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since the pioneering work of casimir @xcite , vacuum energy of quantum fields has been the subject of intense investigations from both experimental and theoretical sides  @xcite .
experimental measurements of the casimir forces , by using an atomic force microscope or micro - electromechanical system , reach the high precision at the level within 1% and agreement with the theoretical prediction is also at the same level at least for zero temperature .
theoretical investigation of the casimir effects extends a variety of fields of physics such as particle physics , atomic physics , astrophysics and cosmology , and condensed matter physics . in particle physics , for example
, the casimir energy of quark and gluon fields inside a hadron makes essential contributions to its mass .
the casimir force offers one of the effective mechanisms for spontaneous compactification of extra spatial dimensions in the kaluza - klein theories .
this paper discuss more theoretical issue , i.e. we examine the consistency between the existence of the casimir energy and the poincar symmetry in the setting of two uniform perfectly - reflecting parallel boundary planes at rest . in this configuration ,
the quantum field theory is invariant under the time - translation , the translations and boosts along the plane , and under the rotation in the plane . as a result of these invariances of the theory
, it seems that , if require the translational invariance of the vacuum ( vanishing total momentum of the field ) , then the vacuum energy should vanish .
this argument has a loophole as expected .
we pay our attention to the representation of symmetries in quantum theory and to the fact that , in order to compare the zero point energies , we have to consider time - dependent hamiltonian connecting the different static configurations .
the paper is organize as follows . in sec .
2 , we set up the problem in the example of a massless scalar field .
3 summarizes the projective representation and linear representations of symmetries in quantum theory . in sec .
4 , an adiabatic process connecting two static configuration is analyzed .
the final section is devoted to the conclusion .
in this sectionwe set up the problem in the case of massless scalar field theory in @xmath3 space with the dirichlet boundary condition @xmath4 on the planes @xmath5 .
this theory is invariant under the translations along the @xmath6directions and the time direction , under the rotation in @xmath7plane , and under the boosts along the @xmath6directions .
these transformations and their compositions form the ( 1 + 2)-dimensional poincar group . denoting the respective generator as @xmath8
, we are given the lie algebra of the poincar group @xmath9 & = [ p_i , h ] = [ j ,   h ] =   0 ,      \label{poi1 } \\
i [ j , p_i ] & = - \epsilon_{ij } p^j , \quad\quad    i [ j , k_i ] =   \epsilon_{ij } k^j , \label{poi2}\\   i   [ k_i , k_j ] & =   \epsilon_{ij }   j ,     \quad\quad\quad i [ k_i , h ] =   p_i , \label{poi3}\\    i [ k_i , p_j ] & =   \delta_{ij } h ,   \label{poi4}\end{aligned}\ ] ] with @xmath10 , the antisymmetric tensor @xmath11 and summation for the repeated indices being implied .    since the field @xmath12 satisfies the dalembert equation and the dirichlet boundary condition , @xmath12 can be expanded as @xmath13 where the expansion coefficient @xmath14 satisfies the ( 1 + 2)-dimensional klein - gordon equation @xmath15 with @xmath16 .
i.e. each mode @xmath14 forms ( 1 + 2)-dimensional scalar field with the mass @xmath17 .
the lagrangian density @xmath18 for the ( 1 + 3)-dimensional field @xmath12 is written as the sum of the lagrangian density @xmath19 for the ( 1 + 2)-dimensional fields @xmath20 : @xmath21 as a result , the ( 1 + 3)-dimensional poincar algebra is decomposed to a direct sum of the ( 1 + 2)-dimensional ones generated by the generators @xmath22 for each @xmath23-th mode .
these generators have the same form of commutation relation as eqs .  . in the following we take up the @xmath23-th mode and drop the upperscript @xmath24 .
we express the generators of the poincar algebra for the @xmath23-th mode in terms of the canonical conjugate pairs @xmath25 , whose dynamics is derive from the lagrangian   and whose commutation relations are given by @xmath26 & = [ \pi(t , \bm{x } ) , \pi(t , \bm{x}^\prime ) ] = 0 ,      \label{ccr1}\\ [ \phi ( t , \bm{x } ) , \pi ( t , \bm{x}^\prime ) ] & = i \delta^2 ( \bm{x } - \bm{x}^\prime )        \label{ccr2}.\end{aligned}\ ] ]    following noether s prescription , we obtain    @xmath27    using the canonical commutation relations  , , we see that eqs .  
are satisfied .
in particular , the vacuum expectation values of eq .
@xmath28 = h \quad ( i = 1 , 2 ) \label{poi4i}\ ] ] gives @xmath29 \left| 0\right\rangle } = { \left\langle 0\right|h \left| 0\right\rangle } .
\label{h0}\ ] ] if we require the translational invariance of the vacuum , @xmath30 , then eqs .   and give @xmath31 which means @xmath32 in contradiction to eq .  .
usually , in field theories without boundaries , with the aid of the arbitrary additive constant inherent in the definition of the hamiltonian , @xmath33 is redefined to satisfy @xmath34 , which , in turn , seems to mean the nonexistence of casimir energy .
this is nothing but the inconsistency sketched out in sec . 1 .
in the following sections , we show that this apparent inconsistency disappears if we note the following two points :    * in quantum theory , symmetries are represented _ projectively _ in general , and represented linearly if certain condition is satisfied . * in the casimir effect , comparison between the vacuum energies of two different configuration are made : in the setting we are considering , two configuration of the plates are e.g. @xmath35 and @xmath36 .
since the system should be described by a single hamiltonian , we are to consider time - dependent hamiltonian connecting @xmath35 and @xmath36 .
in sec . 2 we describe the lie algebra of poincar group as eqs .  . if the group is linearly represented , i.e. represented by a homomorphism from the group to linear operators , then the lie algebra is nothing but the commutator algebra of the generators .
however , as is well - known , in quantum theory , symmetry group @xmath37 is represented projectively in general  @xcite : unitary operators @xmath38 form a linear representation up to phase factor , which means @xmath39 setting @xmath40 without loss of generality and expanding @xmath41 around @xmath42 , we get , from eq .  , the algebra of the generators , wherein there appear central charges corresponding to the phase factor @xmath43 . the associative law of the products of @xmath41 s @xmath44 gives some constraints , called the cocycle condition , on the the phases @xmath45 . if we multiply @xmath41 by a phase factor @xmath46 and redefine @xmath47 as @xmath41then the phase becomes @xmath48 . in most cases ,
this redefinition of @xmath41 could makes the phase factor to disappear  @xcite
.    in our case of ( 1 + 2)-dimensional poincar group , the algebra of the generators @xmath8 has the central charges in the righthand side of eq .   : for example , @xmath49 & =   p_i + c_{0 , 0i } ,         \label{projkh}\\   i [ k_i , p_j ] & =   \delta_{ij } h     + c_{j , 0i } ,    \label{projkp}\\   i [ j , p_i ] & = - \epsilon_{ij } p^j + c_{i , 12}.   \label{projjp}\end{aligned}\ ] ] the cocycle condition for the central charges is @xmath50 from these equations , we get @xmath51 thus , we can eliminate @xmath52 and @xmath53 by redefining @xmath54 and @xmath55 as @xmath56 and @xmath33 , respectively .
other central charges disappear by similar redefinitions of @xmath57 and @xmath58 .    the choice of arbitrary additive constant in the definition of the hamiltonian , mentioned in sec .
2 , corresponds to the elimination of the central charge @xmath53 .    as for the setting of two parallel plates discussed in this paper ,
even if we eliminate the central charge in one configuration , there remains non - zero central charge in the other configuration .
the casimir energy is the difference between the vacuum energies of two different configurations , in our case @xmath35 and @xmath36 .
the hamiltonian of the @xmath23-th mode scalar field given by @xmath59 has the mass parameter @xmath60 and hence become time - dependent when connecting these two configurations .
we set the time - dependence of @xmath61 as , for example , @xmath62 as shown in fig . 1 .    , width=283 ]    in the regions @xmath63 and @xmath64 , the hamiltonian , denoted as @xmath65 and @xmath66 respectively , is time - independent , and hence the system is invariant under the infinitesimal poincar transformations .
we denote the vacuum state of @xmath65 as @xmath67 and the energy eigenvalue as @xmath68 : @xmath69    in the region @xmath70 , the state @xmath67 is not necessarily an eigenstate of the hamiltonian @xmath71 .
however , because the excited states of @xmath71 given by eq .
consist of quanta with the mass @xmath72 , excitation energy above the ground state is greater than @xmath73 .
hence we can invoke the adiabatic theorem : if we take @xmath74 large enough , then the state @xmath67 remains the ground state of @xmath71 , which means in particular @xmath75    now , we consider , in the regions @xmath63 and @xmath76 , the consistency of invariance under the infinitesimal poincar transformations and the existence of the casimir energy . we take the generators for the translations and the boosts as @xmath77 the generators @xmath78
are independent of @xmath61 and are defined by normal ordered product , while the generators @xmath71 given by eq .   and @xmath79 are dependent on @xmath61 and hence are not normal - ordered . straightforward calculation of the left - hand side of eq .   for @xmath80 shows that @xmath81 = h(l ) - e(l ) ,   \label{kp}\ ] ] where @xmath82 is an @xmath61-dependent constant . from eqs .
, , , we see that , if we choose @xmath83 and @xmath84 , then the translational invariance of the vacuum @xmath67 @xmath85 and the commutator algebra with the central charges of poincar generators @xmath86 = h(l_0 ) - e_0   \label{k(l0)}\\ i [ k_{i}(l_1 ) , p_i ] = h(l_1 ) - e_1 \label{k(l1)}\end{gathered}\ ] ] are compatible , in contrast to eq .  .
by adding constant @xmath68 to @xmath71 , we could redefine @xmath71 so as to eliminate central charge in eq .
, i.e. we could reproduce eq .   in the region @xmath63 . then
, in the region @xmath76 , there remains central charge @xmath87 in eq .  .
in this paper , we have confirmed the consistency between the existence of the casimir energy and translational invariance of the vacuum of the poincar invariant massless scalar field in the configuration of two parallel boundary plates .
the points are :    * since , in the casimir effect , comparison between the vacuum energies of two different static configuration are made , we are to consider time - dependent hamiltonian connecting these static configurations . *
even if we could choose the additive constant of the hamiltonian so as to make the representation of the poincar group linear ( no central charge in the algebra ) in the one configuration , the representation of the group become projective ( nonzero central charge in the algebra ) in the other .
a few comments are order .
first , the additive constants such as @xmath82 in eq .
are divergent and the discussion in this paper is formal : we should investigate the energy density @xmath88 ( the eigenvalue of the hamiltonian density @xmath89 ) instead of the total energy @xmath82 ( the eigenvalue of the hamiltonian @xmath90 ) . in particular , we are to consider , instead of eq .  , the commutator between @xmath78 and the boost operator denisity @xmath91 , which may suffer from the singular schwinger terms appearing in the commutator among the components of stress tensor @xcite .
second , we have focused on the @xmath23-th mode @xmath20 in most of the present paper .
we should sum up all of the modes and treat the resulting divergence using some regularization .
detailed study of these two points will be a subject of further research .
we thank the anonymous referee for noticing the importance of the expectation values and commutators of stress tensor and the related references .
one of the authors ( m. m. ) thanks c. hattori , m. matsuda , and t. matsuoka for discussion .
the casimir effect is a physical manifestation of zero point energy of quantum vacuum . in a relativistic quantum field theory , poincar symmetry of 
 the theory seems , at first sight , to imply that non - zero vacuum energy is inconsistent with translational invariance of the vacuum . in the setting of two uniform boundary plates at rest , quantum fields outside the plates have ( 1 + 2)-dimensional poincar symmetry . 
 taking a massless scalar field as an example , we have examined the consistency between the poincar symmetry and the existence of the vacuum energy . 
 we note that , in quantum theory , symmetries are represented projectively in general and show that the casimir energy is connected to central charges appearing in the algebra of generators in the projective representations .    * casimir energy between two parallel plates + and projective representation of poincar group *    takamaru akita @xmath0 and mamoru matsunaga @xmath1    _ @xmath0department of physics engineering , mie university , + tsu 514 - 8507 , japan + @xmath2college of liberal arts and sciences , mie university , + tsu 514 - 8507 , japan _