Polarization fields in the positronium atom undergoing emission or absorption of
optical photons
O. N. Gadomski and S. G. Moiseev
Ul’yanovsk State University, 432700 Ul’yanovsk, Russia
Submitted 20 March 1997
Zh. E
´
ksp. Teor. Fiz. 113, 471–488 February 1998
This paper solves the problem of the interaction of an electron and positron via the field of soft
and hard photons with emission or absorption of a real photon. The interaction is interpreted
as a third-order QED effect in the coordinate representation. The role of intermediate states with
positive and negative frequencies is studied. A general expression is derived for the matrix
elements of the operator of the effective electron–positron interaction energy for different types
of quantum transitions. The expression makes it possible to calculate the probabilities of
the corresponding transitions in the nonrelativistic approximation. Electric dipole transitions in
the positronium atom accompanied by emission absorption of an optical photon are
investigated. Two-particle wave functions of the positronium atom are used to introduce the
concept of polarization fields inside the positronium atom. It is found that the polarization fields
depend on the coordinates and time and on the choice of the pair of states between which a
quantum transition with emission or absorption of a photon takes place. © 1998 American
Institute of Physics. S1063-77619800602-7
1. INTRODUCTION
Polarization fields play an important role in shaping vari-
ous optical processes. For instance, in the classical optics of
insulators such a polarization field is the electric dipole
field,
1
which allows not only for an explaination of reflection
and refraction of light but also makes it possible to derive in
a rigorous manner the Lorenz–Lorentz formula for the re-
fractive index. As shown in Refs. 2 and 3, establishing the
nature of polarization fields in insulators requires using an
approach based on QED first- and second-order effects. This
makes it possible to study various types of quantum transi-
tion in the field of virtual and real photons and to allow for
the orbital and spin degrees of freedom of the atomic elec-
trons, the various types of intermediate states in the spectrum
of interacting atoms, and the retardation effect for atoms lo-
cated at arbitrary distances from one another. A considerable
achievement here is the development of a method for obtain-
ing new types of integral equations that describe the propa-
gation of photons in a medium with allowance for different
quantum-transition types electric dipole, quadrupole, mag-
netic dipole, spin, etc. in the spectrum of atoms.
4,5
Here
allowance is made for electron and positron polarization
fields, which correspond to intermediate states of the atoms
with positive and negative energies, respectively. An integral
field equation in the electric dipole approximation has been
used to study the laws of reflection and refraction of light in
quantum
6
and nonlinear
4
optics, to build a theory of a non-
linear refractive index,
4
and to theoretically predict the near-
field effect with allowance for a discrete distribution of the
atoms near the the observation point.
4,7
. In contrast to Refs.
1–7, our paper studies the role of polarization fields inside an
individual atom.
Drake
8
examined the interaction of two atomic electrons
in a helium-like atom as a QED third-order effect. It was
found that with allowance for the orbital and spin degrees of
freedom and intermediate states with positive and negative
energies, third-order effects lead only to relativistic correc-
tions to the photon emission and absorption probability.
The present paper is devoted to the theory of quantum
transitions between the levels of the positronium atom in-
duced by soft optical or radio-frequency photons. The need
for such a theory derives from the use of precision methods
of radio and optical spectroscopy in the physics of the posi-
tronium atom, research into the annihilation process in
highly excited states,
9–11
and the study of the possibility of
markedly changing the kinetic characteristics of the annihi-
lation process in the field of an optical laser.
12
. Obviously,
the interaction of the positronium atom and the photon field
is largely determined by the coupling constants in the effec-
tive Hamiltonian. Hence we pay special attention to these
constants by interpreting the optical transition between the
levels of the positronium atom as a QED third-order effect.
We find that these effects explain the one-photon processes
of emission and absorption in the positronium atom due to
the Coulomb electron–positron interaction and induced po-
larization fields. We also find that calculations of the prob-
ability per unit time of spontaneous photon emission yield
the same result if we use the single-particle wave functions
of the positronium atom of Refs. 10 and 13 or the two-
particle wave functions derived in the present paper. In other
words, thanks to the specific properties of the positronium
atom, third-order effects are most evident in such an atom, in
contrast to the case of helium-like atoms.
8
As noted earlier, in this paper we use the two-particle
wave functions of the positronium atom. First, this makes it
possible to study the various schemes of quantum transitions
JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS VOLUME 86, NUMBER 2 FEBRUARY 1998
259 1063-7761/98/86(2)/11/$15.00 © 1998 American Institute of Physics