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IMSc/2006/6/17
On obtaining classical mechanics from quantum mechanics
Ghanashyam Date1, ∗
1The Institute of Mathematical Sciences,
CIT Campus, Chennai-600 113, INDIA
Abstract
Constructing a classical mechanical system associated with a given quantum mechanical one,
entails construction of a classical phase space and a corresponding Hamiltonian function from the
available quantum structures and a notion of coarser observations. The Hilbert space of any quan-
tum mechanical system naturally has the structure of an infinite dimensional symplectic manifold
(‘quantum phase space’). There is also a systematic, quotienting procedure which imparts a bundle
structure to the quantum phase space and extracts a classical phase space as the base space. This
works straight forwardly when the Hilbert space carries weakly continuous representation of the
Heisenberg group and one recovers the linear classical phase space R2N. We report on how the pro-
cedure also allows extraction of non-linear classical phase spaces and illustrate it for Hilbert spaces
being finite dimensional (spin-j systems), infinite dimensional but separable (particle on a circle)
and infinite dimensional but non-separable (Polymer quantization). To construct a corresponding
classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian.
The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of
semiclassicality and tangentiality.
PACS numbers: 04.60.Pp,98.80.Jk,98.80.Bp
∗Electronic address: shyam@imsc.res.in
1

Developing a semi-classical approximation to quantum dynamics is in general a non-trivial
task. Intuitively, such an approximation entails an adequate class of observable quantities
(eg. expectation values of self-adjoint operators) whose time evolution, dictated by quantum
dynamics, is well approximated by a classical Hamiltonian evolution. Roughly, the adequate
class refers to (say) basic functions on a classical phase space (symplectic manifold) with a
Hamiltonian which is a function of these basic functions. The accuracy of an approximation
is controlled by how well the classically evolved observables stay close to the quantum evolved
ones within a given precision specified in terms of bounds on quantum uncertainties. Having
a description of the quantum framework as similar as possible to a classical framework is
obviously an aid in developing semi-classical approximations.
Such a description is indeed available and is referred to as geometrical formulation of
quantum mechanics [1]. The quantum mechanical state space, a projective Hilbert space,
is naturally a symplectic manifold, usually infinite dimensional (finite dimensional for spin
systems). Furthermore, dynamics specified by a Schrodinger equation is a Hamiltonian
evolution. This is true for all quantum mechanical systems. In addition, there is also a
systematic quotienting procedure to construct an associated Hamiltonian system (usually
of lower and mostly finite dimensions) which views the quantum state space as a bundle
with the classical phase space as its base space. This works elegantly when the quantum
Hilbert space is obtained as the weakly continuous representation of a Heisenberg group.
Generically these are separable Hilbert spaces and the extracted classical phase spaces are
linear, R2n.
Quantum mechanical Hilbert spaces however arise in many different ways. For example,
the (kinematical) Hilbert space of loop quantum cosmology carries a non-weakly continuous
representation of the Heisenberg group and is non-separable. For examples such as particle
on a circle and spin systems, one does not even have the Heisenberg group. A semi-classical
approximation is still needed for such systems. Likewise, in classical mechanics (even for
finitely many degrees of freedom), the classical phase space is not necessarily linear (eg
the cylinder for particle on a circle, reduced phase spaces of constrained systems etc). It
is important to develop a quotienting procedure to construct such, possibly non-linear,
classical phase spaces from more general quantum state spaces. In this work we develop
such a procedure and illustrate it for three examples: arbitrary spin-J system, particle on a
circle and Bohr or polymer quantization appearing in loop quantum cosmology (LQC). This
2

takes care of the kinematical aspects.
To construct an associated classical dynamics one has also to obtain a Hamiltonian func-
tion (an effective Hamiltonian) on the classical phase space. This is done by choosing a
section of the bundle and obtaining the effective Hamiltonian on the base space as a pull
back of the quantum mechanically defined one. An effective Hamiltonian so defined, depends
on the section chosen. One can now construct two trajectories on the classical phase space:
(a) projection of a quantum trajectory (i.e. trajectory in the quantum state space) onto the
base space and (b) a trajectory in the base space, generated by the effective Hamiltonian
function. In general, i.e. for arbitrary sections, these two trajectories do not coincide. They
do so when the section is tangential to the quantum trajectories (equivalently when the
section is preserved by quantum dynamics). Since the classical states are obtained from ex-
pectation values (via the quotienting procedure), for the classical trajectories to reflect the
quantum one, within a certain approximation, it is necessary that the quantum uncertainties
also remain bounded within prescribed tolerances. In other words, the states in the section
should also satisfy conditions of semiclassicality.
In section I, we recall the basic details of the geometric formulation from [1] and describe
the quotienting procedure in a general setting. In general the classical phase is obtained as
a sub-manifold of the base space of the bundle. The general procedure is then illustrated
with three examples in the three subsections.
Section II contains a discussion of dynamics and the conditions for developing semi-
classical approximation. The classical phase space being not the same as the base space in
general, puts a first requirement that the projection of sufficiently many quantum trajectories
onto the base space should actually be confined to the sub-manifold of classical phase space.
For constructing a Hamiltonian dynamics on the classical phase space, a section over the
classical phase space needs to be chosen which provides an embedding of the classical phase
space into the quantum state space. For having a useful semiclassical approximation the
section has to satisfy two main conditions of semiclassicality and tangentiality. The state
space of polymer quantization brings out further points to pay attention to while developing
a semi-classical approximation.
In the last section we conclude with a summary and a discussions.
3

I. KINEMATICAL SET-UP
This section is divided in five subsections. The first one recalls the symplectic geometry
of the quantum state space. This is included for a self contained reading as well as for fixing
the notation, experts may safely skip this section. More details are available in [1, 2]. The
second one describes the quotienting procedure in some generality (but still restricted to
finitely many classical degrees of freedom) to get candidate classical phases spaces which
could also be non-linear. The next three sub-sections illustrate the quotienting procedure
for the examples of spin-j system, particle on a circle and isotropic, vacuum loop quantum
cosmology.
A. Symplectic geometry of (projective) Hilbert space
Let H be a complex Hilbert space, possibly non-separable and let P be the corresponding
projective Hilbert space: the set of equivalence classes of non-zero vectors of the Hilbert
space modulo scaling by non-zero complex numbers. Equivalently, if S denotes the subset
of normalized vectors, then P = S/phase equivalence. Unless stated otherwise, the Hilbert
space is assumed to be infinite dimensional.
Any complex vector space can be viewed as a real vector space with an almost complex
structure defined by a bounded linear operator J, J2 = −I defined on it and multiplication
by a complex number a+ ib being represented as a+bJ . The Hermitian inner product of the
complex Hilbert space can then be expressed in terms of a symmetric and an anti-symmetric
quadratic forms, G(·, ·) and Ω(·, ·) respectively. Explicitly,
〈Ψ,Φ〉 := 12~G(Ψ,Φ) + i2~Ω(Ψ,Φ). (1)
The non-degeneracy of the Hermitian inner product then implies that G and Ω are (strongly)
non-degenerate. The real part G will play no role in this paper.
For a separable Hilbert space, let |en〉 denote an orthonormal basis so that we have
|Ψ〉 =
∑
n ψn|en〉, ψn ∈ C. Writing ψn := xn + iyn and using the definitions of G,Ω,
one can see that 12~G(Ψ,Ψ′) =
∑
n xnx
′n + yny′n and 12~Ω(Ψ,Ψ′) =
∑
n xny
′n − ynx′n.
Viewing (xn, yn) as (global) ‘coordinates’ on H one can see that G corresponds to the
infinite order identity matrix while Ω corresponds to the infinite dimensional analogue of
the canonical form of a symplectic matrix. The coordinates are Darboux coordinates with
4

yn, xn as generalized coordinates and momenta, respectively. A change of basis effected by
a unitary transformation just corresponds to an orthosymplectic transformation, analogue
of the result that U(n,C) is isomorphic to OSp(2n,R). A separable Hilbert space can thus
be viewed as the N → ∞ form of the usual phase space R2N . A non-separable Hilbert space
does not admit a countable basis and an identification as above is not possible. Much of the
finite dimensional intuition from R2N can be borrowed for separable Hilbert spaces.
The real vector space can naturally be thought of as a (infinite dimensional) manifold
with tangent spaces at each point being identified with the vector space itself. Explicitly, a
tangent vector |Φ〉 at a point Ψ ∈ H acts on real valued functions, f(Ψ) : H → R as,
|Φ〉|Ψ(f) := limǫ→0
f(Ψ+ǫΦ)−f(Ψ)
ǫ (2)
Clearly, every vector in H can be viewed as a constant vector field on H viewed as a
manifold. The Ω introduced above is then a non-degenerate 2-form on H which is trivially
closed and hence defines a symplectic structure on H. This immediately allows one to define,
for every once differentiable function f(Ψ), a corresponding Hamiltonian vector field, Xf ,
via the equation: Ω(Xf , Y ) = Y (f) ∀ vector fields Y . Non-degeneracy of Ω implies that
Hamiltonian vector fields are uniquely determined. The Poisson bracket between two such
functions, f, g, is then defined as: {f, g}q := Ω(Xf , Xg).
Next, for every self adjoint operator Fˆ , we get a non-constant vector field XF |Ψ :=
− i~Fˆ |Ψ〉 and in analogy with the Schrodinger equation, it is referred to as a Schrodinger
vector field. This vector field turns out to be a Hamiltonian vector field i.e. there exist
a function f(Ψ) such that Ω(XF , Y ) = df(Y ) := Y (f) for all vector fields Y . From the
definition of XF and of Ω in terms of the inner product, it follows that Ω(XF , Y )|Ψ =
〈FˆΨ, Y |Ψ〉 + 〈Y |Ψ, FˆΨ〉. This is exactly equal to Y (f) for all vector fields Y provided we
define f(ψ) := 〈Ψ, FˆΨ〉. Thus, every self-adjoint operator defines a Schrodinger vector field
which is also Hamiltonian with respect to the symplectic structure and the corresponding
‘Hamiltonian function’ is the expectation value (up to the norm of Ψ) of the operator. This
function is quadratic in its argument and is invariant under multiplication of Ψ by phases.
We will mostly be concerned with such quadratic functions. For any two such quadratic
functions, f(Ψ) := 〈Ψ, FˆΨ〉, g(Ψ) := 〈Ψ, GˆΨ〉, the Poisson bracket {f, g}q = Ω(XF , XG)
evaluates to 〈 1i~
[
Fˆ , Gˆ
]
〉. It follows that the quantum mechanical evolution of expectation
5

value functions is exactly given by a Hamiltonian evolution:
d
dtf(Ψ) =
d
dt〈Ψ, FˆΨ〉 = 〈Ψ, (i~)
−1
[
Fˆ , Hˆ
]
Ψ〉 = {f(Ψ), h(Ψ)}q (3)
This shows how the quantum dynamics in a Hilbert space can be viewed as Hamiltonian
dynamics in an infinite dimensional symplectic manifold.
Quantum mechanical state space consists of rays or elements of the projective Hilbert
space. One can import the Hamiltonian framework to the projective Hilbert space as well
[1, 2]. In the first step restrict attention to the subset S of normalized vectors (the quadratic
functions defined above now exactly become the expectation values). The projective Hilbert
space can then be viewed as equivalence classes of the relation: Ψ ∼ Ψ′ ⇔ Ψ′ = eiαΨ. There
is a natural projection ρ : Ψ ∈ S → [Ψ] ∈ P and the natural inclusion i : S → H. With
the inclusion map we can pull back the symplectic form Ω to S on which it is degenerate.
Furthermore, the degenerate subspace gets projected to zero under ρ∗ and hence, there exist
a non-degenerate 2-form ω on P such that i∗Ω = ρ∗ω. This is also closed and thus endows
the projective Hilbert space with a symplectic structure [1, 2]. The quadratic functions
defined on H are automatically phase invariant and thus project down uniquely to functions
on P. In particular, we also get Poisson brackets among the projected quadratic function
satisfying {f, g}q = ω(Xf , Xg). In this manner, one obtains a Hamiltonian description of
quantum dynamics.
B. Quotienting Procedure
Now consider a set of self-adjoint operators Fˆi, i = 1, · · · , n with the corresponding
quadratic functions fi : P → R. Define an equivalence relation on P which identifies
two states if they have the same expectation values, fi, of all the Fˆi operators. Let Γ de-
note the set of equivalence classes. The values xi = fi(Ψ) naturally label the points of Γ.
We will assume that Γ can be viewed as a region in Rn with xi serving as coordinates. In
other words, the topology and manifold structure on Γ is assumed to be inherited from the
standard ones on Rn. The manifold structure of the quantum phase space is discussed in
[2]. There is a natural projection π : P → Γ.
We have two natural subspaces of TΨP: (a) the vertical subspace VΨ consisting of vectors
which project to zero i.e. π∗vΨ = 0 ∈ Tπ(Ψ)Γ and (b) the subspace (VF )Ψ which is the span of
6

the Hamiltonian vector fields {Xfi, i = 1, · · · , n}. For notational simplicity, we will suppress
the suffix Ψ on these subspaces. The vertical vectors naturally annihilate functions which
are constant over the fibre and are thus tangential to the fibre.
Let V⊥F , denote the ω-complement of VF i.e. v ∈ V⊥F implies that ω(Xfi, v) = v(fi) = 0 ∀ i.
Since fi are constant over a fibre, every vertical vector satisfies this condition and thus
belongs to V⊥F . Are there vectors in V⊥ which are not vertical? This is a little subtle. It is
easy to see that a vector in V⊥F annihilates all functions polynomial in fi. One can consider
a function algebra generated by fi with suitable restrictions on fi (and hence on Fˆi) and
completed with some suitable norm. Then all elements of such a function algebra will also
be annihilated by elements of V⊥F . All these functions are of course constant over the fibre. If
the class of functions annihilated by the vertical vectors coincides with the function algebra,
then the subspace V⊥F coincides with the vertical subspace otherwise V ⊂ V⊥F . We will ignore
such fine prints and simply assume that appropriate choices can be made so that the vertical
subspace is the ω-complement of VF : V = V⊥F . Using this identification, we will refer to
V⊥F as the vertical subspace. At this stage, we do not know if VF is symplectic subspace
or not. If and only if VF is symplectic (i.e. ω restricted to VF is non-degenerate), one has
(i) VF ∩ V⊥F = {0} and (ii) TΨ(P) = VF ⊕ V⊥F . If, however, VF is not symplectic, then the
intersection of the symplectic complements is a non-trivial subspace and the vector sum of
the two is a proper subspace of the tangent space1. In the symplectic case, it is appropriate
to refer to VF as the horizontal subspace. In the non-symplectic case, this terminology for
VF is in-appropriate since there are tangent vectors which are not in VF + V⊥F , are also
intuitively ‘horizontal’ or transversal to the fibre. In general, we will refer to vector fields
valued in VF as basal vector fields.
These two cases are precisely distinguished by the non-singularity and singularity of the
matrix of Poisson brackets, Aij := ω(Xfi, Xfj) = {fi, fj}q respectively. This is because, if
the matrix Aij is singular, then there exist linear combinations, Ya := αiaXfi, a = 1, · · · , m,
of vectors of VF which are in V⊥F (i.e. αiaAij = 0 has non-trivial solutions) and thus have a
non-trivial intersection. Clearly, VF is not a symplectic subspace in this case.
Suppose now that in addition, Fˆi are closed under the commutator or equivalently fi
1 This follows by noting that for any non-trivial subspace, W , of a symplectic space, V , (W + W⊥)⊥ =
W ∩W⊥ and V ⊥ = {0}.
7

are closed under {·, ·}q. Then, independent of the (non-)singularity of Aij , the following
statements are true by virtue of the Poisson bracket closure property of fi : (i) commutator
of two vertical vector fields is a vertical vector field and hence the vertical subspaces define
an integrable distribution on P (this of course does not need the closure property); (ii) the
commutator of two basal vector fields is a basal vector field because of the Poisson bracket
closure and hence the VF subspaces also define an integrable distribution on P; (iii) finally,
the commutator of a vertical vector field and a basal vector field is a vertical vector field since
ω(Xfi, [v,Xfj ]) = [v,Xfj ](fi) = v({fj, fi}q) = 0. Thus, LvX ∈ V⊥F , ∀X ∈ VF , ∀v ∈ V⊥F .
The last equality follows from the closure property of the Poisson brackets. Due to this last
property, the push-forward π∗ of X ∈ VF from any point along a fibre, gives the same (i.e. a
well defined) vector field on Γ. The vectors Ya project to the ‘zero’ vector field on Γ. Thus
π∗(VF ) is a subspace of dimension (n−m) of the tangent space of Γ.
One can make the projection π∗ explicit in the following manner. Consider projection of
Xfi. By definition, for functions f(xi) on Γ, [π∗Xfi ](f) = Xfi(π∗f). The pull-back function
is constant over the fibre and it depends on Ψ only through fi(Ψ). Hence,
Xfi(π∗f) = ∂π
∗(f)
∂fj Xfi(fj) = {fi, fj}q
∂f
∂xj = Aij ∂∂xj f. (4)
Thus, ξi := [π∗Xfi] = Aij ∂∂xj . It follows immediately that [π∗Ya] = αiaAij ∂∂xj = 0 as noted
above. This also implies that only (n − m) of these vectors in T~x(Γ) are independent.
Noting that Aij being linear combinations of the fi’s, are constant along the fibres and thus
descend to Γ as the same linear combinations of xi. Computing the commutator of the ξi
directly and using the Jacobi identity satisfied by the structure constants, one can verify
that the projected vectors fields, π∗Xfi are also closed under commutators (with the same
structure constants). Thus, π∗(VF ) define an integrable distribution on Γ. The integral
sub-manifolds are candidate classical phases spaces, Γcl, with a symplectic form α (defined
below) satisfying, ω = π∗α.
Let ZI := βiIXfi , I = 1, · · · , (n −m) be independent linear combinations of the vector
fields Xfi which are closed under the commutator bracket and Aij is nonsingular on their
span. Their projections are given as ζI := [π∗ZI ] = βiIξi and are tangential to the integral
sub-manifolds. If, in addition, the ζI ’s commute, then the parameters of their integral
curves provide local coordinates on these integral sub-manifolds. In some cases (see the
examples discussed in the subsections), the integral sub-manifolds of Γ are defined by m
8

equations, φa(xi) = constant, with the functions satisfying ζI(φa) = 0 ∀I. The integral sub-
manifolds then are embedded sub-manifolds. In general, however, one only gets immersed
sub-manifold.
Now we would like to define a symplectic form on Γcl. This can be done in two steps.
Let s : Γ → P be a section. From this one gets the pull-back, ω˜ := s∗ω on Γ. This is a
closed two form but degenerate if Aij is degenerate. Since any Γcl is a sub-manifold, we also
get a closed two form α on Γcl via the pull-back of the inclusion map, α := i∗ω˜ = i∗ ◦ s∗ω.
Explicitly,
α(ζI , ζJ) = ω˜(i∗ζI , i∗ζJ) = ω˜(ζI , ζJ) = ω(s∗ζI , s∗ζJ) . (5)
Since π∗s∗ζI = ζI , one sees that s∗ζI = ZI + vI for some vI ∈ VF ∩ V⊥F . Clearly, ω(ZI +
vI , ZJ + vJ) = ω(ZI , ZJ) and α is well defined. That ω(vI , vJ) = 0 follows because both
vectors are vertical as well as basal.
The 2-form α also turns out to be independent of the section chosen. To see this, observe
that Lie derivative of ω along a vertical vector field v, when evaluated on basal vector fields
X, Y , vanishes:
[Lvω](X, Y ) = [d(ivω)](X, Y ) = X(ω(v, Y ))− Y (ω(v,X))− ω(v,LXY ) = 0 . (6)
Here we used the facts that the vertical and the basal spaces are symplectic complements
and that the commutator of basal vectors is basal. Thus, the 2-form α is well defined,
independent of section, closed since it is a pull-back of a closed form and non-degenerate
because ω is non-degenerate on the subspace spanned by ZI ’s. The definition of α is extended
to all vector fields on the integral sub-manifolds by linearity. Thus each of the sub-manifolds
is now equipped with a symplectic structure and is a candidate classical phase space which
will be generically denoted as Γcl. Note that the symplectic structure on the integral sub-
manifolds is independent of the section chosen in the intermediate steps. It however, depends
on the particular Γcl, via the inclusion map.
If Aij is non-singular, Γ itself is the classical phase space. Consider the usual case of
{Fˆi} = {Qˆa, Pˆa, Iˆ, a = 1, · · · , m} forming the Heisenberg Lie algebra. The corresponding
quadratic functions are qa(Ψ), pa(Ψ) and the constant function with value 1. The number of
corresponding Hamiltonian vector fields are however one less, since X81′ = 0. Furthermore,
the space Γ is a single hyper-plane in R2m+1. We can thus take Γ = R2m and focus only on
the non-trivial functions. The matrix of Poisson brackets is then non-singular. There are
9

no vectors which are both basal and vertical and Γ itself is the classical phase space. This
case is discussed in detail in [1, 2].
In summary: In this subsection, we have seen that for every Lie algebra defined by the
self adjoint operators Fˆi, one can develop a natural quotienting procedure and construct a
classical phase space Γcl. The classical phase space is in general an immersed sub-manifold
of the space of equivalence classes, Γ. This procedure is capable of yielding linear as well
as non-linear classical phase spaces. The linear versus non-linear cases are distinguished by
the (non-)singularity of the matrix of Poisson brackets, Aij. The vector fields on Γ which
connect different Γcl are projections of vector fields which are valued in TΨ(P)− (VF +V⊥F ).
The choice of the basic operators is naturally made if the Hilbert space itself is obtained as
a representation of the corresponding Lie group (as one would do in the reverse process of
quantization). If however, the Hilbert space is not so chosen, then the choice is to be made
appealing to the purpose of constructing a classical phase space. Our reason for constructing
a classical phase space has been to develop a semiclassical approximation and this involves
dynamics as well. Thus, the choice of algebra will be dictated by the quantum dynamics
and its semiclassical approximation sought.
The general procedure given above is illustrated in three simple examples in the next
three subsections.
C. Spin-j system
The Hilbert space is complex 2j + 1 dimensional or real (4j + 2) dimensional and the
projective Hilbert space is CP2j . As basic operators we choose three hermitian matrices
Si satisfying [Si, Sj] = i~ǫijkSk. We also have the relation, 〈Ψ|
∑
Sˆ2i |Ψ〉 = j(j + 1)~2.
Furthermore, 〈Sˆ2i 〉 = (xi)2 + ∆S2i ≥ (xi)2. Therefore, r2 :=
∑
(xi)2 ≤ j(j + 1)~2.
The equivalence relation is defined in terms of Si(Ψ) := Ψ†SiΨwhich gives a 3 dimensional
Γ and VS subspaces of tangent spaces of the CP 2j. Correspondingly, the symplectic structure
gives ω(XSi, XSj) = {Si, Sj}q = ǫijkSk. Clearly VS cannot be a symplectic subspace and
indeed Y := Si(Ψ)XSi is also a vector field valued in V⊥S . Denoting the coordinates on Γ by
xi = Si, the projections are given by, ξi := π∗XSi = ǫijkxk∂j . These projected vectors are
not independent. Let ~βI , I = 1, 2 be any two, 3 dimensional vectors and define ζI := βiIξi.
Their commutator is given by [~x · ~βJ βiI∂i − ~x · ~βI βiI∂i]. Clearly, if the two vectors ~βI are
10

orthogonal to the ‘radial vector’ ~x, then the two vector fields commute. Let us further choose
the two vectors ~βI to be mutually orthogonal in anticipation. The two dimensional integral
sub-manifolds are defined by φ(~x) = φ(
∑
xixi) = constant, which are 2-spheres as expected
for Γcl. Note that this is true for all spins.
Let us compute the induced symplectic structure. The 2-form α is defined by,
α(ζI , ζJ) := βiIβjJ ω˜(ξi, ξj) = βiIβjJω(XSi, XSj ) = βiIβjJǫijkSk = ~r · ~βI × ~βJ . (7)
Thus we have several spheres of radii r as candidate classical phases spaces and the
normalization of the induced symplectic structure depends on both the particular sphere as
well as arbitrary magnitudes of the two ~βI . The arbitrariness due to the magnitudes can be
fixed by the choice of coordinates provided by the integral curves of the commuting vector
fields. The normalization of βI can be deduced by requiring ζ1 := ∂θ, ζ2 := ∂φ where θ, φ
are the usual spherical polar tangles. This requirement together with orthogonality of ~r, ~βI ,
fixes the normalizations completely and leads to α(∂θ, ∂φ) = r sin(θ).
For the case of j = 1/2, the Hilbert space is four (real) dimensional, the subset of
normalized vectors, S is the three dimensional sphere and the state space P is the two
dimensional sphere. The space Γ itself becomes two dimensional, the fibre over each point
of Γ is zero dimensional. Thus there are no vertical vectors and the subspace VS is two
dimensional and symplectic. Thus Γcl = Γ holds.
D. Particle on a circle
In this case the Hilbert space is the space of periodic (say) square integrable complex
functions on the circle. We have three natural operators forming a closed commutator
sub-algebra:
〈φ|n〉 := 1√
2π
einφ, n ∈ Z basis functions (8)
ĉos|n〉 := 1
2
{|n+ 1〉+ |n− 1〉} ; (9)
ŝin|n〉 := 1
2i {|n+ 1〉 − |n− 1〉} ;
p̂|n〉 := ~n|n〉
[
ĉos, ŝin
]
= 0 (10)
11

[ĉos, p̂] = −i~ ŝin
[
ŝin, p̂
]
= i~ ĉos
ĉos ĉos + ŝin ŝin = I (11)
Thus, for Fˆi we choose the ĉos, ŝin and pˆ operators and denote the corresponding quadratic
functions fi(Ψ) by cos(Ψ), sin(Ψ) and p(Ψ). The Poisson brackets among these is obtained
via {fi, fj}q = ω(Xfi, Xfj) = 〈 1i~[Fˆi, Fˆj]〉. From the (11) it is easy to see that
〈cos2〉+ 〈sin2〉 = 〈cos〉2 + 〈cos〉2 + (∆cos)2 + (∆sin)2 = 1. (12)
The space Γ defined as the set of equivalence classes of rays having the same expectation
values of the three basic operators, is the region in R3 : (−1 < x < 1,−1 < y <
1,−∞ < z < ∞) where x = cos, y = sin, z = p and 0 ≤ x2 + y2 < 1. Notice that the
uncertainties in the trigonometric operators are also bounded from above.
It is easy to see that the span of the three Hamiltonian vector fields on the projective
Hilbert space, Xcos, Xsin, Xp also contains a vector field, Y = cosXcos + sinXsin, which is
valued in V⊥F . From π∗Xfi = {fi, fj}q ∂∂xj it follows that the projection of these vectors fields
to Γ are given by,
π∗Xcos = − y
∂
∂z , π∗Xsin = x
∂
∂z , π∗Xp = − x
∂
∂y + y
∂
∂x , π∗Y = 0 . (13)
Clearly all three projected vectors at any point of Γ are not independent and the ‘radial’
vector field x∂x + y∂y on Γ is not a projection of any vector field valued in VF + V⊥F . As
an example, the two independent vectors ζ1 := π∗(−sinXcos + cosXsin) and ζ2 := π∗Xp
commute. The two dimensional vector spaces spanned by these, define a two dimensional
integrable distributions on Γ. Introducing the usual polar coordinates in the x − y plane,
x := rcosφ, y := rsinφ, the two vector fields can be expressed as ζ1 = r2∂z, ζ2 = −∂φ.
The integral sub-manifolds are defined by the integral curves of these vector fields and are
characterized by r = constant(6= 0). Thus the classical phase space manifolds are cylinders
as expected.
To obtain a symplectic structure α, on any of these cylinders, define α(ζ1, ζ2) :=
ω˜(−sinXcos + cosXsin, Xp) = −sin{cos, p} + cos{sin, p} = sin2 + cos2 = r2. This is very
much like the usual symplectic structure on a cylinder. Indeed, the polar coordinates intro-
duced suggest that φ and pφ := z/r2 be identified as the usual canonical variables on the
cylinder.
12

For r = 0, of course one does not have a classical phase space (the cylinder degenerates
to a line). Since we are interested in constructing classical phases spaces with a view to
constructing a semiclassical approximation, we exclude the degenerate case. This also means
that we exclude the fibres over the line (x = 0 = y). These fibres consists of all normalized
states of the form |Ψ〉 = ∑n∈Z ψn|n〉 with the coefficients satisfying the condition that if
Ψn 6= 0 for some n then Ψn±1 = 0. Analogous considerations will be more relevant in the
next example.
In both the examples above, the Hilbert space has been separable and the classical phase
spaces are obtained as embedded sub-manifolds of Γ. Both these features will change in the
next example.
E. Isotropic loop quantum cosmology
This is an example with a non-separable Hilbert space [3, 4] but in many ways still
similar to the particle on a circle example. We can use the so-called triad representation
whose basis states are labelled by eigenvalues of the self-adjoint triad operator p̂ which take
all real values with corresponding eigenstates being normalized. Unlike the particle on a
circle example, the Hilbert space is not made up of periodic (or quasi-periodic) functions
on a circle, but consists of almost periodic functions of a real variable c. The Hilbert space
carries a non-weakly-continuous unitary representation of the Heisenberg group such that
there is no operator ĉ generating translations of eigenvalues of p̂. However, exponentials of
ic are well defined operators. In this regard, this is similar to the particle on a circle. Thus
we can define:
〈c|µ〉 := 1√
2π
eiµc, µ ∈ R basis functions (14)
ĉosλ|µ〉 :=
1
2
{|µ+ λ〉+ |µ− λ〉} ; (15)
ŝinλ|µ〉 :=
1
2i {|µ+ λ〉 − |µ− λ〉} ;
p̂|µ〉 := ~µ|µ〉
[
ĉosλ, ŝinλ
]
= 0 (16)
[ĉosλ, p̂] = −i~ λ ŝinλ[
ŝinλ, p̂
]
= i~ λ ĉosλ
13

ĉosλ ĉosλ + ŝinλ ŝinλ = I (17)
The label λ indicates that we can define such ‘trigonometric’ operators for every non-
zero real number λ. Since ĉos−λ = ĉosλ and ŝin−λ = −ŝinλ, we will take λ > 0. We will
denote by Lλ, the Lie algebra defined by (16) for any fixed λ. In the circle example also
we could define such operators, but the periodicity would restrict the λ to integers. The
trigonometric operators for different λ’s of course commute. The Hilbert space carries a
reducible representation of the Lie algebra Lλ for every λ while every (countable) subspace
Ha,λ, spanned by vectors of the form {|a + kλ〉, k ∈ Z}, gives an irreducible representation
of Lλ, for every a ∈ [0, λ).
The presence of λ leads to ζ1 = λr2∂z, ζ2 = −λ∂φ. The integral curves are defined by
constant r and φ˙ = −λ. The integral curves of ζ2 are clearly closed and we can choose the
scale of ζ2 so that the curve parameter itself is φ, i.e. redefine ζ2 := −λ−1ζ2. We see that
the classical phase space is the cylinder as in the previous example.
But the expected classical phase space is supposed to be R2 whose Bohr quantization
leads to the non-separable quantum Hilbert space. How does one see the linear classical
phase space?
Now, unlike the previous example where the trigonometric operators must induce shifts
by integers to respect the periodicity, here we have more possibilities. We could consider
enlarging the set of basic operators by including ĉosλ, ŝinλ for n distinct values of λ. Thus
our space of equivalence classes, Γ will be (2n + 1) dimensional. One can see easily that
there are now (2n−1) independent vectors in VF which will project to zero vector on Γ and
we will be left with exactly two independent vector fields, closed under commutator, on Γ.
Explicitly, these vector fields can be chosen as: ζ1 = π∗(
∑
i λi{−sinλiXcosλi +cosλiXsinλi}) =∑
i λi{(xi)2 + (yi)2}∂z and ζ2 = π∗Xp =
∑
i{−λi(xi∂yi − yi∂xi)}. There is freedom available
in defining ζ1, but ζ2 has freedom of only overall scaling. In the (xi, yi) planes, we can
introduce the polar variables (ri, φi) and see that the integral curves of ζ1 are along the
z-direction while those of ζ2 are defined by the equations: ri = constanti( 6= 0), φ˙i = −λi
i.e. winding curves on an n-torus, T n. Evidently, for λi with irrational ratios, these integral
curves are non-periodic. Thus we get a map of R into the T n which however cannot be an
embedding since the induced topology on the winding curve is not the standard topology
on R. One has only an immersion. Combining with the curve parameter of ζ1, one has an
14

immersion of R2 and this is adequate to define the symplectic form α. In effect we obtain
the classical phase space which is topologically R2 and is immersed in an R × T n ⊂ Γ.
The symplectic form is computed as before and leads to, α(ζ1, ζ2) =
∑
i λ2i r2i . Denoting
the curve parameter for the vector field ζ2 by Q and defining its canonically conjugate
variable as P := z/(∑i λ2i r2i ), one has the usual symplectic form on R2.
If all the λi happen to be rational numbers (or rational multiples of a single irrational
number), then the integral curves of ζ2 are closed and one would obtain the classical phase
space to be the cylinder (or a covering space thereof). Clearly, to obtain the planar phase
space, one must have at least two λ’ whose ratio is an irrational number with the correspond-
ing ri 6= 0.
Here we find an example where depending upon the choice of basic operators or more
precisely the parameter(s) λi, we can obtain two different topologies for the classical phase
space. We also see that although the choice of basic operators can vary widely, one always
obtains a two dimensional classical phase space which is generically R2.
All these mathematical procedures of constructing ‘classical’ phase spaces from the quan-
tum one become physically relevant for studying semiclassical approximation(s) only when
a dynamics is stipulated and attention is paid to the uncertaintities in the basic operators
Fˆi. These ultimately decide which choice of the Lie algebra of Fˆi is useful. We address these
in the next section.
II. DYNAMICS AND SEMICLASSICALITY
In this section, we discuss how a quantum evolution taking place in the quantum phase
space can generate a Hamiltonian evolution on a classical phase space constructed in the
previous section. This will naturally involve a choice of a ‘section’. While, every section
will generate a classical dynamics, further conditions have to be imposed for the classical
dynamics to be a good approximation to the quantum dynamics. These are the conditions
of ‘semiclassicality’ and ‘tangentiality’. The fact that the classical phase of the isotropic cos-
mology arises as an immersed manifold in the base space, introduces further considerations
which are discussed next.
So far we just constructed a ‘classical’ phase space, Γcl, selecting a sub-algebra of self-
adjoint operators and using the naturally available symplectic geometry of the quantum
15

state space. This is completely independent of any dynamics. Consider now a quantum
dynamics specified by a self-adjoint Hamiltonian operator, Hˆ. The bundle structure allows
one to project any quantum trajectory generated by XH , to a trajectory in the base space
Γ.
In the general case where a Γcl is a submanifold of Γ, the first problem is that projected
trajectories may not even be confined to Γcl i.e. π∗(XH) /∈ π∗(VF ). Notice that XH is not
‘constant’ along the fibres (LvXH is not vertical) and therefore we need to specify the points
on the fibres. If there are no quantum trajectories whose projections remain confined to some
Γcl, then the choice of the Lie algebra of Fˆi used in the quotienting procedure, in conjunction
with the Hamiltonian, is inappropriate for developing a semiclassical approximation and one
has to look for a different choice. Let us assume that one has gone through this stage and
found a suitable Lie algebra so that there are at least some quantum trajectories whose
projections are confined to Γcl. Note that when Γcl = Γ, as in the usual case of Heisenberg
Lie algebra, this issue does not arise at all.
The second issue is whether there are ‘sufficiently many’ quantum trajectories whose
projections are confined to Γcl. Sufficiently many would mean that projection of these
trajectories will be at least an open set of Γcl. This in turn allows the possibility that at
least a portion of the classical phase space can be used for a semiclassical approximation.
Let us also assume this.
The third issue is whether the projected trajectories are generated by a Hamiltonian
function on Γcl and whether such a Hamiltonian function can be constructed from the
quantum Hamiltonian function H(Ψ) = 〈Ψ|Hˆ|Ψ〉 already available on the quantum state
space. This function however is not constant along the fibres and cannot be ‘projected’
down to the base space. One has to choose a section: s : Γ → P, π ◦ s = id. There are
infinitely many choices possible. However, for every such choice, we can define a function
H˜ : Γ → R as the pull back of 〈Ψ|Hˆ|Ψ〉|s(Γ). The restrictions of these functions to any of
the sub-manifolds, Γcl would be candidate effective Hamiltonians, Heff .
In this regard, we would like to note a simple fact. Let s : Γ → P be a sec-
tion (possibly local) and let XH be a Hamiltonian vector field on P, H(Ψ) := 〈Ψ|Hˆ|Ψ〉.
Let ξ := π∗(XH |s(Γ)) , H˜(xi) := s∗(H|s(Γ)). The (symplectic) 2-forms are related as
ω := π∗ω˜ ↔ ω˜ = s∗ω. Then it is true that ξ is a Hamiltonian vector field (on Γ) with
16

respect to ω˜, with H˜ as the corresponding function. This follows as,
ω˜(ξ, ξi) = ω(XH , Xfi) = Xfi(H) = π∗(Xfi)(H˜) = ξi(H˜) ∀ i. (18)
Since any Γcl is a sub-manifold of Γ we have natural projection (restriction) and inclusion
maps so that we obtain the restriction of ξ to tangent spaces of Γcl as the Hamiltonian
vector field with i∗H˜ as the corresponding function. The net result is that for any section
and any Hamiltonian vector field, we can always obtain a classical Hamiltonian description.
Since there are infinitely many sections, we have infinitely many classical dynamics induced
from a given quantum dynamics. Is any of these dynamics a ‘good’ approximation to the
quantum dynamics? For this we have to invoke further conditions on the sections.
The expectation value functions are naturally observables and our restriction to expec-
tation values of a subset of the observables Fˆi, amounts to saying that in a given situation
and with a given experimental capability, we can discern the quantum dynamics only in
terms of these few ‘classical’ variables. With better experimental access, we may need more
such variables. Generically, these observables also have quantum uncertainties and pro-
vided these uncertainties are smaller than the observational precision, we can ignore them,
thereby justifying a classical description. Obviously, such a property will not be exhibited by
all quantum states and not for all times. Consequently, one first defines a quantum state to
be semi-classical with respect to a set of observables Fˆi provided the uncertainties (∆Fi)Ψ,
are smaller than some prescribed tolerances δi. For such a notions to be observationally
relevant/useful, semi-classical states so defined must evolve, under quantum dynamics, into
semi-classical states for sufficiently long durations. Since we identified a classical phase space
using equivalence relation specified by a set of functions fi(Ψ) := 〈Ψ|Fˆi|Ψ〉, closed under
the {·, ·}q, these are naturally the candidate observables with respect to which we can define
semiclassicality of a state.
On each fibre then, we could find subsets for which the uncertainties would be smaller
than some specified tolerances δi 2. Quantum states within these subsets would be semi-
classical states. Let us assume that we identify such bands of semiclassicality on each fibre.
Our sections should intersect each fibre within its semi-classical band(s). This still leaves
2 The tolerances could be prescribed to vary over different regions of Γ. For instance, for larger expectation
values, one could have less precision and thus permit larger values of δi.
17

an infinity of choices and is still not enough to have the classical trajectories generated by
an effective Hamiltonian to approximate the projections of quantum trajectories.
That the projection of a Hamiltonian vector field restricted to s(Γ) is also a Hamiltonian
vector field on Γcl follows independent of whether or not the Hamiltonian vector field XH
is tangential to the section. However, tangentiality of the Hamiltonian vector field XH to a
section is necessary so that projection of a quantum trajectory in P, generated by XH , gives
the corresponding classical trajectory (in Γ) generated by ξH˜ . If tangentiality fails, then the
pre-image of a classical trajectory, will not be a single quantum trajectory i.e. pre-images of
nearby tangent vectors of a classical trajectory will belong to different quantum trajectories.
Once XH is tangential to s(Γ), the classical evolution in Γ will mirror the quantum evolution
(and by assumption made at the beginning, classical evolution will be confined to Γcl).
Thus, to develop a semi-classical approximation what is needed is to guess or identify a
classical phase space Γcl such that sufficiently many quantum trajectories project into Γcl,
select criteria of semiclassicality and choose a suitable (and possibly local) section which is
tangential to the XH and lies within a semi-classical band. The classical phase space is
obtained via the quotienting procedure while the effective classical Hamiltonian is obtained
as the pull-back of 〈Hˆ〉. Note that we have not assumed that the Hamiltonian operator is an
algebraic function of the basic operators chosen in the quotienting procedure. The effective
Hamiltonian however is always a function on Γcl (and H˜ is a function on Γ) by construction.
Even if the Hamiltonian operator is an algebraic function of the basic operator, the H˜ will
not be an algebraic function of the expectation values of basic operators.
Suppose now that we have chosen a section (over Γ) satisfying all the conditions above.
In the general case where Γcl 6= Γ, we have to restrict to the section over a Γcl i.e. restrict
to those states in the section which project into Γcl. Secondly, in general we will not have
operators corresponding to the canonical coordinates in Γcl and hence for the semiclassical
criteria we have to use only the basic operators used in obtaining Γ. The LQC example,
reveals implications of these aspects. We will see that the Hamiltonian operator also needs to
have non-trivial dependences on all the basic operators chosen for the quotienting procedure
and the semiclassicality criterion also needs to be phrased differently.
Recall that to construct the classical phase space R2 (as opposed to a cylinder), one
needs to use at least two sets of trigonometric operators with labels λ, λ′ such that (i)
λ′/λ is irrational and (ii) r, r′ both being non-zero. If a state in a section has support
18

only on a lattice generated by λ (say), or a subset thereof, one can see immediately that
cosλ′ = 0 = sinλ′ and hence r′ = 0. Therefore such states in a section will not represent
points in the classical phase space (but will represent points in a cylinder). Let us then
assume that our section consists of states having support on (sub-)lattices generated by
both λ, λ′. To be definite, let us take, |Ψa,λ〉 to be a normalized vector which is a linear
combination of vectors of the form |a+kλ〉, k ∈ Z, a ∈ (0, λ) and likewise |Ψb,λ′〉. Let a state
in a section be |Ψ〉 := (|Ψa,λ〉+ |Ψb,λ′〉)/
√
2. We are now guaranteed to have the expectation
values of pˆ, ĉosλ, ŝinλ, ĉosλ′ , ŝinλ′ to determine a unique point of the classical phase space, R2.
Furthermore, for any basic trigonometric operator with a label λ′′, with λ′′ irrational multiple
of λ, λ′, its expectation value in the state |Ψ〉 will be zero and the uncertainty will be 1/2. For
the uncertainty, we note the identities, ĉos2λ′′ = (1 + ĉos2λ′′)/2 and ŝin2λ′′ = (1− ĉos2λ′′)/2.
We would like to see evolution of these quantities generated by a self-adjoint Hamiltonian
operator, Hˆλ0 which is a function only of pˆ, ĉosλ0 , ŝinλ0 for some fixed λ0. The quantum
evolution of 〈cosλ′′〉, 〈sinλ′′〉 will be given by the expectation values of the commutators of
the corresponding trigonometric operator with the Hamiltonian operator,
The Hamiltonian acting on |Ψ〉 will generate vectors with labels kλ+ lλ0 and kλ′ + lλ0.
The trigonometric operator acting on 〈Ψ| on the other hand will generate vectors with
labels kλ± λ′′ and kλ′ ± λ′′. If λ0, λ, λ′, λ′′ are all incommensurate, then the inner product
of these states will be zero and hence the expectation value of the commutator will be
zero. Consequently, expectation values of these trigonometric operators will not evolve. By
the same logic, their uncertainties also will not evolve. If the Hamiltonian operator is a
sum of a function of pˆ alone plus a function of the trigonometric operator alone (as could
happen for polymer quantization of usual systems [4]), then of course expectation values and
uncertainties of trigonometric operators with both labels λ, λ′, will evolve, irrespective of
λ0. Barring this exception, the only way to get a non-trivial evolution is to have λ0 to equal
λ or λ′. Suppose we choose λ = λ0 (The quantum Hamiltonian is given and we can choose
the states in the section so as to develop a semiclassical approximation). Now, the evolution
of 〈ĉosλ0〉 and 〈ŝinλ0〉 will be non-trivial and so also the evolution of the corresponding
uncertaintities. However, the evolution of corresponding quantities for λ′ will still be trivial!
In effect, the projection of the quantum trajectories on the classical phase space will not be
non-periodic curves.
Thus, to have a non-trivial classical evolution (in Γcl), one will have to have (a) trigono-
19

metric operators with an incommensurate set of λ’s (at least two), (b) a section satisfying
semiclassicality and tangentiality, (c) states (in a section) involving (sub-)lattices correspond-
ing to these λ’s and (d) the quantum Hamiltonian also involving trigonometric operators
with the chosen set of λ’s. Note that this is a statement about developing a semiclassi-
cal approximation and does not imply or indicate any inconsistency of the choice of the
Hamiltonian operator at the fundamental quantum level.
Having multiple incommensurate λ’s also affects the uncertaintities in the trigonometric
operators. Continuing with the choice of just λ, λ′, one sees that,
(∆cosλ)2Ψ =
1
4 +
1
4〈Ψa,λ|ĉosλ|Ψa,λ〉
2 + 12(∆cosλ)
2
Ψa,λ (19)
The first term comes from |Ψb,λ′〉 piece of the wave function. Similar expressions are obtained
for the other three other trigonometric operators. Notice that these uncertainties are always
larger than 1/4. If we choose a state based on N incommensurate λ’s, then the first term
changes to (N − 1)/(2N), the coefficient of the second term changes to (N − 1)/N2 while
the coefficient of the third term changes to 1/N . For large N , the uncertaintities become
1/2. This is irrespective of the details of the states. Thus, if the semiclassicality criteria
required uncertainties in the trigonometric operators to be small (recall that uncertainties
are bounded above by 1 (12)), then no state involving several incommensurate λ’s will
satisfy the criterion of semiclassicality. Since for N = 1 (single λ), only the last term
survives, a way out is to require the uncertainties in ĉosλ, ŝinλ in the states of the form
|Ψa,λ〉 to be smaller than prescribed tolerances. This extra feature arises because the classical
phase space Γcl, is immersed in a complicated way in the base space Γ and one does not
have operators corresponding to the canonical coordinates on Γcl which could be used in
formulating semiclassicality criteria.
III. DISCUSSION
In this work, we have explored a particular strategy of developing a semiclassical ap-
proximation, namely, systematically, constructing a classical Hamiltonian system using the
available quantum structures such that at least some quantum motions can be faithfully
viewed as classical motions within certain precisions. This was done by exploiting the sym-
plectic structure of the quantum state space which is typically infinite dimensional. One
20

first constructs a (finite) dimensional classical phase space by a quotienting procedure which
views the quantum state space as a bundle over a base space Γ. There are several such
phase spaces one can construct depending on the choice of a Lie algebra of basic operators.
Generically, one gets classical phase space, Γcl as a submanifold of the base space. So far,
typically the special case wherein Γcl = Γ has been analysed in the literature, eg [1, 2, 4, 5].
We have given a generalization of the procedure. The main difference that occurs in the
general case is that the vertical subspace and the space spanned by the Hamiltonian vector
fields corresponding to the basic operators, have a non-trivial intersection and also together
they do not span the tangent space of the bundle. We illustrated the general procedure for
three different types of quantum systems - none resulting from a weakly continuous rep-
resentation of the Heisenberg group. Note that if the view of semiclassical approximation
mentioned above is to have a general enough validity, then it is necessary to be able to
construct non-linear phase spaces as well and the quotienting procedure given here shows
that it is possible.
A useful semiclassical approximation however, cannot be developed without reference
to dynamics. Construction of effective classical dynamics induced from the quantum one
required the choice of a section of the bundle. Once a section (any section) is chosen, one can
immediately define an effective Hamiltonian (and other effective functions) on the classical
phase space and a corresponding classical dynamics. However, this associated classical
dynamics will be a poor approximation to the underlying quantum mechanics unless the
sections are further restricted. We discussed the necessary conditions on the section. The
two main conditions are those of semiclassicality and tangentiality. However, when Γcl 6= Γ,
one needs additional condition namely, there should be sufficiently many quantum motions
which will project to curves in Γcl (and not just in Γ) and that these comprise a (local)
section.
The polymer state space brought out further features. To have a non-trivial motion on Γcl,
one needs to enlarge the set of basic operators to include trigonometric operators with a set of
incommensurate λ’s, use states which are based on (sub-)lattices generated by these λ’s and
have the Hamiltonian operator also depend on trigonometric operators with many λ’s, except
when the Hamiltonian operator has a additively separated dependence on the trigonometric
operators and the operator pˆ. Furthermore, the semiclassicality criteria also needs to be
applied to the trigonometric operators such that the uncertaintities for the operators labelled
21

by λ are computed with states based on (sub-) lattice generated by the same label λ. In effect,
one chooses states based on the various λ− lattices and chooses a linear combination of these
states to form a (local) section. How to choose the set of incommensurate λ’s is not clear
at this stage. A detailed construction of a semiclassical approximation for LQC, following
the steps discussed here, needs to be done. The new Hamiltonian operator proposed in [6]
looks promising since it is self-adjoint and also naturally connects triad labels in a specific
irregular lattice. A detailed analysis of the original LQC Hamiltonian of [3], with coherent
states based on a single λ−lattice is available in [5].
There is an alternate way to develop a semiclassical approximation [7]. This also uses the
geometric view of quantum mechanics and looks at the Hamiltonian flow on the quantum
phase space directly. For the special case of Hilbert space carrying the weakly continuous
representation of the Heisenberg group, one can introduce the so-called Hamburger momenta
variables to introduce suitable (adapted to the bundle structure) coordinates on the quantum
state space. The exact quantum dynamical equations can then be viewed as the Hamilton’s
equations of motion. Depending upon the quantum Hamiltonian function, evolution of most
of these variables could decouple from those of a smaller (finite) set of variables. This smaller
set of variables would then constitute a classical approximation i.e. be thought of as classical
degrees of freedom. In effect, the evolution of the remaining ‘quantum degrees of freedom’,
control how the uncertaintities in the classical degrees of freedom evolve. Violation of semi-
classicality can then be viewed as coupling of the evolutions of the classical and the quantum
degrees of freedom. This method would be more useful to track when the quantum evolution
can exit the semiclassical bands signalling break down of semiclassical approximation. So far
this method has been available in the context of Schrodinger quantization. Such an explicit
description of quantum evolution probably has to be done on a case by case basis.
Clearly, within the restrictions provided by semiclassicality and tangentiality, one can
imagine different approximation schemes which will systematically construct a better and
better semi-classical approximation. A natural way to phrase such a procedure is to for-
mulate it in terms of a family of sections. Thus one may begin with a section as giving
leading classical approximation and systematically change it to improve the approximation.
The usual perturbative approach can be viewed as beginning with a section defined by “free
particle states” and adding corrections to it get the new section closer to tangentiality. This
would be an interpretation of inclusion (or computation) of quantum corrections.
22

One also encounters a situation wherein new degrees of freedom are excited beyond a
certain threshold scale. This will mean that merely changing sections will not ensure tan-
gentiality and semiclassicality. In the language of [7], some of the ‘quantum degrees of
freedom’ have to be thought of as new ‘classical degrees of freedom’. One has to include
further ‘basic’ operators and use a new quotienting procedure. Now one can repeat the anal-
ysis and it is certainly conceivable that the true dynamics does satisfy the two requirements
with these additional variables defining sections.
Thus the geometrical view point shows that from a quantum perspective, inadequacy of a
classical approximation can arise in two ways – inappropriate choice of section which could
be improved perturbatively in some cases and/or inadequate choice of basic variables used
in the quotienting procedure.
The present analysis leaves out two important classes of systems: finite dimensional
constrained systems and field theories with or without constraints. We hope to return to
these in the future.
Acknowledgments
Part of this work was completed during a visit to IUCAA, Pune in May. The warm
hospitality is gratefully acknowledged. I would like to thank Alok Laddha for discussions.
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