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Hindawi Publishing Corporation Research International Volume 2014, Article ID 187432, 5 pages http://dx.doi.org/10.1155/2014/187432

Research Article Vector Potential and the Quantum

Constantin Meis

National Institute for Nuclear Science and Technology, Centre d’Etudes de Saclay (CEA), 91191 Gif-sur-Yvette, France

Correspondence should be addressed to Constantin Meis; [email protected]

Received 17 January 2014; Revised 4 June 2014; Accepted 4 June 2014; Published 19 June 2014

Academic Editor: Ali Hussain Reshak

Copyright © 2014 Constantin Meis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the quantization of the vector potential amplitude of the electromagnetic to a single state starting from the fundamental link equations between the classical electromagnetic theory and the quantum mechanical expressions. The resulting wave- formalism ensures a coherent transition between the classical electromagnetic wave theory and the quantum representation. A quantization constant of the photon vector potential is defined. A new quantum vacuum description results directly in having very low density. The calculated rate and Lambs shift for the 𝑛𝑆 states of the hydrogen are in agreement with . This low energy might be compatible with recent astrophysical observations.

1. Introduction In this paper, we enhance the quantization procedure of the electromagnetic field vector potential20 [ ]downtoa It is well known [1–5] that the Hamiltonian radiation in quan- single photon level in order to better understand the nature tum electrodynamics (QED) writes of the photon and that of the vacuum state. The resulting 1 𝐻=∑ ℏ𝜔 (𝑁̂ + ), quantum vacuum field is a very low energy density medium 𝑘 𝑘𝜆 2 (1) that can be compatible with the astronomical observations. It 𝑘,𝜆 is also applied in this paper for the calculation of the spon- ̂ + where ℏ is Planck’s reduced constant (ℎ/2𝜋) and 𝑁𝑘𝜆 =𝑎𝑘𝜆𝑎𝑘𝜆 taneous emission transition rates as well as the of + 𝑛𝑆 isthenumberofphotonoperatorwith𝑎𝑘𝜆 and 𝑎𝑘𝜆 being the states in the showing the consistency the annihilation and creation operators of a 𝑘 mode and 𝜆 and complementarity with QED. polarization photon, respectively, with angular frequency 𝜔𝑘. In the absence of , (1)leadstotheQEDsingularity of infinite [5–7] ∑𝑘,𝜆 ℏ𝜔𝑘/2.Ithasbeen demonstrated that when considering only the QED quantum 2. Background Theory of the Vector Potential vacuum, even with “reasonable” frequency cut-offs, the corre- Amplitude of a Single Photon sponding energy density is many orders of magnitude greater than actually observed [8, 9]inuniverse.Thissituationhas We will not consider, here, for the vacuum energy the been called catastrophe entailing the ∑ ℏ𝜔 /2 necessity of new theoretical developments. singularity 𝑘,𝜆 𝑘 obtained from (1)intheabsenceof Indeed, in the last two decades, various studies, develop- photons since this term drops when using the “normal ing different models10 [ –19], have been carried out in order ordering” for the photon creation and annihilation operators to interpret the recent astrophysical observations which are in the quantization process [5, 6, 21]. Furthermore, it has been in contradiction with the quantum electrodynamics (QED) demonstrated [22]thattheCasimireffectcanverywellbe vacuum representation [5–8]. Those studies have revealed the calculated without referring at all to this term. Hence, we start necessity of developing a different approach for the electro- from the fundamental relations defining in a given volume magnetic vacuum representation. 𝑉 the transition between the classical electromagnetic wave 2 Physics Research International vector potential amplitude and the corresponding quantized eliminating by this way the parameter 𝑉 in all calculations field operators [1–7]usedinQED: involving the square of the vector potential. The expression (5) is valid whatever the shape of the volume 𝑉 is, provided ℏ that the dimensions of the last one are much bigger than the 𝐴𝑘𝜆 󳨀→ √ 𝑎𝑘𝜆, 2𝜀0𝜔𝑘𝑉 electromagnetic field wavelengths [2, 5, 25]. (2) Let us now have a close look at the vector potential ampli- ℏ tude of a single photon. Therein, we recall that a dimensional 𝐴∗ 󳨀→ √ 𝑎+ , 𝑘𝜆 2𝜀 𝜔 𝑉 𝑘𝜆 analysis of the general expression of the vector potential 0 𝑘 amplitude of an electromagnetic field shows that it should 𝜔𝑘 where 𝜀0 is the vacuum electric permittivity. be proportional to the angular frequency .Noticethat It is very important to recall that the above link equations this is not a hypothesis but a mathematical deduction from result from the energy density equivalence as expressed by Maxwell’s equations. 𝑘 the classical electromagnetic wave theory, based on Maxwell’s Consequently, for the vector potential of a mode equations, and the representation of photons in QED. In fact, photon, we should have [20] the energy density of 𝑁 photonsinagivenvolume𝑉 equals 𝛼0𝑘 (𝜔𝑘)∝(Constant ×𝜔𝑘)󳨀→𝜉𝜔𝑘. (6) the mean energy density over a period 2𝜋/𝜔𝑘 of the classical electromagnetic field of a plane wave with angular frequency So, the expression of the photon vector potential may now 𝜔𝑘 according to the relation [9–12] be written in a general plane wave expression

2 2 ⃗ ⃗ 1 󵄨 󵄨 1 󵄨 󵄨 𝑖(𝑘⋅𝑟−𝜔⃗ 𝑘𝑡+𝜑) ∗ −𝑖(𝑘⋅𝑟−𝜔⃗ 𝑘𝑡+𝜑) ⟨ (𝜀 󵄨𝐸⃗ (𝑟,⃗ ) 𝑡 󵄨 + 󵄨𝐵⃗ (𝑟,⃗ ) 𝑡 󵄨 )⟩ 𝛼⃗𝜔 (𝑟,⃗ ) 𝑡 =𝜉𝜔𝑘 [𝜀𝑒̂ + 𝜀̂ 𝑒 ] (7) 2 0󵄨 𝜔 󵄨 𝜇 󵄨 𝜔 󵄨 𝑘 0 period whose propagation occurs within a period 𝑇 over a wave- 2 2 2 ⃗ =⟨4𝜀0𝜔 𝐴0 (𝜔) sin (𝑘⋅𝑟−𝜔𝑡)⟩⃗ (3) length 𝜆 and repeated successively along the propagation axis. period 𝛼 (𝑟,⃗ 𝑡) It has been demonstrated [20]that 𝜔𝑘 satisfies the wave 2 2 𝐸 (𝜔) 𝑁ℎ𝜔 propagation equation =2𝜀0𝜔 𝐴0 (𝜔) = = , 𝑉 𝑉 2 ⃗ 2 1 𝜕 ⃗ ⃗ ∇ 𝛼⃗𝜔 (𝑟,⃗ ) 𝑡 − 𝛼⃗𝜔 (𝑟,⃗ ) 𝑡 =0, (8) where 𝐸𝜔(𝑟,⃗ 𝑡) and 𝐵𝜔(𝑟,⃗ 𝑡) are the electric and magnetic 𝑘 𝑐2 𝜕𝑡2 𝑘 fields, respectively, of the plane wave, 𝐴0(𝜔) is the amplitude where 𝑐 is the velocity of light in vacuum, as well as a linear of the vector potential, 𝐸(𝜔) is the photons energy, and 𝜇0 is the vacuum magnetic permeability. It is worth noticing that differential equation the volume 𝑉 is an external parameter and the link relations 𝜕 𝑁=1 𝑖𝜉 𝛼⃗𝜔 (𝑟,⃗ ) 𝑡 = 𝛼̃0 𝛼⃗𝜔 (𝑟,⃗ ) 𝑡 , (9) (2)areobtainedfor ,thatis,forasinglephotonstate. 𝜕𝑡 𝑘 𝑘 The fact that the photon vector potential depends on an ⃗ external arbitrary parameter 𝑉 is rather puzzling. wheretheamplitudeoperatorwrites𝛼̃0 =−𝑖𝜉𝑐∇. Considering that the vector potential generally consists of The mathematical correspondence between the pairs a superposition of 𝑘 modes and 𝜆 polarizations, it writes with {𝐸(𝜔), ℏ} ↔0 {𝛼 (𝜔), 𝜉} for a single photon is quite obvious. 𝛼 (𝑟,⃗ 𝑡) the well-known expression Following the expression (7), 𝜔𝑘 writes with a wave- particle formalism 𝐴⃗ (𝑟,⃗ ) 𝑡 𝑖(𝑘⋅⃗ 𝑟−𝜔⃗ 𝑡+𝜑) ∗ ∗ −𝑖(𝑘⋅⃗ 𝑟−𝜔⃗ 𝑡+𝜑) 𝛼⃗ (𝑟,⃗ ) 𝑡 =𝜔 [𝜉𝜀̂ 𝑒 𝑘 +𝜉 𝜀̂ 𝑒 𝑘 ] 𝜔𝑘𝜆 𝑘 𝑘𝜆 𝑘𝜆 ℏ 𝑖(𝑘⋅⃗ 𝑟−𝜔⃗ 𝑡+𝜃) + ∗ −𝑖(𝑘⋅⃗ 𝑟−𝜔⃗ 𝑡+𝜃) = ∑√ [𝑎 𝜀̂ 𝑒 𝑘 +𝑎 𝜀̂ 𝑒 𝑘 ], 2𝜀 𝜔 𝑉 𝑘𝜆 𝑘𝜆 𝑘𝜆 𝑘𝜆 ⃗ 𝑘,𝜆 0 𝑘 =𝜔𝑘Ξ𝑘𝜆 (𝜔𝑘, 𝑟,⃗ 𝑡)

(4) ⃗ ⃗ 𝑖(𝑘⋅𝑟−𝜔⃗ 𝑘𝑡+𝜑) ∗ + ∗ −𝑖(𝑘⋅𝑟−𝜔⃗ 𝑘𝑡+𝜑) 𝛼̃𝑘𝜆 =𝜔𝑘 ⌊𝜉𝑎𝑘𝜆𝜀̂𝑘𝜆𝑒 +𝜉 𝑎𝑘𝜆𝜀̂𝑘𝜆𝑒 ⌋ where 𝜀̂𝑘𝜆 is a complex unit vector of polarisation and 𝜃 is a ̃ + phase parameter. =𝜔𝑘Ξ𝑘𝜆 (𝜔𝑘,𝑎𝑘𝜆,𝑎𝑘𝜆), Recent experiments [23, 24] demonstrated the individu- (10) ality and indivisibility of photons. However, it is obvious that the link equations used above where, for the quantum mechanical formalism, we have used + for the vector potential amplitude do not result in a well- the creation and annihilation operators 𝑎𝑘𝜆 and 𝑎𝑘𝜆,respec- defined description of individual photons but they are rather tively, for a 𝑘 mode and 𝜆 polarization photon. valid for a large number of photons in a big volume 𝑉.This The mathematical representation of the wave and particle is precisely the reason why in QED the discrete summation expressions of the vector potential amplitude operator writes over the modes is often replaced by a continuous summation [20] in the space ⃗ ∗ ∗ ⃗ wave:{𝛼̃0 =−𝑖𝜉𝑐∇; 𝛼̃ =𝑖𝜉 𝑐∇} , 𝑉 𝑉 0 ∑ 󳨀→ ∫ 4𝜋𝑘2𝑑𝑘 = ∫ 𝜔2𝑑𝜔 (11) 8𝜋3 2𝜋2𝑐3 (5) :{𝛼̃ =𝜉𝜔𝑎 ; 𝛼̃∗ =𝜉∗𝜔 𝑎+ }. 𝑘 particle 0𝑘𝜆 𝑘 𝑘𝜆 0𝑘𝜆 𝑘 𝑘𝜆 Physics Research International 3

An approximate value of the quantization constant 𝜉 has been time dependent perturbation theory using the well-known obtained previously [26]. Consider formalism [1, 3, 5]: 𝑒 𝑡 󵄨 󵄨 (1) 󵄨 ⃗ 󵄨 𝑐𝜔 (𝑡) =− ∫ [⟨Ψ𝑓,𝑛𝑘𝜆 󵄨𝜉𝜀𝑘𝜆⃗ ⋅ ∇𝑎𝑘𝜆󵄨 Ψ𝑖,0⟩ 󵄨 󵄨 ℏ −25 −1 2 𝑘 𝑚 󵄨 󵄨 󵄨𝜉󵄨 ∝ = 1.747 10 , (12) 𝑒 0 󵄨 󵄨 4𝜋𝑒𝑐 Volt m s 𝑖(𝜔 −𝜔 −𝑛 𝜔 )𝑡󸀠 ×𝑒 𝑖 𝑓 𝑘𝜆 𝑘 (15) where 𝑒 is the charge. Appropriate experiments could 󵄨 󵄨 +⟨Ψ ,𝑛 󵄨𝜉∗𝜀∗⃗ ⋅ ∇𝑎⃗ + 󵄨 Ψ ,0⟩ attribute a more precise value to 𝜉. 𝑓 𝑘𝜆 󵄨 𝑘𝜆 𝑘𝜆󵄨 𝑖

−𝑖(𝜔 −𝜔 −𝑛 𝜔 )𝑡󸀠 󸀠 ×𝑒 𝑖 𝑓 𝑘𝜆 𝑘 ]𝑑𝑡 . Ξ̃ 3. The 0𝑘𝜆 Quantum Vacuum Field Effects Since 𝑎𝑘𝜆|Ψ𝑖,0⟩ = 0, using Heisenberg’s equation of According to the relations in (10), for 𝜔𝑘 =0,thevector [5, 21], the scalar product of the last equation corre- 𝑘 + potential as well as the energy of a mode photon vanishes. sponding to the creation operator 𝑎𝑘𝜆 writes However, the field Ξ𝑘𝜆 does not vanish but is reduced to the 𝑟⃗ vacuum field represented, here, as a quantum mechanical 󵄨 ∗ ∗ ⃗ + 󵄨 ∗ ∗ 𝑖𝑓 ⟨Ψ𝑓,𝑛𝑘𝜆 󵄨𝜉 𝜀⃗ ⋅ ∇𝑎 󵄨 Ψ𝑖,0⟩=−𝜉 𝛿1,𝑛 𝑚𝑒𝜔𝑖𝑓 𝜀⃗ ⋅ operator: 󵄨 𝑘𝜆 𝑘𝜆󵄨 𝑘𝜆 𝑘𝜆 ℏ (16) ̃ 𝑖𝜑 ∗ + ∗ −𝑖𝜑 Ξ =𝜉𝑎 𝜀̂ 𝑒 +𝜉 𝑎 𝜀̂ 𝑒 . 𝑟𝑖𝑓⃗ =⟨Ψ𝑓|𝑟|Ψ⃗ 𝑖⟩ 𝜔𝑖𝑓 =𝜔𝑖 −𝜔𝑓 𝛿1,𝑛 0𝑘𝜆 𝑘𝜆 𝑘𝜆 𝑘𝜆 𝑘𝜆 (13) with , ,and 𝑘𝜆 being the Kronecker’s delta symbol. Considering the expression 𝜉∝(ℏ/4𝜋𝑒𝑐)from (12), one Ξ̃ Thus, 0𝑘𝜆 is a real entity of the vacuum state having the gets the spontaneous emission rate in the elementary solid −1 2 dimensions Volt m s corresponding to a vector potential angle 𝑑Ω. Consider per angular frequency and implying an electric nature of the 2 ̃ 1 𝑒 3 󵄨 󵄨2 quantum vacuum. Obviously, Ξ0 is a dynamic entity capa- 𝑊 = 𝜔 󵄨𝑟⃗ 󵄨 𝑑Ω. (17) 𝑘𝜆 𝑖𝑓 3ℏ𝑐3 4𝜋𝜀 𝑖𝑓 󵄨 𝑖𝑓 󵄨 ble of inducing electronic transitions in matter since it is 0 𝑎+ 𝑎 described by a function of 𝑘𝜆 and 𝑘𝜆 operators. This result is in full agreement with previous QED cal- culations [1, 3, 21] and shows that the spontaneous emission 𝑎+ 3.1. Spontaneous Emission. In a rigorous calculation, the is mainly due to the creation operator 𝑘𝜆,which,here,is Hamiltonian interaction between an atomic electron with involved in the quantum vacuum expression. Consequently, ⃗ ⃗ Ξ̃ dipole moment 𝑑=𝑒𝑟⃗ and the vacuum writes 𝐻int =−𝑑⋅ 0𝑘𝜆 istheskeletonofeveryphoton.Accordingtotherelations ⃗ (𝑖/ℏ)[𝐴(𝑟,⃗ 𝑡), ∑𝑘,𝜆(1/2)ℏ𝜔𝑘], which vanishes as in the semi- in (10), a photon appears to be a vacuum perturbation classical description. The physical reason is that, in QED, the generated when the last one interacts with matter conferring + 𝜔 Ξ̃ vacuum state is not described by a function of 𝑎𝑘𝜆 and 𝑎𝑘𝜆 an angular frequency 𝑘 to 0𝑘𝜆 . operators and, consequently, a Hamiltonian interaction for the spontaneous emission cannot be defined. Consequently, 3.2. Lamb Shift of the 𝑛𝑆 LevelsoftheAtomicHydrogen. as it is well known, the spontaneous emission in QED results Various calculations of the energy shift of the 𝑛𝑆 levels of + from the noncommutation between the 𝑎𝑘𝜆 and 𝑎𝑘𝜆 operators thehydrogenatomhavebeencarriedout,eachonebyconsid- rather than from a direct interaction between a bounded ering the vacuum effect on the atomic orbitals through differ- electron and the energy of the zero level photons ∑𝑘,𝜆 ℏ𝜔𝑘/2. ent aspects, mass [27], vacuum electric field Conversely, for the description of the electron-vacuum perturbation [28], Feynman’s gas interpretation [29], Stark interaction, a “vacuum action” operator corresponding to a shift5 [ ], and so forth. Here, we give a complementary ̃ Hamiltonian interaction per angular frequency between the approach based on the Ξ0 vacuum field representation. Ξ̃ 𝑚 𝑘𝜆 vacuum field 0𝑘𝜆 and an atomic electron of mass 𝑒 and In a first approximation, the displacements of charge 𝑒 cannowbedefinedasfollows: theelectronboundedstatescanbeseenasatopologicaleffect of the vacuum radiation pressure exerted upon the electronic 𝑒 orbitals. In fact, the motion of a bounded state electron with 𝐻 =−𝑖ℏ Ξ̃ ⋅ ∇.⃗ ̃ 𝜔𝑘 0𝑘𝜆 (14) charge 𝑒, whose energy is 𝐸𝑛𝑙𝑗,inthevacuumfieldΞ0 can be 𝑚𝑒 𝑘𝜆 characterised by Bohr’s angular frequency: 𝐸 𝑅 Thus, the amplitude of the transition probability per 𝜔 = 𝑛𝑙𝑗 = ∞ , 𝑛𝑙𝑗 2 (18) angular frequency between an initial state |Ψ𝑖,0⟩ with total ℏ 𝑛 ℏ 𝐸 =ℏ𝜔 |Ψ ⟩ energy 𝑖 𝑖, corresponding to an atom in the state 𝑖 where 𝑛, 𝑙,and𝑗 are the quantum numbers of the correspond- 𝐸 |...,0 ,...,0 󸀠 󸀠 ,...⟩ with energy 𝑖 in vacuum 𝑘𝜆 𝑘 𝜆 ,andafinal ing orbital entailing the rise in the electron frame of a vector state |Ψ𝑓,𝑛𝑘𝜆⟩ with total energy 𝐸𝑓,𝑛 =ℏ𝜔𝑓 +𝑛𝑘𝜆ℏ𝜔𝑘,rep- 𝑘𝜆 potential amplitude resenting the atom in the state |Ψ𝑓⟩ with energy 𝐸𝑓 =ℏ𝜔𝑓 in the presence of 𝑛𝑘𝜆 photons, can be expressed in first order 𝛼0(𝑛𝑙𝑗) =𝜉𝜔(𝑛𝑙𝑗) (19) 4 Physics Research International

−10 corresponding to “vacuum photons” with energy with 𝑎0 = 0.53 10 mbeingthefirstBohrradius;thefre- quency of the energy shift writes 4𝜋𝑒𝑐𝜉𝜔𝑛𝑙𝑗 =ℏ𝜔𝑛𝑙𝑗. (20) 9𝑅4 𝑎3 ̃ 1 ∞ 0 Hence, due to its periodic motion in the field Ξ0 ,an ]𝑛𝑆 ≈ . (28) 𝑘𝜆 𝑛3 16𝛼𝜋2𝑐3ℏ4 electron experiences in its frame vacuum photons whose pressure per unit surface writes [25] We obtain the Lamb shifts frequencies for the atomic 𝑛𝑆 󵄨 (𝑅,𝐿)󵄨2 hydrogen levels: 𝑑𝑃 ( ) = ∑ 𝜀0󵄨𝐸 󵄨 𝑑Ω, vac 󵄨 0 󵄨 (21) 𝜆 1𝑆: ∼7.96 GHz (8.2) , 2𝑆: ∼1000 MHz (1040) , (29) |𝐸(𝑅,𝐿)| 𝐿 3𝑆: ∼297 (303) , 4𝑆: ∼124 where 0 is the electric field amplitude of the left ( )and MHz MHz right hand (𝑅) polarized vacuum photons seen by the electron. with the experimental values given in parenthesis. The From classical electrodynamics, we have [1, 25] agreement with the experiment is quite satisfactory for such asimplecalculation. 󵄨 (𝑅)󵄨2 2 󵄨 (𝑅)󵄨2 󵄨 (𝑅)󵄨2 󵄨𝐸 󵄨 =𝜔 (󵄨𝐴 󵄨 + 󵄨𝐴 󵄨 ) 󵄨 0 󵄨 𝑛𝑙𝑗 󵄨 𝑥0 󵄨 󵄨 𝑦0 󵄨 (22) 3.3. Astrophysical Observations and the Quantum Vacuum 2 󵄨 (𝐿)󵄨2 󵄨 (𝐿)󵄨2 2 2 2 =𝜔 (󵄨𝐴 󵄨 + 󵄨𝐴 󵄨 )=𝜔 (2𝜉 𝜔 ). Catastrophe. It has been pointed out that recent astronomical 𝑛𝑙𝑗 󵄨 𝑥0 󵄨 󵄨 𝑦0 󵄨 𝑛𝑙𝑗 𝑛𝑙𝑗 observations revealed the vacuum energy density in the The summation over the two circular polarizations and universe to be many orders of magnitude [5, 8]lowerthan over the whole solid angle 4𝜋 gives the total vacuum radiation that predicted by QED. In fact, the studies on the pressure and the arising constraints have stimulated the creation of new models to remedy the theoretical cosmology difficulties. 4 ℏ𝜔𝑛𝑙𝑗 The modelling of the dark energy as a fluid has led tothe 𝑃 ( ) =4𝜋𝜀 (4𝜉2𝜔4 )≈ . (23) vac 0 𝑛𝑙𝑗 4𝛼𝜋2𝑐3 classical phantom description [10] which was revealed to be instable from both quantum mechanical and gravitational Hence, if the electron subsists in the effective volume pointsofview.Someproposals,amongmanyothers,basedon 𝑉 ( ) 𝑛𝑙𝑗 𝑛𝑙𝑗 eff of the orbital, the corresponding energy shift due phantom cosmology have introduced more consistent dark to the vacuum radiation pressure writes energy models by introducing quantum effects in general rel- ℏ𝜔4 ativity [11], nonlinear -matter interaction [12], revised 𝑛𝑙𝑗 gravity [13], and effective phantom phase and holography 𝛿𝐸 =𝑃 (vac) 𝑉𝑛𝑙𝑗 (eff) ≈ 𝑉𝑛𝑙𝑗 (eff) . (24) 4𝛼𝜋2𝑐3 [14]. On the other hand, various studies [15–17]onthecontri- bution of the quantum vacuum state to the dark energy have A delicate operation consists of defining 𝑉(𝑛𝑙𝑗)(eff).The𝑛𝑆 electronic orbitals, for example, are characterised by a density been carried out based on both QCD (quantum chromody- probability distribution decreasing smoothly with an expo- namics) and QED (quantum electrodynamics) descriptions. nential expression. Furthermore, for the excited electronic 𝑛𝑆 However, it has been pointed out by many authors that our states, the radial wave functions contain negative values and understanding of the QCD vacuum state still remains very the electron probability density distribution involves a signif- elusive. Conversely, QED vacuum offers a more comprehen- icant space area with zero values. Hence, in the case of atomic sive framework and there is a general scientific effort to hydrogen for the spherically symmetrical electronic orbitals analyse the contribution of the electromagnetic field lower 𝑛𝑆, in order not to take into account the space regions where energy level to the dark energy. the probability density is zero, the effective volume may be For a finite universe, which we may consider behaving as Ξ⃗ (𝜔 , 𝑟,⃗ 𝑡) written in a first approximation: a cosmic cavity, 𝑘𝜆 𝑘 gets very small but not zero. In fact, for 𝜔𝑘 =(2𝜋𝑐/𝜆𝑘)→0, the wavelength 𝜆𝑘 of a 𝑘 mode 1 4 󵄨 󵄨3 photon tends to cosmic dimensions while the energy and the 𝑉(𝑛𝑆) (eff) ≈ 𝜋󵄨⟨𝜓𝑛𝑠 |𝑟| 𝜓𝑛𝑠⟩󵄨 (25) 𝑛 3 vector potential tend to zero. Thus, we may assume that the universe is filled by a superposition of standing waves or and, following the last two equations, the corresponding fre- photons with cosmic dimensions corresponding a very low quency of the energy shift energy density. Under these conditions, considering the cos- 4 𝛿𝐸(𝑛𝑆) 𝜔 1 󵄨 󵄨3 mological estimations on the extend of the universe, an ] = ≈ 𝑛𝑆 󵄨⟨𝜓 |𝑟| 𝜓 ⟩󵄨 . 𝑛𝑆 2 3 󵄨 𝑛𝑠 𝑛𝑠 󵄨 (26) extremely high number of cosmic standing waves, roughly of ℎ 6𝛼𝜋 𝑐 𝑛 100 the order of 10 , could correspond to the measured back- 2 −9 3 Putting 𝜔𝑛𝑆 =𝑅∞/𝑛 ℏ with 𝑅∞ = 13.606 eV and con- ground vacuum energy density which is close to 10 J/m . sideringthecubeofthemeanvalueofthedistanceinthe 1 hydrogen electronic orbitals 𝑛𝑆 , 4. Conclusions 󵄨 󵄨3 2 3 3 󵄨 ∗ 3 󵄨 3 3𝑛 |⟨𝑟⟩| = 󵄨∫ Ψ 𝑟Ψ 𝑑 𝑟󵄨 =𝑎( ) (27) The difficulty raised in QED by the infinite energy of thezero 𝑛,𝑙=0 󵄨 (𝑛,𝑙=0) (𝑛,𝑙=0) 󵄨 0 2 point energy can be lifted by considering the quantization of Physics Research International 5

the electromagnetic field vector potential to a single photon [13]M.C.B.Abdalla,S.Nojiri,andS.D.Odintsov,“Consistent 𝛼 (𝜔 )=𝜉𝜔 modified gravity: dark energy, acceleration and the absence of 0𝑘 𝑘 𝑘. This fact is not a model but results directly from Maxwell’s equations. The issued theoretical devel- cosmic doomsday,” Classical and ,vol.22,no. opment is in agreement with the experimental evidence 5,pp.L35–L42,2005. accordingtowhichthephotonisrevealedtobeanintegral [14] E. Elizalde, S. Nojiri, S. Odintsov, and P. Wang, “Dark energy: and indivisible particle with simultaneous wave-particle vacuum fluctuations, the effective phantom phase, and hologra- properties. We have thus obtained both wave and particle phy,” D,vol.71,no.10,ArticleID103504,2005. representations of the vector potential whose amplitude is [15] J. A. Frieman, M. S. Turner, and D. Huterer, “Dark energy and proportional to a constant 𝜉. 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Nojiri and S. D. Odintsov, “Gravity assisted dark energy dominance and cosmic acceleration,” Physics Letters B: Nuclear, and High-Energy Physics,vol.599,no.3-4, pp.137–142,2004. Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Condensed Matter Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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