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Vector Potential Quantization and the Quantum Vacuum

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Vector Potential Quantization and the Quantum Vacuum Hindawi Publishing Corporation Physics Research International Volume 2014, Article ID 187432, 5 pages http://dx.doi.org/10.1155/2014/187432 Research Article Vector Potential Quantization and the Quantum Vacuum 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 field to a single photon state starting from the fundamental link equations between the classical electromagnetic theory and the quantum mechanical expressions. The resulting wave-particle 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 energy density. The calculated spontaneous emission rate and Lambs shift for the states of the hydrogen atom are in agreement with quantum electrodynamics. This low energy quantum vacuum state 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 =∑ ℏ (̂ + ), quantum vacuum field is a very low energy density medium 2 (1) that can be compatible with the astronomical observations. It , is also applied in this paper for the calculation of the spon- ̂ + where ℏ is Planck’s reduced constant (ℎ/2) and = taneous emission transition rates as well as the Lamb shift of + isthenumberofphotonoperatorwith and being the states in the hydrogen atom showing the consistency the annihilation and creation operators of a mode and and complementarity with QED. polarization photon, respectively, with angular frequency . In the absence of photons, (1)leadstotheQEDsingularity of infinite vacuum energy [5–7] ∑, ℏ/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 the quantum vacuum catastrophe entailing the ∑ ℏ /2 necessity of new theoretical developments. singularity , 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. the transition between the classical electromagnetic wave 2 Physics Research International vector potential amplitude and the corresponding quantized eliminating by this way the parameter 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 is, provided ℏ that the dimensions of the last one are much bigger than the → √ , 20 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 ∗ → √ + , 2 analysis of the general expression of the vector potential 0 amplitude of an electromagnetic field shows that it should where 0 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. 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 photonsinagivenvolume equals 0 ()∝(Constant ×)→. (6) the mean energy density over a period 2/ of the classical electromagnetic field of a plane wave with angular frequency So, the expression of the photon vector potential may now according to the relation [9–12] be written in a general plane wave expression 2 2 ⃗ ⃗ 1 1 (⋅−⃗ +) ∗ −(⋅−⃗ +) ⟨ ( ⃗ (,⃗ ) + ⃗ (,⃗ ) )⟩ ⃗ (,⃗ ) = [̂ + ̂ ] (7) 2 0 0 period whose propagation occurs within a period over a wave- 2 2 2 ⃗ =⟨40 0 () sin (⋅−)⟩⃗ (3) length and repeated successively along the propagation axis. period (,⃗ ) It has been demonstrated [20]that satisfies the wave 2 2 () ℎ propagation equation =20 0 () = = , 2 ⃗ 2 1 ⃗ ⃗ ∇ ⃗ (,⃗ ) − ⃗ (,⃗ ) =0, (8) where (,⃗ ) and (,⃗ ) are the electric and magnetic 2 2 fields, respectively, of the plane wave, 0() is the amplitude where is the velocity of light in vacuum, as well as a linear of the vector potential, () is the photons energy, and 0 is the vacuum magnetic permeability. It is worth noticing that time differential equation the volume is an external parameter and the link relations =1 ⃗ (,⃗ ) = ̃0 ⃗ (,⃗ ) , (9) (2)areobtainedfor ,thatis,forasinglephotonstate. The fact that the photon vector potential depends on an ⃗ external arbitrary parameter is rather puzzling. wheretheamplitudeoperatorwrites̃0 =−∇. Considering that the vector potential generally consists of The mathematical correspondence between the pairs a superposition of modes and polarizations, it writes with {(), ℏ} ↔0 { (), } for a single photon is quite obvious. (,⃗ ) the well-known expression Following the expression (7), writes with a wave- particle formalism ⃗ (,⃗ ) (⋅⃗ −⃗ +) ∗ ∗ −(⋅⃗ −⃗ +) ⃗ (,⃗ ) = [̂ + ̂ ] ℏ (⋅⃗ −⃗ +) + ∗ −(⋅⃗ −⃗ +) = ∑√ [ ̂ + ̂ ], 2 ⃗ , 0 =Ξ (, ,⃗ ) (4) ⃗ ⃗ (⋅−⃗ +) ∗ + ∗ −(⋅−⃗ +) ̃ = ⌊̂ + ̂ ⌋ where ̂ is a complex unit vector of polarisation and is a ̃ + phase parameter. =Ξ (,,), 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 and ,respec- defined description of individual photons but they are rather tively, for a mode and polarization photon. valid for a large number of photons in a big volume .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 wave vector space ⃗ ∗ ∗ ⃗ wave:{̃0 =−∇; ̃ = ∇} , 0 ∑ → ∫ 42 = ∫ 2 (11) 83 223 (5) :{̃ = ; ̃∗ =∗ + }. particle 0 0 Physics Research International 3 An approximate value of the quantization constant has been time dependent perturbation theory using the well-known obtained previously [26]. Consider formalism [1, 3, 5]: (1) ⃗ () =− ∫ [⟨Ψ, ⃗ ⋅ ∇ Ψ,0⟩ ℏ −25 −1 2 ∝ = 1.747 10 , (12) 0 4 Volt m s ( − − ) × (15) where is the electron charge. Appropriate experiments could +⟨Ψ , ∗∗⃗ ⋅ ∇⃗ + Ψ ,0⟩ attribute a more precise value to . −( − − ) × ] . Ξ̃ 3. The 0 Quantum Vacuum Field Effects Since |Ψ,0⟩ = 0, using Heisenberg’s equation of According to the relations in (10), for =0,thevector motion [5, 21], the scalar product of the last equation corre- + potential as well as the energy of a mode photon vanishes. sponding to the creation operator writes However, the field Ξ does not vanish but is reduced to the ⃗ vacuum field represented, here, as a quantum mechanical ∗ ∗ ⃗ + ∗ ∗ ⟨Ψ, ⃗ ⋅ ∇ Ψ,0⟩=− 1, ⃗ ⋅ operator: ℏ (16) ̃ ∗ + ∗ − Ξ = ̂ + ̂ . ⃗ =⟨Ψ||Ψ⃗ ⟩ = − 1, 0 (13) with , ,and being the Kronecker’s delta symbol. Considering the expression ∝(ℏ/4)from (12), one Ξ̃ Thus, 0 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 Ω. Consider per angular frequency and implying an electric nature of the 2 ̃ 1 3 2 quantum vacuum. Obviously, Ξ0 is a dynamic entity capa- = ⃗ Ω. (17) 3ℏ3 4 ble of inducing electronic transitions in matter since it is 0 + described by a function of and operators. This result is in full agreement with previous QED cal- culations [1, 3, 21] and shows that the spontaneous emission + 3.1. Spontaneous Emission. In a rigorous calculation, the is mainly due to the creation operator ,which,here,is Hamiltonian interaction between an atomic electron with involved in the quantum vacuum expression. Consequently, ⃗ ⃗ Ξ̃ dipole moment =⃗ and the vacuum writes int =−⋅ 0 istheskeletonofeveryphoton.Accordingtotherelations ⃗ (/ℏ)[(,⃗ ), ∑,(1/2)ℏ], 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 + Ξ̃ vacuum state is not described by a function of and an angular frequency to 0 .
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