MA
THEMA
TICS:
J.
M.
COOK
or
mk
1,
is
aneven
entire
function
of
order
1
and
type
p
<ir,
which
can
be
shown
to
be
O(z')
as
z

through
real
values;
it
is
real
on
the
real
axis,
and
changes
sign
at
all
integers
not
of
the
form
mk
or
mk
1.
By
a
theorem
of
Paley
and
Wiener,6
F(z)
=
fo
cos
zu
h(u)
du,
where
h(u)
belongs
to
L2.
Write
g(u)
=
h(u)
cos
'/2u;
then
F(n
1/2)
=
P
g(u)
cos
(n
'/2)U
sec1/2u
du

(_
1)n
Jtp
g(7r

u)
sin
(n
1/2)u
csc
'/2u
du.
Thus
(
1)'F(n
1/2)
is
the
nth
partial
sum
of
the
Fourier
seriesof
the
evenfunction
which
is
irg(7r

u)
for7r

p
<
u
<7r
and
zero
for
0
_
u
<r

p.Furthermore,
(
1)
IF(n
1/2)
>
0
when
n
is
not
a
multiple
of
k,
and
so
for
a
sequence
of
integers
of
density
arbitrarily
close
to
1,if
k
is
large
enough.
I
On
saitfort
peu
de
choses
sur
I'approximation
oriente6
dans
les
espaces
fonctionnels
r'ticules :
Favard,
J.,
Remarques
sur
I'approximation
des
fonctionscontinues,
Acta
Sci.
Math.,
Szeged,
12,
part
A,101104
(1950).
2
Fejer,
L.,
Gestaltliches
uber
die
Partialsummen
und
ihrer
Mittelwerte
bei
derFourierreihe
und
der
Potenzreihe,
Z.
angew.
Math.
u.
Mech.,
13,
8088
(1933).
3
Levinson,
N.,
Gap
and
DensityTheorems,
New
York,
1940,
chap.
II.
Levinson,
N.,
op.
cit.,
chap.
VII.
5
Duffin,
R.
J.,
and
Schaeffer,
A.
C.,
Power
Series
with
Bounded
Coefficients,
Am.
J.
Math.,
67,
141154
(1945).
6
Paley,
R.
E.
A.
C.,
and
Wiener,
N.,
Fourier
Transforms
in
the
ComplexDomain,
New
York,
1934,
p.13.
THE
MATHEMATICS
OF
SECOND
QUANTIZATION*
BY
J.
M.
COOK
DEPARTMENT
OF
MATHEMATICS,
UNIVERSITY
OF
CHICAGO
Communicated
by
M.
H.
Stone,
May
16,
1951
In
order
to
make
a
beginning
on
the
problem
of
constructinga
mathe
matically
rigorous
foundation
for
quantum
field
theory,
we
define
the
annihilation
and
creation
operators
and
the
particle
and
fieldobservables
as
transformations
on
Hilbert
space,
and
investigate
their
domains,
adjoints,
commutation
relations,
normality
and
other
properties.
The
resulting
formalism
is
given
a
physical
interpretation
which
is
illustrated
by
applications.
The
state
of
an
elementary
particle
is
represented
by
a
point
in
the
Hilbert
space
T
and
an
observable
by
theoperator
A
on
9S.
Then
the
VOL.
37,
1951
417
MA
THEMA
TICS:
J.
M.
COOK
stateof
a
system
of
n
such
particles
is
represented
by
apoint
in
the
tensor
product
9
=
9
...
®
and
theobservable
by
(ZA8('
...
®A6(i
n))
,
where
T
is
the
closure'
of
T.
In
field
theory,
j
Ej
e(n)
n
=0
co
n
is
the
state
space2
and
Q(A)
=
ED
(A'(
',)
.
(.A.
A1i
n))
the
cor
n
=
0
i
=
1
responding
observable.
Q(A)
exists
when
A
is
densely
defined,
closed
and
linear.
It
is
defined
to
be
0
on
T('),
the
onedimensional
space
of
no
particle
states.
Q
preserves
commutation,
order
and
adjoint
relations,
and
normality.
If
His
selfadjoint,
thenexp
(iQl(H))Q2(A)
exp
(iQ(H))
=
Q(exp
(iH)A
exp
(iH)).Under
certain
conditions
(e.g.,
when
A
and
B
are
bounded),
Q(aA
bB)
=
(aQ(A)
bQ(B))
and
Q [A
B])
=
[Q(A),
Q(B)]f
(where
[A,
B]
=
AB
BA).
If
P
is
a
projection,
the
eigenvalues
of
Q(P)
are
the
occupation
numbers
m
of
P9.
In
the
cor
responding
eigenstates,
exactly
m
of
the
particles
have
theproperty
P.
With
the
spectral
theorem,
this
gives
us
the
standardenergy
expressions
hk2kckNk
except
that
there
is
no
infinite
nullpoint
energy
I2kwk/2
to
be
subtracted.3
A
similar
result
holds
for
the
nullpoint
momentum,
no
artificial
summation
to
zero
being
necessary.
To
every
permutation
7r
in
the
symmetric
group
HIn
corresponds
the
unitary
operator
UT
n
9?(n)
defined
by
UTl
...
n
=
(1)
)...
0
T(n)
These
operators
U,T
generate
a
ring
9,,
isomorphic
to
the
groupalgebra
of
H..
For
Gn
e
9.
and
e
9E,
we
construct
the
densely
defined,
closed,linear
transformations
co(+)
=
(E
EIGn)(4)®),
w*(
)
=
((O
)*(E
DGn*)n
n O
n==
0
on
a,
where
(+X®)
maps
(n)
into
9?(n
+
1)
by
(4)0)4)o1
...
04)n
=
4)41
04
)..
The
mapping
w
obeys
the
rules
wc(+)
=
w*(*)*,
((4))
=
o(aO
b )
(aw(4))
b
()) ),
co*(a)
bo)
=
(a*w
*(4)
b*o*( )),
exp
(iQ(H))w(O)
exp
(iQ(H))
=
w(exp
(iH)O),
exp
(iQ(H))w*(4)
exp
(iQ
(H))
w
*(exp
(iH)
)),
[Q[A],
co(4)]
co(A
4),
and
[Q2(A),
Co*(4))]
=

W*(A*4)).
The
center
of
,,
is
spanned
by
a
set
of
orthogonal
projections
P
=
nT(7r)
U/fn
indexed
by
the
characters
r
of
H,n.
The
alternating
and
symmetric
characters
an
and
Sn
give
us
the
subspaces
W
=
Pan)
a
n
co
and
5
=
(E
P,)
of
antisymmetric
and
symmetric
wavefunctions.
n
=
0
Setting
G,,
equal
to
ViPan
or
IP,,,
we
getthe
creation
operators
'0a(4))
or
wc(4))
and
annihilation
operators
Wa*(4))
or
w
o*(),
on
or
e
(to
which
they
must
be
restricted),
for
the
FermiDirac
or
BoseEinstein
cases,
respectively.
Both
Wa(4)
and
Oa*(4))
are
bounded:
jJW,a(9b)
=
Wa*(4b))H
=
qb.
The
domains
of
co(4)
and
ws*(4)
are
the
same
as
that
418
PROC.
N.A.
S.
MA
THEMA
TICS:
J.
Al.
COOK4
of
VQ(P[,1)
(on
i),
where
P[,]
is
the
projection
of
51
on
the
subspace
[4)]
spanned
by
4.
If
01,
2
e)
?,
we
define
412*
to
be
the
transformation
on
9)
such
that
( 102*)O
=
(4),
2) 1.
Then
Wa(4))a*( )
=
Q(4)*),
a*(#)w(4)
=
(4),
/
Q(4)*),
and
(w(4))w*(#))
=
2(4#*),
(ws*( )wS(4))
=
(4),
)I
Q(00*)i
from
which
it
follows,
if
[A,
B]+
=
AB
BA,
that
[Wia())
Wa*(4')]+
=
[ws()),
ws*(O)] 
=
4,O
)I,
and
[Wa(4)),
Wa( )]+
=
[wh(),
Ws(4')]P
=
[Wa*(4)),
Wa*( )]+
=
[cs*(4)),
Ws*( )]
0.
If
{4)}
is
an
orthonormal
basis
of
9?,
then
wCa(0i)
}
is
irreducible
on
2l
and
{
W°(s
is
irreducible
on
ei.
By
purely
formalmanipulations
it
can
be
seen
that
Q(A)
corresponds
to
the
expression2
2m,
nC(4m)(An),
Om)W*()n)
if
o
is
Wa
or
ws.
The
operators
i(ws(4))

ws*(4)))/V2
and
(co5(4)
cos*(4)))/X/0
are
essentiallyselfadjoint,'
so
their
closures
p(q5)
and
q(4)
exist
and
are
self
adjoint.
The
commutation
relations
[q(4)),
q(4,)]
=
[p(o),
p( )]
=
((4,,
4)

(4),
4))I/2
and
[q(o),
p(4 )]
=
i(( ,4)
(4),
))I/2
reduce
to
the
standard
ones
when
4
and
6
are
elements
of
an
orthonormal
basis.
Timedependent
commutation
relations
enable
us
to
avoidthe
singular
Dirac
5function
and
the
JordanPauli
invariantDfunction.Field
quantities
have
physical
meaning
only
as
averages
over
a
region.
Point
dependence
introduces
divergencies
into
the
mathematics,
so
p
and
q
here
depend
onelements
of
Hilbertspace
(asdistributions
with
respect
to
which
theaverages
are
taken)
rather
than
points
of
Euclidean
3space
E3.
The
formalism
is
illustrated
by
a
derivation
of
the
Yukawapotential,
and
by
the
following
completely
rigorizable,
relativistically
invariant,
divergencefree
(as
far
as
it
goes)
derivation
of
Maxwell's
equations:
A
photon
is
represented
by
6
in
5)
=
5?2(E3)
®l4,
where
S4
is
a
4dimensional
Hilbert
space.
If
k,
=
ih(bl/x),
etc.,
and
k
=
V/k
2
ky2
k
*2
then
the
Hamiltonian
is
=
ck
@e.
The
Lorentz
group
acts
on
time
dependent
elements
of
5)
by
having
exp
(
(itc/h)k)4),
for
4
e
62(E3),
transform
like
a
scalar,
and
the
orthonormal
basis
P1,
P2,
P3,
P4
of
S4
trans
form
contragrediently
to
x,
y,
,
ct.
The
fourvector
wavefunctions
come
inpairs,k
and
,
as
co
and
contravariant
components
for
the
same
par
ticle.
Expectation
values
are
written
(A ,
,).
We
restrict
photons
to
besuch
that
their
covariant
wavefunctions
must
be
in
the
subspace
P
of
all;
=
41ip1
420P2
4)3Op3
4)40P4(4i
E
V2(E3))
for
which
(b/1x)4i
(6/by)42
(6/6z)43

c1(6/bt)q64
=
0.
This
eliminates
thephysical
influence
of
longitudinal
and
scalar
componentsfrom
expectation
values,
and
leaves
only
two
effective
polarization
states.
They
are
perpendicular
to
the
direction
of
motion
of
the
photon
and
have
the
desired
spin
proper
ties.
Now
let
P(4,)
=
i(w5(#)Xw*(#))7V
and
Q( )
=
(wo( )
w*(^))
/V2.
Then
if
4
e
22(E3)
is
arealvaluedfunction
on
E3
for
taking
419
OL.
37,
1951
MATHEMATICS:
FUGLEDE
AND
KADISON
PROC.N.
A.
S.
fieldvalue
averages,
thecovariant
fourpotential
operators
are
Ai(o)
=
tQ(V/H1'Spi),
i
=
1,
2,
3,
and
t(o)
=
kQ(VHiY®p4).
Averages
over
mutually
spacelikeregions
commute.
The
total
energy
of
the
field
is
Q2(H)
c(Al(si)
A2(s2)
Aa(s3)
4(s4)),
where
s1,
S2,S3,
s4
is
any
real,
squareintegrable,
contravariant
fourcurrent
density.
Expectation
values
of
(V
..A
.
ci(b/8t)c)(4)
are
always
zero
(on
photons),
and
(02A
)
(4)

c'(si,
4))I,
i
=
1,
2,
3,
(Q24)(4)
c1(S4,
0)I,
so
Maxwell's
equations
are
satisfied.
A
photon
is
polarized
parallel
to
its
electric
vector
and
per
pendicular
to
its
magnetic
vectorboth
perpendicular
to
its
momentum.
Its
energy
satisfies
Planck's
relation
E
=
hp,
where
v
is
the
frequency
of
the
induced
field.
*This
note
summarizes
a
longer
paper
submitted
for
publication
elsewhere.
It
was
written,
with
the
continuing
advice
of
Prof.
I.
E.
Segal,
for
presentation
to
the
Depart
ment
of
Mathematics
of
the
University
of
Chicago
in
partialfulfillmentof
requirements
for
the
Ph.D.
Most
of
the
work
was
done
while
under
contract
with
the
Office
of
Naval
Research.
1
Stone,
M.
H.,
Linear
Transformations
in
Hilbert
Space,
Am.
Math.
Soc.
Coll.
Publ.,
XV,
New
York,
1932.
2
Fock,
V.,
Zeits.
f.
Phys.,
75,
622647
(1932).
3Wentzel,
G.,
Einfuhrung
in
die
Quantentheorie
der
Wellenfelder,
FranzDeuticke,
Vienna,
1943.
ON
A
CONJECTURE
OF
MURRAY
AND
VON
NEUMANN
By
BENT
FUGLEDE
AND
RICHARD
V.
KADISON*
THE
INSTITUTE
FOR
ADVANCED
STUDY
Communicated
by
John
von
Neumann,
April
7,
1951
1.
Introduction.In
this
note
the
authorspresent
a
proof
of
a
conjec
ture
of
F.
J.
Murray
and
J.
v.
Neumann'
concerning
normalcy
offactors.
A
ring
of
operators2
CR
is
said
to
be
normal
if
each
subring
3
of
CR
coincides
with
the
set
of
operators
in
CR
each
of
which
commutes
withevery
operator
in
W'M,
where
W'
is
the
ring
of
operators
in
CR
each
of
which
commutes
with
every
operator
in
S.
In
symbols,
normalcy
requires
that
('aY)a'R
=
8
for
each
subring
3
of
CR.
The
center
of
a
normal
ring
(R
consistsof
the
operators
a
I,
a
complex
(put
3
=
{a
I});
i.e.,
MR
is
a
factor.
J.
v.
Neu
mann
proved8
thatthe
factor
(B
of
all
bounded
operators
is
normal.
The
question
of
which
factors
are
normal
was
raised
by
F.
J.
Murray
and
J.
v.
Neumann
(R.O.
I,
p.
185).
They
showed
that
all
factors
in
case
I
(the
discrete
case)
are
normaland
exhibited
examples
of
nonnormal
factorsin
case
II
(the
continuous
case).
Their
later
results
establish
the
non
normalcy
of
each
member
of
a
restricted
class
of
factors
in
case
II,
viz.,
420