The symplectic egg in classical and quantum mechanics

Abstract

Symplectic geometry is the language of Classical Mechanics in its
Hamiltonian formulation, and it also plays a crucial role in Quantum
Mechanics. Symplectic geometry seemed to be well understood until 1985,
when the mathematician Gromov discovered a surprising and unexpected
property of canonical transformations: the non-squeezing theorem.
Gromov's result, nicknamed the ``principle of the symplectic camel,''
seems at first sight to be an abstruse piece of pure mathematics. It
turns out that it has fundamental--and unsuspected--consequences in the
interpretations of both Classical and Quantum Mechanics, because it is
essentially a classical form of the uncertainty principle. We invite the
reader to a journey taking us from Gromov's non-squeezing theorem and
its dynamical interpretation to the quantum uncertainty principle,
opening the way to new insights.

Full text

The Symplectic Egg in Classical and Quantum
Mechanics
Maurice A. de Gosson
University of Vienna
Faculty of Mathematics, NuHAG
Nordbergstr. 15, 1090 Vienna
October 27, 2012
Abstract
Symplectic geometry is the study of canonical transformations. It
is the language of Classical Mechanics in its Hamiltonian formulation,
and it also plays a crucial role in Quantum Mechanics. Symplectic
geometry seemed to be well understood until 1985, when the mathe-
matician M. Gromov discovered a surprising and unexpected property
of canonical transformations: the non-squeezing theorem. Gromov’s
result seems at …rst sight to be an abstruse piece of pure mathematics,
but it turns out that it has fundamental consequences in the inter-
pretations of both Classical and Quantum Mechanics, because it is
essentially a classical form of the uncertainty principle. We invite the
reader to a journey taking us from Gromov’s non-squeezing theorem
and its dynamical interpretation to the quantum uncertainty principle
in phase space, opening the way to new discoveries.
Prologue
Take an egg –preferably a hard boiled one, and cut it in half along its middle
using a very sharp knife. The surface of section will be roughly circular and
have area r2(see Fig.1). Next, take a new egg of same size, and cut it
this time along a line joining the egg’s tops, again as shown in Fig1. This
time we get an elliptic surface of section with area R2larger than that
of the disk we got previously. So far, so good. But if you now take two
maurice.de.gosson@univie.ac.at
1

symplectic eggs, and do the same thing, then both sections will have exactly
same area! Even “worse”, no matter along which plane passing through
the center of the egg you cut, you will always get sections having the same
area! This is admittedly a very strange property, which you probably never
have experienced (at least in a direct way) in everyday life. But what is a
symplectic egg? The eggs we are cutting are metaphors for ellipsoids; an
ellipsoid is a round ball that has been deformed by a linear transformation
of space, i.e. a transformation preserving the alignment of three, or more,
points. In mathematics such transformations are represented by matrices.
Thus the datum of an ellipsoid is the same thing as the datum of a ball and
of a matrix. What we call a symplectic egg is an ellipsoid corresponding
to the case where the matrix is symplectic (we’ll de…ne the concept in a
moment). The reason for which the only symplectic egg you have seen on
your breakfast table is ‡at –a fried egg!– is that the number of rows and
columns of a symplectic matrix must always be even. Since we are unable
to visualize things in dimension three or more, the only symplectic eggs
that are accessible to our perception are two dimensional. But what is a
symplectic matrix? In the case of smallest dimension two, a matrix
S=a b
c d(1)
is symplectic if it has determinant one:
ad bc = 1:(2)
In higher dimensions, 4, 6, 8, etc. there are many more conditions: for
instance 10 if the dimension is 4, 21 if it is 6, and n(2n+ 1) if it is 2n. We
will write these conditions explicitly in section 2.1.
So far, so good. But where do symplectic eggs come from, and what are
they good for? Let me …rst tell you where symplectic matrices come from.
They initially come from the study of motion of celestial bodies, which is
really rich in mathematical concepts, some of these going back to the obser-
vations of Tycho Brahe, and the work of Galileo Galilei and Johannes Kepler
(these were the “Giants” on the shoulder’s of which Isaac Newton stood).
But the notion of symplectic matrix, or more generally that of symplectic
transformation, did really have a long time to wait until it appeared explic-
itly and was recognized as a fundamental concept. It was implicit in the
work of Hamilton and Lagrange on classical and celestial mechanics, until
the word “symplectic” was …nally coined by the mathematician Hermann
Weyl in his book The Classical Groups, edited in 1939, just before World
2

War II. But still then, as Ian Stewart reminds us in his Nature article The
Symplectic Camel [29], it was a rather ba- ing oddity which presumably
existed for some purpose –but which? It was only later agreed that the pur-
pose of symplectic transformations is dynamics, that is the study of motion.
Let me explain this a little bit more in detail: if we have a physical system
consisting of “particles”(sand corns, planets, spacecraft, or quarks) it is eco-
nomical from both a notational and computational point of view to describe
their motion (that is, their location and velocity) by specifying a phase space
vector, which is a matrix consisting of only one column. For instance, if we
are dealing with one single particle with coordinates (x; y; z )and momentum
(px; py; pz)(the momentum of a particle is just its velocity multiplied by its
mass m) the phase space vector will be the column vector whose entries are
x; y; z; px; py; pz. If we have a large number Nof particles with coordinates
(xi; yi; zi)and momenta (pxi; pyi; pzi)the phase space vector will be obtained
by …rst writing all the position coordinates and thereafter the momentum
coordinates in corresponding order, their momenta. These vectors form the
phase space of our system of particles. It turns out that the knowledge of
a certain function, the Hamiltonian (or energy) function, allows us to both
predict and retrodict the motion of our particles; this is done by solving
(exactly, or numerically) the Hamilton equations of motion, which are in
the case n= 1 given by
dx
dt =@H
@px
,dpx
dt =@H
@x (3)
(and similar relations for the other coordinates). Mathematically these equa-
tions are just a fancy way to write Newton’s second law F=ma. That is,
knowing exactly the positions and the momenta at some initial time, we are
3

able to know what these are going to be at any future time (we can actually
also calculate what they were in the past). The surprising, and for us very
welcome fact is that the transformation which takes the initial con…guration
to the …nal con…guration is always a symplectic transformation! These act
on the phase vectors, and once this action is known, we can determine the
future of the whole system of particles, and this at any moment (mathemati-
cians would say we are in presence of a “phase space ‡ow”). The relation
between symplectic transformations and symplectic matrices is that we can
associate a symplectic matrix to every symplectic transformation: it is just
the Jacobian matrix of that transformation.
The symplectic egg is a special case a deep mathematical theorem dis-
covered in 1985 by the mathematician Gromov [13], who won the Abel Prize
in 2010 for his discovery (the Abel Prize is the equivalent of the Nobel Prize
in mathematics). Gromov’s theorem is nicknamed the “principle of the sym-
plectic camel” [4, 7, 8, 29], and it tells us that it impossible to squeeze a
symplectic egg through a hole in a plane of “conjugate coordinates” if its
radius is larger than that of the hole. That one can do that with an ordinary
(this time uncooked) egg is easy to demonstrate in your kitchen: put it into
a cup of vinegar (Coca Cola will do as well) during 24 hours. You will then
be able to squeeze that egg through the neck of a bottle without any e¤ort!
The marvelous thing with the symplectic egg is that it contains quantum
mechanics in a nutshell... er ... an eggshell! Choose as radius p~, the square
root of Planck’s constant hdivided by 2. Then each surface of section will
have radius of h=2. In [7, 10] I have called such a tiny symplectic egg a
quantum blob. It is possible –and in fact quite easy if you know the rules
of the game–to show that this is equivalent to the uncertainty principle of
quantum mechanics. The thing to remember here is that a classical property
(i.e. a property involving usual motions, as that of planets for instance),
here symbolized by the symplectic egg, contains as an imprint quantum
mechanics (or is it the other way around?). The analogy between “classical”
and “quantum”can actually be pushed much further, as I have shown with
Basil Hiley [11]. But this, together with the notion of emergence, is another
story.
Some of the ideas presented here are found in my Physics Reports paper
[12] with F. Luef; they are developed and completed here in a di¤erent way
more accessible to a general audience.
4

1 Notation and terminology
Position and moment vectors will be written as column vectors
0
B
@
x1
.
.
.
xn
1
C
Aand 0
B
@
p1
.
.
.
pn
1
C
A
and the corresponding phase vector is thus
x
p= (x; p)T= (x1; :::; xn;p1; :::; pn)T
where the superscript Tindicates transposition. The integer nis unspeci-
…ed; we will call it the number of degrees of freedom. If the vector (x; p)T
denotes the phase vector of a system of Nparticles, then n= 3Nand the
numbers x1; x2; x3, (resp. p1; p2; p3) can be identi…ed with the positions
x; y; z (resp. the momenta px; py; pz) of the …rst particle, x4; x5; x6, (resp.
p4; p5; p6) with those of the second particle, and so on. This is not the only
possible convention, but our choice has the advantage of making formulas in-
volving symplectic matrices particularly simple and tractable. For instance,
the “standard symplectic matrix”is here J=0Id
Id0where Idis the
nnidentity matrix and 0the nnzero matrix. Note that
J2=Id,JT=J1=J: (4)
2 The Symplectic Egg
“One should not increase, beyond what is necessary, the num-
ber of entities required to explain anything”(William of Ockham,
alias Doctor Invincibilis)
2.1 Symplectic matrices
Let Sbe a (real) matrix of size 2n. We say that Sis a symplectic matrix if
it satis…es the condition
STJS =J: (5)
The standard symplectic matrix Jis itself symplectic since we have JTJ J =
J2J=Jin view of the properties (4). The de…nition above is, admittedly,
somewhat abrupt! Where does it come from? Let us make a little geometric
5

digression. The usual way to measure relative positions in our everyday
world consists in using a metric. For instance, in Euclidean geometry (which
is the most commonly used), the metric is associated to the inner product:
if u= (u1; :::; um),v= (v1; :::; vm)are two vectors in a m-dimensional
space, the inner product is uv=u1v1+   +umvm; the length (or norm)
of the vector uis then juj=puu=u2
1++u2
m. When studying
Euclidean geometry one is interested in linear transformations preserving
length, or, which amounts to the same, the inner product. Representing
such a transformation by its matrix, say M, the condition M u Mv =uv
is equivalent to MTM u v=uvthat is to MTM=I(the identity
matrix): linear transformations preserving the Euclidean metric are thus
the well-known orthogonal transformations studied in elementary textbooks.
In symplectic geometry, one is not interested in calculating lengths, but one
rather focuses on the notion of area. Instead of an inner product, one de…nes
asymplectic (or skew) product. It can only be de…ned on even-dimensional
linear spaces, e.g. the phase space of classical mechanics. In this case it is
customary to de…ne the symplectic product of two vectors z= (x; p)and
z0= (x0; p0)by
z^z0= (z0)TJz: (6)
Notice that it does not make sense to de…ne the “symplectic length” of
a vector by the formula jzj=pz^z, because we always have z^z= 0.
However, the number z^z0has a simple geometric interpretation: in position
and momentum coordinates we have
z^z0=px0p0x=
n
X
j=1
pjx0
jp0
jxj(7)
which we can rewrite as
z^z0=
n
X
j=1 
xjx0
j
pjp0
j:(8)
Thus, up to the sign, the symplectic product z^z0is the sum of the algebraic
areas of the parallelograms spanned by the projections of the vectors z; z0
on the planes xj; pjof conjugate coordinates.
Now, as Euclidean geometry is the study of transformations preserving
the inner product, symplectic geometry is the study of those linear trans-
formations preserving the symplectic product. Now, this condition, for a
matrix S, reads Sz ^Sz0=z^z0that is, taking the de…nition above into
account, (Sz0)TJSz = (z0)TJz. Since (Sz0)TJSz = (z0)TSTJSz we get
6

(z0)TSTJSz = (z0)TJz for all vectors z; z0, that is STJ S =J; and this is
precisely condition (5) de…ning a symplectic matrix. One often writes
(z; z0) = z^z0(9)
and calls the function asymplectic form. Note the formal similarity be-
tween the de…nitions of orthogonal and symplectic transformations:
Mis orthogonal () MTIM =I
Sis symplectic () STJS =J:
One passes from the …rst to the second by replacing the identity Iwith the
standard symplectic matrix J.
Assume now that we write the matrix Sin block form
S=A B
C D(10)
where A; B; C; D are matrices of size n. It is a simple exercise in matrix alge-
bra to show that condition (5) is equivalent to the the following constraints
on the blocks A; B; C; D
ATC=CTA,BTD=DTB, and ATDCTB=Id:(11)
Notice that the two …rst conditions mean that both products ATCand
BTDare symmetric. These conditions collapse to the identity ad bc = 1
in (2) when n= 1: in this case A; B ; C; D are the numbers a; b; c; d so that
ATC=ac and BTD=bd; the condition ATDCTB=Idreduces to
ad bc = 1.
The product of two symplectic matrices is a symplectic matrix: if S
and S0satisfy (5) then (SS0)TJ SS0=S0T(STJ S)S0=S0TJ S0=J. Also,
symplectic matrices are invertible, and their inverses are symplectic as well:
…rst, take the determinant of both sides of STJS =Jwe get det(STJ S) =
det J; since det J= 1 this is (det S)2= 1 hence Sis indeed invertible.
Knowing this, we rewrite STJS =Jas J S = (S1)TJ, from which follows
that (S1)TJS1=JSS1=Jhence S1is symplectic. The symplectic
matrices of same size thus form a group, called the symplectic group and
denoted by Sp(2n). An interesting property is that the symplectic group is
closed under transposition: if Sis a symplectic matrix, then so is ST(to
see this, just take the inverse of the equality (S1)TJS1=Jwhich yields
SJ [(S1)T]1=SJ [ST=Jnoting that J1=Jand [(S1)T]1=ST).
7

Since this means that a matrix is symplectic if and only if its transpose is,
inserting STin (5) and noting that (ST)T=Swe get the condition
SJ ST=J. (12)
Replacing S=A B
C Dwith ST=ATCT
BTDTthe conditions (11) are
thus equivalent to the set of conditions:
ABT=BAT,CDT=DCT,ADTBCT=Id:(13)
One can obtain other equivalent sets of conditions by using the fact that
S1and (S1)Tare symplectic.
It is very interesting to note that the inverse of a symplectic matrix is
S1=DTBT
CTAT:(14)
It is interesting because this formula is very similar to that giving the inverse
db
c a of a 22matrix a b
c dwith determinant one. The inversion
formula (14) suggests that in a sense symplectic matrices try very hard to
mimic the behavior of 22matrices: symplectic geometry is in essence an
aerial geometry, as already noticed in the discussion following formula (8).
A major manifestation of this property will be discussed below, when we
study Gromov’s non-squeezing theorem.
We will see that this is actually the essence of symplectic geometry, and
at the origin of the symplectic egg property!
A last property of symplectic matrices: recall that when we wanted to
show that a symplectic matrix always is invertible, we established the iden-
tity (det S)2= 1. From this follows that the determinant of a symplectic
matrix is a priori either 1or 1. It turns out –but there is no really elemen-
tary proof of this– that we always have det S= 1 (see for instance §2.1.1
in [7] where I give one proof of this property; Mackey and Mackey’s online
paper [17] give a nice discussion of several distinct methods for proving that
symplectic matrices have determinant one.
Conversely, it is not true that any 2n2nmatrix with determinant one
is symplectic when n > 1. Consider for instance
M=0
B
B
@
0 0 0
0 1= 0 0
0 0 0
0 0 0 1=
1
C
C
A(15)
8

where 6= 0; this matrix trivially has determinant one, but the condition
ADTBCT=Idin (13) is clearly violated unless =1. Another simple
example is provided by
M=R() 0
0R()
where R()and R()are rotation matrices with angles 6=(this coun-
terexample generalizes to an arbitrary number 2nof phase space dimensions
replacing R()and R()by arbitrary but distinct rotations in the xand p
spaces, respectively).
For a detailed exposition of symplectic matrices, with complete proofs,
see Chapter 2 in [7].
2.2 The …rst Poincaré invariant
In what follows (t),0t2, is a loop in phase space: we have (t) =
x(t)
p(t)where x(0) = x(2),p(0) = p(2); the functions x(t)and p(t)are
supposed to be continuously di¤erentiable. By de…nition, the …rst Poincaré
invariant associated to (t)is the integral
I() = I
pdx =Z2
0
p(t)T_x(t)dt: (16)
The fundamental property –from which almost everything else in this paper
stems– is that I()is a symplectic invariant. By this we mean that if we
replace the loop (t)by the a new loop S(t)where Sis a symplectic matrix,
the …rst Poincaré invariant will keep the same value: I(S) = I(), that is
I
pdx =IS
pdx: (17)
The proof is not very di¢ cult if we carefully use the relations characteriz-
ing symplectic matrices (see Arnol’d [3], §44, p.239, for a shorter but more
abstract proof). We will …rst need a di¤erentiation rule for vector-valued
functions, generalizing the product formula from elementary calculus. Sup-
pose that
u(t) = 0
B
@
u1(t)
.
.
.
un(t)
1
C
A,v(t) = 0
B
@
v1(t)
.
.
.
vn(t)
1
C
A
9

are vectors depending on the variable tand such that each component uj(t),
vj(t)is di¤erentiable. Let Mbe a symmetric matrix of size nand consider
the real-valued function u(t)TMv(t). Its derivative is given by the formula
d
dt u(t)TMv(t)= _u(t)TMv(t) + u(t)TM_v(t)(18)
(we are writing _u; _vfor du=dt,dv=dt as is customary in mechanics); for a
proof I refer you to your favorite calculus book.
Let us now go back to the proof of the symplectic invariance of the …rst
Poincaré invariant. Writing the symplectic matrix Sin block form A B
C D
the loop S(t)is parametrized by
S(t) = Ax(t) + Bp(t)
Cx(t) + Dp(t),0t2:
We thus have, by de…nition of the Poincaré invariant,
I(S) = Z2
0
(Cx(t) + Dp(t))T(A_x(t) + B_p(t))dt;
expanding the product in the integrand, we have I(S) = I1+I2where
I1=Z2
0
x(t)TCTA_x(t)dt +Z2
0
p(t)TDTB_p(t)dt
I2=Z2
0
x(t)TCTB_p(t)dt +Z2
0
p(t)TDTA_x(t)dt:
We claim that I1= 0. Recall that CTAand DTBare symmetric in view
of the two …rst equalities in (11); applying the di¤erentiation formula (18)
with u=v=xwe have
Z2
0
x(t)TCTA_x(t)dt =1
2Z2
0
d
dt(x(t)TCTAx(t))dt = 0
because x(0) = x(2). Likewise, applying (18) with u=v=pwe get
Z2
0
p(t)DTB_p(t)dt = 0
hence I1= 0 as claimed. We next consider the term I2. Rewriting the
integrand of the second integral as
x(t)TCTB_p(t) = _p(t)TBTC x(t)T
10

(because it is a number, and hence equal to its own transpose!) we have
I2=Z2
0
_p(t)TBTCx(t)Tdt +Z2
0
p(t)TDTA_x(t)dt
that is, since DTA=Id+BTCby transposition of the third equality in
(11),
I2=Z2
0
p(t)T_x(t)dt +Z2
0p(t)TBTCA _x(t) + _p(t)TBTCAx(t)dt:
Using again the rule (18) and noting that the …rst integral is precisely I()
we get, DTAbeing symmetric,
I2=I() + Z2
0
d
dt p(t)TBTCAx(t)dt:
The equality I(S) = I()follows noting that the integral in the right-hand
side is
p(2)TBTCAx(2)p(0)TBTCAx(0) = 0
since (x(2); p(2)) = (x(0); p(0)).
The observant reader will have observed that we really needed all of the
properties of a symplectic matrix contained in the set of conditions (11);
this shows that the symplectic invariance of the …rst Poincaré invariant is a
characteristic property of symplectic matrices.
2.3 Proof of the symplectic egg property
Let us denote by BRthe phase space ball centered at the origin and having
radius R. It is the set of all points z= (x; p)such that jzj2=jxj2+jpj2R2.
What we call a “symplectic egg”is the image S(BR)of BRby a symplectic
matrix S. It is thus an ellipsoid in phase space, consisting of all points zsuch
that S1zis in the ball BR, that is jS1zj2R2. Let us now cut S(BR)
by a plane jof conjugate coordinates xj; pj. We get an elliptic surface j,
whose boundary is an ellipse denoted by j. Since that ellipse lies in the
plane jwe can parametrize it by only specifying coordinates xj(t),pj(t)
all the other being identically zero; relabeling if necessary the coordinates
we may as well assume that j= 1 so that the curve jcan be parametrized
as follows:
j(t) = (x1(t);0;;0; p1(t);0;;0)T
11

for 0t2with x1(0) = x1(2)and p1(0) = p1(2). Since xk(t)=0
and pk(t)=0for k > 1the area of the ellipse is given by the formula
Area(1) = Z2
0
p1(t) _x1(t)dt
=
n
X
k=1 Z2
0
pk(t) _xk(t)dt
=I1
pdx
hence Area(1) = I(1). Since the inverse matrix S1is also symplectic, we
have I(1) = I(S11). But the loop S11bounds a section of the ball BR
by a plane (the plane S1j) passing through its center. This loop is thus
a great circle of BRand the area of the surface S11is thus exactly R2,
which was to be proven.
We urge the reader to notice that the assumption that we are cutting
S(BR)with a plane of conjugate coordinates is essential, because it is this
assumption that allowed us to identify the area of the section with action.
Here is a counterexample which shows that the property does not hold for
arbitrary sections of S(BR). Take, for instance
S=0
B
B
@
10 0 0
020 0
0 0 1=10
0 0 0 1=2
1
C
C
A,1>0,2>0;and 16=2(19)
so that S(BR)is de…ned by the inequality
1
1
x2
1+1
2
x2
2+1p2
1+2p2
2R2:
The section of S(BR)by the x1; p1plane is the ellipse obtained by setting
x2= 0 and p2= 0;that is
1
1
x2
1+1p2
1R2:
This elliptic section has area (R2p1p1=1) = R2as predicted. If we in-
stead intersect S(BR)with the x2; p1plane (which is not a plane of conjugate
variables), we get the ellipse
1
1
x2
1+2p2
2R2;
12

the latter has area (R2p1=2)which is di¤erent from R2since 16=2.
The assumption that Sis symplectic is also essential. Assume that we
scramble the diagonal entries of the matrix Sabove, getting the new matrix
S0=0
B
B
@
10 0 0
020 0
0 0 1=20
0 0 0 1=1
1
C
C
A.
The matrix S0still has determinant one, but it is not symplectic (cf. the
matrix (15)). The section S0(BR)by the x2; p2plane is here the ellipse
1
1
x2
1+2p2
1R2
with area R2p1=26=R2:
3 The Symplectic Camel
The property of the symplectic camel is a generalization of the property of
the symplectic egg to arbitrary canonical transformations; it reduces to the
latter in the linear case.
3.1 Gromov’s non-squeezing theorem: static formulation
As we mentioned in the Prologue, the property of the symplectic egg is
related to a deep topological result, the “non-squeezing theorem”of Gromov
[13] published in 1985. To understand it fully we have to introduce the notion
of canonical transformation [3, 5, 6, 7]. A canonical transformation is an
invertible in…nitely di¤erentiable mapping
f:x
p! x0
p0
of phase space on itself whose inverse f1is also in…nitely di¤erentiable and
such that its Jacobian matrix
f0(x; p) = @(x0; p0)
@(x; p)
is symplectic at every point (x; p). A symplectic matrix S=A B
C Dau-
tomatically generates a linear canonical transformation by letting it act on
13

phase space vectors x
p! Sx
p: it is an invertible transformation (be-
cause symplectic matrices are invertible), trivially in…nitely di¤erentiable,
and its Jacobian matrix is Sitself. Phase space translations, that is map-
pings x
p! x+x0
p+p0
are also canonical: their Jacobian matrix is just the identity Id0
0Id,
which is trivially symplectic. By composing linear canonical transformations
with translations one obtains the class of all a¢ ne canonical transformations.
Here is an example of a nonlinear canonical transformation: assume that
n= 1 and denote the phase space variables by rand 'instead of xand p;
the transformation de…ned by (r; ')! (x; p)with
x=p2rcos ',p=p2rsin ',0' < 2; (20)
has Jacobian matrix
f0(r; ') = 1
p2rcos '1
p2rsin '
p2rsin 'p2rcos '!
which has determinant one for every choice of rand '. The transformation
fis thus canonical, and can be extended without di¢ culty to the multi-
dimensional case by associating a similar transformation to each pair (xj; pj).
It is in fact a symplectic version of the usual passage to polar coordinates
(the reader can verify that the latter is not canonical by calculating its
Jacobian matrix); it can also be viewed as the simplest example of action-
angle variable [3, 5, 6].
We will see in a moment why canonical transformations play such an
important role in Physics (and especially in classical mechanics), but let us
…rst state Gromov’s theorem:
Gromov’s theorem: No canonical transformation can squeeze a ball BR
through a circular hole in a plane jof conjugate coordinates xj; pj
with smaller radius r < R .
This statement is surprisingly simple, and one can wonder why it took
such a long time to discover it. There are many possible answers. The
most obvious is that all known proofs Gromov’s theorem are extremely dif-
…cult, and make use of highly non-trivial techniques from various parts of
14

pure mathematics, so the result cannot be easily derived from elementary
principles. Another reason is that it seems, as we will discuss below, to
contradict the common conception of Liouville’s theorem, and was therefore
unsuspected!
So, what is the relation of Gromov’s theorem with our symplectic eggs,
and where does its nickname “principle of the symplectic camel”come from?
The denomination apparently appeared for the …rst time in Arnol’d’s paper
[4]. Recalling that it is stated in the Scriptures that
...Then Jesus said to his disciples, ‘Amen, I say to you, it will be
hard for one who is rich to enter the kingdom of heaven. Again
I say to you, it is easier for a camel to pass through the eye of a
needle than for one who is rich to enter the kingdom of God’
the biblical camel is here the ball BR, and the eye of the needle is the hole
in the xj; pjplane! (For various interpretations of the word “camel”see the
comments following E. Samuel Reich’s New Scientist paper [24] about or
“symplectic camel”paper [8].)
Let us next show that the section property of the symplectic egg is indeed
a linear (or a¢ ne) version of Gromov’s theorem. It is equivalent to prove
that no symplectic egg S(BR)with radius Rlarger than that, r, of the
hole in the xj; pjplane can be threaded through that hole. Passing S(BR)
through the hole means that the section of the symplectic egg by the xj; pj
plane, which has area R2, is smaller than the area r2of the hole; hence
we must have Rr.
3.2 Dynamical interpretation
The reason for which canonical transformations play an essential role in
Physics comes from the fact that Hamiltonian phase ‡ows precisely consist
of canonical transformations. Consider a particle with mass mmoving along
the x-axis under the action of a scalar potential V. The particle is subject
to a force F=d
dx V(x). Since F=mdv=dt =dp=dt (Newton’s second
law), the equations of motion can be written
dx
dt =p
m,dp
dt =dV
dx . (21)
Introducing the Hamilton function
H(x; p) = 1
2mp2+V(x)
15

this system of di¤erential equations is equivalent to Hamilton’s equations of
motion dx
dt =@H
@p ,dp
dt =@H
@p :(22)
We will more generally consider the n-dimensional version of (22) which
reads dxj
dt =@H
@pj
,dpj
dt =@H
@xj
,1jn: (23)
(In mathematical treatments of Hamilton’s equations [3, 5, 6, 7] the function
Hcan be of a very general type, and even depend on time t). In either case,
these equations determine –as any system of di¤erential equations does– a
‡ow. By de…nition, the Hamiltonian ‡ow is the in…nite set of mappings H
t
de…ned as follows: suppose we solve the system (23) after having chosen
initial conditions x1(0); :::; xn(0) and p1(0); :::; pn(0) at time t= 0 for the
position and momentum coordinates. Denote the initial vector thus de…ned
x(0)
p(0). Assuming that the solution to Hamilton’s equations at time t
exists (and is unique), we denote it by x(t)
p(t). By de…nition, H
tis just the
mapping that takes the initial vector to the …nal vector:
x(t)
p(t)=H
tx(0)
p(0):(24)
As time varies, the initial point describes a curve in phase space (often called
a “‡ow curve”or “Hamiltonian trajectory”).
The essential fact to remember is that each mapping H
tis a canonical
transformation; Hamiltonian ‡ows are therefore, in particular, volume pre-
serving: this is Liouville’s theorem [3, 5, 6]. This property easily follows
from the fact that symplectic matrices have determinant one. Since it is not
true that every matrix with determinant one is symplectic, as soon as n > 1
volume preservation also holds for other transformations, and is therefore
not a characteristic property of Hamiltonian ‡ows; see Arnold [3], Ch.3, §16
for a discussion of this fact. The thing to observe is that volume preserva-
tion does not imply conservation of shape, and one could therefore imagine
that under the action of a Hamiltonian ‡ow a subset of phase space can be
stretched in all directions, and eventually get very thinly spread out over
huge regions of phase space, so that the projections on any plane could a
priori become arbitrary small after some time t. In addition, one may very
well envisage that the larger the number nof degrees of freedom, the more
16

that spreading will occur since there are more directions in which the ball
is likely to spread! This possibility, which is ruled out by the symplectic
camel as we will explain below, has led to many philosophical speculations
about Hamiltonian systems. For instance, in his 1989 book Roger Penrose
([21], p.174–184) comes to the conclusion that phase space spreading sug-
gests that “classical mechanics cannot actually be true of our world”(p.183,
l.–3). Our discussion of Gromov’s theorem shows that Hamiltonian evolu-
tion is much less disorderly than Penrose thought. To see this, consider
again our phase space ball BR. Its orthogonal projection (or “shadow”) on
any two-dimensional subspace of phase space is a circular surface with
area R2. Suppose now that we move the ball BRusing a Hamiltonian
‡ow H
tand choose for the plane jof conjugate coordinates xj; pj.
The ball will slowly get deformed, while keeping same volume. But, as a
consequence of the principle of the symplectic camel, its “shadow” on any
plane jwill never decrease below its original value R2(as illustrated in
Fig.2)! Why is it so? First, it is clear that if the area of the projection
of f(BR)on on a plane xj; pj(fa canonical transformation) will never be
smaller than R2, then we cannot expect that f(BR)lies inside a cylinder
(pjaj)2+ (xjbj)2=r2if r < R. So is the “principle of the symplectic
camel” stronger than Gromov’s theorem? Not at all, it is equivalent to it!
Here is a simple proof. We assume as in section 2.3 that j= 1; this does
not restrict the generality of the argument. Let 1be the boundary of the
projection of f(BR)on the x1; p1plane; it is a loop encircling a surface 1
with area at least R2. That surface 1can be deformed into a circle with
same area using an area-preserving mapping of the x1; p1plane; call that
mapping f1and de…ne a global phase space transformation fby the formula
f(x1; p1; x2; p2; :::::; xn; pn)=(f1(x1; p1); x2; p2; :::; xn; pn)
(we are using in this formula, for obvious reasons of readability, an ordering
of the position and momentum variables di¤erent from the standard one).
Calculating the Jacobian matrix it is easy to check that the matrix fis a
canonical transformation, hence our claim. For a more detailed discussion
of this and related topics see [8, 12].
3.3 The symplectic camel and Newton’s second law
Recall that we derived Hamilton’s equations for a particle moving in a force
…eld F=d
dx V(x)by writing down the equations of motion in the form
mdx
dt =p,dp
dt =dV
dx .
17

The observant reader will have noticed that these two equations are just
one way to express Newton’s second law (it is, by the way, the correct
formulation: the formula “force =mass acceleration”, due to Birkho¤, is
circular!). More generally for a system of Npoint-like particles moving in
three-dimensional physical space, Newton’s second law would be
mdxj
dt =pj,dpj
dt =dV
dxj
;
or, equivalently, Hamilton’s equations (23) with 1jn= 3N. Thus,
for Hamiltonian systems, Gromov’s non-squeezing theorem just expresses a
very deep and invisible property of Newton’s second law!
4 Quantum Blobs
What’s in a name? That which we call a rose by any other
name would smell as sweet.
Romeo and Juliet, Act 2, Scene 2 (W. Shakespeare)
By de…nition, a quantum blob is a symplectic egg with radius R=p~:
The section of quantum blob by a plane of conjugate coordinates is thus
~=1
2h. We will see that quantum blobs qualify as the smallest units
of phase space allowed by the uncertainty principle of quantum mechanics.
We begin with a very simple example illustrating the basic idea, which is
that a closed (phase space) trajectory cannot be carried by an energy shell
18

smaller (in a sense to be made precise) than a quantum blob. As simple as
this example is, it allows us to recover the ground energy of the anisotropic
quantum harmonic oscillator.
4.1 The harmonic oscillator
The fact that the ground energy level of a one-dimensional harmonic oscil-
lator
H=p2
x
2m+1
2m!2x2
is di¤erent from zero is heuristically justi…ed in the physical literature by the
following observation: since Heisenberg’s uncertainty relation pxx1
2~
prevent us from assigning simultaneously a precise value to both position
and momentum, the oscillator cannot be at rest. To show that the lowest
energy has the value 1
2~!predicted by quantum mechanics one can then
argue as follows: since we cannot distinguish the origin (x= 0; p = 0) of
phase space from a phase plane trajectory lying inside the double hyperbola
pxx < 1
2~, we must require that the points (x; p)of that trajectory are such
that jpxxj  1
2~; multiplying both sides of the trivial inequality
p2
x
m! +m!x22jpxxj  ~
by !=2we then get
E=p2
x
2m+1
2m!2x21
2~!
which gives the correct lower bound for the quantum energy. This argument
can be reversed: since the lowest energy of an oscillator with frequency !
and mass mis 1
2~!, the minimal phase space trajectory will be the ellipse
p2
x
m~!+x2
(~=m!)= 1
which encloses a surface with area 1
2h. Everything in this discussion imme-
diately extends to the generalized anisotropic n-dimensional oscillator
H=
n
X
j=1
p2
j
2mj
+1
2mj!2
jx2
and one concludes that the smallest possible trajectories in xj; pjspace are
the ellipses
p2
j
mj~!j
+x2
j
(~=mj!j)= 1.
19

By the same argument as above, using each of the Heisenberg uncertainty
relations
pjxj1
2~(25)
we recover the correct ground energy level
E=1
2~!1+1
2~!2++1
2~!n
as predicted by standard quantum theory [18]. In addition, one …nds that,
the projection of the motion on any plane of conjugate variables xj; pjwill
always enclose a surface having an area at least equal to 1
2h. In other
words, the motions corresponding to the lowest possible energy must lie on
a quantum blob!
These considerations suggest a strong relationship between quantum
blobs and the uncertainty principle.
4.2 Quantum blobs and uncertainty
The Heisenberg inequalities (25) are a weak form of the quantum uncer-
tainty principle; they are a particular case of the more accurate Robertson–
Schrödinger [23, 26] inequalities
(pj)2(xj)2(xj; pj)2+1
4~2(26)
(see Messiah [18] for a simple derivation). Here, in addition to the standard
deviations xj,pjwe have the covariances (xj; pj)which are a measure-
ment of how much the two variables xj; pjchange together. (We take the
opportunity to note that the interpretation of quantum uncertainty in terms
of standard deviations goes back to Kennard [15]; Heisenberg’s [14] initial
formulation was much more heuristic). Contrarily to what is often believed
the Heisenberg inequalities (25) and the Robertson–Schrödinger inequali-
ties (26) are not statements about the accuracy of our measurements; their
derivation assumes on the contrary perfect instruments (see the discussion
in Peres [22], p.93). Their meaning is that if the same preparation proce-
dure is repeated a large number of times on an ensemble of systems, and is
followed by either by a measurement of xj, or by a measurement of pj, then
the results obtained have standard deviations xj,pj; in addition these
measurements need not be uncorrelated: this is expressed by the statistical
covariances (xj; pj)appearing in the inequalities (26).
It turns out that quantum blobs can be used to give a purely geometric
and intuitive idea of quantum uncertainty. Let us …rst consider the case
20

n= 1, and de…ne the covariance matrix by
 = x2(x; p)
(p; x) p2:(27)
Its determinant is det  = (p)2(x)2(x; p)2, so in this case the Robertson–
Schrödinger inequality is the same thing as det  1
4~2. Now to the geo-
metric interpretation. In statistics it is customary to associate to the
so-called covariance ellipse: it is the the ellipse de…ned by
1
2(x; p)1x
p1:(28)
The area of this ellipse is 2pdet , that is, by (27):
Area() = 2(p)2(x)2(x; p)21=2
and the inequality det  1
4~2is thus equivalent to Area()~=1
2h.
We have thus succeeded in expressing the rather complicated Robertson–
Schrödinger inequality (26) in terms of the area of a certain ellipse. In
higher dimensions the same argument applies, but contrarily to what com-
mon intuition suggests, the Robertson–Schrödinger inequalities will not be
expressed in terms of volume (which is the generalization of area to higher
dimensions), but again in terms of areas –namely those of the intersections
of the conjugate planes xj; pjwith the covariance ellipsoid
 = (x; x) (x; p)
(p; x) (p; p):(29)
Here (x; x);(x; p), etc. are the nnblock-matrices ((xi; xj))1i;jn,
((xi; pj))1i;jnetc. Notice that the diagonal terms of are just the
variances x2
1; :::; x2
n; p2
1; :::; p2
nso that (29) reduces to (27) for n= 1.
De…ning the covariance ellipsoid as above, one then proves that the
inequalities (26) are equivalent to the property that the intersection of 
with the planes xj; pjis at least 1
2h. These inequalities are saturated (i.e.
they become equalities) if and only if these intersections have exactly area
1
2h, that is, if and only if is a quantum blob! The proof goes as follows (for
a detailed argument see [8, 12]): one …rst remarks, using a simple algebraic
argument that if is non-singular the Robertson–Schrödinger inequalities
are equivalent to the following condition of the covariance matrix, due to
Narcowich [19] and often used in quantum optics (see [2, 27, 28] and the
references therein):
21

The eigenvalues of the Hermitian matrix  + i~
2Jare non-
negative (which we write for short: + i~
2J0).
One shows that this condition implies that the covariance matrix is de…nite
positive, and hence invertible. The next step consists in noting that in view
of Sylvester’s theorem from linear algebra the leading principal minors of
the matrix
 + i~
2J=(x; x) (x; p) + i~
2I
(p; x)i~
2I(p; p)
are non-negative. This applies in particular to the minors of order 2so that
we must have 
x2
j(xj; pj) + i~
2
(pj; xj)i~
2p2
j0;
expanding the determinant on the left side, this condition is precisely the
Robertson–Schrödinger inequality (26).
As we have seen, the fact that the covariance ellipsoid is cut by the
conjugate coordinate planes along ellipsoids with areas 1
2himplies the
Robertson–Schrödinger inequalities. This is thus a geometric restatement of
the quantum uncertainty principle; we can rephrase it as follows:
Every quantum covariance ellipsoid contains a quantum blob,
i.e. a symplectic egg with radius p~.When this ellipsoid is
itself a quantum blob, the Robertson–Schrödinger inequalities are
saturated.
This statement can be extended in various ways; in a very recent paper
[9] we have applied this geometric approach to the quantum uncertainty
principle to the study of partial saturation of the Robertson–Schrödinger
inequalities for mixed quantum states. We show, in particular, that partial
saturation corresponds to the case where some (but not all) planes of con-
jugate coordinates cut the covariance ellipsoid along an ellipse with exactly
area 1
2h; this allows us to characterize those states for which this occurs
(they are generalized Gaussians, more precisely the “squeezed states”famil-
iar from quantum optics).
Another important thing we will unfortunately not be able to discuss
in detail because of length limitations, is the following: everything we have
said above still holds true if we replace the sentence “planes of conjugate
coordinates xj; pj” with “symplectic planes”. A symplectic plane is a two-
dimensional subspace of phase space which has the property that if we re-
strict the symplectic form (9) to pairs of vectors in the symplectic plane,
22

then we again obtain a symplectic form. For instance, it is easy to check
that the xj; pjare symplectic planes (but those of coordinates xj; pk,j6=k,
or xj; xk, or pj; pkare not). One proves [3, 7] that every symplectic plane
can be obtained from any of the xj; pjplanes using a symplectic transfor-
mation, and that such transformations take symplectic planes to symplectic
planes This implies, in particular, that the Robertson–Schrödinger inequali-
ties (26) are covariant under symplectic transformations: if one de…nes new
coordinates x0; p0by (x0; p0)T=S(x; p)T,Sa symplectic matrix, then if
(pj)2(xj)2(xj; pj)2+1
4~2
we also have
(p0
j)2(x0
j)2(x0
j; p0
j)2+1
4~2:
On the other hand, it is moderately di¢ cult (but we will not do it here)
to show that the Robertson–Schrödinger inequalities do not retain their
form under changes of coordinates that are not symplectic, so that linear
symplectic transforms are the only linear transforms which preserves the
uncertainty principle.
We mention that there are possible non-trivial generalizations of the
uncertainty principle, using new results in symplectic topology, for instance
in [1] Gromov’s theorem (in the linear case) is extended to projections on
symplectic subspaces with dimension greater than 2. They …nd that the
volume of the projections are conserved during linear Hamiltonian motions.
These results certainly deserve to be investigated further, since they lead to
important “quantum universal invariants”which have not yet been studied.
4.3 Coarse-graining by quantum blobs
There is a very interesting and deep relation between the geometric notion of
quantum blob, and the Wigner formalism (for detailed studies of the latter
see [7, 16] and the numerous references therein).
It is customary in quantum statistics to “coarse-grain” phase space in
“quantum cells”which are cubes with volume (ph)2n=hn(see the seminal
paper [20]). These cells do not have any symmetry under general symplec-
tic transformations: while such a transformation preserves volume, a cube
will in general be distorted into a multidimensional polyhedron. But what
is more striking is the comparison of volumes. Since a quantum blob is
obtained from a ball with radius p~by a symplectic, and hence volume-
preserving, transformation its volume is hn=n!2nwhich is n!2nsmaller than
that of a quantum cell. This is a huge number. For instance, in the case
23

of the physical three-dimensional con…guration space this leads to a factor
of 48. In the case of a macroscopic system with n= 1023 this fact be-
comes unimaginably large. This is in strong contrast with the fact that
the orthogonal projection of a quantum blob on any plane xj; pjof conju-
gate coordinates (or, more generally, on any symplectic plane) is an ellipse
with area equal to ~=h=2. The coarse graining of phase space by quan-
tum blobs has several advantages, which I have discussed in [10]. Here is
one: a quantum blob S(Bp~)is the set of all points z= (x; p)such that
jS1zj  p~; equivalently (S1)TS1zTz~. Set G= (S1)TS1and
consider the phase space Gaussian
(z) = 1
~ne1
~zTGz
One shows [7, 16] that (z)is the Wigner transform of the generalized
coherent state
(x) = 1
~n=4(det X)1=4e1
2~xTMx
where Mis a symmetric complex matrix of the type M=X+iY whose
real part Xis positive de…nite. The matrices Xand Ycan be determined
in terms of the matrix Gby solving the equation
X+Y X1Y Y X1
X1Y X1=G(30)
in Xand Y. This shows that there is a one-to-one correspondence between
quantum blobs and coherent states. For instance, if Sis the identity, in
which case the quantum blob is just the ball Bp~, formula (30) yields X=I
and Y= 0 so that (x) = (~)nejxj2=2~, which is the …ducial coherent
state initially introduced by Schrödinger [25] in 1926. Our quantum blobs
can thus be viewed as the phase space pictures of Gaussian states.
5 Conclusion
In these days the angel of topology and the devil of abstract
algebra …ght for the soul of each individual mathematical domain
(H. Weyl, 1939)
This quotation from the mathematician Hermann Weyl goes straight
to the point, and applies to Physics as well: while algebra (in the large)
has dominated the scene of quantum mechanics for a very long time (in
fact, from its beginning: think about Heisenberg’s “matrix mechanics”), we
24

are witnessing a slow but steady emergence of geometric ideas, and to a
“symplectization of Science”. Not only do these geometric ideas add clarity
to many concepts, but they also lead to new insights. This is what we had
in mind while writing the present paper.
Acknowledgement 1 The present work has been supported by the Austrian
Research Agency FWF (Projektnummer P20442-N13).
Acknowledgement 2 I wish to thank the referees for valuables comments
and suggestions. Also, many thanks to my son Sven for having drawn the
pictures in this paper.
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