Equiprobability vector and symmetry- broken
1．The definition
of equiprobability vector and symmetry- broken
In a coordinate system
, there is a vector
with the starting point
and
the starting pointis fixed always. In short time
interval
,
the direction of this vector
is
changed anytime and appeared in any direction equal probably, then
it is formed a group of vectors
（N=1, 2, ……）
in.
We define the group
of vectors as a set of equiprobability.
Or said, in short time interval
, a group of vectors （N=1,
2, ……）are appeared, their starting
pointis fixed all
along, and they go in different direction at any time, each direction is equiprobability,
we define these group of vectors as a set of equiprobability.
It should be noted that 
should be a very and very
small interval. 
In
the case of the light quantum system we are talking about, the speed of light
quantum is 300,000 Kilometers per second. 
The time intervals are possible.
In the issues related to the light quantum system we are discussing, the
speed of light quantum is 300,000 kilometers per second. Inside the
particle, the direction of the speed of light quantum changes very, very
quickly due to mutual collision.
They can be considered as a set of vectors with
equal probability.
When
the length of all equiprobability （N=1，2
，……）are
equal each other, 
, or
at a certain time
interval, we define
this
group of vectors as
a symmetrical equiprobability. When the length of all equiprobability （N=1，2
，……）aren’t
equal each other, 
,
or
at a
certain time interval,
we define this group of vectors as a broken equiprobability.
The
coordinate system is
a stationary coordinate system. When the coordinate system 
is stationary with the
coordinate system
relatively, a
symmetrical equiprobability in the coordinate system still is a symmetrical equiprobability
as observing from the coordinate system. If the coordinate system is moving relative to the
coordinate system with
a velocity
, the
symmetrical equiprobabilityin the coordinate system
becomes broken when observing from
the coordinate system,
then. This time
is equal to a vector
which direction is identical or contrary to
. That is said a symmetrical equiprobability
in the coordinate
system is
broken when observing from the coordinate system. This kind of
broken is called as relative motion broken.
In a curvature space the
distribution of matter is asymmetrical, a coordinate system is a static coordinate
system, and the coordinate system is a
local coordinate system of the
coordinate system. There
is a symmetrical equiprobability vector in the local coordinate system, but when
observing from
the coordinate system
that becomes. That
is because of the normal curvature of curves in different directions is different.
Then this symmetrical equiprobability vectors in the coordinate system is broken when observing
in the coordinate system. This kind of broken is called as the
relative curvature broken.（The curvature space
referred to here is the curvature space of General Relativity.）
A
coulomb field centered on the origin of a stationary system
. An electron revolves around the
center. The
directions of angular momentum
are equal probably. 
. They are symmetrical equiprobability
vectors. If you introduce some magnetic material, then 
in this
magnetic field. At this case, the symmetrical equiprobability vectors have relative curvature broken.
The light quantum of a particle is moving on a tangent plane of distributing curvature surface of the light quantum system. During the distributing curvature surface is a spherical surface, its velocity vectors are symmetrical equiprobability vectors. During the distributing curvature surface isn’t a spherical surface, the velocity vectors are no longer to be symmetrical equiprobability vectors, so the relative curvature broken are happened.
The
isospin of particle is revolved to wind a linear axis through center of the
particle. The axis and the isospin angular momentum are symmetrical
equiprobabilities
respectively in space. If the outfield doesn’t exist,
, is a symmetrical equiprobability in
space. Once particle is moving or
there is an outside field, then
, the symmetrical equiprobability
is broken. In these case
. For a particle, 
, the particle has an isospin
magnetic moment externally.
2. Express real and imaginary numbers as equiprobability vectors.
In
a complex plane, the X-axis is the real axis, and the Y-axis is the imaginary
axis. But in space, we can use X-axis to represent real number, we can use the Y-axis
or Z-axis to represent a real number too. We also can take an arbitrarily
-axis as a real axis, then
the imaginary axis should be
distributed on the perpendicular plane of-axis. Similarly, any
vector
in space is to
represent the imaginary axis, the real axis should be an
equiprobability axis distributed
on the perpendicular plane of vector.
You have to think of specifies axes as an equiprobability axes, only by clarifying the concept of equiprobability axis and equiprobability vector can we completely represent the complex space.
For
example: An electron rotates around the Coulomb field, the image of the
electron state function is the spherical Bessel function. But the actual trajectory
of the electron is cloud of electrons. Only if one axis, such as the Z-axis is
regarded as the equiprobability vector axis, so set up the coordinate system
and image, that the spherical Bezier function bedome three-dimensional, and
reflects the actual trajectories of electron motion. 
For functions, we usually have to use the equiprobability
axis to construct the coordinates system and image, in order to reflect the actual
trajectories of motion and distribution of particle.
3. Vectorization of a symmetrical equiprobability vectors.
For
a set of
symmetrical equiprobability
vectors, we tend to divide them into positive and negative vectors. It’s like a
number line, once the origin
is
determined, the right side of the axis is
positive and the left side is negative. For symmetrical equiprobability vectors,
the same is true. Once the origin and the plane that goes through this
point are determined,
the top half of the plane is positive and the bottom half of the plane is
negation. Take the
moving light quantum for example. The light quantum is
going to the plane and moving in the direction above the plane （
）are the input
energy
and momentum. The light quantum is out of the plane and moving in the direction
below the plane（
）are the output energy and
momentum. Look at their energy and momentum in both cases, they have different
effects. So they should be calculated separately. To calculate the energy and
momentum of their vibrations, you often only need to take half space.
Take
the spin of a light quantum for example. The direction of
pointing up the plane, its
spin angular momentum is positive. The direction of
pointing below the plane, its spin angular
momentum is negative.
A
set of symmetrical equiprobability vectors are divided into two parts in
opposite directions: （）and （）for integration
respectively.
Then they are formed two vectors. We just only take one of two vectors, such an operation is defined as the vectorization of symmetric equiprobability vectors.
However, if the integration is divided into two parts, the interface between the two parts can be chosen according to the actual situation. Often takes the direction of a certain section. The direction of the 0-axis of the integral is the vertical direction of the interface passing through the 0 point andThe interface of the integral summation range. Another way is that the direction of the 0-axis of the integral can be taken according to the actual situation. It can be parallel to the direction of another known vector, or parallel to the vectorized vector direction of another set of symmetric equal-probability vectors. In this case, the direction of the interface of the summation range is perpendicular to the 0-axis of the integration. According to the latter method, the vector directions of the two sets of symmetric equal-probability vectors can be synchronously the same after vectorization. This is what is called "probability synchronization" between the two vectors. "Probability synchronization:" is an important topic in our course.
The movement direction and momentum direction of any light quantum are
arbitrary and are a set of symmetric vectors with equal probability.
But when two
light quantum collide
, it is necessary to realize the vectorization
of the symmetric equal probability vector.
One light quantum is the incident particle and the other light quantum is
the outgoing particle. For each light quantum vectorization,the integral interface of the
momentum is perpendicular to the collision direction, and
the direction of
the 0-axis of the integral is the collision direction.
Sometimes the direction of the collision of two
photons is always taken as the direction of the x-axis or the tangent
direction of a certain curved surface.
Not only through vectorization of two symmetric equiprobability vectors, they can have the same direction synchronously, but also through vectorization of several, even a large group symmetric equiprobability vectors, they can all have the same direction synchronously.
Take
the spin of a particle for example, a particle contains n
number light quantum mass
points (
).The spin angular
momentum of every mass
point is
, it is a vector revolving to wind an
equiprobable linear axis through itself. can be realized a set of synchronism
vectors. When they are vectorization, their direction can be identical. The
spin angular momentum of the entire particle can be considered uniform. The
direction of spin
angular momentums of total particle can be thought of as
. Because probability
synchronization,may
be thought of in a definite direction.
The vectorization for
symmetrical equiprobable vector is similar with a column matrix or a matrix. They
have the flexibility to choose the principal axis of coordinate system. However，it
can represent some vector or other physical properties of a particle.A matrix or column matrix
can often be used as a representation for the vectorization of an equiprobability
vector. Why is it possible to represent physical quantities with matrices in
quantum mechanics? It is synchronization for a symmetric equiprobability vector
can be represented by a matrix.
4．Vectorization after a symmetrical equiprobable vector is broken
When the symmetrical equiprobable
vector is broken, it forms a special directional vector. This special direction
generally just is the direction of the broken.
The
coordinate system is
a static coordinate system. The coordinate system is a subsystem.
The velocity of
coordinate system is
along
axis relatively to
coordinate system. A
set of
symmetrical equiprobable velocity vector
in coordinate system, When observe from the coordinate system, the symmetrical
equiprobable velocity vector is broken.
The coordinates of the light
quantum points
in the are
，
，
. The coordinates
of the light
quantum points in the are
，
，
.
According
to the special theory of relativity：
，
，
,
。
，
。
，
。
A
light quantum point is moving relatively to coordinate system
，the
velocity is 
. is a set of symmetrical
equiprobable vector. Suppose the angle
between and axis
is 
. plane
,
the angle between the projection of onto the 
plane and the 
axis is 
, then we get
，
，
.
The coordinate system
is moving in the x-direction
with the coordinate systemat velocity. According to the Lorentz transformation,
when
observing from the coordinate system, the velocities
，
. 
, 
, 
are the velocities of a light quantum when
observing from the
coordinate system, They are equiprobable vector. Sum of equiprobable
vector, 
，
，
. 
, 
, 
are the velocities of the light
quantum
point when observing from the coordinate system. When 
is the same value, and 
is integrated from 
，
，
.
，
；
，
，
,
，take
，
。
，
，after
the calculation can be obtained：
，
，and ，.
This means that after the Lorentz
transformation and the Vectorization of equiprobable vector, the direction of
the resulting vector is still along the 
axis.
From this we can get: If the
coordinate system is
moving relative to the coordinate system at
, when observing from the static
coordinate system, a
set of symmetrical equiprobable vectors in the coordinate system becomes broken. Through Lorentz
transformation and vectorization, it
becomes a directional velocity vector, its
direction is identical with the moving velocity of the coordinate system .
The same result can be obtained if you change the velocity vector of light quantum to the vibrational velocity vector.
We discuss the
spin of light quantum：The coordinate system is a static coordinate
system. The center of particle is at point. 
is a light
quantum mass point on the
distribution surface of light quantum system. Let’s draw a coordinate system 
through point, its axis 
is
parallel to the axis 
of the coordinate
system. The
coordinate system 
is
rotates with respect to the coordinate system at rotational
angular velocity
. Suppose the direction
of is in the
direction of. 
is slightly less
than the speed of light
.
Take the mass point
of light quantum in,
its spin
angular velocity is
,
is a set of
symmetrical equiprobable vector in particle.
Let
the Angle between and
axis is. The Angle between the
projection of 
onto
the plane and
the axis is
. The components of 
in
three axis of the
coordinate system are
、
，
.
，
，
.
The coordinate
system has
rotational angular velocity
relative to the coordinate system. is in the direction of .
In terms of
rotation, according to the Lorentz transformation, if you look at it in the static
coordinate system,
we can get:
，
.
.
、
、
、
are spin angular
velocity of light quantum pointand its components when observing from the static
coordinate system.
Compare
the equalities 
、
、
with the equalities 
、
、
at the preceding part of
this text. If you change the spin variable to the translational variable, then 
、、
are completely similar
to、
、. So let’s calculate formulas
、、to be the same as the
、、 formulas, the resulting broken
vector can be viewed as a rotation velocity vector, which direction is in the
same direction as the rotation
velocity of the coordinate system
with respect to coordinate system
.
Draw
the following conclusions: The coordinate system translates or rotates with respect to
the coordinate system.
Let look at it in the coordinate system, a symmetric equiprobable translational or
rotational velocity vector in the system is broken. After Lorentz transformation and
Vectorization of symmetrical equiprobable vector, a velocity vector or a rotational velocity vector is formed. It moves in the same direction as the
coordinate
system to translate
or to rotate with respect to the coordinate system.
The same result can be obtained, if the spin angular velocity vector of the particle is replaced by the spin vibration velocity vector.
Introduction and Contents 引言和目录