The photon wave function ${\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$ for a free k-mode photon with angular frequency ${\omega }_k$ and circular polarization left (L) or right (R), corresponding respectively to spin $\pm \hslash$, is expressed as follows [19] [20]

${\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)={\left(\frac{{\epsilon }_0{\omega }_k}{\hslash }\right)}^{1/2}\left[{\alpha }_{0k}\left({\omega }_k\right)\left({\hat{\epsilon }}_{k\left(L,R\right)}{\text{e}}^{i\left(\stackrel{\to }{k}\cdot \stackrel{\to }{r}-{\omega }_{k}t+\theta \right)}+{\hat{\epsilon }}_{k\left(L,R\right)}^{*}{\text{e}}^{-i\left(\stackrel{\to }{k}\cdot \stackrel{\to }{r}-{\omega }_{k}t+\theta \right)}\right)\right]$ (1)

where ${\hat{\epsilon }}_{k\left(L,R\right)}$ is the circular polarization complex unit vector, $\stackrel{\to }{k}$ the wave-vector with amplitude $\left|\stackrel{\to }{k}\right|=2\pi /{\lambda }_k$, ${\lambda }_k$ is the wavelength of the mode k and θ a phase parameter.

${\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$ satisfies Maxwell’s propagation equation in vacuum,

${\stackrel{\to }{\nabla }}_k^2{\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}{\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)=0$ (2)

Helmholtz equation,

${\stackrel{\to }{\nabla }}_k^2{\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)+k^2{\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)=0$ (3)

and the photon vector potential - energy quantum equation,

$i\left(\begin{array}{c}\xi \\ \hslash \end{array}\right)\frac{\partial }{\partial t}{\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)=\left(\begin{array}{c}{\tilde{\alpha }}_{0k}\\ {\tilde{H}}_k\end{array}\right){\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$ (4)

expressing both the first (energy) and second (vector potential) quantization of the electromagnetic field corresponding to k-mode photons with energy $E_k\left({\omega }_k\right)=\hslash {\omega }_k$ and vector potential amplitude ${\alpha }_{0k}\left({\omega }_k\right)=\xi {\omega }_k$.

The operators involved in (4) write

$\left(\begin{array}{c}{\tilde{\alpha }}_{0k}\\ {\tilde{H}}_k\end{array}\right)=-i\left(\begin{array}{c}\xi \\ \hslash \end{array}\right)c{\stackrel{\to }{\nabla }}_k$ (5)

where ${\stackrel{\to }{\nabla }}_k$ merely means that the gradient acts upon the mode k.

The vector potential quantization constant is

$\xi =\frac{\hslash }{4\pi ec}$ (6)

with $\hslash$ Planck’s reduced constant, c the speed of light in vacuum and e the electron-positron elementary charge. Note that ξ can be negative or positive [14] - [16] .

The function ${\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$ issues [21] from the photon vector potential function ${\stackrel{\to }{\alpha }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$ with the quantized amplitude ${\alpha }_{0k}\left({\omega }_k\right)=\xi {\omega }_k$

${\stackrel{\to }{\alpha }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)=\xi {\omega }_k\left({\hat{\epsilon }}_{k\left(L,R\right)}{\text{e}}^{i\left(\stackrel{\to }{k}\cdot \stackrel{\to }{r}-{\omega }_{k}t+\theta \right)}+{\hat{\epsilon }}_{k\left(L,R\right)}^{*}{\text{e}}^{-i\left(\stackrel{\to }{k}\cdot \stackrel{\to }{r}-{\omega }_{k}t+\theta \right)}\right)={\omega }_k{\stackrel{\to }{\Xi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$ (7)

Overcoming the point photon concept, ${\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$ is normalized with respect to the photon quantization volume V_k, an intrinsic topological property issued from both the density of states theory and the energy normalization process of a plane electromagnetic wave to a single photon [19]

$V_k=\left(\frac{\hslash }{2{\epsilon }_0{\xi }^2}\right){\omega }_k^{-3}=4\alpha {\lambda }_k^3$ (8)

The normalization condition is satisfied with respect to the quantization volume

${\displaystyle {\int }_{V_k}{\left|{\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)\right|}^2{\text{d}}^3{r}^{\prime }}=1$ (9)

Obviously, according to (1) and (7), the photon wave function ${\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$ depends fundamentally on the main function of the vector potential ${\Xi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$

${\stackrel{\to }{\Phi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)={\left(\frac{{\epsilon }_0{\omega }_k^3}{\hslash }\right)}^{1/2}{\stackrel{\to }{\Xi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$ (10)

It is straightforward to show that the mean values of the relativistic massless particle Hamiltonian ${\tilde{H}}_k=-i\hslash c{\stackrel{\to }{\nabla }}_k$ and the momentum operator ${\stackrel{\to }{p}}_k=-i\hslash {\stackrel{\to }{\nabla }}_k$ in the single photon state, respecting the quantization volume V_k, give the single photon energy and momentum respectively [20]

$\left|\langle {\Phi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)|{\tilde{H}}_k{|{\Phi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)\rangle }_{V_k}\right|=\left|\langle {\Phi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)|-i\hslash c{\stackrel{\to }{\nabla }}_k{|{\Phi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)\rangle }_{V_k}\right|=\hslash {\omega }_k$ (11)

$\left|\langle {\Phi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)|{\stackrel{\to }{p}}_k{|{\Phi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)\rangle }_{V_k}\right|=\left|\langle {\Phi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)|-i\hslash {\stackrel{\to }{\nabla }}_k{|{\Phi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)\rangle }_{V_k}\right|=\frac{\hslash {\omega }_k}{c}$ (12)

The precession of the quantized vector potential around the propagation axis at the angular frequency ${\omega }_k$ with circular polarization gives birth to orthogonal electric ${\stackrel{\to }{\epsilon }}_k$ and magnetic ${\stackrel{\to }{\beta }}_k$ fields whose amplitudes are proportional to $\xi {\omega }_k^2$ and the corresponding magnitudes, that is the square root of the sum of the squares of the components, are obtained directly from Maxwell’s equations [19] [20]

$\begin{array}{l}\Vert {\stackrel{\to }{\epsilon }}_k\Vert =\Vert -\partial {\stackrel{\to }{\alpha }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)/\partial t\Vert =\sqrt{2}\left|\xi \right|{\omega }_k^2\\ \Vert {\stackrel{\to }{\beta }}_k\Vert =\Vert \stackrel{\to }{\nabla }\times {\stackrel{\to }{\alpha }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)\Vert =\sqrt{2{\epsilon }_0{\mu }_0}\left|\xi \right|{\omega }_k^2\end{array}$ (13)

with ${\mu }_0$ the vacuum magnetic permeability.

In addition, the energy density for a single photon state as a pointless particle is readily obtained and is identical in both classical and quantum representations

$\begin{array}{c}{W}_k={\left|{\Phi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)\right|}^2\left|\langle {\Phi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)|{\tilde{H}}_k{|{\Phi }_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)\rangle }_{V_k}\right|\\ =\frac{1}{2}\left({\epsilon }_0{\Vert {\stackrel{\to }{\epsilon }}_k\Vert }^2+\frac{1}{{\mu }_0}{\Vert {\stackrel{\to }{\beta }}_k\Vert }^2\right)=\frac{\hslash {\omega }_k}{V_k}=2{\epsilon }_0{\xi }^2{\omega }_k^4\end{array}$ (14)

showing the self-consistency of the developed formalism.

We recall that the zero-point energy in QED issues from the harmonic oscillator Hamiltonian and represents a constant. Therefore, it commutes with all quantum mechanics Hermitian operators corresponding to observables and consequently is not responsible for the vacuum effects such as the Lamb shift and the spontaneous emission. In fact, the last effects are calculated in QED by using the photon creation and annihilation operators [22] . In addition, it has been demonstrated that the Casimir effect is due to Lorentz’s forces of source fields and not to the zero-point energy [23] - [26] . Finally, the density of the zero-point energy conflicts with recent well-validated astrophysical observations by 120 orders of magnitude leading to the well-known vacuum catastrophe riddle, a real cosmological problem [27] - [30] .

In fact, the zero-point energy of the electromagnetic field issues from the fundamental mathematical ambiguity consisting of replacing commuting classical canonical variables of position and momentum by the non-commuting corresponding quantum mechanics operators [12] [15] . Consequently, the zero-point energy cannot represent a real physical state and the question arises how to complement the normal ordering Hamiltonian of the electromagnetic field with a realistic vacuum description.

For that purpose, let us analyze what happens to the photon functions at zero frequency. For ${\omega }_k=2\pi c/{\lambda }_k\to 0$, that is for ${\lambda }_k\to \infty$, the photon energy, vector potential, electric and magnetic fields vanish. However, the final state does not correspond to a perfectly empty space since the fundamental function ${\stackrel{\to }{\Xi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$ of the vector potential (7) still subsists and is expressed in both classical and quantum formalisms [16] - [18] [20] [31]

${\stackrel{\to }{\Xi }}_{0\left(L,R\right)}=\xi \left[{\hat{\epsilon }}_{\left(L,R\right)}{\text{e}}^{i\theta }+{\hat{\epsilon }}_{\left(L,R\right)}^{*}{\text{e}}^{-i\theta }\right]$ (15a)

${\tilde{\Xi }}_{0\left(L,R\right)}=\xi \left[{\hat{\epsilon }}_{\left(L,R\right)}{a}_{k\left(L,R\right)}{\text{e}}^{i\theta }+{\hat{\epsilon }}_{\left(L,R\right)}^{*}{a}_{k\left(L,R\right)}^{+}{\text{e}}^{-i\theta }\right]$ (15b)

where in the quantum expression (15b) we have used the creation $a_{k\left(L,R\right)}^{+}$ and annihilation $a_{k\left(L,R\right)}$ non-Hermitian operators respectively for a single k-mode photon with circular polarization.

The expressions (15) represent the electromagnetic field ground state corresponding to the electromagnetic vacuum, a zero-energy field with electric potential nature filling all of the space.

For the first time, the classical electromagnetic theory is endowed with a vacuum state and QED with a vacuum representation depending on the photon creation and annihilation operators, complementing the normal ordering Hamiltonian.

Obviously, the phase parameter θ in the expressions (15) may take any value and consequently the electromagnetic vacuum is composed of all possible states ${\stackrel{\to }{\Xi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$

${\stackrel{\to }{\Xi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)=\xi \left({\hat{\epsilon }}_{k\left(L,R\right)}{\text{e}}^{i\left(\stackrel{\to }{k}\cdot \stackrel{\to }{r}-{\omega }_{k}t+\varphi \right)}+{\hat{\epsilon }}_{k\left(L,R\right)}^{*}{\text{e}}^{-i\left(\stackrel{\to }{k}\cdot \stackrel{\to }{r}-{\omega }_{k}t+\varphi \right)}\right)$ (16a)

${\tilde{\Xi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)=\xi \left({\hat{\epsilon }}_{k\left(L,R\right)}{a}_{k\left(R,L\right)}{\text{e}}^{i\left(\stackrel{\to }{k}\cdot \stackrel{\to }{r}-{\omega }_{k}t+\varphi \right)}+{\hat{\epsilon }}_{k\left(L,R\right)}^{*}{a}_{k\left(L,R\right)}^{+}{\text{e}}^{-i\left(\stackrel{\to }{k}\cdot \stackrel{\to }{r}-{\omega }_{k}t+\varphi \right)}\right)$ (16b)

The electromagnetic vacuum states ${\stackrel{\to }{\Xi }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$, which can be called kenons (from κενo = vacuum), involve all the characteristic physical parameters for any k-mode of the electromagnetic field, that is wave vector, frequency and spin (helicity, corresponding to circular polarization).

It is extremely important noting that the kenons result naturally from the quantized vector potential without stating any postulates or advancing any hypothesis.

From the fundamental photon vector potential - energy quantum Equation (4) we obtain the equation governing the kenons [31]

$i\frac{\partial }{\partial t}\left[\begin{array}{c}{\stackrel{\to }{\Xi }}_{k\lambda }\left(\stackrel{\to }{r},t\right)\\ {\tilde{\Xi }}_{k\lambda }\left(\stackrel{\to }{r},t\right)\end{array}\right]=\left(\begin{array}{c}{\omega }_k\\ {\tilde{\omega }}_k\end{array}\right)\left[\begin{array}{c}{\stackrel{\to }{\Xi }}_{k\lambda }\left(\stackrel{\to }{r},t\right)\\ {\tilde{\Xi }}_{k\lambda }\left(\stackrel{\to }{r},t\right)\end{array}\right]=\left[\begin{array}{c}{\stackrel{\to }{\alpha }}_{k\lambda }\left(\stackrel{\to }{r},t\right)\\ {\tilde{\alpha }}_{k\lambda }\left(\stackrel{\to }{r},t\right)\end{array}\right]$ (17)

where the angular frequency operator acting upon each mode k is ${\tilde{\omega }}_k=-ic{\stackrel{\to }{\nabla }}_k$.

The physical interpretation of Equation (17) is that real photons with vector potential ${\stackrel{\to }{\alpha }}_{k\left(L,R\right)}\left(\stackrel{\to }{r},t\right)$ are generated by the action of the angular frequency operator ${\tilde{\omega }}_k$ upon the kenons. In other words, photons are oscillations (precessions) of the electromagnetic vacuum states. Hence, the electromagnetic vacuum is composed of an infinite sea of kenons involving all modes k with circular polarizations.

An interaction Hamiltonian between kenons and electrons can be readily defined [14] [15] [20]

$H_{int}=\frac{\hslash e}{m_{e}c}{\tilde{\Xi }}_{k\left(L,R\right)}{\tilde{\omega }}_k=-i\hslash \frac{e}{m_e}{\tilde{\Xi }}_{k\left(L,R\right)}{\stackrel{\to }{\nabla }}_k$ (18)

characterizing the spontaneous emission effect following which a kenon state is transformed to a real photon under the action of the angular frequency operator.

As about the Lamb shift, it is estimated exactly with the well-known method in QED since the calculation is based on the photon creation operator $a_{k\lambda }^{+}$.

In addition, it is straightforward to demonstrate that every particle accelerated in the electromagnetic vacuum with an acceleration γ experiences the Fulling-Davies-Unruh temperature [18]

$T_H=\frac{\hslash }{2\pi c{k}_B}\left|\stackrel{\to }{\gamma }\right|$ (19)

where $k_B$ is Boltzmann’s constant.

Finally, it has been shown that the fluctuations of the electromagnetic vacuum yield an energy density ~10⁻¹⁰ J·m⁻³, compatible with the astrophysical observations [30] . This result issues from the fact that the photon electric field magnitude is proportional to the square of the frequency [17] [20] [32] .

Thus, the electromagnetic vacuum (16) is expressed in both classical and quantum formalisms complementing the quantum representation by associating a zero-energy vacuum field to the normal ordering Hamiltonian overcoming the zero-point energy singularity.