Excitation of a particle with interna! structure moving near an ideal wall

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1 Remita Mezicana de Fúica 39, Suplemento 2 (1993) 1' Excitation of a particle with interna! structure moving near an ideal wall A. KLIMOV* Instituto de Física, Universidad Nacional Aut6noma de México Apartado postal , México, D. F. México ABSTRACT. nfluence of boundary conditions on the excitation of the rotating detector is studied. It is shown that there is a motion regime in which the presence of an ideal wall could change noticeable the response function of the detector. RESUMEN. Se estudia la influencia de las condiciones de frontera en la excitación de un detector rotante. Se muestra que existe un régimen de movimiento en el cual la presencia de una pared ideal cambiarla de una manera notoria la función de respuesta del detector. PACS: w; x Existence of zero-point f1uctuations of quantized fields leads to the set of observable efreets. One of the most striking examples is the Casimir force which arises between objects with finite (dielectrics) or infinite (conductors) polarizability. Also the motion of bodies, interacting with a quantized field may cause quantum radiation [1]. Quantum radiation of bodies with an internal structure may be accompanied by the transitions from one energy level to the other. Systems with internal degrees of freedom interacting with quantized field are usually called detectors [1-3). It is well known that the excitation of a detector moving in vacuum can arise only in the case of nonuniform motion. For example a detector moving with constant acceleration a has a thermal distribution over energy levels with efrective temperature T= ha. However the temperature T = 1 K corresponds to the acceleration a ~ cm/c 2 making very difficult to use macroscopic objects as detectors of vacuum f1uctuations. As the most acceptable objects for this purpose one can consider elementary partides and nuclei, having some "interna!" quantum states (quantum numbers), for example spin projection S, on some axis Z [41. So there are two kind of quantum efrects: 1. The interaction of a nonmoving detector with sorne macroscopic object and the Casimir force between them (obviously, that static interaction does not cause excitation of the detector, it could just shift the energy of its bound state); 'On leave from Karpov Institute of Physica! Chemistry, Moscow. 21fC

2 EXCITATION OF A PARTICLE The excitation of a detector in the process of his motion and quantum radiations connected with it. We will consider both effects together in one example, namely, a uniformly rotating detector in a hah space bounded with an ideal (perfectly conducting) wall. The problem of the excitation of a uniformly rotating detector interacting with scalar and electromagnetic fields in free space has been studied in a number of articles [2,5,61. For simplicity, let us consider the interaction ofthe detector with a scalar field ~(x). The boundary condition imposed on the field ~(x) is: ~(t,x = O,y, z) = O, when the position of the wall is the plane with x = O. The Hamiltonian of interaction in the monopole approximation has the following form Wint = J dxm(x)~(x), where m(x) is the operator of monopole momentum of the detector, x = (t, x, y, z). The transition probability between energy levels of the detector In) -> 1m) can be written down (in the second order of perturbation theory) as follows [1]: Here D nm = (mlm(x)ln) is the detector function, which does not depend upon the character of motion. The response function of the detector F(E) is defined as the Fourier transformation of the Wightman function D+(x, x') = (OI~(x)~(x')IO), F(E) =1:d(r - r') exp( -ie(r - r'»)d+(x,x'), (1) where r is the proper time of the detector. The response function F(E) doesn't depend upon the internal structure of the detector. The positive frequency Wightman function D+(x, x') has the following form for the detector in half space 1 1 D+(x x') = D+ - D+ = , O I 471'2 [r: _ (t _ t')2] '2 [r~ - (t - t')2], (2) where r~ = (x 'f X')2 + (y - y')2 + (z - Z')2. According to the formulas (1) and (2) we willlook for the response function in the form F(E) = Fo(E) - F (E). The trajectory of the detector for the uniform rotation case can be written as x = {-yr,xo + Rcos(í!r),Rsin(í!r)} where r = tl is a proper time, í! = Wo is a proper frequency. This trajectory corresponds to the rotation with the frequency Wo and radius R.

3 150 A. KLJMOV Substituting the explicit expression for detector's trajectory in the formula (2) and making the Fourier transformation (1) we get for the wall-independent part of the response function Fo(E) the following formula (3) where A = E In, (3= wor, ko = [Al+ 1. In the case of ultrarelativistic motion "1» can obtain the following approximation for the expression (3) Fo(A) wo{32 = -- exp( -l/a) 4""Il/ 1 one (4) where l/2 = 12/Uh)2, which coincides with the previously obtained in a different way approximation [51. For the case of uniformly accelerated detector the response function F.(E) has the form of thermal distribution F.(E) = E{ 271"(exp(27I"E/a) _ 1)}- which take in an ultrarelativistic case the form analogous to Eq. (4) (for a suitable range of E): Fa - a (271")2exp(-27I"E/a). As far as concerns the wall-dependent part of response function F (E), it depends periodically upon time T = T + T'. We will average F over the period of rotation ij = 271" /wo: 1 r~ (F (E,T)) =:a Jo dtf (E,T). Assuming that the radius of the rotation is much more less then the distance from the centre of the rotation to the wall: Xo» R, we get F (A) =...2:... [f:(-i)ksin(2xowo(k - A))ff({3(k - A)) + O(R/XO)]. (5) "IXo k=ko Usually (F (Al) is much more less then Fo(A). So we could say that the influence of boundary conditions on the excitation of the rotating detector is quite small (usual situation in the problems connected with Casimir forces). But there is a "resonant" case, when the influence of boundary, though being rather small, could be considerable. The

4 EXCITATION OF A PARTICLE , o ~ - 00 ::-02 -O'" ( -O 6 -O 8-10 O-O O 2 O -1 O () OH O I 2 I" H 2 O :2 () 2 8 :J O A FIGURE 1. Response function F (A) (in units,, -'xo) for "resonant" case: XoWo = 2.5". {3= [ o ;; o o ~ , 2-15 o o o 2 DA o 6 o B 1 o o A :1o FIGURE 2. Response function F (A) (in units,, -'xo) for "non-resonant" case: XoWo = 2". {3 = resonant condition corresponds to the case when the time for the signal passing to the wall and back equals to the half period of motion of the detector: A 2xo = 7'ó(2N + 1) N = Wo Under that condition we can take the rapidly oscillating sinus term out of the sumo It allows us to rewrite (5) in the form F[es(A) = ~ sin(2xowoa) 2: Jl ({3(k - A)); Xo k=k o <X>

5 152 A. KLIMOV in contrast with the "non-resodant" case id which the sum in Eq. (5) will be very small due to rapid oscillatiods of sin(2xowo(k - A)). Graphics of F (A) for "resodadt" add "non-resonant" cases are presented id Figs. 1 and 2, respectively, where (3 = 0.95, and XoWo = ".( ) for resonant and XoWo = 2". for non-resonant cases. One can see that the amplitud e of oscillations in "resonant" case is much more greatly despite the distance from the wall in "non-resonant" case is less. Comparing response functions for free add half spaces we see that the value of walldependent part F (E) is about 3% of the wall-independent part Fo(E) in the "resonant" case. The author is grateful to R. Jauregui and S. Hacyan for useful discussions and critical remarks. This work is supported by CONACYT and Instituto de Física UNAM. REFERENCES 1. N. Birrel, P. Davies, Quantum fields in cutwd space, CUP (1982). 2. S. Takagi, Prog. Theor. Phys. Suppl. 88 (1987) V. Ginzburg, V. Frolov, Proc. Lebedev Phys. Inst. 197 (1989). 4. J.S. Bell, J.M. Leinaas, Nucl. Phys. B212 (1983) S. Kim, K. Soh, J. Yee, Phys. Rev. D35 (1987) S. Hacyan, A. Sarmiento, Phys. Rev. D40 (1989) 2641.