Dynamics of a magnetic monopole in matter, Maxwell equations in dyonic matter and detection of electric dipole moments
For a monopole, the analogue of the Lorentz equation in matter is shown to be f = g(H-v×D). Dual-symmetric Maxwell equations, for matter containing hidden magnetic charges in addition to electric ones, are given. They apply as well to ordinary matter if the particles possess T-violating electric dip...
Gespeichert in:
| Datum: | 2001 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
|
| Schriftenreihe: | Вопросы атомной науки и техники |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/79436 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Dynamics of a magnetic monopole in matter, Maxwell equations in dyonic matter and detection of electric dipole moments / X. Artru, D. Fayolle // Вопросы атомной науки и техники. — 2001. — № 6. — С. 110-114. — Бібліогр.: 3 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-79436 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-794362015-04-02T03:01:58Z Dynamics of a magnetic monopole in matter, Maxwell equations in dyonic matter and detection of electric dipole moments Artru, X. Fayolle, D. Electrodynamics of high energies in matter and strong fields For a monopole, the analogue of the Lorentz equation in matter is shown to be f = g(H-v×D). Dual-symmetric Maxwell equations, for matter containing hidden magnetic charges in addition to electric ones, are given. They apply as well to ordinary matter if the particles possess T-violating electric dipole moments. Two schemes of experiments for the detection of such moments in macroscopic pieces of matter are proposed. 2001 Article Dynamics of a magnetic monopole in matter, Maxwell equations in dyonic matter and detection of electric dipole moments / X. Artru, D. Fayolle // Вопросы атомной науки и техники. — 2001. — № 6. — С. 110-114. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 14.80.Hv 03.50.De 11.30.Er http://dspace.nbuv.gov.ua/handle/123456789/79436 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Electrodynamics of high energies in matter and strong fields Electrodynamics of high energies in matter and strong fields |
| spellingShingle |
Electrodynamics of high energies in matter and strong fields Electrodynamics of high energies in matter and strong fields Artru, X. Fayolle, D. Dynamics of a magnetic monopole in matter, Maxwell equations in dyonic matter and detection of electric dipole moments Вопросы атомной науки и техники |
| description |
For a monopole, the analogue of the Lorentz equation in matter is shown to be f = g(H-v×D). Dual-symmetric Maxwell equations, for matter containing hidden magnetic charges in addition to electric ones, are given. They apply as well to ordinary matter if the particles possess T-violating electric dipole moments. Two schemes of experiments for the detection of such moments in macroscopic pieces of matter are proposed. |
| format |
Article |
| author |
Artru, X. Fayolle, D. |
| author_facet |
Artru, X. Fayolle, D. |
| author_sort |
Artru, X. |
| title |
Dynamics of a magnetic monopole in matter, Maxwell equations in dyonic matter and detection of electric dipole moments |
| title_short |
Dynamics of a magnetic monopole in matter, Maxwell equations in dyonic matter and detection of electric dipole moments |
| title_full |
Dynamics of a magnetic monopole in matter, Maxwell equations in dyonic matter and detection of electric dipole moments |
| title_fullStr |
Dynamics of a magnetic monopole in matter, Maxwell equations in dyonic matter and detection of electric dipole moments |
| title_full_unstemmed |
Dynamics of a magnetic monopole in matter, Maxwell equations in dyonic matter and detection of electric dipole moments |
| title_sort |
dynamics of a magnetic monopole in matter, maxwell equations in dyonic matter and detection of electric dipole moments |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Electrodynamics of high energies in matter and strong fields |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79436 |
| citation_txt |
Dynamics of a magnetic monopole in matter, Maxwell equations in dyonic matter and detection of electric dipole moments / X. Artru, D. Fayolle // Вопросы атомной науки и техники. — 2001. — № 6. — С. 110-114. — Бібліогр.: 3 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT artrux dynamicsofamagneticmonopoleinmattermaxwellequationsindyonicmatteranddetectionofelectricdipolemoments AT fayolled dynamicsofamagneticmonopoleinmattermaxwellequationsindyonicmatteranddetectionofelectricdipolemoments |
| first_indexed |
2025-07-06T03:29:08Z |
| last_indexed |
2025-07-06T03:29:08Z |
| _version_ |
1836866652894920704 |
| fulltext |
DYNAMICS OF A MAGNETIC MONOPOLE IN MATTER,
MAXWELL EQUATIONS IN DYONIC MATTER
AND DETECTION OF ELECTRIC DIPOLE MOMENTS
X. Artrua, D. Fayolleb
aInstitut de Physique Nucléaire de Lyon
IN2P3-CNRS & Université Claude Bernard, France
e-mail: x.artru@ipnl.in2p3.fr
bLaboratoire de Physique Corpusculaire de Clermont
IN2P3-CNRS & Université Blaise Pascal, France
For a monopole, the analogue of the Lorentz equation in matter is shown to be f = g(H-v×D). Dual-symmetric
Maxwell equations, for matter containing hidden magnetic charges in addition to electric ones, are given. They
apply as well to ordinary matter if the particles possess T-violating electric dipole moments. Two schemes of
experiments for the detection of such moments in macroscopic pieces of matter are proposed.
PACS: 14.80.Hv 03.50.De 11.30.Er
1. INTRODUCTION
The question of which classical macroscopic fields
exert a force on a magnetic monopole of charge g in
matter is still controversial [1]. For the static force, the
formula
f = gH (1)
instead of f = gB, is generally accepted. However, for
the velocity dependent force, there is no consensus
between f = -gv×E and f = -gv×D (we use rationalized
equations with c=ε0=µ0=1). A more general problem is
to generalize the macroscopic Maxwell equations to the
dual-symmetric matter. The atoms or molecules of such
a matter would be made not only of electrically, but also
of magnetically charged particles. Thus they can possess
• electric dipole moments coming from decentered
electric charges as well as spinning magnetic
charges
• magnetic moments comming from spinning electric
charges as well as decentered magnetic charges.
After a rederivation and a discussion of Eq.(1), we will
present below a consistent solution for the velocity-
dependent force and the dual-symmetric Maxwell
equations in matter, using simple physical arguments.
We will consider only isotropic matter and assume that
its electric and magnetic polarizations P and M are
linear in D and B (or E and H). It will appear that our
equations can also take into account the electric dipole
moments (e.d.m.) of the ordinary fermions generated by
T-violating interactions, and we will propose two kinds
of possible measurements of the e.d.m. in macroscopic
matter.
2. STATIC FORCE ON A MONOPOLE IN
MATTER
If the force acting on a monopole in matter were f =
gB, a monopole following a closed magnetic line of a
permanent magnet could gain energy at each turn,
providing a perpetual motion of the first species. This is
an argument for chosing f = gH, whose curl is zero for a
static system.
One might object that the monopole can gain energy
at each turn at the expense of the magnetic energy
stored by the magnet and will eventually erase the
magnetization of the metal. This is indeed what happens
when a monopole is circulating through a super-
conducting loop : the varying flux of the monopole field
through the loop produces a counter-electromotive force
which damps the supercurrent. However, in the case of
a ferroelectric annulus, the magnetized state has the
lowest energy and the annulus cannot yield any energy
to the monopole.
Another argument for (1) comes from the (gedan-
ken) following experiment: Let us measure the force on
a magnetic charge immersed in a ferrofluid. The latter is
a practical realization of a liquid magnetizable matter.
No static frictional force can perturb the measurement.
We protect the monopole from the fluid by a waterproof
box. This should not change the result~; anyway the
physical monopole is probably dressed by a swarm of
ordinary particles. In the absence of the monopole, we
denote by B ≡ µH the field outside the box and by
Bbox = Hbox the field inside the box. The fields coming
110 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, 110-114
from the monopole will be denoted by a prime. Let us
consider two shapes of box (Fig.1):
a) the box is elongated parallel to B. Then Bbox = H
and the measured force is f = gH;
b) the box is flattened perpendicular to B.
Then Bbox = B. The force acting on the pole is f1 = gB,
which is different from case a). On the other hand, in
front of the box the total field Btot = B + B´ is larger than
behind. The magnetic grains of the ferrofluid are
therefore attracted toward this region and build a
hydrostatic pressure which pushes the box backwards.
Quantita-tively, the force acting on one grain of
magnetic moment m in the nonuniform field Btot is
tot
jij
tot
ijji BmBmf ∂=∂= (2)
since ∇×Btot = 0 for a static system. Repeated indices
are summed over. The resulting macroscopic force by
unit volume is
Fig. 1. Force acting on a monopole in a ferrofluid.
(a) elongated box parallel to the field (b) flattened box
perpendicular to the field
)BB(
2r3
tottot
i
tot
ji
tot
j
i BM
d
dF ⋅∂χ=∂=
where Mtot = M+M´ = χ(B+B´)is the magnetization
density and χ=(µ-1)/µ. This field of force builds the
pressure
p=(χ/2)Btot.Btot=(χ/2)(B2+B´2)+M.B´
The first two terms are symmetrical about the box and
exert no net force on it. The last term gives
f2=-∫(M.B´)dS=-∫(B´.dS)M=-gM
Here dS is the vector representation of a surface element
of the box and is directed outward. Permuting dS and M
was allowed because they are parallel in the region
where B’ is important. The last equality comes from
Gauss theorem for magnetic charges. Adding f1 and f2
one recovers the result (1):
f=f1+f2=g(B-M)=gH. (3)
Most probably, (1) can be generalized to any shape of
box. Thus, the relevant field which drives a monopole in
matter is H. It is the field found in a parallel elongated
cavity, as for the force f = eE driving an electric charge
inside a dielectric. Mnemonic: this kind of cavity allows
the test charge to follow the force without touching the
matter.
Eq. (1) allows trapping a (not too heavy) monopole
in the pole of a permanent magnet, where the lines of
gH converge from all directions. This would not be true
for gB.
MICROSCOPIC INTERPRETATION
Atomic magnetic dipoles are often pictured as
microscopic loops of electrical current. Then B appears
as the average of the microscopic field b over semi-
macroscopic volumes sufficiently large compared to the
atomic scale. The work of b along a straight line L is
therefore
g∫Lb dl = g∫LB dl
In contrast, the work along a line L’ which avoids
passing through the loops is
g∫L b dl = g∫L H dl .
Eq. (1) implies that the monopole avoids passing
through the microscopic current loops or more likely
that the loops move to "dodge" the monopole. This was
of course the case with the ferrofluid, but in solid matter
the atoms cannot escape from the monopole trajectory.
Does it means that (1) is false if the monopole goes
through an atom? Not necessarily. At the approach of
the pole, the electron wave functions are deformed and,
if the monopole is sufficiently slow, they return
adiabatically to the ground states. Thus no energy is
exchanged between the monopole and the atom, as if the
loop "dodges" the monopole.
3. VELOCITY-DEPENDENT FORCE
In vacuum, the analog of the Lorentz force for a
moving monopole is f = – gv×E. Accordingly, a piece
dl of wire carrying a current I* of magnetic charges is
subjected to the dual Laplace force df = – I*dl×E.
Following the ferrofluid example, we consider a wire
protected by a waterproof tube in a liquid dielectric
(Fig. 2):
a) the tube is flattened perpendicular to D = εE. Then
Etube=D and the measured force is
df = – I*dl×D.
b) the tube is flattened parallel to D = εE. Then Etube=E.
The force acting on the wire is df1 = – I*dl×E. On the
other hand, on the right of the tube, the total field
D + D’ is larger than on the left. The polar molecules
are therefore attracted toward this region, building an
excess of pressure which pushes the tube toward the
left. Calculations like those between Eqs. (2) and (3)
give the thrust df2 = – I*dl×P, where P is the
macroscopic electric polarization. In total,
df=df1+df2=– I*dl×(E+P) (4)
111
is equivalent to (4). Thus the field acting on a wire of
magnetic current is D. It is the field found in a
perpendicular flattened cavity, as for the Laplace force
df = Idl×B on an ordinary current in a magnetized
matter.
Mnemonic: this cavity allows the wire to follow the
force without touching the matter.
For a moving monopole, (4) becomes
f = – gv×D. (4')
Fig. 2. Force acting on a wire carrying the dual
current I*, in a dielectric liquid. The tube is flattened
(a) perpendicular (b) parallel to the field
MICROSCOPIC INTERPRETATION
In a dielectric, E is the average of the microscopic
field e over volumes sufficiently large compared to the
molecular scale. The work of e when the wire sweeps a
flat surface S is
I*∫∫Se dS = I*∫∫SE dS
In contrast, the work of e along a surface S’ which
avoids cutting the dipole molecules is
I*∫∫S e dS = I*∫∫S D dS
Eq.(4) implies that a moving dual wire avoids cutting
the dipole molecules, or that the molecules "dodge" the
wire. This has a well-defined topological meaning. Let
us recall however that this wire was introduced to make
the problem time-independent. For a moving monopole,
there is no swept surface and the topological
interpretation is lost.
Gathering (1) and (4'), the total force on a magnetic
charge is
f = g (H – v×D). (5)
This result does not take into account dissipation and
holds only for sufficiently slow monopoles, such that
atoms and molecules evolve adiabatically under the
influence of the monopole field.
4. MAXWELL EQUATIONS IN DYONIC
MATTER
We consider matter containing magnetic charges ±g
bound in magnetically neutral molecules, in addition to
ordinary particles. These molecules pass magnetic
dipoles of the form gr, building a macroscopic magnetic
polarization p* (r is the north-south charge separation).
If they have spin, they also possess electric dipoles of
the form γ*S building a macroscopic electric
polarization m* (γ* is the "giro-electric" ratio). p* and
m* are dual respectively to the polarization p and the
magnetization m built by the ordinary particles. The
dual-symmetric Maxwell equations for the space
average of the microscopic fields are
jje-b t δ+=∂×∇
δ ρρ +=⋅∇ e
*j*jb-e t δ+=∂×− ∇
**e δ ρρ +=⋅∇
Here (ρ, j) is the external ordinary charge-current
density and (δρ, δj) the induced one, given by
δρ = –∇.p
δj = ∇×m + ∂tp
Similarly, for the magnetic charge analogues,
δρ*= –∇.p*
δj* = ∇×m*+∂tp*
From these equations, we can write the dual-symmetric
Maxwell equations in matter:
∇×H–∂tD = j
∇.D = ρ
–∇×E-∂tB = j*
∇.B = ρ* (6)
where H = b – m, D = ē + p as usual, but E = ē – m*
and B = b + p*. We see that E and B can no more be
interpreted as the spatial averages of the microscopic
fields. In that sense they are no more "fundamental"
than D and H. In fact the dual of E is not B but H,
whereas the dual of B is D. The usual relations
D = E + P, B = H + M (7)
are recovered, defining
P ≡ p + m*, M ≡ m + p* (8)
It means that the microscopic nature of the dipole is
forgotten at the level of the macroscopic Maxwell
equations. Only their long range fields in 1/r3 are
relevant. As in ordinary matter, E and H are the fields
found in elongated cavities parallel to the respective
fields, whereas D and B are found in perpendicular flat
cavities.
5. THE DYONIC PERMITTIVITY-
PERMEABILITY MATRIX
We assume that the polarizations P and M respond
linearly to the macroscopic fields D and B.
[ ] [ ]
=
=
H
E
B
D
M
P
'χχ (9)
112
with [1+χ´]≡[1–χ]–1. In ordinary matter χ11=χe=χe´/ε ,
χe´≡ε-1; χ22=χm=χm´/µ, χm´≡µ-1, and χ12=χ21=0. In a
matter containing only one species of dyon (e,g)and
antidyon (– e, – g) bound in polar molecules, P = p and
M = p* are linked by
ge
*pp = ,
j
i
j
i
B
p
gD
p
c ∂
∂=
∂
∂ 11
(10)
wherefrom
[ ]
= 2
2
geg
egeC teχ . (11)
An analogous matrix, with e ↔ g, is obtained with
dipoles coming from spinning dyons (m/e = m*/g). We
note that [χ] and [χ´] are symmetrical matrices. This
remains true for a mixture of different species of
molecules.
Thus, the usual relations D = εE, B = µH are
replaced by
[ ]
−=
B
D
H
E
χ1 (12)
The speed of light is
c = (det[1-χ])1/2 (cvac.≡ 1) (13)
Whatever they come from, the nondiagonal elements of
[χ] violate P- and T- symmetries, since they connect
vector to pseudovectors. However PT is conserved.
6. ENERGY-MOMENTUM TENSOR
The various components of the energy-momentum
tensor Θµν can be derived from energy and momentum
conservation in simple physical systems. Let us suppose
that the whole space is filled with dual-symmetric
matter. To get Θi0 (energy flow) and Θij (momentum
flow) one considers a sandwich made of three slab-like
regions of the z coordinate, R1 = [– a, 0], R2 = [0, b] and
R3 = [b, b + a]. R1 carries uniform electric and magnetic
charge-current densities, {ρ, j; ρ*, j*} and R3 carries
the opposite densities, such that the fields vanish outside
the sandwich. Solving (6) and (12) with appropriate ρ
, j; ρ*, j*, any kind of uniform field configuration
{E, D; H, B} can be obtained in R2. These fields are
linearly attenuated in R1 and R3. In R3 a power
r3dtd
dW
= E.j + H.j* (14)
is dissipated and a force
r
f
3d
d
= ρE + j × B + ρ*H – j* × D (15)
is exerted per unit of volume. The same quantities per
unit of area (integrated over z in R3) give Θz0 and Θzi in
R2.
To get Θ00 (energy density) and Θ0i (momentum
density) one has to "rotate" the sandwich in the 4-
dimensional space-time, replacing z by t and slabs by
time-slices or "epoch" T1, T2, T3.During T1 the (3-
dimensional) space is filled with uniform current
densities j and j*, which progressively build uniform
fields according (6) and (12). The second epoch is
current-free and the uniform fields remain constant. The
last epoch destroys the fields with opposite currents.
Integrating (14) and (15) over t in T3 give Θ00 and Θ0i in
T2. This method is detailled in [2]. One obtains
−−Θ×
×⋅+⋅
=Θ jijiij HBEDδ
µ ν
00
)(2/1
HE
BDBHDE
(16)
as in ordinary matter.
The Dirac condition in matter. One way to derive the
Dirac condition between an electron and a monopole
is to quantize the joint angular momentum of their fields
which are
34 r
erD
π
= , 34 r
erB
π
=
where r (resp. r´) is the distance from the charge (resp.
the pole) to the observation point. According to (15) the
momentum density is Θ0i
= (D×B)i, from which one gets
the angular momentum
J = ∫∫∫d3r r×(D×B)=(eg/4π) n̂ (17)
where n̂ is the unit vector from the charge toward the
pole. The usual Dirac condition eg = 2nπħ is obtained
from the quantization rule J. n̂ = n ħ/2. Note that if the
momentum density were E×H, as sometimes advocated
(see the discussion in [3]), the Dirac condition in
medium would not be consistent with that in vacuum.
7. APPLICATION TO THE SEARCH FOR AN
ELECTRIC DIPOLE MOMENT
The dual-symmetric formalism applies as well to the
case were the electron (or the nucleus) possesses an
electric dipole moment (e.d.m.) d
= γ*S in addition to
the usual magnetic moment m = γ*S .Then we have m/
γ= m*/γ* and a nondiagonal [χ] matrix element is
generated, like with dyonic molecules (Eqs.10-11):
[ ]
+=
mm
mme
r
rr
χχ
χχχχ
2
(18)
where r ≡ γ*/γ. χm = (η-1)/η comes from the spinning
electrons and χe = (ε-1)/ε from polar molecules.
A nonzero χ may be generated in another way~: the
e.d.m. tends to align the spin of an electron along the
internal electric field of a polar molecule. It couples m
to p. Here we consider only the first mechanism.
Eq.(18) suggests two possible measurements of r:
a) In Fig.3a, a cylinder of magnetizable, but
insulating, material is immersed in a large magnetic
field B0. The inside field B induces a small electric
polarization P = χm r B and an electric field E.If the
cylinder is much broader than high, we have B = B0,
D ≅ 0 and E ≅ – P. More generally one has
〈E〉 = – xχm r B0 (19)
where the coefficient x < 1 depends on the container
geometry. Let us take a cubic container of size L.
Between the top and the bottom, we can measure a
potential difference U = EL. The ratio between the
stored electrostatic energy W = (1/2)ε E2L3 and the
magnetic one W0 = (1/(2µ)) 2
0B L3 is
W/W0 = εµ (xχmr)2. (20)
in terms of common units and r16 = 1016r, we have
tesla
B
metre
Lrx
volt
U
m
0
16
8103 χ−⋅= ,
113
2
0
3
2
16
228105.2
⋅= −
tesla
B
metre
Lrx
eV
W
mε χ .
[useful relations are:1tesla = 3.108 volt/metre,
1(tesla)2 (metre)3 = 0.8.106joule, 1eV = 1.6.10-19 joule ≅
104 kelvin = 5.106 ħ/metre and γe = eħ/me =4⋅10−11 e⋅cm].
Let us assume r = 1016, which corresponds to an
electron e.d.m. of 2.10-27 e × cm, ε ~1, χm ~ 0.5 and
x ~ 0.5. For a field of 1 tesla, and a cube of 1 meter, a
potential difference of about 0.5.10-8 volt is obtained,
W ~ 10-9 eV ~ 10-5 kelvin. The voltmetre has to be cooled
at least to this temperature to prevent thermal noise.
Fig. 3. Scheme of e.d.m. search in macroscopic
matter. (a) container in a magnetic field; a small
potential difference is measured with the voltmetre V.
(b) container in an electric field; a small magnetic flux
is measured with a SQUID
b) In Fig. 3b the same container is put in a large
electric field E0. Using the second matrix of (9)
[ ]
−
≅
mm
m
r
r
''
'1
'
χε χ
ε χε
χ , (18′)
with χm´≡ µ – 1,we predict a small magnetization M = ε
χ´m rE. If the cylinder is much higher than broad, we
have E ≅ E0, H ≅ 0 and B ≅ M. For a cubic container
we assume
B ≅ – x ε χ´m r E0 (21)
with x ~ 0.5. This field can be measured by a SQUID
encircling the container. The phase shift of the wave
function in one loop is
ϕ =eL2B/ħ
=0.5.10-4x ε χm´r16
metrevolt
E
metre
L
/105
0
2
.
This phase can be multiplied by a large number of turns
around the cylinder. The ratio between the output
(magnetostatic) energy W and the input (electric) one W0
is still given by (20), but W and W0 are typically 105
times smaller and the temperature must be much lower
than in case a).
8. CONCLUSION
We have given arguments that the macroscopic
fields acting on magnetic charges and currents are H
and D. Comparing with electric charges and currents,
one has a unified mnemonic principle: in each case, the
acting field is the one found in a parallel-elongated
(resp. flat-perpendicular) cavity in which a charge (resp.
current wire) can follow the force without touching the
medium. In a classical microscopic picture, a monopole
avoids passing through the microscopic current loops
and a dual current wire avoids cutting the dipole
molecules. Quantum mechanically, it means that the
perturbation of the atoms and molecules lying on the
trajectory of the monopole is adiabatic. This should be
the case at low enough velocity in a liquid. The
monopole will be presumably accompanied by a swarm
of atoms magnetically (or electrically, for a dyon)
bound to it. In a solid, such a swarm could forbid the
monopole to move without producing cracks.
The dual-symmetric Maxwell equations in matter
are formally unchanged, but E and B can no more be
interpreted as the spatial averages of the microscopic
fields. The duality correspondance is E → H and D → B.
When dyons are present, or when ordinary particles
possess electric dipole moments, ε and µ are replaced
by a permittivity-permeability matrix [1-χ] whose
nondiagonal elements violate the P- and T- symmetries
(but not PT). The energy momentum tensor is also
unchanged. The usual Dirac condition eg = 2nπħ is
obtained provided the momentum density is D × B.
These results have been obtained under the hypothesis
that P and M are linear in the fields.
As an application of the dual-symmetric formalism,
two possible measurements of the electron e.d.m. have
been suggested. They are at the limit of the present
technological possibilities. However, mechanisms like
the m – p coupling in a polar molecule mentioned in
Sect. 7 might enhance the signal.
REFERENCES
1. Y. Hara. Electromagnetic force on a magnetic
monopole // Phys. Rev. 1985, v. A32, p. 1002-1006.
2. D. Fayolle. Dynamique d'un monopôle magnétique
dans la matière, LYCEN/9961, July 1999
(unpublished).
3. F.N.H. Robinson. Electromagnetic stress and
momentum in matter // Phys. Rep. 1975, v. C16,
p. 313-354.
114
|