article_eng12.DVI Mathematical Problems of Computer Science 32, 86{95, 2009. M axwell E lectr odynamics Subjected to Quantum Vacuum Fluctuations A s h o t S . Ge vo r kya a n d A r a ks ya A . Ge vo r kya n 1 Institute for Informatics and Automation Problems of NAS of RA. 1Yerevan State University g ashot@sci.am Abstract The propagation of electromagnetic waves in vacuum is considered taking into ac- count quantum °uctuations in the limits of Maxwell-Langevin (ML) equations. For a model of "white noise" °uctuations, using ML equations, the second order partial di®erential equation is found which describes the quantum distribution of virtual pho- tons in vacuum. It is proved that in order to satisfy observed facts, the Lamb Shift etc, the virtual photons should be quantized in nonperturbed vacuum. For a model of the reverse harmonic quantum oscillator, the quantum distribution of photons is obtained precisely. It is shown, that the quantized virtual photons having negative en- ergies, in toto (approximately 85 percent) are condensed on the energy level absolute value of which is minimal. It is proved that the extension of Maxwell electrodynam- ics with inclusion of vacuum quantum ¯eld °uctuations may be constructed on 6D space-time continuum with 2D compacti¯ed subspace. The problem of propagation of various types electromagnetic waves in vacuum is investigated. Their in°uence on the refraction index of vacuum is studied. Refer ences [1 ] W . E . L a m b , Jr ., a n d R . C. R e t h e r fo r d , " Fin e S t r u c t u r e o f t h e H yd r o g e n A t o m b y a Mic r o wa ve Me t h o d " , P hys. R ev., vo l. 7 2 , 2 4 1 ( 1 9 4 7 ) . [2 ] P . W . Milo n n i, R . J. Co o k a n d M. E . Go g g in , " R a d ia t io n P r e s s u r e fr o m t h e V a c u u m : P h ys ic a l In t e r p r e t a t io n o f t h e Ca s im ir Fo r c e " , P hys. R ev., A vo l. 3 8 , 1 6 2 1 ( 1 9 8 8 ) . [3 ] R . L . Fo r wa r d , " E xt r a c t in g E le c t r ic a l E n e r g y fr o m t h e V a c u u m b y Co h e s io n o f Ch a r g e d Fo lia t e d Co n d u c t o r s " , P hys. R ev., B vo l. 3 0 , 1 7 0 0 ( 1 9 8 4 ) . [4 ] A . D . S a kh a r o v, " V a c u u m Qu a n t u m Flu c t u a t io n s in Cu r ve d S p a c e a n d t h e Th e o r y o f Gr a vit a t io n , D o kl. A ka d . N a u k. S S S R ( Sov. P hys. - D okl., vo l. 1 2 , 1 0 4 0 ( 1 9 6 8 ) . S e e a ls o d is c u s s io n in C. W . Mis n e r , K . S . Th o r n e a n d J. A . W h e e le r , Gravitation ( Fr e e m a n , S a n Fr a n c is c o , 1 9 7 3 ) , p . 4 2 6 . [5 ] T. W . Ma r s h a ll, " R a n d o m E le c t r o d yn a m ic s " , P r o c . R o y. S o c ., A vo l. 2 7 6 , p . 4 7 5 ( 1 9 6 3 ) . [6 ] T. H . B o ye r , " R a n d o m e le c t r o d yn a m ic s : Th e t h e o r y o f c la s s ic a l e le c t r o d yn a m ic s wit h c la s s ic a l e le c t r o m a g n e t ic z e r o -p o in t r a d ia t io n " . P h ys . R e v., vo l. 1 1 , p . 7 9 0 -8 0 8 ( 1 9 7 5 ) . [7 ] B . H a is c h , A . R u e d a , a n d H . E . P u t h o ®, " In e r t ia a s a z e r o -p o in t -̄ e ld L o r e n t z fo r c e " , P h ys . R e v., A vo l. 4 9 , p . 6 7 8 -6 9 4 ( 1 9 9 4 ) . 8 6 A. Gevorkyan and Ar. Gevorkyan 8 7 [8 ] L . d e la P e n a , a n d A . M. Ce t t o , " Th e Qu a n t u m D ic e : A n In t r o d u c t io n t o S t o c h a s t ic E le c t r o d yn a m ic s " , D o r d r e c h t : K lu we r ( 1 9 9 6 ) . L . d e la P e n a , a n d A . M. Ce t t o , ( " Co n t r ib u t io n fr o m s t o c h a s t ic e le c t r o d yn a m ic s t o t h e u n d e r s t a n d in g o f qu a n t u m m e c h a n ic s " . a r X iv: Ja n 2 0 0 5 qu a n t -p h / 0 5 0 1 0 1 1 4 Ja n 2 0 0 5 . [9 ] T. H . B o ye r , " A B r ie f S u r ve y o f S t o c h a s t ic E le c t r o d yn a m ic s ," in Fo u n d a t io n s o f R a d ia - t io n Th e o r y a n d Qu a n t u m E le c t r o d yn a m ic s , e d it e d b y A . O. B a r u t ( P le n u m , N e w Y o r k, 1 9 8 0 ) S e e a ls o t h e ve r y r e a d a b le a c c o u n t " Th e Cla s s ic a l V a c u u m ," in S c ie n t i¯ c A m e r ic a n , p . 7 0 ( A u g u s t 1 9 8 5 ) [1 0 ] Je a n -L u c Ca m b ie r , " In e r t ia l Ma s s fr o m S t o c h a s t ic E le c t r o d yn a m ic s " in : Ma r c G. Millis ( e t a l.) : F rontiers of P ropulsion Science. p . 4 2 3 -4 5 4 , A m e r ic a n In s t . o f A e r o n a u t ic s a n d A s t r o n a u t ic s , R e s t o n ( 2 0 0 9 ) . [1 1 ] A . S . Ge vo r kya n , " E xa c t ly s o lva b le m o d e ls o f s t o c h a s t ic qu a n t u m m e c h a n ic s wit h in t h e fr a m e wo r k o f L a n g e vin -S c h r e o d in g e r t yp e e qu a t io n " , Analysis and applications. E d s . b y B a r s e g ia n G. a n d B e g e h r H ., N A TO S c ie n c e p u b lic a t io n s , p p . 4 1 5 -4 4 2 , K lu we r , ( 2 0 0 4 ) . سùëí»ÉÇ ¿É»Ïïñá¹ÇݳÙÇÏ³Ï³Ý Ñ³ßíÇ ³éÝí³Í ùí³Ýï³ÛÇÝ Ø³ùëí»ÉÇí³ÏáõÙÇ ýÉáõÏïáõ³ódzݻñÁ ². ¶¨áñ·Û³Ý ¨ ²ñ. ¶¨áñ·Û³Ý ²Ù÷á÷áõÙ øí³Ýï³ÛÇÝ ¹³ßïÇ ï»ëáõÃÛ³Ý ßñç³Ý³ÏÝ»ñáõÙ í³ÏáõÙÁ Çñ»ÝÇó Ý»ñϳ۳óÝáõÙ ¿ ï»Õ, áñï»Õ ³éϳ »Ý ¿Ý»ñ·»ïÇÏ ³Ù»Ý ï»ë³Ï Ù³ëÝÇÏÝ»ñ ¨ ¹³ßï»ñÇ ÃéÃéáõÙÝ»ñ: ²ÛÉ Ëáëù»ñáí, í³ÏáõÙÁ µÝáõó·ñíáõÙ ¿ ýǽÇÏ³Ï³Ý ã³÷»ñáí ¨ ϳéáõóí³Íùáí, áñÁ ϳ½ÙáõÙ¿ µ³ó³ë³Ï³Ý ¿Ý»ñ·»ïÇÏ ÙÇç³í³ÛñÁ, áñÁ ﻽»ñùáõ٠ó÷³ÝóáõÙ ¿ ³Ù»Ýáõñ»ù: ºÃ» ùí³Ýï³ÛÇÝ ¹³ßïÇ ï»ëáõÃÛáõÝÁ ËáïáñáÙÝ»ñÇ ÙÇçáóáí ϳñáճݳñ ×ß·ñÇï Ýϳñ³·ñ»ñ »ñ¨áõÛÃÝ»ñÁ, ³å³ í³ÏáõÙÇ Ñ³ïÏáõÃÛáõÝÝ»ñÁ ÝÙ³Ý ÏÉÇÝ»ÇÝ ùí³Ýï³ÛÇÝ Ý»ñ¹³ßÝ³Ï ï³ï³Ý³ÏÇ Ñ³ïÏáõÃÛáõÝÝ»ñÇÝ: Ø»Ýù ³é³çÇÝ ³Ý·³Ù ³Ûë ËݹÇñÁ ¹Çï³ñÏ»É »Ýù سùëí»É–ȳÝŨ»Ý ïÇåÇ å³ï³Ñ³Ï³Ý ¹Çý»ñ»ÝóÇ³É Ñ³í³ë³ñáõÙÝ»ñÇ ßñç³Ý³ÏÝ»ñáõÙ: ¸³ Ù»½ Ñݳñ³íáñáõÃÛáõÝ ¿ ïí»É í³ÏáõÙáõÙª ùí³Ýï³ÛÇÝ µ³ßËÙ³Ý Ñ³Ù³ñ ½³ñ·³óÝ»É Ï³Ýáݳíáñ ï»ëáõÃÛáõÝ ³é³Ýó û·ï³·áñÍ»Éáõ ËáïáñáõÙÝ»ñÇ Ù»Ãá¹Ý»ñÁ: ì»ñçÇÝ Ñ³Ý·³Ù³ÝùÁ Ñݳñ³íáñáõÃÛáõÝ ¿ ïí»É Ù³Ýñ³Ù³ëÝáñ»Ý ѻﳽáï»É í³ÏáõÙÇ ¿É»ÏïñáÙ³·ÝÇë³Ï³Ý µ³Õ³¹ñÇ íÇ׳ϳ·ñáõÃÛáõÝÁ ¨ ϳéáõóí³ÍùÁ: سëݳíá- ñ³å»ë óáõÛó ¿ ïñí³Í, áñ ùí³Ýï³ÛÇÝ í³ÏáõÙÇ Ý»ñ³éáõÙÁ Ù³ùëí»ÉÇ ¿É»Ïïñá¹ÇݳÙÇϳÛÇ áõñí³·ÍáõÙ Ýϳñ³·ñíáõÙ ¿ »ñÏáõ Éñ³óáõóÇã ã³÷»ñáí, áñáÝù ÏáÙå³ÏïýÇϳóí³Í »Ý: ²ñï³ùÇÝ ¹³ßïáõÙ ·ïÝíáÕ í³ÏáõÙÇ ùí³Ýï³ÛÇÝ µ³ßËÙ³Ý Ñ³Ù³ñ ëï³óí³Í ¿ üáÏÏ»ñ– äɳÝÏÇ ïÇåÇ Ñ³í³ë³ñáõÙ ¨ ϳéáõóí³Í »Ý í³ÏáõÙÇ µ»ÏÙ³Ý óáõóÇãÝ»ñÁ: òáõÛó ¿ ïñí³Í, áñ Ýñ³Ýù ϳñáÕ »Ý ÷áËí»É ³ñï³ùÇÝ ¹³ßïÇ ³½¹»óáõÃÛ³Ý ï³Ï: