j
(8)
If
A(I)S'=5,
with
I,
S
disjoint,
then
5'C.
I+5,
dimS~&
dimS'~&
dimI+dimS.
(C)
If
L
is
a
vector
not
in
5,
then
A({I.
))S'=5
if
and
only
if
either
S'=S+{L)
or
there
exist
numbers
Ni
I„
with
S'=
{Li+NiL,
Ls+NsL,
,
L,
+N,
L),
where
S=
{I
i
~
I
r).
(These
two
alternatives
represent
the
possibilities
dimS'=
dimS+1
and
dimS'=
dimS,
respectively.
)
(D)
If
I
and
5
are
disjoint
subspaces
then
5'+5+I
if
and
only
if
there
is
some
subspace
5"(
5
such
that
A(I)S'=S".
LProof:
The
subspace
5"
is
just
{Li'
L,
').
g
(E)
If
S,
Si,
Ss
are
disjoint
subsPaces,
and
A(Si)S'=
S,
then
A(Ss)5"=5'
if
and
only
if
A(St+5&)5"=S.
t
Proof:
We
can
write
S'={Li+Li',
,
L„+L,
')
where
5=
{Li.
.
L,
),
Li'.
.
L„'eSi.
Then
A(Ss)5"=5'
means
that
S"=
{Li+Li'+Li",
,
L,
+L,
'+L,
"},
where
Li"
.
L„"eSs.
Also,
A(Si+Ss)5"=5
means
that
5"=Li+Li"',
,
L„+L„"'
where
Li"'
L„"'eSi+Ss.
The
most
general
L;"'eSi+Ss
may
be
written
L,
"'=
L
+L;",
so
these
statements
are
equivalent.
J
ACKNOWLEDGMENTS
It
is
a
pleasure
to
thank
Professor
N.
M.
Kroll,
Professor
T.
D.
Lee,
Professor
S.
B.
Treiman,
and
Professor
A.
S.
Wightman
for
discussions
on
this
work
and
related
matters.
I
particularly
wish
to
thank
Pro-
fessor
Wightman
for
his
many
valuable
suggestions,
and
for
enabling
this
paper
to
satisfy
Salam's
criterion.
bIote
to
be
added
iN
proof
For
the
sake
of
ma.
t—
hematical
rigor,
the
de6nition
in
Equation
(III-1)
of
the
class
A
requires
a
slight
modification.
The
coeKcients
P
should
be
taken
as
functions
of
the
individual
vectors
L&,
12,
~
and
not
only
of
the
subspaces
{L&,
le,
.
}.
The
proof
in
Sec.
IV
that
if
fed„then
freA„sis
then
correct,
with
no
changes
(except
minor
notational
ones)
required.

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- Fall '05
- Energy, Radiation, asymptotic behavior, J Li