Path Integral Quantization of the Aharonov–Bohm–Coulomb
System in Momentum Space
De-Hone Lin1
Institute of Electro-Physics, National Chiao Tung University, Hsinchu 30043, Taiwan
Received November 13, 2000; revised February 5, 2001
The Coulomb system with a charge moving in the fields of Ahanorov and Bohm is quantized via path integral in momentum space. Due to the dynamics of the system in momentum space being in curve space, our result not only gives the Green function of this interesting system in momentum space but provides the second example to answer an open problem of quantum dynamics in curved spaces posed by DeWitt in 1957: We find that the physical Hamiltonian in curved spaces does not contain the Riemannian scalar curvature R. °C2001 Academic Press
I. INTRODUCTION
Half a century ago, Feynman proposed the method of the path integral to describe quantum dynam-ics [1, 2]. It provides us with a global approach for studying quantum dynamdynam-ics via fluctuating paths. Feynman’s method has been successfully applied to diverse areas of physics [3, 4]. Nevertheless, so far almost all the exact results for the dynamic problems of particles moving in the external potentials given by the path integral are obtained in position space [5, 6]. In this paper, we calculate the path integral of the Aharonov–Bohm–Coulomb (A-B-C) system in momentum space. Our procedures can serve a further example for treating quantum dynamics in momentum space by path integrals. Because the dynamics of the system in momentum space is in curved space, it may give an answer to an old question in quantum dynamics in curved spaces proposed by DeWitt in 1957: The Hamil-tonian in curved spaces should not contain an additional term of the Riemannian scalar curvature R [7]. Otherwise, the level spacing of the energy spectra will be changed. The phenomenon was first observed by Kleinert in Ref. [8] who pioneered the treatment of the Coulomb system in momentum space. Here we provide the second example to confirm his result.
II. PATH INTEGRAL QUANTIZATION OF THE A-B-C SYSTEM IN MOMENTUM SPACE With the space-time transformation, a stable path integral representation for the quantum Green function of a charge particle moving in the external electromagnetic fields from xa to xbis given by
[8, 10, 11] G(xb, xa; E)= Z ∞ 0 d S Z D f (λ)8 [ f (λ)] Z D3x(λ) exp{−A E[x, ˙x]} (2.1) 1E-mail: [email protected]. 95 0003-4916/01 $35.00
Copyright°C2001 by Academic Press All rights of reproduction in any form reserved.
with the classical action AE[x, ˙x] = Z λb λa dλ · ˙x2(λ) 2 f (λ)− ieA(x) · ˙x(λ) + f (λ)(V (x) − E) ¸ , (2.2) where S is defined as S = Z λb λa dλf (λ), (2.3)
in which f (λ) is an arbitrary dimensionless fluctuating scale variable, and 8[ f (λ)] is some convenient gauge-fixing functional. The only condition on8[ f (λ)] is that
Z
D f (λ)8[ f (λ)] = 1. (2.4) In Eq. (2.1), natural units with h-= c = m = 1 are used. We see that if 8[ f (λ)] is taken as the delta functionalδ[ f − 1], the representation reduces to Feynman’s original path integral of a relativistic particle. The path integral representation arises from the continuous limit of theλ-sliced expression
G(xb, xa; E )≈ Z ∞ 0 d S N Y n=0 · Z d fn8( fn) ¸ 1 (2π²bfb)3/2 N Y n=1 · Z ∞ −∞ d3xn (2π²nfn)3/2 ¸ exp©−ANEª (2.5) with theλ-sliced action
AN E = N X n=0 · (xn− xn−1)2 2²nfn − ieA(xn)· (xn− xn−1)+ ²nfn(V (xn)− E) ¸ , (2.6)
where²n = λn− λn−1,λb = λN,λa = λ0, xa = x(λ0), and xb = x(λN). In the A-B-C potential
problem that we consider, the vector and scalar potentials are defined as
A(x)= 2g−yˆex+ x ˆey
x2+ y2 , V (r )= −
α
r, (2.7)
where ˆex,ystand for the unit vector along the x, y axis. Let’s first analyze the influence of A-B effect
on the G(xb, xa; E). Introducing the azimuthal angle around the A-B tube
ϕ(x) = arctan(y/x), (2.8)
the components of the vector potential can be expressed as
Ai = 2g∂iϕ(x). (2.9)
The associated magnetic field lines are confined to an infinitely thin tube along the z-axis,
where x⊥ stands for the transverse vector x⊥ = (x, y). Note that the derivatives in front of ϕ(x) commute everywhere, except at the origin where Stokes’ theorem yields
Z
d2x(∂x∂y− ∂y∂x)ϕ(x) =
I
dϕ = 2π. (2.11)
The magnetic flux through the tube is defined by the integral
Ä =
Z
d2x B3. (2.12)
This shows that the coupling constant g is related to the magnetic flux by
g = Ä
4π. (2.13)
Inserting Ai = 2g∂iϕ(x) into the action of Eq. (2.2), the magnetic interaction takes the form
Amag= iµ0
Z S
0
dλ ˙ϕ(λ), (2.14)
whereϕ(λ) = ϕ(x(λ)), ˙ϕ = dϕ/dλ, and µ0is the dimensionless number
µ0= −2eg. (2.15)
The minus sign is a matter of convention. Since the particle orbits are present at all times, their worldlines in space-time can be considered as being closed at infinity, and the integral
k= 1
2π
Z S
0
dλ ˙ϕ(λ) (2.16)
is the topological invariant with integer values of the winding number k. The magnetic interaction is therefore purely topological; its value is given by
e−Amag= ei e Rλb
λadλA(x)·˙x(λ) = e−iµ02kπ. (2.17)
Due to the nature of topological interaction, the influence of the A-B effect may be considered after the dynamics of the Coulomb system in the momentum space is carried out. To perform the path integral of the Coulomb system in the momentum space, we note that the representation of the path integral in Eq. (2.1) has the phase space version
G(pb, pa; E )= Z ∞ 0 d S Z D f 8[ f ] Z D3p 2π Z D3x exp{−A [p, x, f ]}, (2.18)
where the action
A [p, x, f ] =
Z S
0
withH is the classical Hamiltonian. For the Coulomb system under consideration, H = p2/2−α/r.
In order to obtain a stable path integral in this case, the gauge fixing functional8[ f ] is chosen as [8]
8[ f ] ≈ N Y n=0 · 1 rn ¸ exp ( − N X n=0 ²n 2r2 n · fn− x2n µ p2n 2 − E ¶¸2) . (2.20)
With the functional measure
Z D f ≈ N Y n=0 ·Z ∞ −∞ d fn √ 2π/²n ¸ , (2.21)
the gauge conditionRD f 8[ f ] = 1 is automatically satisfied. Combining Eq. (2.20), we have the action in the path integral
A[p, x, f ] = Z S 0 dλ " i ˙p· x +1 2x 2 µ p2 2 − E ¶2 + f2 2r2 − fα r # . (2.22)
We see that the path integrals over f and x in this equation are Gaussian and can be performed. The time-sliced path integral in f gives us a factor
exp ( N X n=0 ²nα2 2 ) . (2.23)
The integrand of the time-sliced path integral associated with x reads now
N Y n=0 23(2π/² n)3/2 ¡ p2 n+ κ2 ¢3 exp ( −1 2 N X n=0 (24pn)2 ²n ¡ p2 n+ κ2 ¢2 ) , (2.24)
where we have definedκ2≡ −2E. If the measure of the path integral is defined as in Dewitt’s paper
[7], the path integral becomes the form
G(pb, pa; E)≈ Z ∞ 0 d S 2 3(2π)3 (2π²a)3/2 ¡ p2 a+ κ2 ¢3 N Y n=1 " Z ∞ −∞ 23d3pn (2π²n)3/2 ¡ p2 n+ κ2 ¢3 # exp©− ANEª. (2.25) The sliced action reads
AN E = N X n=0 " 1 2 (24pn)2 ²n ¡ p2 n+ κ2 ¢2 − ²nα2 2 # . (2.26)
Here we perform the stereographic projection onto a unit sphere in the four dimensions
z= 2κp
p2+ κ2, z=
p2− κ2
p2+ κ2, z
Then, Eq. (2.26) becomes AN E[ˆz]= N X n=0 · 1 2 (4ˆzn)2 ²nκ2 −²nα2 2 ¸ (2.28)
and the measure in Eq. (2.26),
Z ∞ 0 d S (2π) 3 (2πκ2² a)3/2 N Y n=1 · Z ∞ −∞ d4ˆzn (2πκ2² n)3/2 ¸ , (2.29)
where ˆz= (z, z) is the four-dimensional unit vector. By defining a pseudomass µ = 1/κ2, we find
the momentum path integral
G(pb, pa; E)= (2π)3
Z ∞
0
d S K (pb, pa; S) (2.30)
with the time sliced pseudo-propagator
K (pb, pa; S)≈ 1 (2π²a/µ)3/2 N Y n=1 · Z ∞ −∞ d4ˆzn (2π²n/µ)3/2 ¸ exp©−ANE[ˆz]ª. (2.31)
This would be the path integral of DeWitt. It should not lead to the correct spectrum for this problem. However, it was shown by Kleinert [5] that the measure of path integrals in curved space is not simply the product of invariant integralsQnRdqn
√
g(qn). By performing a non-holonomic mapping from
flat to curved space, he found that the measure receives a contribution explicitly,
Aeff= −
Z S
0
dλ R
6µ (2.32)
in which R is the Riemannian scalar curvature. For a sphere of radius r in D dimensional space, R is given by (D− 1)(D − 2)/r2, so that the measure correction for an unit sphere in four-dimensional
space gives a contribution
e−Aeff = e RS
0dλR/6µ= e RS
0dλ1/µ= eκ2S. (2.33)
The path integral (2.31) had been solved in Refs. [5, 9] by
K (pb, pa; S)= ∞ X n=1 n2 2π2Pn(cosϑ)e −(κ2n2−α2)S/2 , (2.34)
where Pn(cosϑ) = sin nϑ/n sin ϑ with ϑ being the angular between the four vectors ˆzband ˆza:
cosϑ = ˆzb· ˆza= ¡ pb2− κ2¢¡p2a− κ2¢+ 4κ2pb· pa ¡ p2 b+ κ2 ¢¡ p2 a+ κ2 ¢ . (2.35)
We now complete the integration on S in Eq. (2.30) and obtain the exact Green function of the Coulomb system in momentum space
G(pb, pa; E)= (2π)3 ∞ X n=1 n2 2π2Pn(cosϑ) 2 2E n2+ α2. (2.36)
We see that the poles display the correct energy levels of hydrogen spectra
En = −
α2
2n2, n= 1, 2, 3, . . . . (2.37)
This path integral derivation was first given by Kleinert in [8].
Here is the place that we can go over to the A-B-C system. Let us define the new quantum number
nr = n − 1 and rewrite (2.36) as G(pb, pa; E)= (2π)3 ∞ X nr=0 (nr + 1) 2π2 2 2E (nr + 1)2+ α2 Cl1(cosϑ), (2.38) where C1
l(cosϑ) = sin(nr + 1)ϑ/ sin ϑ are the Gegenbauer polynomials [12, p. 218]. By defining
the new variables
u= ¡ pb2− κ2¢ ¡ p2 b+ κ2 ¢, v = ¡ pa2− κ2¢ ¡ p2 a+ κ2 ¢, (2.39)
with the addition theorem of the Gegenbauer polynomials [12, p. 223], C1
l(cosϑ) has the expansion
Cn1 r(cosϑ) = C 1 nr ¡ u· v + (1 − u2)1/2(1− v2)1/2cos4ϑ¢ = nr X l=0 4l(2l+ 1)0(nr− l + 1)02(l+ 1) 0(nr + l + 2) × (1 − u2)l/2(1− v2)l/2Cl+1 nr−l(u)C l+1 nr−l(v)Pl(cos4ϑ), (2.40)
where4ϑ is the angular between the unit vectors ˆpb, ˆpain the momentum space. Since the Legendre
function has the expansion into the spherical harmonic
Pl(cos4ϑ) = 4π (2l+ 1) l X k=−l Ylk( ˆpb)Ylk∗( ˆpa), (2.41)
the exact Green’s function becomes
G(pb, pa; E)= (2π)3 ∞ X nr=0 nr X l=0 l X k=−l (nr+ 1) 2π2 2 2E(nr+ 1)2+ α2 ×4l(2l+ 1)0(nr − l + 1)0(l + k + 1)0(l − k + 1) 0(nr+ l + 2) × (1 − u2)l/2(1− v2)l/2Cl+1 nr−l(u)C l+1 nr−l(v)P (k,k) l−k (cos2b)P (k,k) l−k (cos2a)
Here, (2, 8) are the angular of the polar coordinates of the total momentum vector in momentum space, and the relation between the associated Legendre polynomial Pk
l (x) and the Jacobi function
P(k,k) n (x) [10] Plk(cos2) = (−1)k0(1 + k + l) 0(1 + l) (cos2/2 sin 2/2) kP(k,k) l−k (cos2), (2.43)
has been used for obtaining Eq. (2.42) via
Ylk( ˆp)= Ylk(2, 8) = (−1)k · 2l+ 1 4π (l− k)! (l+ k)! ¸1/2 Plk(cos2)ei k8. (2.44)
To proceed, we change summation indices by defining nr− l = n, and a further change of the index
of summation replaces l with q by defining l− k = q. We obtain from (2.42)
G(pb, pa; E)= (2π)3 ∞ X n=0 ∞ X q=0 ∞ X k=−∞ (n+ q + k + 1) 2π2 2 2E (n+ q + k + 1)2+ α2 ×4q+k[2(q+ k) + 1]0(n + 1)0(q + 1)0(q + 2k + 1) 0(n + 2(q + k) + 2) × (1 − u2)(q+k)/2(1− v2)(q+k)/2Cq+k+1 n (u)C q+k+1 n (v)P (k,k) q (cos2b)Pq(k,k)(cos2a)
× (cos 2b/2 cos 2a/2 sin 2b/2 sin 2a/2)k· ei k(8b−8a). (2.45)
We are now prepared to include the influence of the A-B potential in (2.17). With the help of Poisson’s summation formula [5, p. 124], ∞ X k=−∞ f (k)= Z ∞ −∞d y ∞ X n=−∞ e2πnyif (y), (2.46)
we obtain for the Green function of the A-B-C system in momentum space
G(pb, pa; E )= (2π)3 ∞ X n=0 ∞ X q=0 ∞ X k=−∞ (n+ q + |k + µ0| + 1) 2π2 2 2E (n+ q + |k + µ0| + 1)2+ α2 ×4(q+|k+µ0|)[2(q+ |k + µ0|) + 1]0(n + 1)0(q + 1)0(q + 2|k + µ0| + 1) 0(n + 2(q + |k + µ0|) + 2) × µ 2κpb p2 b+ κ2 ¶q+|k+µ0| · µ 2κpa p2 a+ κ2 ¶q+|k+µ0| × Cq+|k+µ0|+1 n µ p2b− κ2 p2 b+ κ2 ¶ · Cq+|k+µ0|+1 n µ p2a− κ2 p2 a+ κ2 ¶ × P(|k+µ0|,|k+µ0|) q (cos2b)· Pq(|k+µ0|,|k+µ0|)(cos2a)
We see that the energy spectra are determined by the poles
En,q,k= −
α2
2(n+ q + |k + µ0| + 1)2.
(2.48) This agrees with the result in Refs. [10, 11]. The wave functions can be read off by giving the form of the spectral representation
G(pb, pa; E )= ∞ X n=0 ∞ X q=0 ∞ X k=−∞ " π3¡p2 b+ κ 2¢2¡p2 a+ κ 2¢2 2κ5(n+ q + |k + µ 0| + 1)2 # 1 E− En,q,k ×23κ5(n+ q + |k + µ0| + 1) π2 · 4 (q+|k+µ0|) ×[2(q+ |k + µ0|) + 1]0(n + 1)0(q + 1)0(q + 2|k + µ0| + 1) 0(n + 2(q + |k + µ0|) + 2) ס 1 p2b+ κ2¢2 · 1 ¡ p2 a+ κ2 ¢2 · µ 2κpb p2 b+ κ2 ¶q+|k+µ0| · µ 2κpa p2 a+ κ2 ¶q+|k+µ0| × Cq+|k+µ0|+1 n µ pb2− κ2 pb2+ κ2 ¶ · Cq+|k+µ0|+1 n µ p2 a− κ2 p2 a+ κ2 ¶ × P(|k+µ0|,|k+µ0|) q (cos2b)· Pq(|k+µ0|,|k+µ0|)(cos2a)
× (cos 2b/2 cos 2a/2 sin 2b/2 sin 2a/2)|k+µ0|· ei k(8b−8a),
=X∞ n=0 ∞ X q=0 ∞ X k=−∞ " π3¡p2 b+κ 2¢2¡p2 a+κ 2¢2 2κ5(n+q + |k +µ 0| + 1)2 # 1 E− En,q,k 9n,q,k(pb)9n∗,q,k(pa)+· · · . (2.49) From this we identify the wave functions as
9n,q,k(p)= 2√2κ5 π 2 (q+|k+µ0|)(n+ q + |k + µ 0| + 1)1/2 × µ [2(q+ |k + µ0|) + 1]0(n + 1)0(q + 1)0(q + 2|k + µ0| + 1) 0(n + 2(q + |k + µ0|) + 2) ¶1/2 × 1 ( p2+ κ2)2 · µ 2κp p2+ κ2 ¶q+|k+µ0| · Cq+|k+µ0|+1 n µ p2− κ2 p2+ κ2 ¶ × P(|k+µ0|,|k+µ0|)
q (cos2) · (cos 2/2 sin 2/2)|k+µ0|· e
i k8. (2.50)
The normalization condition
Z
d3p9n,q,k(p)9n∗,q,k(p)= 1 (2.51)
can be easily checked by using the recursion formula
zCnν(z)= 1
2(n+ ν)
£
and the following orthogonality relations of the Jacobi function [5, p. 383] Z −1 −1 d x(1− x) α(1+ x)βP(α,β) n (x)P (α,β) m (x)= 2α+β+1 α + β + 2n + 1 0(α + n + 1)0(β + n + 1) n!0(α + β + n + 1) δm,n, (2.53) as well as the Gegenbauer polynomials [5, p. 378],
Z 1 −1d xC λ n(x)Cλm(x)(1− x 2 )λ−1/2= π2 1−2λ0(n + 2λ) n!(λ + n)02(λ) δm,n. (2.54)
It is of interest to evaluate the average values of the square of the momentum in the various quantum states: hP2i = Z ∞ 0 Z π 0 Z 2π 0 P2d P sin2 d2 d89n,q,k(p)P29n∗,q,k(p). (2.55)
With the help of Eqs. (2.53) and (2.54), we find
hP2i = κ2=
µ α
n+ q + |k + µ0| + 1
¶2
. (2.56)
This quantity characterizes the modified circular Bohr orbit of the A-B-C system. For the hydrogen atom it specifics the square momentum of the electron in a circular Bohr orbit with the same total quantum number. Indeed, forµ0 = 0, when there is no A-B effect, Eq. (2.50) reduces to the wave
functions of the pure Coulomb system
9˜n,l,k(p)= 2√2κ5 π 2 l0(l + 1) s 4π ˜n0(˜n − l) 0(˜n + l + 1) × 1 ( p2+ κ2)2 · µ 2κp p2+ κ2 ¶l · Cl+1 ˜n−l−1 µ p2− κ2 p2+ κ2 ¶ · Ylk(2, 8), (2.57)
where we have defined ˜n = n + l + 1 (˜n = 1, 2, 3, . . . , l = 0, 1, 2, . . .). Wave functions were first obtained by Podolanski and Pauling by carrying out the Fourier transformation on the wave function of the position space in Ref. [13]. From the Schr¨odinger equation in the momentum space, the Coulomb system was solved by Fock [14, 15]. The average values of the square are given by
hP2i = κ2=
µα
˜n
¶2
. (2.58)
This special case of (2.56) is the same as for a circular Bohr orbit, as discussed in the classical paper of Pauling [13].
III. CONCLUSION
We have derived the exact Green function of the A-B-C system in momentum space from path integrals. From this, the wave functions as well as the energy spectra have been extracted. Since the
dynamics of the system in momentum space is on a curved manifold, our discussion offers another piece of evidence in answering the historic question of different Hamiltonians in curved spaces.
ACKNOWLEDGMENTS
I thank the referee for his critical reading, marks, and comment. The author has benefited from discussion with Dr. P. G. Luan.
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