Vector potential quantization and the photon wave-particle representation

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1 Vector potentil quntiztion nd the photon wve-prticle representtion Constntin Meis, Pierre-Richrd Dhoo To cite this version: Constntin Meis, Pierre-Richrd Dhoo. Vector potentil quntiztion nd the photon wve-prticle representtion. Journl of Physics: Conference Series, IOP Publishing, 16, 738, pp.199. <1.188/ /738/1/199>. <insu > HAL Id: insu Submitted on 1 My 16 HAL is multi-disciplinry open ccess rchive for the deposit nd dissemintion of scientific reserch documents, whether they re published or not. The documents my come from teching nd reserch institutions in Frnce or brod, or from public or privte reserch centers. L rchive ouverte pluridisciplinire HAL, est destinée u dépôt et à l diffusion de documents scientifiques de niveu recherche, publiés ou non, émnnt des étblissements d enseignement et de recherche frnçis ou étrngers, des lbortoires publics ou privés.

2 Vector potentil quntiztion nd the photon wve-prticle representtion C Meis 1* nd P R Dhoo 1 CEA Scly. Ntionl Institute for Nucler Science nd Technology, Université Pris Scly Gif-sur-Yvette, Frnce. LATMOS /IPSL, UVSQ Université Pris-Scly, UPMC Univ. Pris 6, CNRS, F-788, Guyncourt, Frnce. E-mil: constntin.meis@ce.fr Abstrct. The quntiztion procedure of the vector potentil is enhnced t single photon stte reveling the possibility for simultneous representtion of the wve-prticle nture of the photon. Its reltionship to the quntum vcuum results nturlly. A vector potentil mplitude opertor is defined showing the prllelism with the Hmiltonin of mssless prticle. It is further shown tht the quntized vector potentil stisfies both the wve propgtion eqution nd liner time-dependent Schrödinger-lie eqution. Introduction We nlyse first the fundmentl lin between the electromgnetic wve theory nd quntum electrodynmics (QED) [1-3]. In the clssicl description issued from Mxwell s equtions the energy density of n electromgnetic wve with electric nd mgnetic fields E( r, nd B( r, respectively is 1 1 E( B( (1) where nd re the electric permittivity nd mgnetic permebility of the vcuum. In the cse of monochromtic plne wve the electric nd mgnetic fields re proportionl to the vector potentil mplitude A () nd the energy density writes 4 A ( ) sin ( r ) () t whose men vlue over period, tht is over wvelength, becomes time nd spce independent A ( ) (3) In the quntum theory the energy density for N photons with ngulr frequency in volume V is N W Q (4) V where h / is Plnc s reduced constnt.

3 In order to lin the clssicl nd quntum description it is generlly imposed for N = 1, thus for single photon stte, the reltions (3) nd (4) to be equl. In this wy, the vector potentil mplitude is A ( ) (5) V It is worth noting tht s result of this procedure n externl rbitrry prmeter V hs been introduced in the lst eqution which is supposed to express nturlly the photon vector potentil mplitude, n intrinsic physicl property. Nevertheless, this eqution is used to define the fundmentl lin reltions between the clssicl nd quntum theory of light through the definition of the vector potentil mplitude opertors for photon where nd A A * V V re the nnihiltion nd cretion opertors respectively for -mode nd. -polriztion photon with ngulr frequency Vector potentil in QED It is useful to exmine how the lst reltions re used in QED clcultions [1-4]. The vector potentil opertor writes generlly s superposition of the vector potentils of ll the -modes nd polriztion photons with polriztion vector ˆ (6) A(, V i r t * i r t ˆ e ˆ e (7) The discrete summtion over the modes is generlly replced by continuous one over the ngulr frequencies following the trnsformtion issued from the density of sttes theory, V c 3 d (8) where c is the velocity of light in vcuum nd tes two vlues corresponding to the Left nd Right hnd circulr polriztions. Consequently, for ll the clcultions involving the squre of the mplitude of the vector potentil this mthemticl opertion helps to eliminte the volume prmeter V. Howeve the reltion (8) hs been obtined under the condition tht ll the photons wvelengths re much smller thn the chrcteristic dimensions of the volume V. Single photon stte vector potentil nd the quntum vcuum The methodology presented bove gives quite physicl results when considering system of photons within volume with dimensions much bigger compred to their wvelengths. Howeve the experimentl evidence [1-4] hs shown tht single photon is n indivisible entity with definite energy nd momentum. Despite of this the reltion (6) gives no precise informtion on its vector potentil mplitude.

4 Now, the energy density of the electromgnetic wves in the clssicl description s well s in QED 4 depends on the fourth power of the ngulr frequency [1,3]. Consequently, the reltion (3) entils utomticlly tht the vector potentil mplitude is proportionl to. Indeed, the unit nlysis of the generl solution of Mxwell s equtions for the vector potentil shows tht it is inversely proportionl to time, thus proportionl to n ngulr frequency. Consequently, for - mode photon the vector potentil mplitude cn be written s [4,5] where is constnt. (9) Thus, the fundmentl physicl quntities chrcterizing both the wve nd prticle nture of single photon: energy nd momentum (prticle), vector potentil mplitude nd wve vector (wve), re ll relted to the ngulr frequency s follows [4,5] p E c / c (1) In the plne wve representtion the vector potentil for -mode nd polriztion photon cn be expressed over period nd repeted successively long the propgtion xis s i r t (, ) ˆ, r t e cc (, (11) with cc the complex conjugte nd hs to stisfy the wve propgtion eqution 1,, (1) c t leding to c, (13) The lst expression entils tht the vector potentil mplitude of the photon cn be expressed s n opertor ~ i c (14) which is quite symmetricl to the reltivistic Hmiltonin opertor for mssless prticle H ~ ic (15) Applying the vector potentil mplitude opertor (14) upon the vector potentil expression (11) we get finlly (, ) ~ i, r t, t (16)

5 This is quntum eqution for the vector potentil with quntiztion constnt nlogue to Schrödinger s eqution for the energy with quntiztion constnt h. Hence, we get the coupled eqution for photon in non-locl representtion with the vector potentil s photon wve function ~ i ; ~, t H (17) In first pproximtion the vlue of the constnt hs been evluted to be [6,7] / 3 8 FS c 4 ec Volt m where FS =1/137 is the Fine Structure constnt nd e is the electron chrge. 1 s (18) Now, t very low frequencies the wvelength tends to infinity nd the vector potentil to zero but the function (, composing the vector potentil does not vnish nd tends to unique expression for ll modes () () ˆ e i cc Consequently, in bsence of photons is rel field with mplitude hving electric units nd permeting ll spce. Thus, it cn be chrcterized s component of the quntum vcuum. This mens tht the electromgnetic wves, tht is photons, re oscilltions of the vcuum field (). Conclusion nd discussion We hve seen here tht the quntiztion of the vector potentil mplitude enhnced t single photon stte (9) complements the fundmentl reltions of the photon (1) nd leds to the coupled eqution (17) for which the vector potentil (11) with the quntized mplitude behves s rel wve function. The quntiztion constnt of the photon vector potentil mplitude hs electric essence nd derives from the quntum vcuum. Consequently, the vcuum is not se of photons, which leds to the well-nown QED singulrity of infinite vcuum energy [8] (so clled quntum vcuum ctstrophe), but the electromgnetic wves (photons) re wves of the quntum vcuum se which is () composed of rel potentil field. The reltion (17) indictes tht vibrtion of the vcuum field t n ngulr frequencygives rise to photon with vector potentil mplitude nd energy. References [1] Grrison J C nd Chio R, Quntum Optics, Oxford University Press (8). [] Milonni P W, The quntum vcuum, Acdemic Press Inc. (1994). [3] Ryder L H, Quntum field theory, Cmbridge University Press (1987). [4] Meis C, Light nd Vcuum, World Scientific (15). [5] Meis C, Found. Phys.,7, (1997) 865. [6] Meis C, Phys. Essys, 1, 1 (1999). [7] Meis C, Phys. Res. Int. Vol.14, ID (14). [8] Weinberg S, Rev. Mod. Phys. 61, (1989) 1. (19)