casuality, haque

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523RESONANCE June 2014

GENERAL ARTICLE

Causality in Classical Physics

Asrarul Haque

KeywordsNewtonian mechanics, Max-wellian electrodynamics, causal-

ity, actionat-a-distance, spatialand temporal nonlocality.

Asrarul Haque is an

Assistant Professor at theDepartment of Physics,

BITS Pilani, Hyderabad

campus. His field of

research is nonlocal

quantum field theories and

theoretical high energy

physics.

Classical physics encompasses the study of phys-ical phenomena which range from local (a point)to nonlocal (a region) in space and/or time. Wediscuss the concept of spatial and temporal non-locality. However, one of the likely implicationspertaining to nonlocality is non-causality. Westudy causality in the context of phenomena in-volving nonlocality. An appropriate domain of space and time which preserves causality is iden-tied.

1. Introduction

Classical physics (Newtonian mechanics and Maxwellianelectrodynamics) deals with the space- and/or time-vary-ing physical phenomena of massive point particles andthe electromagnetic eld. The physical happenings inclassical physics are ordered in time. What ensures thecorrect chronological order? It is causality. Causality, ingeneral, refers to the fact that event E 1 ( r, t ) must occurbefore in time (i.e., earlier) than event E 2 ( r , t > t ) if E 1inuences E 2 . For instance, the scalar potential V (r, t )due to an arbitrarily moving point charge is

V (r, t ) = 14 0

( r , t |r r |c )| r r |

d3 r ;

(c is the speed of light in vacuum).Charge density ( r , t |r r |c ) as a cause precedes po-tential V (r, t ) as an observable effect. The question nowarises: how long does it take for the inuence to reachE 2 from E 1 ? The instantaneous inuence from an eventE 1 ( r, t ) to an event E 2 ( r , t ) is not desirable, as our ex-perience tells us. In fact, there exists a minimum time


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GENERAL ARTICLE

Figure 2.(a) A system of N stationaryparticles placed at a distanceR each interacting via an ac-tion at a distance of range R.

By nonlocality wemean that an event in

addition to relativelynearby events can

influence the suitablydistant events as

well. Therefore,nonlocality impliesthat the state of a

particle at time t canbe determined by itsposition ( ) x t as well

as all possible timederivatives of itsposition such as

in the sense that D has now gotten support over a region| t t | in time through E . By nonlocality we meanthat an event in addition to relatively nearby events caninuence the suitably distant events as well. Therefore,

nonlocality implies that the state of a particle at timet can be determined by its position x(t ) as well as allpossible time derivatives of its position such as x(t ), x(t),...x (t),... .

2. Action-at-a-Distance

Interactions in classical physics (Newtonian mechanics)involve action at a distance, which simply means thatinteraction can occur between any two distinct spatialpoints instantly . We shall illustrate the concept of actionat a distance by considering two simple systems: onediscrete and the other continuous. Consider a system of N stationary particles separated by a distance R each(as shown in Figure 2a). Suppose the particles interactvia an action at a distance of range, say, R . Action at adistance implies that the force between the ith and j thparticles

F ij =

F ij (r i (t i ), r j (t j )) t i

= t j = t ,depends on the position of the ith particle as well as the j th particle at the same time. Thus all the particles hap-pen to interact at the same time. This is possible pro-vided the interaction propagates instantaneously (i.e.,with innite speed) across all possible spatial separa-tions in the system. This is what conicts with causalitywhich requires a nite speed of propagation of interac-tion.

x ( t ), x ( t ), ...x ( t ),... .


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GENERAL ARTICLE

What is the form ofspatio-temporaldependence of the

relation A( k, ) =

B ( k, ) C ( k, ) ?

We now consider three physical quantities, namely A,B and C that vary in space and time. Suppose these

quantities describe some physical phenomenon in whichthe rst order spatial and temporal derivatives of theirphases happen to be the same. Then all the three phys-ical quantities will have the same wave vector and fre-quency dependence as A( k, ), B ( k, ) and C ( k, ). Wewish to study the causal structure of the generic form of the ubiquitous equation

A( k, ) = B ( k, ) C ( k, ), (8)

which embodies spatio-temporal nonlocality in classi-

cal physics. What is the form of spatio-temporal de-pendence of the above relation? A point in wave vec-tor/frequecy space corresponds to a region in the cong-uration/time space: the former is the reciprocal spaceof the latter. Such correspondence is established viaFourier transform which is based on the fact that anygood function can be built out of a superposition of sine/cosine functions. The Fourier transform (FT) of

A( r, t ) and inverse Fourier transform (IFT) of A( k, )are dened as follows:

FT A( r, t ) = A( k, )

=+

dt(2 )1/ 2

d3r(2 )3/ 2

A( r, t )e i( k.r t ) , (9)

IFT A( k, ) = A( r, t )

=+

d3k(2 )3/ 2

d(2 )1/ 2

A( k, )e i( k.r t ) . (10)


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Figure 4. A macroscopic di-pole moment of size d cre-ated by a sinusoidal varyingelectric field of wavelength .

When the electricfield possesses acharacteristic lengthscale such aswavelength ( ), then

for l , spatial nonlocality is not important.

E

E


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Suggested Reading

[1] J D Jackson, Classical Electrodynamics , John Wiley & Sons, New York,

pp.330333, 2003.

[2] H M Nussenzveig, Causality and Dispersion Relations , Academic press,

New York, pp.316, 1972.

[3] D I Blokhintsev, Space and Time in the Microworld , D. Reidel Publishing

Company, Dordrecht-Holland, pp.191199, 1973.

Address for Correspondence Asrarul Haque

B209Department of Physics

BITS PilaniHyderabad Campus

Jawahar NagarShameerpet Mandal

Hyderabad 500 078, India.Email:

[email protected]

form of the equation A( k, ) = B ( k, ) C ( k, ) whichembodies spatio-temporal nonlocality and appears atseveral places and in various contexts in classical physics:the motion of a forced damped harmonic oscillator, aux-

iliary equations (electromagnetic constitutive relationsfor space-time dependent elds), etc. The phenomenagoverned by the said equation might occur over the do-main of speeds ranging from the small speeds ( v

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