PH
YSI
CAL
R
EVI
EW
VOLUM
E
74,
NUMBER
10
NOVEM
BER
15,
1948
Quantum
Electrodynamics.
I.
A
Covariant
Formulation
JULIAN
SCHWINGER
Harvard
University,
Cambridge,
Massachusetts
(Received
July
29,
1948)
Attempts
to
avoid
the
divergence
difhculties
of
quan-
tum
electrodynamics
by
mutilation
of
the
theory
have
been
uniformly
unsuccessful.
The
lack
of
convergence
does
in-
dicate
that
a
revision
of
electrodynamic
concepts
at
ultra-
relativistic
energies
is
indeed
necessary,
but
no
appreciable
alteration
of
the
theory
for
moderate
relativistic
energies
can
be
tolerated.
The
elementary
phenomena
in
which
divergences
occur,
in
consequence
of
virtual
transitions
involving
particles
with
unlimited
energy,
are
the
po-
larization
of
the
vacuum
and
the
self-energy
of
the
elec-
tron,
e6ects
which
essentially
express
the
interaction
of
the
electromagnetic
and
matter
fields
with
their
own
vacuum
fluctuations.
The
basic
result
of
these
fluctuation
inter-
actions
is
to
alter
the
constants
characterizing
the
prop-
erties
of
the
individual
fields,
and
their
mutual
coupling,
albeit
by
infinite
factors.
The
question
is
naturally
posed
whether
all
divergences
can
be
isolated
in
such
unob-
servable
renormalization
factors;
more
specifically,
we
in-
quire
whether
quantum
electrodynamics
can
account
unambiguously
for
the
recently
observed
deviations
from
the
Dirac
electron
theory,
without
the
introduction
of
fundamentally
new
concepts.
This
paper,
the
first
in
a
series
devoted
to
the
above
question,
is
occupied
with
the
formulation
of
a
completely
covariant
electrodynamics.
Manifest
covariance
with
respect
to
Lorentz
and
gauge
transformations
is
essential
in
a
divergent
theory
since
the
use
of
a
particular
reference
system
or
gauge
in
the
course
of
calculation
can
result
in
a
loss
of
covariance
in
view
of
the
ambiguities
that
may
be
the
concomitant
of
infinities.
It
is
remarked,
in
the
first
section,
that
the
customary
canonical
commutation
relations,
which
fail
to
exhibit
the
desired
covariance
since
they
refer
to
field
variables
at
equal
times
and
different
points
of
space,
can
be
put
in
covariant
form
by
replacing
the
four-dimensional
surface
t=const.
by
a
space-like
surface.
The
latter
is
such
that
light
signals
cannot
be
propagated
between
any
two
points
on
the
surface.
In
this
manner,
a
formulation
of
quantum
electrodynamics
is
constructed
in
the
Heisenberg
repre-
sentation,
which
is obviously
covariant
in
all
its
aspects.
It
is
not
entirely
suitable,
however,
as
a
practical
means
of
treating
electrodynamic
questions,
since
commutators
of
field
quantities
at
points
separated
by
a
time-like
in-
terval
can
be
constructed
only
by
solving
the
equations
of
motion.
This
situation
is
to
be
contrasted
with
that
of
the
Schrodinger
representation,
in
which
all
operators
refer
to
the
same
time,
thus
providing
a
distinct
separation
between
kinematical
and
dynamical
aspects.

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