Institut für Theoretische Physik,
Freie Universität Berlin
Arnimallee 14, D - 14195 Berlin
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  and further discussed by many authors  -, so that it has become conference  and textbook material , the relativistic problem has remained open -- for particles of spin zero as well as spin-1/2 . 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 
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  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. ). Then we perform a Kustaanheimo-Stiefel transformation (Eq. (13.101) in Ref. ). 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. )
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. ).
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. ). The integration over y yields
(compare Eq. (9.64) in Ref. . See also p. 139 in Ref. ).
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. ), the last equation can be rewritten as
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. ), 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. ).
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  appeared in which a new Chapter 19 on Path Integrals and Relativistic Particle Orbits discusses the above issue in more detail.