such as [14, 15] , sheds little light on this issue. We discuss several examples where it is possible to argue that quantum "decoherence" produces exactly this type of limit on the range of validity of large N truncations.
I. INTRODUCTION
The paper is arranged as follows. The general framework which allows us to treat many theories with a classical limit in a uniform fashion is outlined in section II. This material is largely taken from Ref. [4] . Section III describes the particular class of operators we will consider, and examines the structure of their coherent state equal time correlators. Section IV presents the resulting time evolution equations and discusses error propagation. These general results are applied to the examples of point particle quantum mechanics, and a general Ncomponent vector model, in section V. For point particle quantum mechanics, we argue that the decoherence time generically scales ash −1/2 , while for vector models it should scale as N 1/2 .
I. INTRODUCTION
A brief concluding discussion follows.
II. COHERENCE GROUP AND COHERENT STATES
The following slightly abstract framework is applicable to typical large N limits (including O(N) or U(N) invariant vector models, matrix models, and non-Abelian gauge theories), as well as theh → 0 limit of ordinary quantum mechanics [4] .
II. COHERENCE GROUP AND COHERENT STATES
Consider a quantum theory depending on some parameter χ (such ash or 1/N). The Hilbert space (which may depend on χ) will be denoted H χ . The quantum dynamics is governed by a
II. COHERENCE GROUP AND COHERENT STATES
Hamiltonian which we will write as (h/χ)Ĥ χ . This rescaling of the Hamiltonian will prove to be convenient, and makes the Heisenberg equations of motion take the form
II. COHERENCE GROUP AND COHERENT STATES
The following assumptions are a set of sufficient conditions implying that the χ → 0 limit is a classical limit.
II. COHERENCE GROUP AND COHERENT STATES
Assume there is a Lie group G (called the coherence group) which, for every value of χ, has a unitary representation on H χ , G χ = {D χ (u) : u ∈ G}. The states generated by applying elements of the coherence group to some (normalized) base state |0 χ ∈ H χ ,
II. COHERENCE GROUP AND COHERENT STATES
are called coherent states. The coherence group acts on these states in a natural way,
II. COHERENCE GROUP AND COHERENT STATES
We assume that the coherence group G χ acts irreducibly on the corresponding Hilbert space H χ . In other words, no operator (except the identity) commutes with all elements of the coherence group. This condition automatically implies that the set of coherent states form an over-complete basis for the Hilbert space H χ . It also implies that any operator acting on H χ may be represented as a linear combination of elements of the coherence group.
II. COHERENCE GROUP AND COHERENT STATES
For any operatorÂ acting in H χ , we define its symbol A χ (u) as the set of coherent state expectation values, A χ (u) = u|Â|u χ , u ∈ G. We assume that the only operator whose symbol vanishes identically is the null operator. Thus, distinct operators have different symbols, which means that any operator can, in principle, be completely reconstructed solely from its diagonal matrix elements in the coherent state basis.
II. COHERENCE GROUP AND COHERENT STATES
Classical observables will be associated with operators that remain non-singular as χ goes to zero, that is, whose coherent state matrix elements, u|Â|u
II. COHERENCE GROUP AND COHERENT STATES
Two coherent states |u and |u ′ are termed classically equivalent (we will write u ∼ u ′ )
II. COHERENCE GROUP AND COHERENT STATES
if in the χ → 0 limit, one can not distinguish between them using only classical operators,
II. COHERENCE GROUP AND COHERENT STATES
i.e., lim χ→0 A χ (u) = lim χ→0 A χ (u ′ ) for all classical operatorsÂ. We assume that the overlap between any two classically inequivalent coherent states decreases exponentially with 1/χ in the χ → 0 limit.
II. COHERENCE GROUP AND COHERENT STATES
Under these assumptions, one may show that the χ → 0 limit of this theory truly is a classical limit [4] . The assumptions hold for O(N) or U(N) invariant vector models, matrix models, and gauge theories [4] . The quantum dynamics reduces to classical dynamics on a phase space Γ given by a coadjoint orbit of the coherence group. Formally, points in Γ correspond to equivalence classes of coherent states,
II. COHERENCE GROUP AND COHERENT STATES
symplectic structure on the phase space is completely determined by the Lie algebra structure of the coherence group. The classical Hamiltonian is just the χ → 0 limit of the coherent state expectation of the quantum Hamiltonian,
II. COHERENCE GROUP AND COHERENT STATES
To have sensible classical dynamics this limit must exist, i.e.,Ĥ χ must be a classical operator.
II. COHERENCE GROUP AND COHERENT STATES
(This is why it was convenient the rescale the Hamiltonian byh/χ.) The classical action is
II. COHERENCE GROUP AND COHERENT STATES
Both the classical Hamiltonian (3) and the action (4) depend only on the equivalence class of the coherent state |u , 2 and thus do define sensible dynamics on the classical phase space.
II. COHERENCE GROUP AND COHERENT STATES
The preceding discussion is just a formalization of the usual picture of a classical limit. A quantum mechanical wave packet, with a width of order χ 1/2 , behaves classically in the χ → 0 limit, and may be associated with a point in the classical phase space. The equations of motion that govern the classical dynamics are just coherent state expectations of the original quantum evolution equations.
III. COHERENT STATE EXPECTATIONS
As noted earlier, the irreducibility of the coherence group implies that all operators may be (formally) constructed from the generators of the coherence group. Consequently, for characterizing the structure, and time evolution, of any state, one may focus attention on equal-time expectation values of products of coherence group generators.
III. COHERENT STATE EXPECTATIONS
Let g denote the Lie algebra of the coherence group G. Let {e i } be a basis of g. The commutator of basis elements defines the structure constants, [e i , e j ] = if k ij e k . The generators e i themselves are not classical operators, but rather are 1/χ times classical operators. For convenience, letx i denote the rescaled generator which is a classical operator,x i ≡ χe i .
III. COHERENT STATE EXPECTATIONS
Consider the coherent state expectation value of the monomialx i 1x i 2 · · ·x i k . We would like to find an expansion of this expectation value in powers of χ. A convenient representation for our purposes involves subtracted expectations
III. COHERENT STATE EXPECTATIONS
2 For the action, this is true up to temporal boundary terms which do not affect the dynamics. 3 To simplify notation, we will omit the superscript "(k)" when this can cause no confusion; for example, we will write g ij for g (2) ij , etc. The same remark applies to the connected expectations discussed below. where · · · denotes an expectation in some coherent state, and x i ≡ x i are the expectations of the rescaled generatorsx i . Subtracted and un-subtracted expectations are related by
III. COHERENT STATE EXPECTATIONS
where the n-tuples (l 1 , l 2 , ..., l n ) are ordered subsets of {i 1 , ..., i k }. (There is no g (1) term since
III. COHERENT STATE EXPECTATIONS
Alternatively, one may expand in terms of connected expectations,
III. COHERENT STATE EXPECTATIONS
The difference, illustrated graphically in Fig. 1 , is that expansions in terms of connected expectations involve products of all possible 'contractions', while the terms in the expansion in subtracted expectations have only one string of generators 'contracted'. The difference between subtracted and connected expectations first arises with four generators. Explicitly,
III. COHERENT STATE EXPECTATIONS
kl + x i x k g
III. COHERENT STATE EXPECTATIONS
jl + x i x l g
III. COHERENT STATE EXPECTATIONS
jk + x k x l g
III. COHERENT STATE EXPECTATIONS
ij + x j x l g
III. COHERENT STATE EXPECTATIONS
ik + x j x k g (2) il
III. COHERENT STATE EXPECTATIONS
jkl + x j g
III. COHERENT STATE EXPECTATIONS
ikl + x k g
III. COHERENT STATE EXPECTATIONS
ijl + x l g
III. COHERENT STATE EXPECTATIONS
ijk + g
III. COHERENT STATE EXPECTATIONS
The coherent state overlap u|u 
III. COHERENT STATE EXPECTATIONS
or s
III. COHERENT STATE EXPECTATIONS
···m··· .
III. COHERENT STATE EXPECTATIONS
By considering which connected expectations contribute to g (k) , one may easily see 4 that subtracted expectations fall off roughly half as fast as the connected ones, 
III. COHERENT STATE EXPECTATIONS
Now consider an operator V that can (at least formally) be expanded in powers of genera-
III. COHERENT STATE EXPECTATIONS
. The largest number of connected diagrams n occurs when all s (mi) are s (2) (except for one, if k is odd, which is s (3) ).
III. COHERENT STATE EXPECTATIONS
form are well behaved for χ → 0, and so are good classical operators. Using (6),
III. COHERENT STATE EXPECTATIONS
This can be packaged in an even more concise form,
III. COHERENT STATE EXPECTATIONS
where summation on repeated indices is implied, andV
III. COHERENT STATE EXPECTATIONS
number obtained by replacing each generator in V by its coherent state expectation. Here we
III. COHERENT STATE EXPECTATIONS
When acting on a string of generators, ordered derivatives produce a sum of products of expectations of the generators which remain after deleting the indicated generators, provided these appear (not necessarily contiguously) somewhere within the string in the order specified by the derivative. For example,
III. COHERENT STATE EXPECTATIONS
2 , and
III. COHERENT STATE EXPECTATIONS
In the χ → 0 limit, coherent state expectations of the (rescaled) generatorsx i turn into coordinates x i on the classical phase space and (classical) operators acting on H χ become functions on phase space, V =V + O(χ). For finite χ, the successive terms in (12) precisely characterize the corrections to this classical limit.
IV. TIME EVOLUTION
Since operators are completely determined by their symbols, to study the time dependence of any observableÂ it is sufficient to take the coherent state expectation value of its Heisenberg equation of motion (1),
IV. TIME EVOLUTION
. In this case, the ordering does not matter, and the n! is needed to make up for over-counting.
IV. TIME EVOLUTION
In other words, we assume that the initial state is precisely some coherent state |u , and wish to determine the subsequent time evolution. To do so, we will first find an expansion, in powers of χ, for the expectation of the commutator of classical operators.
A. Symbols of Commutators
Consider classical operators A and B which (as in Section III) may be written as power series in the generators,
A. Symbols of Commutators
Using our previous result (12), we find
A. Symbols of Commutators
We see from (15) that, to leading order, products of classical operators factorize, AB = A B + O(χ).
A. Symbols of Commutators
Using the expansion (15) , and the reduction formulas for operator derivatives (13), one can evaluate the commutator. A generic term in the result will be
A. Symbols of Commutators
The last term in the sum (16) is either g
A. Symbols of Commutators
whether k is even or odd. Using (10) to reduce the differences (g
A. Symbols of Commutators
··· ) yields the final form for the expectation of the commutator of classical operators. The leading term is precisely the Poisson bracket on the classical phase space, while subsequent terms involve successively higher expectations g (k) . Displaying subleading O(χ) and O(χ 2 ) terms explicitly, one finds
B. Equations of Motion
To determine the evolution to order O(χ 3 ), we need the time derivatives of x i (t), g ij (t), and g ijk (t). Take the commutator of products of generators with the Hamiltonian and subtract the disconnected parts to find:
B. Equations of Motion
Recall that, through third order, there is no difference between the subtracted and connected correlators. Only the disconnected parts of the fourth order correlators appearing in Eq's. (18) and (19) are needed, since
B. Equations of Motion
. If equations only accurate to O(χ 2 ) are desired, then all terms in Eq's. (18)- (20) involving third (or higher) order correlators, as well as products of second order correlators, may be dropped.
7
Given these equations of motion for the connected expectations of generators, one can use (12) to describe the dynamics of any classical operator in terms of its symbol. IfV =V ({x i })
7
is a (time-independent) function of the generators, then its time-dependent expectation value, at next-to-next-to-leading order, is given by
7
where x i (t), g ij (t) = s ij (t), and g ijk (t) = s ijk (t) are to be obtained by integrating Eq's. (18)- (20) forward in time, using
C. Error Accumulation
To any given order in χ, we have a system of non-linear, first-order, ordinary differential equations. Initial conditions are imposed by specifying x i (t=0) = u|x i |u and s
C. Error Accumulation
, and the equations forṡ (k) (t) involve only terms of order χ k−1 and higher, we still formally have
C. Error Accumulation
However, as the truncated equations of motion are integrated forward in time, errors accumulate; it is important to understand the rate of growth of this truncation error. 7 The resulting next-to-leading order equations are simply
C. Error Accumulation
and
C. Error Accumulation
We are dealing with a system of equations which we can write asẏ i = F i (y) + G i (y) where {y i (t)} are the variables in our problem (that is, the x i 's and s
C. Error Accumulation
··· 's), F (y) represents the terms we keep, and G(y) stands for everything thrown away by the truncation. Let y 0 (t) be the solution to the above equation with G ≡ 0, and solve perturbatively, y(t) = y 0 (t) + ǫ(t) with ǫ small. Linearizing about y(t) = y 0 (t), we havė
C. Error Accumulation
where
C. Error Accumulation
, and we have dropped O(ǫ 2 ) terms. This linearized system of equations is easy to solve (at least formally). For t > 0,
C. Error Accumulation
Here, T denotes time ordering (with smaller times on the right). If f (t) and g(t) are globally bounded during the time evolution, ||f (t)|| ≤f , ||g(t)|| ≤g, where || · · · || is some appropriate norm, then a crude estimate of the deviation of the true solution from the approximation is
C. Error Accumulation
Of course for t small, errors grow linearly and ||ǫ(t)|| ≤ ||ǫ(0)|| (1+f t) +gt + O(t 2 ); with a truncation good to order χ k at t = 0, both ||ǫ(0)|| andg will be O(χ k+1 ).
C. Error Accumulation
In a general treatment, it is hard to do better than the crude bound (25). In dynamical systems with only a few degrees of freedom, there typically are "regular" portions of phase space where perturbations grow only linearly with time [17] . However, it is not at all clear that this is applicable to the truncated quantum dynamics represented by Eq. (23).
C. Error Accumulation
In simple examples discussed in the following section, we will find that for times of order should always expect such decoherence to eventually set in.
V. EXAMPLES
We will discuss two examples of theories to which the preceding general results may be applied: the usual semi-classical limit of point particle quantum mechanics, and the large N limit of O(N) invariant vector models. For brevity of presentation, we will display explicitly only the first corrections to the leading classical approximation, but we emphasize that it is completely straightforward to include yet higher order corrections, such as the O(χ 2 ) terms displayed in Eq's. (18)- (20).
V. EXAMPLES
A.h → 0 Quantum Mechanics
V. EXAMPLES
Consider ordinary point particle quantum mechanics, in one dimension for simplicity. The coherence group G is the Heisenberg group, generated by {e i } = {x/h,p/h, 1/h}. The formal parameter that controls how close the theory is to the classical limit is, of course, χ =h. The rescaled generators of the coherence group,x i =he i , include the positionx and momentump operators whose expectations will serve as classical phase space coordinates. The Heisenberg group, acting on a fixed Gaussian base state, generates conventional coherent states {|p, q }, with wave functions given (up to an overall phase) by
V. EXAMPLES
We have arbitrarily chosen units such that our Gaussian base state has equal variance inx and p. Consider a Hamiltonian of the typical formĤ = 1 2p
V. EXAMPLES
2 + V (x), where, for simplicity, we have set the particle mass to unity. The equations of motion are, of course,
V. EXAMPLES
We are interested in the time evolution of x(t), p(t), and the connected correlators g xx (t), g xp (t) = g * px (t), and g pp (t), all to orderh. From equations (18) and (19) we find:
V. EXAMPLES
subject to the initial conditions
V. EXAMPLES
ih. Notice that to this order, det
V. EXAMPLES
is a constant of the motion, and Eq's. (28) are equivalent to a Gaussian variational ansatz [18] (where one approximates the wave packet by a Gaussian with a time-dependent centroid and width). However, if we went to next-tonext-to-leading order inh it would become clear that our setup is different. For positive times, higher moments will not be given by simple algebraic expressions in terms of the centroid and variance, and the details of evolution will depend on the shape of the potential. 
V. EXAMPLES
Because the potential is quadratic these are exact. Equally simple is an inverted harmonic 8 In our Gaussian initial state, all connected correlators higher than second order vanish at time zero, 
V. EXAMPLES
In both of these examples, the time-evolution of the variances are independent of x 0 and p 0 .
V. EXAMPLES
As one would expect, they oscillate (with twice the natural frequency) in the case of the simple harmonic oscillator, and grow (exponentially) for the inverted oscillator.
V. EXAMPLES
As a more complicated example, consider the problem of small oscillations in a weakly anharmonic potential,
V. EXAMPLES
The moment equations (28) becomë
V. EXAMPLES
We will solve these perturbatively; the two small parameters are βq 2 and βh. We will work to first order in βh (since we have omitted O(h 2 ) terms in the moment equations), and will display explicit results through second order in βq 2 . In principle, one could work to any order in βq 2 desired.
V. EXAMPLES
In order to keep our error estimates simple, we will treat the time as O(1) (in units where the natural frequency is unity). This means we need not worry about the appearance of secular 9 We choose the curvature of the potential at the minimum to equal unity, so that our chosen coherent states have the natural width for the unperturbed potential. This ensures that the resulting dynamics (such as oscillations of g (2) ) are not merely reflecting purely harmonic oscillations.
V. EXAMPLES
terms -terms which grow as powers of t -and may solve Eq's. (31) strictly perturbatively in the naive fashion. A straightforward calculation, with the initial conditions x(0) = q, p(0) = 0,
V. EXAMPLES
, and g xp = g * px = (cos 3t − cos t) − 18 t sin 3t + 93 t sin t − 54 t 2 cos t
V. EXAMPLES
with
V. EXAMPLES
Examining the secular terms in Eq's. (32) and (33), one sees that terms of order β k are accompanied by at most k powers of t. This is a general result. It implies that our stated condition that the time be O(1) is needlessly restrictive. For small βq 2 and βh, the perturbative expansions (32) and (33) are actually valid in the wider domain |βq 2 t| ≪ 1 and |βht| ≪ 1, provided a factor of t is included with each factor of βq 2 or βh in the error estimates.
V. EXAMPLES
It is instructive to compare this treatment with the result of a perturbative quantum mechanical calculation. Using the brute-force approach of first finding perturbed eigenstates and energy levels, and then evaluating the time-dependent expectation value x(t) by projecting the initial coherent state onto individual eigenstates and summing the resultant contributions, a rather tedious calculation using both wavefunctions and energies correct to O(β) leads to 
V. EXAMPLES
In other words, x(t) shows damped harmonic behavior, with a shifted q-dependent frequency, and with an amplitude which decays significantly on the time scale
V. EXAMPLES
This implies that on this time scale, the initially well localized wavepacket has dispersed so much that its probability distribution is spread out over most of the classically allowed region. Although it will have no bearing on our discussion, it is interesting to note that on yet longer time scales, when t is near 2π/(3βh) or integer multiplies thereof, the exponential factors in Eq. (36) return to near unity, implying that the time-dependent state has "reassembled" itself into a recognizable wavepacket oscillating in the potential. 12 Presumably, this is a reflection of the fact that this is an integrable single degree of freedom system.
V. EXAMPLES
The existence of the decoherence time scale (39) has important consequences for the utility of any truncated moment expansion, such as Eq's. (18) (19) (20) . If the wavepacket has spread to such 11 Of course, the fact that the amplitude of oscillations in the mean position x(t) decays on the decoherence time scale t d cannot mean that the wavepacket has come to rest at the bottom of the potential while remaining a well-localized wavepacket, as this would violate energy conservation. In the semi-classical regime under discussion, the position of the initial wavepacket is significantly displaced from the minimum of the potential, q 2 ≫h, implying that the total energy is large compared to the zero-point energy. Therefore, a negligible mean position at large times necessarily indicates that the wavepacket has spread so much that its probability density, at any late time, is delocalized over the entire classically allowed region, and no longer "sloshes" backand-forth to any significant extent. Within the classically allowed region, energy conservation implies that the wavefunction must have substantial variations on scales far smaller than the (square root of the) variance in position -which will be comparable to the width of the classically allowed region.
V. EXAMPLES
an extent that it is significantly sampling all of its classically allowed region, while necessarily retaining structure on smaller scales, then the formal hierarchy of connected correlators, s (k) ∼ h k−1 , will have broken down. Higher order moments will not be small compared to lower order ones. Consequently, the moment expansion presented in the previous section can only be useful for times which are small compared to the decoherence time t d .
V. EXAMPLES
The 1/ √h dependence of the decoherence time (39) may also be seen in another very simple example. Consider the free evolution of a coherent state in the absence of any potential. As is well known, the width of the wavepacket grows without bound. The evolution equations forx andp are, of course, trivial,p(t) =p(0), andx(t) =x(0) +p(0) t. Hence,x 2 (t) =
V. EXAMPLES
, and so for our initial Gaussian coherent state (with equal variance in x and p),
V. EXAMPLES
Here also, we see that for times of orderh −1/2 the hierarchy of correlators s (k) (t) ∼ O(h k ) no longer holds.
V. EXAMPLES
We believe this to be a general result. Whenever a semiclassical system exhibits dispersion, the decoherence time is expected to scale ash −1/2 , and truncations of the moment hierarchy equations (18) (19) (20) will only be accurate for times small compared to the decoherence time.
B. Vector Models

Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
vectors. For simplicity, we will assume that the degrees of freedom are all bosonic, 13 and divided into a set of canonical coordinates {x 13 Extending this discussion to U (N ) invariant fermionic models is completely straightforward. invariant states, and all physical operators can be constructed from the basic bilinearŝ
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
It will be convenient to regardx 
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Viewed as matrices,Â andĈ are symmetric, whileB is non-symmetric. The individual components ofÂ,B, andĈ are all Hermitian operators acting on H N .
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
We will take the Hamiltonian to have the general form
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
The overall factor of N (given our scaling of coordinates and momenta by 1/ √ N) is exactly what is needed to ensure that the N → ∞ limit is a classical limit in the framework of section II.
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
The potential energy function V (A) may be any chosen scalar-valued function of a symmetric matrix A. The kinetic energy takes the simple form 
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
where V (r 2 ) is now a function of just a one variable. 
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
[Here ∇ is a lattice forward difference operator, dot products denote the implicit sum over 
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Returning to the general discussion, a straightforward calculation shows that the commutators of the basic bilinears are
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
14 In terms of coordinates and momenta which have not been rescaled by
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
In other words, the commutators ofÂ,B, andĈ (as well as justÂ andB) close and these operators generate a Lie algebra. 15 The appropriate coherence group which will create suitable O(N)
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
invariant coherent states may be taken to be the group generated by (anti-Hermitian linear combinations of) the operators {Â αβ } and {B αβ }. Enlarging the coherence group by including theĈ αβ operators among the generators is equally acceptable, but unnecessary. The group generated by {Â αβ } and {B αβ } alone satisfies all the conditions for producing an over-complete set of coherent states which behave classically as N → ∞. Including theĈ αβ operators among the generators enlarges the coherence group, but has no effect whatsoever on the resulting manifold of coherent states.
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Acting on an initial Gaussian base state, the coherence group generates a set of coherent states {|z }, where z is a complex symmetric m × m matrix, with positive definite real part, which may be used to uniquely label an individual coherent state. The position space wavefunctions of these coherent states are given by
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
It will be convenient to decompose the matrix z into its real and imaginary parts by writing
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
so that a = (z+z * ) −1 and ω = i 2 (z−z * ). Both a and ω are real symmetric matrices, and a 15 The Lie algebra structure constants are f
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Bµν A αβ C γδ = δ µα δ νγ δ βδ + δ µβ δ νγ δ αδ + δ µα δ νδ δ βγ + δ µβ δ νδ δ αγ , and f 
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
The variances of these operators in the coherent state |z are
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Given our choice of Hamiltonian (43), the operator equations of motion for the basic bilinears are
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Here, V ′ is shorthand for the variation of V (A) with respect to the symmetric matrix A,
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
and is defined so that δV (A) = tr (V ′ δA).
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
17 16 For aesthetic reasons, we set g
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
17 Note that with this definition, the matrix variation V ′ reduces to an ordinary variational derivative in the case of a single vector (m=1).
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Applying the general results (18) and (19) [actually, only (21) is needed] to the case at hand, one finds in a straightforward fashion the following equations, valid to next-to-leading order in 1/N, for the time evolution of the expectation values and variances of basic bilinears,
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
together with
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Here
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
, etc. Of course, g BA γδ,αβ = g AB αβ,γδ * and so on, since the basic bilinearŝ A αβ ,B αβ , andĈ αβ are all Hermitian.
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
As they stand, the (truncated) moment equations (54) and (55) are highly redundant. This is because the operatorsÂ αβ ,B αβ , andĈ αβ are not independent when acting on the O(N) invariant Hilbert space H N . For many purposes, it is preferable to reduce the evolution equations to a smaller set of independent observables. To see the redundancy, it is convenient first to note that the actions ofB andÂ on any coherent state |z are related,
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Hence, the coherent state expectation value ofÂ −1B is directly related to that ofÂ −1 ,
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
In a similar fashion, the coherent state expectation value ofĈ may be expressed as 
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
then (57) 18 The following discussion assumes that coherent state matrix elements ofÂ −1 exist, which requires N > m+1.
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
showing that the operators {Ĉ αβ } are not independent ofÂ andB [when acting on O(N)
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
invariant states]. Inverting the definition (59) to expressB in terms ofΩ,
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
and using this, plus the Hermiticity ofΩ, allows one to rewrite expression (60) forĈ aŝ 
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
If the complex symmetric matrix z parameterizing coherent states is separated into real and imaginary parts by writing z = 
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
[The first equality was previously noted in Eq. (50).]
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Re-expressing the quantum equations of motion (52) in terms of the independent canonically conjugate operators gives
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
where the "effective" radial potential
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
equals the original potential energy augmented by a "centrifugal potential".
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
One may directly evaluate the evolution equations for expectations and variances of the canonically conjugate operatorsÂ andΩ, or equivalently (and rather tediously) rewrite the previous equations (54) and (55) in terms ofÂ andΩ. Either way, one finds
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
together with
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Initial conditions corresponding to a given coherent state |z (with z = 
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
The next-to-leading order evolution equations (67) In the special case (44) of a single vector (corresponding to a point particle moving in an N-dimensional spherically symmetric potential) one may drop all the indices and the next-toleading order evolution equations become
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
with initial conditions given by A(0) = a, Ω(0) = ω, and
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
From Eq's. (70) and (71) one may again see that to next-to-leading order, the determinant of the variance matrix on the left-hand side of (71) is a constant of the motion, det
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
To this order, our method gives exactly same predictions as the Gaussian approximation of [18] . One may, of course, systematically extend the treatment to higher order in 1/N simply by specializing the next-to-next-to-leading order results in section IV.
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
The evolution equations (70) in this single-vector case may be cast in a more transparent form by defining radial position and momentum operators viâ
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
or equivalentlyr
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
These operators are canonically conjugate,
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
and a short exercise rewriting the quantum equations of motion (65) yields
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
where
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
and U ′ eff = dU eff /dr. This is a well-known result: s-wave dynamics in an N-dimensional central potential is equivalent to one-dimensional quantum dynamics in an effective radial potential U eff containing an additional "centrifugal" potential
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
which is non-vanishing in all dimensions other than 1 and 3 [21, 22] . As seen in the commutation relations (74), the parameter 1/N plays the role ofh so that the large N limit is precisely equivalent to the semiclassical limit of ordinary one-dimensional quantum mechanics.
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
The next-to-leading order evolution equations (70) for the coherent state expectation values and variances of A and Ω may be easily be converted to equivalent next-to-leading order equations for expectations and variances of p and r. One finds,
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
Through next-to-leading order, these evolutions equations are identical to the evolution equations (28) for the usual semiclassical limit. 19 The initial variances differ, however, due to the differing shapes of the initial wavepackets (26) and (48). For our large N coherent states, 
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
[and once again det g (2) (t) = O (N −3 )]. The form of this variance matrix (including, for example, the growth in the variance g rr with increasing r) reflects the fact that the underlying O(N) invariant coherent state wavefunctions are not constant width one-dimensional Gaussians, but rather N-dimensional Gaussians centered at the origin with variable width. Hence, the position of the peak in the resulting radial probability distribution is positively correlated with the width of the radial probability distribution about this peak.
Consider an O(N) invariant theory whose fundamental degrees of freedom form O(N)
For any given choice of the potential, one may integrate the five equations (77) to study the non-equilibrium dynamics of thermalization or hydrodynamic transport, as the relevant time scales for these processes scale like N in the large N limit [16] . We hope that future work will shed light on these issues. 

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