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an important step in better understanding casimir forces and their application to nano - scale environments is the nature and influence of the geometries used .
if we consider a collection of small bodies and their interactions , a quantitative understanding is necessary to address any issue of applications .
one can calculate the forces between bodies directly by using a stress tensor evaluated on the body in question .
this is an experimentally accessible prediction of the theory e.g. three body forces can be measured between dielectric spheres  @xcite for critical casimir forces .
the choice of stress tensor in media is not unique and depends on what consistency criteria is used .
typically the differences will show up both in the scale of the forces and the higher order curvature corrections , and one may view this as a way of finding the correct low energy description .    to address the nature of such interactions between bodies  @xcite casimir interaction energies
have been evaluated for collections of compact objects interacting with different force carrying fields ( electromagnetic and scalar ) .
the approach taken has been to evaluate a suitable energy functional integral using a t - matrix , whereby an interaction energy can be deduced , normalised with respect to their energy when separated at infinity .    in this
talk i summarise recent work we have done on calculating the casimir force on a single sphere , in an @xmath0-sphere setup  @xcite . by using a multiple scattering approach to evaluate essentially the classical scattering green s function of the configuration , we are able to evaluate the force directly on the sphere .
the total path length plays a fundamental role and what leads one to a loop description  @xcite .
similar loop descriptions have appeared before in  @xcite where a semi - classical type expression of the density of states is found , and in  @xcite a worldline description for scalar fields is given .
the question we are addressing is how to calculate the force on a particular sphere as a result of all the interactions with the remaining spheres in a particular static configuration .
the casimir force on a sphere ( given by the ball @xmath1 ) in the @xmath2-direction due to the effects of the @xmath0-sphere system is given by @xmath3 the stress tensor we choose is the standard vacuum expression ( which is consistent with the lorentz force law  @xcite ) @xmath4 where @xmath5 and it is understood that we are taking the limit for the initial and final points .
we then need to evaluate the scattering correlation functions ( whilst dropping the direct modes of propagation ) @xmath6 and similarly for magnetic fields . to construct the scattering two point function
we write the fields in a mode decomposition  @xcite of spherical vector wave functions that are centred on each sphere centre . then by applying the standard continuity equations at each of the spheres surfaces ,
one can calculate the out modes in terms of the in modes and scattering ( mie ) coefficients . assuming that the background in which we are evaluating this is filled with quantum noise such that the noise - current two point function is non - zero we find for the @xmath0-body force on a sphere ( suppressing the @xmath7 indices ) @xmath8&=&-(-1)^n\frac{\hbar } { 4\pi } r[1 ]   \im \int_{0}^{\infty}d\omega k \coth ( \hbar \omega /k_bt)\langle\mathbf{1}| [ \alpha^{1}(\omega r[1 ] ) \nonumber \\   & & \times   \sum_{i=2}^{n}a^{1,i}(\mathbf{r}[1,i ] ) \cdot \alpha^{i } ( \omega r[i ] ) \cdots   \cdots \sum_{j=2}^{n}a^{i , j}(\mathbf{r}[i , j ] ) \cdot \alpha^{j } \nonumber \\ & & \times \sum^n_{j=2}\nabla_{\mathbf{r}[j,1 ] } a^{j,1}(\mathbf{r}[j,1 ] ) ] j(kr[1 ] ) h^{+}(kr[1])w(\omega r[1 ] ) |\mathbf{1}\rangle .
\nonumber \\ & \equiv &   -(-1)^n\frac{\hbar } { 4\pi } \sum^{n}_{i=2 } \nabla^{last}_{\mathbf{r}[i,1]}\int_{0}^{\infty}d\omega   \cot \left(\frac{\hbar \omega}{k_bt}\right ) \mathcal{z}[\alpha , a , w].\end{aligned}\ ] ] here , @xmath9)$ ] are the mie scattering coefficients in the @xmath7 basis for sphere @xmath10 with radius @xmath11 $ ] ; @xmath12)$ ] are the translation matrices mapping the te and tm vector wave functions between spheres @xmath10 and @xmath2 ; the vectors @xmath13 give the truncation in the @xmath14 angular momentum quantum number ( leading to a multipole type expansion ) ; and the @xmath15 is just the collection of the four different contributions that make up the stress tensor , together with the two necessary bessel functions evaluated on the spheres surface . in the last line
a wick rotation to imaginary frequencies has been performed and the @xmath16-function has been defined for later reference .
note the explicit form of the translation matrices involve exponentials of the inter - sphere separations  @xcite and thus it is the _ total path length _ that plays the key role in understanding the variables of the system . for simple setups ( e.g. two and three sphere systems ) we can evaluate equation  ( [ eq : nsphereforce2 ] ) in different perturbative regimes e.g. retarded or non - retarded limits using static values for the permittivities ( see  @xcite for explicit evaluations and force plots of two and three sphere systems ) .
following on from the observation that equation  ( [ eq : nsphereforce2 ] ) involves the total path length in the form of a loop , in  @xcite an attempt is made to develop this further .
it is similar to the path integrals used in  @xcite and  @xcite where a fictitious time is introduced as well as mass scale set equal to unity and an appearance of planck s constant in the particle action . as pointed out in  @xcite ,
it is misleading to call this a semi - classical evaluation because of the absence of the dimensionful planck s constant .
concurrent with this is the absence of a mass or length scale with which to define a dimensionless action .    in  @xcite a symbolic expression for the @xmath16-function
was found that features the loop structure in an explicit manner @xmath17 & = & \sum_{\mathcal{c}_x } \langle\mathbf{f}^s_x|   \exp \left[-\oint_{\mathcal{c}_x}dq^i\hat{d}_i(q,\omega ) \right ] |\mathbf{i}^s_x\rangle \nonumber \\ & = & \sum_{\mathcal{c}_x}\mathbf { tr}\left (   \exp \left[-\hat{s}_r(\mathcal{c}_{x } , \gamma , \omega ) \right ] \right)_{\{\mathbf{i}^s_x,\mathbf{f}^s_x\}}\end{aligned}\ ] ] where @xmath18 is the initial and final point of the loop , and @xmath19 and @xmath20 are the initial and final states ( i.e. boundary conditions imposed on the eigenfunctions used to represent in and out modes ) .
the generator @xmath21 of translations and the loop @xmath22 implicitly depend on the background potentials .
the connection @xmath23 is formed from the background potentials in which the field propagates .
one now needs to find a representation of this object .
in fact it can be given a path integral representation , albeit a classical one i.e. no @xmath24 featuring anywhere .
one first needs to invert the helmholtz operator ( here partial derivatives have been promoted to covariant derivatives w.r.t .
the permittivity and permeability ) @xmath25^{-1}| y \rangle \nonumber \\ & = & \int_0^{\infty}d\tau \langle z |e^{-\tau [ c^2/\omega^2\nabla_{\epsilon}\wedge\nabla_\mu\wedge + \mathbf{1}\cdot]}| y \rangle.\end{aligned}\ ] ] introducing a world line metric @xmath26 to implement the helmholtz equation on physical states , and an integration over the @xmath27 operators ( again acting on physical states ) , together with an integration over paths one finds for the scalar version of the helmholtz operator ( i.e. two potentials but no spatial indices ) @xmath28 = \int^{1}_{0 } d\tau \sum^{\infty}_{n=1 } \langle\mathbf{f}_x | \oint_{q[0]=q[1]=x } [ dq]\sqrt{\det [ \omega^2/c^2(\epsilon \cdot \mu ) ] } \nonumber \\
\int [ d\hat{\nabla } ] [ de]\exp \left(-n \int^{\tau}_{0}dt [ \dot{q } \cdot \hat{\nabla}+e(\hat{\nabla}_{\epsilon}\cdot \hat{\nabla}_{\mu}+1)]\right ) |\mathbf{i}_x\rangle,\end{aligned}\ ] ] and @xmath29 \sim \sum_{loops } \mathcal{z}[\omega , x].\ ] ] if we perform first the integral over the world line metric , the helmholtz equation is implemented .
performing the integral over the derivatives returns a configuration space path integral which would require gauge fixing the world - line metric .
the classical equations of motion then lead to closed geodesics defining the loops .
integration over the frequency provides part of the sum over paths , whilst the tau integration gives the windings of the loops .
in this talk i have summarised recent work we have done on calculating casimir forces between spheres using a multiple scattering approach .
the total closed path length plays a key role in understanding the calculated forces and leads to a loop description . by considering the origin of the translation coefficients , together with a path integral representation of the helmholtz operator ,
one is able to reformulate the loop integral as a sum over of all possible loops .
j.  b. wishes to thank stefan buhmann , stefan scheel , alex crosse , rachele fermani and john gracey for numerous helpful and constructive discussions , and the organisers of qfext09 for the opportunity to present this material .
this work was supported by the scala programme of the european commission .
a.  bulgac , p.  magierski and a.  wirzba , phys .
d * 73 * 025007 ( 2006 ) .
j.  babington and s.  scheel " , arxiv:0909.3285 [ quant - ph ] .
j.  babington , arxiv:0909.3315 [ quant - ph ] .
m.  schaden and l.  spruch , phys .
a * 58 * , 2 , ( 1998 ) .
m.  schaden , phys .
rev a * 73 * , 4 , 042102 , ( 2006 ) .
a summary of recent calculations of casimir forces between a collection of @xmath0-dielectric spheres is presented . 
 this is done by evaluating directly the force on a sphere constructed from a stress tensor , rather than an interaction energy . a loop integral formulation is also discussed where we rewrite the expressions for the force in terms of loop integrals for the effective classical propagation of the electric and magnetic fields .