Download Geometry of the Magnetic Monopole M. K. Fung The seminal paper

CHINESE JOURNAL OF PHYSICS
VOL. 49, NO. 4
August 2011
Geometry of the Magnetic Monopole
M. K. Fung
Department of Physics, National Taiwan Normal University,
Taipei, Taiwan 116, R.O.C.
(Received October 25, 2010)
It is shown that the vector potential of a magnetic monopole can be obtained from consideration of the Berry phase in a new perspective. The two-component complex wavefunction
can be characterized by a single CP1 projective coordinate w on the projected compactified
complex plane. So the vector potential on the sphere is expressed solely in terms of w. The
projective structure is more evident in this consideration. We also attempt to demonstrate
that the wavefunctions can be coupled to yield a monopole with a doubling of the magnetic
charge in a simple case. Generalization from a complex phase to a quaternionic phase is
discussed.
PACS numbers: 14.80.Hv
I. PHASE FACTOR AND THE MAGNETIC MONOPOLE
The seminal paper of Dirac [1] had established the importance of the concept of
the non-integrable phase factor in the modern theory of the magnetic monopole. The
non-integrable phase factor is relevant in quantum mechanics since the description of the
complex wavefunction in quantum mechanics is unique except for a complex phase factor.
When the phase factor is made to be space-time dependent we are then in the realm of the
theory of abelian gauge fields, i.e., Maxwell fields.
The magnetic monopole is spherical symmetric. If we define the vector potential Aµ
over the sphere, the vector potential is bound to have a singularity. This implies that we
cannot cover the coordinates on the sphere with a single patch of a coordinate system.
Such was the idea of Wu and Yang [2], who introduced the fibre bundle concept into the
theory of the magnetic monopole where transition functions are utilized to sew up different
patches of the coordinate systems. In this viewpoint it is possible to work out the singular
vector potential [3, 4] in the context of the Hopf map.
On the other hand a time varying Hamiltonian gives rise to the existence of a Berry
phase for the time varying wavefunction [5, 6]. There is a neat way to get the vector
potential of the magnetic monopole via the Berry phase formula. We present the details of
such an argument since this is relevant for our subsequent exposition.
Consider a matrix H of the form
(
)
x3
x1 − ix2
H=
,
(1)
x1 + ix2 −x3
where x21 + x22 + x23 = 1. The coordinates x1 , x2 , and x3 lie on the unit sphere. We can
http://PSROC.phys.ntu.edu.tw/cjp
877
c 2011 THE PHYSICAL SOCIETY
⃝
OF THE REPUBLIC OF CHINA
878
GEOMETRY OF THE MAGNETIC MONOPOLE . . .
VOL. 49
write the matrix as
H=
3
∑
xi σi ,
(2)
i=1
where σi are the three Pauli matrices. People usually look upon H as the Hamiltonian of a
time varying unit-strength magnetic field interacting with the spin. Rather, we would look
upon this representation as a Dirac way of taking the square root of the sphere. In this
way, it is readily generalized to higher dimensional analogues.
The eigenvalue problem
H|ψ⟩ = λ|ψ⟩ ,
(3)
admits two eigenvalues of ±1. As the coordinates of the sphere evolve with time the
eigenvalue remains unchanged. This is a kind of isospectral problem. We work with the
λ = +1 eigenvalue. The corresponding normalized eigenvector is
 √

|N ⟩+ = 
1+x3
2

√x1 +ix2
2(1+x3 )
.
(4)
Notice that the wavefunction is singular at x3 = −1, i.e., it is singular along the −x3 axis.
In terms of the spherical coordinates (sin θ cos ϕ, sin θ sin ϕ, cos θ) the normalized eigenvector
takes the form
(
)
cos 2θ
|N ⟩+ =
.
(5)
sin 2θ eiϕ
Using a simple Berry-phase argument we can obtain the vector potential one-form A as
iA =+⟨N |d|N ⟩+ ,
(6)
and the normalization of the wavefunction ensures that A is real. Employing the above
wavefunction explicitly we get
θ
A = sin2 dϕ .
2
(7)
Taking the exterior derivative once more, we get the field strength two-form F as
F = dA =
1
sin θ dθdϕ, ,
2
which integrates over the sphere to give 2π.
We can get another gauge equivalent wavefunction
(
)
cos 2θ e−iϕ
′
|N ⟩+ =
,
sin 2θ
(8)
(9)
VOL. 49
M. K. FUNG
879
which is singular along the +x3 axis giving
θ
A′ = − cos2 dϕ .
2
(10)
The difference A′ − A is a closed one-form, i.e.
A − A′ = dϕ .
(11)
We regard the vector potential as that from a basic magnetic charge of strength g. If we
couple the magnetic vector potential to an electric charge, and require that the phase for
dϕ is unobservable for a total of angle of 2π we would have
eg
= 1.
2π
(12)
This is just the Dirac quantization condition.
From the above consideration we see that we start with a basic unit of the monopole.
The half-string singularity comes as a consequence. It would be nice to examine an alternative way of calculating the monopole vector potential [8], in which we seek for a solution
with a spherical symmetric monopole vector potential, which is possible only up to modulo
a gauge transformation. The half-line singularity must be implemented in order to get the
Dirac quantization condition.
Now we turn to consider the λ = −1 eigenvalue. Note that the eigenvectors for
different eigenvalues are orthogonal. In explicit spherical coordinate form the normalized
eigenvector is
(
)
− sin 2θ e−iϕ
|N ⟩− =
,
(13)
cos 2θ
which has a string of singularity along the −x3 axis. Another gauge equivalent form, with
the half string singularity along the x3 axis is given by
(
)
sin 2θ
′
|N ⟩− =
.
(14)
− cos 2θ eiϕ
As expected this yields a magnetic monopole with the opposite magnetic charge.
II. PROJECTIVE STRUCTURE
Now we are in a position to study the Berry-phase formula of the magnetic monopole
from a new perspective. The wavefunction is indeed defined only up to a complex number
multiplication. Although we have a two-component wavefunction the physics only depends
on the complex projective coordinates. Since a complex number has magnitude and phase,
880
GEOMETRY OF THE MAGNETIC MONOPOLE . . .
VOL. 49
the gauge field is for the phase, and there is a scale invariance for the magnitude. Mathematically, it is convenient to write the two-component wavefunction as


1
|P⟩+ =  x1 + ix2  .
(15)
1 + x3
We can employ the stereographic projection of the Riemman sphere (x1 , x2 , x3 ) to the
complex plane w from the south pole with the formula
w=
x1 + ix2
.
1 + x3
(16)
This stereographic projection gives
w̄w =
1 − x3
.
1 + x3
(17)
A useful mathematical entity is the chordal distance ξ(w, w′ ) which relates the Euclidean
distance |w−w′ | on the complex plane to the chordal distance between the two corresponding
points on the Riemann sphere. Geometrically [7] we can derive the following formula
2|w − w′ |
√
.
ξ(w, w′ ) = √
1 + w̄w 1 + w̄′ w′
(18)
The infinitesimal chordal distance can be obtained by taking the limit of the above expression or by directly computing the infinitesimals as
dx21 + dx22 + dx23 =
4|dw|2
.
(1 + w̄w)2
We can write the wavefunction in terms of the stereographic coordinates as
( )
1
|P⟩+ =
.
w
(19)
(20)
Note that this wavefunction has a singularity at w = ∞, i.e., at the south pole. In mathematical parlance, for the complex projective space CP1 the projective coordinate w characterizes a magnetic monopole configuration. The wavefunction |P⟩+ is not normalized, and
we need a modified formula [9] for the Berry-phase:
iA =
+⟨P|d|P⟩+
− (d+⟨P|)|P⟩+
.
2+⟨P|P⟩+
(21)
The right hand side is explicitly an imaginary expression, and this eases our calculation,
since we can just compute the imaginary contribution. Carrying out the calculation, we
get in this representation the vector potential one-form A as
A=
w̄dw − (dw̄)w
.
2(1 + w̄w)
(22)
VOL. 49
M. K. FUNG
881
From the vector potential formula we see that the coordinate w and eiδ w, where eiδ is a
constant phase, will yield the same vector potential. In particular w and −w will give an
identical vector potential. Taking the exterior differentiation once more we get the field
strength two-form F as
F =
dw̄dw
.
(1 + w̄w)2
(23)
This is just half the infinitesimal chordal area on the Riemann sphere. So this is just a
restatement of the result obtained using spherical coordinates.
We can take the other solution
(
)
1/w
|P ′ ⟩+ =
,
(24)
1
which is singular at w = 0, i.e., at the north pole. Hence the coordinates w and 1/w will
yield gauge equivalent vector potentials. The gauge equivalent vector potential one-form A
is
−(1/w)dw + (1/w̄)dw̄
,
(25)
A=
2(1 + w̄w)
giving the same field strength two-form Eq. (21).
For the oppositely charged monopole we prefer to write the wavefunction as
)
(
) (
1
1
w
|P⟩− =
=
,
−
−1/w̄
w̄w
which is singular along the x3 axis, or we can take
(
) ( w̄w )
−w̄
−
′
,
|P ⟩− =
=
w
1
1
(26)
(27)
which is singular at the −x3 axis. Thus essentially the coordinates w and w̄ will yield
1 − x3
w
magnetic monopoles with opposite sign. As w̄w =
, we can readily see that
is
1 + x3
w̄w
just the coordinate of the stereographic projection from the north pole.
III. COMPOSITION OF MONOPOLES
In this section we would like to demonstrate how to compose two wavefunctions to
get “higher spin” magnetic monopoles. The natural way is to take the direct product of
the the basic wavefunctions, adopting the symmetrized wavefunction. For instance, we take
the symmetric direct product of two |P⟩+ to get


1
√
(28)
|ψ⟩++ =  2w .
w2
882
GEOMETRY OF THE MAGNETIC MONOPOLE . . .
VOL. 49
After some reduction, the resulting vector potential one-form can be calculated to be
A=
w̄dw − dw̄w
,
1 + w̄w
which has double the magnetic charge.
On the other hand we can combine one |P⟩+ and one |P ′ ⟩+ to get


1/w
√
|ψ ′ ⟩++ =  2  .
w
(29)
(30)
The resulting vector potential one-form is
iA =
(w̄w − 1)(dw/w − dw̄/w̄)
,
2(1 + w̄w)
(31)
which is singular along the whole x3 axis. In spherical coordinates this vector potential
one-form becomes
A = − cos θ dϕ .
(32)
Schwinger [10] had discussed such a vector potential while stressing that there is a singularity string on the whole x3 axis. In our way we suppose that the Dirac monopole is
more fundamental. Higher spin composition can be effectively carried out via the Majorana way [11, 12]. It should be noticed that the construction of monopole harmonics via
composition [13] is a consequence of the above concept.
IV. DISCUSSION
There are various ways to contemplate the physics of the magnetic monopole, for
example, the non-integrable phase, the fibre bundle formulation, the spherical symmetry
Lie derivative method, or the Berry phase, etc. Of these the Berry phase consideration
starts with a basic two-component wavefunction. Hence the minimal Dirac quantization is
built into the theory.
The usual approach in the Berry phase study is to start with a normalized wavefunction and proceed with the calculation of the vector potential. In this paper we give
another way of going with the unnormalized wavefunction. We specify one component of
the two-component wavefunction to be unity, while the other component will be described
by a projective coordinate on the compactified complex plane, stereographically projected
from the Riemann sphere. In this way the projective structure will be made evident in this
viewpoint. We find that this coordination offers easier calculation. The identification of
a gauge equivalent vector potential is through the projective invariance of the wavefunction. We also demonstrate that a simple coupling of the wavefunctions yields the expected
doubling of the magnetic charge.
VOL. 49
M. K. FUNG
883
The magnetic monopole problem can be generalized by replacing the abelian complex
phase with an nonabelian SU (2) phase, i.e., instead of considering the complex variable
we can go for the quaternion analogue. So we have a Yang monopole [14] with SO(5)
symmetry. The projected configuration is just the instanton solution [15] in Euclidean 4
dimensions. It is interesting to note that the instanton solution was first discovered via
its self-dual and anti-self-dual property. Of course, on the projected plane we have the
conformal symmetry rather than the spherical symmetry on the sphere. There are multiinstanton solutions [16], and it would be a rather cumbersome task to unravel the symmetry
involved with the concept of wavefunction coupling.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
P. A. M. Dirac, Proc. Roy. Soc. A133, 60 (1931).
T. T. Wu and C. N. Yang, Phys. Rev. D12, 3845 (1975).
M. Minami, Prog. Theor. Phys. 62, 1128 (1978).
L. H. Ryder, J. of Phys. A13, 437 (1980).
M. V. Berry, Proc. Roy. Soc. A392, 45 (1984).
B. Simon, Phys. Rev. Lett. 51, 2167 (1983).
G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable, Vol.
1, (P. Noordhoof Ltd., Groningen, 1960).
M. K. Fung, Chin. J. Phys. 29, 191 (1991).
B. Zumino, Preprint UCB/PTH-87/13, LBL-23056 (1987).
J. Schwinger, Phys. Rev. 144, 1087 (1966).
E. Majorano, Nuovo Cimento 9, 43 (1932).
R. Penrose, Shadows of the Mind, (Cambridge Univ. Press, Cambridge, 1984).
M. K. Fung, Chin. J. Phys. 40, 490 (2002).
C. N. Yang, J. Math. Phys. 19, 320 (1978).
A. A. Belavin, A. M. Polyakov, A. S. Schwartz, and Yu. S. Tyupkin, Phys. Lett. 59B, 85 (1975).
M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld, and Yu. I. Manin, Phys. Lett. 65A, 185 (1978).