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a r X i v :q u a n t -p h /9912045v 1 9 D e c 1999

Quantum Particle-Trajectories and Geometric Phase

M.Dima

F.B.-8Physik,Universit¨a t Wuppertal,Gau βstr.20,Wuppertal D-42097,

GERMANY

-Submitted to Phys.Lett.-

“Particle”-trajectories are de?ned as integrable dx μdp μ=0paths in projective space.Quantum states evoluting on such trajectories,open or closed,do not delocalise in (x,p )projec-tion,the phase associated to the trajectories being related to the previously known geometric (Berry)phase and to the Classi-cal Mechanics action.High Energy Physics properties of states evoluting on “particle”-trajectories are discussed.

Quantal wave-packet revival [1]is the periodic re-assembly of a state’s localised structure along a classi-cally stable orbit.The phenomenon has been observed

experimentally in Rydberg atoms [2]as well as in one-atom masers [3],and prompts the question whether such revival is possible also for states evoluting on open trajec-tories,similarly to classical point-particles.It is shown in this Letter that integrable dx μdp μ=0trajectories in projective space do provide such a context,the aspect being related to Di?erential Geometry [4],independently on the existence of a Hamiltonian.

The revival of quantal wave-packets is related to the concept of geometric phase [5]introduced by Berry.Berry [6]has shown that additionally to a Hamiltonian induced dynamic phase,a quantum state evoluting in pa-rameter space on a trajectory that returns to the initial state acquires an extra phase termed geometric phase.Subsequent analysis has generalised the context in which the phenomenon occurs,lifting the restriction of adia-baticity [7],cyclicity and unitarity [8].An important step was made by the kinematic approach [9],which demon-strated that the Hamiltonian is not needed in de?ning ge-ometric phase,and underlined the native geometrical na-ture of the quantity by relating it to the Bargman invari-ants [10,11].The acquirment of a geometrical phase by quantum states evoluting on closed trajectories in param-eter space has been veri?ed experimentally in neutron interference [12],in two photon states produced in spon-taneous parametric down-conversion [13],etc .The latter paper makes the important remark that experiments re-lated to non-locality vis `a vis Bell inequalities [14]and the Berry phase are connected,non-locality in Quantum Mechanics being pointed out as a consequence of com-pleteness as early as 1948,by Einstein [15].

The fundamental assumption [16]of this Letter is the commutation relation [x μ,k ν]?=?ig μν·1between the

wave-vector k μand coordonate x μoperators.The lat-ter act as tangent space vectors on the manifold,action

revealed by the translation operators U ?x def

=e +i ?x μk

μand U ?k def

=e ?i ?k μx μ:

U ??x x μU ?x =x μ+?x μ

U ??k k μU ?k =k μ+?k μ(1)

respectively |x U

?x ?

|x +?x and |k U

?k

?|k +?k .Given an arbitrary reference state |ψref ,a set of translated states can be de?ned as:

|ψ(ξ,κ) def

=U ?k U ?x |ψref

(2)

with correspondingly translated state averages:

x μ ψ(ξ,κ)= x μ ref +?x μ=ξμ k μ ψ(ξ,κ)= k μ ref +?k μ=κμ

(3)

It is important to note that the spread of these states is identical to that of |ψref regardless the translation (?x,?k ):

δx μψ(ξ,κ)=δx μ

ref =const.δk μψ(ξ,κ)=δk μref =const.

(4)

The interchange of U ?x and U ?k in the de?nition of

|ψ(ξ,κ) leads to a state corresponding in projective space [7,11]to the same point,the two states di?ering only by a phase factor:

U ?x U ?k =e +i ?x μ?k μ

U ?k U ?x

(5)

The situation is better evidentiated by the comparison of |ψref to its image transported around the ?x →?k →??x →??k quantum loop:

U loop =U ??k U ??x U ?k U ?x =e

?i ?x μ?k

μ

·1(6)

respectively around an arbitrary quantum loop:

U loop = loop

U dk U dx =e ?i k μdx μ

·1

=e

+i

x μdk μ

·1(7)

In both cases the state acquires a geometrical phase pro-portional to the(x,k)area enclosed by the loop in pro-jective space.It is noteworthy to point out that should this phase be zero,the anholonomy[8]hold preventing the realisation of a proper(x,k)coordinate system on the (Hilbert)representation space disappears.It could seem from the above that the concept of geometric phase is de-?nable only for closed(x,k)paths,however,generalising equation(5)to continuous open paths:

U open=final

initial U dk U dx=e?i f i kμdxμ·U?k U?x(8)

and holding the initial and final states apart at?xed displacements(?x,?k),a path dependent phase can be de?ned-arbitrary up to a path independent gauge[17]?eldΦ(x,k):

S def=? f i kμdxμ=+ f i xμdkμ(9)

The above relation supports a class of canonical trans-formations-such as Q=k,P=?x,consistent with

[xμ,kν]

?=?igμν·1and x|k =(2π)?2e?ixμkμ-that

identify the geometrical phase as the Classical Mechan-ics action[18].Assuming that|ψref can evolute on two neighbouring paths via a beam-splitter like mechanism, the interference in the?nal state is destructive unless δS=0(for remote trajectories:δS=2nπ),respec-tively the extremal action condition.Paths satisfying the extremal action condition at each point-or equiva-lently dxμdkμ=0in equation(6)-preserve constructive interference along the path,and are termed“particle”-trajectories.

In the beginings of Quantum Mechanics it was puzzling that quantum phenomenae could not be formulated in (x,p)space,as in Classical Mechanics.These attempts failed due to the non-zero commutator of the coordinate

and momentum operators[xμ,pν]

?=?iˉh gμν·1,and

are best summarised by the Heisenberg inequalityδxμ·δpν≥ˉh

2ˉh

accumulates a phase factorπ,as seen from equation(7).For manifolds of greater dimension this phase may vanish due to reciprocal phase compensation among dimensions. For an Euclidian metric it can be shown that this is re-alised only by trajectories on a sphere.The Minkowski metric however,allows for non-trivial solutions of the n+1pairs of canonically conjugate variables-(Q,P) plus the temporal dimension(t,E).To have thus a proper (x,k)grid on the manifold two conditions must be met: 1.-necessary condition:dxμdkμ=0 P AT H

This relation de?nes locally a coordinate system

and it is better known than apparent in physics.

For example in the case of wave-packet propaga-

tion,requiring the constituent waves to move in

sync yields the condition v g= ?kω,which can be

re-written as v g·d k= ?kω·d k=dω,respectively:

dt·dω?d x·d k=0(11)

For point-particles,the work-energy relation dE=

F d x=d x·d p/dt,can be likewise re-written as:

dt·dE?d x·d p=0(12) 2.-su?cient condition:d2x=0and d2k=0 P AT H

This relation conditions path integrability,neces-

sary for the path independent de?nition of an(x,k)

coordinate system on the manifold.It is a global

condition,with the solution[19]:

kμkμ=±k2C

dxμ

k C

(13)

that makes the important connection between kine-

matics and dynamics.Due to the inertia of

the initial condition,“cross-over”trajectories from

kμkμ>0to kμkμ<0are ruled out,±k2C being a

characteristic of the trajectory.Initial states lying

on the light-cone cannot“fall”onto kμkμ>0or

kμkμ<0solutions due to the gradient of the dif-

ferential equation parallel to the sheet of the light-

cone.The kμkμ=±k2C relation is known in physics in the form of E=c

?spreadless transport of states:

δx μψ(ξ,κ)=δx μ

ref =const.δk μψ(ξ,κ)=δk μref =const.

(16)

?x -k evolution [20]

equations:

d x μ

k C

d x μ d k μ =0

path

k μ d x μ =extremal

(17)

?path type constraint:

k μ k μ =±k 2

C

k μk μ =±k 2

C

?δk μδk μ(18)

?“particle”-grid to physically meaningful state-average contact condition:

x μ ψ(ξ,κ)=ξμ k μ ψ(ξ,κ)=κμ

(19)

Although no physical interpretation has been assumed

for k ,it is evident now that it can be associated to what is known experimentally as 4-momentum:k μ=ˉh p μ/c .Properties related to the geometric phase have been dis-cussed mostly in the context of low energy phenomenae,the following being dedicated to aspects related to High Energy Physics.

Quantum states travelling on “particle”-trajectories k μ k μ =const.have two associated constants of mo-tion:

m 20def =

ˉh 2c 2

k μk μ

(20)

respectively the rest and bare mass of the state,related

to each other by the spread of the state in k -space:

m 20

=m 2bare

?ˉh

2m bare

?

ˉh √m bare c

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and F.Wilczek,(World Scienti?c,Singapore,1989);S.I.Vinitsky,V.L.Dubovik,B.L.Markovski and Yu.P.Stepanaski,http://www.wendangku.net/doc/5f36bc8bd0d233d4b14e69a0.htmlp.33,403(1990).

[6]M.V.Berry,Proc.R.Soc.London,A392,45(1984).[7]Y.Aharonov and J.Anandan,Phys.Rev.Lett.,58,1593

(1987).

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(1988).

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(1993),228,269(1993).

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A60,3397(1999).

[12]B.E.Allman,H.Kaiser,and S.A.Werner;A.G.Wagh

and V.C.Rakhecha;J.Summhammer,Phys.Rev.A56,4420(1997).

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(1997).

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[16]To be published.In essence it is possible to arrive at

the x μand k μoperators and their commutation relation solely on grounds related to the separability of states and Di?erential Geometry,without prior knowledge of their physical equations.

[17]Gauge Theory of Elementary Particle Physics ,Ta-Pei

Cheng and Ling-Fong Li,(Oxford Univ.Press,1988).[18]Mathematical Methods of Classical Mechanics ,V.I.

Arnold,(Springer,New York,1978).[19]For dx =0the dx μdk μ=0relation requires that dk be

“perpendicular”to dx ,respectively dk μ=C μν⊥ων,with ωan arbitrary 1-form not “parallel”to the unit “norm”

direction vector n μdef =dx μ/ dx ,and C μν⊥=g

μν

?n μn νa tensor that selects the “perpendicular”component to dx μ.To be integrable,dx and dk must be closed forms:

d 2x μ=0

d 2k μ=d C μν⊥∧ων+C μν

⊥dων=0

(25)

where d C μν⊥=?C μρ⊥dn ρn ν?n μdn ρC ρν⊥.The d 2

k =0condition becomes:

n μ(dn ρ∧d C ρν⊥ων)=C μν⊥ dων?n ρ

ωρ∧dn ν

(26)

The left hand side proportional to n and the right hand side “perpendicular”to n imply:

C μν⊥dn μ∧ων=0

(27)

condition having the following solutions:(i)-C μν⊥ων=0,(ii)-C μν

⊥dn μ=0and (iii)-dn μ∧ων=antisymmetric .Solution (i)is equivalent to dk =0,solution (ii)restricts dn “parallel”to n -impossible in view of n μn μ=±1,thus the only viable solution is (iii),ων=k C ·dn νwhere

k C is a scalar ?eld.From the right hand side of equation (26)equal to zero and the arbitrary orientation of dn with

respect to n ,the scalar ?eld ΛC =2π/k C has to be a con-stant (known as the “Compton wavelength”).Therefore

dk μ=k C ·C μν⊥dn ν,respectively in view of n μn μ

=±1:dk μ=k C dn μ,or k μ=k C n μ+(const.)μ.In the eigen-system of reference of the trajectory the dx μdk μ=0

condition is simply dk ′

0=0,hence the constant in the solution above must be zero,and:

k μ=k C ·n μ=k C ·

dx μ

β2?1with β≥1,and C a tensor that

selects the parallel component to the boost.

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