We present theoretical developments expressing the physical characteristics of a single photon in conformity with the experimental evidence. The quantization of the electromagnetic field vector potential amplitude is enhanced to a free of cavity photon state. Coupling the Schrödinger equation with the relativistic massless particle Hamiltonian to a symmetrical vector potential relation, we establish the vector potential - energy equation for the photon expressing the simultaneous wave-particle nature of a single photon, an intrinsic physical property. It is shown that the vector potential can be naturally considered as a real wave function for the photon entailing a coherent localization probability. We deduce directly the electric and magnetic field amplitudes of the cavity-free single photon, which are revealed to be proportional to the square of the angular frequency. The zero-energy electromagnetic field ground state (EFGS), a quantum vacuum real component, issues naturally from Maxwell’s equations and the vector potential quantization procedure. The relation of the quantized amplitude of the photon vector potential to the electron-positron charge results directly showing the physical relationship between photons and electrons-positrons that might be at the origin of their mutual transformations. It becomes obvious that photons, as well as electrons-positrons, are issued from the same quantum vacuum field.
- single photon
- vector potential
- photon wave-particle equation
- photon wave function
- photon electric field
- electromagnetic field ground state
- electron-positron charge
During the last decades, an impressive technological development has been achieved permitting the manipulation of single photons with a high degree of statistical accuracy. However, despite the significant experimental advances, we still do not have a clear physical picture of a single photon state universally accepted by the scientific community, especially involved in quantum electrodynamics. In this chapter, based on the present state of knowledge, we make a synthesis of the physical characteristics of a single photon put in evidence by the experiments, and we advance theoretical developments for its representation. Accordingly, the concept of the wave-particle nature of a single photon becomes physically comprehensive and in agreement with the experimental evidence.
However, before advancing in the theoretical developments, we consider that it is important starting with a brief historical review on the efforts carried out previously for understanding the nature of light while simultaneously making a synthesis of the main experimental results which are of crucial importance for the comprehension of the birth of the photon concept.
The very first scientific publications on the nature of light are due to ancient Greeks who believed light is composed of corpuscles [1, 2]. Around 300 BC Euclid published the book
In the year of 1670, Newton revived the ideas of ancient Greeks and advanced the theory following that light is composed of corpuscles that travel rectilinearly . Ten years later, Huygens developed the principles of the wave theory of light [1, 4, 5]. Huygens’ wave theory was a hard opponent to Newton’s corpuscle concept. In the beginning of the nineteenth century, Young obtained experimentally interference patterns using different sources of light and explained some polarisation observations by assuming that light oscillations are perpendicular to the propagation axis [1, 6]. Euler and Fresnel explained the diffraction patterns observed experimentally by applying the wave theory . In 1865, Maxwell published his theory on the electromagnetic waves establishing the relations between the electric and magnetic fields and showing that light is composed of electromagnetic waves . A few years later, Hertz confirmed Maxwell’s theory by discovering the long-wavelength electromagnetic radiation [1, 7]. Thus, at the end of the nineteenth century, the scientific community started to accept officially the wave nature of light replacing Newton’s theory.
Nevertheless, new events supporting the particle nature of light occurred in the beginning of the twentieth century. Stefan and Wien discovered the direct relationship between the thermal radiation energy and the temperature of a black body [8, 9]. However, the emitted radiation energy density as a function of the temperature calculated by Rayleigh failed to describe the experimental results at short wavelengths. Scientists had given the name “UV catastrophe” to this problem revealing the necessity of a new theoretical approach. Planck managed to establish the correct energy density expression for the radiation emitted by a black body with respect to temperature, in excellent agreement with the experiment . For that purpose, he assumed that the bodies are composed of “oscillators” which have the particularity of emitting the electromagnetic energy in “packets” of
In 1902, Lenard pointed out that the photoelectric effect, discovered by Hertz 15 years earlier , occurs beyond a threshold frequency of light and the kinetic energy of the emitted electrons does not depend on the incident light intensity. Based on Planck’s works, Einstein proposed a simple interpretation of the photoelectric effect assuming that the electromagnetic radiation is composed of quanta with energy
Thus, the photoelectric effect and Compton scattering have been initially considered as the undoubtable demonstrations of the particle nature of light and historically were the strongest arguments in favour of the light quanta concept, which started to be universally accepted, and Lewis introduced the word “photon”, from the Greek word
Therein, it is extremely important to mention that Wentzel in 1926  and Beck in 1927 , as well as much later Lamb and Scully in the 1960s , demonstrated that the photoelectric effect can be interpreted remarkably well by only considering the wave nature of light, without referring to photons at all . Furthermore, the Compton scattering has been fully interpreted by Klein and Nishina in 1929  also by considering the electromagnetic wave nature of light without invoking the photon concept. On the other hand, Young’s experiment, initially presented as the most convincing argument for the wave nature of light, was applied by Taylor at very low intensities to demonstrate the particle concept of light . Indeed, much later Jin et al.  published an excellent theoretical interpretation of Young’s diffraction experiments based only on the particle representation of light.
Thus, the picture on the nature of light in the 1930s was rather confusing since both opposite sides defending the wave or the particle nature advanced equally strong arguments. Hence, Bohr, inspired by de Broglie’s thesis on the simultaneous wave character of particles, announced the
The development of lasers  in the 1960s and the revolutionary parametric down-convertion techniques [23, 24] in the 1970s, have made it possible to realise conditions in which, with a convenient statistical confidence, only a single photon may be present in the experimental apparatus. In this way, the double-prism experiment  realised in the 1990s contradicted for the first time Bohr’s
According to the experimental investigations, it has been always stated that a photon has circular, left or right, polarisation with spin and cannot be conceived along the propagation axis within a length shorter than its wavelength . Indeed, since Mandel’s experiments in the 1960s [27, 28], all the efforts to localise precisely a single photon remained fruitless yielding the conclusion is that a photon cannot be better localised than within a volume of the order of the cube of its wavelength [29, 30]. Furthermore, Grangier et al. demonstrated experimentally the indivisibility of photons [19, 31], while in recent years the entangled state experiments [29, 30] have shown that the photon should be locally an integral entity during the detection procedure but with a real non-local wave function.
The lateral expansion of a single photon, considered locally as an indivisible entity, was always an intriguing part of physics. With the purpose of studying the lateral expansion of the electromagnetic rays, Robinson in 1953  and Hadlock in 1958  carried out experiments using microwaves crossing small apertures and deduced that no energy is transmitted through apertures whose dimensions are smaller than roughly ∼
Thus, the experiments have shown that the single photon is not a point and cannot be localised at a coordinate, as stated by Einstein, while it can exhibit both the wave and particle natures in the same experimental conditions contradicting Bohr’s mutual exclusiveness. However, quantum electrodynamics (QED) has been developed during the 1930s to 1960s based upon the point particle model for the photon [36, 37, 38, 39]. In fact, the point photon concept has permitted to establish an efficient mathematical approach for describing states before and after an interaction processes [19, 39, 40, 41], but it is naturally inappropriate for the description of the real nature of a single photon.
Finally, what we can essentially draw out by summing up the experimental evidence is that a single photon is a minimum, local, indivisible part of the electromagnetic field with precise energy
In what follows, we present first the standard theoretical representation of the electromagnetic field quantization resulting in photons, and next we proceed to recent advances based on the vector potential quantization enhanced to a single photon state.
2. The electromagnetic field vector potential
2.1 Reality of the vector potential
where is the scalar potential, as well as the magnetic field:
In 1949, Ehrenberg and Siday were the first to put in evidence the influence of the vector potential on charged particles  deducing that it is a real physical field. Ten years later, Aharonov and Bohm re-infirmed the influence of the vector potential on electrons in complete absence of electric and magnetic fields . That was confirmed experimentally by Chambers , Tonomura et al. , and Osakabe et al.  demonstrating without any doubt the reality of the vector potential field end its direct influence on charges.
From a theoretical point of view , the behaviour of a particle with charge
with as Planck’s reduced constant.
If the solenoid is extremely long along the
The Schrödinger equation for a charged particle outside the solenoid, where the vector potential is not zero, writes in complete absence of any other external potential:
with given by Eq. (4). The solutions of the last equation are the wave functions:
where is the solution of Schrödinger’s equation in absence of the vector potential:
The exponential part of the wave function of Eq. (6) entails that two particles have equal charge and mass moving both outside the solenoid at the same distance from the axis, but the first in the same direction with the vector potential and the second in the opposite direction will suffer a phase difference:
Interference patterns for electrons in analogue conditions have been observed experimentally [44, 45, 46] demonstrating that the vector potential is a real physical field and interacts directly with charged particles in complete absence of magnetic and electric fields and of any other potential.
2.2 The radiation vector potential: classical to quantum link
The vector potential, being a real field, is considered as the fundamental link between the electromagnetic wave theory issued from Maxwell’s equations and the particle concept in quantum electrodynamics (QED) [19, 36, 39]. We will show analytically how this link is established.
In the case of a monochromatic plane wave with angular frequency , the electric and magnetic fields are proportional to the vector potential amplitude :
is a unit vector perpendicular to the propagation axis,
is the wave vector along the propagation axis, and
is the wavelength of the mode
The mean value over a period, thus over a wavelength, is time independent:
Note that the last equation expressing the mean energy density over a period of the mode
On the other hand, in the quantum description, the energy density for a number
In order to link the classical to the quantum description [4, 9, 19], the classical mean energy density over a period, expressed by Eq. (13), is imposed to be equivalent to the quantum mechanics expression of Eq. (14) for
getting the vector potential amplitude for a single
The last relation is the fundamental link between the classical and quantum theory of light which is used to define in QED the vector potential amplitude operators for a single photon [19, 26, 29, 36, 37, 38, 39, 40, 41]:
are, respectively, the annihilation and creation non-Hermitian operators for a
Therein, it is worth noting that an external arbitrary volume parameter
3. Electromagnetic field quantization and the photon description
3.1 Harmonic oscillator representation of the electromagnetic field
The energy of the electromagnetic field in a volume
Replacing in Eq. (17) the vector potential amplitude and its conjugate by the relations of the vector potential amplitude operators defined in Eq. (16), we get the “normal ordering” radiation Hamiltonian corresponding to the order of the creation and annihilation operators:
and the “anti-normal ordering” Hamiltonian corresponding to the order
where we have used the fundamental commutation relation in quantum electrodynamics:
In Dirac’s representation the eigenfunctions take the simple expression
, and the action of the creation and annihilation operators of a single
The successive action of both operators in the normal order corresponds to the photon number Hermitian operator
having the eigenvalue
representing the number of
In this representation the normal and anti-normal ordering radiation Hamiltonians write, respectively:
We obtain a harmonic oscillator Hamiltonian for the electromagnetic field by considering the mean value of the normal ordering and anti-normal ordering Hamiltonians:
Although we have no experimental facts showing the harmonic oscillator nature of a single photon, this representation has been adopted since the 1930s .
In a different way, a harmonic oscillator representation for the electromagnetic field can be obtained by the intermediate of the canonical variables of position and momentum . For that purpose we introduce the definitions expressing the vector potential amplitude and its complex conjugate with respect to and [19, 29, 41]:
Introducing the last expressions in Eq. (17), we get the electromagnetic field energy:
where the (+) sign is obtained when Eq. (17) is considered initially to be in the “normal order”, , and the (−) one when in the “anti-normal order” .
With the purpose of establishing a harmonic oscillator representation for the electromagnetic field, it is generally considered that in Eq. (26), because and are simply canonical variables, getting the energy of an ensemble of harmonic oscillators:
and putting , one gets the harmonic oscillator Hamiltonian for the radiation field:
At that level it is important to note that, for a harmonic oscillator of a particle with mass and momentum , with canonical variables of position and momentum , the transition from the classical expression of energy:
to the quantum mechanics Hamiltonian:
Consequently, the harmonic oscillator Hamiltonian for a particle of mass
Conversely, this is not the case for the electromagnetic field [19, 29, 39] because commutations between the canonical variables and occur during the mathematical transition from Eq. (17) to Eq. (26). It is then considered that in order to obtain Eq. (27) just before replacing the canonical variables by the corresponding quantum mechanics operators. Therein, it is important to remark that Heisenberg’s commutation relation is a fundamental concept of quantum mechanics, which should not be ignored when replacing classical variables by the corresponding quantum mechanics operators . In fact, without dropping in Eq. (26) and replacing the canonical variables by the corresponding quantum operators of Eq. (28), we get naturally the same normal ordering and anti-normal ordering radiation Hamiltonians as in Eq. (23):
Obviously, as frequently quoted [2, 19, 39], the fundamental mathematical ambiguity consisting of cancelling the commuting classical variable term before the substitution by non-commuting quantum mechanics operators leads to the harmonic oscillator Hamiltonian for the electromagnetic field.
In fact, since no experiment has yet demonstrated that a single photon is a harmonic oscillator, the main reason for considering the electromagnetic field as an ensemble of harmonic oscillators lies in the importance of the zero-point energy (ZPE) issued in absence of photons from the eigenvalue of Eq. (29) corresponding to the vacuum energy:
Nevertheless, the zero-point energy is very important because it is considered to be the basis for the explanation of the vacuum effects such as the spontaneous emission, the Lamb shift and the Casimir effect. However, as pointed out by many authors [19, 26, 39, 41], it is important to underline that the explanation of the spontaneous emission and the Lamb shift in QED is not due to Eq. (33) but precisely to the commutation properties of the photon creation and annihilation operators, and , respectively. It has to be emphasized that in quantum mechanics theory Eq. (33), being a constant, commutes with all Hermitian operators corresponding to physical observables and consequently has absolutely no influence to any quantum process.
Regarding the Casimir effect, it is often commented that caution has to be taken concerning the interpretation of its physical origin because it has been demonstrated by different methods [48, 49, 50] that it can be easily explained using classical electrodynamics without invoking at all the zero-point energy.
Hence, in view of the above, the normal ordering Hamiltonian is the one mainly used in QED, casting aside the vacuum singularity issued from the harmonic oscillator formalism, while the zero-point energy issued from the harmonic oscillator Hamiltonian is principally useful in the classical formalism for the interpretation of the vacuum effects [2, 19, 39, 47].
3.2 Electromagnetic field vector potential quantization in QED
We have analysed in Section 3.1 the electromagnetic field energy quantization according to the harmonic oscillator representation. Now, we will analyse the vector potential field quantization following the second quantisation process.
where is the Kronecker delta, is the Dirac delta function, and
with and .
For then and Eq. (37) writes:
Switching now to Heisenberg’s representation:
Considering the scalar potential to be constant, the electric field is:
The last expressions represent in a given volume
The amplitudes in Eqs. (42) and (43) have been obtained using the density of state theory and are valid only on the condition of Eq. (44). Furthermore, the boundary conditions of the electromagnetic waves considered in cavities and waveguides impose the wave vectors
The last equations represent the vector potential and the electric field of a large number of modes
4. Quantized vector potential of a single photon
We have seen in Section 3.1 that according to the energy quantization procedure, a
4.1 Photon vector potential amplitude and quantization volume
As mentioned in Section 2.2, the classical expression of the mean energy density over a period for a single electromagnetic mode
From a theoretical point of view, this is also compatible with the density of state theory according to which the spatial volume corresponding to a single state of the quantized field is proportional to [19, 29, 39, 41].
Indeed, it is well established experimentally that the energy density radiated by a dipole is proportional to entailing from Eq. (12) that the vector potential amplitude is normally proportional to [4, 5, 7].
This result is gauge independent since it concerns the natural units of the vector potential.
According to the previous considerations, for a free single
where, following to the above analysis, the amplitude writes:
We can evaluate [2, 53] by using Eqs. (49) and (50) in Eq. (13) and normalising the energy to that of a single photon, , by integrating over a wavelength along the propagation direction while taking into account the experimental results on the lateral expansion of the photon [32, 33, 34, 35, 56]. We get:
is the fine structure constant and
Thus, the characteristic volume of a free single photon writes in agreement with Eq. (47):
4.2 Photon classical-quantum (wave-particle) physical properties
For a free
We can now express the quantum properties of the photon, energy, momentum, and spin by integrating the classical electromagnetic expressions over the volume and by using the vector potential amplitude obtained in Eq. (50), linking in this way the classical (wave) to the quantum (particle) representations . The energy writes:
According to the classical electromagnetic theory, the spin can be written through the electric and magnetic field components; hence, using again the circular polarisation, we get:
where we have taken the mean value obtained for a single photon state .
The fact that the quantum properties, energy, momentum, and spin, of the photon can be expressed through the classical electromagnetic fields integrated over the volume
We can now obtain Heisenberg’s uncertainty relation for position and momentum using
The fundamental properties of the photon, energy , momentum , and wave vector , are complemented by the vector potential amplitude expressing its electromagnetic nature:
we directly deduce from Eq. (62) the
The energy and vector potential uncertainties with respect to time are intrinsic physical properties of the wave-particle nature of the photon.
4.3 Photon wave-particle equation and wave function
Obviously, the photon vector potential function expressed in Eq. (49) satisfies the wave propagation equation in vacuum issued from Maxwell’s equations:
It is worth remarking the symmetry between the pairs
for a single photon characterising, respectively, the particle (energy) and electromagnetic wave (vector potential) natures, having in mind that the energy corresponds to the integration of the single-mode electromagnetic field energy density over the volume
Now, when considering the propagation of a
In fact, from a theoretical point of view, for a photon propagating in the
Notice that the momentum uncertainty along the propagation axis is expressed through the uncertainty over the inverse of the wavelength.
Considering now the vector potential function with the quantized amplitude
as a real wave function for the photon, then when a
Obviously, the shorter the wavelength of the photon, the higher the localization probability in agreement with Heisenberg’s uncertainty and the experimental evidence.
4.4 Electromagnetic field ground state, photons, and electrons-positrons
The photon vector potential is composed of a fundamental function times the angular frequency and writes in the classical (wave) and quantum (particle) formalisms:
In this way, the general equation for the vector potential of the electromagnetic wave considered as a superposition of plane wave modes writes:
and that of a large number of cavity-free photons in quantum electrodynamics is:
According to Eqs. (55) and (62), for all the physical properties of the photon vanish entailing that the photon exists only for a non-zero frequency of the vector potential oscillation. However, the zero-frequency level does not correspond to perfect inexistence because the fundamental field does not vanish for but reduces to involving the amplitude and the general expression of the polarisation vectors [63, 64] and writes in the classical and quantum representations:
Combination of the expression
to the fine structure constant definition
permits to draw directly the electron-positron elementary charge
, a fundamental physical constant, which now is expressed exactly through the EFGS amplitude
Using again Eq. (51) and recalling that the electron mass may be written as
is the Bohr magneton, we deduce that the electron mass is also expressed as a function of the EFGS amplitude
entailing that the mass derives also from the EFGS and is proportional to the charge square.
Equations (50), (74), and (75) show the strong physical relationship between photons and electrons-positrons which are all related directly to the EFGS through the amplitude
In this chapter we have presented recent theoretical developments complementing the standard formalism with the purpose of describing a single photon state in conformity with the experiments. We resume below the principal features.
The quantization of the vector potential amplitude
, a real physical entity, for a single free of cavity
A single photon, as a local three-dimensional entity of the electromagnetic field, is absorbed and emitted as a whole and propagates guided by the non-local vector potential function (Eq. (49)), which appears to be a natural wave function for the photon satisfying both the propagation equation (Eq. (65)) and the
Finally, the electromagnetic field ground state (EFGS) at zero frequency, a real quantum vacuum component, issues naturally from the vector potential wave function putting in evidence that photons are oscillations of the vacuum field. Furthermore, the electron-positron charge and mass are directly proportional to the vector potential amplitude quantization constant showing the strong physical relationship with the photons. Obviously, the origin of the mechanisms governing the transformations of photons to electrons-positrons and inversely lies in the nature of the electromagnetic field ground state.
Ronchi V. The Nature of Light: An Historical Survey. Cambridge (MA) USA: Harvard University Press; 1970
Meis C. Light and Vacuum. 2nd ed. Singapore: World Scientific; 2017
Newton I. Opticks. New York: Dover; 1952
Saleh BEA, Teich MC. Fundamentals of Photonics. New York: John Wiley & Sons; 2007
Read FH. Electromagnetic Radiation. New York: John Wiley & Sons; 1980
Born M, Wolf E. Principles of Optics. Cambridge (MA) USA: Harvard University Press; 1999
Jackson JD. Classical Electrodynamics. New York: John Wiley & Sons; 1998
Planck M. The Theory of Heat Radiation. New York: Dover; 1959
Bransden BH, Joachain CJ. Physics of Atoms and Molecules. 2nd ed. London: Longman; 2003
Mickelson AA et al. Conference on Michelson-Morley Experiment. Astrophysical Journal. 1928; 68:341
Hertz H. Ueber sehr schnelle electrische Schwingungen. Annalen der Physik. 1887; 33:983
Einstein A. In: Stachel J, Cassidy DC, Renn J, Schulmann R, editors. The collected papers of Albert Einstein. New Jersey: Princeton University Press; 1987
Bohr N, Kramers HA, Slater JC. The quantum theory of radiation. Philosophical Magazine. 1924; 47:785
Taylor GI. Interference fringes with feeble light. Proceedings of the Cambridge Philosophical Society. 1909; 15:114
Compton AH. A Quantum Theory of the Scattering of X-rays by Light Elements. Physical Review. 1923; 21:483
Wentzel G. Zur Theorie des photoelektrischen Effekts. Zeitschrift für Physik. 1926; 40:574
Beck G. Zur Theorie des photoeffekts. Zeitschrift für Physik. 1927; 41:443
Lamb WE Jr, Scully MO. Polarization, Matière et Rayonnement, Volume Jubilaire en L’honneur d’Alfred Kastler. Paris: French Physical Society, Press Universitaires de France; 1969
Garrison JC, Chiao RY. Quantum Optics. New York: Oxford University Press; 2008
Klein O, Nishina Y. über die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen. Zeitschrift für Physik. 1929; 52:853
Jin F, Yuan S, De Raedt H, Michielsen K, Mayiashita S. Corpuscular Model of Two-Beam Interference and Double-Slit Experiments with Single Photons. Journal of the Physical Society of Japan. 2010; 79(7):074401
Schawlow A, Townes C. Infrared and optical masers. Physical Review. 1958; 112(6):1940-1949
Klyshko DN, Penin AN, Polkovnikov BF. Parametric luminescence and light scattering by polaritons. JETP Letters. 1970; 11:05
Burnham DC, Weinberg DL. Observation of simultaneity in parametric production of optical photon pairs. Physical Review Letters. 1970; 25(2):84
Ghose P, Home D. The two-prism experiment and wave-particle duality of light. Foundations of Physics. 1996; 26(7):943
Akhiezer AI, Berestetskii BV. Quantum Electrodynamics. New York: Interscience; 1965
Mandel L. Configuration-Space Photon Number Operators in Quantum Optics. Physical Review. 1966; 144:1071
Pfleegor RL, Mandel L. Interference of Independent Photon Beams. Physical Review. 1967; 159(5):1084
Grynberg G, Aspect A, Fabre C. Quantum Optics. New York: Cambridge University Press; 2010
Auletta G. Foundations and Interpretation of Quantum Mechanics. Singapore: World Scientific; 2001
Grangier P, Roger G, Aspect A. Experimental evidence for a photon anticorrelation effect on a beam splitter: A new light on single-photon interferences. Europhysics Letters. 1986; 1(4):173
Robinson HL. Diffraction patterns in circular apertures less than one wavelength. Journal of Applied Physics. 1953; 24(1):35
Hadlock RK. Diffraction Patterns at the Plane of a Slit in a Reflecting Screen. Journal of Applied Physics. 1958; 29:918
Hunter G, Wadlinger RLP. Photons and Neutrinos as Electromagnetic Solitons. Physics Essays. 1989; 2:158
Hunter G. Quantum uncertainties. NATO ASI Series B. In: Proceedings of NATO Advanced Research Workshop; June 23–27, 1986. p. 331
Feynman R. The Strange Theory of Light and Matter. New Jersey: Princeton University Press; 1988
Heitler W. The Quantum Theory of Radiation. Oxford: Clarendon Press; 1954
Ryder LH. Quantum field theory. Cambridge: Cambridge University Press; 1987
Milonni PW. The Quantum Vacuum. London: Academic Press Inc.; 1994
Mittleman MH. Introduction to the Theory of Laser-Atom Interactions. New York: Plenum Press; 1982
Weissbluth M. Photon-Atom Interactions. London: Academic Press Inc.; 1988
Ehrenberg W, Siday RE. The refractive index in electron optics. Proceedings of the Physical Society. Section B. 1949; 62:8-2
Aharonov Y, Bohm D. Significance of electromagnetic potentials in the quantum theory. Physics Review. 1959; 115:485-491
Chambers RG. Shift of an electron interference pattern by enclosed magnetic flux. Physical Review Letters. 1960; 5:3-5
Tonomura A et al. Evidence for aharonov-bohm effect with magnetic field completely shielded from electron wave. Physical Review Letters. 1986; 56:792
Osakabe N et al. Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor. Physical Review A. 1986; 34:815
Welton TA. Physics Review. 1948; 74:1157
Milonni PW. Physical Review A. 1982; 25:1315
Schwinger J, DeRaad LL Jr, Milton KA. Annals of Physics. 1978; 115:1
Ezawa H, Nakamura K, Watanabe K. The Casimir force from Lorentz's. Frontiers in Quantum Physics. In: Lim SC, Abd-Shukor R, Kwek KH, editors. Proceedings of the 2nd International Conference; Kuala Lumpur, Malaysia; July 9–11, 1997. Springer; 1998. pp. 160-169
Tannoudji CC, Diu B, Laloë F. Mécanique Quantique. Vol. III. Paris: EDP Sciences; 2017
Walther H, Varcoe BTH, Englert B-G, Becker T. Reports on Progress in Physics. 2006; 69(5):1325
Meis C. Photon wave-particle duality and virtual electromagnetic waves. Foundations of Physics. 1997; 27:865
Meis C, Dahoo PR. Vector potential quantization and the photon intrinsic electromagnetic properties: Towards nondestructive photon detection. International Journal of Quantum Information. 2017; 15(8):1740003
Meis C, Dahoo PR. Vector potential quantization and the photon wave function. Journal of Physics: Conference Series. 2017; 936:012004
Jeffers S, Roy S, Vigier JP, Hunter G. The Present Status of the Quantum Theory of Light. Boston: Kluwer Academic Publishers; 1997
Rybakov YP, Saha B. Soliton model of atom. Foundations of Physics. 1995; 25(12):1723-1731
Bialynicki-Birula I. Photon wave function. In: Wolf E, editor. Progress in Optics XXXVI. Amsterdam: Elsevier; 1996
Chandrasekar N. Quantum mechanics of photons. Advanced Studies in Theoretical Physics. 2012; 6(8):391-397
Sipe JE. Photon wave functions. Physical Review A. 1995; 52:1875
Smith BJ, Raymer MG. Photon wave functions, wave packet quantization of light and coherence theory. New Journal of Physics. 2007; 9:414
Khokhlov DL. Spatial and temporal wave functions of photons. Applied Physics Research. 2010; 2(2):50-54
Meis C. Vector potential quantization and the quantum vacuum. Physics Research International. 2014; 2014:187432
Meis C. The electromagnetic field ground state and the cosmological evolution. Journal of Physics: Conference Series. 2018; 1141:012072
Meis C, Dahoo PR. The single photon state, the quantum vacuum and the elementary electron-positron charge. American Institute of Physics: Conference Proceedings. 2018; 2040:020011