Electromagnetic wave representation as a superposition of the quantized vector potential field of single photons

Abstract

Without stating postulates or advancing any hypothesis, we show that a physical representation for the photon results directly from Maxwell’s theory once the vector potential amplitude is normalized at a single photon level. Electromagnetic waves are composed of a superposition of single photon states with circular polarizations, each extended over a wavelength and occupying a specific quantization volume. This representation is valid at any frequency and consequently can be applied for the modeling of advanced devices ranging from X-Rays and UV applications to fiber optics, microwave antennas and radio waves emitters.

Keywords

Electromagnetic waves photons quantized vector potential field electromagnetic field

1. Introduction

The modeling of electromagnetic waves is traditionally based on the spatial and temporal representation of the electric and magnetic fields, $\vec{E}(\vec{r},t)$ and $\vec{B}(\vec{r},t)$ respectively. However, these fields derive directly from the vector potential $\vec{A}(\vec{r},t)$ according to the well-known equations $\vec{E}(\vec{r},t)=-\partial \vec{A}(\vec{r},t)/\partial t$ and $\vec{B}(\vec{r},t)=\vec{{\nabla}}\times \vec{A}(\vec{r},t)$ , where we have considered as is generally the case that the scalar potential ${\Phi}(\vec{r},t)$ of the electromagnetic field is spatially constant $\vec{{\nabla}}{\Phi}(\vec{r},t)=0$ [1].

During almost 80 years after the formulation of Maxwell’s equations the vector potential $\vec{A}(\vec{r},t)$ was considered to be a pure function with no physical substance and an intermediate mathematical tool for the calculation of the electric and magnetic fields.

Ehrenberg and Siday were the first to put in evidence experimentally the direct influence of the vector potential on charged particles and concluded that it is a real physical field [2]. Much later, Bohm and Aharonov re-infirmed theoretically the influence of the vector potential on electrons in complete absence of electric and magnetic fields [3]. Chambers first [4], followed by Tonomura [5] and Osakabe [6] demonstrated experimentally the reality of the vector potential studying its influence on electron beams. Today, we know that the vector potential is a real physical field and all the interaction Hamiltonians between the electromagnetic field and charges are described in quantum electrodynamics (QED) through the vector potential Hermitian operator [7]. Furthermore, recent applications have shown the efficiency of the formulation based on the vector potential in order to deduce the magnetic field in specific physical situations such as magnetorheological fluids [8].

On the other hand, since the years of sixties it has been demonstrated that a single k-mode photon is not a point and extends over the wavelength 𝜆_k along the propagation axis [9–11] while its lateral expansion is roughly ∼𝜆_k∕4 [11–14]. These concepts are still valid today and consequently a single photon cannot be better localized than in a volume proportional to the cube of its wavelength ${\lambda}_{k}^{3}$ [15–17].

We deal here with the theoretical representation of the electromagnetic field based on the vector potential by considering the enhancement of its amplitude quantization to a single photon state and taking into account the experimental evidence for its spatial extension. The developments offer the flexibility of modeling the electromagnetic field throughout the whole spectrum at any frequency.

2. The single photon vector potential amplitude and the quantization volume

In QED the vector potential operator writes with the well-known expression [16–18] $\begin{eqnarray}\displaystyle \tilde{A}(\vec{r},t)=\mathop{\sum }_{k,{\lambda}}\sqrt{\frac{\hbar}{2{\varepsilon}_{0}{\omega}_{k}V}}[{\alpha}_{k{\lambda}}\hat{{\varepsilon}}_{k{\lambda}}e^{i(\vec{k}\cdot \vec{r}-{\omega}_{k}t)}+{\alpha}_{k{\lambda}}^{+}\hat{{\varepsilon}}_{k{\lambda}}^{\ast }e^{-i(\vec{k}\cdot \vec{r}-{\omega}_{k}t)}] & & \displaystyle\end{eqnarray}$ (1) where a_k𝜆 and $a_{k{\lambda}}^{+}$ are the annihilation and creation non-Hermitian operators respectively for a single k-mode and 𝜆 polarization photon, with angular frequency ω_k and wavelength 𝜆_k.

${\stackrel{\frown}{\varepsilon}}_{k\lambda}$ is the polarization complex unit vector with 𝜆 taking only two values corresponding to circular left and right. $\hbar$ is Planck’s reduced constant, ϵ₀ the vacuum electric permittivity and V the quantization volume.

The expression (1) is only valid for an ensemble of k-mode photons whose wavelengths are much shorter than the characteristic dimensions of the volume V, which is an external parameter, that is ∀𝜆_k ≪ V^1∕3. Obviously, the term $\sqrt{\hbar/2{\varepsilon}_{0}{\omega}_{k}V}$ does not represent the vector potential amplitude of a single k-mode photon and we may naturally wonder what could be the single photon quantization volume and the corresponding vector potential amplitude.

2.1. The single photon quantization volume from the density of states theory

According to the density of states theory the number of k-mode photons dn (ω_k) in the quantization volume V in the frequency interval between ω_k and ω_k + dω_k writes [16,17] $\begin{eqnarray}\displaystyle dn({\omega}_{k})=4V\frac{{\omega}_{k}^{2}}{{\pi}^{2}c^{3}}d{\omega}_{k} & & \displaystyle\end{eqnarray}$ (2) where c is the speed of light in vacuum.

We have also taken into account the two values of spin ± $\hbar$ , corresponding to circular left and right polarization, as well as the two possible propagations ± z along the same axis z.

All the possible states in the volume V are occupied. Thus, $\begin{eqnarray}\displaystyle n({\omega}_{k})=4V\frac{{\omega}_{k}^{3}}{3{\pi}^{2}c^{3}}. & & \displaystyle\end{eqnarray}$ (3) Considering n (ω_k) =1 the corresponding quantization volume for a single k-mode photon is $\begin{eqnarray}\displaystyle V_{k}=\left(\frac{3}{4}{\pi}^{2}c^{3}\right){\omega}_{k}^{-3}. & & \displaystyle\end{eqnarray}$ (4) With ${\omega}_{k}^{-3}={\lambda}_{k}^{3}/(2{\pi}c)^{3}$ the last expression is in agreement with the experimental evidence that the single photon quantization volume is proportional to ${\lambda}_{k}^{3}$ , we get $\begin{eqnarray}\displaystyle V_{k}=\frac{3}{32{\pi}}{\lambda}_{k}^{3}\simeq 0.029∼{\lambda}_{k}^{3}. & & \displaystyle\end{eqnarray}$ (5)

2.2. The single photon quantization volume from the energy normalization. The electron-positron elementary charge from the photon vector potential

In a different way, in order to get a more precise result, we can normalize the electromagnetic energy of a single plane-wave mode to that of a single photon $\begin{eqnarray}\displaystyle \int {\varepsilon}_{0}{\omega}_{k}^{2}{\alpha}_{0k}^{2}({\omega}_{k})∼[\hat{{\varepsilon}}_{k{\lambda}}e^{i({\omega}_{k}t-\vec{k}\cdot \vec{r}+{\phi})}+\hat{{\varepsilon}}_{k{\lambda}}^{\ast }e^{-i({\omega}_{k}t-\vec{k}\cdot \vec{r}+{\phi})}]^{∼2}d^{3}r=\hbar{\omega}_{k} & & \displaystyle\end{eqnarray}$ (6) where 𝛼_0k(ω_k) represents the single k-mode photon vector potential amplitude and $\hat{{\varepsilon}}_{k{\lambda}}^{\ast }$ the complex conjugate of the polarization unit vector.

The Eq. (6) holds at any instant t if the k-mode photon polarization unit vector $\hat{{\varepsilon}}_{k{\lambda}}$ has two orthogonal components, $\hat{e}_{1}$ and $\hat{e}_{2}$ , such as $\hat{e}_{1}\cdot \hat{e}_{2}=0$ and $\hat{{\varepsilon}}_{k{\lambda}}={\sigma}_{k1}\hat{e}_{1}+{\sigma}_{k2}\hat{e}_{2}$ with |σ_k1|² + |σ_k2|² =1. Obviously, these conditions are satisfied naturally by circular left (L) and right (R) polarization unit vectors $\hat{{\varepsilon}}_{L,R}=\frac{1}{\sqrt{2}}(\hat{e}_{1}\pm i\hat{e}_{2})$ , getting $\begin{eqnarray}\displaystyle \int 2{\varepsilon}_{0}{\omega}_{k}^{2}∼{\alpha}_{0k}^{2}({\omega}_{k})d^{3}r=\hbar{\omega}_{k}. & & \displaystyle\end{eqnarray}$ (7) Now, a dimension analysis of the general solution of the vector potential obtained from Maxwell’s equations [1] yields that it is proportional to a frequency, so as we can write the vector potential amplitude 𝛼_0k(ω_k) of a single k-mode photon as $\begin{eqnarray}\displaystyle {\alpha}_{0k}({\omega}_{k})={\xi}{\omega}_{k} & & \displaystyle\end{eqnarray}$ (8) with 𝜉 a constant to be determined through the normalization procedure.

As quoted above, the experimental evidence has shown that it is impossible to consider a single k-mode photon along the propagation axis within a length smaller than its wavelength 𝜆_𝜅 [7,9] while its lateral spatial extension is roughly ∼𝜆_k∕4 [11–14]. Considering the propagation along the z axis, with 𝜌 the lateral coordinate, the integration of (7) within the limits 0 ≤ z ≤𝜆_k and 0 ≤𝜌≤𝜂𝜆_k, where 𝜂 ∼1∕4 characterizes the experimental approximate radial extension, the Eq. (7) gives [18] $\begin{eqnarray}\displaystyle 2{\varepsilon}_{0}{\omega}_{k}^{2}{\alpha}_{0k}^{2}({\omega}_{k})\left(\frac{{\eta}^{2}}{2}{\lambda}_{k}^{3}\right)=(2{\pi}c)^{3}{\varepsilon}_{0}{\xi}^{2}{\eta}^{2}{\omega}_{k}=\hbar{\omega}_{k}. & & \displaystyle\end{eqnarray}$ (9) Hence, $\begin{eqnarray}\displaystyle {\eta}{\xi}=\pm \frac{1}{(2{\pi})^{3/2}}\sqrt{\frac{\hbar}{{\varepsilon}_{0}c^{3}}}. & & \displaystyle\end{eqnarray}$ (10) Instead of using the approximate experimental value of 𝜂 ∼1∕4 we can now try to get a more precise result. In fact, Eq. (9) writes with a slight rearrangement $\begin{eqnarray}\displaystyle 4{\pi}∼c∼\left(\frac{1}{4{\pi}∼c}2{\varepsilon}_{0}{\omega}_{k}^{2}{\alpha}_{0k}({\omega}_{k})\left(\frac{{\eta}^{2}}{2}{\lambda}_{k}^{3}\right)\right){\alpha}_{0k}({\omega}_{k})=4{\pi}∼c∼Q∼{\alpha}_{0k}({\omega}_{k})=\hbar{\omega}_{k} & & \displaystyle\end{eqnarray}$ (11) where Q has charge dimension.

Using (8) and (10) we get $\begin{eqnarray}\displaystyle Q=\frac{1}{4{\pi}∼c}2{\varepsilon}_{0}{\omega}_{k}^{2}{\alpha}_{0}({\omega}_{k})\left(\frac{{\eta}^{2}}{2}{\lambda}_{k}^{3}\right)=\pm \frac{{\eta}}{2}\sqrt{{\varepsilon}_{0}h∼c} & & \displaystyle\end{eqnarray}$ (12) where h is Planck’s constant.

It is worth remarking that when introducing in (12) the approximate experimental value for 𝜂 ∼1∕4 yields $\begin{eqnarray}\displaystyle Q\simeq \pm 1.6∼10^{-19}C & & \displaystyle\end{eqnarray}$ (13) which is the electron-positron charge, a physical constant, appearing here naturally [18,20].

We draw that the physical origin of the elementary charge is strongly related to the photon vector potential [18–20]. Now, we can define more precisely 𝜂 and 𝜉 by replacing Q in ((12)) by the elementary charge e and using the fine structure constant 𝛼 = e²∕4πϵ₀ $\hbar$ c = 1∕137, we get $\begin{eqnarray}\displaystyle {\eta}=\sqrt{8{\alpha}} & & \displaystyle\end{eqnarray}$ (14) and from Eq. ((10)) we obtain the vector potential amplitude normalization constant $\begin{eqnarray}\displaystyle {\xi}=\pm \frac{1}{(2{\pi})^{3/2}}\sqrt{\frac{\hbar}{8{\alpha}{\varepsilon}_{0}c^{3}}=}\pm \frac{\hbar}{4{\pi}|e|c}=\pm 1.747∼10^{-25}∼\text{Volt}∼\text{m}^{-1}\text{s}^{2}. & & \displaystyle\end{eqnarray}$ (15) The single k-mode photon quantization volume writes now $\begin{eqnarray}\displaystyle V_{k}=\frac{{\eta}^{2}}{2}{\lambda}_{k}^{3}=4{\alpha}{\lambda}_{k}^{3}\simeq 0.029∼{\lambda}_{k}^{3} & & \displaystyle\end{eqnarray}$ (16) which is equivalent to ((5)).

Consequently, the helicoidally distributed vector potential field of a single photon state over a wavelength occupies roughly 3% of the ${\lambda}_{k}^{3}$ volume and has a wave front section $\begin{eqnarray}\displaystyle S_{k}=8{\pi}{\alpha}{\lambda}_{k}^{2}\simeq 0.18∼{\lambda}_{k}^{2}. & & \displaystyle\end{eqnarray}$ (17) The last relations are useful for technological applications like wave-guides and fiber optics transmissions or Faraday grid shielding etc.

Finally, using the fine structure constant 𝛼 in Eq. (15) it is straightforward to obtain the electron-positron charge depending uniquely on the photon vector potential amplitude quantization constant 𝜉 [18–20] $\begin{eqnarray}\displaystyle e=\pm (4{\pi})^{2}{\alpha}\frac{{\xi}}{{\mu}_{0}}. & & \displaystyle\end{eqnarray}$ (18) This demonstrates that the elementary charge is strongly related to the photon vector potential amplitude and puts the basis for understanding the physical mechanism of the electron-positron transformations to photons and inversely.

3. The single photon wave-function and wave-particle properties

Following the above analysis, the k−mode photon vector potential function writes now with the quantized amplitude $\begin{eqnarray}\displaystyle \vec{{\alpha}}_{k{\lambda}}(\vec{r},t)={\xi}{\omega}_{k}(\hat{{\varepsilon}}_{{\lambda}}e^{i({\omega}_{k}t-\vec{k}\cdot \vec{r}+{\phi})}+cc). & & \displaystyle\end{eqnarray}$ (19) It is straightforward to show that ${\alpha}_{k{\lambda}}(\vec{r},t)$ satisfies the wave propagation equation in vacuum $\begin{eqnarray}\displaystyle \vec{{\nabla}}^{2}\vec{{\alpha}}_{k{\lambda}}(\vec{r},t)-\frac{1}{c^{2}}\frac{\partial ^{2}}{\partial t^{2}}\vec{{\alpha}}_{k{\lambda}}(\vec{r},t)=0 & & \displaystyle\end{eqnarray}$ (20) as well as the vector potential-energy (wave-particle) equation $\begin{eqnarray}\displaystyle i\left(\begin{array}{@{}c@{}}{\xi}\\ \hbar\end{array}\right)\frac{\partial }{\partial t}\vec{{\alpha}}_{k{\lambda}}(\vec{r},t)=\left(\begin{array}{@{}c@{}}\tilde{{\alpha}}_{0k}\\ \tilde{H}_{k}\end{array}\right)∼\vec{{\alpha}}_{k{\lambda}}(\vec{r},t) & & \displaystyle\end{eqnarray}$ (21) where $\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}\tilde{{\alpha}}_{0k}\\ \tilde{H}_{k}\end{array}\right)=-i\left(\begin{array}{@{}c@{}}{\xi}\\ \hbar\end{array}\right)c\vec{{\nabla}}_{k} & & \displaystyle\end{eqnarray}$ (22) are the photon vector potential amplitude $\tilde{{\alpha}}_{0k}$ operator and the relativistic massless particle Hamiltonian $\tilde{H}_{k}$ respectively [18,20–22].

Evidently, $\vec{{\alpha}}_{k{\lambda}}(\vec{r},t)$ is a natural wave function for the photon that can be suitably normalized and the localization probability is proportional to $\begin{eqnarray}\displaystyle |\vec{{\alpha}}_{k{\lambda}}(\vec{r},t)|^{2}\propto {\xi}^{2}{\omega}_{k}^{2}. & & \displaystyle\end{eqnarray}$ (23) Thus, the higher the frequency the better the localization possibility, in agreement with the experimental evidence.

A single k-mode photon is not a point and the particle properties, energy, momentum and spin are carried by a “wave-corpuscle” and are expressed in terms of the quantization volume V_k. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle \text{Energy}:E_{k}=\int _{V_{k}}2{\varepsilon}_{0}{\alpha}_{0k}^{2}{\omega}_{k}^{2}d^{3}r=\int _{V_{k}}2{\varepsilon}_{0}{\xi}^{2}{\omega}_{k}^{4}d^{3}r=2{\varepsilon}_{0}{\xi}^{2}{\omega}_{k}^{4}V_{k}=\hbar{\omega}_{k}\end{eqnarray}$ (24) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle \text{Momentum}:\vec{P}_{k}=\int _{V_{k}}{\varepsilon}_{0}\vec{{\varepsilon}}_{k{\lambda}}\times \vec{{\beta}}_{k{\lambda}}d^{3}r={\varepsilon}_{0}(\sqrt{2}{\omega}_{k}{\alpha}_{0k})\left(\frac{1}{c}\sqrt{2}{\omega}_{k}{\alpha}_{0k}\right)V_{k}\frac{\vec{k}}{|\vec{k}|}=\hbar\vec{k}\end{eqnarray}$ (25) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle \text{Spin}:|\vec{S}|=\int _{V_{k}}{\varepsilon}_{0}\vec{r}\times (\vec{{\varepsilon}}_{k{\lambda}}\times \vec{{\beta}}_{k{\lambda}})d^{3}r=\pm {\varepsilon}_{0}\left(\frac{c}{{\omega}_{k}}\right)(\sqrt{2}{\omega}_{k}{\alpha}_{0k})\left(\frac{1}{c}\sqrt{2}{\omega}_{k}{\alpha}_{0k}\right)V_{k}=\pm \hbar.\end{eqnarray}$ (26) Where we have used the electric $\vec{{\varepsilon}}_{k{\lambda}}=-\partial \vec{{\alpha}}_{k{\lambda}}/\partial t$ and magnetic $\vec{{\beta}}_{k{\lambda}}=\vec{{\nabla}}\times \vec{{\alpha}}_{k{\lambda}}$ fields components of a single mode k with circular polarization [21]. We have also considered in ((26)) that the mean value of the length in a single k-mode photon state is c∕ω_k [23–25].

The single photon quantization volume V_k is an intrinsic property and defines the minimum space in which the quantized vector potential and the resulting electric and magnetic fields rotate at the frequency ω_k over a wavelength 𝜆_k along the propagation axis. It also ensures the link with the particle representation showing the consistency of the above formalism.

In fact, when introducing the particle operators of the single photon vector potential amplitude, expressed through the creation and annihilation non-Hermitian operators $\begin{eqnarray}\displaystyle \tilde{{\alpha}}_{0k}={\xi}{\omega}_{k}a_{k{\lambda}};\quad \tilde{{\alpha}}_{0k}^{\ast }={\xi}{\omega}_{k}a_{k{\lambda}}^{+}∼ & & \displaystyle\end{eqnarray}$ (27) into the quantum electrodynamics position $\hat{Q}_{k{\lambda}}$ and momentum $\hat{P}_{k{\lambda}}$ operators [16,17], $\begin{eqnarray}\displaystyle \hat{Q}_{k{\lambda}}=\sqrt{{\varepsilon}_{0}V_{k}}(\tilde{{\alpha}}_{0k}+\tilde{{\alpha}}_{0k}^{\ast })\quad \hat{P}_{k{\lambda}}=-i{\omega}_{k}\sqrt{{\varepsilon}_{0}V_{k}}(\tilde{{\alpha}}_{0k}-\tilde{{\alpha}}_{0k}^{\ast }) & & \displaystyle\end{eqnarray}$ (28) then Heisenberg’s commutation relation, a fundamental concept in quantum theory, is obtained straightforward $\begin{eqnarray}\displaystyle [\hat{Q}_{k{\lambda}},\hat{P}_{k^{\prime }{\lambda}^{\prime }}]=-i{\varepsilon}_{0}{\omega}_{k^{\prime }}^{2}{\omega}_{k}\sqrt{V_{k}V_{k^{\prime }}}[({\xi}∼a_{k{\lambda}}+{\xi}^{\ast }a_{k{\lambda}}^{+}),({\xi}∼a_{k^{\prime }{\lambda}^{\prime }}-{\xi}^{\ast }a_{k^{\prime }{\lambda}^{\prime }}^{+})]=i\hbar{\delta}_{kk^{\prime }}{\delta}_{{\lambda}{\lambda}^{\prime }}. & & \displaystyle\end{eqnarray}$ (29) This is a direct demonstration that the physical origin of Heisenberg’s uncertainty for photons lays on the spatial extension of the single states expressed by the quantization volume V_k.

Consequently, we may now resume all the above theoretical and experimental elements as follows.

A single k-mode photon is a localized wave-particle (wave-corpuscle) extending over a wavelength 𝜆_k. It has circular polarization (left or right) with corresponding spin ± $\hbar$ , total energy $\hbar$ ω_k, momentum $\hbar$ $\vec{{k}}$ , a wave front $S_{k}=8{\pi}{\alpha}{\lambda}_{k}^{2}$ and quantization volume $V_{k}=4{\alpha}{\lambda}_{k}^{3}$ . It propagates as a whole rotating along the propagation axis at an angular frequency ω_k and it is guided by the non-local vector potential wave-function ${\alpha}_{k{\lambda}}(\vec{r},t)$ with quantized amplitude 𝛼_0k(ω_k) = 𝜉ω_k.

Thus, the electromagnetic field can be expressed as an ensemble of successive wave-corpuscle packets of single k-mode states with a general vector potential wave function operator expressed as a superposition of all the individual wave functions whose amplitude is now independent on any external volume V, $\begin{eqnarray}\displaystyle \tilde{A}(\vec{r},t)=\mathop{\sum }_{k}\mathop{\sum }_{{\lambda}=1}^{2}{\xi}{\omega}_{k}[{\alpha}_{k{\lambda}}\hat{{\varepsilon}}_{k{\lambda}}e^{i(\vec{k}\cdot \vec{r}-{\omega}_{k}t)}+{\alpha}_{k{\lambda}}^{+}\hat{{\varepsilon}}_{k{\lambda}}^{\ast }e^{-i(\vec{k}\cdot \vec{r}-{\omega}_{k}t)}] & & \displaystyle\end{eqnarray}$ (30) where the summation runs over all the modes k and the two circular polarizations 𝜆.

4. Conclusion

A full quantum mechanical description of the electromagnetic waves can been developed based on the single wave-corpuscle modes with quantized vector potential amplitude 𝜉ω_k, quantized energy $\hbar$ ω_k, precise momentum $\hbar$ $\vec{k}$ and spin ± $\hbar$ corresponding to circular polarization, each guided by a non-local vector potential wave function. Heisenberg’s uncertainty lays in the spatial extension of the single modes expressed through the quantization volume. Finally, the energy normalization process yields a direct physical relation between the elementary charge and the single photon vector potential.

We have advanced no assumptions, made no approximations and imposed no limits. Consequently, the established formalism is valid for the modeling and simulation of any electromagnetic waves device, for any frequency interval and for any photon number.

References

Jackson

J.D.

, Classical Electrodynamics, Wiley, 1998.

Ehrenberg

and Siday

R.E.

, . The refractive index in electron optics.Proc. Phys. Soc. B62 (1949), 8.

Aharonov

and Bohm

, Significance of electromagnetic potentials in the quantum theory.Phys. Rev.115(3) (1959), 485.

Chambers

R.G.

, Shift of an electron interference pattern by enclosed magnetic flux.Phys. Rev. Lett.5(1) (1960), 3.

Tonomura

, Evidence for aharonov-bohm effect with magnetic field completely shielded from electron wave, Phys.Rev. Lett.48(21) (1982), 1443.

Osakabe

, Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor, Phys. Rev. A34(2) (1986), 815.

Akhiezer

A.I.

and Berestetskii

B.V.

, Quantum Electrodynamics, Interscience, NY, 1965.

Versaci

, A magneto-thermo-static study of a magneto-rheological fluid damper: A finite element analysis, IEEE Transactions on Magnetics57(1) (2021), 1–10, Art no. 4600210.

Mandel

, Configuration-space photon number operators in quantum optics.Phys. Rev.144 (1966), 1071.

10.

Pfleegor

R.L.

and Mandel

, Interference of independent photon beams.Phys. Rev.159(5) (1967), 1084.

11.

Robinson

H.L.

, Diffraction patterns in circular apertures less than one wavelength.J. Appl. Phys.24(1) (1953), 35.

12.

Hadlock

R.K.

, Diffraction patterns at the plane of a slit in a reflecting screen.J. Appl. Phys.29 (1958), 918.

13.

Hunter

and Wadlinger

R.L.P.

, Photons and Neutrinos as Electromagnetic Solitons.Physics Essays2 (1989), 2.

14.

Hunter

, “Quantum uncertainties” NATO ASI Series B pp 331 Proceedings of NATO Advanced Research Workshop, June 23–27, 1986.

15.

Auletta

, Foundations and Interpretation of Quantum Mechanics, World Scientific, 2001.

16.

Garrison

J.C.

and Chiao

, Quantum Optics, Oxford University Press, 2008.

17.

Weissbluth

, Photon-Atom Interactions, Academic Press, Inc., 1988.

18.

Meis

, Light and Vacuum, 2nd edn, World Scientific, Singapore, 2017.

19.

Meis

and Dahoo

P.R.

, The single photon state, the quantum vacuum and the elementary electron-positron charge. American Institute of Physics Publications, Conf. Proc., 2040, 020011, 2018.

20.

Meis

, Quantized field of single photons, in: Single Photon Manipulation, IntechOpen, London, 2019, Xia Keyu (ed.), Nanjing University, China. doi:10.5772/intechopen.88378, ISBN: 978-1-83880-353-7. Open Access. https://www.intechopen.com/online-first/quantized-field-of-single-photons.

21.

Meis

and Dahoo

P.R.

, Vector potential quantization and the photon intrinsic electromagnetic properties: Towards nondestructive photon detection.Int. J. Quantum Information15(8) (2017), 1740003.

22.

Meis

, The vacuum electromagnetic energy density - Towards a physical comprehension of the vacuum energy scale problem, Int. J. Mod. Phys. A35(25) (2020), 2050153.

23.

Rybakov

and Saha

, Soliton model of atom.Found. Phys.25(12) (1995), 1723–1731.

24.

Meis

, Photon wave-particle duality and virtual electromagnetic waves.Found. Phys.27 (1997), 865–873.

25.

Meis

and Dahoo

P.R.

, Vector potential quantization and the photon wave function.J. Phys. Conf. Series936 (2017), 012004.