Communications
in
Commun. math. Phys.
65,
189-201
(1979)
Mathematical
Physics
© by
Springer-Verlag
1979
Geometrization
of
Quantum Mechanics
T.
W. B.
Kibble
Blackett Laboratory, Imperial College, London
SW7
2BZ,
England
Abstract. Quantum mechanics
is
cast into
a
classical Hamiltonian
form
in
terms
of a
symplectic
structure,
not on the
Hubert space
of
state-vectors
but on
the
more physically relevant
infinite-dimensional
manifold
of
instantaneous
pure states. This geometrical structure
can
accommodate generalizations
of
quantum
mechanics,
including
the
nonlinear relativistic models recently
proposed.
It is
shown that
any
such generalization
satisfying
a few
physically
reasonable conditions would reduce
to
ordinary quantum mechanics
for
states
that
are
"near"
the
vacuum.
In
particular
the
origin
of
complex structure
is
described.
1.
Introduction
Geometrical ideas, especially symplectic structures, have come
to
play
an
increas-
ingly
important role
in
classical mechanics
[1,
2].
The
geometry
of the
classical
phase space
Γ
also underlies
the
geometrical quantization programme
of
Souriau
[3, 4]
(see
also Kostant
[5]).
Moreover
it is
known that quantum dynamics
can be
expressed
in
terms
of a
Hamiltonian structure
on the
Hubert space
ffl of
state-
vectors, where
the
imaginary part
of the
scalar product
defines
a
symplectic
structure
[6].
However
if one is
seeking
an
axiomatic
basis
for
quantum mechanics,
it
seems
better
to
start
from
structures
of
direct physical significance,
as in the
operational
approach
of
Haag
and
Kastler
[7] and
others
[8, 9] or the
work
on the
geometry
of
quantum logics
[10,
11].
It
is
pointed
out in
Sect.
2
that this
can be
achieved
by a
slight modification
of
the
Hamiltonian formalism.
We
have
to
consider
not the
Hubert
space
tff
itself
but the
manifold
Σ of
"instantaneous pure
states"
which
is
(essentially
but not
quite)
a
projective Hubert
space.
This formalism
is
closely akin
to the
work
of
Mielnik
[12]
on the
geometry
of the
space
of
quantum states.
It
provides
a
convenient
framework
within
which
to
discuss possible generalizations
of
quan-
tum
mechanics.
In
particular
it can
readily accommodate
the
relativistic model
theories proposed
in a
recent paper
[13],
or at
least
a
large class
of
them.
It is
190 T. W. B.
Kibble
worth noting that
by
formulating
the
theory
on Σ
rather than
Jtf
we
automatically
ensure that
it
satisfies
the
scaling property shown
in
[13]
to be
necessary
for a
consistent
measurement theory
(see
also
[14]).
From
this point
of
view,
the
essential
difference
between
classical
and
quantum
mechanics lies
not in the set of
states (save
for the
infinite
dimensionality)
nor in
the
dynamic evolution,
but
rather
in the
choice
of the
class
of
observables, which
is
far
more restricted
in
quantum than
in
classical mechanics.
One
motivation
for the
present
work
is the
possibility that
it
might
be of use in
the
unification
of
quantum mechanics with general relativity.
The
idea that this
unification
must
be an
essentially geometric
one,
so
long championed
by
Einstein
in
his
search
for a
unified
theory,
has
recently been coming
back
into favour.
It
seems natural that
as a
prerequisite quantum mechanics itself should
be
cast
in
geometrical language. Moreover there
are
good
reasons
for
seeking
to
generalize
it,
to
free
it
from
the
restrictions
of
linearity just
as
general relativity
has
freed
space-time
from
the
limitations
of flatness
[12,
15].
The
geometrical structure
described
in the
present paper
can
easily
be
generalized
to
allow
the
space
of
quantum
states
to be an
arbitrary infinite-dimensional symplectic manifold.
Obviously,
any
viable generalization
of
quantum mechanics must reduce
to
that theory
for a
wide range
of
phenomena.
One of the
main aims
of
this paper
is
to
discuss
how
this might happen.
I
shall show,
on the
basis
of
some rather natural
physical assumptions, that
all the
main features
of
conventional quantum
mechanics would emerge naturally
for
states that
are in a
suitable sense near
the
vacuum, near enough
to be
represented
by
vectors
in the
tangent space.
In
Sects.
2 and 3 we
recall
the
main features
of
symplectic structures
and
Hamiltonian dynamics,
and
introduce
the
quantum phase space
Σ. For
con-
ventional quantum mechanics,
Σ is
essentially
a
projective complex Hubert space
more precisely,
a
dense subspace thereof. However
the
formalism
can be
applied
to
a
much more general case,
in
which
Σ is a
real
infinite-dimensional
manifold
with
(weak) symplectic structure.
The
dynamics
is
discussed
in
Sect.
3 in
terms
of a
Hamiltonian
function
E on Σ. For
ordinary quantum mechanics
E is the
expectation value
of the
Hamiltonian operator,
and the
evolution equation
reproduces
Schrodinger's
equation.
The
concept
of a
symmetry
is
examined
in
Sect.
4
with particular emphasis
on
the
space-time symmetries associated with
the
Poincare
group. Although
an
eventual
aim is the
unification
of
quantum theory with gravity,
for the
present
a
flat
space-time
is
assumed.
So too is the
existence
of a
unique vacuum state
v. The
tangent space
T
v
to Σ at the
vacuum plays
an
important role.
The aim of
Sect.
5 is to
show that
for
states
close enough
to the
vacuum
to be
represented
by
vectors
in the
tangent space,
the
formalism necessarily reduces
to
ordinary linear quantum mechanics.
In
particular,
I
shall show how, although
Σ is
a
real manifold,
the
complex structure
of
quantum mechanics would naturally
appear
on the
tangent space.
The
conclusions
are
summarized
in
Sect.
6 and a
number
of
unresolved
questions described.
2.
The
Quantum Phase Space
Axiomatic
treatments
of
quantum mechanics
often
begin with
the set of all
mixed
states
or
ensembles
[7,
11,
16,
17].
Pure states
are
regarded merely
as
extremal
Geometrization
191
elements
of
this convex set. However
we
shall deal exclusively with
the set
ίf
of
pure states,
and
assume
at the
outset that
all
others
can be
expressed
as
(presumably countable) mixtures
of
them.
Moreover
instead
of the
Heisenberg-picture
approach
we
shall
use the
Schrodinger
picture.
We
assume that, with respect
to a
given choice
of
time axis
each pure state
can be
represented
by a
path,
or
"history"
in a
space
Σ of
"instantaneous pure
states".
For
convenience,
we
reserve
the
term
state
for an
element
of
Σ;
elements
of
£f
will
be
called
histories.
Of
course,
the
dynamics
establishes
a
one-to-one correspondence between elements
of Σ at a
specified
time
and
elements
of
ίf.
Although
the
formalism will
be
more general,
it
will
be
useful
to
begin
by
discussing
the
structure
of Σ in the
special case
of
standard
quantum mechanics.
Indeed this
is the
only space
we
shall actually need
in
this paper because
the
generalized models considered
use the
same
set of
instantaneous states
and
differ
from
the
standard theory only
in the
dynamics. Later, however, more general
theories will
be
considered.
In
classical mechanics,
Σ is of
course
the
phase space
Γ,
normally
a
finite-
dimensional
manifold,
but in
quantum mechanics
it is
infinite-dimensional.
Essentially
it is a
projective Hubert space,
the set of
rays
in a
complex Hubert
space
Jtf
of
state vectors. However there
are
technical problems associated with
the
fact
that
the
Hamiltonian operator
H is
unbounded.
We
have
to
choose
between
two
alternative formalisms, each with
its
peculiar
advantages
and
disadvantages.
One is to
work with
the
rays
of
^
itself
and
accept that
the
Hamiltonian vector
field
which specifies
the
dynamics
is
defined only
on a
dense
subspace
[6].
The
other, which
we
shall
use
here,
is to
work
from
the
start with
a
dense
subspace
ff of ffl
equipped with
a
finer
topology that makes
H
continuous.
The
chief drawback
of
this second alternative
is
that
Σ
possesses
only
a
weak
symplectic structure (see below).
If we
were trying
to
prove existence theorems
for
solutions
of the
time-evolution equation, this might
be a
major
defect,
but for our
present purposes
it is not
important.
Let us
assume then that there
is a
dense linear subspace
$f
QΪ
3F
which
forms
a
common invariant domain
for all the
operators
we
wish
to
consider, including
in
particular
the
Hamiltonian
and
other symmetry generators.
For
example,
in a
nonrelativistic many-body theory
we may
take
3C
to be the
subspace
of
states
whose wave
functions
belong
to
some
test-function
space,
say the
Schwartz space
£f
(see
for
instance
[18]).
In a
field
theory
we
might
take
it to be the
subspace
generated
by
applying some
algebra
of
observables
to the
vacuum state.
Physically,
it is
only states
in
this subspace that
we can
actually prepare,
so
JΓ
is of
more direct physical relevance than
#?..
The
space
jf
is
assumed
to be
equipped with
a
topology
finer
than that
of
J^
defined
for
example
by a
countable
family
of
norms [18], which makes
H and the
other symmetry generators continuous
on
Jf.
It is
possible
to do
this
in
such
a way
that
Jf
becomes
a
Frechet-Schwartz
space.
Now
let
Jf
° be the set of all
nonzero
vectors
in
JΓ,
and let
<C
υ
be the
multiplicative
group
of all
nonzero complex numbers. Then
we
choose
as our
standard quantum phase space
the
projective space
Z
=
JΓ°/<C°,
which
we
shall
regard
as a
real manifold.
192 T. W. B.
Kibble
For the
general
theory,
all we
shall assume
is
that
Σ is a
real
infinite-
dimensional
paracompact
manifold modelled
on
some Frechet-Schwartz
space
Ύ*.
What this means
[19]
is
that
Σ
is
a
Hausdorff
topological space that
can be
covered
by a
countable
family
of
open sets
U
Λ
on
each
of
which
a
homeomorphism
φ
α
is
defined
to an
open
set in
^
and
that whenever
U
α
nU
β
=t0,
the map
φ
α
φ
β
l
restricted
to
φ
β
(U
α
nU
β
)
is
twice continuously
differentiable.
In a
general
infinite-
dimensional
non-Banach
space,
the
definition
of
differentiability
is
quite proble-
matic
[20].
However
in the
particular
case
of
Frechet-Schwartz spaces, there
is a
perfectly
serviceable
definition
[21]
which allows
the
construction
of
C
k
manifolds
(see
also
[22,
23]).
It
might
not be
unreasonable
to
require that
Σ be a
C°°
manifold,
but we
shall
not
need that assumption here.
In
addition
to its
manifold structure,
Σ is
required
to
have
a
weak
symplectic
structure.
This means
[6]
that there
is
defined
on Σ a
closed two-form
ω
which
is
weakly non-degenerate
in the
sense that
if
ω
u
(X
9
Y)
=
0 for all
tangent vectors
YE
T
U
Σ,
then^
=
0.
This
form
can be
used
to
"lower"
indices
[1].
It
defines
a map
X\-^X
]>
from
the
tangent
bundle
TΣ to the
cotangent bundle
T*Σ
:
if X is any
tangent vector
(or
vector
field)
the
corresponding cotangent vector
(or
differential
one-form)
is
X*
=
iχω,
(1)
where
i
x
is the
interior product (evaluation
function)
defined
by
(Note that there
is a
difference
of a
factor
2
between conventions used here
and in
references
[1, 2,
6].
I
follow
the
conventions
of
Choquet-Bruhat
et
al.
[24]).
The map
Xt-*X*
is
injective.
However,
ω is
only weakly nondegenerate
in the
sense that
Xt-^
17
is not in
general
bijective.
There
may be
one-forms that
are not
images
of any
vector
field.
In
the
special case
of
ordinary quantum mechanics,
ω is
defined
in
terms
of the
inner
product.
(As
well
its
finer
topology
JΓ
retains
the
inner product
<
,
> of
Jf
.)
To
define
ω we
need
to
introduce
a
local coordinate system
in Σ. Let π be the
canonical projection
of
JΓ°
onto
Σ.
Thus
if u is any
nonzero vector
in
JΓ,
πu = u
denotes
the
corresponding state.
Let v be any
normalized vector
in
ctf.
Then
in a
neighbourhood
of
πveΣ
we can
represent states uniquely
by
vectors
u in the
hyperplane
Thus
we
have
a
homeomorphism
φ
from
this neighbourhood
to an
open
set in
JΓ
V
.
We
then
define
ω at πv by
ω
πv
(X,y)
=
2Im<(^,<^y>
(2)
for
any
X,
Ye
T
πv
Γ,
or
more generally
at any
point
in
this coordinate patch
by
where
X,
YE
T
U
Σ
and
P
u
is the
orthogonal projector onto
the
subspace
normal
to u.
Geometrization
193
This two-form
ω
is
obviously skew-symmetric
and
weakly
nondegenerate.
But
while
ω is
readily seen
to be
closed,
it is not
exact.
It is
easy
to
construct
a
closed
surface
over which
its
integral
is
nonzero, although locally
ω = — dθ
with
Im
(dφu,
φu)
This
is
another interesting
difference
between quantum
and
classical mechanics.
Classically,
the
phase space
has the
structure
of a
cotangent bundle
and
ω=—dθ
where
θ is the
canonical one-form
θ =
Σpdq.
No
such form exists globally
on the
quantum phase space
Σ.
3.
Dynamics
on the
Quantum Phase
Space
Let
us now
consider
the
specification
of
dynamics
on Σ,
assumed
to be a
real
C
2
Frechet-Schwartz
manifold equipped with
a
weak symplectic structure.
The
dynamics
is
described
by a
Hamiltonian
flow,
that
is to say a
one-
parameter group
of
diffeomorphisms
τ
t
:Σ-+Σ
(with
τ
0
= l and
τ
t+s
=
τ
t
°τ
s
)
which
preserve
the
symplectic structure,
i.e.
τ*ω
=
ω.
(4)
The
τ
f
depend continuously
and
differentiably
[21]
on t and can
thus
be
written
τ
t
= exp tK ,
where
K is a
C
1
vector
field
on
Σ,
which
of
course also leaves invariant
the
symplectic
structure
:
£
κ
ω
=
0,
(5)
where
L
κ
is the
corresponding
Lie
derivative.
(Had
we
adopted
the
alternative
approach mentioned
in
Sect.
2 we
could
not
have required
τ
t
to be
differentiable
in
t
and
would have
had to
allow
K to be
defined
only
on a
dense subset
of
Σ,
see
[6].)
A
history,
i.e.
an
element
of
^
is an
integral curve
of
this
flow in
Σ,
a
curve
c
everywhere
tangent
to
K
:
^
=*,,„,
w
or
equivalently
φ + ί) =
τ
f
φ).
(7)
By
virtue
of the
identity
L
κ
=
i
κ
d
4-
dί
κ
and the
fact
that
ω is
closed,
it
follows
from
(5) and the
definition
(1)
that
=
0,
(8)
i.e.
X
b
is
closed.