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Magnetic monopoles with no strings
Nuclear Physics B162 (1980) 385-396 0 North-Holland Publishing Company
MAGNETICMONOPOLESWITHNOSTRINGS A.P. BALACHANDRAN Physics Department,
l)
Syracuse University, Syracuse, New York 13210,
USA
G. MARMO 2, Istituto di Fisica Teorica, Universitb di Napoli, Mostra d’oltremare
Pad. 19, 80125 Napoli,
Italy 3, and Physics Department,
Northeastern
University, Boston, Massachusetts 02215,
USA
B.S. SKAGERSTAM TH Division, CERN, Switzerland
A. STERN l) Physics Department,
Syracuse University, Syracuse, New York 13210,
USA
Received 17 April 1979
Previous treatments of the charge-monopole system have used either a lagrangian singular along the Dirac string or a multiple valued action integral. Here we formulate an action principle for this system which involves no strings and is single valued. This is achieved by writing the lagrangian directly in terms of the variables of a suitable fibre bundle. Canonical quantization is carried out in a simple way and known results are recovered.
1. Introduction In this paper, we study the description
of a charged
non-relativistic
particle
in a
treatment of this system [ 11, the lagrangian is singular along the “Dirac string”. The presence of this unphysical singularity leads to technical difficulties in the quantization of this system. An alternative magnetic
monopole
field. In the customary
l) Supported by the US Department of Energy. 2, Supported by I stituto Nazionale di Fisica Nucleare, Italy. 3, Permanent address. 385
386
A.P. Balachandran et al. /Magnetic
monopoles
approach which avoids the singular string works with a multiple valued action integral [2]. Here we will formulate an action principle which involves no strings and is single valued. We will also quantize the system and recover its known quantum mechanics. No technical difficulties are encountered during the calculation. The lagrangian formalism is treated in sect. 2. In contrast to earlier formulations, the lagrangian L depends on a degree of freedom associated with the gauge group U(1). Under a gauge transformation, L changes by a time derivative of a function. As a result, this gauge degree of freedom does not occur in the equations of motion. The latter are the usual equations. We show however that it is impossible to globally eliminate the gauge degree of freedom from L. In sect. 3, the canonical quantization of L is carried out in a simple and unambiguous way. In sect. 4, which is slightly more technical in nature, the structure and symmetry properties of the system are explained in terms of the fibre-bundle picture. In our approach, the wave functions for the charge monopole system are defined on a nontrivial U(1) bundle over the space of relative coordinates. It is emphasized that (smooth) wave functions of this sort do not in general project down to (smooth) wave functions on the configuration space. Under the action of an element u E U(l), the wave functions on the bundle change by a phase which depends only on u. As a result, it is shown in this section that the expectation value of any observable constructed out of coordinates and velocities is independent of this phase. Thus a physical interpretation becomes possible. We summarize the general nature of such quantum theoretic possibilities where the wave functions are not defined on the configuration space B of observable coordinates. The relation of such quantum systems to classical systems with only local lagrangians over B is discussed. [The system is said to have a local lagrangian over B if there is a lagrangian for each coordinate neighborhood of B. (It will of course depend on velocities as well in this neighborhood.) It has a global lagrangian if there& a single lagrangian for all of B.] The material in much of this section is known to some physicists *. It is presented here for completeness. Some time ago, Miller [4] studied a formulation of the charge-monopole system which has important points of similarity to our work l*. Unlike him, however, we avoid the Kaluza-Klein approach and are much closer to the conventional treatment. Recently, Friedman and Sorkin [6] have independently developed a treatment of this system which is similar to ours. In another paper [7], we will report on the supersymmetric version of the present work. It describes a charged spin-half particle in the field of a magnetic monopole.
l
l See, in this connection, refs. [2,3]. * The connection form he uses is the same as ours. This form is also discussed
in [5]
387
A.P. Balachandran et al. f Magnetic monopoles
2. The lagrangian Let x = (x,, x2, xs) denote the relative coordinate the charge-monopole system. We write
and m the reduced mass of
3 x = r2,
r=
I( C
1
i=
x?)‘/~I.
(2.1)
Let uol be the Pauli matrices and let g E SU(2). [More precisely, (g} is the twodimensional tmitary irreducible representation of SU(2).] Then we parametrize .? in terms of g by setting l ’ X = o&?, =go,g+
(2.2) .
(2.3)
In the lagrangian below, the basic dynamical variables are r and g. So wherever P occurs, it is to be regarded as written in terms of g. The lagrangian is
1
d 2 t ni Tr 03gtg L = $rn z (~-2) [ = irnf2
+ irnr’
Tr X‘2 + ni Tr a3gtg .
(2.4)
Here r and g are functions of the time t and the dot denotes time differentiation. We have used the identity i. k = 0 in writing (2.4). The relation of the constant n -to the product eg of electric and magnetic charges will emerge below. The resemblance of this lagrangian and the lagrangian for a particle in interaction with a Yang-Mills field [9] may be noted. Variation of r leads to mi:=m/..-$2.
(2.5)
Since g E SU(2), its most general variation is induced by an arbitrary element of its Lie algebra: 6g = ia, e,g ,
E, real .
(2.6)
Hence h_t=i[o.e,Tk],
6 Tru3gtQ,=iTru.&.
u . E = a,e, ,
(2.7) w3)
* The charge and the monopole are not allowed to occupy the same position at any given time (see [8,2] in this connection). Hence the point r = 0 is excluded from the set of allowed x.
388
A.P. Balachandran et al. /Magnetic
Variation
monopoles
of g thus leads to
Tr u.
E i
{-ii[$ rnr*k]+n_?} =0
(2.9)
(In deriving this, we have used the identity Tr A [B, C] = Tr B [C, A] .) Here the bracketed expression is a linear combination of Pauli matrices and E, are arbitrary. Therefore
+ni} =0 . $ {-$i[i, rnr*i-] This can be rewritten
%I
eolprx&+)
(2.10)
in the form + nz?,] = 0 .
(2.11)
One recognizes (2.5) and (2.11) as the usual equations system with * --
for the charge-monopole
n=eg.
(2.12)
47r We note that _? and hence x are invariant g
+gei03@
,
But L is not invariant
under the U(1) gauge transformation (2.13)
8 = e(t).
under this transformation:
L+L-2nb
(2.14)
So it is not possible to write L in terms of x and i. Since J? depends on two parameters and g on three parameters, there is an extra gauge degree of freedom present non-trivially in L. It does not affect the equations of motion since the change in L in (2.14) is by the time derivative of a function. We can try to eliminate this additional degree of freedom by fixing a gauge. That is, for each X, we can try to find one group element g(x) with the property i = g($) a,gQ?)t
.
(2.15)
But it is well-known that there is no such g(J.!) which is defined for all _A!and is continuous in i. Thus it is impossible to manipulate L to a global lagrangian which depends only on x. It is of course possible to find a g(% which fails to be continuous only at one point, say at the south pole [i = (0, 0, -l)] . Then L becomes a monopole lagrangian
* With this identification, the equations PLYi = (Zg/4n) Eij@+k/r3.
of motion
can be written
in the form
A.P. Balachandran et al. /Magnetic
monopoles
389
with a string singularity along the negative third axis *. Alternatively, we can cover the two-sphe_re S2 = {i} by two-coordinate patches U, and U2 and find two group elements gj(X) whi!h are defined and continuous on Ui and which fulfill (2.15). Substitution of g&X) in (2.4) leads to lagrangians Li defined on Ui. In the intersection region U, f7 U2, L 1 - L2 is the total time derivative of a function of 2. Such.a formulation is thus the non-relativistic analogue of the work of Wu and Yang [2].
3. Canonical quantization 3.1. The canonical formalism It is useful to have a method to treat group elements for setting up the canonical formalism. Such a method has been developed elsewhere [9]. We recall it briefly here Let g be parametrized by a set of variables < = (El, g2, &) so that g = g(f). The t’s can be the Euler angles. [The functional form of g(l;) is not important for us.] We regard L as a function of r, i, l and .& We first note a preliminary identity. We can write ezO~‘~&I Differentiating
=dH41 ,
t(o) = 5 .
(3.1)
on ea and setting E = 0, we find, (3.2)
(3.3) It is proved in ref. [9] that detN#O.
(3.4)
Now the coordinates ket (PB) relations t&&1=
.& and their conjugate momenta
71, fulfill the Poisson brac-
{%JTp)=O, (3.5)
ts,, np 1 = &p . It follows easily from (3.2) that {J,, g) = i&g {J,,
g-’
) =
,
-ig-‘io,
,
lLetg(~)=~{a!+1/01[03,X]},(~=l[2(1+-P3)~~’~ easy to verify that it is in SU(2) and fulfills (2.15). which is a conventional form of the interaction.
(3.6) I. This g is ill-defined a; f = (0, 0, - 1). It is Also ni Tr o3g(X)ti(X) = ne3$$$(1 + f3),
A.P. Balachandran et al. /Magnetic
390
monopoles
where J, = -fqNpol
.
(3.7)
It may also be shown [9] that
{J,, Jp > = ‘cxavJ, .
(3.8)
Since N is non-singular [eq. (3.4)], we can work with J, instead of the momenta rr,. The PB’s involving J, and g are simple and do not depend on the parametrization we use for g. Below, we will therefore be eliminating rr, in favour of Jol_.Note that in view of (2.3) and (3.6), J, is just the generator of spatial rotations on X. For the lagrangian (2.4), the momentum pr conjugate to r is given by pr=mi,
(3.9)
while IT, is given by IT, = $rnr2 Tr _&,i+ Here a, = a/a&.
ni Tr u3 gt &g .
(3.10)
Hence, by (3.7) and (3.2),
J,=-$zr”
Trk
[ioa,x]
tnTrio,X
= ecvavxp rn$ t n& . From (3.9) and (3.1 l), we see that there is one primary constraint #=&J,
-!I,
(3.11)
l, (3.12)
which vanishes weakly: (3.13)
l#J=O. The Hamiltonian
is
H=p,+n&-L+v@ P2 1 =-.!I_ +[J&J, - n2] t I@, 2m 2mr2
(3.14)
where v is a Lagrange multiplier. The constraint @is rotationally invariant: (4, J, } = 0. Hence (4, H) x 0 and there are no secondary constraints. Since u is arbitrary, we see from (3.14) that only those variables with (weakly) zero PB’s with @have a well-defined time evolution. These are the first-class variables of the theory. Only such variables are of physical interest. We will call them observables.
* Constrained hamiltonian dynamics is discussed in [lo].
A.!‘. Balachandran et al. /Magnetic
One set of observables is the angular momentum
391
monopoles
generators
J C!’
(3.15)
Further from (3.6) (2.2) and (2.3) we find, M g] =
{hJaj g) = it& .
(3.16)
Hence those functions of g which are invariant under the gauge transformation (2.13) are also observables. But these are precisely functions of i. Thus we have also the observables I (3.17) x, . We get a complete set of observables by adjoining to (3.15) and (3.17) the observables (3.18)
C Pr . This set is subject to the one constraint
4 = 0.
3.2. Quantum mechanics The algebra of observables in quantum theory is obtained as usual by identifying the Poisson bracket with t-i) times the commutator bracket. A representation of this algebra can be constructed as follows. Let us ignore the constraint for a moment and regard the wave functions as functions of r and g: (3.19)
II, = NC g)
The coordinates are diagonal in this representation of wave functions in view of (2.2) and (2.3). The momentum P,. acts as the usual differential operator on $J. The operators J, are the differential operators which represent the elements aa, in the left regular representation of SU(2). In particular, [eieJa$]
(r, g) = $(r, e-ie~0~/2g)
We can now take the constraint &J,$
(3.20)
into account by imposing the condition
= nJ/
(3.21)
on the wave functions. +(r
.
ge-ie0312
)
In view of (3.20), (2.2) and (2.3), this means =
NC
s>
eien
The scalar product on the wave functions
@&)~*(cdxk&
(G,x)=Id+0
(3.22)
.
SU(2)
where dp is the invariant measure on SU(2).
is (3.23)
392
A.P. Balachandran et al. /Magnetic
monopoles
Let {Di(g)} be the representation of SU(2) with angular momentum tions $ with finite norm have the expansion [ 1 I]
j. Wave func
(3.24) Here DLO are the matrix elements of Di in the conventional basis with the third component of angular momentum diagonal. The constraint (3.22) means that in (3.24), only those a:, with cr = -n are nonzero. Thus @_,,
fixed n
1,
is a basis for expansions of the form (3.24). Since u is necessarily integral or half-integral, condition
(3.25)
we also find the Dirac quantization
2n = integer .
(3.26)
In (3.25), j and p are half-integral if 2n is odd and integral if 2n is even. The quantum mechanics outlined here is essentially equivalent to conventional treatments. (See also subsect. 4.2 in this connection.)
4. Final remarks Much of the discussion in this section is known to some physicists ded here for completeness.
[3]. It is inclu-
4.1. Fibre bundles, symmetries Let Q denote the set {x } of relative coordinates Then
with the origin x = 0 removed.
Q= R’ XS2, R’ = {rlr>O}, s2= {a}.
(4.1)
The two-sphere S2 admits non-trivial U(1) bundles. It is known that the description of magnetic monopoles requires such bundles [ 121. They can be constructed as follows: regard SU(2) as a U( 1) bundle over S2 where we take the action of U( 1) to be g +ge”@
.
Hence the projection g ego3g+
(4.2) map to S2 is
=x .
(4.3)
A.P. Balachandran et al. /Magnetic
monopoles
393
Now the cyclic group Z,
= {zk = eios2nk’Mlk = 0, 1, 2, . . .. M - 1 } ,
M>l,
(4.4)
has an action (4.5)
g+gk
which commutes with the projection go,g+
= bk)
03(gzk)+
(4.3): (4.6)
The quotient L,,,, of SU(2) by this action of ZM is thus a U(1) bundle over S*. A function on LM can be regarded as a function f on SU(2) which is Zw invariant: fkzk)
(4.7)
=f@.
The lagrangian L in (2.4) is written on (the tangent bundle of) R’ X SU(2). As explained in sect. 2, it can not be globally written in terms of x and i. In view of (3.22) and (4.7), the wave functions of the charge-monopole system for a given n can be regarded as wave functions on R’ X L12,1. (They can in general be regarded as wave functions on any R’ X LM such that 2n/M is an integer.) We note that the action, g+gog,
g? go E SU(2) 9
(4.8)
of SU(2) on itself corresponds to a spatial rotation of x by (2.3). This action commutes with that of Z,,,, and hence is well defined on LM. Thus we have a natural lift of spatial rotations from S* to LM. There is a similar lift of spatial reflection defined by g+gp, P= exp[iino,]
.
(4.9)
Clearly under P, i--i.
(4.10)
[Note that such a lift is not unique since for instance ur can be replaced by cos Bar + sin 00~ without changing (4.10).] Since PZMP-’ = ZM, spatial reflection is also well-defined on L,,,. However, the lagrangian is not invariant under g + gP if n f 0. So, as is known, this system violates P if n # 0. 4.2. Physical interpretation For n # 0, a (continuous) wave function J, on R’ X SU(2) (just as the lagrangian L) can not in general be written as a (global, continuous) function on Q due to
394
A.P. Balachandran et al. /Magnetic
(3.2) l. The physical interpretation Consider the operator u(e) = exp [i&J,]
monopoles
of this system thus merits some attention,
.
(4.11)
We have [u(e) $1 (r, g) = $(r, ge-‘eos/2)
= $(r, g) eien .
(4.12)
The observables commute with $ and hence with u(e). Unlike $, they are thus exclusively defined in terms of coordinates and velocities x and i [CL (3.15) (3.17) and (3.18)]. It follows that the specification of the observables determines the state only up to an action of V(0). This does not cause difficulties in interpretation since the expectation value of any observable A is insensitive to this ambiguity:
(A A$) = (u(e) ti,Au(e)
+) .
(4.13)
In view of (4.12), it may seem that this ambiguity is like the possibility of multiplying wave functions by a phase in ordinary quantum mechanics. Note, however, that there is an important difference. In the latter, the wave functions are global functions of x, here they are not. [Of course, as for the lagrangian, we can find local wave functions on coordinate neighborhoods of Q. They are related by a gauge transformation on the overlap of these neighborhoods (cf: ref. [3]).] 4.3. Global and local wave functions Our study of the charge-monopole system suggests the following general remarks on the domain of definition of wave functions [3]. Suppose our classical physics is in terms of coordinate and velocities of a base space B (Here B is the analogue of Q). Then the wave functions $Yneed not be globally defined on B. They can, for instance, be defined on a principal fibre bundle E over B with structure group G = {g}. They should of course fulfill a constraint of the form ti(Pg)=%)+(P),
PEE.
(4.14)
Here $ has values in a Hilbert space which carries a unitary representation {cl(g) } of G. If the observables commute with (I(g), then we can interpret observations in this quantum system in terms of observations of the original classical system. When the l
We see from (2.15) that theg(XJ defined in the footnote at the end of sect. 2 approaches different elements of U(1) = {eta3 } as jZ -+ (0, 0, -1). Hence, if J, (r, e”3’) = 0, J, [r, g(X)] is continuous on S* and vanishes at 4 = (0, 0, -1). The usual set D of wave functions on Q with zeros on the Dirac string are of this sort. The Hilbert space H on Q is conventionally defied with a scalar product of the form (3.23) where dM now is the rotationally invariant measure on S*. Since D is dense in H, all wave functions (including those which do not vanish on the Dirac string) can be approximated arbitrarily well (in norm) by functions taken from D. Nonetheless, due to (3.22), J, [r, g(x)] is not well-defined at2 = (0, 0, -1) when $(r, ei03’) + 0, even if $J(r, g) is a smooth
function
of g.
A.P. Balachandran et al. f Magnetic monopoles
395
bundle is trivial [E = B X G] , then it admits a global section (or gauge). Then we can construct global wave functions on B and gain no new physics by working with E. But when E is not a trivial bundle, attempts to write down global wave functions on B may introduce singularities in the theory like the Dirac string. Thus it is more appropriate to work with E. [Alternatively, we can work with local sections (and hence wave functions which are only locally defined) and transition functions [3] .] There are proposals in the literature [ 131 that singular gauge transformations and the singularities they induce in fields are important for a proper understanding of gauge theories. It may be that such singularities can be eliminated and the mathematics made more transparent (as for the charge-monopole system) by formulating the problem in terms of suitable fibre bundles. 4.4. Local and global lagrangians In sect. 1, we explained the meaning of a system with a local lagrangian. We now indicate the relevance of such a system to the preceding discussion. Consider such a system S with lagrangians Lj for coordinate neighborhoods Ui of B (we assume that no Lj leads to a constrained hamiltonian dynamics, that is, to a degenerate symplectic form). The quantization of S has to be performed for each Li separately. It leads to wave functions $i defined over Ui. In the overlap region Ui n Vi, there are also transition functions which are determined by Li - Li. The charge-monopole system is seen to be an example of such an S. The treatment of this system by Wu and Yang [2,3] is along such lines. In some cases, it is possible to regard B as the base manifold of a principal fibre bundle E and find a global lagrangian over E which gives the equations of motion. (This in turn leads to wave functions defined globally on E.) This is what we did for the charge-monopole system. Elsewhere we will discuss other (physically relevant) examples of this sort.
G. Marmo is grateful to the Physics Department hospitality during the spring of 1979.
of Northeastern
University
for
References [l]
V.I. Strazhev and L.M. Tomil’chik, Fiz. Elem. Chastits At. Yad. 4 (1973) Part. Nucl. 4 (1973) 78); P. Goddard and D.I. Olive, Rep. Prog. Phys. 41 (1978) 91. [2] T.T. Wu and C.N. Yang, Phys. Rev. 14 (1976) 437. [3] T.T. Wu and C.N. Yang, Nucl. Phys. B107 (1976) 365; S.J. Avis and C.J. Isham, Cargese lectures (1978), and references therein. [4] J.G. Miller, J. Math. Phys. 17 (1976) 643; D. Maison and S.J. Orfanidis, Phys. Rev. D15 (1976) 3608.
187 (Sov. J.
396
A.P. Balachandran et al. /Magnetic
monopoles
[S] A. Trautman, Int. J. Theor. Phys. 16 (1977) 561; J. Nowakowski and A. Trautman, J. Math. Phys. 19 (1978) 1100. [6] J.L. Friedman and R.D. Sorkin, Phys. Rev. D, to be published. [ 71 A.P. Balachandran, G. Marmo, B.S. Skagerstam and A. Stern, CERN preprint TH27 1 lCERN (1979). [8] D. Rosenbaum, Phys. Rev. 147 (1966) 891. [9] A.P. Balachandran, S. Borchardt and A. Stern, Phys. Rev. D17 (1978) 3247. [lo] P.A.M. Dirac, Lectures on quantum mechanics, Belfer Graduate School of Science Monographs Series no. 2 (Yeshiva University, New York, 1964); E.C.G. Sudarshan and N. Mukunda, Classical dynamics: a modern perspective (Wiley, New York, 1974); A. Hanson, T. Regge and C. Teitelboim, Constrained hamiltonian systems (Accademia Nazionale dei Lincei, Rome, 1976). [ 1 l] J. Talman, Special functions, a group theoretic approach (Benjamin, New York, 1968), p. 92.
[ 121 T.T. Wu and C.N. Yang, Phys. Rev. D12 (1975) 3845. [13]
H.C. Tze and Z.F. Ezawa, Phys. Rev. D14 (1976) 1006; Z.F. Ezawa, Phys. Rev. D18 (1978) 2091; F. Englert, Cargese Lectures (1977); F. Englert and P. Windey, Les Houches Lectures (1978); Nucl. Phys. B135 (1978) 529; Phys. Reports 49 (1979) 173; G. ‘t Hooft, Nucl. Phys. B138 (1978) 1; Utrecht preprint (1979); M. Wadati, H. Matsumoto and H. Umezawa, Phys. Lett. 73B (1978) 448; Phys. Rev. D18 (1978) 1192; H. Matsumoto, P. Sodano and H. Umezawa, Phys. Rev. D19 (1979) 511, and references therein; G. Mack and V.B. Petkova, Desy preprint DESY 78/69 (1978).
MAGNETICMONOPOLESWITHNOSTRINGS A.P. BALACHANDRAN Physics Department,
l)
Syracuse University, Syracuse, New York 13210,
USA
G. MARMO 2, Istituto di Fisica Teorica, Universitb di Napoli, Mostra d’oltremare
Pad. 19, 80125 Napoli,
Italy 3, and Physics Department,
Northeastern
University, Boston, Massachusetts 02215,
USA
B.S. SKAGERSTAM TH Division, CERN, Switzerland
A. STERN l) Physics Department,
Syracuse University, Syracuse, New York 13210,
USA
Received 17 April 1979
Previous treatments of the charge-monopole system have used either a lagrangian singular along the Dirac string or a multiple valued action integral. Here we formulate an action principle for this system which involves no strings and is single valued. This is achieved by writing the lagrangian directly in terms of the variables of a suitable fibre bundle. Canonical quantization is carried out in a simple way and known results are recovered.
1. Introduction In this paper, we study the description
of a charged
non-relativistic
particle
in a
treatment of this system [ 11, the lagrangian is singular along the “Dirac string”. The presence of this unphysical singularity leads to technical difficulties in the quantization of this system. An alternative magnetic
monopole
field. In the customary
l) Supported by the US Department of Energy. 2, Supported by I stituto Nazionale di Fisica Nucleare, Italy. 3, Permanent address. 385
386
A.P. Balachandran et al. /Magnetic
monopoles
approach which avoids the singular string works with a multiple valued action integral [2]. Here we will formulate an action principle which involves no strings and is single valued. We will also quantize the system and recover its known quantum mechanics. No technical difficulties are encountered during the calculation. The lagrangian formalism is treated in sect. 2. In contrast to earlier formulations, the lagrangian L depends on a degree of freedom associated with the gauge group U(1). Under a gauge transformation, L changes by a time derivative of a function. As a result, this gauge degree of freedom does not occur in the equations of motion. The latter are the usual equations. We show however that it is impossible to globally eliminate the gauge degree of freedom from L. In sect. 3, the canonical quantization of L is carried out in a simple and unambiguous way. In sect. 4, which is slightly more technical in nature, the structure and symmetry properties of the system are explained in terms of the fibre-bundle picture. In our approach, the wave functions for the charge monopole system are defined on a nontrivial U(1) bundle over the space of relative coordinates. It is emphasized that (smooth) wave functions of this sort do not in general project down to (smooth) wave functions on the configuration space. Under the action of an element u E U(l), the wave functions on the bundle change by a phase which depends only on u. As a result, it is shown in this section that the expectation value of any observable constructed out of coordinates and velocities is independent of this phase. Thus a physical interpretation becomes possible. We summarize the general nature of such quantum theoretic possibilities where the wave functions are not defined on the configuration space B of observable coordinates. The relation of such quantum systems to classical systems with only local lagrangians over B is discussed. [The system is said to have a local lagrangian over B if there is a lagrangian for each coordinate neighborhood of B. (It will of course depend on velocities as well in this neighborhood.) It has a global lagrangian if there& a single lagrangian for all of B.] The material in much of this section is known to some physicists *. It is presented here for completeness. Some time ago, Miller [4] studied a formulation of the charge-monopole system which has important points of similarity to our work l*. Unlike him, however, we avoid the Kaluza-Klein approach and are much closer to the conventional treatment. Recently, Friedman and Sorkin [6] have independently developed a treatment of this system which is similar to ours. In another paper [7], we will report on the supersymmetric version of the present work. It describes a charged spin-half particle in the field of a magnetic monopole.
l
l See, in this connection, refs. [2,3]. * The connection form he uses is the same as ours. This form is also discussed
in [5]
387
A.P. Balachandran et al. f Magnetic monopoles
2. The lagrangian Let x = (x,, x2, xs) denote the relative coordinate the charge-monopole system. We write
and m the reduced mass of
3 x = r2,
r=
I( C
1
i=
x?)‘/~I.
(2.1)
Let uol be the Pauli matrices and let g E SU(2). [More precisely, (g} is the twodimensional tmitary irreducible representation of SU(2).] Then we parametrize .? in terms of g by setting l ’ X = o&?, =go,g+
(2.2) .
(2.3)
In the lagrangian below, the basic dynamical variables are r and g. So wherever P occurs, it is to be regarded as written in terms of g. The lagrangian is
1
d 2 t ni Tr 03gtg L = $rn z (~-2) [ = irnf2
+ irnr’
Tr X‘2 + ni Tr a3gtg .
(2.4)
Here r and g are functions of the time t and the dot denotes time differentiation. We have used the identity i. k = 0 in writing (2.4). The relation of the constant n -to the product eg of electric and magnetic charges will emerge below. The resemblance of this lagrangian and the lagrangian for a particle in interaction with a Yang-Mills field [9] may be noted. Variation of r leads to mi:=m/..-$2.
(2.5)
Since g E SU(2), its most general variation is induced by an arbitrary element of its Lie algebra: 6g = ia, e,g ,
E, real .
(2.6)
Hence h_t=i[o.e,Tk],
6 Tru3gtQ,=iTru.&.
u . E = a,e, ,
(2.7) w3)
* The charge and the monopole are not allowed to occupy the same position at any given time (see [8,2] in this connection). Hence the point r = 0 is excluded from the set of allowed x.
388
A.P. Balachandran et al. /Magnetic
Variation
monopoles
of g thus leads to
Tr u.
E i
{-ii[$ rnr*k]+n_?} =0
(2.9)
(In deriving this, we have used the identity Tr A [B, C] = Tr B [C, A] .) Here the bracketed expression is a linear combination of Pauli matrices and E, are arbitrary. Therefore
+ni} =0 . $ {-$i[i, rnr*i-] This can be rewritten
%I
eolprx&+)
(2.10)
in the form + nz?,] = 0 .
(2.11)
One recognizes (2.5) and (2.11) as the usual equations system with * --
for the charge-monopole
n=eg.
(2.12)
47r We note that _? and hence x are invariant g
+gei03@
,
But L is not invariant
under the U(1) gauge transformation (2.13)
8 = e(t).
under this transformation:
L+L-2nb
(2.14)
So it is not possible to write L in terms of x and i. Since J? depends on two parameters and g on three parameters, there is an extra gauge degree of freedom present non-trivially in L. It does not affect the equations of motion since the change in L in (2.14) is by the time derivative of a function. We can try to eliminate this additional degree of freedom by fixing a gauge. That is, for each X, we can try to find one group element g(x) with the property i = g($) a,gQ?)t
.
(2.15)
But it is well-known that there is no such g(J.!) which is defined for all _A!and is continuous in i. Thus it is impossible to manipulate L to a global lagrangian which depends only on x. It is of course possible to find a g(% which fails to be continuous only at one point, say at the south pole [i = (0, 0, -l)] . Then L becomes a monopole lagrangian
* With this identification, the equations PLYi = (Zg/4n) Eij@+k/r3.
of motion
can be written
in the form
A.P. Balachandran et al. /Magnetic
monopoles
389
with a string singularity along the negative third axis *. Alternatively, we can cover the two-sphe_re S2 = {i} by two-coordinate patches U, and U2 and find two group elements gj(X) whi!h are defined and continuous on Ui and which fulfill (2.15). Substitution of g&X) in (2.4) leads to lagrangians Li defined on Ui. In the intersection region U, f7 U2, L 1 - L2 is the total time derivative of a function of 2. Such.a formulation is thus the non-relativistic analogue of the work of Wu and Yang [2].
3. Canonical quantization 3.1. The canonical formalism It is useful to have a method to treat group elements for setting up the canonical formalism. Such a method has been developed elsewhere [9]. We recall it briefly here Let g be parametrized by a set of variables < = (El, g2, &) so that g = g(f). The t’s can be the Euler angles. [The functional form of g(l;) is not important for us.] We regard L as a function of r, i, l and .& We first note a preliminary identity. We can write ezO~‘~&I Differentiating
=dH41 ,
t(o) = 5 .
(3.1)
on ea and setting E = 0, we find, (3.2)
(3.3) It is proved in ref. [9] that detN#O.
(3.4)
Now the coordinates ket (PB) relations t&&1=
.& and their conjugate momenta
71, fulfill the Poisson brac-
{%JTp)=O, (3.5)
ts,, np 1 = &p . It follows easily from (3.2) that {J,, g) = i&g {J,,
g-’
) =
,
-ig-‘io,
,
lLetg(~)=~{a!+1/01[03,X]},(~=l[2(1+-P3)~~’~ easy to verify that it is in SU(2) and fulfills (2.15). which is a conventional form of the interaction.
(3.6) I. This g is ill-defined a; f = (0, 0, - 1). It is Also ni Tr o3g(X)ti(X) = ne3$$$(1 + f3),
A.P. Balachandran et al. /Magnetic
390
monopoles
where J, = -fqNpol
.
(3.7)
It may also be shown [9] that
{J,, Jp > = ‘cxavJ, .
(3.8)
Since N is non-singular [eq. (3.4)], we can work with J, instead of the momenta rr,. The PB’s involving J, and g are simple and do not depend on the parametrization we use for g. Below, we will therefore be eliminating rr, in favour of Jol_.Note that in view of (2.3) and (3.6), J, is just the generator of spatial rotations on X. For the lagrangian (2.4), the momentum pr conjugate to r is given by pr=mi,
(3.9)
while IT, is given by IT, = $rnr2 Tr _&,i+ Here a, = a/a&.
ni Tr u3 gt &g .
(3.10)
Hence, by (3.7) and (3.2),
J,=-$zr”
Trk
[ioa,x]
tnTrio,X
= ecvavxp rn$ t n& . From (3.9) and (3.1 l), we see that there is one primary constraint #=&J,
-!I,
(3.11)
l, (3.12)
which vanishes weakly: (3.13)
l#J=O. The Hamiltonian
is
H=p,+n&-L+v@ P2 1 =-.!I_ +[J&J, - n2] t I@, 2m 2mr2
(3.14)
where v is a Lagrange multiplier. The constraint @is rotationally invariant: (4, J, } = 0. Hence (4, H) x 0 and there are no secondary constraints. Since u is arbitrary, we see from (3.14) that only those variables with (weakly) zero PB’s with @have a well-defined time evolution. These are the first-class variables of the theory. Only such variables are of physical interest. We will call them observables.
* Constrained hamiltonian dynamics is discussed in [lo].
A.!‘. Balachandran et al. /Magnetic
One set of observables is the angular momentum
391
monopoles
generators
J C!’
(3.15)
Further from (3.6) (2.2) and (2.3) we find, M g] =
{hJaj g) = it& .
(3.16)
Hence those functions of g which are invariant under the gauge transformation (2.13) are also observables. But these are precisely functions of i. Thus we have also the observables I (3.17) x, . We get a complete set of observables by adjoining to (3.15) and (3.17) the observables (3.18)
C Pr . This set is subject to the one constraint
4 = 0.
3.2. Quantum mechanics The algebra of observables in quantum theory is obtained as usual by identifying the Poisson bracket with t-i) times the commutator bracket. A representation of this algebra can be constructed as follows. Let us ignore the constraint for a moment and regard the wave functions as functions of r and g: (3.19)
II, = NC g)
The coordinates are diagonal in this representation of wave functions in view of (2.2) and (2.3). The momentum P,. acts as the usual differential operator on $J. The operators J, are the differential operators which represent the elements aa, in the left regular representation of SU(2). In particular, [eieJa$]
(r, g) = $(r, e-ie~0~/2g)
We can now take the constraint &J,$
(3.20)
into account by imposing the condition
= nJ/
(3.21)
on the wave functions. +(r
.
ge-ie0312
)
In view of (3.20), (2.2) and (2.3), this means =
NC
s>
eien
The scalar product on the wave functions
@&)~*(cdxk&
(G,x)=Id+0
(3.22)
.
SU(2)
where dp is the invariant measure on SU(2).
is (3.23)
392
A.P. Balachandran et al. /Magnetic
monopoles
Let {Di(g)} be the representation of SU(2) with angular momentum tions $ with finite norm have the expansion [ 1 I]
j. Wave func
(3.24) Here DLO are the matrix elements of Di in the conventional basis with the third component of angular momentum diagonal. The constraint (3.22) means that in (3.24), only those a:, with cr = -n are nonzero. Thus @_,,
fixed n
1,
is a basis for expansions of the form (3.24). Since u is necessarily integral or half-integral, condition
(3.25)
we also find the Dirac quantization
2n = integer .
(3.26)
In (3.25), j and p are half-integral if 2n is odd and integral if 2n is even. The quantum mechanics outlined here is essentially equivalent to conventional treatments. (See also subsect. 4.2 in this connection.)
4. Final remarks Much of the discussion in this section is known to some physicists ded here for completeness.
[3]. It is inclu-
4.1. Fibre bundles, symmetries Let Q denote the set {x } of relative coordinates Then
with the origin x = 0 removed.
Q= R’ XS2, R’ = {rlr>O}, s2= {a}.
(4.1)
The two-sphere S2 admits non-trivial U(1) bundles. It is known that the description of magnetic monopoles requires such bundles [ 121. They can be constructed as follows: regard SU(2) as a U( 1) bundle over S2 where we take the action of U( 1) to be g +ge”@
.
Hence the projection g ego3g+
(4.2) map to S2 is
=x .
(4.3)
A.P. Balachandran et al. /Magnetic
monopoles
393
Now the cyclic group Z,
= {zk = eios2nk’Mlk = 0, 1, 2, . . .. M - 1 } ,
M>l,
(4.4)
has an action (4.5)
g+gk
which commutes with the projection go,g+
= bk)
03(gzk)+
(4.3): (4.6)
The quotient L,,,, of SU(2) by this action of ZM is thus a U(1) bundle over S*. A function on LM can be regarded as a function f on SU(2) which is Zw invariant: fkzk)
(4.7)
=f@.
The lagrangian L in (2.4) is written on (the tangent bundle of) R’ X SU(2). As explained in sect. 2, it can not be globally written in terms of x and i. In view of (3.22) and (4.7), the wave functions of the charge-monopole system for a given n can be regarded as wave functions on R’ X L12,1. (They can in general be regarded as wave functions on any R’ X LM such that 2n/M is an integer.) We note that the action, g+gog,
g? go E SU(2) 9
(4.8)
of SU(2) on itself corresponds to a spatial rotation of x by (2.3). This action commutes with that of Z,,,, and hence is well defined on LM. Thus we have a natural lift of spatial rotations from S* to LM. There is a similar lift of spatial reflection defined by g+gp, P= exp[iino,]
.
(4.9)
Clearly under P, i--i.
(4.10)
[Note that such a lift is not unique since for instance ur can be replaced by cos Bar + sin 00~ without changing (4.10).] Since PZMP-’ = ZM, spatial reflection is also well-defined on L,,,. However, the lagrangian is not invariant under g + gP if n f 0. So, as is known, this system violates P if n # 0. 4.2. Physical interpretation For n # 0, a (continuous) wave function J, on R’ X SU(2) (just as the lagrangian L) can not in general be written as a (global, continuous) function on Q due to
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A.P. Balachandran et al. /Magnetic
(3.2) l. The physical interpretation Consider the operator u(e) = exp [i&J,]
monopoles
of this system thus merits some attention,
.
(4.11)
We have [u(e) $1 (r, g) = $(r, ge-‘eos/2)
= $(r, g) eien .
(4.12)
The observables commute with $ and hence with u(e). Unlike $, they are thus exclusively defined in terms of coordinates and velocities x and i [CL (3.15) (3.17) and (3.18)]. It follows that the specification of the observables determines the state only up to an action of V(0). This does not cause difficulties in interpretation since the expectation value of any observable A is insensitive to this ambiguity:
(A A$) = (u(e) ti,Au(e)
+) .
(4.13)
In view of (4.12), it may seem that this ambiguity is like the possibility of multiplying wave functions by a phase in ordinary quantum mechanics. Note, however, that there is an important difference. In the latter, the wave functions are global functions of x, here they are not. [Of course, as for the lagrangian, we can find local wave functions on coordinate neighborhoods of Q. They are related by a gauge transformation on the overlap of these neighborhoods (cf: ref. [3]).] 4.3. Global and local wave functions Our study of the charge-monopole system suggests the following general remarks on the domain of definition of wave functions [3]. Suppose our classical physics is in terms of coordinate and velocities of a base space B (Here B is the analogue of Q). Then the wave functions $Yneed not be globally defined on B. They can, for instance, be defined on a principal fibre bundle E over B with structure group G = {g}. They should of course fulfill a constraint of the form ti(Pg)=%)+(P),
PEE.
(4.14)
Here $ has values in a Hilbert space which carries a unitary representation {cl(g) } of G. If the observables commute with (I(g), then we can interpret observations in this quantum system in terms of observations of the original classical system. When the l
We see from (2.15) that theg(XJ defined in the footnote at the end of sect. 2 approaches different elements of U(1) = {eta3 } as jZ -+ (0, 0, -1). Hence, if J, (r, e”3’) = 0, J, [r, g(X)] is continuous on S* and vanishes at 4 = (0, 0, -1). The usual set D of wave functions on Q with zeros on the Dirac string are of this sort. The Hilbert space H on Q is conventionally defied with a scalar product of the form (3.23) where dM now is the rotationally invariant measure on S*. Since D is dense in H, all wave functions (including those which do not vanish on the Dirac string) can be approximated arbitrarily well (in norm) by functions taken from D. Nonetheless, due to (3.22), J, [r, g(x)] is not well-defined at2 = (0, 0, -1) when $(r, ei03’) + 0, even if $J(r, g) is a smooth
function
of g.
A.P. Balachandran et al. f Magnetic monopoles
395
bundle is trivial [E = B X G] , then it admits a global section (or gauge). Then we can construct global wave functions on B and gain no new physics by working with E. But when E is not a trivial bundle, attempts to write down global wave functions on B may introduce singularities in the theory like the Dirac string. Thus it is more appropriate to work with E. [Alternatively, we can work with local sections (and hence wave functions which are only locally defined) and transition functions [3] .] There are proposals in the literature [ 131 that singular gauge transformations and the singularities they induce in fields are important for a proper understanding of gauge theories. It may be that such singularities can be eliminated and the mathematics made more transparent (as for the charge-monopole system) by formulating the problem in terms of suitable fibre bundles. 4.4. Local and global lagrangians In sect. 1, we explained the meaning of a system with a local lagrangian. We now indicate the relevance of such a system to the preceding discussion. Consider such a system S with lagrangians Lj for coordinate neighborhoods Ui of B (we assume that no Lj leads to a constrained hamiltonian dynamics, that is, to a degenerate symplectic form). The quantization of S has to be performed for each Li separately. It leads to wave functions $i defined over Ui. In the overlap region Ui n Vi, there are also transition functions which are determined by Li - Li. The charge-monopole system is seen to be an example of such an S. The treatment of this system by Wu and Yang [2,3] is along such lines. In some cases, it is possible to regard B as the base manifold of a principal fibre bundle E and find a global lagrangian over E which gives the equations of motion. (This in turn leads to wave functions defined globally on E.) This is what we did for the charge-monopole system. Elsewhere we will discuss other (physically relevant) examples of this sort.
G. Marmo is grateful to the Physics Department hospitality during the spring of 1979.
of Northeastern
University
for
References [l]
V.I. Strazhev and L.M. Tomil’chik, Fiz. Elem. Chastits At. Yad. 4 (1973) Part. Nucl. 4 (1973) 78); P. Goddard and D.I. Olive, Rep. Prog. Phys. 41 (1978) 91. [2] T.T. Wu and C.N. Yang, Phys. Rev. 14 (1976) 437. [3] T.T. Wu and C.N. Yang, Nucl. Phys. B107 (1976) 365; S.J. Avis and C.J. Isham, Cargese lectures (1978), and references therein. [4] J.G. Miller, J. Math. Phys. 17 (1976) 643; D. Maison and S.J. Orfanidis, Phys. Rev. D15 (1976) 3608.
187 (Sov. J.
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[S] A. Trautman, Int. J. Theor. Phys. 16 (1977) 561; J. Nowakowski and A. Trautman, J. Math. Phys. 19 (1978) 1100. [6] J.L. Friedman and R.D. Sorkin, Phys. Rev. D, to be published. [ 71 A.P. Balachandran, G. Marmo, B.S. Skagerstam and A. Stern, CERN preprint TH27 1 lCERN (1979). [8] D. Rosenbaum, Phys. Rev. 147 (1966) 891. [9] A.P. Balachandran, S. Borchardt and A. Stern, Phys. Rev. D17 (1978) 3247. [lo] P.A.M. Dirac, Lectures on quantum mechanics, Belfer Graduate School of Science Monographs Series no. 2 (Yeshiva University, New York, 1964); E.C.G. Sudarshan and N. Mukunda, Classical dynamics: a modern perspective (Wiley, New York, 1974); A. Hanson, T. Regge and C. Teitelboim, Constrained hamiltonian systems (Accademia Nazionale dei Lincei, Rome, 1976). [ 1 l] J. Talman, Special functions, a group theoretic approach (Benjamin, New York, 1968), p. 92.
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