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I~_!:eg·~abL_:_!-l.1.mj~!~Jnian
Systems
and
Interactions
through
Quadratic
Constraints
2 H
AM
B
U
R
G 5
2.
DESY
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•
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0
.:'
·.
Integrable
Hamiltonian
Systems
and
Interactions
through
Quadratic
Constraints
by
K.
Pohlmeyer
II.
Institut
fUr
Theoretische
Physik
der
Universitat
Hamburg, Germany
Abstract:
0
-
invariant
classical
relativistic
field
theories
n
in
one
time
and
one
space
dimension
with
interactions
that
are
entirely
due
to
quadratic
constraints
are
shown
to
be
closely
related
to
integrable
Hamiltonian
systems.
····-·-·---
......................................
_,.
____________
.......................
--·.-
....
__
_j
- I -
I.
Tntroduc
tion
Even
in
one
space
dimension,
relativistically
invariant
classical
field
theories
defining
integrable
Hamiltonian
systems
with
a
non-trivial,
momentum
dependent
scattering
matrix,
are
not
in
oversupply.
Actually,
up
to
equivalence
and
slight
modifications
there
is
only
one
such
model
available,
the
celebrated
sine-Gordon
equation
\),z,3J.
In
this
paper
we
shall
present
a
whole
series
of
non-equivalent
relativistically
invariant
field
theories
in
one
time
and
one
space
dimension,
each
having
a
one
parameter
family
of
Backlund
transformations
and
an
infinite
number
of
known
integrals
-of
motion.
These
conserved
quantities
are
associated
with
covariant
local
conserved
currents
for
which
the
family
of
Back1.und
transformations
serves
as
a
generating
functional.
Further,
each
one
of
these
models
has
non-trivial
momentum
dependent
scattering,
and
possesses
stable
stationary
finite
energy
solutions,
so-called
solitons.
By
a
procedure
explained
below
("reduction"),
the
series
of
new
models
is
obtained
from
0
-
invariant
Lagrangian
field
theories
n
whose
interaction
arises
solely
from
the
condition
that
the
values
of
the
field
functions
be
constrained
to
the
surface
of
a
sphere
(describing
a homogeneous
space
for
0
).
The new
examples
should
be
n
viewed
as
generalizations
(involving
more
and
more
fields)
of
the
sine-Gordon
theory
,
which
corresponds
to
the
chiral
symmetry
group
o
3
(To
o
2
there
corresponds
the
theory
of
a
free
massless
field).
The
connection
with
the
0
-
invariant
chiral
theories
allows
for
. n
a
simple
geometrical
interpretation
of
various
computational
manipulations
in
the
new
models.
For
n~
6
we
set
up
the
linear
- 2 -
eigenvalue
equation
(for
the
characteristic
initial
value
problem),
which
is
the
key
to
the
inverse
scattering
method
Th
,51.
We
determine
the
evolution
of
the
spectral
data
and
thereby
solve
the
characteristic
initial
value
problem.
Conversely,
the
analysis
of
the
new
models
provides
a
significant
first
step
towards
the
complete
description
of
all
finite
energy
solutions
of
the
original
0
-
invariant
chiral
theories
e.g.
n
supplying
for
them
an
infinite
number
of
integrals
of
motion
associated
with
covariant
local
conserved
currents.
Of
particular
interest
is
the
o
4
:::_
SU
(2)
x
SU(2)
-
invariant
chiral
theory,
the
one-space-
lac.
dimensional
version
of
the
non-linear
Cf-
model
t61.
Apropos,
the
original
0
-
invariant
chiral
theories
do
not
possess
n
soliton
solutions.
However,
the
solutions
related
to
the
soliton
solutions
of
the
corresponding
reduced
model
are
expected
to
play
a
special
role.
To
sum
up,
the
alm
of
this
paper
lS
twofold:
i)
furnishing
new
examples
with
the
same
powerful
structure
as
the
one
which
is
at
the
bottom
of
the
sine-Gordon
theory
and
ii)
contributing
to
the
solution
of
theories
with
an
effective
Lagrangian
comprising
the
results
of
current
algebra
(6J.
The
present
communication
grew
out
of
joint
work
with
H.
Lehmann
and
G.
Roepstorff
in
1968 when
the
connection
between
the
chiral
o
3
-
invariant
theory
and
the
sine-Gordon
theory
came
to
light.
"
. "
............
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