H. Kleinert
Institut für Theoretische Physik
Freie Universität Berlin
Arnimallee 14 D - 1000 Berlin 33
Exploiting the recently found extra monopole gauge symmetry which ensures the physical irrelevance of the Dirac strings in electromagnetism with Dirac magnetic monopoles, we formulate a local quantum field theory of charges and monopoles.
1) In a recent note [1] we have pointed out the existence of an extra gauge symmetry in the action describing the electromagnetic forces between a particle of charge e on a worldline L' with a 4-current
and a Dirac magnetic monopole [2] of charge g on a worldline L with a current
The action reads
where 
is
the usual field strength
while 
is what we have called the monopole gauge field,
describing the monopole via
the dual
of the 
-function on
the worldsheet S of the Dirac string
in the following way:
The physically observable field strength is
and the
finiteness of the action 
containing the
monopole gauge field enforces the presence of
a -function in 
on the world surface precisely equal to so that
. This is why
the action is regular and does not contain a square
of a -function as the expression (3) might initially suggest.
The worldline of the monopole is, of course, the boundary
line of the string's worldsheet S, as expressed by Stokes' theorem:
This implies that the monopole gauge field satisfies the equation
2) As noted in [1], the action (3) is invariant under the monopole gauge transformations
with integrable vector
functions 
, which
have the general form
with the sum running over arbitrary choices of 3-volumes V
and
being the -function on these volumes:
The monopole gauge transformations (9) express the freedom of distorting the Dirac strings without changing the boundary lines, as can be seen from the transformation
if V is the volume enclosed by the two surfaces 
and 
with a common boundary line L.
For some monopole gauge transformations the string distortions
are trivial, namely those
of the form
with 
, where 
is the -function on the four-volume 
,
These
do not give any change in since
they are a submanifold of the original
gauge transformations 
.
We may remove them from 
by a gauge-fixing condition
such as
where 
is an arbitrary fixed
unit vector.
The remaining monopole gauge freedom can be used to bring all Dirac strings to a standard
shape so that 
becomes a function of
only the boundary lines L.
In fact, with the above we may always
reach the
axial monopole gauge defined by
To see this we take along the 4-axis and consider the
gauge fixing equations
With (14) we have
and 
could certainly
all be determined if they were arbitrary real functions.
But the same thing is possible
for the restricted class of gauge functions at hand, with the form
(10).
This is seen most easily by approximating the 4-space by a fine-grained
hypercubic
lattice of spacing 
and imagining to be functions defined on the
plaquettes. The -functions correspond to integer-valued functions
on sites
[ 
], on links [ 
],
or plaquettes [ 
], and
the derivatives 
to 
times
lattice differences 
across links. Thus can be written as
with integer 
.
The gauge fixing in (16) with the restricted gauge functions amounts
then to solving a set of integer-valued
equations of the type
with 
. This is always possible as ha sbeen shown with
similar equations
in Ref. [4].
With the gauge being fixed we can solve
Eq. (8) uniquely by
3) If we want to turn the classical theory associated with (3)
into a quantum field theory, we have to take the amplitude
and form the path integral
over all fluctuating grand-canonical ensembles of world lines
L' and worldsheets S. For the world lines it is well known
that such a path integral can be replaced by a functional integral
over a single fluctuating field [3]. In the absence of monopoles
this gives rise, for charged electrons,
to the standard quantum field
theory of electromagnetism (QED) in which the electric interaction
is replaced by the second quantized field action
where 
is the mass of the electron and 
are the
standard Dirac fields of the electron.
4) For the monopoles, the situation is initially much more involved since the path integral is a sum over a grand-canonical ensemble of surfaces S. Up to date, there exists no satisfactory second-quantized field theory which could replace such a sum. The vacuum fluctuations of some non-abelian gauge theory will eventually do the job. Fortunately, however, due to the monopole gauge invariance of the action (3) under (9), most configurations of the surfaces S are physically irrelevant. If we fix the gauge as described above, the monopole gauge field is uniquely given by Eq. (18) and thus depend only on the orbital worldlines L of the monopoles via (2). But then we can rewrite the action with the fixed monopole gauge as
where 
are now
two arbitrary fluctuating fields (i.e., 
is
no longer of the restricted form
implied by (6)].
The latter field plays the role of a Lagrange
multiplyer to enforce the specific gauge relation
(18). The two action terms in which it apears have been denoted by
and 
.
We have omitted a gauge fixing term
for the ordinary electromagnetic gauge since it is standard.
The monopole enters now via the magnetic current coupling
where 
is short for
The ensemble of monopole orbits L can now be turned into a a single fluctuating field as usual. If monopoles are spin 1/2 particles, this obviously replaces [just as in going from (19) to (20)] the magnetic interaction (22) by
where 
is the Dirac field of the monopole.
The total gauge-fixed field action is therefore
Before gauge fixing, the total path integral for the fluctuating theory involves the fields
and the gauge fields 
and .
While the path integrals over the first three fields
are
defined in the standard way as being
the product, over all sites x of the above
specified hypercubic lattice, of integrals
over Grassmann variables 
and c-number variables
,
the latter is of a new type: It is defined as the product, over all plaquettes
of the lattice,
of discrete sums over
all integers with
.
After the gauge fixing, the path integrand
will in general contain Fadeev-Popov determinants.
For the original gauge transformation ,
any standard gauge fixing, for instance
the axial one 
, gives only a trivial constant
Fadeev-Popov determinant which can be ignored.
The same thing happens with the monopole gauge fixing.
At the level of the hypercubic lattice, Kronecker 's
in the path integrand
ensure conditions like (16) which corespond to
Also such conditions produce only trivial
constant Fadeev-Popov determinants [4], so that there is no need to
introduce compensating fermionic ghost fields.
The fluctuations in the gauge-fixed action
(25) involves only path integrals over
ordinary Grassmann fields
and c-number fields 
.
This completes the construction of the quantum field theory of
electric charges and Dirac monopoles [5].
Notice that the dependence of this theory on the monopole gauge
degree of freedom is much more dramatic than in pure QED.
There, it was only a polarization degree of freedom which was made irrelevant by
the electromagnetic gauge transformation .
Here the monopole gauge transformations
(9) reduce the dimensionality of the fluctuations from surfaces S
to lines L.
5) Let us end by remarking that by integrating out the field
in the classical action (3) we obtain the interaction
The third term is the usual electric current-current interaction. The first two terms are seen, via (8), to reduce to the magnetic current-current interaction
The last term is the current-current interaction between magnetic and electric currents. In the axial gauge with (18) it becomes
These are the correct current-current interactions which can
be found in the textbooks [6].
