Institut für Theoretische Physik,
Freie Universität Berlin
Arnimallee 14, D - 14195 Berlin
H. Kleinert
June 12, 1996
The path integral of the relativistic spinless Coulomb system is solved, and the wave functions are extracted from the resulting amplitude.
1) While the path integral of the nonrelativistic Coulomb system has been solved some 15 years ago [1] and further discussed by many authors [2] -[14], so that it has become conference [15] and textbook material [16], the relativistic problem has remained open -- for particles of spin zero as well as spin-1/2 [17]. The purpose of this note is to fill this gap for spin-zero particles.
2) Consider first
a free relativistic spinless particle of mass M.
If 
describes its orbit in D spacetime dimensions
in terms of some parameter 
,
the
classical action reads
where c is the light velocity.
This action cannot be used to set up
a path integral
for the time evolution amplitude since it would not yield the
well-known Green function of a Klein-Gordon field.
An action which serves this purpose
can be constructed
with the help of an
auxiliary fluctuating variable 
and reads [18]
Classically, this action coincides with the original
since it is extremal for
Inserting this back into (2)
we see that
reduces to
.
The action is
invariant under
arbitrary reparametrizations
The action
shares this invariance,
if is simultaneously transformed as
The action
has the advantage of being
quadratic in the orbital variable
. If the physical time
is analytically continued to imaginary values so that the metric
becomes euclidean,
the action looks like that of a
nonrelativistic particle moving as a function of a pseudotime
through a D-dimensional
euclidean spacetime, with a mass depending on .
3)
To set up a path integral, the action has to be pseudotime-sliced,
say at 
.
If 
denotes the
thickness of the nth slice, the sliced
action reads
where 
,
, and
.
A path integral
may be defined as the
limit
of the product of integrals
This can immediately be evaluated, yielding
where the quantity
has the continuum limit
Classically, this is the reparametrization invariant length of a path, as is obvious after inserting (3).
If the amplitude (8) is multiplied by 
,
where 
is the Compton wavelength of the particle,
an integral over all positive L
yields the correct Klein-Gordon amplitude
where 
is
the modified Bessel function.
The result does not depend on the choice of , this being
a manifestation of the reparametrization invariance.
We may therefore write the continuum version of the path integral
for the relativistic free particle
as
where 
denotes
a convenient gauge-fixing functional, for instance
which fixes
to unity everywhere.
To understand the factor 
in the measure
of
(12),
we make use of the
canonical form of
the action (2),
After pseudotime slicing, it reads
At a fixed ,
the path integral has then the usual canonical measure:
By integrating out the momenta, we obtain (7) with the action (6).
4) The fixed-energy amplitude is related to (12) by a Laplace transformation:
where 
denotes the temporal component
and
the purely spatial part of the D-dimensional vector x.
The poles and cut of 
along the
energy axis
contain all information on the bound and continuous
eigenstates of the system.
The fixed-energy amplitude has the path integral
representation
with the action
This is seen by
writing the temporal part of the sliced D-dimensional
action (6)
in the canonical form (14).
By integrating out
the temporal coordinates
in (15), we obtain N 
-functions.
These remove N integrals over
the momentum
variables
, leaving only a single integral over a common 
.
The Laplace transform
(16), finally, eliminates also this integral,
making 
equal to
-iE/c.
In the continuum limit,
we thus obtain the action (18).
5)
The path integral
(17) forms the
basis for studying relativistic particles in an
external time-independent potential 
.
This
is introduced into the path integral
(17)
by simply substituting the energy E by 
.
For an attractive Coulomb potential in D-1=3 spatial dimensions, the above substitution changes the second term in the action (18) to
The associated path integral is calculated with the help of a Duru-Kleinert transformation [1] as follows:
First, we increase
the three-dimensional configuration space
in a trivial way by a dummy forth component
(as in the nonrelativistic case).
The additional variable is eliminated at the end by
an integral 
(see
Eqs. (13.114) and (13.121) in Ref. [16]).
Then we perform a
Kustaanheimo-Stiefel transformation
(Eq. (13.101) in Ref. [16]).
This changes
into
,
with the vector symbol indicating the four-vector nature of
.
The transformed action reads
We now choose the gauge
,
and go from
to a new parameter s
via the time transformation
with
. This leads to the Duru-Kleinert-transformed
action
It describes a particle
of mass 
moving as a function of the ``pseudotime"
s
in a harmonic oscillator potential of frequency
The oscillator possesses an additional
attractive potential
which is conveniently parametrized
in the form of a centrifugal barrier
whose squared angular momentum has the negative value
where 
denotes the finestructure constant 
.
In addition,
there is also a trivial constant potential
If we ignore, for a moment, the
centrifugal barrier 
,
the solution of the path integral can immediately be
written down
(compare Eq. (13.121) in Ref. [16])
where
is the time evolution amplitude
of the four-dimensional
harmonic oscillator.
There are no time-slicing corrections for the same reason as in the three-dimensional case. This is ensured by the affine connection of the Kustaanheimo-Stiefel transformation satisfying
(see the discussion in Section 13.6 of Ref. [16]).
A 
-integration leads to
with the variable
and the parameters
We now use the well-known expansion
and obtain the partial wave decomposition
with the usual notation for Legendre polynomials and spherical harmonics. The radial amplitude is, therefore,
At this place, the additional centrifugal barrier (23) is incorporated via the replacement
(as in Eqs. (8.146) and (14.237) in Ref. [16]). The integration over y yields
(compare Eq. (9.64) in Ref. [16]. See also p. 139 in Ref. [15]).
This fixed-energy amplitude has poles in the Gamma function
whenever
. They
determine the bound-state energies of the
Coulomb system.
Subsequent formulas
can be simplified by introducing the small
positive l-dependent parameter
With it, the pole positions are given by
, with 
,
and the
bound state energies become:
Note the appearance of the plus-minus sign as a characteristic property of energies in relativistic quantum mechanics. A correct interpretation of the negative energies as positive energies of antiparticles is straightforward within quantum field theory; it will not be discussed here.
To find the wave functions,
we approximate
near the poles 
:
with the radial quantum number 
.
In analogy with
a corresponding nonrelativistic equation
(Eq. (13.203) in Ref. [16]),
the last equation can be rewritten
as
where
denotes a modified energy-dependent
Bohr radius. Instead of being 
times the
Compton wave length of the electron 
,
the modified Bohr radius which sets
the length scale of relativistic bound states
involves the energy E instead of the rest energy 
.
With the above parameters, the positive-energy poles in the Gamma function can be written as
Using this behavior and a property of the Whittaker functions (see Eq. (9.80) in Ref. [16]), we write the contribution of the bound states to the spectral representation of the fixed-energy amplitude as
A comparison between the pole terms in (35) and (42) renders the radial wave functions
The properly normalized total wave functions are
The continuous wave functions are obtained in the same way as in the non-relativistic case (see Eqs. (13.211)-(13.219) in Ref. [16]).
This concludes the solution of the path integral of
the relativistic spinless Coulomb system.
Note added in Proof:
While this paper was in the refereeing process,
a second edition of the textbook [16]
appeared in which a new Chapter 19 on Path Integrals and
Relativistic Particle Orbits discusses the above issue in
more detail.
