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A Pseudo-Quantum Triad: Schrödinger's Equation, the Uncertainty Principle, and the Heisenberg Group

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We show that the paradigmatic quantum triad "Schrödinger equation–Uncertainty principle–Heisenberg group" emerges mathematically from classical mechanics. In the case of the Schrödinger equation, this is done by extending the metaplectic representation of linear Hamiltonian flows to arbitrary flows; for the Heisenberg group this follows from a careful analysis of the notion of phase of a Lagrangian manifold, and for the uncertainty principle it suffices to use tools from multivariate statistics together with the theory of John's minimum volume ellipsoid. Thus, the mathematical structure needed to make quantum mechanics emerge already exists in classical mechanics.
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A Pseudo-Quantum Triad: Schrödinger's Equation, the Uncertainty Principle, and the
Heisenberg Group
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2012 J. Phys.: Conf. Ser. 361 012015
(http://iopscience.iop.org/1742-6596/361/1/012015)
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A Pseudo-Quantum Triad: Schr¨odinger’s Equation,
the Uncertainty Principle, and the Heisenberg Group
Maurice A. de Gosson
University of Vienna, Faculty of Mathematics, NuHAG, Nordbergstraße 15, A-1090 Vienna
AUSTRIA
E-mail: maurice.de.gosson@univie.ac.at
Abstract. We show that the paradigmatic quantum triad “Schr¨odinger equation–Uncertainty
principle–Heisenberg group” emerges mathematically from classical mechanics. In the case of
the Schr¨odinger equation, this is done by extending the metaplectic representation of linear
Hamiltonian flows to arbitrary flows; for the Heisenberg group this follows from a careful analysis
of the notion of phase of a Lagrangian manifold, and for the uncertainty principle it suffices
to use tools from multivariate statistics together with the theory of John’s minimum volume
ellipsoid. Thus, the mathematical structure needed to make quantum mechanics emerge already
exists in classical mechanics.
1. Introduction
The Schr¨odinger equation, the uncertainty principle, and the Heisenberg group are three basic
paradigms of Quantum Mechanics. The aim of this contribution is to show that this “triad”
can actually be rigorously (i.e. mathematically) constructed within classical mechanics in
its Hamiltonian formulation; Planck’s constant hthen appears as a scaling parameter whose
dimension is that of an action. This fact is actually very much in accordance with Mackey’s [31]
statement following which quantum mechanics is a refinement of classical mechanics. Quantum
Mechanics can thus only emerge if one can give a physical meaning to this triad, justifying the
need for Planck’s constant h. We will not attempt to do this in the present work.
Here is a short description of what we are going to do:
The Schr¨odinger Equation. According to conventional wisdom, Schr¨odinger’s equation
cannot be derived from classical considerations; to witness Feynman’s declaration: “Where
did that [the Schr¨odinger equation] come from? Nowhere. It came out of the mind
of Schr¨odinger, invented in his struggle to find an understanding of the experimental
observations in the real world. However, at the time when Feynman made this declaration
(1965) it was already pretty well-known among specialists working in harmonic analysis,
that the Schr¨odinger equation is implicit in the metaplectic representation of the symplectic
group; in fact it was already known that Schr¨odinger’s equation could be rigorously derived
for all Hamiltonians which are quadratic in the position and momentum variables. We will
show that this equation can actually be derived for all Hamiltonian functions, not only for
those of the classical type “kinetic energy plus potential”.
The Uncertainty Principle. In Classical Mechanics (CM) it is tacitly assumed that
the physical quantities that are measured have an exact value, that can be in principle
EmerQuM 11: Emergent Quantum Mechanics 2011 IOP Publishing
Journal of Physics: Conference Series 361 (2012) 012015 doi:10.1088/1742-6596/361/1/012015
Published under licence by IOP Publishing Ltd
1
determined. In practice, the numerical results obtained from our measurements will
cluster around this objective exact value, and one then uses statistical methods to obtain
approximations to the exact value. In Quantum Mechanics (QM) the situation is different;
of course there are also measurements errors, but in the limiting case where perfect
measurements is assumed, the results of identical experiments performed on identically
prepared systems are generally not yielding identical results (this is well illustrated by
the discussion in Peres [33], §4-3, of a spin experiment). This quantum indeterminacy is
traditionally expressed in terms of the Uncertainty Principle of QM. We will show that the
Heisenberg inequalities (or their refinement, the Robertson–Schr¨odinger inequalities) are
–at least formally– perfectly classical.
The Heisenberg Group. The theory of the Heisenberg group is of a slightly more
abstract nature. Its “Schr¨odinger representation” is supposed to be a key to quantization,
as explained in Refs. [23, 24]. We will see that the Heisenberg group and the associated
Heisenberg–Weyl operators have a very simple interpretation in terms of the notion of phase
of a Lagrangian manifold, using the properties of the Poincar´e–Cartan invariant familiar
from the theory of the Hamilton–Jacobi equation from classical mechanics.
Acknowledgement
This work has been financed by the Austrian Research Agency FWF (Projektnummer P 23902-
N13). The author wishes to thank the organizers of the Heinz von Foerster Congress in Vienna.
2. Symplectic Mechanics
We gather in this preliminary section the tools from symplectic geometry and Hamiltonian
mechanics we will need; for details and proofs see Refs. [1, 4, 13, 15, 18]; in [15] we have given
a detailed study of the metaplectic group.
2.1. The symplectic group
A symplectic matrix is a real matrix Sof size 2nsatisfying anyone of the equivalent relations
STJS =Jor SJST=J; here STis the transpose of Sand Jis the skew-symmetric matrix
0I
I0where 0 and Iare, respectively, the n×nzero and identity matrices. In particular a
2×2 real matrix a b
c dis symplectic if and only if its determinant ad bc is equal to one; but
it is not true in dimension n > 1 that every unimodular matrix is symplectic. Using the identity
STJS =J(or SJST=J) one easily checks that the symplectic matrices form a group, which
is denoted by Sp(2n, R).
The symplectic group Sp(2n, R) is generated by the matrix Jtogether with all matrices
ML=L10
0LT,VP=I0
P I(1)
where det L6= 0 and P=PT(Land Phaving size n); the matrices ML, and VPare trivially
symplectic. Since the determinants of J,ML, and VPare equal to one, this shows at the
same time that the determinant of a symplectic matrix always is equal to one. This fact implies
Liouville’s theorem on the conservation of phase space volume under Hamiltonian flows, see
Refs. [4, 12].
2.2. The metaplectic representation
The symplectic group is connected and has covering spaces of all orders. Among all these, the
two-fold covering group plays a very special role, because it can be faithfully represented by
EmerQuM 11: Emergent Quantum Mechanics 2011 IOP Publishing
Journal of Physics: Conference Series 361 (2012) 012015 doi:10.1088/1742-6596/361/1/012015
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a group of unitary operators acting on the square integrable functions. This operator group
is called the metaplectic group; we will denote it by Mp(2n, R). The easiest way to describe
Mp(2n, R) is to use generators. Since Mp(2n, R) is a two-fold covering of Sp(2n, R) to each of
the generators J,ML,VPcorrespond two elements of Mp(2n, R), differing by a sign. These
operators b
J,c
ML,b
VPare defined by
b
Jψ(x) = 1
2πi n/2Zeix·yψ(y)dny(2)
c
MLψ(x) = det (Lx) (3)
b
VPψ(x) = ei
2P x·xψ(x); (4)
the argument of det Lin formula (3) is chosen to be 0 or 2πif det L > 0, and πor 3πif det L < 0.
The covering projection Π : Mp(2n, R)Sp(2n, R) is determined by the equalities
Π( b
J) = J, Π(c
ML) = ML, Π( b
VP) = VP.(5)
However, one obtains another perfectly honest covering projection by using inner automorphisms
of the base group Sp(2n, R) (see e.g. de Gosson [13], §6.4.2, for a simple description of this
conjugation relation). This means that given an arbitrary element S0of Sp(2n, R) the mapping
Π0defined by Π0(b
S) = S0Π( b
S)S1
0is also a bona fide covering projection. Notice that the
definition of Π0can be rewritten in the form
Π0(b
S) = Π( b
S0b
Sb
S1
0) , Π0(b
S0) = S0(6)
since Π is a group homomorphism. Choose now S0=M1/ε(it is the symplectic matrix ML
with L= (1/ε)I) and define the projection
Πε(b
S) = M1/εΠ( b
S)Mε= Π(c
M1/εb
Sc
Mε).(7)
One immediately verifies that if we use Πεinstead of Π, the projections formulas (5) become
Πε(b
Jε) = J, Πε(c
Mε
L) = ML, Πε(b
Vε
P) = VP.(8)
the metaplectic operators b
Jε,c
Mε
L,b
Vε
Pbeing defined by
b
Jεψ(x) = 1
2πiε n/2Zei
εx·yψ(y)dny(9)
c
Mε
Lψ(x) = c
MLψ(x) = det (Lx) (10)
b
Vε
Pψ(x) = ei
2εP x·xψ(x).(11)
In QM one chooses ε=~.
2.3. Hamiltonian flows
Let H=H(z) (the “Hamiltonian”) be a smooth function of the variables x, p. Hamilton’s
equations of motion ˙x=pH(x, p) ˙p=−∇xH(x, p) can be written in compact form as
˙z=JzH(z).(12)
The presence of the matrix Jin this formula indicates that the symplectic group might lurk
behind Hamiltonian mechanics. Let us examine this “educated guess” somewhat more in detail.
EmerQuM 11: Emergent Quantum Mechanics 2011 IOP Publishing
Journal of Physics: Conference Series 361 (2012) 012015 doi:10.1088/1742-6596/361/1/012015
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We denote by (fH
t) the associated phase flow: fH
tis the mapping which takes an initial point
z0= (x0, p0) at time t= 0 to the point z= (x, p) at time talong the solution curve to (12)
through z0. That is, z(t) = fH
t(z0) is the solution of (12) with initial condition z(0) = z0.
The importance of the symplectic group in Hamiltonian mechanics comes from the following
fundamental property of the phase flows (fH
t): each mapping fH
tis a canonical transformation;
see [1, 4, 12, 15, 18]. This means that for every z= (x, p) the Jacobian matrix of fH
tcalculated
at z= (x, p) is symplectic:
Df H
t(z) = (x(t), p(t))
(x0, p0)Sp(2n, R).
In particular, when the Hamiltonian function is quadratic in the xand pvariables, that is of
the type
H(z) = 1
2Mz ·z=1
2(x, p)M(x, p)T,(13)
the flow determined by Hamilton’s equations is linear, and consists of the symplectic matrices
SH
t=etJM . As we are going to see below, the study of linear flows is the first step towards a
general derivation of Schr¨odinger’s equation.
3. Schr¨odinger’s Equation
We now proceed to show how the Schr¨odinger equation emerges from Hamiltonian mechanics.
We assume that the Hamiltonians are time-independent, but everything in this Section can be
extended to the time-dependent case at the price of some technical difficulties (see de Gosson
[18], de Gosson and Hiley [19]).
3.1. Quadratic Hamiltonians
Consider first a quadratic Hamiltonian of the type (13). As we observed above, the phase flow can
easily be calculated, and consists of symplectic matrices SH
t. We have of course SH
tSH
t0=SH
t+t0so
the flow is a one-parameter subgroup of Sp(2n, R). When tvaries these matrices describe a curve
in Sp(2n, R) which passes by the identity at the initial time t= 0. Now, the unique path lifting
theorem in the theory of covering groups says that this curve can be lifted to any of the covering
groups of Sp(2n, R) and that this can be made in an unique way if one specifies a point through
which the lifting should pass. Choosing the metaplectic group Mp(2n, R) as the covering group,
with a projection Πε: Mp(2n, R)Sp(2n, R), and imposing to the lifting to pass through the
identity of Mp(2n, R) at time t= 0 we thus obtain a curve of metaplectic operators b
SH
tsuch
that Πε(b
SH
t) = SH
t; one moreover verifies that the group property SH
tSH
t0=SH
t+t0carries over to
these operators: b
SH
tb
SH
t0=b
SH
t+t0. The operators b
SH
tare unitary and act on the square integrable
functions on Rn. For such a function ψ0we set ψ(x, t) = b
SH
tψ0and ask the question “what is
the partial differential equation satisfied by ψ(x, t)?” The answer is: if the initial function ψ0is
at least twice continuously differentiable in the xvariables, then
ψ
∂t (x, t) = b
H(x, x)ψ(x, t) (14)
where b
H(x, x) is obtained from the Hamiltonian function (13) by the formal substitution
(x, p)(x, i~x). Writing M=Hxx Hxp
Hpx Hppthis is
b
H(x, x) = ε2
2Hppx2iεHpx xx+1
2Hxxx2i
2Tr(Hpx) (15)
EmerQuM 11: Emergent Quantum Mechanics 2011 IOP Publishing
Journal of Physics: Conference Series 361 (2012) 012015 doi:10.1088/1742-6596/361/1/012015
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where Tr(Hpx) (the trace) is the sum of the diagonal elements of Hpx.
Thus, if we choose ε=~we obtain Schr¨odinger’s equation; but this particular choice of
parameter has to be motivated by physical considerations once a physical meaning has been
given to the solution ψ. What must be remembered is that the derivation of (14) heavily
depends on the choice of projection Πεof Mp(2n, R) onto Sp(2n, R): each choice of εis a priori
equally good.
We have thus established the existence, for each ε > 0, of a one-to-one and onto
correspondence
Cε: (SH
t)(b
SH
t) (16)
between phase flows arising from quadratic Hamiltonian functions and unitary evolution groups
b
SH
t=eit b
H/ε ,b
H=b
H(x, x). We now make a fundamental observation. Assume that we
make a symplectic change of variables in the Hamiltonian H(z), i.e. that we replace H(z) with
H(S1z) where Sis in Sp(2n, R). Then, the phase flow (fHS1
t) of H(S1z) is obtained from
the flow (fH
t) by conjugation with S:
fHS1
t=Sf H
tS1(17)
(see e.g. Refs. [4, 15, 18, 12, 34]). In the case of quadratic Hamiltonians we have SHS1
t=
SSH
tS1and the group of unitary operators b
SH
tis replaced with b
SHS1
t=b
Sb
SH
tb
S1. It follows
that the correspondence Cεsatisfies the covariance formula
Cε(SSH
tS1)=(b
Sb
SH
tb
S1).(18)
3.2. The general case
We begin by recalling two results from functional analysis. The first is Stone’s theorem [45]
about one-parameter groups of operators (see Refs. [1, 35] for “modern” proofs):
Stone’s Theorem: (i) For every strongly continuous one-parameter group (Ft)of
unitary operators on a Hilbert space L2(Rn)there exists a self-adjoint operator Aon
L2(Rn)such that Ft=eitA/~; in particular Ais closed and densely defined in H.(ii)
Conversely, if Ais a self-adjoint operator on L2(Rn)then there exists a unique one-
parameter unitary group (Ft)whose infinitesimal generator is A, that is Ft=eitA/~.
The second result says that every continuous operator (in a sense we will be precise) has a
distributional kernel:
Schwartz’s kernel theorem: Let Abe a linear operator S(Rn)→ S0(Rn). We
assume that Ais continuous in the sense that if (ψk)is a sequence of functions such
that ψkψfor k→ ∞ in S(Rn)then kin S(Rn). Then there exists a
distribution KA(the kernel of A) such that
(x) = ZKA(x, y)ψ(y)dny(19)
for all functions ψin S(Rn)[the integral in the right hand-side of (19) is interpreted
as a distributional bracket].
Let now Hbe an arbitrary Hamiltonian function; we assume that the phase flow (fH
t)
generated by the Hamilton equations for Hexist for all times t(this is just a technical
requirement, which can be alleviated; see de Gosson and Hiley [19]). We are going to show
that the functor (16) can be extended in a unique way to arbitrary Hamiltonian flows if one
makes the following requirement:
EmerQuM 11: Emergent Quantum Mechanics 2011 IOP Publishing
Journal of Physics: Conference Series 361 (2012) 012015 doi:10.1088/1742-6596/361/1/012015
5
The symplectic covariance relation (18):
Cε(Sf H
tS1)=(b
Sb
FH
tb
S1) (20)
should hold for each Sin Sp(2n, R).
We have again Sf H
tS1=fHS1
tin view of formula (17), and ( b
FH
t) is a strongly continuous
one-parameter group of unitary operators on L2(Rn) which has to be determined.
To the phase flow (fH
t) (that is, equivalently, to the Hamiltonian function H) we associate
the Weyl operator
b
H=1
2πiε n/2ZHε
σ(z)b
Tε(z)d2nz
where b
Tε(z) = e(ˆz,z )is the ε-Heisenberg–Weyl operator (cf. formula (38) and
Hε
σ(z0) = 1
2πiε nZei
εσ(z,z0)H(z0)d2nz0
is the ε-symplectic Fourier transform of H. Since His real it follows from the standard theory
of Weyl operators (see Refs. [15, 18, 29]) that the operator b
His self adjoint and densely defined
on the Schwartz space of rapidly decreasing functions. In view of part (ii) of Stone’s theorem
there exists a unique one-parameter group of unitary operators b
FH
t=eit b
H/ε , so we define
Cε(fH
t)=(b
FH
t)=(eit b
H/ε ).(21)
To show that this is the desired extension of the functor Cεwe have to check that it satisfies the
symplectic covariance property (20) and that its restriction to linear flows (SH
t) is given by (16).
The symplectic covariance property is proven as follows: since Sf H
tS1=fHS1
tthe functor
Cεassociates to (Sf H
tS1) the one-parameter group (eit \
HS1). In view of the symplectic
covariance property of Weyl operators we have \
HS1=b
Sb
Hb
S1and hence
eit \
HS1=eit b
Sb
Hb
S1=b
Seit b
H/ε b
S1
so that Cε(Sf H
tS1) = b
SCε(fH
t)b
S1which is precisely formula (20) we set out to prove. Let us
now show that definition (21) of the functor Cεimplies that we have Cε(SH
t)=(b
SH
t). We must
thus show that the operators b
FH
t=eit b
H/ε are metaplectic when His a quadratic Hamiltonian.
The infinitesimal generator b
Hof b
FH
tsatisfies the Schr¨odinger equation
i~d
dt b
FH
t=b
Hb
FH
t
where b
HWeyl
H. Now, the operator b
H(x, x) defined by formula (15) is precisely the
Weyl quantization of the quadratic Hamiltonian function Hhence the b
FH
tare precisely the
metaplectic operators b
SH
tcorresponding to the lift of the linear flow (SH
t) to Mp(2n, R).
We next have to show that, conversely, for every strongly-continuous one-parameter group b
Ft
of unitary operators on L2(Rn) there exists a Hamiltonian function Hsuch that b
Ft=Cε(fH
t).
This amounts to show that there exists a Weyl operator b
HWeyl
Hsuch that b
Ft=eit b
H/ε . We
begin by remarking that in view of part (i) of Stone’s theorem there exists a unique self-adjoint
operator b
Hsuch that b
Ft=eit b
H/ε for all times t, so all we have to do is to prove that b
His
indeed a Weyl operator; the Weyl correspondence b
HWeyl
Hwill then identify the Hamiltonian
EmerQuM 11: Emergent Quantum Mechanics 2011 IOP Publishing
Journal of Physics: Conference Series 361 (2012) 012015 doi:10.1088/1742-6596/361/1/012015
6
function such that b
Ft=Cε(fH
t). The restriction of b
Hto the dense subspace S(Rn) is continuous
S(Rn)→ S0(Rn) because b
His essentially self-adjoint; it follows that Schwartz’s kernel theorem
applies, and we can thus find Ksuch that
b
Hψ(x) = ZK(x, y)ψ(y)dny. (22)
Let now τbe an arbitrary real number and define the “τ-symbol” Hτof b
Hby the Fourier
integral
Hτ(z) = Zei
εpyK(x+τ y, x (1 τ)y)dny. (23)
One shows (de Gosson [18], Shubin [44]) that the operator b
Hcan be expressed in terms of Hτ
by the formula
b
Hψ(x) = 1
2πiε nZZ ei
εpyHτ((1 τ)x+τ y, p)ψ(y)dnydnp(24)
which is the “pseudo-differential representation of b
Hin terms of its τ-symbol”. We next observe
that if the operator is multiplication by a function a(x) then we must have K(x, y) = a(x)δ(xy)
and hence, by (23),
Hτ(z) = a(x)Zei
εpyδ(y)dny=a(x).
This property, together with the symplectic covariance requirement (20) implies (see Refs.
[43, 50]), that this is only possible if we choose τ=1
2, in which case formula (24) becomes
b
Hψ(x) = 1
2πiε nZZ ei
εpyH1/2(1
2(x+y), p)ψ(y)dnydnp.
But this is exactly the integral representation of the Weyl operator b
HWeyl
H=H1/2, and we
are done.
3.3. The Groenewold and Van Hove “no-go” theorem
It is sometimes claimed that one cannot “quantize classically” arbitrary non-quadratic
observables; to sustain this claim an old (and famous) series of results going back to Groenewold
[21] and van Hove [26, 27] is invoked. But this objection cannot be retained in the present
case: a careful mathematical reading of the Groenewold-van Hove obstruction results reveals
that what is actually proven is the impossibility to quantize observables in such a way that
Dirac’s [9] prescription that Poisson brackets should correspond to commutators is preserved.
And, indeed, such a quantization is a chimera. This state of affairs is very lucidly analysed by
Giulini’s [11] who reminds us that Dirac [9] himself had noted that
...The strong analogy between quantum Poisson brackets [i.e. commutators] and
classical Poisson brackets leads us to make the assumption that the quantum Poisson
brackets, or at any rate the simpler ones of them, have the same values as the
corresponding classical Poisson brackets.
The italics (added) show that Dirac himself was very aware of the fact that it was not
reasonable to expect that the Poisson bracket/commutators correspondence should hold for all
pairs of observables Hand K. The Groenewold–van Hove obstructions thus have nothing to do
with the impossibility of a derivation of Schr¨odinger’s equation. They only show that one has to
go beyond the metaplectic representation if one wants to quantize non-quadratic Hamiltonians.
This is precisely what we have done by extending the correspondence Cεto arbitrary Hamiltonian
flows.
EmerQuM 11: Emergent Quantum Mechanics 2011 IOP Publishing
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4. The Uncertainty Principle
We are following in this section the discussion in de Gosson [16, 17].
4.1. Uncertainty inequalities
Contrarily to what is often believed Heisenberg’s uncertainty principle ∆pjxj1
2~is not
a statement about the accuracy of our measurement instruments; its derivation assumes on
the contrary perfect instruments. The correct interpretation of Heisenberg’s inequalities is the
following (see e.g. Peres [33], p.93): if the same preparation procedure is repeated a large number
of times, and is followed by either by a measurement of xj, or by a measurement of pj, the results
obtained have standard deviations ∆xjand ∆pjsatisfying the Heisenberg inequalities. The same
interpretation is of course true for the stronger Robertson–Schr¨odinger [36, 40] inequalities
(∆pj)2(∆xj)2∆(xj, pj)2+1
4~2(25)
to which the Heisenberg inequalities reduce if one neglects the covariances ∆(xj, pj)2; they are
complemented by the trivial inequalities
pjxk0 if j6=k, ∆pjpk0 , ∆xjxk0.(26)
where θ2
jk =θ2
kj and η2
jk =η2
kj (see Dias et al. [8] and the references therein). In classical
statistical mechanics the situation is somewhat different: due to the inherent inaccuracy of the
measurement apparatus there are uncertainties for all pairs of variables, conjugate or not. This is
actually reminiscent of what happens in noncommutative mechanics (NCQM) where the second
and third inequalities (26) are replaced with
p2
jp2
k∆(pj, pk)2+1
4θ2
jk , ∆x2
jx2
k∆(xj, xk)2+1
4η2
jk .(27)
4.2. The Minimum Volume Ellipsoid method
Performing position and momentum measurements on a large number Kof identical copies
of a system of particles, we get a cloud S={z1, z2, ..., zN},N=nK, of points in R2n. An
efficient method for studying that cloud consists in using the minimum volume ellipsoid (MVE)
method (Rousseeuw [37], Van Aelst and Rousseeuw [42]). Geometrically speaking this method
is an application of John’s theorem [28] which says that a convex set is contained in a unique
ellipsoid with smallest volume. It works as follows: consider a subset {zi1, zi2, ..., zik}of Sof
points in general position. This condition is sufficient and necessary for any ellipsoid containing
these points to have positive volume. The points zi1, zi2, ..., zikdetermine a polyhedron in
R2n; we denote by Kthe convex hull of that polyhedron (it is the smallest subset of R2n
containing {zi1, zi2, ..., zik}). John’s theorem ensures us that there exists a unique ellipsoid J
in R2ncontaining Kand having minimum volume among all the ellipsoids having this property.
Repeating this process for all subsets of the cloud Shaving kelements in general position we get
a family of ellipsoids; by definition the MVE is the one with the smallest volume. To determine
that ellipsoid one proceeds as follows: choose an integer kbetween [N/2] + 1 and N([N/2] the
integer part of N/2); this constant kdetermines the robustness of the resulting estimator; a
common choice is to take k= [(N+ 2n+ 1)/2]. The first step of the procedure is to minimize
the determinant of the matrices Msubject to the condition
#j: (zj¯z)TM1(zj¯z)m2k(28)
where the minimization is over all ¯zR2nand all positive definite symmetric matrices Mof
size 2n. Here mis a fixed constant, chosen so that the MVE estimator is a consistent estimator
for of the covariance matrix for data coming from a multivariate normal distribution, that is
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m=qχ2
2n,α,α=k/N where χ2
2n,α is a chi-square distribution with 2ndegrees of freedom
(see Lopuh¨aa and Rousseeuw [30]). Once the pair (M, ¯z) is determined, the minimum volume
ellipsoid (MVE) is the set of all zin R2nsuch that
(z¯z)TM1(z¯z)m2.(29)
The second step consists in associating to the MVE Ja covariance matrix. For this one has to
choose an adequate value m0for m; denoting the corresponding matrix Mby Σ the MVE is the
ellipsoid
C: (z¯z)TΣ1(z¯z)m2
0(30)
and Σ is then precisely the covariance matrix. We will write that matrix in the form
Σ = ∆(x, x) ∆(x, p)
∆(p, x) ∆(p, p)(31)
where ∆(x, x) = ((∆(xi, xj))1i,jnand so on; writing as is customary ∆(xi, xi) = ∆x2
iand
∆(pi, pi) = ∆p2
iwe have ∆(xi, xj) = ∆(xj, xi), ∆(pi, pj) = ∆(pj, pi), ∆(xi, pj) = ∆(pj, xi)
(observe that Σ is symmetric).
4.3. A condition on the covariance matrix
Let A= (ajk )1j,knand C= (cj k )1j,knbe two real antisymmetric matrices, and B=
(bjk )1j,kna real symmetric matrix. To A, B, C we associate the 2n×2nantisymmetric matrix
Ω = A B
B C
which is the most general form an antisymmetric of size 2ncan have. Assuming Ω invertible, the
bilinear form ωon R2ndefined by ω(z, z0) = (z0)T1zis a symplectic form on R2n. Notice
that when A=C= 0 and B=Iwe have Ω = J=0I
I0. The general case is reduced to
the standard case by noting that there exists a matrix Fsuch that Ω = FTJF ; equivalently F
is a linear canonical transformation (R2n, σ)(R2n, ω ).
Let now Σ be the covariance matrix (31). The matrix
Σ + iΩ = ∆(x, x) + iA ∆(x, p) + iB
∆(p, x)iB ∆(p, p) + iC(32)
is Hermitian since Σ is symmetric and (iΩ)=iΩ, hence the eigenvalues of Σ + i are real.
Assuming in addition that these eigenvalues are nonnegative, Σ + iΩ is semi-definite positive,
which we write Σ + i0. It follows that
∆(x, x) + iA 0 , ∆(p, p) + iC 0.(33)
When n= 1 the covariance matrix is just
Σ = x2∆(x, p)
∆(p, x) ∆p2
and the antisymmetric matrices Θ and Nare zero so that Ω = aJ =0a
a0. The condition
Σ + iΩ = Σ + iaJ 0 is in this case equivalent to
x2+ia ∆(x, p)
∆(p, x) ∆p2ia0
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that is, to the single inequality
x2p2∆(x, p)2+a2
which formally reduces to the Robertson–Schr¨odinger inequality (25) if one chooses ε=~/2.
In higher dimensions one shows, using Sylvester’s criterion which says that a Hermitian
matrix is positive semidefinite if and only if all of its principal minors are nonnegative, that the
condition Σ + i0 implies that
x2
jx2
k∆(xj, xk)2+a2
jk (34)
p2
jp2
k∆(pj, pk)2+c2
jk (35)
x2
jp2
k∆(xj, pk)2+b2
jk .(36)
In particular, if Ω = εJ,ε > 0, these conditions again reduce to the Robertson–Schr¨odinger
type inequalities
x2
jp2
k∆(xj, pk)2+ε2.
In [16, 17] we have given an interpretation of the uncertainty principle (both quantum,
and classical) in terms of the topological notion of symplectic capacity (also see the review
paper by de Gosson and Luef [20]). This approach seems promising since it allows to recast
the Robertson–Schr¨odinger inequalities in a topological form, invariant even under non-linear
canonical transformations. This is an extension of the fact that the Robertson–Schr¨odinger
inequalities are invariant under linear symplectic transformations.
5. Heisenberg Group and Operators
In this Section we show that the Heisenberg group is essentially a classical object associated to
the notion of phase of a Lagrangian manifold (see de Gosson [14, 15]).
5.1. The Heisenberg group
The Heisenberg group Hnis the space R2n+1 equipped with the product
(z1, t1)~(z0, t0) = (z1+z0, t0+t1+1
2σ(z0, z1).(37)
It is intimately related to the Heisenberg–Weyl operators
b
T(z0) = ei
}σz,z0),σ(ˆz, z0) = p0xx0(i~x) (38)
whose explicit action on a function ψ(x) is given by
b
T(z0)ψ(x) = ei
}(p0·x1
2p0·x0)ψ(xx0) (39)
(see e.g. de Gosson [15, 18], Littlejohn [29]). The Heisenberg–Weyl operators do not commute:
b
T(z1)b
T(z0) = ei
}σ(z0,z1)b
T(z0)b
T(z1) (40)
as immediately follows from the property
b
T(z0+z1) = ei
2}σ(z0,z1)b
T(z0)b
T(z1) (41)
which leads to the definition of the Heisenberg group (see Refs. [23, 24]). This traditional
construction gives the impression that some strange algebraic structure which is supposed to be
the quintessence of QM abruptly emerges! We will see that this is not the case, by showing that
formula (39) has a perfectly classical interpretation in terms of the notion of phase.
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5.2. The phase of a Lagrangian manifold
A venerable topic in CM is the Hamilton–Jacobi equation
∂ϕ
∂t +H(x, xφ) = 0 , ϕ(x, 0) = ϕ0(x); (42)
for instance, if His of the classical type kinetic energy plus potential it takes the more familiar
form (see Refs. [4, 12]):
∂ϕ
∂t +
n
X
j=1
1
2m(xφ)2+U(x) = 0 , ϕ(x, 0) = ϕ0(x).
The Hamilton–Jacobi equation allows, among other things, to solve explicitly Hamilton’s
equations for a wide class of systems. The solution (which we assume exists for tin some
open interval surrounding the initial value t= 0), is constructed as follows. Let us denote
by V0the graph of the equation p=xϕ0(x) (i.e. the set of all points (x, xϕ(x)) where x
takes its values in the domain of ϕ). V0is a n-dimensional submanifold of phase space, in fact
a Lagrangian manifold. This means that the symplectic product of any two tangent vectors
Z, Z0at a same point zof V0is equal to zero. In [14] we have called the function ϕ0the phase
of the manifold V0; it is uniquely determined up to a constant. Lagrangian manifolds are the
fundamental geometric objects of classical mechanics in its Hamiltonian formulation (see Refs
[1, 4, 47, 48]; for instance, the invariant tori associated with completely integrable systems are
Lagrangian manifolds (albeit not of the type V0above).
Consider now the time-dependent flow (fH
t) determined by the Hamilton equations for H,
and set Vt=fH
tV0. If |t|is sufficiently small, the manifold Vtwill again be a graph, that is,
there exists a function ϕ(x, t) (the phase of Vt) such that the equation p=xϕ(x, t) represents
Vt(if tis too large, bending of the original manifold V0may lead to the appearance of several
points on Vthaving same projection on x-space). The phase ϕ=ϕ(x, t) (which is precisely the
solution of the Hamilton–Jacobi equation (42)) is explicitly calculated as follows: since Vtis a
graph, for each value xin the domain of ϕthere exits a unique psuch that p=xϕ(x, t), and
the phase space point (x, p) is obtained from a unique point (x(0), p(0)) in V0by the flow (fH
t).
Denoting by Γtthe arc of phase space trajectory joining (x(0), p(0)) in V0to (x, p) in Vtthe
phase ϕof Vtsatisfies
ϕ(x, t) = ϕ0(x(0)) + ZΓt
αH(43)
where
αH=pdx Hdt (44)
is the Poincar´e–Cartan invariant (see Refs. [1, 4, 15]). We now remark that since both
V0and Vtare graphs, the datum of x(0) uniquely determines x(t) and hence the formula
(x, p) = fH
t(x(0), p(0)) can be inverted, expressing x(0) as a function of x(0); we will write
this as
x(0) = (fH
t)1
X(x) (45)
hence the solution ϕ(x, t) is given by
ϕ(x, t) = ϕ0((fH
t)1
X(x)) + ZΓt
αH.(46)
It is customary in theoretical statistical mechanics to identify a physical system with a pair
(ρ, V ) where Vis a Lagrangian manifold and ρa measure (probability density) carried by that
manifold. At time t= 0 we thus have a pair (ρ0, V0) where the Lagrangian manifold V0is the
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graph of p=xϕ0(x). Since V0is uniquely determined by its phase ϕ0up to an inessential
additive constant, we can without any loss of generality identify that system with the pair
(ρ0, ϕ0). Under Hamiltonian evolution (ρ0, ϕ0) will become (ρ, ϕ) where ρis obtained by “push-
forward” of ρ0, that is ρ(z) = ρ0((fz0
t)1z) and ϕ=ϕ(x, t) is given by formula (46). We now
remark that since the variable z= (x, p) in ρ(z) is constrained by the condition p=xϕthe
measure ρis actually only a function of xso we may write ρ0(z) = ρ0,X(x) and
ρ(z, t) = ρX(x, t) = ρ0,X (fH
t)1
Xx.(47)
We next remark that the datum of the pair of real functions (ρX, ϕ) defined on Rnis
mathematically equivalent to the datum of the complex function
µ(x) = ρX(x)e(x)(48)
and the transformation formulas (45), (47) can then collectively be written in the compact form:
µ(x, t) = eiRΓtαHµ0(fH
t)1
X(x)(49)
5.3. Displacement operators
Let us now calculate (ρ, ϕ) in the particular case where His the Hamiltonian
Hz0(z) = σ(z, z0) = px0p0x; (50)
here z0= (x0, p0) will play the role of a translation vector. The function Hz0is called the
“displacement Hamiltonian”, because the solutions of the corresponding Hamilton equations
˙x=x0, ˙p=p0are
x=x(t) = x(0) + tx0,p=p(t) = x(0) + tp0(51)
so the flow (fz0
t,0) determined by Hz0is just the phase space translation with vector tz0=
(tx0, tp0). Let us calculate ϕ(x, t). The arc of trajectory Γtis here the line segment joining
(x(0), p(0)) to (x(0)+ tx0, p(0) +tp0) and the Hamiltonian is σ(z(0) + tz0, z0) = σ(z(0), z0) along
Γtso that
αH= [(p(0) + tp0)x0σ(z(0), z0)] dt
A straightforward calculation then shows that we have
ZΓt
pdx Hz0dt =tp0x(0) + 1
2t2p0x0(52)
and hence, applying formula (46), the phase ϕ(x, t) is given by
ϕ(x, t) = ϕ0(x(0)) + tp0x(0) + 1
2t2p0x0
that is, using the relation x=x(0) + tx0,
ϕ(x, t) = ϕ0(xtx0) + tp0x1
2t2p0x0.(53)
The measure ρt(z) = ρ0((fz0
t)1z) is also easily calculated: since fz0
tis the translation
zz+tz0we immediately get
ρt(z) = ρ0(ztz0).(54)
Let us now choose t= 1 and set T(z0) = fz0
1: it is the phase space translation with vector
z0= (x0, p0). Under the action of T(z0) the complex function µ0(x) = ρ0,X(x)e0(x)becomes a
new function e
T(z0)µ0(x) = µ0(xx0)eiRΓ1αH
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where Γ1= Γt=1. In view of Eqn. (53) we have
ZΓ1
αH=p0x1
2p0x0(55)
and hence
e
T(z0)µ0(x) = µ0(xx0)ei(p0x1
2p0x0)(56)
and this is exactly formula (39) giving the action of the Heisenberg–Weyl operator with ~= 1
on the function ψ=µ0! From this we immediately deduce without any calculation at all that
the operators e
T(z0) satisfy the relations
e
T(z1)e
T(z0) = e(z0,z1)e
T(z0)e
T(z1) (57)
e
T(z0+z1) = ei
2σ(z0,z1)e
T(z0)e
T(z1).(58)
Let us however check, for the sake of completeness, the relation (57) by a calculation of
classical phases (the relation (58) will immediately follow). Let z1be a second fixed point in
phase space; we set out to compare compare the states
e
T(z0+z1)(ρ0, ϕ0) and e
T(z0)e
T(z1)(ρ0, ϕ0).
Both Lagrangian manifolds e
T(z0+z1)V0and e
T(z0)e
T(z1)V0are identical, being both obtained
by a translation with vector z0+z1, but these two states are not the same! First, replacing
(x0, p0) in Eqn. (55) with (x0+x1, p0+p1) the phase of T(z0+z1)V0is
ϕ(z0+z1)(x) = ϕ0(x)+(p0+p1)[x+1
2(x0+x1)].(59)
The phase ϕ(z1,z0)(x) of e
T(z1)he
T(z0)V0iis obtained by adding to the expression (55) of ϕ(z0)(x)
the integral of the Poincar´e form pdxσ(z, z1)dt along the line segment Γ0
1joining (x+x0, p+p0)
to (x+x0+x1, p +p0+p1); this is
ZΓ0
1
pdx σ(z, z1)dt =p1(x+x0) + 1
2p1x1
and hence
ϕ(z1,z0)(x) = ϕ0(x) + p0x+p1(x+x0) + 1
2(p0x0+p1x1) (60)
A straightforward calculation shows that the phases given by Eqns. (59) and (60) differ by the
quantity
ϕ(z1,z0)(x)ϕ(z0+z1)(x) = 1
2σ(z0, z1) (61)
(it is, up to the sign, the area of the triangle with sides the vectors z0, and z1; see Fig. 3 in
Littlejohn [29]); this immediately implies formula (57).
The important thing to remember is that everything in our construction is entirely classical,
without any reference to the canonical commutation relations which are historically at the heart
of the definition of the Heisenberg–Weyl operators and of the Heisenberg group.
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6. Discussion and (Unconcluding) Remarks
Here are a few facts that could be used to illustrate the topics developed in this paper. It is
usually assumed that the quantum wave-particle duality can have no counterpart in classical
physics. It is also said that the probabilistic aspects of quantum dynamics are intrinsic and
to have no possible relation with underlying unresolved dynamical phenomena. However, Yves
Couder and his collaborators (Couder et al. [7]) have recently investigated the properties of a
droplet bouncing on a vibrated bath. In this system, a propagative wave-particle association
appears when the droplet couples to the surface wave it excites. Surprisingly, both a form of
uncertainty and a form of quantization are observed. In these experiments you see textbook
quantum mechanics, but everything is perfectly classical! It would be interesting to interpret
Couder’s results using our constructs. We mention that in a recent work [5] Bartlett et al. study
the reconstruction of a version of QM from classical considerations; this work where the notion
of epistemic restriction plays a key role is certainly closely related to our ideas. In [46] Gerard
’t Hooft contends that “...Quantum mechanics is ‘emergent’ if a statistical treatment of large
scale phenomena in a locally deterministic theory requires the use of quantum operators”. The
fact that Schr¨odinger’s equation is (mathematically speaking) a classical equation contradicts
in some sense this statement, since we are able to associate operators to classical observables,
and these operators have a perfectly classical meaning unless one decides, a posteriori, to give
their characteristics (e.g., their eigenvalues) a precise physical meaning.
To conclude, let us cite Stephen Adler [2]: “...quantum theory is not a complete, final theory,
but is in fact an emergent phenomenon arising from a deeper level of dynamics...” (also see
Bateson [6] who makes similar claims).
References
[1] Abraham R and Marsden J E 1978 Foundations of Mechanics. The Benjamin/Cummings Publishing
Company, 2nd edition
[2] Adler S L 2004 Quantum Theory as an Emergent Phenomenon. The Statistical Mechanics of Matrix Models
as the Precursor of Quantum Field Theory. Cambridge University Press
[3] Anderson P W 4 August 1972 More Is Different. Science, Vol. 177 No. 4047 393–396
[4] Arnold V I 1989 Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, 2nd edition,
Springer-Verlag
[5] Bartlett S D , Rudolph T and Spekkens R W Reconstruction of Gaussian quantum mechanics from Liouville
mechanics with an epistemic restriction. arXiv:1111.5057
[6] Bateson R D 2010 Causal space-Time. Emergent Physics from Causality. CreateSpace ; extended version to
be published by World Scientific.
[7] Couder Y, Boudaoud A, Proti`ere S, Moukhtarb and Fort J E Walking droplets: a form of wave-particle
duality at macroscopic level? , DOI: 10.1051/epn/2010101
[8] Dias N C, de Gosson M, Luef F and Prata J N 2010 A deformation quantization theory for noncommutative
quantum mechanics, J. Math. Phys.51 072101
[9] Dirac P A M 1999 The Principles of Quantum Mechanics. Oxford Science Publications, 4th revised edition
[10] Fort E, Eddi A , Boudaoud A, Moukhtar J and Couder Y 2010 Path-memory induced quantization of classical
orbits. PNAS,107(41), 17515–17520
[11] Giulini D 2003 That Strange Procedure Called Quantisation. Lect. Notes Phys. 361 17–40
[12] Goldstein H 1959 Classical Mechanics. Addison–Wesley (1950), 2nd edition, (1980), 3d edition (2002)
[13] de Gosson M 2001 The Principles of Newtonian and Quantum Mechanics: The need for Planck’s constant,
h. With a foreword by Basil Hiley. Imperial College Press
[14] de Gosson M 2004 On the notion of phase in mechanics, J. Phys. A: Math. Gen., 37(29), 7297–7314
[15] de Gosson M 2006 Symplectic Geometry and Quantum Mechanics. Birkh¨auser, Basel, series “Operator
Theory: Advances and Applications” (subseries: “Advances in Partial Differential Equations”), Vol. 166
[16] de Gosson M 2009 The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg? Found.
Phys.99 194–214
[17] de Gosson M 2010 On the Use of Minimum Volume Ellipsoids and Symplectic Capacities for Studying
Classical Uncertainties for Joint Position-Momentum Measurements. J. Stat. Mech. P11005
[18] de Gosson M 2011 Symplectic Methods in Harmonic Analysis and in Mathematical Physics. Birkh¨auser,
Basel
EmerQuM 11: Emergent Quantum Mechanics 2011 IOP Publishing
Journal of Physics: Conference Series 361 (2012) 012015 doi:10.1088/1742-6596/361/1/012015
14
[19] de Gosson M and Hiley B J 2011 Imprints of the Quantum World in Classical Mechanics. Found. Phys.41
1415–1436
[20] de Gosson M and Luef F 2009 Symplectic Capacities and the Geometry of Uncertainty: the Irruption of
Symplectic Topology in Classical and Quantum Mechanics. Physics Reports 484, 131–179
[21] Groenewold H J 1946 On the principles of elementary quantum mechanics. Physics 12 405–460
[22] Gromov M 1985 Pseudoholomorphic curves in symplectic manifolds. Invent. Math.82 307–47 (1985)
[23] Guillemin V and Sternberg S 1978 Geometric Asymptotics. Math. Surveys Monographs 14,Amer. Math.
Soc., Providence R.I.
[24] Guillemin V and Sternberg S 1984 Symplectic Techniques in Physics. Cambridge University Press, Cambridge,
Mass.
[25] Hall M.J.W and Reginatto M 2002 Schr¨odinger equation from an exact uncertainty principle. J. Phys. A:
Math. Gen. 35 3289
[26] van Hove L 1951 Sur le Probl`eme des Relations entre les Transformations Unitaires de la M´ecanique
Quantique et les Transformations Canoniques de la M´ecanique Classique. em. Acad. Roy. Belg. 26:610
[27] van Hove L1952 Sur certaines repr´esentations unitaires d’un groupe fini de transformations. em. Acad.
Roy. Belg. Classe des Sci. 26(6)
[28] John F 1948 Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented
to R. Courant on his 60th Birthday, January 8, Interscience Publishers, Inc., New York, N.Y. 187–204
[29] Littlejohn R G 1986 The semiclassical evolution of wave packets. Physics Reports 138(4–5), 193–291
[30] Lopuh¨aa H P and Rousseeuw P J 1991 Breakdown points of affine equivariant estimators of multivariate
location and covariance matrices, The Annals of Statistics 19(1) 229–248
[31] Mackey GW 1998 The Relationship Between Classical and Quantum Mechanics. In Contemporary
Mathematics 214,Amer. Math. Soc., Providence, RI
[32] Nelson E 1966 Derivation of the Schr¨odinger Equation from Newtonian Mechanics. Phys. Rev. A,150(4):6,
1079–1085
[33] Peres A 1993 Quantum Theory: Concepts and Methods. Kluwer Academic Publishers
[34] Polterovich L 2001 The Geometry of the Group of Symplectic Diffeomorphisms. Lectures in Mathematics,
Birkh¨auser
[35] Reed M., Simon B 1972 Methods of Modern Mathematical Physics. Academic Press, New York
[36] Robertson H P 1929 The uncertainty principle, Phys. Rev.34 163–164
[37] Rousseeuw P J 1985 Multivariate estimation with high breakdown point. In Grossmann, W, Pflug, G, Vicenze,
I & Wertz, (Eds.), Mathematical Statistics and Applications, Vol. B., Riedel Publishing, Dordrecht the
Netherlands 283–297
[38] Rousseeuw P J and Leroy A M, 1987 Robust regression and outlier detection, John Wiley & Sons, New York
[39] Scheeres D J, Hsiao F Y, Park R S, Villac B F, and Maruskin J M 2006 Fundamental Limits on Spacecraft
Orbit Uncertainty and Distribution Propagation, Journal of the Astronautical Sciences 54 505–523
[40] Schr¨odinger E 1930 Zum Heisenbergschen Unsch¨arfeprinzip, Berliner Berichte 296–303 [English translation:
Angelow, A., Batoni, M.C.: About Heisenberg Uncertainty Relation. Bulg. Journal of Physics,26, nos.5/6,
193–203 (1999), and http://arxiv.org/abs/quant-ph/9903100]
[41] Schr¨odinger E 1926 Quantisierung als Eigenwertproblem, Ann. der Physik,384, 361–376
[42] Van Aelst S and Rousseeuw P J, 2009 Minimum volume ellipsoid, Wiley Interdisciplinary Reviews:
Computational Statistics 1(1) 71–82
[43] Stein EM 1993 Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals,
Princeton University Press
[44] Shubin M A 1978 Pseudodifferential Operators and Spectral Theory, Springer-Verlag, (1987) [original Russian
edition in Nauka, Moskva ]
[45] Stone M H 1930 Linear transformations in Hilbert space, III: operational methods and group theory. Proc.
Nat. Acad. Sci. U.S.A, 172–175
[46] ’t Hooft G 2007 Emergent Quantum Mechanics and Emergent Symmetries. AIP Conf. Proc.957, 154-163 ;
http://arxiv.org/abs/0707.4568
[47] Tulczyjew W 1976 C.R. Acad. Sci. Paris 283, 15–18
[48] Tulczyjew W M 1977 The Legendre transformation, Ann. Inst. H. Poincar´e, Sect. A,27(1), 101–114
[49] Williamson J 1936 On the algebraic problem concerning the normal forms of linear dynamical systems, Amer.
J. of Math.58 141–163.
[50] Wong M W 1998 Weyl Transforms. Springer
[51] Woodhouse N M 1991 Geometric Quantization, second edition. Oxford Science Publications.
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... 136,137 In fact the mathematical structure that allows quantum mechanics to emerge already exists in classical mechanics. 138 Particularly surprising, maybe, is that Schrödinger-Robertson uncertainty principle has an exact counterpart in classical mechanics which can be formulated using some subtle developments in symplectic topology, namely Gromov's non-squeezing theorem and the related notion of symplectic capacity. 139 On the other hand there are unexpected and deep relations between gravity and quantum mechanics, in particular between Einstein-Rosen wormholes and quantum entanglement. ...
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Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.
... The imprints left by quantum mechanics in classical mechanics are more numerous than is usually believed [112,113]. In fact the mathematical structure that allows quantum mechanics to emerge already exists in classical mechanics [114]. Particularly surprising, maybe, is that Schrödinger-Robertston uncertainty principle has an exact counterpart in classical mechanics which can be formulated using some subtle developments in symplectic topology, namely Gromov's non-squeezing theorem and the related notion of symplectic capacity [115]. ...
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Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman-Von Neumann's Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations. A possible interpretation is that classicality requires the notorious hierarchy problem.
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We invite the reader (presumably an upper level undergraduate student) to a journey leading from the continent of Classical Mechanics to the new territories of Quantum Mechanics. We'll be riding the symplectic camel and have William of Occam as travel companion, so no excess baggage is allowed. The first part of our trip takes us from the symplectic egg to Gromov's non-squeezing theorem and its dynamical interpretation. The second part leads us to a symplectic formulation of the quantum uncertainty principle, which opens the way to new discoveries.
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On définit en Mécanique classique le groupe Γ (étroitement lié au groupe des transformations canoniques) qui correspond le mieux au groupe U des transformations unitaires de la Mécanique quantique. On compare les groupes U et Γ et on montre comment l’absence d’isomorphisme entre U et Γ fournit l’explication des ambiguïtés inévitables qui apparaissent dans tout processus de quantification d’un système de la Mécanique classique.
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Foreword.- Preface.- Prologue.- Part I: Symplectic Mechanics.- 1. Hamiltonian Mechanics in a Nutshell.- 2. The Symplectic Group.- 3. Free Symplectic Matrices.- 4. The Group of Hamiltonian Symplectomorphisms.- 5. Symplectic Capacities.- 6. Uncertainty Principles.- Part II: Harmonic Analysis in Symplectic Spaces.- 7. The Metaplectic Group.- 8. Heisenberg-Weyl and Grossmann-Royer Operators.- 9. Cross-ambiguity and Wigner Functions.- 10. The Weyl Correspondence.- 11. Coherent States and Anti-Wick Quantization.- 12. Hilbert-Schmidt and Trace Class Operators.- 13. Density Operator and Quantum States.- Part III: Pseudo-differential Operators and Function Spaces.- 14. Shubin's Global Operator Calculus.- Part IV: Applications.- 15. The Schrodinger Equation.- 16. The Feichtinger Algebra.- 17. The Modulation Spaces Mqs.- 18. Bopp Pseudo-differential Operators.- 19. Applications of Bopp Quantization.- Bibliography.- Index.
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Quantum mechanics is our most successful physical theory. However, it raises conceptual issues that have perplexed physicists and philosophers of science for decades. This 2004 book develops an approach, based on the proposal that quantum theory is not a complete, final theory, but is in fact an emergent phenomenon arising from a deeper level of dynamics. The dynamics at this deeper level are taken to be an extension of classical dynamics to non-commuting matrix variables, with cyclic permutation inside a trace used as the basic calculational tool. With plausible assumptions, quantum theory is shown to emerge as the statistical thermodynamics of this underlying theory, with the canonical commutation/anticommutation relations derived from a generalized equipartition theorem. Brownian motion corrections to this thermodynamics are argued to lead to state vector reduction and to the probabilistic interpretation of quantum theory, making contact with phenomenological proposals for stochastic modifications to Schrödinger dynamics.