Non-Abelian Aharonov-Bohm effect with the time-dependent gauge fields

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by  Seyed Ali Hosseini Mansoori

Non-Abelian Aharonov-Bohm effect with the time-dependent gauge fields
Seyed Ali Hosseini Mansoori1,2 and Behrouz Mirza2
1 Department of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215, USA
2Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran∗
(Dated: February 4, 2016)
We investigate the non-Abelian Aharonov-Bohm (AB) effect for time-dependent gauge fields.
We prove that the non-Abelian AB phase shift related to time-dependent gauge fields, in which
the electric and magnetic fields are written in the adjoint representation of SU(N) generators,
vanishes up to the first order expansion of the phase factor. Therefore, the flux quantization in a
superconductorring does not appear in thetime-dependentAbelian or non-Abelian AB effect.
6
1 I. INTRODUCTION wave solutions [15] and the time-dependent Wu-Yang
0 monopole [16]. Here, we prove that when the non-
2 In1959,Y.AharonovandD.Bohmproposedanexper- Abelian gauge field Aa is a function of spacetime, the

b iment to test the effect of the electromagnetic gauge po- AB phase shift coming from the electric and magnetic
e tential on the quantum wave function [1]. Later, Cham- non-Abelianfields willbe canceledoutupto the firstor-
F
bers performed the proposed experiment and proved der. Our results also show that the ”single-valuedness

3 that the effect did exist [2]. The AB effect is indeed of the wave function” does not constrain the flux of a
a quantum-mechanical phenomenon in which the wave time-dependent magnetic field to be quantized in a su-
h] function of a charged particle traveling around an ex- perconducting ring. It will be interesting to verify this
p tremely long solenoid undergoes a phase shift depending result experimentally.
- on the magnetic field between the paths albeit B = 0 The outline of this paper is as follows. Section II
t
n along the paths themselves [1, 2]. presents a description of the AB phase shift. In Sec-
a Overthepastfewyears,considerableinteresthasbeen tion III, we study the quantization of the magnetic flux
u shown in the AB effect in Abelian gauge fields with a in a superconducting ring. We will show that there is no
q
time-independentmagneticfield. Recently,theABeffect ”single-valuedness” condition for the wave function be-
[
with a time-dependent magnetic field has been investi- cause the phase shift will be zero in a time-dependent

2 gated [4–8] to show that a cancelation of phases occurs AB effect. In Section IV, we generalize the Abelian AB
v in the AB effect with a time-dependent magnetic field. effect to a time-dependent non-Abelian field configura-
4
Strictly speaking, an extra phase coming from the elec- tion. We prove that the AB phase factor remains equal
6
1 tricfield,E =−∂tA,outsidethesolenoidcancelsoutthe to zero up to the first order when considering the time-
8 phaseshiftofthetime-dependentmagneticfield. Theex- varying vector fields. Conclusions will be presented in
0 perimental results of Marton et al. [9], where the effect the last Section.
. ofthetimevariationofthemagneticfieldwasnotseenin
1
0 the interferencepattern,also confirmthe theoriticalpre-
II. TIME-DEPENDENT AB EFFECT FOR
6 dictionof[10],i.e.,anexactcancellationoftheABphase
ABELIAN GAUGE FIELDS
1 shift by means of the phase shift coming from the direct
: Lorentzforce. Inthisframework,thetime-dependentAB
v
i effect can be considered as a type II AB effect. Indeed, The relativistic form of the AB phase factor can be
X thetypeIeffectsareinsituationsthatachargedparticle written as follows:
r ismovingthrougharegionwithoutmagneticandelectric
a e e
fields, while the type II AB effects are when the charged β =exp A dxµ =exp ϕdt−A.dx (1)
particle develops an AB phase passing through a region (cid:20)~I µ (cid:21) (cid:20)~I (cid:21)
of space with non-zero fields [11]. Recently, in Ref. [12],
where,Aµ istheAbeliangaugefieldthatmightbetrans-
authors have shown that type II AB effect due to elec-
formed under the U(1) group as follows:
tromagneticplane wavesvanishesunder some conditions
in terms of the parameters of the system like frequency Aµ →Aµ′ =Aµ+∂µξ (2)
of the electromagnetic wave, the size of the space-time
loop, and amplitude of the electromagnetic wave.
where, ξ is a transformation function of space-time co-
It is, therefore, interesting to study the non-Abelian
ordinates [17, 18]. We may rewrite the phase factor in
ABeffect[13]withatime-dependentmagneticfield. Re-
a 2-form structure by making use of Stokes’ theorem,
cently,theABeffecthasbeenstudiedfortime-dependent
stating that the integralof a differential form ω over the
non-Abelian fields by using two specific, known time-
boundary of some orientable manifold Ω is equal to the
dependent solutions [14] such as the Coleman plane
integralof its exterior derivative dω over the whole of Ω,
which may be expressed as follows:
∗ [email protected]; [email protected]; ω = dω (3)
Z∂Ω ZΩ
2
where,ωanddωarep-formand(p+1)-form,respectively. In this case, the variation of the wave function phase
One couldalso define the 1-formas ω =A=A dxµ and is,
µ
2-form dω =dA as the Faraday 2-form F by:
δα = 2e A.dl = 2e ∇×A.dS (8)
B ~ ~
dA=F = 21Fµνdxµ∧dxν = (4) =H2e B.dS =R2eΦ
~ ~
(E dx+E dy+E dz)∧dt+B dy∧dz
x y z x R
+B dz∧dx+B dx∧dy where,Φisthemagneticfluxandthefactor2eshowsthat
y z
theCooperpairs[24]inthesuperconductorhavecharges
where E and B are the electric and magnetic fields, re- twicethatofanelectron. Inordertomaintainthesingle-
spectively. Therefore, Eq. (1) can be rewritten as in (5) valuedness of the wave function, this phase factor must
below: be equal to 2πn ( n = 1,2,3,...), so that we can obtain
the following quantum flux,
e
β =exp − F dxµ∧dxν (5)
(cid:20) 2~Z µν (cid:21) ~πn hn
Φ = = n=1,2,3,... (9)
n
e 2e
This expression plays a key role in the study of the
AB phase factor when considering the time-dependent Now, we consider a time-dependent magnetic field. Ac-
Abeliangaugefields. Time-dependentABEffectisbased cording to Maxwell’s equation, there is an electric field
on constructing a subspace in a spacetime in which the ( E = −∇ϕ−∂tA) which creates an additional phase
four-vector potential depends on time [19, 20]. Both the factor. Moreover, for this case, the scalar potential is
electricandthemagneticeffectsdependontheparticle’s still zero and the vector potential is a function of time
particular path in this subspace [5]. We assume that the and space. Based on Eq. (4), the relativistic phase shift
magnetic field inside the solenoid is time-dependent so willbe zeroduetothe cancelationofthe magneticphase
that the vector potential A will be time-dependent out- shift due to a phase shift coming from the electric field,
sidethesolenoid. However,basedonMaxwell’sequation, E = −∂tA. As a result, there is no constraint on the
i.e., E = −∂ A, an electric field is also created outside magnetic flux Φ. It will be interesting to design an ex-
t
the solenoid (We have assumed the scalar potential field perimental plan to examine this effect.
ϕtobezero). Thus,fromEqs. (4)and(5),themagnetic
phase factor is obtained by:
IV. TIME-DEPENDENT AB EFFECT FOR
~e [Bxdy∧dz+Bydz∧dx+Bzdx∧dy] (6) NON-ABELIAN GAUGE FIELDS
R = e B(x,t).dS
~
R In section II, we investigated the time-dependent AB
and the electric part of the phase is given by: effect [4, 6, 7], and showed that there is no phase shift
e [E dx∧dt+E dy∧dt+E dz∧dt] (7) in this case. In this section, we will verify the claim that
~ x y z the phase shift of the non-Abelian AB effect is zero for
R =−~e A.dx=− ~e B(x,t).dS time-dependent gauge fields.
H R The concept of the non-Abelian gauge field was first
where, we have replaced the electric field by −∂ A. It
t introduced in 1954 by Yang and Mills [25]. The 4-vector
is clear that the AB phase shift for a time-varying mag-
gauge fields A were introduced with N internal compo-
netic field vanishes. This means that the magnetic AB a
nents labeled by a = 1,2,3,...,N, corresponding to the
phase shift is canceled out by a phase shift coming
N-generatorsofthe gaugegroupclosedunder the follow-
from the Lorentz force associated with the electric field,
ing commutation;
E =−∂ A, outside the solenoid [4].
t
[L ,L ]=iCc L (10)
a b ab c
III. NON-FLUX QUANTIZATION IN
where, the constants Cc are real numbers called struc-
SUPERCONDUCTING RINGS FOR ab
tureconstants. Forsimplicity,weshallusetheshorthand
TIME-DEPENDENT MAGNETIC FIELDS
Aµ =Aµ L ,whichisamatrix. Moreover,Aµ underan
a a a
infinitesimallylocalgaugetransformationcanbe written
Let us now consider a superconducting ring with rigid
in the following form,
walls which is exposed to an external uniform magnetic
field. Assumingaparticleofchargeecompletelyconfined 1
Aµ(x)→Aµ(x)+ ∂µω (x)+C ω (x)Aµ(x) (11)
in the interior shell of the superconducting ring, one can a a g a abc b c
obtaintherelevantenergyeigenvaluesandwavefunctions
[21–23]. However, care must be taken to ensure that where, g and ω are the gauge coupling constant and ar-
the value of the wave function at any given point in the bitrary real functions, respectively. In Maxwell’s U(1)
ringhasthesamevalueasthewavefunctionobtainedby gauge theory, a gauge-invariant field tensor F =
µν
traveling around the ring to return back to the original ∂ A −∂ A is defined, whose components are the elec-
µ ν ν µ
point. In other words, the wave function must have a tric and magnetic fields; in the non-Abelian case, how-
single value at a given point in the ring. ever,suchafieldtensorisnotgaugeinvariantor,indeed,
3
thereisnogauge-invariantfieldtensor. Inordertodefine One can then divide up the above equation into the two
a gauge-invariant field tensor for the non-Abelian gauge magnetic and electric parts. In this way, the phase dif-
fields, the representation must be the adjoint represen- ference associated with the magnetic field terms is given
tation [26]. The following equation satisfies our require- by:
ments,
g Bady∧dz+Badz∧dx+Badx∧dy = (21)
~ x y z
Fµaν =∂µAaν −∂νAaµ+gCabcAbµAcν (12) R (cid:2) g Ba(x,t).dS (cid:3)
~
where, the antisymmetric constants Cabc =−i(Lb)ac are Substituting Ba fromREq. (14) and using the Stoke’s
defined in the adjoint representation. Using the above theorem, we have:
equation,theelectricandthemagneticfieldscanbewrit-
2
ten as in the following equations [17], g g g
Ba(x,t).dS = Aa.dl+ Cabc (Ab×Ac).dS
E =−∇A0−∂ A −gC A A0 (13) ~Z ~I 2~ Z
a a t a abc b c (22)
On the other hand, the electric field part is given by:
1
Ba =∇×Aa+ 2gCabcAb×Ac (14) ~g Exadx∧dt+Eyady∧dt+Ezadz∧dt (23)
R (cid:2) =−g Aa.dl (cid:3)
It is surprising that the AB experiment can also be used ~
H
to examine the existence of non-Abelian gauge fields where, the electric field is E =−∂ A . Finally, one can
a t a
[27, 28]. One can generalize the phase factor of the obtain the phase shift from the 2-form tensor as follows:
Abelian AB effect to the non-Abelian AB one [13] us-
ing the following Relation: g g2
L Fa dxµ∧dxν =+ CabcL (Ab×Ac).dS
2~ aZ µν 2~ aZ
g
β =P exp A (15) (24)
(cid:20) (cid:20)~I (cid:21)(cid:21) Moreover,when considering the 4-vector potential Aµ ≡
a
(A0,Ai) = (0,Ai(x,t)), we can rewrite the second part
where, A = AaL dxµ and P is the path-ordering oper- a a a
µ a of Relation (19) as follows:
ator. This phase factor is quite similar to Wilson loop
[29, 30]. Expanding this phase shift up to the second − g2CabcL AbAcdxµ∧dxν
order, we will have: ~ a µ ν
=−g2CabcL R(Ab)(Ac)dxj ∧dxk
β ≃1+ gL Aadxµ+ (16) ~ a j k
P(g)2 dxµAa ~(x)adHxν.Aµ b(x)L L +... =−2g~2CabcLRa (Ab×Ac).dS (25)
~ µ ν a b
R
H H in which the following wedge product is used
Wewillnowgoontoshowthatthetime-dependentnon-
Abelian AB phase shift vanishes up to the first order,
εijk εijk
while the other orders indicate a non-zero non-Abelian dxj ∧dxk = dxj ⊗dxk ≡ dS . (26)
i
2 2
AB phase factor. Let us consider a 4-vector poten-
tial in the non-Abelian AB effect as Aµa ≡ (A0a,Aia) = Finally, using Eqs. (19), (24), and (25), we arrive at the
(0,Aia(x,t)). Therefore, from Eq. (13), the electric field following interesting result:
will be a non-zero term (E = −∂ A ). Applying the
a t a
Stoke’s theorem (Eq. 3), we will have: gL Aadxµ =0 (27)
~ aI µ
dω =dA=dAaL = ∂ Aa−∂ Aa L dxµ∧dxν (17)
a µ ν ν µ a
(cid:0) (cid:1) Therefore, the phase shift related to the time-dependent
Based on Eq. (12), the above equation can be rewritten non-Abelian AB effect vanishes up to the first order ex-
as: pansion of the phase factor. This is a generally valid
1 result. Forfuture research,itwillbe interestingtoinves-
dAa = Fa dxµ∧dxν −gCabcAbAcdxµ∧dxν (18)
2 µν µ ν tigate the higher order terms of gauge fields. It may be
anticipatedthatallhigherordertermsofgaugefieldswill
where, the factor 1/2 comes from the anti-symmetry
also vanish. This conjecture cannot, however, be proved
propertyofF anddxµ∧dxν [26]. Therefore,thesecond
µν presently.
term of the expansion in Eq. (16) may be replaced with
the following equation:
gL Aadxµ = g L Fa dxµ∧dxν V. CONCLUSION
~ a µ 2~ a µν
−H g2CabcL AbARcdxµ∧dxν (19)
~ a µ ν In this paper, we studied time-dependent Abelian and
R
where, the 2-form tensor can be defined as, non-AbelianAB effects. We showedthat for a supercon-
ductor exposedto a time-varying magnetic field, there is
21Fµνadxµ∧dxν =(Eaxdx+Eaydy+Eazdz)∧dt noconstraintonthemagneticfluxduetothepresenceof
+Ba dy∧dz+Ba dz∧dx+B dx∧dy (20) zero phase factor in both Abelian and non-Abelian AB
x y az
4
effects. We also investigated the non-Abelian AB effect neticfields,theABphasedisappearsuptothefirstorder
and proved that, for time-dependent non-Abelian mag- in the expansion of the phase factor.
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