Submitted
to
Journal
of
Mathematical
Physics
'.
,
.
LBL-4283
Preprint
c.
\
ON
THE
DUALITY
CONDITION
FOR
QUANTUM
FIELDS
Joseph
J.
Bisognano
and
Eyvind
H.
Wichmann
August
1975
Prepared
for
the
U.
S.
Energy
Research
and
Development
Administration
under
Contract
W
-7405-ENG-48
For
Reference
Not
to
be
taken from this room
DISCLAIMER
This document was prepared
as
an account
of
work sponsored by the United States
Government. While this document is believed to contain correct information, neither the
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of
the University of
California, nor any
of
their employees, makes any warranty, express or implied, or
assumes any legal responsibility for the accuracy, completeness, or usefulness
of
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United States Government or any agency thereof, or the Regents
of
the University
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of
the
University
of
California.
• .
t"
. t
" '!i"
..
(1)
On
the
duality
condition
for
quantum
fields.
*)t)
Joseph
J.
Bisognano
Lawrence
Berkeley
Laboratory
University
of
California
Berkele,Y,
California
94720
and
Eyvind
H.
Wichmann
**)
Department
of
Physics
l
University
of
California
Berkeley,
California
94720
Abstract.
I .
A
general
quantum
field
theory
is
considered,
in
which
the
fields
are
assumed
to
be
operator-valued
tempered
distributions.
The
system
of
fields
may
include
any
number
of
boson
fields
and
fermion
fields.
A
theorem
which
relates
'certain
complex
Lorentz
transformations
to
the
TCP-transformation
is
stated
and
proved
•
With
reference
to
this
theorem
duality
conditions
are
considered,
and
it
is
shown
that
such
conditions
hold
under
various
physi-
cally
rea
sonable
a
ssumptions
about
the
fields.
Extensions
of
the
algebras
of
field
operators
are
discussed
with
reference
to
the
duality
conditions.
Local
internal
symmetries
are.
discussed,
and
it
is
shown
that
these
,commute
with
the
Poincare
group
and
with
the
TCP-transforma
tion.
Submitted
for
publication
to
the
Journal
of
Mathematical
Physics.
August
1975
(2)
10
Introduction.
In
an
earlier
publication
1),
hereaf'ter
ref'erred
to
as:
BW
I,
the
authors
have
discussed
the
~uality
condition
f'or
a
Hermitian
scalar
field.
It
is
the
purpose
of'
the
present
paper
to
extend
the
results
in
BW
I
to
a
general
f'ield
theory,
within
the
frame-
work
described
in
the
monographs
by
Streater
and
Wightman
2),
and
by
Jost
3).
We
thus
consider
a
theory
in
which
there
appears
an
arbitrary
set
of'
local
and
relatively
local
spinor-
and
tensor
fields.
Each
field
hes
a
finite
number
of
components,
and
is
assumed
to
be
an
operator-valued
tempered
distribution.
In
contrast
to
the
situation
in
BW
I
we
now
have
to
consider
fermion
fields,
and
their
characteristic
anticommutation
rela-
tions,
mich
necessitates
an
obvious1modification
in
the
defi-
nitions
of
the
duality
conditions.
As
we
shall
see,.
however,
much
of
the
reaaoning
in
BW
I
applies
in
almost
unchanged
f'orm
to
the.
issues
in
the
present
study.
When
this
is
the
case
we
shall
rely
heavily
on
BW
I,
and
not
repeat
arguments
already
given
in
that
paper.
The
nota-
tion
and
terminology
in
BW
I
will
be
followed
whenever
appli-
cable.
We
also
refer
to
BW
I
for
additional
references
to
re-
lated
worlt.
In
Sec.
II
we
review
some
aspects
of
the
geometry
of
Min-
kowski
space,
andwe
also
review
some
well-known
facts
about
the
quantum
mechanical
Poincare
group
and
its
complex
exten-
sion.
In
Sec.
III
we
state
our
assumptions
about
the
quantum
fields,
which
are
more
or
less
standard.
In
these
two
sections
we
also
explain
the
notation
whioh
we
follow
in
the
subsequent
•
•
. '