No need models - the fundamental theoretical physics is a part of classical probability theory (the part that considers the probability of dot events in the 3 + 1 space-time).

1. ZERO.
PROBABILYSTIC FOUNDATION
OF
THEORETYICAL PHYSICS

2. HTTP://VIXRA.ORG/ABS/1111.0051
For a detailed discussion of this PRESENTATION
formulas read In this book
A reference of type [p.X]
means: "on the page X"

3. DOT EVENTS
Camille Pissarro, LA ROUTE DE
This
picture is
drawn with
colored
dots

4. Let A(x) be a dot event in a 3+1 point x
(x = = 〈x₀, x₁, x₂, x₃〉).
Let be a probability function defined on the set
of dot events in the 3+1 space-time.

 xx ,0
Let D be a domain in the 3+1 space-time and
let be a defined in the 3+1 space-time
function such that
 xpA
      DxxAxpxd
D
A 

5.  xpA
Function is invariant under the Lorentz
transformations [p.58]. I call this function an
absolute probability density of event A.
   
  

yxpyd
xxp
xx
A
A
A
,
,
:,
0
0
0
If
then is a probability density of
event A in an instant .
 xxA ,0
0x

6. A probability density of event A in an instant is
not invariant under the Lorentz transformations.
0x
 xxxxdx AkAk ,0 If
then is an average k-coordinate of event A.
And if
Akx
 xxxxdxu AkAkAk ,000  
Aku average k-coordinateThen is an of velocity of
event A.

7.  xxxu AkkA ,: 00, If then is a k-
coordinate of velocity of event A.
kAu ,
I consider events which obey the following
condition: 22
3,
2
2,
2
1, cuuu AAA 
(Traceable events)
Let  3,2,1:,: ,,0,  kforujcj kAAkAAA 
is a probability current vector of event A.
AA jj ,0,

8.  
,1
10
01
:,
00
00
:0,
10
01
:1 4
22
220
22 

















 
Denote:
The Pauli matrices:
.
10
01
:,
0
0
:,
01
10
: 321 











 






 
i
i

9. A set of complex n x n matrices is called a Clifford set
of rank n if the following conditions are fulfilled:
.setofelementsallfor
2
;2,
,
,
CthenC
if
thenCCif
ks
skkssk
skkssksk






C
If n = 4 then a Clifford set either contains 3 matrices
(a Clifford triplet) or contains 5
matrices (a Clifford pentad)

10. [Madelung, E., Die
Mathematischen Hilfsmittel
des Physikers. Springer
Verlag, (1957) p.29].

11. If n = 4 then a Clifford set either contains 3 matrices (a
Clifford triplet) or contains 5 matrices (a Clifford
pentad).
Here exist only six Clifford pentads [pp.59-60]:
one light pentad β:
 
 
   
.
01
10
:,
01
10
:
;3,2,1
0
0
:
22
224
22
220
2
2






















i
kfor
k
kk





12. three chromatic pentads:
the red pentad ζ:
     
,
0
0
:,
0
0
:,
0
0
:
32
233
22
222
12
211






























   
;
0
0
:,
0
0
:
21
124
21
120





















 i
the green pentad :
     
,
0
0
:,
0
0
:,
0
0
:
32
233
22
222
12
211






























   
;
0
0
:,
0
0
:
22
224
22
220





















 i

13.      
,
0
0
:,
0
0
:,
0
0
:
32
233
22
222
12
211































   
;
0
0
:,
0
0
:
23
324
23
320





















 i
the blue pentad:
two gustatory pentads:
the sweet pentad⧋:
     
,
0
0
:,
0
0
:,
0
0
:
23
323
22
222
21
121































   
;
01
10
:,
10
01
:
22
224
22
220













 i

14. the bitter pentad :
     
,
0
0
:,
0
0
:,
0
0
:
23
323
22
222
21
121





 





 





 







iii
   
;
01
10
:,
10
01
:
22
224
22
220













Further we do not consider gustatory pentads since
these pentads are not used yet in the contemporary
physics.

15. For all : ifAA jj ,0,
22
3,
2
2,
2
1, cuuu AAA 
then a 4X1 complex matrix function φ exists which
obeys to the following conditions:
 
  

4
1
4
1
,
*,
4
1
*
,
k s
skss
A
s
ssA
c
j
 
[pp.61—62]
Let a Fourier series for has the following
form:
 xtj ,
    



w p
pwpwjj xtcxt ,, ,,,


16.    xtcxt pwpwjpwj
,:, ,,,,,
 
Let ⟨t; ⟩ be any space-time point.
Denote:
x

xtpwkkA
,,,
: 
the value of function in this point, and:pwk ,,

Let
 
xts
pwssjpwjtj
c
C
,
4
1
3
1
,,,,,
1
: 





  



the value of function  






  
4
1
3
1
,,,,,
1
s
pwssjpwjt
c 



In this point

17. Here and are complex numbers. Hence, the
following set of equations:
kA jC







pwkjpwjk
jk
k
pwkj
zz
CAz
,,,
*
,,,
4
1
,,,
,
is a system of 14 algebraic equation with complex
unknowns
pwkj
z ,,,
This system can be transformed as system of 8 linear real
equations with 16 real unknowns [pp.66—67].

18. This system has solutions in accordance with the
Kronecker-Capelli theorem.
Hence, such complex numbers exist in all
points [pp.65—67]
pwkj
z ,,,
xt,
xt,
Let be the linear operators on a linear space,
spanned of basic functions , such that
pw,

 xtpw
,,








'/'0
',''',
'',,
pporandwwif
ppwwifpw
pwpw



pw
pwpwkjkj zQ
,
,,,,, : Let

19. Therefore, for every function here exists an operator
such that a dependence of on t is described
by the following differential equation [pp.67—68]:
j
kjQ , j
 
k
k s
kjs
s
kjjt Qc    







4
1
3
1
,,
  jkpw
pw
pwkjkj QzQ ,,
,
*
,,,,  

and
 








22
225
10
01
:If

20. then this equation can be transformed to the
following form [pp.81—82]:
 
 
   
 
   
 
 
 
   
 
   
 
 
 
   
 
   
 
 
 
   
 
   
 
0
2
2
2
2
3
1
4
4,
0
0,
5
5
000
3
1
4
4,
0
0,
5
5
000
3
1
4
4,
0
0,
5
5
000
3
1
4
4
0
0
5
5
000











































































k
kkk
k
k
kkk
k
k
kkk
k
k
kkk
k
MMi
ii
ii
MMi
ii
ii
MMi
ii
ii
MMi
ii
ii

21. with real
  3,2,1,0,,,,,,,,, 4,0.4,0,4,0,40  kMMMMMMMM kk
       hkkk
 Because then [p.82]
   
 
       
       
0
4
4,
0
0,
4
4,
0
0,
4
4,
0
0,
4
4
0
0
5
3
0


























iMiMiMiM
iMiMiMiM
ii kkk
k
k

22. 04,0.4,0,4,0,   MMMMMM
Let
then
   
     
04
4
0
0
5
3
0







 iMiMii kkk
k
k
Such equation is called a lepton’s equation [p.83]
 
 4
4, : 
 cjA
Let
 
 0
5, : 
 cjA
A
A
A
A
A
A
j
u
j
u

5,
5,
4,
4, :,: 

23. There [p.83]:
22
5,
2
4,
2
3,
2
2,
2
1, cuuuuu AAAAA 
Thus, only all 5 elements of Clifford Pentada provide a
full set of speed components and, for completeness,
two more "space" coordinates and should be
added to our three .
4x
5x
321 ,, xxx
Coordinates and are not of any events
coordinates. Hence, our devices do not detect of its as
space coordinates.
5x
4x

24. Let    
      3214432105
32154321
,,,,,,exp
,,,:,,,,,~
xxxtMxxxxtMxi
xxxtxxxxxt

 
In this case a lepton equation of moving has the
following shape [p.84]:
   
     
0~
4
4
5
0
3
0
5







 iii
k
kkk
k
For every real and real , , and a
constant exist which obey to following condition
[p.84]:
k k kF kB
1g
 
kkkk YBgF 1
5
5.0 








22
22
120
01
:Ywith

25.  
     
0~5.0 4
4
5
0
3
0
1 






 iiYBgFi
k
kkk
k
this equation of moving is invariant under the following
transformation [pp.85—89]:
 






UFUFF
g
BBB
forxxx
xxxx
xxxx
U
kkkkkkk
~~
:',
1
:'
3,2,1,0:'
2
sin
2
cos:'
,
2
sin
2
cos:'
~~
:'~~
1
4555
5444






That is [p.85]:

26. if is a real function and 3210 ,,, xxxx
 
  















22
22
1exp0
01
2
exp
:
~



i
i
U
Therefore, likes to the B-boson field of the Standard
Model.
kB

27. MASSES

28. Let




















































1
0
0
0
:,
0
1
0
0
:,
0
0
1
0
:,
0
0
0
1
: 4321 
Functions of type :
  knxsx
c
h
i
c
h








 54exp
2
with an integer n and s form orthonormal basis of some
unitary space 𝔍 with scalar productof the following
shape:

29.   




~~:~,~
45

   h
c
h
c
h
c
h
c dxdx
In that case [p.90]  
 
  c
j sAs
A
,~,~
~,~




  3,2,1s
 
  













h
cM
xxxtN
h
cM
xxxtN
4
321
0
321
trunc:,,,
trunc:,,,


Let

30.  
 
 
  



























3214
3215
321
54321
,,,
,,,
exp,,,
,,,,,~
xxxtN
c
h
x
xxxtN
c
h
x
ixxxt
xxxxxt




In that case to high precision:
A Fourier series for is of the following form [p.91]:~
   
      








sn
xtsNxtnN sxnx
c
h
i
xtxxxt
,
45,,
45
exp
,,,,~






31. From properties of 𝛿: in every point : eitherxt

,
  0,,,~
45 xxxt


or integer numbers and exist for which:
0n 0s
     





 405045 exp,,,,~ xsxn
c
h
ixtxxxt


If 2
0
2
0
2
: sn
c
h
m 






then m is denoted mass of ~
.That is for every space-time point: either this point is
empty or single mass is placed in this point [p.91].

32. Under the transformation : U
~
400500
40504050
2
sin
2
cos
2
sin
2
cos
''
xnsxsn
xsxnxsxn















Therefore, a mass is invariant under this
transformation
Mass expressed hypotenuse of a Pythagorean triangle.
Therefore, this invariance requires a fairly large amount
of such triangles from this hypotenuse [pp.91-93].

33. For each natural number n, there exist at
least n different Pythagorean triples with the
same hypotenuse. Sierpinski, 2003, с. 31. —
Dover, 2003. — ISBN 978-0-486-43278-6.
John F. Goehi Jr. TRIPLES, QUARTETS,
PENTADS
: Mathematics teacher, ISSN 0025-5769, Vol.
98, Nº 9, 2005, pág. 580

35. ELECTRO-WEAK
FIELD

36.    
 
       
       
0
4
4,
0
0,
4
4,
0
0,
4
4,
0
0,
4
4
0
0
5
3
0


























iMiMiMiM
iMiMiMiM
ii kkk
k
k
If equation
does not contain the chromatic members then [p.141]:
 
      
0
~~
5.0
~
2
4
1
0
3
0












 e
mimWZiAiei
s
ssss
s 

Here the vector field is similar to the
electromagnetic potential and is similar to
the weak potential.
A
~
WZ
~~


37.  
 }{
~~1
,,,
,
3
1
22
0
2
2,
3
1
22
2


WWW
WWWgW
c
jiji
s ss st







  













,2
,1
,0
W
W
W
:
~
W
Let:
The components of obey to the
Klein-Gordon equation [pp.130--137
Wˆ
with “mass” m = 

3
1
22
02
~~
s
sWWg
c
h
This “mass” is invariant under rotation in the 3-space,
under the Lorentz transformations, and under
global SU(2)-transformations [pp.130--139]

38. But it is not invariant under local SU(2) transformations
and this “mass” depends on the points coordinates.
But these circumstances are
insignificant
for W0,μ, W1,μ, W2,μ since all three particles are very short-
lived (3×10-25 c.)
and a measurement
of masses of these particles is practically possible only at
the point <t ≈ 0; x ≈ 0 > .
And if
2
1
arctg:
g
g
 then
cos
W
Z
m
m  [p.139]

39. QUARKS , GLUONS
and
GRAVITATION

40. It is a chromatic equation of moving [p.146].

41. The mass members of this equation form
the following matrix sum [p.146]:

42. Elements of these matrices are rotated by
transformations which similar to:
There exist only eight of such transformations
These transformations add to chromatic equation
eight more gauge fields:
with some real
constant g3 (similar
to 8 gluons)
[pp.146--.157]

43. Some of these transformations bend space-time so,
as shown by the following figure [pp.161,179]:.
Here function g(t; x1) represents the acceleration of
the curved coordinate system relative to the original system [p.160].
    222
/cosh/, xtxcxtg

{1) (2)
2
/ x



44. Hence, to the right from point C′ and to the
left from poin C the Newtonian gravitation
law is carried out.
AA′ is the Asymptotic Freedom Zone.
CB and B′C′ is the Confinement Zone.

45. Some oscillations of chromatic states bend space-
time as follows [pp.148,164]:
.2sin2cos
'
,2sin2cos
'
xyy
yxx


















Let iyxz  , i.e. .i
rez 
What is DARK MATTER?
''' iyxz 
Because linear velocity of the curved coordinate
system ⟨x′;y′⟩ into the initial system is the
following
⟨x;y⟩
2
1
22
''

























t
x
t
x
v

46. then
t
z
v



' Let function z′ be a holomorphic
function. Hence, in accordance with
the Cauchy-Riemann conditions the
following equations are fulfilled:
.
''
,
''
x
y
y
x
y
y
x
x










Therefore,
 
dzedz i 2
' 

where 2α is an holomorphic function, too.
For example, let     21
2 yxixy
t
 In this case:
          dy
t
xyiyx
idx
t
xyiyx
z 




22
'
[pp.166—167]

47. Let k = y/x. Hence,
    
dy
t
k
y
yiy
k
y
i
dx
t
xkxikxx
z











































 

2
2
exp
exp'
Calculate::
    
 
 
,
242
1
242
1
erf
2
1
exp
2
ikik
t
ikik
t
x
dx
t
xkxikxx


















 



48.  
 ikik
tk
ikik
tk
y
dy
t
k
y
yiy
k
y
i
242
1
242
1
erf
2
1
exp
2
2
2
2
2












































 
 
     
     
   
    





























































2
22
2
2
22
22
222
2
2
22
2
22
3
2
1
erf
1
2
2
1
erf
1
2
2
1
exp2
1
4
2
1
exp2
1
4
28
1'
iki
t
x
tk
ikikt
iki
tk
y
tk
ikitki
ikix
t
iki
tk
ikikx
ikiy
tk
iki
t
iky
iiktit
z



49. For large t:
 
    



















 2
22
22
3
2
1
erf
1
2
28
1'
iki
tk
y
tk
ikitki
iiktit
z

Hence,
    










2
2
1
erf
1
1
8
1
iki
tik
kiv 
Because  cos,tan rxk  then
  
 
     










2
tan2
1
coserf
tan
1
tan1
8
1
ii
t
r
i
iv 



50. For θ:=0.98π,
t=10E4:
Compare with

51. Rotation curve of NGC 6503. The dash-dotted lines are the
contributions of dark matter [K. G. Begeman, A. H. Broeils
and R. H. Sanders, 1991, MNRAS, 249, 523]

52. For θ = 13π/14.
t = 10E4:
Compare with

53. A rotation curve for a typical spiral galaxy. The solid
line shows actual measurements
(Hawley and Holcomb., 1998, p. 390)

54. Hence, Dark Matter and Dark Energy can be
mirages in the space-time, which is curved
by oscillations of chromatic states.
The Dark Energy phenomena is explained jn similar
way

55. CHROME

56. The red state becomes the green state under
rotation of the Cartesian system [168—182].21,xx
Similarly, for other Cartesian rotations, the chrome
of the other quark states changes.
This explains the phenomenon of confinement and
the composition of baryon families

58. No need models - the fundamental
theoretical physics is a part of classical
probability theory (the part that considers
the probability of dot events in the 3 + 1
space-time).