1 Introduction

The Zero Point Field (ZPF) or the field of the quantum vacuum is looked upon by many scientists as the gene of the universe. Sidharth [1] and Haisch, Rueda and Puthoff [2] have shown that the ZPF is the most rudimentary field. Sidharth, resorting to the ZPF in his work [1, 3] elucidated several inexplicable cosmological phenomena. Also, Alfonso Rueda and Bernard Haisch have shown the inertia and gravitation nexus with the vacuum fields by using their quantum vacuum inertia hypothesis [4]. One point of argument is the subject of how the addition of the ZPF leads from classical angular momentum to the quantum mechanical case [5, 6]. It may also be mentioned that the existence of the ZPF is brought out by the celebrated experimental Lambshift [7] of about 1000 Mega cycles/sec and more recently in the Cosmic Radio Wave Background which is a direct consequence of the Lambshift and indeed is a footprint of the all pervading ZPF [8]. This was discovered in the last few years by NASA’s Arcade experiment.

It has been shown in a different context that the subsequent emissions of Hawking quanta near the horizon of a black hole, which are ultimately due to the ZPF, can be interpreted as the quantum jumps among the quantum levels of a black hole. The fundamental consequence is that the black holes seem really to be the “gravitational atoms” of quantum gravity [9–12].

In the light of all this, we further substantiate the fundamental nature of the ZPF. Let us couple it with the electromagnetic field and resort to the Lagrangian formulation of field theory.

2 The Electromagnetic Field and ZPF

In the beginning of our theory we modify the electromagnetic vector potential, such that

$$ \vec{A}^{\prime} = \vec{A}_{0} + \xi \vec{A}_{em} $$
(1)

which in terms of four vectors can also be written as

$$A^{\prime}_{\mu} = A_{\mu0} + \xi A_{\mu}$$

where, \(\vec {A}_{0}\) or (A μ )0 is the vector potential associated with the ZPF [5], A e m is an external electromagnetic field and ξ is constant which later can be neglected. As a consequence of this modification the modified electromagnetic field tensor is given by

$$\begin{array}{@{}rcl@{}} F^{\prime}_{\mu \nu} &=& \frac{\partial A^{\prime}_{\nu}}{\partial x_{\mu}} - \frac{\partial A^{\prime}_{\mu}}{\partial x_{\nu}} \\ &=& \left\{\frac{\partial A_{\nu0}}{\partial x_{\mu}} - \frac{\partial A_{\mu0}}{\partial x_{\nu}}\right\} + \xi \left\{\frac{\partial A_{\nu}}{\partial x_{\mu}} - \frac{\partial A_{\mu}}{\partial x_{\nu}}\right\} \\ &=& (F_{\mu \nu})_{0} + \xi (F_{\mu \nu}) \end{array} $$

where, \((F_{\mu \nu })_{0} = \left \{\frac {\partial A_{\nu 0}}{\partial x_{\mu }} - \frac {\partial A_{\mu 0}}{\partial x_{\nu }}\right \}\) is the field tensor for the ZPF and F μ ν is that of the external electromagnetic field. Thus

$$ F^{\prime}_{\mu \nu} = (F_{\mu \nu})_{0} + \xi (F_{\mu \nu}) $$
(2)

It must be borne in our minds that the effect of the ZPF is extremely faint as we know that the length scale associated with it is the Compton length which is minutely small.

In the course of this approach, looking at (1) we can say that all known quantities associated with the electromagnetic vector potential will be modified. Let us observe the case of the modified magnetic field which will be given by

$$ \vec{B}^{\prime} = \nabla \times \vec{A}^{\prime} $$
(3)

Using (1) we would obtain

$$ \vec{B}^{\prime} = \vec{B}_{0} + \xi \vec{B}_{em} $$
(4)

where, \(\nabla \times \vec {A}_{0} = \vec {B}_{0}\) is the magnetic field emanating from the ZPF, albeit the field \(\vec {B}_{0}\) is faint as we mentioned previously. This magnetic field will induce a magnetic moment which in turn is related to Zitterbewegung in the Compton scale.

Now, the current density associated with the modified vector potential will be given by

$$\frac{j^{\prime}_{\mu}}{c} = \frac{\partial F^{\prime}_{\mu\nu}}{\partial x_{\nu}} $$

Using (2) we would obtain

$$ \frac{j^{\prime}_{\mu}}{c} = \frac{\partial(F_{\mu \nu})_{0}}{\partial x_{\nu}} + \xi \frac{\partial F_{\mu\nu}}{\partial x_{\nu}} $$
(5)

which ultimately gives

$$ j^{\prime}_{\mu} = j_{\mu0} + \xi j_{\mu} $$
(6)

where, J μ0 is the current density concurring with the vector potential. An equation akin to (5) was obtained by Sidharth [13] resorting to the non-commutative nature of space-time. We rewrite (6) as

$$ j_{\mu0} = j^{\prime}_{\mu} + (-\xi) j_{\mu} $$
(7)

Now, we take in account the bilinear densities of the form \(\overline {\psi }{\Gamma }\psi \), where Γ is the product of the gamma matrices. Such densities have definite transformation properties under Lorentz transformations. In this vein the charge-current density for the ZPF is given as: \(j_{\mu 0} = iec \overline {\psi }\gamma _{\mu }\psi \), where the symbols have their usual meanings. Again, in the presence of electromagnetic couplings the Dirac equation reads

$$ \left( \frac{\partial}{\partial x_{\mu}} - \frac{ie}{\hbar c}A_{\mu0}\right) \gamma_{\mu}\psi + \frac{mc}{\hbar}\psi = 0 $$
(8)

Let us rewrite the charge-current density as

$$\begin{array}{@{}rcl@{}} j_{\mu0} &=& \left( \frac{iec}{2}\right) (\overline{\psi}\gamma_{\mu}\psi + \overline{\psi}\gamma_{\mu}\psi) \\ &=& \frac{ie\hbar}{2m} \left[-\overline{\psi}\gamma_{\mu}\gamma_{\nu}\left( \frac{\partial}{\partial x_{\nu}} - \frac{ie}{\hbar c}A_{\nu0}\right)\psi + \left\{\left( \frac{\partial}{\partial x_{\nu}} + \frac{ie}{\hbar c}A_{\nu0}\right)\overline{\psi}\right\}\gamma_{\nu}\gamma_{\mu}\psi\right] \end{array} $$

where, we have used (8) and it’s analog for the adjoint wavefunction \(\overline {\psi }\). If, j μ is split into two parts as

$$j_{\mu0} = j_{\mu1} + j_{\mu2}$$

according to whether or not the summation index ν coincides or not with μ, then we may identify j μ1 as \(j^{\prime }_{\mu }\) and j μ2 as (−ξ)j μ . Thus, we have

$$ j^{\prime}_{\mu} = \frac{ie\hbar}{2m}\left( \frac{\partial\overline{\psi}}{\partial x_{\mu}}\psi - \overline{\psi}\frac{\psi}{\partial x_{\mu}}\right) - \frac{e^{2}}{mc} (A_{\mu0}) \overline{\psi}\psi $$
(9)

and

$$(-\xi) j_{\mu} = \frac{ie\hbar}{2m} \left[-\overline{\psi}\gamma_{\mu}\gamma_{\nu}\frac{\partial}{\partial x_{\nu}}\psi + \frac{\partial\overline{\psi}}{\partial x_{\mu}}\gamma_{\nu}\gamma_{\mu}\psi + \frac{ie}{\hbar c}(A_{\nu0})\overline{\psi}\gamma_{\mu}\gamma_{\nu}\psi + \frac{ie}{\hbar c}(A_{\nu0})\overline{\psi}\gamma_{\nu}\gamma_{\mu}\psi\right]_{\mu \neq \nu}$$
$$ (-\xi) j_{\mu} = - \frac{e\hbar}{2m} \frac{\partial}{\partial x_{\nu}} \overline{\psi}\sigma_{\nu\mu}\psi $$
(10)

This method is analogous to the Gordon decomposition [14] where, the charge current density (j μ ) can be decomposed as

$$j_{\mu} = j^{(1)}_{\mu} + j^{(2)}_{\mu}$$

where, the first term on the right hand side is the convection current due to the moving charge and the second is the current associated with the intrinsic magnetization of the electron.

Therefore, in the non-relativistic limit, if we consider slowly moving electrons then (7) can be looked upon analogously to the Gordon decomposition of j μ into the convection current due to the moving charge and the current associated with the intrinsic magnetization (magnetic dipole density) of the electron. This would imply that the intrinsic magnetization of the electron arises due to the ZPF which in turn will result in the electron acquiring intrinsic spin. We must reiterate that with the usual electromagnetic field too, equations like (9) would hold [14]. However, here, it is the ZPF that is playing that part. This inexplicable intrinsic spin of the electron can be succinctly understood by the ZPF field, which was also concluded by Sidharth [13].

Now, j μ0 associated with the intrinsic magnetization has an inherent significance. When j μ0 interacts with the vector potential A μ through the Hamiltonian: \(\mathcal {H}_{int} = - \frac {j_{\mu 0}A_{\mu 0}}{c}\), the interaction Hamiltonian is given by

$$\begin{array}{@{}rcl@{}} \xi\frac{j_{\mu}A_{\mu0}}{c} &=& \frac{e\hbar}{2mc} \left[ \frac{\partial}{\partial x_{\nu}} \overline{\psi}\sigma_{\nu\mu}\psi\right]A_{\mu0} \\ &=& \frac{e\hbar}{2mc} \frac{\partial A_{\mu0}}{\partial x_{\nu}} (\overline{\psi}\sigma_{\nu\mu}\psi)\\ &=& \frac{e\hbar}{2mc} \left[\frac{1}{2} \frac{\partial A_{\mu0}}{\partial x_{\nu}} (\overline{\psi}\sigma_{\nu\mu}\psi) + \frac{1}{2} \frac{\partial A_{\nu0}}{\partial x_{\mu}} (\overline{\psi}\sigma_{\mu\nu}\psi)\right] \\ &=& \frac{e\hbar}{2mc} \left[\frac{1}{2} \{(F_{\nu\mu})_{0}\}\overline{\psi}\sigma_{\nu\mu}\psi\right] \end{array} $$

where, we have neglected \(\frac {\partial }{\partial x_{\nu }}(\overline {\psi }\sigma _{\nu \mu }A_{\mu 0})\) since it is immaterial when the interaction density is integrated.

Noting that

$$\frac{1}{2} \{(F_{\nu\mu})_{0}\}\overline{\psi}\sigma_{\nu\mu}\psi \approx {\vec{B_{0}}}. ({\psi^{\dag}_{A}{\vec{\sigma}}\psi_{A}})$$

we can say that in the non-relativistic limit, the obtained result regarding the interaction of j μ and A μ0 can account for the spin magnetic moment interaction with the gyromagnetic ratio g=2 in the usual formulation [8]. This value has also been shown by Sidharth [1] and Sachidanandam [15].

Now, in light of the Dirac equation [14], the modified Dirac equation which has been studied by a few authors in recent years [16] and [17] can be written as

$$ \left( \frac{\partial}{\partial x_{\mu}} - \frac{ie}{\hbar c}A_{\mu0}\right) \gamma_{\mu}\psi + \frac{mc}{\hbar}\psi - \gamma^{2}_{\mu}\gamma_{4}\frac{\lambda l}{\hbar^{2}}p^{2} \psi = 0 $$
(11)

where, the extra term \((- \gamma ^{2}_{\mu }\gamma _{4}\frac {\lambda l}{\hbar ^{2}}p^{2} \psi )\) is due to the non-commutative nature of space-time as shown by Sidharth [16, 18] and \(\gamma ^{2}_{\mu } = \delta _{\mu \mu }\) while λ is a small conversion term. Considering the momentum \(p (-i\hbar \frac {\partial }{\partial x})\) as a four vector we would have

$$\begin{array}{@{}rcl@{}}j_{\mu0} &=& \frac{ie\hbar}{2m} \left[-\overline{\psi}\gamma_{\mu}\gamma_{\nu}\left( \frac{\partial}{\partial x_{\nu}} - \frac{ie}{\hbar c}A_{\nu0} + \lambda l\gamma_{\nu}\gamma_{4}\frac{\partial^{2}}{\partial x^{2}_{\nu}}\right)\psi \right.\\ &&\qquad\left.+ \left\{\left( \frac{\partial}{\partial x_{\nu}} + \frac{ie}{\hbar c}A_{\nu0} + \lambda l\gamma_{\nu}\gamma_{4}\frac{\partial^{2}}{\partial x^{2}_{\nu}}\right)\overline{\psi}\right\}\gamma_{\nu}\gamma_{\mu}\psi\right] \end{array} $$

So this time we will be using (11),

$$ j^{\prime}_{\mu} = \frac{ie\hbar}{2m}\left( \frac{\partial\overline{\psi}}{\partial x_{\mu}}\psi - \overline{\psi}\frac{\partial\psi}{\partial x_{\mu}}\right) + \frac{ie\hbar}{2m}(\lambda l\gamma_{\nu}\gamma_{4})\left( \frac{\partial^{2}\overline{\psi}}{\partial x^{2}_{\mu}}\psi - \overline{\psi}\frac{\partial^{2}\psi}{\partial x^{2}_{\mu}}\right) - \frac{e^{2}}{mc} (A_{\mu0}) \overline{\psi}\psi $$
(12)

and

$$ (-\xi) j_{\mu} = - \frac{e\hbar}{2m} \frac{\partial}{\partial x_{\nu}} (\overline{\psi}\sigma_{\nu\mu}\psi) - \frac{e\hbar}{2m}(\lambda l\gamma_{\nu}\gamma_{4}) \frac{\partial^{2}}{\partial x^{2}_{\nu}} (\overline{\psi}\sigma_{\nu\mu}\psi) $$
(13)

We can see that (13) has an extra term . Now, as we have done previously, the interaction Hamiltonian will be given by

$$\begin{array}{@{}rcl@{}} \xi\frac{j_{\mu}A_{\mu0}}{c} &=& \frac{e\hbar}{2mc} \left[ \frac{\partial}{\partial x_{\nu}} \overline{\psi}\sigma_{\nu\mu}\psi\right]A_{\mu0} + (\lambda l\gamma_{\nu}\gamma_{4})\frac{e\hbar}{2mc} \left[ \frac{\partial^{2}}{\partial x^{2}_{\nu}} \overline{\psi}\sigma_{\nu\mu}\psi\right]A_{\mu0}\\ &=& \frac{e\hbar}{2mc} \frac{\partial A_{\mu0}}{\partial x_{\nu}} (\overline{\psi}\sigma_{\nu\mu}\psi) + (\lambda l\gamma_{\nu}\gamma_{4})\frac{e\hbar}{2mc} \frac{\partial^{2}A_{\mu0}}{\partial x^{2}_{\nu}} (\overline{\psi}\sigma_{\nu\mu}\psi)\\ &=& \frac{e\hbar}{2mc} \left[\frac{1}{2} \{(F_{\nu\mu})_{0}\}\overline{\psi}\sigma_{\nu\mu}\psi\right] + (\lambda l\gamma_{\nu}\gamma_{4})\frac{e\hbar}{2mc} \left[\frac{1}{2} \frac{\partial}{\partial x_{\nu}}\left \{\frac{\partial A_{\mu 0}}{\partial x_{\nu}}(\overline{\psi}\sigma_{\nu\mu}\psi) + \frac{\partial A_{\nu 0}}{\partial x_{\mu}} (\overline{\psi}\sigma_{\mu\nu}\psi)\right]\right\} \\ &=& \frac{e\hbar}{2mc} \left[\frac{1}{2} \{(F_{\nu\mu})_{0}\}\overline{\psi}\sigma_{\nu\mu}\psi\right] - (\lambda l\gamma_{\nu}\gamma_{4})\frac{e\hbar}{2mc} \left[\frac{1}{2} \frac{\partial}{\partial x_{\nu}}\{-(F_{\nu\mu})_{0}\}\right](\overline{\psi}\sigma_{\nu\mu}\psi) \\ &=& \frac{e\hbar}{2mc} \left[\frac{1}{2} \{(F_{\nu\mu})_{0}\}\overline{\psi}\sigma_{\nu\mu}\psi\right] - (\lambda l\gamma_{\nu}\gamma_{4})\frac{e\hbar}{2mc} \left[\frac{1}{2} \frac{\partial}{\partial x_{\nu}}\{(F_{\mu\nu})_{0}\}\right](\overline{\psi}\sigma_{\nu\mu}\psi) \end{array} $$

Therefore, we have obtained

$$ \xi\frac{j_{\mu}A_{\mu0}}{c} = \frac{e\hbar}{2mc} \left[\frac{1}{2} \{(F_{\nu\mu})_{0}\}\overline{\psi}\sigma_{\nu\mu}\psi\right] + \frac{e\hbar}{2mc}(-\lambda) \left[\frac{1}{2} (\gamma_{\nu}\gamma_{4})(lj_{\mu0})\overline{\psi}\sigma_{\nu\mu}\psi\right] $$
(14)

where, as before we have neglected \(\frac {\partial ^{2}}{\partial x^{2}_{\nu }}(\overline {\psi }\sigma _{\nu \mu }A_{\mu 0})\) since it is immaterial when the interaction density is integrated. We have already shown that the first term on the right hand side leads us to the spin magnetic moment of the electron and g=2. As for the second term we see that a current density has emanated due to the modified Dirac equation (11). The factor (l j μ0) will generate an extra spin magnetic moment where l is the Compton length and this moment will also contribute to the magnetic field B 0. Following what we did previously we may write from the second term of equation (14),

$$ -\lambda = \left[\frac{\alpha}{2\pi} + f(\alpha)\right] $$
(15)

where, f(α) is some function of the higher powers ofα (fine-structure constant). In (14) we identify λ as (15), the reason of which will be seen below. Now, we have the final form of the g-factor as

$$ g = 2\left[1 + \frac{\alpha}{2\pi} + f(\alpha)\right] = 2[1 - \lambda] $$
(16)

Now, in (15) if we take \(\lambda \approx -\frac {\alpha }{2\pi }\) then we have

$$\lambda \approx -10^{-3}$$

It is remarkable that the correction term to the g=2 factor in (16) has also been suggested by J. Schwinger from a field theoretic point of view. This is the rationale for the identification (15). Now, as we have deduced earlier, the first term on the right hand side of (14) leads to the g=2. Therefore the extra terms in the expression of the g-factor can be inferred as a contribution of the second term of (14).

Now, as an analogue to the electromagnetic vector potential we can write for the vector potential

$$ \vec{A}_{\mu0} = \frac{\mu_{0}}{4\pi} \int \frac{j_{\mu0}(r_{0})}{|\vec{r} - \vec{r_{0}}|} \mathrm{d}v_{0} $$
(17)

or,

$$ \vec{A}_{\mu0} = \frac{\mu_{0}}{4\pi l_{c}} \int j_{\mu0}(r_{0}) \mathrm{d}v_{0} $$
(18)

where, we have considered \(|\vec {r} - \vec {r_{0}}| = l_{c}\) which in our context is the Compton length, a scale important to remember for all the considerations above. With (18) we can derive the vector potential when j μ0(r 0) is known.

3 Discussions

  1. (1)

    For completeness let us find a Lagrangian for the modified field. Let us commence with the the known form of Lagrangian density for the electromagnetic field and use it for the modified field as

    $$\mathfrak{L^{\prime}} = -\frac{1}{4} F^{\prime}_{\mu\nu}F^{\prime}_{\mu\nu} + \frac{(J^{\prime}_{\mu}A^{\prime}_{\mu})}{c}$$

    By use of (1), (2) & (6) we obtain

    $$\begin{array}{@{}rcl@{}} \mathfrak{L^{\prime}} &=& -\frac{1}{4} \left[(F_{\mu\nu})_{0} (F_{\mu\nu})_{0} + \xi^{2} F_{\mu\nu} F_{\mu\nu} + \xi\{(F_{\mu\nu})_{0}F_{\mu\nu} + F_{\mu\nu}(F_{\mu\nu})_{0}\} + \frac{j_{\mu0}A_{\mu0}}{c}\right.\\ &&\qquad\left. + \frac{\xi^{2}j_{\mu}A_{\mu}}{c} + \frac{\xi}{c}\{j_{\mu}A_{\mu0} + A_{\mu}j_{\mu0}\}\right] \end{array} $$
    (19)

    Arranging the terms we get

    $$ \mathfrak{L^{\prime}} = \mathfrak{L}_{0} + \xi^{2}\mathfrak{L}_{em} + \xi\mathfrak{L}_{coupled} $$
    (20)

    where,

    $$\begin{array}{@{}rcl@{}} \mathfrak{L}_{em} = -\frac{1}{4} F_{\mu\nu}F_{\mu\nu} + \frac{(J_{\mu}A_{\mu})}{c}\\ \mathfrak{L}_{0} = -\frac{1}{4} (F_{\mu\nu})_{0}(F_{\mu\nu})_{0} + \frac{j_{\mu0}A_{\mu0}}{c}\\ \mathfrak{L}_{coupled} = -\frac{1}{4} \left\{(F_{\mu\nu})_{0}F_{\mu\nu} + F_{\mu\nu}(F_{\mu\nu})_{0}\right\} + \frac{1}{c}\{j_{\mu}A_{\mu0} + A_{\mu}j_{\mu0}\}\\ \end{array} $$

    Here, \(\mathfrak {L}_{0}\) is the Langrangian for the ZPF and \(\mathfrak {L}_{coupled}\) is the Langrangian due to the coupled state of the external electromagnetic field and the ZPF. The field equations can be derived from the aforesaid Lagrangian to give

    $$\frac{\partial}{\partial x_{\nu}}\left[(F_{\mu\nu})_{0} + \xi F_{\mu\nu})\right] = \frac{j_{\mu0}}{c} + \xi\frac{j_{\mu}}{c}$$

    as expected.

  2. (2)

    Now, we may push our efforts further to enunciate the vector potential regarding the electromagnetic field. The (1) can also be written as

    $$ A_{\mu0} \rightarrow A^{\prime}_{\mu} = A_{\mu0} + \xi (A_{\mu}) $$
    (21)

    Again, the gauge transformation for the vector potential A μ0 is

    $$ A_{\mu0} \rightarrow A^{\prime}_{\mu} = A_{\mu0} + \frac{\partial {\Lambda}}{\partial x_{\mu}} $$
    (22)

    Looking at these two equations we can say that the former can be looked upon as a gauge transformation.

    Now, comparing (21) with equation (22) gives

    $$\xi A_{\mu} = \frac{\partial {\Lambda}}{\partial x_{\mu}}$$

    which gives

    $$ A_{\mu} = \frac{1}{\xi} \frac{\partial {\Lambda}}{\partial x_{\mu}} $$
    (23)

    The above methodology can be looked upon as gauge fixing and the (23) gives us a way to determine the vector potential associated with the external electromagnetic field. Now, let us consider a Lie group G of which Λ is a group element. We may write

    $$ {\Lambda} = G(x_{\mu}) $$
    (24)

    i.e., λ is a function of the x μ ’s. Now, the generators of the group G (dimension = 4) will be given by

    $$X_{\mu} = \frac{\partial {\Lambda}}{\partial {x_{\mu}}}$$

    Comparing this with (23) we have

    $$ \xi A_{\mu} = X_{\mu} $$
    (25)

    Therefore, we can ascertain an astounding conclusion that the phase factor (Λ) necessary for the electromagnetic gauge transformation is the element of the group G of which {ξ A μ } is the generator.

4 Conclusions

We have tried to argue that the ZPF is a fundamental field and has significance in case of the electromagnetic gauge. Our earlier work had also explained several inexplicable phenomena in diverse areas such as in the case of graphene. In particular, it is remarkable that it is closely related to the mysterious minimum conductivity. If the ZPF is construed as a fundamental field then many other phenomena might be elucidated. The gyromagnetic ratio has been successfully interpreted including the extra Schwinger terms in it’s observed value. The phase factor in the gauge transformations of U(1) group has been shown to be the element of a Lie group of which the vector potential ξ A μ (ξ being some constant) is a generator. This result too, might have implications in the proper understanding of the U(1) symmetry.