PHYSICAL
REVIEW
VOLUM
E
ZS,
F
EB
RUAR
Y
15,
1949
Quantum
Electrodynamics.
II.
Vacuum
Polarization
and
Self-Energy
JULIAN
SCH%'INGER
Department
of
Physics,
Harvard
University,
Cambridge,
Massachusetts
(Received
November
1,
1948)
The
covariant
formulation
of
quantum
electrodynamics,
developed
in
a
previous
paper,
is
here
applied
to
two
elementary
problems
—
the
polarization
of
the
vacuum
and
the
self-energies
of
the
electron
and
photon.
In
the
first
section
the
vacuum
of
the
non-interacting
electromagnetic
and
matter
fields
is
covariantly
defined
as
that
state
for
which
the
eigenvalue
of
an
arbitrary
time-like
component
of
the
energy-momentum
four-vector
is
an
absolute
minimum.
It
is
remarked
that
this
definition
must
be
compatible
with
the
requirement
t'hat
the
vacuum
expec-
tation
values
of
a
physical
quantity
in
various
coordinate
systems
should
be,
not
only
covariantly
related,
but
identical,
since
the
vacuum
has
a
significance
that
is
inde-
pendent
of
the
coordinate
system.
In
order
to
construct
a
suitable
characterization
of
the
vacuum
state
vector,
a
covariant
decomposition
of
the
field
operators
into
positive
and negative
frequency
components
is
introduced,
and
the
properties
of
these
associated
fields
developed.
It
is
shown
that
the
state
vector
for the
electromagnetic
vacuum
is
annihilated
by
the
positive
frequency
part
of
the
trans-
verse
four-vector
potential,
while
that
for
the
matter
vacuum
is
annihilated
by
the
positive
frequency
part
of
the
Dirac
spinor
and
of
its
charge
conjugate.
These
de-
fining
properties
of
the
vacuum
state
vector
are
employed
in
the
calculation
of
the
vacuum
expectation
values
of
quadratic
field
quantities,
specifically
the
energy-mo-
mentum
tensors
of
the
independent
electromagnetic
and
matter
fields,
and
the
current
four-vector.
It
is
inferred
that
the
electromagnetic
energy-momentum
tensor,
and
the
current
vector
must
vanish
in
the
vacuum,
while
the
matter
field
energy-momentum
tensor
vanishes
in
the
vacuum
only
by
the
addition
of
a
suitable
multiple
of
the
unit
tensor.
The
second
section
treats
the
induction
of
a
current
in
the
vacuum
by an
external
electromagnetic
field.
It
is
supposed
that
the
latter
does
not
produce
actual
elec-
tron-positron
pairs;
that
is,
we
consider
only
the
phe-
nomenon
of
virtual
pair
creation.
This
restriction
is
introduced
by
requiring
that
the
establishment
and
sub-
sequent
removal
of
the
external
field
produce
no
net
change
in
state
for the
matter
field.
It
is
demonstrated,
in
a
general
manner,
that
the
induced
current
at
a
given
space-time
point
involves
the
external
current
in
the
vicinity
of
that
point,
and
not
the
electromagnetic
potentials.
This
gauge
invariant
result
shows
that
a
light
wave,
propagating
at
remote
distances
from
its
source,
induces
no
current
in
the
vacuum
and
is
therefore
undisturbed
in
its
passage
through
space.

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