Space-time formulation of quantum transitions
T. Petrosky, G. Ordonez, and I. Prigogine
Center for Studies in Statistical Mechanics and Complex Systems, The University of Texas at Austin, Austin, Texas 78712
and International Solvay Institutes for Physics and Chemistry, CP231, 1050 Brussels, Belgium
Received 7 June 2001; published 6 November 2001
In a previous paper we have studied dressed excited states in the Friedrichs model, which describes a
two-level atom interacting with radiation. In our approach, excited states are distributions or generalized
functions in the Liouville space. These states decay in a strictly exponential way. In contrast, the states one
may construct in the Hilbert space of wave functions always present deviations from exponential decay. We
have considered the momentum representation, which is applicable to global quantities trace, energy transfer.
Here we study the space-time description of local quantities associated with dressed unstable states, such as,
the intensity of the photon field. In this situation the excited states become factorized in Gamow states. To go
from local quantities to global quantities, we have to proceed to an integration over space, which is far from
trivial. There are various elements that appear in the space-time evolution of the system: the unstable cloud that
surrounds the bare atom, the emitted real photons and the ‘‘Zeno photons,’’ which are associated with devia-
tions from exponential decay. We consider a Hilbert space approximation to our dressed excited state. This
approximation leads already to decay close to exponential in the field surrounding the atom, and to a line shape
different from the Lorentzian line shape. Our results are compared with numerical simulations. We show that
the time evolution of an unstable state satisfies a Boltzmann-like H theorem. This is applied to emission and
absorption as well as scattering. The existence of a microscopic H theorem is not astonishing. The excited
states are ‘‘nonequilibrium’’ states and their time evolution leads to the emission of photons, which distributes
the energy of the unstable state among the field modes.
DOI: 10.1103/PhysRevA.64.062101 PACS numbers: 03.65.Ta, 32.70.Jz, 32.80.-t
I. INTRODUCTION
As is well known the decay of excited states or unstable
particles leads in the framework of quantum mechanics to
deviations from exponential decay 1. This effect, while
small, leads to some puzzles. Schwinger has written ‘‘ . . .
with the failure of the simple exponential decay law we have
reached, not merely the point at which some approximation
ceases to be valid, but rather the limit of physical meaning-
fulness of the very concept of unstable particle’’ 2. Wigner
has gone so far as to limit the idea of elementary particles to
stable particles 3.
We have presented a solution to this problem in a recent
paper 4, in the framework of our extension of quantum
mechanics to density matrices outside the Hilbert space. In
this extension we have complex spectral representations of
the Liouville–von Neumann operator or Liouvillian L
H
= H , that allow us to rigorously disentangle the exponen-
tial and nonexponential components of the evolution of a
given initial condition. The exponential component corre-
sponds to the dressed excited state or unstable particle, and
dressed photons. The nonexponential component corre-
sponds to dressed correlations. The dressed states and corre-
lations are given by nonfactorizable density matrices outside
the Liouville-Hilbert space. They are related to the bare den-
sity matrices through a transformation . This transforma-
tion is star unitary 4,5, which corresponds to a generaliza-
tion of unitary transformations to unstable systems.
In our recent paper we considered global quantities, such
as, the trace and total energy. To obtain these global quanti-
ties we used the momentum representation. Now we consider
local quantities using the space-time representation. As will
be discussed in Sec. III, the transition from the momentum
representation to the space representation is far from trivial
because of the singularities associated with states outside the
Hilbert space.
In Sec. II we briefly summarize our previous paper. For
simplicity we consider the Friedrichs model in the rotating
wave approximation and in one-dimensional space. We con-
sider in succession stable and unstable excited states. We
describe the Gamow vectors, which correspond to the com-
plex spectral representation of the Hamiltonian. We describe
as well the complex spectral representation of L
H
in the ex-
tended Liouville space that includes distributions. Starting
from this representation, we formulate the dressed unstable
state |
1
0
as well as the dressed photon states and correla-
tions.
In Sec. III we consider the space-time representation of
the decay, starting from the bare excited state. We obtain a
closed form for the field intensity I ( x , t ) at time t.
In our previous paper we have shown that the time evo-
lution, starting from the bare excited state, may be split into
two parts: a slow one Markovian, corresponding to the ex-
ponential decay and emission of the dressed excited state cf.
Eq. 40, and a rapid one non-Markovian associated with
the dressing of the bare state, leading to nonexponential ef-
fects. In the local field intensity I ( x , t ) we may also distin-
guish these two parts. Note that to obtain causality the van-
ishing of the emitted field outside the light cone we have to
combine both the Markovian and non-Markovian compo-
nents of the field. We may not isolate either component as
this would lead to noncausal behavior.
The evolution law of the non-Markovian component de-
pends on the initial conditions. In contrast, the decay law of
PHYSICAL REVIEW A, VOLUME 64, 062101
1050-2947/2001/646/06210121/$20.00 ©2001 The American Physical Society 64 062101-1