Greek Alphabets

In physics, we often meet some notations such as and etc. It turned out that these notations actually are the Greek alphabets. There are 24 number alphabets in the Greek including: Alpha, Beta, Gamma, Delta, Epsilon, Zeta, Eta, Theta, IOTA, Kappa, Lambda, Mu, Nu, Xi, Omicron, Phi, Rho, Sigma, Tau, Upsilon, Pi, Chi, Psi and Omega.  The Greek alphabet has been used to write the Greek language since the end of the 9th BC. This is an oldest written alphabet that still used today. The letters are also used for representing Greek figures (numbers), since the 2nd BC.

It's important to know the spelling of each notation when one will write a physics formula in La-TeX program. Here I present the spelling for each notation of the whole Greek alphabets.

Hopefully useful.

What is a Soliton?

Soliton can be said as one of the integrable solutions of nonlinear differential equations. In a system, it propagates with constant velocity and has localized shape. Collision between solitons does not change their shape. During the collision each phases are shifted for the moment, but their shapes will be recovered immediately in the end of collision. Therefore, this behavior makes the soliton solutions cannot be represented as a linear combination of two solitons solutions.

An example of soliton phenomenon in our life can be seen in Tsunami Disaster. Waves in the disaster is not an ordinary wave, but its an soliton. Solitariness of the waves is one of the characteristic a soliton. Beside of that, when a soliton propagates through an object, the object is not vibrated but would be carried away by the wave.

Admirable Quotations in Physics and Mathematics

Gottfried Wilhelm Leibniz (1646–1716), Theoria cum praxi.

We begin with some quotations which exemplify the philosophical underpinnings of this work.

Johannes Kepler (1571–1630), Astronomia Nova

It is very difficult to write mathematics books today. If one does not take pains with the fine points of theorems, explanations, proofs and corollaries, then it won’t be a mathematics book; but if one does these things, then the reading of it will be extremely boring.

Marvin Schechter, Operator Methods in Quantum Mechanics 5

The interaction between physics and mathematics has always played an important role. The physicist who does not have the latest mathematical knowledge available to him is at a distinct disadvantage. The mathematician who shies away from physical applications will most likely miss important insights and motivations.

Freeman Dyson (born in 1923), From the Preface to Elliott Lieb’s Selecta 6

In 1967 Lenard and I found a proof of the stability of matter. Our proof was so complicated and so unilluminating that it stimulated Lieb and Thirring to find the first decent proof. Why was our proof so bad and why was theirs so good? The reason is simple. Lenard and I began with mathematical tricks and hacked our way through a forest of inequalities without any physical understanding. Lieb and Thirring began with physical understanding and went on to find the appropriate mathematical language to make their understanding rigorous. Our proof was a dead end. Theirs was a gateway to the new world of ideas collected in this book.

Freeman Dyson (born in 1923), The Art in Quantum Field Theory

All through its history, quantum field theory has had two faces, one looking outward, the other looking inward. The outward face looks at nature and gives us numbers that we can calculate and compare with experiments. The inward face looks at mathematical concepts and searches for a consistent foundation on which to build the theory. The outward face shows us brilliantly successful theory, bringing order to the chaos of particle interactions, predicting experimental results with astonishing precision. The inward face shows us a deep mystery. After seventy years of searching, we have found no consistent mathematical basis for the theory. When we try to impose the rigorous standards of pure mathematics, the theory becomes undefined or inconsistent. From the point of view of a pure mathematician, the theory does not exist. This is the great unsolved paradox of quantum field theory.

To resolve the paradox, during the last twenty years, quantum field theorists have become string-theorists. String theory is a new version of quantum field theory, exploring the mathematical foundations more deeply and entering a new world of multidimensional geometry. String theory also brings gravitation into the picture, and thereby unifies quantum field theory with general relativity. String theory has already led to important advances in pure mathematics. It has not led to any physical predictions that can be tested by experiment. We do not know whether string theory is a true description of nature. All we know is that it is a rich treasure of new mathematics, with an enticing promise of new physics. During the coming century, string theory will be intensively developed, and, if we are lucky, tested by experiment.

Hermann Weyl (1885–1930), The Classical Groups

The stringent precision attainable for mathematical thought has led many authors to a mode of writing which must give the reader an impression of being shut up in a brightly illuminated cell where every detail sticks out with the same dazzling clarity, but without relief. I prefer the open landscape under a clear sky with its depth of perspective, where the wealth of sharply defined nearby details gradually fades away towards the horizon.

Steven Weinberg (born in 1933), On the occasion of a conference on the interrelations
between Mathematics and Physics in 1986

I am not able to learn any mathematics unless I can see some problem I am going to solve with mathematics, and I don’t understand how anyone can teach mathematics without having a battery of problems that the student is going to be inspired to want to solve and then see that he or she can use the tools for solving them.

Cheng Ning Yang (born in 1922), Mathematical Intelligencer

In 1983 I gave a talk on physics in Seoul, South Korea. I joked “There exist only two kinds of modern mathematics books: one which you cannot read beyond the first page and one which you cannot read beyond the first sentence. The Mathematical Intelligencer later reprinted this joke of mine. But I suspect many mathematicians themselves agree with me.

Sir Michael Atiyah (born in 1929), Mathematical Intelligencer

The more I have learned about physics, the more convinced I am that physics provides, in a sense, the deepest applications of mathematics. The mathematical problems that have been solved, or techniques that have arisen out of physics in the past, have been the lifeblood of mathematics. . . The really deep questions are still in the physical sciences. For the health of mathematics at its research level, I think it is very important to maintain that link as much as possible.

Richard Feynman (1918–1988), The Feynman Lectures in Physics

Finally, may I add that the main purpose of my teaching has not been to prepare you for some examination – it was not even to prepare you to serve industry or military. I wanted most to give you some appreciation of the wonderful world and the physicist’s way of looking at it, which, I believe, is a major part of the true culture of modern times.

Generating Functional for Free Bosonic Lagrangian

Generating functional for scalar fields in the presence of source J is defined by

Vacuum-to-vacuum transition amplitude can be written as

For free bosonic particle, the lagrangian is

Then, the transition amplitude is

(1)

Let us now simplify the expression by evaluate this

where can be written as D'Alembertian. Exchange the position, so we have

The first term of right side can be vanished by taking at infinity. Then,

Substitute this result to Eq. (1). Thus

To evaluate it more simplify, let us change to

This is the semi-classical approximation that will separate the classical aspect and the quantum aspect. 

Using the fact that

,
then we have

(2)

(3)

This equation have a solution in Fourier representation, that is

(4)

where is called Feynman propagator and must obey

The Fourier representation of this Green's function is

(5)

Subtituting Eq. (3) to Eq. (2)

(6)

Furthermore, substituting Eq. (4) to Eq. (6)

In the case of abbreviation the notation, let

'
thus

The result of the path integral on the kinetic term must be a number, then just called it as N

Therefore, we have generating functional of vacuum in simpler form.

Generating functional for interacting field can be obtained as functional derivatives of this vacuum transition in respect to the external current.

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.