The scalar quantum field uniformly moving plates in three dimensional space is studied. Field equations for Dirichlet boundary conditions are solved exactly. Comparison of the resulting wavefunctions with their instantaneous static counterpart is performed via Bogolubov coefficients. Unlike the one dimensional problem, 'particle' creation as well as squeezing may occur. The time dependent Casimir energy is also evaluated

Hacyan, S.

Jauregui, Roco

Villarreal, Carlos

English

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QUANTUM FIELD BETWEEN MOVING MIRRORS:
A THREE DIMENSIONAL EXAMPLE
S. Hacyam R. J_uregui, C. Villarreal
Instituto de Fisica, U. N. A. M., Apdo. Postal 20-36_,
Mdxico D.F. 01000, Mdxico
Abstract
The scalar quantum field between uniformily moving plates in three dimensional space is
studied. Field equations for Dirichlet boundary conditions are solved exactly. Comparison
of the resulting wavefunctions with their instantaneous static counterpart is performed via
Bogolubov coefficients. Unlike the one dimensional problem, "particle" creation as well as
squeezing may occur. The time dependent Casimir energy is also evaluated.
1 Introduction.
During the last twenty five years, nmch effort has been devoted to tile understanding of quantum
phenomena in systems under the influence of external conditions. In particular, Moore [1] initi-
ated the study of the quantization of the electromagnetic field in a cavity with perfectly reflecting
movable boundaries. Nowadays, it is recognized that this kind of system has several interesting
nonclassical properties. Among them, there is the possibility of producing a nonadiabatic distor-
tion of vacuum state leading to a modification of the field (Casimir) energy [2], along with the
"creation" of photons [3]. It is also possible to obtain nonclassical statistical properties of the
photons inside such a cavity: squcczing [4] and nonthermal distril)utions [5] are expected.
In order to avoid technical complications, most investigations of the field between moving
plates have been restricted to the one dimensional case. However, it is not obvious whether all the
results can be extrapolated to the three dimensional space. In this article, we study the quantum
mechanics of a scalar massless field propagating between two plates which approach or recede each
other with constant speed. The main results which follow are that the boundary conditions on the
moving plates produce squeezed states and a nonzero vacuum expectation value of the particle
number operator. These effects vanish in the one dimensional case [6]. The nonstationary Casimir
energy is also evaluated.
2 Quantum field between the plates
Consider two parallel'plates which are moving with a constant relative velocity.
coordinates for this problem are
t=Tcosh_, z=vsinh_,
The natural
(1)
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https://ntrs.nasa.gov/search.jsp?R=19950016548 2020-06-16T08:04:58+00:00Z
where z and t are the Minkowski coordinates. Taking -cx_ < r, _ < cx_, the Milne coordinates
cover the entire past and future quadrants of the (z, T) plane.
The equation for a massless scalar field _b is:
me- rot r ¢, + 2 + + ¢=0. (2)
Now, the world-line of each plate can be taken as ( = ±(0, where (0 is the speed of each plate
as seen in their center of velocity frame. Dirichlet boundary conditions on the plates take the
simple form ¢(=k(0) = 0. (It is also straightforward to impose Neumann boundary conditions.) It
will be convenient to normalize the field in a box with fixed walls with separation a and b in the
x and y directions
The general solution of the wave equation with the above boundary conditions can be de-
composed as the product of a function of ( and T, and a plane wave solution propagating in the
r =_ {x,y} plane with wave vector k - 7c{n_/a, nv/b }. Explicitely,
_/',,k = N_,sin(kxx) sin( kvy) sin [v( _ + _0)] H_J)(kr), (3)
where
1/2N¢:L = 7r e+_./2" (4)
is a normalization factor, H_ ) are the standard Hankel functions (j = 1,2), k - Ikl, and we
have defined v =_ n_r/2(o, n being a positive integer. In the future region, t > 0, H_2) and H_J )
correspond to modes of positive and negative frequency respectively, while the opposite is true in
the past region, t < 0 (see, e. g.: [7][8]). We will deuote the positive (negative) frequency modes
by ¢+ (¢-).
At this point, we note that the field between plates separated a fixed distance L is given by
¢.,i<= + LI2)], (5)
where w = [k 2 + (n_r/Ij)2] '/2 and the normalization coefficient is now
(6)
This coefficient follows from the scalar product in Minkowski coordinates:
/LI2 .002(Ol, _)Ins = --i J-LI2 dz i dy d.v(¢ 1 _ - ¢_ ), (7)
where the subindex Ins refers to the instantaneous frame: the integration is taken over the volume
enclosed by the fixed box at an arbitrary time t.
Hereafter, the field modes between moving plates will be called dynamical modes, whereas the
modes between fixed plates will be called instantaneous modes. The crucial point is that between
moving plates, the positive frequency modes of the dynamical field are a sum of both positive and
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negative frequency modesof the instantaneousfield betweenfixed plates. In general, any field
mode Cncan be expandedin terms of ¢m as
wheream, and _m. are the Bogolubov coefficients, and the indices m and n describe the set of all
parameters characterizing the modes. In the particular case we are interested in, take O and t/, as
the wave functions describing the fields between fixed and moving plates respectively. More pre-
cisely, consider a pair of plates of fixed separation L = 2 t tanh _0 which coincides instantaneously
at Minkowski time t with a pair of plates moving with relative speed v = tanh(2_0) (Fig. 1). Now,
the Bogolubov coefficients can be'calculated taking the scalar product between the _m and ¢,
modes over the hypersurface t = const, between the plates, (Note that for definitiveness we are
considering the future region t > 0 where the plates are separating but it is straighforward to
adapt the analysis to the past region.) Thus,
+ +
"m,.,k,k' = (O.,k, %,k' (9)
and
-- _m,k,)ln s .flm,n,k,k' (0n, k, + (10)
The integration involved in Eqs. (9) and (10) is to be performed over the hypersurface t = const.
bounded by the plates with separation L = 2 t tanh _'0, with t interpreted as a parameter. At this
point, we note that in any practical case, the velocities of the plates are non relativistic, that is
_'0 << 1. This permits to make a convenient approximation which, together with the change of
variables z = t tanh _, simplifies Eq. (9) to tile form:
a b4 [d H}_)(kt) - iwH}_)(kt) ] ei_t (11)
with an analogous equation for/3. In this nonrelativistic approximation, it is very convenient to use
the asymptotic forms of the Hankel functions which are valid for indices with large magnitudes[10]:
H}2)(kt) __ V/2-/rr(/12 + k2t2) -1/4
exp [-/17r/2 -i(/12 + k2t2)U2 + i/1Arsh(/1/kt) - i7r/4] .
It finally follows that the Bogolubov coefficients can be approximated as
[ .-i(kt) 2 .] e-ni/4+ivArsh(_'/kt)
Om,n,k,k _ "' _kk' (_m,n [1 q'- 4[/1./2 + (kt)213/2j .
(12)
(13)
and
i(kt)2 ] e-'_i/4-2i[_'2+(k')2] '/2+i'A'_h('/k,) (14)/3m,n,k,k' "_ _kk' _rn,n 4[/12 at- (kt)213/2j
It is also worth mentioning that in tlm case when there are no plates, the/3 coefficient turns out
to be null when evaluated as a scalar product over the entire r = const, hyperplane (the interested
reader can check this point using the standard properties of the Hankel functions). This implies
63
that the Milne vacuum is equivalent to the Minkowski vacuum (seeref.[8] for a discussionof this
point).
However,unlike the one dimensional case,the coefficient fl is not null when the field is rec-
tricted between the moving plates; thus the Fock space defined by the dynamical modes {¢,k}
is nontrivialy related to the instantaneous Fock space specified by the fixed modes {¢,k}. which,
in principle, can be interpreted as the number of particles "created" by the motion of the plates;
this point will be further discussed in the next section.
In particular, the dynamical vacuum state 10 >by, has a nonzero expectation value of the
number of "instantaneous" particles. Thus, the dynamical vacuum is a nontrivial distortion of
the instantaneous vacuum state 10 >1,_. Explicitly, the "particle" number density is given by the
distribution function
Pk,., = _ Iflk,. 2 = (kt)4
k,, 16[ v2 + (kt)2] a' (15)
Notice, however, that the real character of this particles is intrinsically related to its measurability.
The Bogolubov coefficients also relate coherent and squeezed sates. A state which is originally
coherent according to a instantaneous configuration of the plates becomes a squeezed state with
variances [2]:
[ 1 . j
and
[ 1 . ]
_.,,.. = Im_ _;,;fl._+ _(_-d.i + _;.) •
In our problem i denotes the set of variables k, n. So that,
(16)
av,,v _ =_ 5u +
+
(kt)' {1+ cos(2[._+ (kt)_]'/_)}
(kt) 2 sin(2[v2 + (kt)2] '/2)
(_ij 4[v2 + (kt)213/2 (17)
and
O'Xi,p j --_ -- (kt)' sin(2[v2 + (kt)2],/2)5'i 16[v2 + (kt)2] a
(kt)_ cos(2[_ + (kt)_]_n)
5'J 4[p2 + (kt)213/2 (18)
Thus, the squeezed ellipse in phase space rotates with its ellipticity vanishing as an inverse power
of time.
3 Casimir effect
Boundary conditions in any given system may alter its ground state. A well known example in
quantum field theory is Casimir effect, i.e., the attractive force between two infinite conducting
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plates in otherwise empty space. A direct consequence of the existence of Casimir forces is that
mantaining even uniform relative motion of a pair of conducting plates requires external forces. It
is also expected that the Casimir energy for nonstationaxy boundaries differs from the stationary
case. In fact, one could think that the creation of particles with distribution (4.3) or the squeezing
(4.5) of originally coherent states take place at tile expense of the Casimir energy between the
plates [2]. Notice, however, that such an interpretation is not obvious because the distribution of
particles (4.3) diverges when integrated over all momenta. This is a consequence of dealing with
idealized conducting plates.
The energy density for our nonstationary problem is given by [11]
lf0_e= - d_2[b+(w, T) + b-(w,T)]7r
where D+ denotes the Fourier transform of Wightman D + functions:
(19)
/_¢ 1 (20)r) = dae"'"D+(v + la, 7 - -_a)
O0LI
1 1 1 1
D± (T + __a, T _ __O) = D_ < OI¢(.r + -_a, x, y, ()¢( r T -_,_, x, y, C) IO > D_. (21)
The free Wightman function in Milne coordinates is given by
D_F(X,X, ) _ 1 147r2 --T 2 -- T'2 + 27T' cosh(_ + (') + (y- if)2 + (x - x') 2 (22)
The boundary conditions in our problem can be easily imposed by image method. So that, for
two infinite plates
D+(x,x') = - 1 _ 147r 2 _ -v 2 - v '2 + 2Tv' cosh(¢ + ¢' - 4(0n) + (y - y,)2 + (x - x') z (23)
n_----OO
The energy of the field between plates per unit area is
1
f]_o d; T EE=-_ _o (24)
When performing the ( integration two different contributions in the energy density arise. The
first one has terms independent of the C0 value. It is formed by the D + term and by the zero
mode term of D-. The Fourier tranform of the latter is the well known w/2 which gives rise
to infinite vacuum expectation value in free space. The second kind of terms correspond to the
Casimir energy per unit area, which is explicitely given by
Ec - 4_r2 (_7) 3 sh4(_n¢o ) + _ sh2(-_n_o )
In the nonrelativistic limit:
 ,0[z 1 1Ec _- 452 (_-r) 3 (2(on) 4 + _ (2_n)2 (26)
65
and the instantaneous separation of the plates is
1 3
L "- 2r_0 - 5T_ (27)
Thus, we recover the static Casimir energy and find the first order correction due to the movement
of the plates.
4 Concluding remarks
From the results obtaiaed above, it is clear that the three dimensional case contains many features
which are not present in one dimension. Roughly speaking, the one dimensional case corresponds
to the limit k - 0 of our formulas, that is, when there are no modes propagating parallel to the
plates.
The first thing to notice is that there is a squeezing of quantum states between the moving
plates, although with peculiar oscillating variances.
The other important result concerns the possibilty of creating "photons". If one believes
the standard interpretation of particle number (see, e. g. [3]), the motion of the plates creates
new particles with a distribution function given by Eq. (4.3). This interpretation is qualitatively
consistent with the change in Casimir energy due to the movement of the plates. In fact, whether
real particles are created is a question which can be settled only when an operational definition of
particle is given, for instance in terms of the interaction of the field with a well defined detector,
e.g. an atom.
The results presented here are still preliminary since we have analized only a scalar field.
The case of an electromagnetic field will be studied in a forthcoming publication. We expect
that by considering a more realistic field, several problems will become clearer. Among them, the
detectability of "created" particles by an incoming atom originally in an ordinary stationary state.
In any case, the problem seems to be sufficiently rich to deserve further considerations.
References
[1] G.T. Moore, J. Math. Phys. 11,2979 (1970).
[2] V. I. Man'ko, Symmetries in Physics, A. Frank and K.B. Wolf Editors, Springer-Verlag,Berlin,
1992, p. 311.
[3] N. Birrel and P. Davies, Quantum Fields in Curved Space, Cambridge University Press,
Cambridge, England, 1982.
[4] V. V. Dodonov, A. B. Klimov and V.I. Man'ko, Phys. Lett. A 149, 225 (1990); J. Sov. Laser
Research 12,439 (1991).
[5] S.Sarkar, J. Phys. A21,971 (1988).
[6] R. I. Baranov, Yu M. Shirokov, Soviet Phys. JETP 26 1199 (1968); S. A. Fulling and P. C.
W. Davies Proc. R. Soc. Lond. A 348,393 (1976).
66
[7] A. di Sessa, J. Math. Phys. 15,1892 (1974); Padmanabhan, Phys. Rev. Lett. 64,2471 (1990).
[8] R. C. Arcuri, B. F. Svaiter and N. F. Svaitcr, Mod. Phys. Lett. A 9,19 (1994).
[9] S. Hacyan, R. Jauregui, F. Soto and C. Villarreal J. Math. Phys. h 23, 2401 (1990).
[10] W. A. Watson, ,4 treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University
Press, Cambridge, England, 1944.
[11] S. Hacyan, h. Sarmiento, G. Cocho and F. Soto, Phys. Rev. D32,914 (1985).
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Quantum field between moving mirrors: A three dimensional example

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