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Derivation of relativistic charge-monopole and monopole-monopole potentials from field theory
ANNALS
OF PHYSICS
148, 135-143 (1983)
Derivation of Relativistic Charge-Monopole and Monopole-Monopole Potentials from Field Theory A. 0. Deparfment
of Physics.
BARUT University
Received
AND
Xu*
Bo-WEI
of Colorado, September
Boulder,
Colorado
80309
28, 1982
Relativistic potentials (generalizing the Breit-potential of quantum electrodynamics) between spin t-electric and magnetic charges are presented, each monopole gj having its own singularity-string along some direction n,. The monopole potentials involve integrations along the singularities. By using suitable gauge transformations and limiting procedures a simple form of the potential independent of nj is derived, if the string connects two monopoles.
I.
INTRODUCTION
In this work we derive the analog for the magnetic charges of the Breit-potential of quantum electrodynamics between electric charges. There are two new types of potentials, namely, between a charge and a monopole, and between different monopoles, or more generally, between two dyons (fields with both electric and magnetic charges). The nonrelativistic vector potential A of a monopole has been known since the original work of Dirac [ 11. A variant of it is known as the Schwinger potential [2]. These were derived recently from field theory as two special cases of a more general class of potentials with N singularity lines [3]. The present work is based on Dirac’s formulation [4] of the theory of charges and monopoles with a single electromagnetic potential A,, having a singularity surface y,(t, o) [string] which is treated as a dynamical variable subject to gauge transformations. This theory can be proved to be equivalent to a Maxwell theory with an additional singular current along the string [3,5]. We treat the case where both the electric and magnetic currents are described by spinorial currents eyly,x and gfy,,x, respectively, and obtain the usual charge-monopole vector potential corrected by the relativistic Dirac matrices a, and the new monopole-monopole potentials with each monopole having its own singularity line. The nonrelativistic Coulomb-like potential g, g,/r between two monopoles is derived by intricate integrals, choice of gauge and limiting procedures (together with its a-dependent relativistic corrections). This potential is usually written down by hand from symmetry arguments, but has never been derived from field theory. In the two-potential-formalism, when a monopole is described by both a * On leave of absence from
the Department
of Modern
Physics,
Lanzhou
University,
Lanzhou.
China.
135 0003-4916/83
$7.50
Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.
136
BARUT
AND
XU
vector potential A and a (dual) scalar potential @such a term may be inserted into $. But in the one-potential approach a monopole is described by a vector potential A only; hence the derivation of the Coulomb potential from a vector potential is not trivial. In the present derivation the string function y,(r, a) and the singular 2-form /i,,. on the string plays a fundamental role, again showing the dynamical significance of the variables u,(r, a), subject to gauge transformations under arbitrary diffeomorphisms (reparametrization of u and r), but necessary to resolve completely many difficulties and objections in the field theory of monopoles [6]. II.
THE
FIELD
EQUATIONS
We start from the basic field equations
(1) describing a number of electric or magnetic charges interacting FE”. For some j, yj and xj might be the same (dyon). Here
via the Dirac field
As is well-known FD cannot be derived from a potential, but we can write [I]
I;;,=$+&A,-II;,,
(2)
where A,,, describes the singularity of the potential A, and will be specified below. We introduce now a new field FM by [5, 31 FMP” ra L”A -8 “PA
(3)
and obtain from (1) and (3) a”Fff” =x
(iejpjy,, vj + a”JL”), j
a”q” = 0.
(4)
and a”A,, = -igXy,x. The new Field FM is a closed two-form, obeys Maxwell’s additional singular current source jiing = G 8’2;“.
(5). equations, but has an
RELATIVISTIC
MONOPOLE-MONOPOLE
137
POTENTIAL
From (3) and (4) with the gauge condition, we have the equation DAu=-~ie,~jy,~j-j~“g,
PAL, = 0,
(6)
with its solution
(7)
+ i dy ~3”D(x - y) x /ij,,,(y). 1^ j III.
THE INTERACTION
LAGRANGIAN
From Eq. (4) the interaction Lagrangian density is given by
= -i 1 dY G - I dY s j,k
ejek
ejWj(X>
v/j(X)
YfiWjCx)
+ i c dy v P-Xi,,(x) z Hence the Lagrangian, of variables, becomes %
combining
dx=-i
dx&
-2
D(x -Y> 9kt.V) YWk(Y)
auDtx -Y)JE(Yjk
PD(x - y) e”(y).
(8)
the two integrals in the middle of (8) by a
x .ik
eiekpj(x)
Y, ‘/j(X)
D(X
-Y>
vk(Y>
change
Y’v/k(Y)
dxdy~eejy/j(x)y,y/j(x)a~o(x-Y)~(y)k i
-i
Y, Vjtx)
I
.ik
dxdy~~~,(x)a’lYD(x-y)/l;;f(y)k. ik
(9)
We shall use in the following retarded Green’s functions
qx-y)=-i 47c&B(x”-yD-lx-YIl
(10)
138
BARUTANDXU
and integrate the b-function over y”. [The calculations function are exactly the same but with two &functions.] For the static states, (9) becomes
I ynt&-‘*j 7
4X
with
Feynmann
Green’s
dx dyIylj(4 r,vjW V,(Y) r“wd~)lllx - Y I
(11) The effective Lagrangian
is then given by
(12)
IV. THE HAMILTONIAN The determination of the potential follows a method used recently to derive the potentials due to (Pauli) anomalous magnetic moment coupling [7]. We define an effective Hamiltonian by
He,= S pj(-ia * Vj + mTbo>t//j + s fj(-ia *
vj
+ m!b)
Xj
-
tLeff)int
(13)
RELATIVISTIC
and the two-body
MONOPOLE-MONOPOLE
interaction
Heff z
operators
8,)
POTENTIAL
139
fi2, r?, by
1 dYWj+ Cx) W: (Y> fi, Vjtx> W/i(Y)
(14) with normalization
J‘dyy/+lp=
1,
\dyX+x= I
1.
In order to evaluate the integrals involving the string variables, we start from the covariant singular differential two-form ,4,,. of the string surface given by [ 1 ]
Arcr,=gCdsdu(Jj,y:-$,.y:)64(x-v),
(15)
where y, = yu(r, a) is the equation of the string in terms of the Lorentz invariants parameters r and u with 4; = ay/& and y’ = 8y/&. We choose a coordinate condition such that [3] yO(7,
a)
= t
(16)
and then set yU(r, UT)= wU(r) + zP(u).
(17)
u+(t) is a timelike vector (position of the monopole at one endpoint of the string), and z.2’ is a space-like vector along the string, ~“(a,) and uU(a2) being the two endpoints. Then g+, is the magnetic current k, (see Eqs. (1) and (5)). Hence we have A,,. = [ k,tx - u) A du,.
(18)
and it can indeed be verified [3] that %A.,,: = k,(a,) - k,(u,). In order to evaluate the potentials explicitly, we choose the string such that u, = (0, (7l - CT)Rn),
ti, = (0, -Rtl),
(19)
where n is a unit 3-vector, u a parameter and R a distance. Then A,,(x)
= ig fi(x
- u) y/x(x - u) A n, da’
(20)
140
BARUT
AND
XU
and (21) where u’ = (n - u) R. Inserting these into Eq. (12) we obtain fi, =x
(-iajVj
1 -aj.
+ mj/j) + x eje,.
j
jk
ak
IX-Y1
47t
1
ti2=-\‘ejg,Elmi y 47c
(22) ’ a(k)
/x-y-c7’nkl
/,
ni(k)
m
1 ,x-yyu~nk,
(‘)knitk’
and fi, = S (-ia . Vj + m!p) - S T i
J du’ du”(aI)j A (niy
1
x (v2Ix+u’nj - y - u”n/, 1 +;~jdu’du”(l)j(nJj
ca’>k
(p’
A
tni)k
1 /x + u’nj - y - upnk)
(l)k
(dk
da’ du”(a,)j A (r~,)~ + v @k z 47z I 1 ama’ /x + u’nj y - u”nk 1 ) (
x
JdU’
du”(l)j
j
k-3’
(O1rn)k
lx
A
(dk
+
u,nj’
y
_
u,,nkI
)
(l)k
(nm)k.
(24)
We now use the relation a,a,[Ix
- y + u’nj - u”nk)]-’
= (3(x - y + u’ni - uNnk)m(x - y + u’nj - u”nj),) 1x - y + u’nj - u”nk 1=5 --d,,/x-y+o’n,-u”n,l-“-~6,,G(x-y+o’n,--o’:n,)
RELATIVISTIC
MONOPOLE-MONOPOLE
141
POTENTIAL
and then shall perform the u’ and UN-integrations. The form of the potential depends whether we have one string going from monopole to infinity, or two straight strings in opposite directions going from monopole to infinity. It has been shown that this is the difference between the Dirac and Schwinger potentials and the corresponding charge quantization conditions, or we could even have more than two strings emanating from the monopole [ 3 1. The charge monopole interaction, with a single string, is given by I(l>j(akXnk)-(l)k(ajX
nk>l
’ ,.(,.-:.
nk)9
(25)
where t = x - y. This agrees with the Dirac-form, corrected with velocity dependent term. The monopole-monopole integrals can be written as
+~!$,fd,,~&,~
3nj’nkaj;Lak*L
+
hj’aknf;h,.L
L
_
2aj. aknj . nk - $ L3
aj . aknj . n,d(L)
T;‘ sjgk
3nj. akaj . Ln, . L 3aj. n,nj , La, . L + LS L5
G
4n
da’ i
do” [
I
2a, . njaj . nk ’ 271 - aj . nkak . njd(L) ; L’ 3 1 LEx-y
+u’nj-a”n,.
(26)
These integrals are rather complicated for arbitrary choices of nj and nk. However, by a gauge transformation we can choose the directions n,j anyway we like. A very convenient choice is, for two monopoles, nj = -nk = n.
We then choose the limits of integrations as da’
142
BARUT
AND
XU
and obtain
(r . n)’ r3 - 2r2r . n - r(r . n)’ + 3(r a n)’ - r?
X
2r-3r.n+@f c
r2
.
I
(27)
11
Finally we take the limit n--t r n. r^=cos~.
E+ 0.
In this limit the final form of the operator H, is
H, = x
(-ia
. Vj + mi”p)
j
Equations (22), (25) and (27) or (28) are the main results of this work and represent relativistic static potentials in the sense that the Born approximations of these potentials will coincide with the scattering amplitudes of the lowest order Feynman exchange diagrams. Tu, Wu and Yang IS] have recently used the charge-monopole and monopole-monopole Hamiltonians and derived Maxwell’s equations in Heisenberg form. They did not include the Breit-terms for both charges and monopoles. We have here related these Hamiltonians to the basic Maxwell-Dirac field theory. Their work could further be extended to include Breit terms.
REFERENCES 1. P. A. M. DIRAC,
Proc.
Roy.
Sot. A 133 (1931),
2. J. SCHWINGER, Phys. Ret). 144 (1966), 1087. 3. A. 0. BARUT. J. Phys. A 11 (1978), 2073; Left. 4. P. A. M. DIRAC, Phyys. Rev. 74 (1948). 817.
60. Math.
Phys.
1 (1977),
367.
RELATIVISTIC
MONOPOLE-MONOPOLE
5. A. 0. BARUT AND G. BORNZIN. Nucl. Phys. B 81 (1974). 417. 6. For a list of these problems and their solutions and further references see A. 0. Parficles Nuclei. 10 (1979), 539. [English transl.] 10, No. (3) (1979). 209. 7. A. 0. BARUT AND B-W. Xu, Phys. Ser. 126 (1982). 129. 8. T. S. Tu, T. T. Wu. AND C. N. YANG. Sci. Sinica 21 (1978). 317.
595/148/1LlO
143
POTENTIAL
BARUT.
SOL.. /.
OF PHYSICS
148, 135-143 (1983)
Derivation of Relativistic Charge-Monopole and Monopole-Monopole Potentials from Field Theory A. 0. Deparfment
of Physics.
BARUT University
Received
AND
Xu*
Bo-WEI
of Colorado, September
Boulder,
Colorado
80309
28, 1982
Relativistic potentials (generalizing the Breit-potential of quantum electrodynamics) between spin t-electric and magnetic charges are presented, each monopole gj having its own singularity-string along some direction n,. The monopole potentials involve integrations along the singularities. By using suitable gauge transformations and limiting procedures a simple form of the potential independent of nj is derived, if the string connects two monopoles.
I.
INTRODUCTION
In this work we derive the analog for the magnetic charges of the Breit-potential of quantum electrodynamics between electric charges. There are two new types of potentials, namely, between a charge and a monopole, and between different monopoles, or more generally, between two dyons (fields with both electric and magnetic charges). The nonrelativistic vector potential A of a monopole has been known since the original work of Dirac [ 11. A variant of it is known as the Schwinger potential [2]. These were derived recently from field theory as two special cases of a more general class of potentials with N singularity lines [3]. The present work is based on Dirac’s formulation [4] of the theory of charges and monopoles with a single electromagnetic potential A,, having a singularity surface y,(t, o) [string] which is treated as a dynamical variable subject to gauge transformations. This theory can be proved to be equivalent to a Maxwell theory with an additional singular current along the string [3,5]. We treat the case where both the electric and magnetic currents are described by spinorial currents eyly,x and gfy,,x, respectively, and obtain the usual charge-monopole vector potential corrected by the relativistic Dirac matrices a, and the new monopole-monopole potentials with each monopole having its own singularity line. The nonrelativistic Coulomb-like potential g, g,/r between two monopoles is derived by intricate integrals, choice of gauge and limiting procedures (together with its a-dependent relativistic corrections). This potential is usually written down by hand from symmetry arguments, but has never been derived from field theory. In the two-potential-formalism, when a monopole is described by both a * On leave of absence from
the Department
of Modern
Physics,
Lanzhou
University,
Lanzhou.
China.
135 0003-4916/83
$7.50
Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.
136
BARUT
AND
XU
vector potential A and a (dual) scalar potential @such a term may be inserted into $. But in the one-potential approach a monopole is described by a vector potential A only; hence the derivation of the Coulomb potential from a vector potential is not trivial. In the present derivation the string function y,(r, a) and the singular 2-form /i,,. on the string plays a fundamental role, again showing the dynamical significance of the variables u,(r, a), subject to gauge transformations under arbitrary diffeomorphisms (reparametrization of u and r), but necessary to resolve completely many difficulties and objections in the field theory of monopoles [6]. II.
THE
FIELD
EQUATIONS
We start from the basic field equations
(1) describing a number of electric or magnetic charges interacting FE”. For some j, yj and xj might be the same (dyon). Here
via the Dirac field
As is well-known FD cannot be derived from a potential, but we can write [I]
I;;,=$+&A,-II;,,
(2)
where A,,, describes the singularity of the potential A, and will be specified below. We introduce now a new field FM by [5, 31 FMP” ra L”A -8 “PA
(3)
and obtain from (1) and (3) a”Fff” =x
(iejpjy,, vj + a”JL”), j
a”q” = 0.
(4)
and a”A,, = -igXy,x. The new Field FM is a closed two-form, obeys Maxwell’s additional singular current source jiing = G 8’2;“.
(5). equations, but has an
RELATIVISTIC
MONOPOLE-MONOPOLE
137
POTENTIAL
From (3) and (4) with the gauge condition, we have the equation DAu=-~ie,~jy,~j-j~“g,
PAL, = 0,
(6)
with its solution
(7)
+ i dy ~3”D(x - y) x /ij,,,(y). 1^ j III.
THE INTERACTION
LAGRANGIAN
From Eq. (4) the interaction Lagrangian density is given by
= -i 1 dY G - I dY s j,k
ejek
ejWj(X>
v/j(X)
YfiWjCx)
+ i c dy v P-Xi,,(x) z Hence the Lagrangian, of variables, becomes %
combining
dx=-i
dx&
-2
D(x -Y> 9kt.V) YWk(Y)
auDtx -Y)JE(Yjk
PD(x - y) e”(y).
(8)
the two integrals in the middle of (8) by a
x .ik
eiekpj(x)
Y, ‘/j(X)
D(X
-Y>
vk(Y>
change
Y’v/k(Y)
dxdy~eejy/j(x)y,y/j(x)a~o(x-Y)~(y)k i
-i
Y, Vjtx)
I
.ik
dxdy~~~,(x)a’lYD(x-y)/l;;f(y)k. ik
(9)
We shall use in the following retarded Green’s functions
qx-y)=-i 47c&B(x”-yD-lx-YIl
(10)
138
BARUTANDXU
and integrate the b-function over y”. [The calculations function are exactly the same but with two &functions.] For the static states, (9) becomes
I ynt&-‘*j 7
4X
with
Feynmann
Green’s
dx dyIylj(4 r,vjW V,(Y) r“wd~)lllx - Y I
(11) The effective Lagrangian
is then given by
(12)
IV. THE HAMILTONIAN The determination of the potential follows a method used recently to derive the potentials due to (Pauli) anomalous magnetic moment coupling [7]. We define an effective Hamiltonian by
He,= S pj(-ia * Vj + mTbo>t//j + s fj(-ia *
vj
+ m!b)
Xj
-
tLeff)int
(13)
RELATIVISTIC
and the two-body
MONOPOLE-MONOPOLE
interaction
Heff z
operators
8,)
POTENTIAL
139
fi2, r?, by
1 dYWj+ Cx) W: (Y> fi, Vjtx> W/i(Y)
(14) with normalization
J‘dyy/+lp=
1,
\dyX+x= I
1.
In order to evaluate the integrals involving the string variables, we start from the covariant singular differential two-form ,4,,. of the string surface given by [ 1 ]
Arcr,=gCdsdu(Jj,y:-$,.y:)64(x-v),
(15)
where y, = yu(r, a) is the equation of the string in terms of the Lorentz invariants parameters r and u with 4; = ay/& and y’ = 8y/&. We choose a coordinate condition such that [3] yO(7,
a)
= t
(16)
and then set yU(r, UT)= wU(r) + zP(u).
(17)
u+(t) is a timelike vector (position of the monopole at one endpoint of the string), and z.2’ is a space-like vector along the string, ~“(a,) and uU(a2) being the two endpoints. Then g+, is the magnetic current k, (see Eqs. (1) and (5)). Hence we have A,,. = [ k,tx - u) A du,.
(18)
and it can indeed be verified [3] that %A.,,: = k,(a,) - k,(u,). In order to evaluate the potentials explicitly, we choose the string such that u, = (0, (7l - CT)Rn),
ti, = (0, -Rtl),
(19)
where n is a unit 3-vector, u a parameter and R a distance. Then A,,(x)
= ig fi(x
- u) y/x(x - u) A n, da’
(20)
140
BARUT
AND
XU
and (21) where u’ = (n - u) R. Inserting these into Eq. (12) we obtain fi, =x
(-iajVj
1 -aj.
+ mj/j) + x eje,.
j
jk
ak
IX-Y1
47t
1
ti2=-\‘ejg,Elmi y 47c
(22) ’ a(k)
/x-y-c7’nkl
/,
ni(k)
m
1 ,x-yyu~nk,
(‘)knitk’
and fi, = S (-ia . Vj + m!p) - S T i
J du’ du”(aI)j A (niy
1
x (v2Ix+u’nj - y - u”n/, 1 +;~jdu’du”(l)j(nJj
ca’>k
(p’
A
tni)k
1 /x + u’nj - y - upnk)
(l)k
(dk
da’ du”(a,)j A (r~,)~ + v @k z 47z I 1 ama’ /x + u’nj y - u”nk 1 ) (
x
JdU’
du”(l)j
k-3’
(O1rn)k
lx
A
(dk
+
u,nj’
y
_
u,,nkI
)
(l)k
(nm)k.
(24)
We now use the relation a,a,[Ix
- y + u’nj - u”nk)]-’
= (3(x - y + u’ni - uNnk)m(x - y + u’nj - u”nj),) 1x - y + u’nj - u”nk 1=5 --d,,/x-y+o’n,-u”n,l-“-~6,,G(x-y+o’n,--o’:n,)
RELATIVISTIC
MONOPOLE-MONOPOLE
141
POTENTIAL
and then shall perform the u’ and UN-integrations. The form of the potential depends whether we have one string going from monopole to infinity, or two straight strings in opposite directions going from monopole to infinity. It has been shown that this is the difference between the Dirac and Schwinger potentials and the corresponding charge quantization conditions, or we could even have more than two strings emanating from the monopole [ 3 1. The charge monopole interaction, with a single string, is given by I(l>j(akXnk)-(l)k(ajX
nk>l
’ ,.(,.-:.
nk)9
(25)
where t = x - y. This agrees with the Dirac-form, corrected with velocity dependent term. The monopole-monopole integrals can be written as
+~!$,fd,,~&,~
3nj’nkaj;Lak*L
+
hj’aknf;h,.L
L
_
2aj. aknj . nk - $ L3
aj . aknj . n,d(L)
T;‘ sjgk
3nj. akaj . Ln, . L 3aj. n,nj , La, . L + LS L5
G
4n
da’ i
do” [
I
2a, . njaj . nk ’ 271 - aj . nkak . njd(L) ; L’ 3 1 LEx-y
+u’nj-a”n,.
(26)
These integrals are rather complicated for arbitrary choices of nj and nk. However, by a gauge transformation we can choose the directions n,j anyway we like. A very convenient choice is, for two monopoles, nj = -nk = n.
We then choose the limits of integrations as da’
142
BARUT
AND
XU
and obtain
(r . n)’ r3 - 2r2r . n - r(r . n)’ + 3(r a n)’ - r?
X
2r-3r.n+@f c
r2
.
I
(27)
11
Finally we take the limit n--t r n. r^=cos~.
E+ 0.
In this limit the final form of the operator H, is
H, = x
(-ia
. Vj + mi”p)
j
Equations (22), (25) and (27) or (28) are the main results of this work and represent relativistic static potentials in the sense that the Born approximations of these potentials will coincide with the scattering amplitudes of the lowest order Feynman exchange diagrams. Tu, Wu and Yang IS] have recently used the charge-monopole and monopole-monopole Hamiltonians and derived Maxwell’s equations in Heisenberg form. They did not include the Breit-terms for both charges and monopoles. We have here related these Hamiltonians to the basic Maxwell-Dirac field theory. Their work could further be extended to include Breit terms.
REFERENCES 1. P. A. M. DIRAC,
Proc.
Roy.
Sot. A 133 (1931),
2. J. SCHWINGER, Phys. Ret). 144 (1966), 1087. 3. A. 0. BARUT. J. Phys. A 11 (1978), 2073; Left. 4. P. A. M. DIRAC, Phyys. Rev. 74 (1948). 817.
60. Math.
Phys.
1 (1977),
367.
RELATIVISTIC
MONOPOLE-MONOPOLE
5. A. 0. BARUT AND G. BORNZIN. Nucl. Phys. B 81 (1974). 417. 6. For a list of these problems and their solutions and further references see A. 0. Parficles Nuclei. 10 (1979), 539. [English transl.] 10, No. (3) (1979). 209. 7. A. 0. BARUT AND B-W. Xu, Phys. Ser. 126 (1982). 129. 8. T. S. Tu, T. T. Wu. AND C. N. YANG. Sci. Sinica 21 (1978). 317.
595/148/1LlO
143
POTENTIAL
BARUT.
SOL.. /.