More research for the second quantization theories.

For the structure of particle, there has been the quark and the string vibration theory etc. But author think the finally essential structure of particle, it is going to be back to the second quantization in higher quantum- theory yet. Please refer to “High Quantum Theory” P, Roman.

1．More research for the second quantization theories.

In the higher quantum mechanics, the essential ideas of second quantization are: Think of the state function  as generalized coordinates of the degrees of freedom in matter field, and think of the space coordinate regarded as the indexes of the generalized coordinate.

Take the space coordinate is regarded as the indexes of the generalized coordinate. That is each  corresponds to a unit of this matter wave, or to say it is regarded as a‘multi-particle’. That is to say, think of the matter- wave as a‘multi-particle system’ , each  corresponds to a multi-particle in ‘multi-particle system’. The generalized coordinate state function is through of one unit element of ‘multi-particle system’. 

Take matter- wave as a‘multi-particle system’, Each multi-particle in the ‘multi-particle system’  is the basic element unit that constitutes the matter- wave.  

The Lagrange function of particle is  .

The Lagrange's equation is: .

The Lagrange function is reflected the energy of a particle. It is the energy accumulation of infinite number of single multi-particle in‘multi-particle system’. are the integral variables ,

 is a carrier that correspond to . Each multi-particle corresponds to a load body of energy and momentum at any moment.

One way to think about it, that the quantum field theory is formed on the basis of the second quantization theory, through which is derived the Lagrangian function of the  whole particle, then combined with the quantum equation and quantized to them.

In this paper, we further study the multi-particle in the second quantization theory. We studied their distribution, density, motion and so on. The further study of the second quantization theories is aimed at finding a way to study the internal structure of particles.

Base on the above analysis we can define the‘multi-particle system’ in the second quantization theories as the light quantum system. Think of the “multi-particle” as a light quantum. Because the light quantum and the “multi-particle” are a minimum load body of matter waves, the light quantum system has the characteristics of ‘multi-particle system’.   

The light quantum systems and ‘multi-particle system’ all are waves at the micro field. The speed of light systems are the same as the speed of the matter waves. The light quantum has the characteristics of light in dimension, infinitesimal mass, energy, and so on. The light quantum is a load body of energy and momentum, as them as the“multi-particle”. So we have shown in the introduction that the microscopic matter that makes up a particle is the light quantum.

The light quantum system is not only a wave, a stable matter wave, but also a particle. That means it has a stable distribution. It shows that the light quantum has stable density, energy and momentum distribution. This distribution is dynamic without chaos. To take electron cloud in the coulomb field for example. The electron  cloud is distributing on the equipotential surface of the state function, and is moving in the direction to tangent to the equipotential surface. Therefore, it is reasonable to assume, that the light quantum also should distribute and move on the direction to tangent to the equipotential surface of state function. Only in this way can let the free particle remains independent and stable. But unlike the electron cloud,   they are not coulomb field. They are the elastic collision and spin of light quantum in the particle.

The distribution surface of light quantum is a series of curved surface that surround the center of the particle. The velocity of the light quantum is equal probability vector on the tangent plane of the distribution surface.

We can examine the distributing equipotential surface of light quantum system by using either the metric method of flat metric or with curved metric. (please see……). When examining with flat metric, the distributing equipotential surfaces are a series of flat plain, also it can be approximated as a series of spherical surface with a very large radius.  When examining with curved metric, the distributing equipotential surfaces are a series of curved surface. They have different curvatures depending on their position. The normal curvature of different curves through a same point of equipotential surface is also different too.

At every point on the distribution surface there exists a conversion function between the curve metric and flat metric: ， Among them 、are the curvature radius and normal curvature of a point on a curve respectively. (Please see the article : )

The velocity of the light quantum is an equal probability vector. When the surface is a plane or a spherical surface, it is the symmetric equal probability vector. When the distribution surface is other functional surface the speed vector of light quantum is the broken probability vector. (Please see the article :  )

2. The distributing equipotential curved surfaces of light quantum system should be minimal curved surfaces.   

The distributing equipotential curved surfaces of light quantum system should be minimal curved surfaces. First explain the definition of equipotential surfaces. For a stable system, the light quantum always moves on a designated area of equal position, and is not going to jump to the surface of another distribution.

Define a small region around any point of the distributing equipotential curved surface of light quantum. During the motion of the light quantum, the position of the point  and small region on the curved surface will change. The point  and small region  go to  and small region.The density of the light quantum at point  is, the area of the small region  is . The density of the light quantum at

point  is, the area of the small region  is.The number of light quantum in the small region is .

The number of light quantum in the small region is. On the distributing equipotential curved surface, the interface

of the region does not change during the motion of the light quantum. . The resulting is. It shows that in the process of light quantum motion, the number of light quantum particles in region is always unchanged.

The subscript of variables， in the article denote，, they are readings of the flat metric.

The number of light quantum in a small region remains constant during the movement of light quantum on the equipotential curved surface.

Take ，There is  when variation, occurs. There are .  In the course of the motion of the light quantum, there are always on the equipotential curved surface said above.

A local coordinate system  is established round point on the distributing equipotential curved surface. Take a surface element  round the point. The area of the surface element is. Suppose the function of the distributing equipotential curved surfaces is . The

number of light quantum in the surface element is. .. The average curvature of the curved surface is, it is the functions of,. As oscillate the number of light quantum changes with the normal direction of the surface. In the vibrating process, it is only

changed along with the normal direction. Direction can also be represented as . In the vibrating process, the distributing curved surfaces of the light quantum are equipotential surfaces all along.

In vibration, the distributing curved surfaces of the light quantum are wave front. The light quantum may not move out the specified region. But they may change depending on the normal direction of the surface during oscillation. During vibration, may change with the normal direction of the surface.

 , .

On the equipotential surfaces, there are  said above.

. . . On the equipotential surfaces, the light quantum vibrates in thedirection. For a stable particle, is the

center of its vibration, the position of corresponds to. For a steady vibration, , ，

This leads to ，, . It can be concluded that the distribution surface of the light quantum system is

a minimal surface according to differential geometry.

According to differential geometry, there’s a rule for minimal surface: .，are the principal curvature of the curve ,respectively. The total curvature of the surface is.

3. The generalized coordinate of the degree of freedom of material field should be the multidimensional generalized coordinates.

In second quantization theories,  is used as the generalized coordinate of one dimension. In order to describe the characteristic of a vector in material field, the generalized coordinates of the degree of freedom of the material field should be multidimensional generalized coordinates.

must be promoted to..When,  corresponds to the momentum of the material fields. When, correspond to the probability density of matter waves, even the traditional state function. In addition, four other generalized coordinates should be defined to describe the spin vibration.

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