Stationary, static, spherically symmetric solutions of the Maxwell-Dirac system, treated as classical fields, have been found which are localised and normalisable. The solutions apply to any bound energy eigenvalue in the range 0 < E < m, where m is the bare mass in the Dirac equation. A point charge of any magnitude and either sign may be placed at the origin and the solutions remain well behaved and bound. However, no such central charge is necessary to result in a bound solution. As found previously by Radford, the magnetic flux density is equal to that of a monopole at the origin. However, no monopole is present, the magnetic flux being a result of the dipole moment distribution of the Dirac field. The Dirac field magnetic dipole moment is aligned with the magnetic flux density and so the resulting magnetic self-energy is negative. It is this which results in the states being bound (E < m). The case which omits any central point charge is therefore a self-sustaining bound state solution of the Maxwell-Dirac system which is localised, normalisable, and requires no arbitrarily added “external” features (i.e., it is a soliton). As far as the author is aware, this is the first time that such an exact solution with a positive energy eigenvalue has been reported. However, the solution is not unique since the energy eigenvalue is arbitrary within the range 0 < E < m. The stability of the solution has not been addressed.

1.
L.
Gross
, “
The Cauchy problem for the coupled Maxwell-Dirac equations
,”
Commun. Pure Appl. Math.
19
,
1
(
1996
).
2.
J.
Chadam
, “
Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac system in one space dimension
,”
J. Funct. Anal.
13
,
495
(
1973
).
3.
M.
Flato
,
J. C. H.
Simon
, and
E.
Taflin
, “
On global solutions of the Maxwell-Dirac equations
,”
Commun. Math. Phys.
112
,
21
(
1987
).
4.
V.
Georgiev
, “
Small amplitude solutions of the Maxwell-Dirac equations
,”
Indiana Univ. Math. J.
40
,
845
(
1991
).
5.
M.
Esteban
and
E.
Sere
, “
Stationary states of the nonlinear Dirac equation: A variational approach
,”
Commun. Math. Phys.
171
,
323
(
1995
).
6.
M.
Esteban
,
V.
Georgiev
, and
E.
Sere
, “
Stationary solutions of the Maxwell-Dirac and Gordon-Dirac equations
,”
Calculus Var. Partial Differ. Equations
4
,
265
(
1996
).
7.
N.
Bournaveas
, “
Local existence for the Maxwell-Dirac equations in three space dimensions
,”
Commun. Partial Differ. Equations
21
,
693
(
1996
).
8.
M.
Flato
,
J. C. H.
Simon
, and
E.
Taflin
, “
Asymptotic completeness global existence and the infrared problem for the Maxwell-Dirac equations
,”
Mem. Am. Math. Soc.
127
(
606
),
1
-
311
(
1997
).
9.
P.
D’Ancona
and
S.
Selberg
, “
Global well-posedness of the Maxwell-Dirac system in two space dimensions
,”
J. Funct. Anal.
260
,
2300
(
2011
).
10.
A.
Comech
and
D.
Stuart
, “
Small amplitude solitary waves in the Dirac-Maxwell system
,” e-print arXiv:1210.7261 (
2012
).
11.
A.
You
and
Y.
Zhang
, “
Global strong solution to Maxwell-Dirac equations in 1 + 1 dimensions
,”
Nonlinear Anal.
98
,
226
(
2014
).
12.
G. P.
Legg
, “
On group invariant solutions to the Maxwell-Dirac equations
,” M.Sc. thesis,
University of Tasmania
,
2007
.
13.
M.
Wakano
, “
Intensely localized solutions of the Dirac-Maxwell field equations
,”
Prog. Theor. Phys.
35
,
1117
(
1966
).
14.
A. G.
Lisi
, “
A solitary wave solution of the Maxwell-Dirac equations
,”
J. Phys. A: Math. Theor.
28
,
5385
(
1995
).
15.
C. S.
Bohun
and
F. I.
Cooperstock
, “
Dirac–Maxwell solitons
,”
Phys. Rev. A
60
,
4291
(
1999
).
16.
C. J.
Radford
, “
Localised solutions of the Dirac-Maxwell equations
,”
J. Math. Phys.
37
,
4418
(
1996
).
17.
C. J.
Radford
and
H. S.
Booth
, “
Magnetic monopoles electric neutrality and the static Maxwell-Dirac equations
,”
J. Phys. A: Math. Gen.
32
,
5807
(
1999
).
18.
W.
Gordon
, “
Der Strom der Diracschen Elektronentheorie
,”
Z. Phys.
50
,
630
(
1928
).
19.
S. M.
Inglis
and
P. D.
Jarvis
, “
Algebraic inversion of the Dirac equation for the vector potential in the non-Abelian case
,”
J. Phys. A: Math. Theor.
45
,
465202
(
2012
).
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