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the development of a single - atom maser allows a detailed study of the atom - field interaction@xcite-@xcite .
the realization of a single - atom maser has been made possible due to the enormous progress in the construction of superconducting cavities together with the laser preparation of highly excited atoms called rydberg atoms .
resonant effects@xcite associated to the coupling of atoms with strong radiofrequency field has been observed . in particular , it has been demonstrated that the spontaneous emission rate of an atom inside the cavity is different from its value in free space@xcite-@xcite .
this effect can be discussed from several different approaches , e.g. , attributed it to a change of the spectral density of the modes of the vacuum radiation field due to the cavity s resonating structure .
the theoretical understanding of these effects by making use of perturbation theory requires the calculation of very high - order terms .
it makes the standard feynman diagram technique practically unreliable in the case . on the other side ,
the nonlinear character of the problem involved in realistic situations can not be ignored simply . a naive solution of this difficulty may be to assume that under certain conditions , the coupled atom - radiation field system can be approximated by a system composed of a harmonic oscillator coupled linearly to the radiation field through some effective coupling constants .
thus , a significant number of works have been sparked to the study of cavity qed .
another motivation to study the radiation field system in cavity comes from the cosmology .
astronomical observations@xcite-@xcite have provided plenty of supports to the cosmological principle@xcite , which states that the universe is spatially homogeneous and isotropic on large scales .
this principle implies that we can build up a comoving coordinate system , in which the spatial part has a maximal symmetry . in general
, there are three types of global symmetry for the three - dimensional maximally symmetric space , _ i.e. _ , @xmath1 , @xmath2 and @xmath3 .
usually , they are called the flat , closed and open universe , respectively . in order to understand the evolution of the universe , the whole distribution and activity of matters ( including radiations ) in space have to be taken into account . and we would like to point out that , if properties of matters in space as a whole are to be studied , the difference of global symmetries should not be neglected . in particular , we know that the early epoch of the universe was governed by the radiation fields
. therefore , a careful study of radiation fields in a compact space is necessary .    in free space ,
it is well - known that the radiation fields can be described by an ideal gas model ( a thermodynamic system consisting of free massless particles ) .
however , one can not extend this idea to the case of compact space straightforwardly . in order to investigate the quantum statistical properties of radiation fields in a compact space , we begin with constructing of the representation of one - particle states .
we notice that the space - time @xmath4 is a conformally deformed de sitter space - time .
this space - time has a translation symmetry in the time direction and six rotation symmetries in the spatial part .
that is to say , the isometry group of the space - time @xmath4 is @xmath5 . by making use of the representation theory of this group
, we get a representation of one - particle states as well as the corresponding dispersion relation for the radiation field .
the density of state with respect to the energy spectrum is continuous in the case of the compact space with a large radius .
it should be pointed out that the obtained density of state is different from that of a black body in free space .
the geometrical parameter dependence of statistical properties of the radiation field system in a compact space can be obtained explicitly .
it has been a goal of cavity qed for a long time .
this paper is organized as follows . in section 2
, we will discuss the kinematics of the radiation fields in the compact space @xmath6 .
the hilbert space of free photons as well as the corresponding dispersion relations is presented .
the bose - einstein statistics of the radiation fields is intensively studied in section 3 .
statistical properties of the radiation fields , which is remarkably different from that in free space , are shown .
we then give the conclusions and remarks in section 4 .
the metric of the compact space - time @xmath4 is of the form @xmath7 where @xmath8 denotes the radius of the sphere @xmath6 .
the spatial part of the metric , @xmath9 , can be expanded with euler angular variables @xmath10 or with a unit vector in the four - dimensional euclidean space @xmath11 the isometry group of the space - time is @xmath5 .
infinitesimal elements of the isometry group can be expressed in terms of the following generators @xmath12 here @xmath13 denotes the hamiltonian and @xmath14 are angular momentums .
the motion of a free photon is described by the klein - gordon equation @xmath15 here the dalembertian operator @xmath16 is an invariant of @xmath0 @xmath17 it is convenient to introduce the set of operators @xmath18 and @xmath19 as@xcite @xmath20 and their linear combinations , @xmath21 then , one can separate the @xmath22 lie algebra @xmath23 into two parts , which commutates with each other @xmath24 = 0~,\nonumber\\ & & [ \hat{j}_{\alpha},~\hat{j}_{\beta } ] = i\epsilon_{\alpha\beta\gamma}\hat{j}_{\gamma}~,\nonumber\\ & & [ \hat{s}_{\alpha},~\hat{s}_{\beta } ] = i\epsilon_{\alpha\beta\gamma}\hat{s}_{\gamma}~,\nonumber\\ & & \frac{1}{2}\hat{m}^{ij}\hat{m}_{ij } = \hat{p}^2+\hat{l}^2= 2(\hat{j}^2+\hat{s}^2)~.\end{aligned}\ ] ] in other words , the @xmath2 group can be expressed locally as a direct product of two @xmath25 groups .
nevertheless , it is easy to see that , @xmath26 and then , we obtain the representation of the @xmath22 universal lie algebra @xmath27 @xmath28 where @xmath29 and @xmath30  .
the representation of the translation group @xmath31 is of the form @xmath32 now we can write the wave - function @xmath33 into the form @xmath34 denotes the degrees of freedom of spin . the corresponding dispersion relation for this eigen - state
is given by eqs .
@xmath35 and @xmath36 @xmath37 the hilbert space of a free photon can therefore be parameterized as @xmath38 statistical properties of radiation fields ------------------------------------------    to conveniently study statistical properties of the radiation fields in a compact space , one should compute the density of state for energy spectrum first . the dispersion relation ( [ dispersion ] ) can be rewritten into the following form @xmath39r^{-2}=k^2-r^{-2}~,\ ] ] where we have introduced @xmath40 for convenience .    for a given energy level @xmath41 ,
the value of @xmath42 ( or @xmath43 ) is determined .
there exist @xmath44 eigen - states at this energy level resulted by summing over the index parameters @xmath45 and @xmath46 .
if the scale of the radius @xmath8 is large enough , one can regard the spectrum of energy and @xmath42 as a continuous distribution .
then , we get a measure of states with respect to the energy spectrum , which is just the density of state , @xmath47 the density of state is obviously different from that of the ordinary black - body radiation system in free space , which is proportional to the square of energy ( @xmath48 ) .
the photons obey the bose - einstein statistics .
the mean number of photons on an eigen - state with energy @xmath41
is@xcite @xmath49 where @xmath50 is the temperature of this system to be zero and hence @xmath51 . ] .
immediately , we obtain a modified planck distribution for the radiation fields in the compact space @xmath6 , @xmath52 the distribution function represents the amount of radiation energy in the spectral interval @xmath53 and a unit volume . in picture * 1 * , we give a sketch for this modified planck distribution comparing with the ordinary one in free space .
[ pic ]     ( real line , @xmath54 ) and the ordinary distribution of a black - body radiation system in free space ( dashed line).,width=377,height=264 ]    the total number of photons of the radiation system can be expressed as @xmath55 when the parameter @xmath56 runs to infinity , we recover the familiar result of the black - body radiation system in free space . in the follows
, we will calculate quantities by expanding them in terms of @xmath57 to see clearly the deviation from those got in free space .
the thermodynamic quantity of total energy is @xmath58 where @xmath59 ( @xmath60 ) is the riemann zeta - function .
the free energy and thermodynamic potential can also be calculated easily , @xmath61 we know that the volume of the three - dimensional sphere with radius @xmath8 is @xmath62 .
hence the radiation pressure is @xmath63 furthermore , by making use of the relations among thermodynamic quantities @xmath64 we can obtain the entropy of radiation fields @xmath65    in an adiabatic process experienced by the radiation fields , the entropy does nt change .
so the product @xmath56 can be viewed as a constant in the process .
the total number of photons is also kept invariant in the adiabatic process . but the other thermodynamic quantities will vary with the geometrical parameter @xmath8 as follows , @xmath66 to discuss the equation of state of this radiation system , we now calculate the density of thermal energy , @xmath67 the equation of state of the radiation fields in the compact space can be read out from eqs .
( [ total - pressure ] ) and ( [ density - energy ] ) , @xmath68 it is obvious that , in the case of a large radius of the compact space , the ordinary equation of state for the radiation fields in free space can be recovered .
a good understanding of the resonant effect and spontaneous emission rate difference in cavity requires first to know the full space dependence of the cavity - induced damping and level shifts .
however , we know that it is a really difficult mathematical task . thus , many useful approximation methods have been developed . as a first step on the way of getting an exact solution of the problem , we suggested working on a compact space of finite size instead of a cavity . by making use of the representation theory of the @xmath5 group
, we studied the kinematics for a free photon propagating in the space - time @xmath69 .
the explicit geometrical parameter dependence of statistical properties of the radiation fields was presented .
results showed remarkable differences from that of the black - body radiation in free space .
the resonant effects associated to the coupling of atoms with strong radiofrequency field as well as the spontaneous emission rate can also be computed in a straightforward way .
it is expected that the nonlinear character of the coupled atom - radiation field may be attributed to the geometrical parameters of the compact space .
these results will be published in a forthcoming paper .
in fact , the statistical properties of radiation fields in a compact space are crucial to the evolution of the early universe . in the early universe ,
radiation field is the dominant matter
. a bit difference of statistical properties of the radiation system can generate observable effects on our present cosmology .
all of these studies are in progressing .
* acknowledgement : * + one of us ( c.b.g ) would like to thank dr .
yu - qi li and xiao - jun wang for useful discussion and suggestions .
the work was supported partly by the natural science foundation of china . c. b. guan is supported by grants through the icts ( ustc ) from the chinese academy of sciences .
we discuss radiation fields in a compact space of finite size instead of that in a cavity for investigating the coupled atom - radiation field system . 
 representations of @xmath0 group are used to give a formulation for kinematics of the radiation fields . 
 the explicit geometrical parameter dependence of statistical properties of radiation fields is obtained . 
 results show remarkable differences from that of the black - body radiation system in free space .    
 pacs numbers :  42.50.pq ,  87.50.jk ,  11.25.mj  . 
 + keywords :  cavity qed ,  radiation fields ,  compact manifold  . 
 +     +   +    * statistical properties of radiation fields + in a compact space *    zhe chang + _ institute of high energy physics , chinese academy of sciences _ + _ 
 p.o.box 918(4 ) , 100039 beijing , china _ + 
 cheng - bo guan + _ interdisciplinary center for theoretical study _ 
 + _ university of science and technology of china , 230026 hefei , china _