Quantum mechanics is only a generalization of the Maxwell equation with local gauge transformation

• Introduction

        No one can deny the successful of quantum mechanics. Its describe kindly microscopic physical problems that cannot be described by classical theory. Planck-Einstein's idea about quantization of energy made universe never look same again. The idea generates classical observables become quantum operators. As one learns in the textbooks, the postulate is the starting point to derive Schrodinger equation etc. Nevertheless, our understanding on the underlying above-mentioned idea has unfortunately not been at the satisfactory level. For instance, this quantization was made only to fulfill experimental facts, one cannot explain why it was postulated.

      Historically, Dirac equation derived by using correspondence principle that allow classical observables replace with quantum operators

      (1)

To improve the lacks of Schrodinger and Klein-Gordon equations that has been found earlier, this equation must linear and fulfill relativistic energy relation for free particle as

      (2)

Then, the Dirac equation can be written in form as

where denotes a wave function. Based on fact that there are both spin up and spin down with positive and negative energy, must be had four components. Consequently, and should be a matrices. The energy relation in the derivation can be replaced by non-relativistic classical Hamiltonian to obtain Schrodinger equation in same way.

        As can be seen in the historical derivation, quantization relation in Eqs. (1) is always be a basic assumption of derivation equation of motions in quantum mechanics. In this work, different point of view will be used to derive Dirac equation. Quantum mechanics will derive from Maxwell's Lagrangian density that invariant to local gauge transformation, not from Plank-Einstein's postulate. Using Maxwell equation as more fundamental principle and using field theory as tool to generate fermion field. Field theory is powerful tools to describe dynamical of elementary particles. Moreover, field theory with certain symmetry breaking has been used to describe protein folding.

• New Point of View

       The model was proposed by T. Fujita, S. Kanemaki, and S. Oshima (1). As previously, this work starting from Maxwell equation. This equation used as a guide to constructing the Lagrangian density for fermion fields. Local gauge transformations is imposed as prerequisite that an interaction is allowed to put in the Lagrangian.

        Maxwell equations in relativistic form can be written as

       (3)

where is an external vector current density. In order to fulfill Lorentz invariance, must be written as

For the case of no external current , we can simplify the Maxwell equation with Coulomb gauge fixing . Then, the equation for gauge field can be obtained as

         (4)

This equation like a quantization equation in quantum mechanics. In other word, Eq. (4) is a quantized equation for gauge field. Therefore, the Maxwell equation knows how to quantize an equation of motion.

        The Lagrangian density that reproduces Eq. (3) is

         (5)

where is the gauge file and   represents electromagnetic field strength. Assuming Lagrangian density in Eq. (5) must be invariant of local gauge transformations

Then, Lagrangian density of fermion can be written as

Fermion particle that described in the Lagrangian still mass-less. But actually, fermion is massive. Therefore, mass term should be generated to complete this journey. Mass term must be Lorentz scalar and must invariant under local gauge transformation. Thus, it should be described as

Minus sign in the mas term is appropriated with historical result. Mass term of field does not needed, because it will interpret as photon particle that actually mass-less. Finally, we arrive at the Lagrangian density of a relativistic fermion which couples with the electromagnetic fields

          (6)

This Lagrangian density is identical with Lagrangian in quantum electrodynamics (QED) that gives free Dirac equation for particle and anti-particle, and Maxwell equation with external current. One can derive the equation of motions using the Euler-Lagrange equations,

          (7)

The results by substituting Eq. (6) to Eqs. (7) in each terms of fields respectively are

          (8)

         (9)

        (10)

where Eqs. (8, 9, 10) are Dirac equations for particle, anti-particle, and Maxwell equation respectively. Then, the famous Schrodinger equation can be derived from the Dirac equation in Eq. (8}) using Foldy-Wouthuysen-Tani transformation in non-relativistic limit.

       The important thing that must be underlined in this work is the derivation of Eq. (6) did not use first quantization that was postulated by Planck-Einstein. The quantization can be derived directly from Dirac equations, like derivation of Eq. (4). Thus, the quantization wont be a fundamental principle anymore. Instead, Maxwell equation with local gauge condition is more fundamental principle. Therefore, we can derive quantum mechanics from the first principle. Quantum mechanics appears more elegance.

• No Boson Fields

       Klein-Gordon equation contains some basic problems. It has been repeatedly discussed in 80 years. In this work, the equation should be discussed again. The presence of the new concept causes addition problems with the equation.

      Historically, same as Dirac equation, Klein-Gordon equation was derived by using correspondence principle Eqs. (1) from relativistic free energy relation Eq. (2), that is

      (11)

where $\phi$ should be a scalar field. In other word, Eqs. (1) be a fundamental idea to derive it. Whereas, in new concept, the ideas are only a consequence of Dirac equation. Beside of that, scalar field in Eq. (11) was used to describe dynamical of boson particles. Unfortunately, scalar field has only one component. Therefore boson with negative energy forbidden to exist. This case different with Dirac field which has four components of fields that allows anti-particle to exist. Thus Klein-Gordon equation failed to account negative energy state. In addition, Klein-Gordon field does not have corresponding field in non-relativistic limit. Therefore, there are no powerful reason to hold the Klein-Gordon equation. But if this equation is disappear, how about the fate of the boson particles?

        Fermion and anti-fermion particles was described successfully by Dirac equations. Although using this new concept of derivation there are no significant problem about it. Before this new concept, T. Fujita et.al. was shown that scalar boson fields are composed of fermions (2). Only one that cannot describe by fermion field, that is gauge boson. Fortunately, mass-less gauge boson can be described directly from field that appear naturally from Maxwell equation. To describe massive gauge boson, other interactions and certain symmetry breaking must be employed in the theory. But be careful, as was proposed in (2, 3), the real scalar field becomes unphysical field. This indicates that the Higgs mechanism should be reconsidered. Therefore, historical view of the symmetry breaking should be corrected (4).

• Conclusion

       First quantization that usually be a starting point to derive equation of motions in quantum mechanics, no longer be a fundamental concept. The starting point of derivation shift to Maxwell equation. Using local gauge invariant, Lagrangian density that generates Dirac equations can be found. The first quantization procedure has been known in advance from Maxwell equation. Therefore, the quantization relation can be obtained directly from Dirac equation. In other word, Maxwell equation is more fundamental principle than first quantization.

       A consequence of the new theory of first quantization makes the Klein-Gordon equation is no longer exist to describe boson particle. Historically, its failed because one freedom of scalar field cannot explain negative energy state. To fix this problems, boson particle is proposed as composite of fermion fields.

     Therefore, we have new interpretations that can be completed the beautiful of quantum mechanics. I hope this new concept can be a gate to yield more interesting interpretations in quantum world.