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# Reappraisal of QED fundamentals

Constantin Meis

► To cite this version:

Constantin Meis. Reappraisal of QED fundamentals. International journal of quantum foundations, 2025, 11 (2), pp.42-56. cea-04986314

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Submitted on 11 Mar 2025

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Original Paper

# Reappraisal of QED Fundamentals

Constantin Meis

National Institute for Nuclear Science and Technology, CEA – Saclay, 91191 Gif-sur-Yvette, France.
E-Mail: constantin.meis@cea.fr

Received: 15 September 2024 / Accepted: 1 February 2025 / Published: 1 March 2025

**Abstract:** The Quantum Electrodynamics fundamental drawbacks are revisited and analyzed. Those concern the vector potential amplitude operator, the electromagnetic field harmonic oscillator Hamiltonian with the associated zero-point energy singularity and the electron-vacuum interaction Hamiltonian. Without stating postulates or advancing any hypothesis it is shown that the QED difficulties can be readily overcome without compromising any of the achievements by enhancing the quantization of the vector potential amplitude to a single photon level. A photon wave function is defined as a probability amplitude normalized within an intrinsic quantization volume and satisfying Schrödinger's equation for both the photon energy and vector potential. Ensuing the single photon vector potential quantization the electromagnetic vacuum appears naturally complementing the normal ordering Hamiltonian without involving singularities.

**Keywords:** QED fundamental singularities, electromagnetic field quantization, photons, photon wave function, photon electric field, electromagnetic field Hamiltonian, zero-point energy, quantum vacuum

## 1. Introduction

All the 50 years of conscious brooding have brought me no closer to answer the question: "What are light quanta?" Of course today every rascal thinks he knows the answer, but he is deluding himself.
--- Albert Einstein

QED is without any doubt among the most successful theories developed in modern physics for the description of light-matter interactions at the atomic and molecular scale. However, despite the numerous achievements and the impressive precision of some theoretical predictions compared to the experiments, Einstein's statement is still valid today because we are unable formulating a complete and satisfactory physical description of a single photon state. Furthermore, the QED theory involves some well-known fundamental aspects leading to ambiguities and singularities [1-3]. For many decades, the last ones had not really a direct impact to the majority of the applications until recent astrophysical observations on the cosmic expansion have shown that the quantum vacuum energy density is finite and extremely low contradicting the singularity of the zero-point energy in QED and in quantum field theory in general [4-10]. Hence, the well validated astrophysical investigations generated an increasing interest last years towards the comprehension of the real nature of the quantum vacuum looking for new representations overcoming the well-known singularity of the zero-point energy issued from the harmonic oscillator Hamiltonian.
On the other hand, in order to define the vector potential amplitude operators of a single photon in QED the classical and quantum energy density equivalence has been considered involving an external volume which is not an intrinsic photon physical property [1-3,11]. Although the vector potential natural units are proportional to a frequency, the defined single photon vector potential amplitude in QED is proportional to the inverse square root of the frequency yielding unphysical consequences. Thus, the dependence on frequency of the photon vector potential and that of the photon electric field are conflictual entailing a fundamental physical incoherence. Finally, the well-known difficulties encountered for establishing a photon wave function are mainly due to the photon point particle concept yielding major constraints for establishing a position operator [12-17].

In what follows we first discuss extensively the origin of the above QED fundamental drawbacks and next we show how the enhancement of the vector potential amplitude to a single photon level permits to remedy all these theoretical inconveniencies.

## 2. Fundamental ambiguities in QED theory

### 2.1. Photon vector potential amplitude operators

In order to define the vector potential amplitude operators for a single k-mode photon with angular frequency $\omega_k$, the mean value over a period of the energy density $\langle W \rangle_T$ of a monochromatic plane wave is put equal to the energy density of a single photon considered in a cavity with volume $V$ [1,2,11]

$$ \langle W \rangle_T = 2\epsilon_0\omega_k^2|a_{0k}(\omega_k)|^2 = \frac{\hbar\omega_k}{V} \rightarrow a_{0k}(\omega_k) = \sqrt{\frac{\hbar}{2\epsilon_0\omega_k V}} \quad (1) $$

where $\alpha_{0k}(\omega_k)$ is the vector potential amplitude of the k-mode electromagnetic plane wave, $\hbar$ is Planck's reduced constant and $\epsilon_0$ is the vacuum electric permittivity.

Based on the classical-quantum energy density equivalence imposed above, the vector potential amplitude operators for a single photon are defined in QED [1,2,18] by associating the k-mode and $\lambda$-polarization photon annihilation and creation non-Hermitian operators, $a_{k\lambda}$ and $a_{k\lambda}^+$ respectively, to the established expression of the vector potential amplitude

$$ \tilde{a}_{k\lambda} = \sqrt{\frac{\hbar}{2\epsilon_0\omega_k V}} a_{k\lambda} \quad ; \quad \tilde{a}_{k\lambda}^* = \sqrt{\frac{\hbar}{2\epsilon_0\omega_k V}} a_{k\lambda}^+ \quad (2) $$

We note that the left hand side of the classical-quantum energy density equivalence in (1) is totally independent on any external parameter while this is not the case for the right hand side which depends on the external volume $V$. Therein, it is extremely important to underline that the right hand side of the relation (1) represents the energy density of the system “photon in a cavity” and not precisely that of a “single photon state”.

Furthermore, the relation (1) raises another fundamental question. In fact, the general solution of the vector potential issued from Maxwell's equation writes with the well-known expression [19,20]

$$ \vec{A}(\vec{r}, t) = \frac{\mu}{4\pi} \int \frac{\vec{j}(\vec{r}', t - |\vec{r} - \vec{r}'|/c)}{|\vec{r} - \vec{r}'|} d^3 r' \quad (3) $$

where $\vec{j}$ is the current density, $c$ is the speed of light in vacuum and $\mu$ is the magnetic permeability.

A careful unit analysis of (3) shows that the vector potential is physically proportional to a frequency. This is in contradiction with the vector potential amplitude established in (1) which is inversely proportional to the square root of the frequency.

In addition, using (1) the electric field amplitude of a single photon writes with the well-known expression [1,2,18]
$$|\vec{\epsilon}_k| = \sqrt{\frac{\hbar\omega_k}{2\epsilon_0 V}} \qquad (4)$$

Comparison between equations (1) and (4) leads to the unphysical conclusion that the higher the photon frequency the lower its vector potential amplitude and the higher its electric field. This result is conflictual and incompatible with the electromagnetic theory entailing that the energy density equivalence (1), set roughly a century ago, has to be reappraised. We will see later that the solution to these ambiguities lies precisely in the volume parameter V.

## 2.2 Origin of the harmonic oscillator Hamiltonian of the electromagnetic field

The total energy of a multimode electromagnetic field considered in a volume V writes in classical theory [1,2]

$$E_{EM} = \sum_{k,\lambda} 2\epsilon_0 \omega_k^2 V |\vec{a}_{k\lambda}(\vec{r}, t)|^2 \qquad (5)$$

where $\vec{a}_{k\lambda}(\vec{r}, t)$ is the vector potential of a single plane wave k-mode with $\lambda$-polarization. Replacing in (5) the single k-mode photon vector potential amplitude operators defined in (2) we get the following two expressions for the quantum mechanics Hamiltonian of the electromagnetic field

$$\tilde{H}_{NO} = \sum_{k,\lambda} a_{k\lambda}^{+} a_{k\lambda} \hbar \omega_k \rightarrow E_{NO} = \sum_{k,\lambda} n_{k\lambda} \hbar \omega_k \qquad (6)$$

$$\tilde{H}_{ANO} = \sum_{k,\lambda} a_{k\lambda} a_{k\lambda}^{+} \hbar \omega_k = \sum_{k,\lambda} (a_{k\lambda}^{+} a_{k\lambda} + 1) \hbar \omega_k \rightarrow E_{ANO} = \sum_{k,\lambda} (n_{k\lambda} + 1) \hbar \omega_k \qquad (7)$$

where $\tilde{H}_{NO}$ corresponds to the normal ordering Hamiltonian and $\tilde{H}_{ANO}$ to the anti-normal ordering one. $E_{NO}$ and $E_{ANO}$ are the corresponding energy eigenvalues while $n_{k\lambda}$ is the total number of k-mode and $\lambda$-polarization photons. Also, for $\tilde{H}_{ANO}$ we have used the fundamental commutation relation $[a_{k\lambda}, a_{k'\lambda}^{+}] = \delta_{kk'}\delta_{\lambda\lambda'}$.

The mean value of the last two Hamiltonians yields a harmonic oscillator Hamiltonian which is generally introduced in QED [1,2,21] as the natural Hamiltonian of the electromagnetic field

$$\tilde{H}_{EM} = \frac{1}{2}(\tilde{H}_{NO} + \tilde{H}_{ANO}) = \sum_{k,\lambda} (a_{k\lambda}^{+} a_{k\lambda} + \frac{1}{2}) \hbar \omega_k \rightarrow E = \sum_{k,\lambda} (n_{k\lambda} + \frac{1}{2}) \hbar \omega_k \qquad (8)$$

Following this operation the electromagnetic field is considered to be composed of single mode harmonic oscillators corresponding to single photon states.

It is of crucial importance to mention here that we have absolutely no physical reason based on the experimental evidence to consider the mean value of the normal and anti-normal ordering Hamiltonians in order to obtain a harmonic oscillator one [2,21]. As we will see later this arbitrary mathematical operation leads to unphysical results compared to the observations.

Now, a different way is also often used in order to establish a harmonic oscillator Hamiltonian for the electromagnetic field. It is based on the canonical variables of position $Q_{k\lambda}$ and momentum $P_{k\lambda}$. The vector potential amplitude and its complex conjugate can be expressed through the position and momentum variables as follows [1,2,18]

$$a_{k\lambda} = \frac{(\omega_k Q_{k\lambda} + i P_{k\lambda})}{2\omega_k \sqrt{\epsilon_0 V}} \quad ; \quad a_{k\lambda}^{*} = \frac{(\omega_k Q_{k\lambda} - i P_{k\lambda})}{2\omega_k \sqrt{\epsilon_0 V}} \qquad (9)$$

Using (9) in the classical electromagnetic field energy (5) we get the following expression

$$E_{EM} = \frac{1}{2} \sum_{k,\lambda} (P_{k\lambda}^2 + \omega_k^2 Q_{k\lambda}^2) \pm i \omega_k [Q_{k\lambda}, P_{k\lambda}] \qquad (10)$$

where the (+) sign is obtained with the normal ordering, $a_{k\lambda}^{*} a_{k\lambda}$, and the (-) one with the anti-normal ordering, $a_{k\lambda} a_{k\lambda}^{*}$.

At that level, as $Q_{k\lambda}$ and $P_{k\lambda}$ are simply canonical variables and not quantum mechanics operators it is generally considered in the literature that the commutation term in (10) vanishes imposing $[Q_{k\lambda}, P_{k\lambda}] = 0$. In this way, a harmonic oscillator expression is obtained for the classical electromagnetic field energy

$$E_{EM} = \frac{1}{2} \sum_{k,\lambda} (P_{k\lambda}^2 + \omega_k^2 Q_{k\lambda}^2) \qquad (11)$$
Replacing in the last equation the canonical variables of position and momentum with the
corresponding quantum mechanics Hermitian operators [1,2,18]

$$
\tilde{P}_{k\lambda} = i \sqrt{\frac{\hbar\omega_k}{2}} (a_{k\lambda}^{+} - a_{k\lambda}) \quad ; \quad \tilde{Q}_{k\lambda} = \sqrt{\frac{\hbar}{2\omega_k}} (a_{k\lambda}^{+} + a_{k\lambda}) \qquad (12)
$$

we get a harmonic oscillator Hamiltonian for the electromagnetic field

$$
\tilde{H}_{EM} = \sum_{k,\lambda} (a_{k\lambda}^{+} a_{k\lambda} + \frac{1}{2}) \hbar \omega_{k} \rightarrow E = \sum_{k,\lambda} (n_{k\lambda} + \frac{1}{2}) \hbar \omega_{k} \quad (13)
$$

It is of high importance to underline here that, for a particle harmonic oscillator with mass $m \neq 0$ and momentum $\vec{p} = md\vec{q}/dt$, with the canonical variables of position $Q = |\vec{q}|\sqrt{m}$ and momentum $P = |\vec{p}|/\sqrt{m}$ the transition from the classical expression of energy

$$
E = \frac{1}{2} (P^2 + \omega^2 Q^2) \tag{14}
$$

to the quantum mechanics Hamiltonian

$$
\tilde{H} = \frac{1}{2} (\tilde{P}^2 + \omega^2 \tilde{Q}^2) = \hbar \omega \left(a^+ a + \frac{1}{2}\right) \quad (15)
$$

where $\tilde{Q} = \sqrt{\hbar/2\omega}(a^+ + a)$ and $\tilde{P} = i\sqrt{\hbar\omega/2}(a^+ - a)$, is direct and needs no commutation operations between the canonical variables $P$ and $Q$ [2,22]. Consequently, for a massive particle the harmonic oscillator Hamiltonian obtained in (15) is a direct result expressing a real physical state, e.g. phonons in solid state physics [23].

Conversely, in the case of the electromagnetic field a commutation operation appears
between the canonical variables $Q_{k\lambda}$ and $P_{k\lambda}$ during the transition from (5) to (10).
Consequently, cancelling the commutation term $[Q_{k\lambda}, P_{k\lambda}]$ in (10) in order to obtain (11) just
before replacing the canonical variables by the corresponding quantum mechanics operators is
not mathematically correct. It is extremely important to underline here that Heisenberg's
commutation relation $[\tilde{Q}_{k\lambda}, \tilde{P}_{k'\lambda'}] = i\hbar\delta_{k'k}\delta_{\lambda'\lambda}$ is a fundamental concept of quantum mechanics
which has not to be ignored when replacing classical canonical variables by the corresponding
quantum mechanics operators.

Hence, keeping the commutation term $[Q_{k\lambda}, P_{k\lambda}]$ in (10) and then substituting the
classical variables by the corresponding quantum operators (12) we obtain exactly the same
normal ordering and anti-normal ordering Hamiltonians as in (6) and (7)

$$
\tilde{H}_{EM}^{(+)} = \frac{1}{2} \sum_{k,\lambda} (\tilde{P}_{k\lambda}^2 + \omega_k^2 \tilde{Q}_{k\lambda}^2) + i\omega_k [\tilde{Q}_{k\lambda}, \tilde{P}_{k'\lambda'}] \rightarrow \tilde{H}_{NO} = \sum_{k,\lambda} a_{k\lambda}^{+} a_{k\lambda} \hbar \omega_k \rightarrow E_{NO} = \\
\sum_{k,\lambda} n_{k\lambda} \hbar \omega_k \tag{16}
$$

$$
\tilde{H}_{EM}^{(-)} = \frac{1}{2} \sum_{k,\lambda} (\tilde{P}_{k\lambda}^2 + \omega_k^2 \tilde{Q}_{k\lambda}^2) - i\omega_k [\tilde{Q}_{k\lambda}, \tilde{P}_{k'\lambda'}] \rightarrow \tilde{H}_{ANO} = \sum_{k,\lambda} (a_{k\lambda}^+ a_{k\lambda} + 1)\hbar\omega_k \rightarrow E_{ANO} = \\
\sum_{k,\lambda} (n_{k\lambda} + 1)\hbar\omega_k \tag{17}
$$

Obviously, as quoted be many authors, the mathematical ambiguity consisting of
suppressing the classical variables commutation term $[Q_{k\lambda}, P_{k\lambda}]$ just before replacing the
classical variables by the non-commuting quantum mechanics operators $[\tilde{Q}_{k\lambda}, \tilde{P}_{k'\lambda'}] =$
$i\hbar\delta_{k'k}\delta_{\lambda\lambda'}$ leads to the harmonic oscillator Hamiltonian [2,24,25]. It comes out that considering
the mean value of the normal and anti-normal ordering Hamiltonians implies inadvertently the
drop of Heisenberg's uncertainty relation.

It is clear that it is impossible to get a harmonic oscillator Hamiltonian for the
electromagnetic field without taking arbitrarily the mean value of the normal and anti-normal
ordering Hamiltonians or without neglecting Heisenberg's uncertainty [2,21].

2.3 *The zero-point energy singularity and the physical consequences*

In total absence of photons, that is for $n_{k\lambda} = 0$ in (8) or (13) we get the well-known
zero-point energy of the electromagnetic field corresponding to the electromagnetic quantum
vacuum
$$E_{ZPE} = \sum_{k,\lambda} \frac{1}{2} \hbar \omega_k \qquad (18)$$

whose energy density in any volume V in space is infinite

$$\rho_{ZPE} = \frac{1}{V} \sum_{k,\lambda} \frac{1}{2} \hbar \omega_k \rightarrow \infty \qquad (19)$$

This is the principal singularity in QED theory, the energy density of the zero-point energy is infinite at any point in space [26].

Equation (19) is a pure mathematical result, issued from the eigenstates of the harmonic oscillator Hamiltonian and should normally represent a real physical state yielding tremendous cosmological consequences [10]. Despite of this, recent well-validated astrophysical observations upon the cosmic acceleration estimated the vacuum energy density to be finite and extremely low, roughly $\rho_{ZPE}^{obs} \sim 10^{-10} Jm^{-3}$ contradicting the theoretical statement resulting from the harmonic oscillator Hamiltonian [4-10]. Following the observational evidence we are now constrained to find a finite value for the zero-point energy density.

Our knowledge of the QCD vacuum state is extremely elusive but QED offers a readily comprehensive framework for estimating the zero-point energy density. Hence, transforming the discrete summation in (19) into a continuous one, according to the density of states theory, we obtain an integration over the frequencies which is also infinite [26]

$$\rho_{ZPE} = \frac{\hbar}{2\pi^2 c^3} \int_{0}^{\infty} \omega^3 d\omega \rightarrow \infty \qquad (20)$$

We can now get a finite value by upper limiting the integration. In fact, it is generally considered that the higher possible photon frequency beyond which the energy density transforms the photon into a black hole should be roughly ~$10^{43}$Hz corresponding to Planck's energy $10^{19}$GeV. Thus, replacing the upper limit of the integration in (20) by the frequency corresponding to Planck's energy we get a theoretical value for the electromagnetic quantum vacuum energy density $\rho_{ZPE}^{th} \sim 10^{110} Jm^{-3}$. Considering the zero-point energy contribution of all the other known fields does not changes radically this result which is in disagreement with the experimental observations by 120 orders of magnitude [7,9,10]. This is the worst theoretical prediction in the history of science and has been called by many authors "vacuum catastrophe". Consequently, the experimental evidence implies that the zero-point energy issued from the harmonic oscillator Hamiltonian does not correspond to a real physical state.

In reality, there is no experimental evidence that a single photon is a harmonic oscillator entailing that there is no physical reason for considering the mean value of the normal and anti-normal ordering Hamiltonians in order to obtain a harmonic oscillator one [2,21]. This is also the reason why in QED the normal ordering Hamiltonian is principally used casting aside the zero point energy singularity.

On the other hand, it is also widespread that the zero-point energy represented by equation (18) is at the origin of the quantum vacuum effects such as the spontaneous emission, the Lamb shift or even the Casimir effect. Therein, we recall that the spontaneous emission effect and the Lamb shift are calculated in QED based on the properties of the single photon creation $a_{k\lambda}^{+}$ and annihilation $a_{k\lambda}$ operators without invoking at all the zero point energy term $\sum_{k,\lambda} \frac{1}{2} \hbar \omega_k$ which is a constant and commutes with all quantum mechanics operators. Indeed, the interaction Hamiltonian between an electron and the quantum vacuum vanishes

$$H_{\text{int}} = -e \vec{r} \cdot \frac{i}{\hbar} \left[ \tilde{A}(\vec{r}, t), \sum_{k,\lambda} \frac{1}{2} \hbar \omega_k \right] = 0 \qquad (21)$$

entailing that the zero-point energy term has absolutely no physical impact to the spontaneous emission and the Lamb shift.

Regarding the well-known Casimir effect it has been demonstrated by many studies using different approaches [27-30] that it is physically due to Lorentz's forces or source fields and not to the zero-point energy.
Thus, as often quoted in the literature, skipping the zero-point energy by considering only the normal ordering Hamiltonian do not compromise any of the achievements related to the calculation of the vacuum effects in QED [2,29,30].

To conclude, the zero point energy term $\sum_{k,\lambda} \frac{1}{2} \hbar \omega_k$ issued from the harmonic oscillator Hamiltonian of the electromagnetic field is obtained by considering without any valuable reason the mean value of the normal and antinormal ordering Hamiltonians and neglecting Heisenberg's uncertainty. In addition, this term is not involved in the vacuum effects in QED and conflicts strongly with the astrophysical observational evidence. Consequently, the vacuum concept in QED has to be reappraised and we may reasonably wonder which physical vacuum representation depending on the creation and annihilation operators without involving singularities could be appropriate for being associated to the normal ordering Hamiltonian.

### 3. Remedying QED fundamental drawbacks

In what follows, without stating postulates or axioms, we enhance the quantization of the vector potential to a single photon level and we show that all the above ambiguities and singularities in the QED formalism are naturally overcome.

#### 3.1. Photon quantization volume, an intrinsic geometrical property.

It is obvious that for the equation (1) to hold physically the parameter $V$ should correspond to the characteristic quantization volume of a single photon state. Such a spatial dependence of the photon may be deducted directly from the density of states theory. In fact, the number of photons $dn(\omega)$ in a given volume $V$ in the frequency interval between $\omega$ and $\omega + d\omega$ writes [1,2]

$$dn(\omega) = 4V \frac{\omega^2}{\pi^2 c^3} d\omega \qquad (22)$$

where we have considered the two photon spin states, corresponding classically to circular left and right helicity, and the two possible directions along the same propagation axis. Integrating (22) implies that a quantization volume can be assigned to a single $k$-mode photon

$$V_k = \left(\frac{3}{4}\pi^2 c^3\right) \omega_k^{-3} \qquad (23)$$

This is compatible with the theoretical concept that a photon extends over a wavelength [2,3,31] and also with the experimental observations which have shown that its lateral dimension is proportional to the quarter of the wavelength [32-35] entailing, that a photon can only be localized within a volume proportional to the cube of the wavelength [1-3,18]. This experimental fact is well established since Mandel's experiments [36,37] until now [2,18]. Hence, introducing (23) in (1) we get directly the vector potential amplitude for a single $k$-mode photon

$$\alpha_{0k}(\omega_k) = \pm \frac{1}{\pi} \sqrt{\frac{2\hbar}{3\epsilon_0 c^3}} \omega_k = \xi \omega_k \qquad (24)$$

The established photon vector potential amplitude issued from the density of states theory is now proportional to the frequency and thus compatible with the natural units of the vector potential. An equivalent result has already been deducted previously applying a different methodology [38,39]. It is worth noting that $\xi$ has both positive and negative values confirming that the photon is its own antiparticle as always suspected in the particle theory.

Although the integration of (22) does not yield a pure integer for the photon number, the vector potential amplitude constant obtained above by the density of states theory is an excellent approximation. A precise value of $\xi$ can be obtained by normalizing the mean energy of a single mode plane electromagnetic wave over a period to the single photon energy [40,41]
$$E_k = \left\langle \int \epsilon_0 \omega_k^2 \xi^2 \omega_k^2 \left[ \hat{\epsilon}_{k(L,R)} e^{i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} + \hat{\epsilon}_{k(L,R)}^* e^{-i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} \right]^2 d^3 r \right\rangle_T = \hbar \omega_k \quad (25)$$

where L,R corresponds to left or right circular polarization respectively.

We draw the expression of the photon vector potential amplitude constant $\xi$

$$\xi = \pm \frac{1}{(2\pi)^{3/2}} \sqrt{\frac{\hbar}{8\alpha\epsilon_0 c^3}} = \frac{\hbar}{4\pi ec} \qquad (26)$$

where $\alpha = e^2/(4\pi\epsilon_0)\hbar c \approx 1/137$ is the fine structure constant and e the electron-positron charge.

Introducing $\alpha_{0k}(\omega_k) = \xi\omega_k = (\hbar/4\pi ec)\omega_k$ in (1) we get the photon quantization volume issued now from the energy normalization

$$V_k = \left( \frac{\hbar}{2\epsilon_0 \xi^2} \right) \omega_k^{-3} = 4\alpha\lambda_k^3 \qquad (27)$$

Note that (23), issued from the density of states, and (27), issued from the energy normalization yield the same numerical result $V_k \approx 0.029 \lambda_k^3$ showing the self-consistency of the calculation.

Therein, it is important noting that the electron-positron charge, a fundamental physical constant, issues naturally during the energy normalization procedure and is directly related to the photon vector potential amplitude quantization constant $\xi$ [40,42,43]

$$e = (4\pi)^2 \alpha \xi / \mu_0 \qquad (28)$$

where $\mu_0$ is the vacuum magnetic permeability.

The last expression relates the photon vector potential amplitude constant to the positive-negative elementary charge which puts the basis for understanding the mechanisms of the mutual transformations photons $\leftrightarrow$ electron-positrons as well as the vacuum polarization. It is worthy investigating whether the electrons and positrons, more generally leptons and antileptons, have an internal structure composed of photons standing states in specific spatial configurations (toroidal or other) as already suggested previously [44].

## 3.2. Single photon electric and magnetic fields.

We can now express the vector potential of an electromagnetic k-mode in classical and quantum representations using the defined vector potential amplitude in (24)

$$\vec{\alpha}_{k(L,R)}(\vec{r}, t) = \xi \omega_k \left[ \hat{\epsilon}_{k(L,R)} e^{i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} + \hat{\epsilon}_{k(L,R)}^* e^{-i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} \right] = \omega_k \vec{\Xi}_{k(L,R)}(\vec{r}, t) \quad (29)$$

$$\vec{\alpha}_{k(L,R)}(\vec{r}, t) = \xi \omega_k \left[ \hat{\epsilon}_{k(L,R)} a_{k(L,R)} e^{i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} + \hat{\epsilon}_{k(L,R)}^* a_{k(L,R)}^+ e^{-i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} \right] = \omega_k \vec{\Xi}_{k(L,R)}(\vec{r}, t) \qquad (30)$$

The precession of the quantized vector potential around the propagation axis at the angular frequency $\omega_k$, generates orthogonal electric $\vec{\epsilon}_k$ and magnetic $\vec{\beta}_k$ fields [25,40,45,46] whose magnitudes, that is the square root of the sum of the squares of the components, are obtained straightforward

$$\vec{\epsilon}_k = -\frac{\partial}{\partial t} \xi \omega_k \left( \hat{\epsilon}_{k(L,R)} e^{i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} + \hat{\epsilon}_{k(L,R)}^* e^{-i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} \right) \rightarrow \|\vec{\epsilon}_k\| = \sqrt{2} |\xi| \omega_k^2 \quad (31)$$

$$\vec{\beta}_k = \vec{\nabla} \times \xi \omega_k \left( \hat{\epsilon}_{k(L,R)} e^{i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} + \hat{\epsilon}_{k(L,R)}^* e^{-i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} \right) \rightarrow \|\vec{\beta}_k\| = \sqrt{2\epsilon_0\mu_0} |\xi| \omega_k^2 \quad (32)$$

Note that $\vec{\epsilon}_k$ and $\vec{\beta}_k$ are proportional to the square of the frequency and independent on any external volume parameter. It is also worth noting that the proportionality of the photon electric field amplitude to the square of the frequency may naturally lead to a vacuum energy density compatible with recent astrophysical observations on the cosmic acceleration [47].

Experimental proposals have already been advanced previously [45] for validating the dependence of the photon electric field on the square of the frequency, which could also be an indirect validation for the vector potential dependence on frequency. As the linear Stark shift is directly proportional to the applied electric field, while the quadric Stark shift to the square
of the electric field, various configurations of quantum systems may be chosen for that purpose. Thus, high resolution laser spectroscopy coupled to microwave spectroscopy might be used for the measurement of the Stark shifts induced by the mean electric field of photons upon the energy levels of selected quantum systems, atoms, ions or molecules. Finally, the established photon electric and magnetic fields satisfy Maxwell's equations

$$ \vec{\nabla} \times \vec{\varepsilon}_k = - \frac{\partial \vec{\beta}_k}{\partial t} \qquad \vec{\nabla} \cdot \vec{\varepsilon}_k = 0 \qquad (33) $$

$$ \vec{\nabla} \times \vec{\beta}_k = \varepsilon_0 \mu_0 \frac{\partial \vec{\varepsilon}_k}{\partial t} \qquad \vec{\nabla} \cdot \vec{\beta}_k = 0 \qquad (34) $$

considered in vacuum and in absence of charges.

### 3.3. The photon quantization volume and Heisenberg's uncertainty

The single photon vector potential amplitude operators now write

$$ \tilde{\alpha}_{k\lambda} = \xi \omega_k \alpha_{k\lambda} \quad ; \quad \tilde{\alpha}_{k\lambda}^* = \xi \omega_k \alpha_{k\lambda}^+ \qquad (35) $$

On the other hand, the position and momentum Hermitian operators are expressed respectively as follows [1,2,18]

$$ \tilde{Q}_{k\lambda} = \sqrt{\varepsilon_0 V_k} (\tilde{\alpha}_{k\lambda} + \tilde{\alpha}_{k\lambda}^*) \quad ; \quad \tilde{P}_{k\lambda} = -i\omega_k \sqrt{\varepsilon_0 V_k} (\tilde{\alpha}_{k\lambda} - \tilde{\alpha}_{k\lambda}^*) \qquad (36) $$

Using now (27) and (35) in the last expressions of position and momentum operators (36) it is straightforward to demonstrate Heisenberg's commutation relation

$$ [\tilde{Q}_{k\lambda}, \tilde{P}_{k'\lambda'}] = -i\varepsilon_0\omega_k^2\omega_k\sqrt{V_kV_{k'}}[(\xi a_{k\lambda} + \xi a_{k\lambda}^+), (\xi a_{k'\lambda'} - \xi a_{k'\lambda'}^+)] = ih\delta_{kk'}\delta_{\lambda\lambda'} \qquad (37) $$

showing that the position-momentum uncertainty for the photon, an intrinsic physical property, lies on the characteristic quantization volume.

Indeed, it is now easy to express the particle properties of the photon, energy and momentum, through the classical electromagnetic functions integrated over the quantization volume $V_k$ and considering circular polarization

$$ E_k = \int_{V_k} 2\varepsilon_0 \alpha_{0k}^2(\omega_k) \omega_k^2 d^3r = \int_{V_k} 2\varepsilon_0 \xi^2 \omega_k^4 d^3r = 2\varepsilon_0 \xi^2 \omega_k^4 V_k = \hbar\omega_k \qquad (38) $$

$$ \vec{p}_k = \int_{V_k} \varepsilon_0 \vec{\varepsilon}_k \times \vec{\beta}_k d^3 r = \varepsilon_0 (\sqrt{2\xi}\omega_k^2) (\sqrt{2\varepsilon_0\mu_0\xi\omega_k^2}) V_k \vec{k} / |\vec{k}| = \hbar \vec{k} \qquad (39) $$

where for the momentum we have used the photon electric and magnetic fields established above in (31) and (32).

Obviously, the photon energy and momentum are not carried by a point particle but by a wave-corpuscle with a precise quantization volume [25,40]. Consequently, the photon is not a point or a singularity and constitutes a particular case in the standard particle model. The photon quantization volume also provides a physical explanation why a photon position operator cannot be defined and can only be expressed through the quantization volume as in relation (36).

Finally, it is important underlying that even though photons are not points it has been demonstrated experimentally that they are indivisible [2,18,48], carrying energy and momentum and consequently they behave exactly as Einstein has predicted [49].

### 3.4. The photon wave function

It can be easily shown that the vector potential function $\vec{\alpha}_{k(L,R)}(\vec{r}, t)$ with the quantized amplitude expressed in (29) satisfies Maxwell's propagation equation in vacuum

$$ \vec{\nabla}_k^2 \vec{\alpha}_{k(L,R)}(\vec{r}, t) - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \vec{\alpha}_{k(L,R)}(\vec{r}, t) = 0 \qquad (40) $$

as well as Schrödinger's equation with the relativistic massless particle Hamiltonian

$$ i\hbar \frac{\partial}{\partial t} \vec{\alpha}_{k(L,R)}(\vec{r}, t) = \tilde{H}_k \vec{\alpha}_{k(L,R)}(\vec{r}, t) \qquad (41) $$
where $\tilde{H}_k = \hbar\tilde{\omega}_k = -i\hbar c\vec{\nabla}_k$, with $\tilde{\omega}_k$ the angular frequency operator $\tilde{\omega}_k = -ic\vec{\nabla}_k$, where $\vec{\nabla}_k$ is merely denoted to act upon the mode $k$.

Also, $\vec{\alpha}_{k(L,R)}(\vec{r}, t)$ satisfies a Schrödinger-like equation for the vector potential amplitude

$$i\xi \frac{\partial}{\partial t} \vec{\alpha}_{k(L,R)}(\vec{r}, t) = \tilde{\alpha}_{0k} \vec{\alpha}_{k(L,R)}(\vec{r}, t) \quad (42)$$

with the vector potential amplitude operator which also writes $\tilde{\alpha}_{0k} = \xi \tilde{\omega}_k = -i\xi c\vec{\nabla}_k$.

The last two symmetrical equations can be coupled together leading to the photon vector potential-energy equation, a unique quantum equation expressing both the first (energy) and second (vector potential) quantization of the electromagnetic field [25,40,46]

$$i \left(\frac{\xi}{\hbar}\right) \frac{\partial}{\partial t} \vec{\alpha}_{k(L,R)}(\vec{r}, t) = \begin{pmatrix} \tilde{\alpha}_{0k} \\ \tilde{H}_k \end{pmatrix} \vec{\alpha}_{k(L,R)}(\vec{r}, t) \quad (43)$$

The quantization constants $\xi$ and $\hbar$ characterize the quantization of the photon vector potential amplitude $\alpha_{0k}(\omega_k) = \xi\omega_k$ and energy $E_k(\omega_k) = \hbar\omega_k$ respectively.

A photon wave function related to the energy density of a single photon has already been established previously based also on the vector potential function [41]. Now, we will establish a photon wave function as a probability amplitude according to the quantum mechanics principles. Indeed, in quantum theory the wave function normalization is generally carried out in such a way that the integration of the square modulus of the wave function within a given volume expresses the localization probability in the said volume.

According to these concepts, as the experiments have shown that a single photon is localizable only within a volume proportional to $\lambda_k^3$, we may establish a wave function for the photon based precisely on the vector potential function $\vec{\alpha}_{k(L,R)}(\vec{r}, t)$ and by taking into account the photon quantization volume $V_k$ for the normalization. Consequently, the photon wave function can be defined as follows [50,51]

$$\vec{\Phi}_{k(L,R)}(\vec{r}, t) = \left(\frac{\varepsilon_0 \omega_k}{\hbar}\right)^{1/2} \left[ \alpha_{0k}(\omega_k) \left( \hat{\varepsilon}_{k(L,R)} e^{i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} + \hat{\varepsilon}_{k(L,R)}^* e^{-i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} \right) \right] \quad (44)$$

where the photon spin is expressed through the helicity with left or right circular polarizations corresponding to the values $\pm\hbar$.

The normalization pre-factor has been chosen in such a way that

$$|\vec{\Phi}_{k(L,R)}(\vec{r}, t)|^2 = \frac{\varepsilon_0 \omega_k}{\hbar} \left[ \xi^2 \omega_k^2 \left( |\hat{\varepsilon}_{k(L,R)}|^2 + |\hat{\varepsilon}_{k(L,R)}^*|^2 \right) \right] = \frac{2\varepsilon_0 \xi^2 \omega_k^3}{\hbar} = \frac{1}{V_k} \quad (45)$$

entailing that the normalization condition is satisfied with respect to the quantization volume

$$\int_{V \le V_k} |\vec{\Phi}_{k(L,R)}(\vec{r}, t)|^2 d^3 r' = 1 \quad (46)$$

Therefore, when a photon is emitted in vacuum at an instant $t_0$ at the coordinate $r_0$ the probability to be localized at $t$ around $r=r_0+ct$ on the propagation axis is normalized with respect to the quantization volume $V_k$.

Furthermore, $\vec{\Phi}_{k(L,R)}(\vec{r}, t)$ satisfies Maxwell's propagation equation in vacuum

$$\vec{\nabla}_k^2 \vec{\Phi}_{k(L,R)}(\vec{r}, t) - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \vec{\Phi}_{k(L,R)}(\vec{r}, t) = 0 \quad (47)$$

as well as the photon vector potential-energy quantum equation

$$i \left(\frac{\xi}{\hbar}\right) \frac{\partial}{\partial t} \vec{\Phi}_{k(L,R)}(\vec{r}, t) = \begin{pmatrix} \tilde{\alpha}_{0k} \\ \tilde{H}_k \end{pmatrix} \vec{\Phi}_{k(L,R)}(\vec{r}, t) \quad (48)$$

Consequently, $\vec{\Phi}_{k(L,R)}(\vec{r}, t)$ is a real wave function for the photon involving the spin through the helicity corresponding to circular polarizations, and normalized with respect to the quantization volume $V_k$.

Obviously, the photon wave function $\vec{\Phi}_{k(L,R)}(\vec{r}, t)$ is defined at an instant $t$ at the coordinate $r$. However, the photon as an integral wave-corpuscle may only be conceived within a volume $V_k$ around $r$ as expressed in (46). In addition, its physical properties are obtained by
integration of the wave properties over $V_k$ according to (38) and (39). Consequently, in the case of the photon, the mean values of the energy and momentum operators can be calculated by considering the quantization volume $V_k$. Hence, using the established photon wave function the mean value of the relativistic massless particle Hamiltonian in a single photon state writes

$$ 
\begin{aligned}
\left|\left\langle\Phi_{k(L,R)}(\vec{r}, t) | \tilde{H}_k | \Phi_{k(L,R)}(\vec{r}, t)\right\rangle_{V_k}\right| &= \left|\left\langle\Phi_{k(L,R)}(\vec{r}, t) | -i\hbar c \vec{\nabla}_k | \Phi_{k(L,R)}(\vec{r}, t)\right\rangle_{V_k}\right| \\
&= \epsilon_0 \xi^2 \omega_k^4 V_k \left|\left(e^{i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} \hat{\epsilon}_{k(L,R)} + e^{-i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} \hat{\epsilon}_{k(L,R)}^*\right) \left(e^{i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} \hat{\epsilon}_{k(L,R)} - e^{-i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} \hat{\epsilon}_{k(L,R)}^*\right)\right| \\
&= \epsilon_0 \xi^2 \omega_k^4 V_k \left|\left(e^{2i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)} - e^{-2i(\vec{k} \cdot \vec{r} - \omega_k t + \theta)}\right)\right| \\
&= (2\epsilon_0 \xi^2 \omega_k^4) V_k \left|\sin 2(\vec{k} \cdot \vec{r} - \omega_k t + \theta)\right| \qquad (49)
\end{aligned}
$$

The phase function $|\sin 2 (\vec{k} \cdot \vec{r} - \omega_k t + \theta)|$ varies from zero to one, which physically means that when $|\sin 2 (\vec{k} \cdot \vec{r} - \omega_k t) + \theta| = 1$ corresponds to the total space of the quantization volume $V_k$ so that the mean value of the Hamiltonian gives precisely Planck's single photon energy $|\langle\Phi_{k(L,R)}(\vec{r}, t)|\tilde{H}_k|\Phi_{k(L,R)}(\vec{r}, t)\rangle_{V_k}| = \hbar\omega_k$.

With the same token, the momentum operator yields Einstein's single photon momentum

$$ \left|\left\langle\Phi_{k(L,R)}(\vec{r}, t) | \vec{p}_k | \Phi_{k(L,R)}(\vec{r}, t)\right\rangle_{V_k}\right| = \left|\left\langle\Phi_{k(L,R)}(\vec{r}, t) | -i\hbar \vec{\nabla}_k | \Phi_{k(L,R)}(\vec{r}, t)\right\rangle_{V_k}\right| = \frac{\hbar\omega_k}{c} \quad (50) $$

It is also straightforward to calculate the energy density for a single photon state

$$ W_k = |\Phi_{k(L,R)}(\vec{r}, t)|^2 \left|\left\langle\Phi_{k(L,R)}(\vec{r}, t) | \tilde{H}_k | \Phi_{k(L,R)}(\vec{r}, t)\right\rangle_{V_k}\right| = \frac{\hbar\omega_k}{V_k} = 2\epsilon_0\xi^2\omega_k^4 \quad (51) $$

which, by considering the photon electric and magnetic fields established in (31) and (32), is exactly equal to the classical expression

$$ W_k = \frac{1}{2} (\epsilon_0 |\vec{\epsilon}_k|^2 + \frac{1}{\mu_0} |\vec{\beta}_k|^2) = 2\epsilon_0 \xi^2 \omega_k^4 \quad (52) $$

We have thus demonstrated the perfect equivalence between the classical and quantum energy density for a single photon reappraisiing the relation (1).

Note also, that the obtained single photon energy density is proportional to the fourth power of the frequency as in the case of the radiation emitted by a classical oscillating dipole [19,20]. Furthermore, a Poynting vector can now be readily defined for a single k-mode photon

$$ \vec{S}_k = \frac{1}{\mu_0} \vec{\epsilon}_k \times \vec{\beta}_k = W_k c \frac{\vec{k}}{|\vec{k}|} \quad (53) $$

To conclude, the photon wave function $\vec{\Phi}_{k(L,R)}(\vec{r}, t)$ expressed in (44) represents a probability density normalized with respect to the intrinsic photon quantization volume $V_k$. In addition, it satisfies Maxwell's propagation equation in vacuum, Schrödinger's equation with the relativistic massless particle Hamiltonian and a Schrödinger-like equation for the vector potential amplitude [50,51].

Excellent studies have been published previously on the intriguing photon wave function [12-17]. However, none of those considers the quantization of the vector potential amplitude at a single photon state, although it has been demonstrated experimentally that the photon, in the sense of Einstein's concept, is a local indivisible quantum of the electromagnetic field. Consequently, the energy quantization should imply physically the single photon vector potential quantization.

From the experimental point of view, the investigation of a cavity free photon wave function should not be an easy task. Nonetheless, the measurement of the minimum quantization volume, used for the wave function spatial normalization, would be of great importance. Thus, the minimization of microwave cavities having homothetic dimensions with the experimental values [32-35] for the single mode wavelength ($\lambda_k$) and lateral extension ($\sim \lambda_k/4$) might be useful.
In the IR frequency range, the optimization of the shorter photoluminescence lifetime of a quantum dot by using various dimensions of micro cavities [52,53] might also be used for that purpose.

### 3.5. The electromagnetic vacuum

According to (24), (31), (32) and (38), for $\omega_k = 0$ the photon vector potential, electric and magnetic fields as well as the energy vanish. This physical state lays beyond the Bohm-Aharonov effect in which the electric and magnetic fields are absent but a vector potential subsists in space. In fact, a photon, as an integral physical entity, can only be conceived for a non-zero frequency which corresponds to the precession of the vector potential amplitude with circular polarization over a wavelength along the propagation axis. However, the zero frequency state does not correspond to an intangible empty space because in absence of precession the vector potential amplitude per frequency still subsists. In fact, the principal function $\mathcal{E}_{k(L,R)}$ of the vector potential as expressed in (29) and (30) does not cancel and writes in both classical and quantum representations

$$ \vec{\mathcal{E}}_{0(L,R)} = \xi (\hat{\varepsilon}_{(L,R)} e^{i\theta} + \hat{\varepsilon}_{(L,R)}^* e^{-i\theta}) \quad (54) $$

$$ \tilde{\mathcal{E}}_{0(L,R)} = \xi (\hat{\varepsilon}_{(L,R)} a_{k(L,R)} e^{i\theta} + \hat{\varepsilon}_{(L,R)}^* a_{k(L,R)}^+ e^{-i\theta}) \quad (55) $$

Hence, in absence of energy, of vector potential as well as of electric and magnetic fields, the zero frequency state $\mathcal{E}_{0(L,R)}$ represents the electromagnetic field ground state, which corresponds to the electromagnetic vacuum. Obviously, $\mathcal{E}_{0(L,R)}$ is a real field permeating all of space, involving the circular polarization information and having electric potential nature with positive and negative components. It is worth underlying that it has both classical and quantum representations entailing the introduction of the new fundamental concept that the electromagnetic vacuum can also be defined in classical theory.

It is clear that the phase parameter $\theta$ can take any values and consequently the electromagnetic vacuum is composed of all possible states $\mathcal{E}_{k(L,R)}(\vec{r}, t)$

$$ \vec{\mathcal{E}}_{k(L,R)}(\vec{r}, t) = \xi \left( \hat{\varepsilon}_{k(L,R)} e^{i(\vec{k} \cdot \vec{r} - \omega_k t + \phi)} + \hat{\varepsilon}_{k(L,R)}^* e^{-i(\vec{k} \cdot \vec{r} - \omega_k t + \phi)} \right) \quad (56) $$

$$ \tilde{\mathcal{E}}_{k(L,R)}(\vec{r}, t) = \xi \left( \hat{\varepsilon}_{k(L,R)} a_{k(R,L)} e^{i(\vec{k} \cdot \vec{r} - \omega_k t + \phi)} + \hat{\varepsilon}_{k(L,R)}^* a_{k(L,R)}^+ e^{-i(\vec{k} \cdot \vec{r} - \omega_k t + \phi)} \right) \quad (57) $$

From the photon vector potential – energy equation we get the equation governing the vacuum states

$$ i \frac{\partial}{\partial t} \left[ \begin{matrix} \vec{\mathcal{E}}_{k(L,R)}(\vec{r}, t) \\ \tilde{\mathcal{E}}_{k(L,R)}(\vec{r}, t) \end{matrix} \right] = \tilde{\omega}_k \left[ \begin{matrix} \vec{\mathcal{E}}_{k(L,R)}(\vec{r}, t) \\ \tilde{\mathcal{E}}_{k(L,R)}(\vec{r}, t) \end{matrix} \right] = \left[ \begin{matrix} \tilde{\alpha}_{k(L,R)}(\vec{r}, t) \\ \tilde{\alpha}_{k(L,R)}(\vec{r}, t) \end{matrix} \right] \quad (58) $$

The physical meaning of equation (58) is that real photons emerge from the vacuum states under the action of the angular frequency operator $\tilde{\omega}_k = -ic\vec{\nabla}_k$.

Consequently, using the electromagnetic vacuum we can now demonstrate the calculation of the spontaneous emission probability lifting the well-known QED ambiguity related to the electron-vacuum interaction Hamiltonian. We recall that the Hamiltonian of an atom in the presence of an electromagnetic field writes in QED formalism within the dipole approximation [1,2]

$$ H_{tot} = H_{atom} + H_{EM} + H_{int} = \sum_i \hbar \omega_i |\Psi_i\rangle\langle\Psi_i| + \sum_{k,\lambda} \hbar \omega_k \left( a_{k\lambda}^+ a_{k\lambda} + \frac{1}{2} \right) - \vec{D} \cdot \vec{E}(\vec{r}, t) \quad (59) $$

where $|\Psi_i\rangle$ in $H_{atom}$ represent the atomic quantum levels with respective energies $\hbar\omega_i$, $H_{EM}$ is the electromagnetic field harmonic oscillator Hamiltonian including the zero-point energy $\sum_{k,\lambda} \frac{1}{2} \hbar\omega_k$, $\vec{D} = e\vec{r}$ is the dipole moment of the bound electron and $\vec{E}(\vec{r}, t)$ is the radiation electric field involved in the interaction Hamiltonian $H_{int}$.
The electromagnetic radiation electric field $\vec{E}(\vec{r}, t)$ is calculated from Heisenberg's equation of motion

$$ \vec{E}(\vec{r}, t) = -\frac{\partial \vec{A}(\vec{r}, t)}{\partial t} = \frac{i}{\hbar} [\vec{A}(\vec{r}, t), \sum_{k,\lambda} \hbar \omega_k (a_{k\lambda}^{\dagger} a_{k\lambda} + \frac{1}{2})] \quad (60) $$

As generally quoted in the literature, the spontaneous emission rate in QED is calculated neglecting the zero-point energy term in the electromagnetic field Hamiltonian [1,2]. Furthermore, the zero-point energy is a constant so that $[\vec{A}(\vec{r}, t), \sum_{k,\lambda} \frac{1}{2} \hbar \omega_k] = 0$ in the radiation electric field $\vec{E}(\vec{r}, t)$ calculated by (60) and consequently the vacuum term has absolutely no contribution to the interaction Hamiltonian used to calculate the spontaneous emission in QED.

Surprisingly, with the aim of calculating the interaction between an electron and the quantum vacuum the zero-point energy is casted aside in both Hamiltonians $H_{EM}$ and $H_{int}$.

As mentioned previously, this well-known ambiguity in QED is due to the fact that the spontaneous emission is calculated based on the properties of the photon creation and annihilation operators, $a_{k\lambda}^{\dagger}$ and $a_{k\lambda}$ respectively, and not by using the zero-point energy. In addition, it comes out from (60) that, since the vacuum state in QED is not described by a function of $a_{k\lambda}^{\dagger}$ and $a_{k\lambda}$ operators, an electron-vacuum interaction Hamiltonian cannot be defined.

Conversely, the electromagnetic vacuum states expressed by $\tilde{\mathcal{E}}_{k(L,R)}(\vec{r}, t)$ in (57) do depend on the single photon creation and annihilation operators. Consequently, it is straightforward to define an interaction Hamiltonian per mode between an electron and the vacuum

$$ H_{int} = \frac{\hbar e}{m_e c} \tilde{\mathcal{E}}_{k(L,R)} \tilde{\omega}_k = -i\hbar \frac{e}{m_e} \tilde{\mathcal{E}}_{k(L,R)} \vec{\nabla}_k \quad (61) $$

where $m_e$ is the electron mass.

Using (61) and applying first order time dependent perturbation theory it has been shown that the spontaneous emission rate is readily calculated with similar procedure as in QED theory [39,40]. The Lamb shift is calculated as usually using the photon creation operator, while it can be readily demonstrated that any particle accelerated in the electromagnetic vacuum field experiences the Unruh temperature [43]. Finally, based on the fact that the single photon electric field amplitude is proportional to the square of the frequency, it can be shown that the electromagnetic vacuum fluctuations reproduce a vacuum energy density comparable to the astrophysical observations [47]. Following these developments, the pic of the electromagnetic vacuum fluctuations should be located in the THz frequencies and therefore a detailed investigation of the James Webb Space Telescope could give an answer to that issue.

Consequently, the electromagnetic vacuum complements the normal ordering Hamiltonian and confers a real physical state to the vacuum expressed in both classical and quantum theories.

As discussed above, a k-mode photon is generated from the application of the angular frequency $\omega_k$ upon the corresponding vacuum k-mode state $\tilde{\mathcal{E}}_{k(L,R)}(\vec{r}, t)$. Thus, it becomes clear that the electromagnetic vacuum is the real physical basis of the photon and of its wave function

$$ \vec{\Phi}_{k(L,R)}(\vec{r}, t) = \left(\frac{\varepsilon_0 \omega_k^3}{\hbar}\right)^{1/2} \vec{\mathcal{E}}_{k(L,R)}(\vec{r}, t) \quad (62) $$

This leads to the interesting conclusion that when a k-mode photon with polarization $\lambda$ is created in space the electromagnetic vacuum is depleted by the corresponding k-mode and $\lambda$-polarization vacuum state. We may consider that immediately after this process the new photon created at the same frequency carries an opposite polarization laying down new possibilities for understanding entanglement.
**4. Conclusions**

The vector potential quantization at a single photon level remedies the QED fundamental drawbacks. Thus, the photon vector potential amplitude operators as well as the electric and magnetic field amplitudes are defined independently on any external volume parameter. Also, the single photon electric and magnetic fields amplitudes in vacuum are directly proportional to the square of the frequency.

Regarding the principal singularity related to the zero-point energy, it can be eliminated by considering the normal ordering Hamiltonian associated to the electromagnetic vacuum which has both classical and quantum representations. The last one is a zero-energy universal field expressed through the photon creation and annihilation operators and permits to describe the interaction between the electrons and the vacuum states. This removes the ambiguity related to the zero-point energy in the interaction Hamiltonian for the vacuum effects without compromising any of the achievements in QED. The concept of the harmonic oscillator Hamiltonian for the electromagnetic field, resulting from the unjustified mean value of the normal and anti-normal order Hamiltonians, has to be appraised.

In addition, the density of states theory ascribes an intrinsic quantization volume to a single photon. The energy normalization of a single mode plane electromagnetic wave over a period confirms precisely the photon quantization volume issued from the density of states theory. Hence, the quantum properties of a single photon, energy and momentum, are not carried by a point but by a wave-corpuscle characterized by the quantization volume whose spatial dimensions are proportional to the cube of the wavelength as confirmed experimentally. Heisenberg's uncertainty of the photon position and momentum is due precisely to its intrinsic quantization volume.

Consequently, single photons are neither point particles nor harmonic oscillators and they constitute a particular case in the standard particle model.

A photon wave function is established normalized with respect to the quantization volume and satisfying Schrödinger's equation for the energy and the vector potential as well as Maxwell's propagation equation in vacuum. Furthermore, it depends directly on the electromagnetic vacuum states envisioning new horizons for understanding entanglement.

Finally, we have advanced some ideas for carrying out specific experiments in order to investigate the frequency dependence of the photon electric field and vector potential as well as for the definition of the intrinsic photon volume for the wave function normalization. Certainly, these questions are open and any other experimental proposals are welcome towards that issue.

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