Zero-Point Energy of a Dilute Dielectric Ball in the Mode Summation Method

Abstract

A bstractIn the (" 1 " 2 ) 2 {approxi m ati on the C asi m i renergy ofa di l ute di el ectri c bal l i s deri ved usi ng a si m pl e and cl ear m ethod of the m ode sum m ati on. T he addi ti on theorem for the B esselfuncti ons enabl es one to present i n a cl osed form the sum over the angul ar m om entum before the i ntegrati on over the i m agi nary frequenci es. T he l i near i n (" 1 " 2 ) contri buti on i nto the vacuum energy i s rem oved by an appropri ate subtracti on. T he rol e of the contact term s used i n other approaches to thi s probl em i s el uci dated. 12. 20. D s,03. 70. + k,78. 60. M q,42. 50. Lc Typeset usi ng R EV T E X I. IN T R O D U C T IO NT he progress i n cal cul ati on ofthe C asi m i r energy i s rather sl ow . In hi s pi oneer paper [ 1]i n 1948 H .B.G .C asi m i r cal cul ated the vacuum el ectrom agneti c energy for the m ost si m pl e boundary condi ti ons,for two paral l elperfectl y conducti ng pl ates pl aced i n vacuum . D i el ectri c properti esofthe m edi a separated by pl ane boundari esdo notadd new m athem ati caldi cul ti es [ 2] . H owever the rst resul t on the cal cul ati on ofthe C asi m i r energy for the non-at boundari es was obtai ned onl y i n 1968. By com puter cal cul ati ons,l asted 3 years, T .H .Boyer found the C asi m i r energy ofa perfectl y conducti ng spheri calshel l [ 3] . A ccount ofdi el ectri c and m agneti c properti es ofthe m edi a i n cal cul ati ons ofthe vacuum energy for non at i nterface l eads to new pri nci paldi cul ti es or,m ore preci sel y,to a new structure of di vergenci es.T he cal cul ati on ofthe C asi m i r energy i n a speci alcase,w hen both the m ateri alm edi a have the sam e vel oci ty ofl i ght,proves to be,from the m athem ati calstand poi nt,exactl y the sam e as for perfectl y conducti ng shel l s pl aced i n vacuum and havi ng the shape ofthe i nterface between these m edi a [ 4{14] .T he rst attem pt to cal cul ate the C asi m i r energy ofa di el ectri c com pact bal lhas been undertaken by K .A .M i l ton i n 1980 [ 15] .A nd onl y justrecentl y thevacuum el ectrom agneti c energy ofa di l ute di el ectri c bal lwas found 1 [ 16{20] . T he l i ght vel oci ty i s di sconti nuous on the surface of such a bal l . In R ef.[ 21]the anal ysi s of the di vergenci es, w hi ch appear i n cal cul ati on ofthe C asi m i r energy ofa di el ectri c bal l ,has been carri ed out by determ i ni ng the rel evant heat kernelcoe ci ents. T he rol e of di spersi on i n thi s probl em i s now under study al so [ 8, 19, 22{25] .U nder these ci rcum stances i t i s of a i ndubi tabl e i nterest to devel op new m ethods for cal cul ati ng the vacuum energy fornon-atboundari esw i th al l owance forthe m ateri alcharacteri sti cs ofthe m edi a. It i s thi s ai m that i s pursued i n the present paper.Practi cal l y al lthe cal cul ati ons ofthe C asi m i r energy for the boundari es w i th spheri cal orcyl i ndri calsym m etri es use the uni form asym ptoti c expansi on forthe Besselfuncti ons. In pl ace ofthi s we are em pl oyi...

Casimir energy of a dilute dielectric ball with uniform velocity of light at finite temperature

“…The calculations were based on the method developed in [14]. This method have since been used in other related problems at zero temperature in [17,18,19]. We expect that these cases now too admit an exact treatment, along similar lines as presented here.…”

Casimir effect for a dilute dielectric ball at finite temperature

“…(2.12) in Ref. [9] one arrives at the following inference. In order to get the free energy at finite temperature by making use of the substitution (3.10), one integration by parts should preliminary be done in Eq.…”

“…The permittivities ε α , α = 1, 2 are assumed to be independent of the frequency ω (dispersion is ignored) and of the temperature T . Following the same way as in calculations of the Casimir energy at zero temperature [9] and making use of Eqs. (2.6), (2.8), and (3.4), we obtain the internal energy of a dielectric ball in the form…”

“…In order to get the free energy at finite temperature by making use of the substitution (3.10), one integration by parts should preliminary be done in Eq. (2.12) in [9]. Thus proceeding from the well justified equations for the free energy (2.2) and for the internal energy (2.6) at finite temperature, one can escape necessity to solve the problem: which energy (free or internal) is obtained on the substitution (3.10) in the initial integral representation for the Casimir energy at zero temperature [16,24].…”

“…[9] by the mode summation method, is the possibility for its straightforward generalization to the finite temperature. It is this problem that will be considered in the present paper.…”