Radiation-reaction force on a moving mirror

(1)

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Radiation-reaction force on a moving mirror

Claudia Eberlein

To cite this version:

Claudia Eberlein. Radiation-reaction force on a moving mirror. Journal de Physique I, EDP Sciences,

1993, 3 (11), pp.2151-2159. �10.1051/jp1:1993237�. �jpa-00246859�

(2)

Classification Physics Abstracts

03.70 05.40 11.10E

Radiation-reaction force

_on _a

moving mirror

Claudia

Eberlein(*)

School of Mathematical and Physical Sciences, University of Sussex, Brighton BNI 9QH, England

(Received 12 March 1993, accepted in final form 22 July1993)

Abstract. The radiation-reaction force

F(t)

acting _on _a moving mirror in _a one-dimensional model is obtained by appeal to the fluctuation-dissipation theorem. Causality is found to pro- vide the link between the dissipative and reactive forces; it is implemented by requirements of

analyticity for the Fourier transform of the _response function that links the _mean force

F(t)

to the velocity

fl(t)

of the mirror. Particular attention is given to the issue of the divergent _mass

renormalization arising from the reactive part ^of the force, detected in

some earlier treatments and denied in others.

1. Introduction.

Quantum theory predicts

radiation from

moving

mirrors 11, 2], as well _as from

moving

di- electrics [3], ⁱⁿ consequence of their interaction with

zero-point

fluctuations of _a

quantized

field.

Conversely,

the

(macroscopic) body,

mirror _or

dielectric,

is

subject

to _a back-reaction from this radiation, I-e- experiences a radiation-reaction force.

Knowing

the spectrum of the

radiation _one can

readily

evaluate the _mean force

F(t) acting

_on the

body. However,

since the calculation of the spectrum

(see

_[3] for _a one-dimensional scalar

model)

_proves

quite

_a

strenuous

task,

it is

tempting

to use the

fluctuation-dissipation

theorem [4, 5] which

yields F(t)

to first order in the

velocity fl(t) (or

its

derivatives)

from the correlation function of the

fluctuating

force _on _a

stationary body;

this has been done _or touched _on in _a series of recent papers [6 or

7-9], respectively.

Though straightforward

at first

sight,

the

application

of the

fluctuation-dissipation

theorem to this kind of

problem

bears _a number of subtleties

easily leading

to specious conclusions.

So,

for instance, ^the

dissipative

_part of the force _on _a

moving

mirror is

manifest,

whereas its reactive part, ^which

gives

rise to _a _mass renormalization in one-dimensional models, is

missed if

divergences

_are ^handled

improperly

and the no-cut-off limit is taken

prematurely.

This observation

gains special

interest in view of the different results for the reactive part of the force

given

in the literature

[1-3, 7-9], although

the

dissipative

part ^is not

subject

to

disagreement.

The present _paper is intended to serve as a consistency ^check on previous results

(*) Present address: University of Illinois, Dept of Physics, Il10 West Green Street, Urbana IL 61801-3080,

U-S-A-

(3)

2152 JOURNAL DE PHYSIQUE I N°11

[3] and to

point

out the niceties in the interrelation of the

dissipative

and the reactive parts of the force. For the

example

of the force _on _a

slowly moving

dielectric

half-space

in _a _one-

dimensional scalar model it is shown

explicitly

that

causality intrinsically

connects

dissipative

and reactive parts.

The dielectric is

supposed

to be

non-dispersive,

and hence

non-absorptive;

its material _prop- erties _are described

by

_a

single

parameter the refractive index _n, which is chosen _as _an

arbitrary

constant.

Although

the limit _n _~ _oc of _a

perfectly reflecting

mirror is aimed at in the end

results,

_one should stick to finite _n in

formulating

the

problem

and in any intermediate steps ^of

calculating

the force. The reader to whom this _seems overcautious _or _even unfounded is reminded of the fact that _no Hamiltonian formalism exists for _a

moving (perfect)

mirror

since the system

changes

from _one Hilbert _space into another each time the mirror _moves

[10],

whereas any derivation of the

fluctuation-dissipation

theorem

indispensably

involves Hamil- tonians for the

unperturbed (here: stationary)

and for the

perturbed (here: moving)

system [4, 5]. ^As demonstrated elsewhere [3], any

ambiguities

can be

sidestepped by keeping

the

refractive index _n finite

everywhere

but in the end results.

2. Force correlation function for the

stationary

mirror.

The mirror at rest represents the

unperturbed

_system, which is

governed by

_some time-

independent

Hamiltonian Ho-

Having

in mind the

application

of the

fluctuation-dissipation

theorem, which is based _on standard

time-dependent perturbation theory,

_one has to

require that, subsequently,

the

moving

mirror is described

by

the same, still

time-independent

Ho

Plus

a

perturbation V(t)

that carries all

explicit time-dependence

of the system and

depends

_on

a small parameter, ^which ⁱⁿ ^fact ^will ^be ^the

velocity fl(t).

The correlation function for the random

fluctuating

force exerted

by

the field _on the

stationary

mirror

(or

to zeroth order _on the

moving mirror)

is

routinely

defined

by

C(t t')

=

101)iF(t),F(t')i10) -101F(t)10)101F(t')10)

,

(2.1)

where the force operator

F(t)

is _a

Heisenberg

operator and is related to the

time-independent Schroedinger

operator F

by

F(t)

=

e'~°~~~~°~Fe~'~°~~~~°~

Although

the

dependence

of the force operator F and of the

unperturbed

Hamiltonian

Ho

on the field variables is

specific

to the

particular

system

investigated, quite generally

the

fluctuation-dissipation

theorem links the force correlation function

C(t -t')

for the

unperturbed

system to the

dissipative

_part of the expectation value

F(t)

of the force operator F in the

perturbed

state

(to

first order in the

perturbation parameter).

The

implementation

of the above formulae for the _case of the mirror

might

_appear rather

trivial,

yet ^is not in fact _as _one will notice when

trying

to

identify

Ho- The one-dimensional scalar model used in the

following

is defined

by

the requirements that the

(classical

_or

quantum)

field

4l(x, t) satisfy

the _wave equation

e4l 4l"

= o

(2.2)

in the rest-frame of the dielectric

(but non-magnetic) medium,

which is described

solely by

the dielectric constant _e, and that acceleration stresses on the

moving

medium be

disregarded.

Standard

Lagrangean theory

establishes the momentum H

conjugate

to 4l,

H = e 4l

(2.3)

(4)

The

quantization

of the system is conducted

by imposing

the

(equal-time)

commutation relations

w(xi

,

n(x')

= i&(x xi

on the

canonically conjugate

field operators ^4l ^and ^H. ^The Hamiltonian for the field in the presence of _a stationary ^dielectric

half-space occupying

_x >

(

reads

with the dielectric constant

e(xii)

= @(i

x)

₊

_n2@(x 1)

featuring

the refractive index _n of the medium. Hamilton's

equations

of _motion

obviously reproduce

the _wave

equation (2.2)

and the

defining

relation for H

(2.3),

_as

they

must. The Hamiltonian H is in fact _very familiar from standard

electromagnetism,

when _one

regards

^4l'

and H _as

loosely analogous

to the

electromagnetic

B and D. The term "dielectric"

frequently

used here also refers to this

analogy.

To _see the

difficulty

in the

application

of the

fluctuation-dissipation

theorem _to this system

note that when the

half-space

_starts

moving

( becomes _a

time-dependent

parameter, and therefore H is not

time-independent

_anymore, _so cannot _serve _as the

unperturbed

Hamiltonian Ho

The d4nouement is _to accommodate the

time-dependence

of H in _a suitable

time-dependent unitary

transformation of the _states and operators in the

Schroedinger equation.

Such _a trans-

formation is

provided by

the

unitary

translation

operator(~) U(((t))

₌

exp[

I

((t)

x

fC~

dx

H(x) 4l'(x)],

whose action could be

interpreted

either _as

always re-displacing

the

moving body

to the

origin

_or _as

making

the coordinate system travel with the

body.

For the

moving half-space

this leads to the effective interaction

Hamiltonian(~) V(t)

=

fl(t) /~

_dx

^~)~)(()~~

,

(2.4)

which adds to the

unperturbed

Hamiltonian

Ho-Ldzlll+~'~(zl

Note that Ho ₌

Ut(((t)) HU(((t))

involves

only

the dielectric _constant for the

half-space

at

position (

₌ o and is

time-independent

_as

required.

In other

words,

for the present ^model ^of

a

moving

mirror _one _can be _sure that the

application

^of ^the

fluctuation-dissipation

theorem is

wholly unproblematic

and

legitimate.

Expressing

the field operators ^{4l and} H in terms of

photon

annihilation and creation operators

by

_means of standard normal-mode

expansion,

_so _as to

diagonalize

the Hamiltonian Ho to

Ho =

fj~~

dk _w

(a(ak

₊

1/2),

allows _one to rewrite the correlation function

(2.I) by

inserting

a

complete

set of

photon

states in between the two force operators; _one

gets(~):

C(t t')

=

/~

_dk

/~

_dk'

cos [(w

+w')(t t')] [(o[F[k, k')[~

(l)Since

the details of this _are not essential for the reasoning in the present article, the interested reader is referred to [3] ^for these; _as regards the unitary translation operator U(((t)) and its utilization, especially to section 4 of reference [3].

(2)F is quadratic in the fields and therefore quadratic in the _ak and

a(

_operators, whence the only transitions

from the _vacuum F _can induce _go to the _vacuum _or to _a two-photon state.

JOURNAL DE PHYSIQUE -T 3 N"I NOVEMBER 'Q93

(5)

The fluctuation spectrum, ^I-e- ^the ^Fourier ^transform ^{of the} correlation function

C(T),

reads

x(fly

₌

j~

_dT

C(T) e'"~ (2.5a)

x(fly

=

j /~

_dk

j~

_dk'

j(0jFjk, k'jj~ jd(w

+ w' fly + &(w + w' ₊

fljj (2.sbj

The force operator ^F on the

half-space

located at _x >o is

given by ill, Chap. 6],

[3,

appendix A.4]

~

/2 ^~'~~~~

^~~'~~

Although

this

expression

is

merely quoted here,

it should be

emphasized

that its derivation

requires

_some _care if the

half-space

is _a dielectric and _not _a

perfect

mirror. The mechanical force

on the

(rigid) body

is obtained from the surface _stress due to the

uncoupled

field

(I.e.

^exclusive

of the

polarization

field inside the

dielectric) by subtracting

the momentum transferred from the

uncoupled

to the

polarization

field.

Only

for _a

perfect

conductor is the force

given directly by

the surface stress of the field since the

uncoupled

field inside the conductor is

suppressed

and cannot deliver _any momentum to the

polarization

field.

Normal-mode

expansion

of the field allows _one to evaluate the matrix element

(o[F[k, k') and, subsequently,

the spectrum

(2.5b)

x(fl)

=

(() ~ lfll~e~°'"'

,

(2.7)

where _a is _a

cut-off(~)

_on the _sum of the

frequencies

of the two

photons.

Although

not needed in the _course of the argument and recorded for

completeness only:

the correlation function reads

~~~~ ~

n

~ ^~

~2

~~

(T ~ia)4

' ~~ ~~

which bears

comparison

with the correlation function _for _the

electromagnetic

_pressure fluctuations at _one and the _same

point

_on _a

plane perfect

mirror

bounding

_a three-dimensional

half-space

[12].

3.

Response

function and

fluctuation-dissipation

theorem.

The _mean force

F(t)

_on the

moving half-space

is

given by

the

expectation

value of the force operator

(2.6)

in the state that has evolved from the _vacuum state under the influence of the

interaction Hamiltonian

(2.4).

To first

order,

_one _can write

Fit)

=

/~

_dT

_G(t

_T)

_fllT)

,

(3.1)

I-e- consider

F(t)

_as _a linear _response to the

perturbation V(t),

equation

(2.4)(4).

(~) The need for _a cut-off is not apparent here, since the expression (2.7) has _an obvious, mathematically well- defined limit for _a _- 0. However, the correlation function (2.8) does not admit _a physically sensible no-cut-off limit. Furthermore, in the following section the finite frequency cut-off will turn out the essential ingredient in the dispersion relations for the _response f~nction.

(4) Since V(t) is parametrized by the velocity p(t) +

I(t)

_(rather _than _by _the displacement ((t)), the force is

more conveniently considered _as _a _response to p(t) rather than to ((t), which will become clear by looking at the dispersion relations for the Fourier transforms of the response functions (cf. footnote 6).

(6)

The

fluctuation-dissipation

theorem involves the Fourier transform of the _response function

r(n)

=

/°'

dT

G(T) ei"T (3.2)

Independently

of _any

dynamical

detsdls of the system,

r(fl)

must have the

following properties:

I)

r(-fl)

=

r*(fl),

I-e-

Rer(fl)

is _an _even and Im

r(fl)

_an odd

function,

since

G(T)

must be real _as _seen in

equation (3.I);

it) r(fl)

is

analytic

in the _upper

half-plane

Imfl _> o,

by

virtue of Titchmarsh's theorem [13,

14],

since

G(T)

is

causal,

I-e- vanishes _on the

negative

real axis _T < o;

furthermore,

if

r(fl)

is

square-integrable

_on the real axis then

Rer(fl)

and

Imr(fl)

_are connected

by dispersion

relations

(Hilbert transformation) according

to

Rer(fl)

₌ P

/~

dw~~~~)~

,

(3.3a)

7r -~x>

Ld

Imr(nj

₌ P

)°~ ^dw~~~ljl ^(3.3bj

7r -~x>

Ld

The

fluctuation-dissipation

theorem [4, 5] relates the _power spectrum

x(fl)

to the Fourier transform of the _response

function;

for _a _response function

linking

force _to

velocity

it

states(~):

Rer(fl)

=

x(fl) (3.4)

(fl(

As the

dissipative

force _as well _as the

velocity

_are odd under

time-reversal, ReT(fl)

stands for the

dissipative

part of the response. From

(2.5b)

_one obtains

Rer(fl)

=

-I /~

_dk

/~

dk' ~~~~~~~'

'~~~

[d(w +w'-

fl)

₊

d(w

^+w'+

fl)]

2

_~ -~

w + w

Then

fornJal application

of the

dispersion

relation

(3.3b) yields

It must be

emphasized

that this is

definitely only formal

in the _sense that

equations (3.3a,b)

are

only

valid if

r(fl)

is

square-integrable.

However, the

square-integrability

^of

r(fl) (and

with it the convergence of the

w-integrals

in the

dispersion relations) depends

_on the

specific

matrix

element

[(o[F[k, k')[~

and has therefore to be checked in each

particular

case.

For the dielectric

half-space equation (2.7) implies

Rer(fl)

=

(fi) fl~e~"'"' (3.5a)

n

127r

the

dispersion

relation

(3.3b) yields

Im

r(n)

=

~

sgn(n) (2

'"' n2e-'n'Ei(ajnj) n2e'"'Ei(ajnj) _(3.5b)

n 127r _a

(~)If

_one had chosen to write the force _as _response to the displacement, F(t)

=

f)

dr G(t r)((r), then the fluctuation-dissipation theorem involving the Fourier transform r(Q) of G(Q) _assumes the more familiar

(zero-temperature) form

Imf(Q)

= sgn(Q)x(Q), since plainly

f(Q)

= -ion(Q).

(7)

(For

the definition and the

properties

of the

exponential integrals

see e-g- Ref.

[15].)

As

regards

^the

square-integrability

of

r(fl)

in

(3.5a,b),

there is

obviously

_no

problem

with

Rer(fl)

since it falls

exponentially

as

(n(

- cc. A short calculation,

using only

standard methods of

asymptotic approximation,

shows that the

expression

in the

curly

brackets in

(3.5b)

for

Imr(fl)

behaves like

-4/(a~ _(n()

_as

_(n(

- cc. Hence

r(fl)

is indeed

square-integrable,

and satisfies the

dispersion

relations

(3.3a,b) just

_as

they

_are

written(~).

4.

Dissipative

and reactive forces in the no-cut-off limit.

In the no-cut-off limit _o _- 0

equations (3.5a,b)

entail

Rer(fl) #

^'~

fl~, _(4.la)

n

~

127r

~~~~~~ ~ ~

n

~

^~

1~2a

~ ~~'~~~

Substitution into

~ ~

F(t)

= dr dfl

r(fl) e~~~~~~~)fl(r)

,

(4.2)

27r

~co ~co

which follows from

equation (3.I)

and the inverse of

equation (3.2), yields

for the force _on the dielectric

half-space

~

~~~~ ^~

n

~7r~~~~ 1~2a~~~~

~~'~~

For _a

perfect

mirror, which _can be modelled

by

infinite refractive index _n, the

n-dependent prefactor

_goes to I, and the force

equals

the

expression

in the

curly

brackets. The first term,

proportional

to

fl(t),

is the

dissipative (or "frictional")

force _on the mirror; the second term,

proportional

to

fl(t),

is

divergent

^for _zero ^cut-off _a and

gives

rise to _a renormalization of the mirror's inertial _mass. As

easily

understood from

equation (4.2)

and illustrated

by

the force

(4.3) ensuing

^from equations

(4.la,b),

_powers

(-I fl)P

in

r(fl),

the Fourier transform of the response

function, engender

terms

fl(P)(t), pth

time-derivatives of the

velocity,

in the _mean force

F(t). So, Rer(fl)

in

(4.la) brings

about the

dissipative force,

and

Imr(fl)

in

(4.lb)

leads _to the mass-renormalization term.

Coincidentally,

the force _on _a

slowly moving

classical point

charge [16],

handled

by

classical

electromagnetism (in

three

dimensions),

is

equivalent

to

equation (4.3),

apart ^from a different

prefactor featuring

the

squared

value of the

charge.

This

analogy

has wide

implications,

as for

instance the

replacement

of Newton's

equation

of motion for the mirror

by

the Lorentz-Dirac

integro-differential

equation _[16] in order _to avoid _run-away solutions. Here the

parallelism

is called _upon

merely

to corroborate the _mass renormalization

required

for the mirror. It is well-known that the electron _mass also gets

renormalized,

in consequence of the interaction between electron and

electromagnetic field;

_moreover, the argument for the electron _can be based _on the

fluctuation-dissipation

theorem [17],

similarly

to the present ^lines ^of

reasoning

for the one-dimensional mirror.

The derivation of the force

(4.3)

_as

presented,

where _a finite

frequency

cut-off is maintained

throughout

^the calculation of real and

imaginary

part ^of

r(fl),

calls for _some

warning

about

(6)By

contrast, the Fourier transform r(n) of the _response function with respect to the displacement (cf.

footnote 5) is not square-integrable, because at infinity Rer(n) tends to a non-zero constant. The subtraction of such _a constant amounts to writing the dispersion relations (3.3a,b) for [r(n) r(0)]/n instead of for r(n).

(8)

the no-cut-off limit.

Suppose

_one wanted to

apply dispersion

relations in the no-cut-off

limit,

I-e- try to determine

Imr(fl)

from

Rer(fl)

₌

[(n I)/n]~ l/(127r) fl~, equation (4.la),

which follows from

(3A)

and

(2.7)

for _zero cut-off-

Generally,

if

r(fl)

is not

square-integrable,

_one must amend the

dispersion

relations

(3.3a,b) by appropriate

subtractions [14].

So,

if

Rer(fl) asymptotically approaches

_a

(second-order) polynomial,

then _one needs to subtract this

polynomial

and to write the

dispersion

relation

(3.3b)

for the function

[r(fl) r(0)

fl

r'(0) (fl~/2) r"(0)]/fl~,

which

yields

Im

r(n)

= Im

r(o)

₊ n Im

r'(o)

₊ Im

r"(o)

fl~

j" ^Rer(uJ) ^Rer(0) ^uJRer'(0) ^~ Rer"(0)

7r

~

_~

~~°

(uJ

fl)

uJ3

This

dispersion

relation has three free parameters,

Imr(0), Imr'(0),

and Im

r"(0),

which _can be determined

only through

additional

physical

information about the function

r(fl).

In the present case Re

r(fl)

is not

just

bounded

by

_a

polynomial,

but is itself _a

polynomial.

So subtraction of this

polynomial simply gives

_zero for the remainder. The

analytic

continuation of _a function

vanishing along

_a finite line segment ^is ^of course zero

everywhere

in the

complex plane. Nevertheless,

_one must not

jump

to the conclusion that

Imr(fl)

is _zero; all that _can be discovered from

dispersion

relations without cut-off is that

Imr(fl)

is _a pure

polynomial

of _up to second order in fl.

Only

the

dispersion

relations for

r(fl)

with finite

frequency

cut- off tell that in fact

Imr(0)

₌ 0

=

Imr"(0),

while

Imr'(0)

₌

[(n I)/n]~ l/(67r~a).

One concludes that this kind of

approach (disregarding

_any

physical

input

beyond

the

dispersion

relations without

cut-off)

is

incapable

by construction of

yielding

sufficient information about the reactive part of the

force,

_or

consequently

about the _mass renormalization.

There is however also _a

pedestrian's argument

^for

finding

the reactive part ^of ^the ^force in the no-cut-off

limit,

which

dispenses

with

dispersion

relations. If

Rer(fl)

₌

[(n I) /n]~

l

/(127r)

fl~

then,

in the

light

^{of the} ^remarks ^below

(4.3),

the _response function

G(t r)

must contain _a term

[(n I) /n]~

l

/(127r)

b(~)

(t r).

But this is acausal since it is _non-zero for times _r _> t, even if

only

in _an infinitesimal interval. So one has to allow for

causality by

hand and put the upper

integration

limit in

(3.I)

to t. Since this eliminates half of the

delta-peak

from under the

integral

_one needs to

multiply by

two in order to retain the correct result for the

dissipative force,

which then

yields

the correct

expression

F(t)

=

'~ dr

b(~)(t r) fl(r)

n

~

67r

/~~

On

integrating by

parts ^twice one encounters a surface term

proportional

to

b(0) fl(t),

and one

is left with

F(t)

_#

(~

~

b(0) fl(t)

+

/~

^dT

^b(t

T)

#(T)

,

n 67r

_~

where

again only

half of the

delta-peak

lies under the

integral. Eventually,

the Lorentzian

representation

of the delta function

b(r)

= lim

/~~

a-0 a + r~

gives b(o)

=

1/(a7r);

_so _one _recovers the

expression (4.3)

for the force.

However,

the

simplistic

reasoning

just

presented

could do _more harm than

good:

it proves fatal to

mingle

_any of the

(9)

above

rough-and-ready

arguments ^with ^the

scrupulous

calculation where the finite

frequency

cut-off is

kept

until the last stage. ^In

particular,

the _upper

integration

limit in the _response relation

(3.I)

_must not at a wrong

point

be

replaced by

t.

A comparison ^of the force

(4.3)

with results in

previous publications

_on this

subject

is _now in order. In reference [3]

exactly

the _same

expression

is obtained

directly

from the

(perturbatively calculated)

radiated

photon

spectrum. ^The ^forces on

perfect

and _on

partially

transparent

mirrors derived in references

[7-9]

^have

dissipative

terms that

fully

_agree with

(4.3),

but lack

terms

proportional

to

fi(t), although requirements

of

causality

_seem to

explicitly

enter the

reasoning.

Also the force _on _a

perfect

mirror found _in references _11, 2] possesses

only

the

dissipative

term, ^and no mass-renormalization term.

Since the derivation of the force _on _a

perfect

one-dimensional mirror

given

in references _11,2]

rests on field-theoretical

techniques,

_mere

comparison

with the present

approach

cannot reveal

the _reason for the absence of mass-renormalization terms; ^further

investigation

will not be

attempted

here.

A _more detailed

comparison

is

possible

to the results of references

[7-9],

which also

employ

the

language

^of ^linear _response

theory.

So, for instance

just

above equation

(6a)

in [9] ^it ^is claimed that the Fourier transform xo(uJ]

(here: r(fl),

cf. footnote

(~))

^of ^the

displacement

response "scales _as uJ3 at low

frequencies".

In the present notation, this would claim that

r(fl)

is

proportional

to

fl~,

from which then had to be concluded,

along

the lines of the

paragraph following equation (4.3) above,

that there is _no term

proportional

to

fl(t)

in the force.

Indeed,

no such term is

given

in references

[7-9] (at

_zero

temperature).

However,

in references [7, 8]

causality

is found to lead to _a difference in the _mass of the mirror if

coupled

to _or

uncoupled

from the

field,

which the authors of [7, 8] refer _to _as "low-"

or

"high-frequency

mass". This difference

equals II (67r~o),

which is

precisely

the amount

given by

the mass-renormalization term in

equation (4.3).

So

far,

the present writer has _not

managed

to find out

why

the model

employed

in references

[7-9]

^leads to such _a _mass difference but _not to a reactive part in the force.

Finally,

_a few remarks

concerning moving

mirrors in three dimensions. With reasonable

effort,

_one _can work out the

time-averaged

mean-square fluctuations of the forces _on _a

piston

in _an infinite

plane [18],

_on _a

sphere

_[19] and _even _on _a flat disk

[20],

^which ^is

practically equivalent

to

calculating

the correlation functions of the fluctuations in

time(~)

if the time-

averaging

function has been chosen to be _a Lorentziau _as in references

[18-20]. However,

it turns out that the

simplicity

of the calculation and _as well _as of the end result for the _mean force

on a

moving

one-dimensional mirror is

peculiar

to _one

dimension;

in the no-cut-off limit the spectrum of the fluctuations _is

given by just

_a _power of the

frequency

fl. As known from field

theory

this is _a consequence of the conformal invariance of the field

equations

for _a massless scalar field in _one space and _one time dimension. In

simpler

words: in _more than _one dimension every

object

has _a certain cross-section that the force is distributed _over. Hence _a

length

scale

(other

than the

cut-off)

enters the

problem,

in comparison to whose inverse the

frequency

fl in the fluctuation spectrum

x(fl)

_can be small _or

large.

Then

x(fl)

is a

complicated function,

even for

simple geometries,

and _can at best be

approximated by (different)

_powers of fl for _very small _or _very

large frequencies.

This

implies

that the

expression

for the force is _not _as

simple

as in equation

(4.3)

for the one-dimensional mirror.

Generally,

it

depends

not

only

_on various derivatives of the

velocity

at the present time t but also on

integrals

_over the

history

before t. Nevertheless, under certain assumptions the expressions for the force _can be

approximated

so as to

depend only

_on derivatives of the current

velocity;

but such

approximations

_are not

(7)The

calculation of the force correlation function at different points on the surface of the object under consideration is far _more demanding; it is done for _an infinite plane mirror in reference [12], but _seems beyond manageability for spheres or even disks.

(10)

straightforward

to discover. In reference _[2]

dissipative

and reactive terms on an infinite three- dimensional mirror, but

only

when

interacting

with _a scalar

field,

have been found. _In references [6, 21] ^the ^issue ^of

dissipative

forces _on mirrors

moving through

the Maxwell _vacuum in three

dimensions has been

addressed,

but reactive forces _are _not considered. The present ^writer

hopes

to tackle the three-dimensional

problem

in _a

forthcoming publication.

Acknowledgments.

It is _a

pleasure

to

acknowledge helpful

discussions with Gabriel

Barton,

_an

illuminating

remark

by

Martin

Reuter, correspondence

with

Marc-Thierry

Jaekel and

Serge Reynaud,

and financial support ^from ^the ^Commission ^{of the}

European

Communities.

References

ill

Fulling S. A, Davies P. C. W., Proc. R. Sac. London A 348

(1976)

393.

[2] Ford L. H, Vilenkin A., Phys. Rev. D 25

(1982)

2569.

[3] ^Barton G, Eberlein C., Ann. Phys.

(1993)

to appear.

[4] Landau L. D, Lifschitz E. M., Statistical Physics, part 1, 3rd ed.

(Pergamon,

1985) _pp. 122-126.

[5] Col~en-Tannoudji C., Dupont-Roc J, Grynberg G., Atom-Photon Interactions

(Wiley,

1992) _com- plement AIV.

[6] Braginsky V. B, Kl~alili F. Ya., Phys. Lent. A161

(1991)

197.

[7] Jaekel M. T, Reynaud S., J. Phys. I France 3

(1993)

1.

[8] Jaekel M. T, Reynaud S., Phys. Lent. A 167

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[9] Jaekel M. T, Reynaud S., Phys. Lent. A 172

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[10] Moore G., J. Math. Phys. Ii

(1970)

2679.

[iii

Van Bladel J., Relativity and Engineering

(Springer, 1984).

[12] ^Barton G., J. Phys. A 24

(1991)

5533.

[13] Titcl~marsh E. C., Introduction to the Theory of Fourier Integrals, 2nd Ed.

(Clarendon,

1948)

section V.

[14]

Hilgevoord

J., Dispersion Relations and Causal Description

(North-Holland, 1960)

chapter 3.

[15] ^Handbook ^of Mathematical Functions, ^M. Abramowitz, I. A. Stegun Eds.

(US

Govt. Printing Office,

1964).

[16] ^Rol~rlich F., Classical Charged ^Particles

(Addison-Wesley, 1965).

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[20] ^Eberlein C., J. Phys. A 25

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