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Zero Point Energy and Dirac Equation

 

 

Zero Point Energy and Dirac Equation

 

H. Javadia and F. Forouzbakhshb

 

a Invited professor of Faculty of Science at Azad Islamic University in Tehran campuses.,

Tehran, Iran

 

Javadi_hossein@hotmail.com

b Academic Researcher, Office of the Vice Chancellor for Research and Technology

University of Tehran., Tehran, Iran

fforouz@ut.ac.ir

 April.03.2007

 

Abstract

Zero Point Energy (ZPE) describes the random electromagnetic oscillations that are left in a vacuum after all other energy has been removed.

One way to explain this is from the uncertainty principle from quantum physics that implies that it is impossible to have an absolutely zero energy condition.

In this article, ZPE explained by using the Dirac equation.

 

Keywords: Zero point energy, Dirac equation, Hawking radiation, graviton, photon, pair production, renormalization, color-charge, magnetic-color, photon, spin.

 

 Contents;

1 Introduction

2 Photon and gravitational field

3 Color-charges and color-magnet

4 Gravitons

5 Definition of a CPH

6 Principle of the CPH theory

7 CPH and Cyclic group

8 Zero Point Energy

9 Hawking Radiation

10 Graviton Spin

References;

 

1 Introduction

Zero Point Energy (ZPE), or Vacuum Fluctuation energy are terms used to describe the random electromagnetic oscillations that are left in a vacuum after all other energy has been removed.

The concept of zero-point energy was proposed by Albert Einstein and Otto Stern in 1913, which they originally called "residual energy" or Nullpunktsenergie. All quantum mechanical systems have a zero point energy. The term arises commonly in reference to the ground state of the quantum harmonic oscillator and its null oscillations. In cosmology, the vacuum energy is taken to be the origin of the cosmological constant. Experimentally, the zero-point energy of the vacuum leads directly to the Casimir effect[1], and is directly observable in nanoscale devices.

One way to explain this is from the uncertainty principle from quantum physics that implies that it is impossible to have an absolutely zero energy condition.

In this article, according to gravitational blue-shift it has been tried to put into consideration the Mössbauer effect and Pound-Rebka experiments and interaction between gravity and photon from Higgs field point of view[2]. Blue-shift and Mössbauer effect indicates clearly that three different Higgs bosons cause increasing photon mass while they have the electromagnetic specifications. These Higgs bosons are called positive and negative color-charges and color-magnet which will be explained by using the Dirac equation.

 

2 Photon and gravitational field

Let's take a new look at the behavior of a photon in gravitational field; it can help us in resolving the vacuum energy.

The electromagnetic fields around a "ray of light" are electromagnetic waves not static fields. Relativistically, the electromagnetic field generated by a photon is much stronger than the associated gravitational field.

When a photon falls in gravitational field, energy (mass) of it increases. According to , the gravity force works on photon, so the mass (energy) of photon increases. But energy of photon depends on its electric field and magnetic field. So, a part of gravity work converts to electric energy and the other part of gravity work converts to magnetic energy. How can it be explained by Higgs boson which it shows how particles acquire mass? And also, according to Higgs boson what happens to blue-shift?

 

2 Photon and gravitational field

Let's take a new look at the behavior of a photon in gravitational field; it can help us in resolving the vacuum energy.

The electromagnetic fields around a "ray of light" are electromagnetic waves not static fields. Relativistically, the electromagnetic field generated by a photon is much stronger than the associated gravitational field.

When a photon falls in gravitational field, energy (mass) of it increases. According to W=Dmc2, the gravity force works on photon, so the mass (energy) of photon increases. But energy of photon depends on its electric field and magnetic field. So, a part of gravity work converts to electric energy and the other part of gravity work converts to magnetic energy. How can it be explained by Higgs boson which it shows how particles acquire mass? And also, according to Higgs boson what happens to blue-shift?

 

3 Color-charges and color-magnet

The frequency of photons changes in a gravitational field that given by;

 

 

 The minus sign is for red-shift (photon leaves gravitational field), and plus sign is for blue-shift (photon falls in gravitational field). 

Where:  is the 2nd and  is the 1st frequency of photon, G is the constant gravitational law, M is the body mass. r is distance between body and photon, c is the speed of light, E is the photon energy, h is Planck constant.

During a photon is falling in gravitational field, it acquires energy equal as  that separates into three parts; a part behaves like positive electrical field and another part that behaves like negative field, which they annihilate each other in structure of photon (photon is neutral) and the third part that behaves like magnetic field.

In quantum mechanics theory, every field is quantized. Also, force is described as energy per distance that shown by:

 

If we consider this equation from point of quantum mechanics, the graviton enters to the structure of photon where graviton carries gravitational force. As a result, a graviton disappears and the energy of photon increases. Similarly, red-shift has opposite effect. As photon escapes from gravitational field, its frequency shifts to red and its entire energy converts to gravitons. How we can describe interaction between photon and gravitons in sub-quantum scale such as in structure of photon?

Another considerable item is virtual particles, that the uncertainty principle in the form:

 

  

implies that in the vacuum one or more particles with energy above the vacuum may be created for a short time . So, any analysis of ZPE should be able to explain particles production even Hawking radiation in a strong gravitational field such as black hole. To do this best way is using the Dirac equation that of the first time showed the pair particle and anti-particle.

The Dirac equation can be written as:

 

 

where i=1, 2, 3, m is the rest mass of the electron, c is the speed of light, p is the momentum operator, x andt are the space and time coordinates respectively, and is a four-component wave function.

A convenient (but not unique) choice of α�s is:

 

 

 

 

 

The Dirac equation describes the probability amplitudes for a single electron. This is a single-particle theory; in other words, it does not account for the creation and destruction of the particles.

In modern physics for massless particles, in the (4) equation m = 0,  then recover the relationship between energy and momentum for massless particles given by:

 

 

Now let�s do change our definition about rest mass of particles. As we know some particle such as photon never seem at rest condition in all reference frame.

According relativity, they have mass that come of their energy. For example a photon has a mass that given by:

 

 

So, there are two kinds particles in physics;

  • Some particles like photon move with the speed of light c, in all inertial reference frame, only. Let�s call these kind the N. R. or Never at Rest condition Particles.

  • Other particles like electron move with speed v that always v<c in all inertial reference frame.

According above definition photon and graviton are NR particles, electron and proton are particles.

Suppose a photon with NR mass m and energy  at high h falls toward the earth relative an inertial reference frame on the surface of earth. Its frequency increases of  to , lot gravitons enter to structure of photon that. The problem is that how many gravitons enter into photon to change the least photon energy (minimum of )?

Thus, according to this expression, for calculate the number of gravitons of  and explain them properties, suppose photon with frequency  is formed of n1 elements and with frequencycontains n2 elements. These elements are not same, because they behave with deferent properties. Let�s propose a 4x1 matrix as following;

 

 

This matrix must satisfy in following relation;

 

 

Now we should calculate A, B, C and D that they satisfy with properties of photon. When gravity works on a photon, gravitons enter into photon and intensity of electric field increases. Photon has no electrical effect; therefore A and B must carry electric effect around the photon with opposite effect. So, according relative between intensity of electric and magnetic fields E=cB we can write;

 

 

 

There H+is positive color-charge and H- is negative color-charge. Also, In above relation c is a pure number that is relative E and B in electromagnetic. So, let�s show , then the above relation becomes to:

 

 

When lots H+ enter into photon, intensity of positive electric field increases. According to Maxwell equations, intensity of magnetic field increases, too. Also, element C must carries magnetic effect around positive color-charges and same reason is for D element. Therefore C and D are same with opposite direction. So, according relationship between intensity of electric and magnetic field we can write;

 

 

Then the matrix (7) becomes to following form the named CPH matrix;

 

 

According to above expression, we are able define the least magnitude of a photon. A tiny energy photon contains some positive color-charge H+, negative color-charge H-, right rotation color-magnet Hm and left rotation color-magnet -Hm that showed in CPH matrix (relation 9). This tiny energy comes of following limit;

 

Energy of a CPH (ECPH) defines in relation (12). 

Now we are on a place that guesses some provable conceptions about photon and a new definition of graviton.

 

4 Gravitons

Many physicists believe the graviton does not exist, at least in the simplistic manner in which it is envisioned. Superficially speaking, quantum gravity using the gauge interaction of a spin-2 field (graviton) fails to work like the photon and other gauge bosons do.

Maxwell's equations always admit a spin-1, linear wave, but Einstein's equations rarely admit a spin-2, linear wave, and when they do it is only perturb active and not exact. But string theory showed graviton is a 3/2 spin particle.

However, if photon is made of gravitons, then graviton does not obey of 2-spin. So, let�s continue with definition of CPH and Principe of CPH and then return to properties of graviton.

 

5 Definition of a CPH

What is a CPH? Creative Particles of Higgs or CPH is an existence unit of nature. Another words, everything is made of CPH. Therefore, a CPH is appropriately referred to as the unit of nature. Although, this is not meant to be the particle as it has been referred to in physics.

A CPH is a particle with constant NR mass,  that moves with a constant magnitude speed of in any inertial reference frame, that c is speed of light. According to the relation mass-energy (relation 11), the NR mass of a CPH is defined relative to photon's NR mass;

 

 

The relationship between energy and momentum for NR mass CPH given by;

 

 

Relation (12) shows in any interaction between two (or more) CPH, energy of ever CPH is constant. So, when does decrease, it takes spin. Other word, in any inertial reference frame and Cartesian components;

 

 

 

When;

 

there is no different between boson and fermions, in this case CPH carries gravity force and behaves like fermion. So, there are color-charges, only.

 When;

 

 

For particles like electron or quarks;

 

 

other bosons do occurrence. For example reconsider to pair production, before pair production there is a photon, only, but after pair production there are an electron (fermion), a positron (anti-fermion) and virtual photon (boson) that carries electric force. So, we can write:

 

 

According a CPH with spin, calls graviton, so space is full of CPH. When density of CPH increases in space, their distance decreases, then they feel and absorb each other. Suppose two CPH are moving on x axis direction and absorb each other, their path change without decreasing the magnitude of . According relation (12), we are able to construct an operator  which rotates CPH by angle  about the x-axis (toward z-axis or y-axis) in position space. Also, we can construct which rotates the CPH by  about the x-axis in spin space. We would expect such an operator to take the form;

 

 

Thus, according to this expression, two CPH rotated each other, they can not have same direction spin. They spin with opposite direction, if positive color-charge has up spin, then negative color-charge must takes down spin. In general their spin must satisfy of Pauli matrix. A great result is this fact that spin of graviton is one-half.   

 

 

6 Principle of the CPH theory

CPH is a unit tiny energy with constant NR mass () that moves with a constant magnitude of speed equal  , in all inertial reference frames. Any interaction between CPH and the other existing particles presents a momentum of inertia I where the magnitude of   is constant and never changes. Therefore,

 

 

 

Based on principal of CPH, a CPH has two types energy generated by its movement within its inertial frame. One is transfer and the other spin. In physics, we present energy summation (both kinetic and potential) by Hamiltonian equation and energy difference by LaGrange. Therefore, in case of CPH, we use Hamiltonian to describe the summation of energy generated by transfer and spin as following:

 

 

Where T is transferring and S is spinning energy of a CPH respectively. Since speed and mass of CPH is constant, then . CPH produces energy and energy produces Matter and Anti-Matter. In fact everything has been formed of CPH.

 

 

7 CPH and Cyclic group

As we explained in section 3, gravitons in interaction with each other convert to color-charges and color-magnetic. Also, when CPH has spin, it calls graviton. Now we can define a cyclic group for electric field that generates by graviton.

So,  given by;

 

Suppose 2k color-charges    combined and move in space. So, there are two electric fields with opposite sign in space. About each field does form a magnetic field that produces by graviton. According sign of these fields, direction of magnetic is different, so their elements are same. Therefore a cyclic group given by:

 

 

According above groups and CPH matrix (relation 9) we can explain zero point energy. 

 

 

8 Zero Point Energy

Space is full of gravitons. Gravitons interact with each other and convert to color-charges. Interaction between gravitons depends to their density of graviton  in volume. Energy production given by:

 

 

If we combine relations (4) and (9), then we are able explain the mechanism of ZPE. Some gravitons with same NR mass   convert to color-charges, two electric fields do form. These fields neuter each other. But positive color-charges repel each other, same action happens for negative color-charges. So, when intensity of color-charges grows, about each fields (negative and positive fields) a magnetic field do form. Magnetic field keeps electric field. This mechanism is explainable by Larmor radius (gyro-radius or cyclotron radius) that given by;

 

 

Where  is the gyro-radius, m is the mass of the charged particle,  is the velocity component perpendicular to the direction of the magnetic field, is the charge of the particle, and B is the constant magnetic field.

Defines the radius of the circular motion of a charged particle in the presence of a uniform magnetic field. when color-charges change in structure of photon, then magnetic-color changes too. Therefore charge particles don�t decay.

However, we should change m with  in relation (4), then we can write;

 

 

 

Consider to n depends to density of graviton in space (relation 20). In general  depends to CPH matrix and it given by:

 

 

 

Dirac equation for space given as;

 

 

 

 In quantum mechanics of plane waves of specific spin shows that a general field can always be written in terms of photons with a simple spin state and a general spatial wave function. Thus the fundamental entity, the photon can be considered quite generally to be a plane wave with a circularly polarized spin piece (Any field can be built from this basic ingredient).  For simplicity consider a photon traveling in the x direction or consider the direction of the photon as choosing the coordinate axis so that x point along the photons momentum. Every element in photon (relation 24), moves with momentum same as photon.

 

9 Hawking Radiation

In a simplified version of the explanation, Hawking predicted that energy fluctuations from the vacuum causes the generation of particle-antiparticle pairs near the event horizon of the black hole. One of the particles falls into the black hole while the other escapes, before they have an opportunity to annihilate each other. The net result is that, to someone viewing the black hole, it would appear that a particle had been emitted.

How Hawking radiation is explainable by equation ZPE (relation 24)? For resolving this problem, there are three considerable items in black hole;

  • Intensity of graviton is so high around a black hole.

  • Gravitons convert to photons speedy.

  • Dirac equation shows how photon produces matter and anti-matter.

According relations (20) and (24), space around black hole produces high energy photon that its energy is enough for pair production. Dirac equation results for total energy given by;

 

 

 

Compare relation (25) with equation (24), energy equation can be written as;

 

 

In black hole n growth speedy, so  is comparable with total mass of particle and anti-particle. The problem is conversation of momentum that it is resolvable by interaction between photon and existence of heavy particles around black hole, especially in photon sphere. This looking not need to time that the uncertainty shows in relation (3). So, pair production is a common projection around a black hole.

 

10 Graviton Spin

Relations (26) and pair production is considerable to thinking of graviton spin. Root of relation (26) given by;

 

 

Pair production shows a photon with spin one, produces two particles (electron and positron). Relation (27) results an electron with spin � (and a positron with spin �) is formed of gravitons. Compare this expression with quarks and proton. A one-half spin (proton) made of three quarks with � spin. In high speed (c or greater than c) graviton (color-charges) behaves like fermions (electron). So, we should reconsider our ideas about spin of graviton. Relation (27) shows graviton has a one-half spin. 

The quantum loop calculation comes with a very big problem. In order to properly account for all virtual processes in the loops, one must integrate over all possible values of momentum, from zero momentum to infinite momentum. But these loop integrals for a particle of spin J in D dimensions take the approximate form;

 

 

 

If the quantity 4J+ D - 8 is negative, then the integral behaves fine for infinite momentum (or zero wavelength, by the de Broglie relation.). If this quantity is zero or positive, then the integral takes an infinite value, and the whole theory threatens to make no sense because the calculations just give infinite answers.

The world that we see has D=4, and the photon has spin J=1. So for the case of electron-electron scattering, these loop integrals can still take infinite values. But the integrals go to infinity very slowly, like the logarithm of momentum and it turns out that in this case, the theory can be renormalized so that the infinities can be absorbed into a redefinition of a small number of parameters in the theory, such as the mass and charge of the electron.

Quantum electrodynamics was a renormalize able theory; this was regarded as a solved relativistic quantum theory.

One big problem was that classical gravitational waves carry spin J=2, so one should assume that a graviton, the quantum particle that carries the gravitational force, has spin J=2. But for J=2, 4J -8+D=D, and so for D=4, the loop integral for the gravitational force would become infinite like the fourth power of momentum, as the momentum in the loop became infinite. And that was just hard cheese for particle physicists, and for many years the best people worked on quantum gravity to no avail*.

According graviton converts to color-charge in interaction with other particles and it has a J=1/2, so, in 4J-8+D, J=1/2, we have, then for our universe D=4, so;

 

 

 and relation (28) is renormalize able simply.

 

 

References;

1- Richard Cushman. and. Wilberd van der Kallen. .(2005) A new interpretation for the mass of a classical relativistic particle, Elsevier Journal, Physics Letters Volume 24, Issue 3, May 2006, Pages 230-234

2- Ravindran V. (2006) �Higher-order threshold effects to inclusive processes in QCD�, Elsevier Journal, Physics Letters B, Available online 11 July 2006

Perelstein M. (2006) �Little Higgs models and their phenomenology�, Elsevier

3- Journal, Progress in Particle and Nuclear Physics, Available online June 2006

4- Gupta G. P. and. Lal K. C.(1971) �On the equivalence of gravitational red shift and temperature shift in the Mssbauer effect�, Elsevier Journal, Physics Letters A Volume 36, Issue 5, September 1971, Pages 421-422

5 - Peter S. Riseborough , E-book , Advanced Quantum Mechanics March 14, 2007

6- Stefan Waner, E-book, Introduction to Differential Geometry & General Relativity

4th Printing January 2005

7- http://www.daviddarling.info/encyclopedia/C/Casimir.html

8- * advanced string theory  http://superstringtheory.com/basics/basic3a.html

9- F. Smarandache, V. Christianto, Fu Yuhua, R. Khrapko, J. Hutchison, E-book, Unfolding the Labyrinth: Open Problems in Physics, (University of Microfilm International)


[1] - Casimir, H. G. B. "On the attraction between two perfectly conducting plates." Proc. Con. Ned. Akad. van Wetensch B51 (7): 793-796 (1948).

[2] - General relativity forces Masataka Mizushima Department of Physics, University of Colorado, Boulder, CO 80309, USA November 2004. 30 November 2005-15 December 2005, Pages e369-e378

 

 

 

 

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Biography

Since 1962 I doubted on Newton's laws. I did not accept the infinitive speed and I found un-vivid the laws of gravity and time.

I learned the Einstein's Relativity, thus I found some answers for my questions. But, I had another doubt of Infinitive Mass-Energy. And I wanted to know why light has stable speed?

 


 

 


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