Table Of ContentPhysical events and quantum field theory
without Higgs
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0
0
G. A. Quznetsov
2
n [email protected], [email protected], [email protected]
a
J
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1 Abstract
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All physics events are expressed by particles which similar to well-
v
9 knownelementary particles -leptons, quarksandgauge bosons. Higgs
3 is not necessary.
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2
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1 Denotations:
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9
/
h c = 1: the light velocity in vacuum;
p
α β : conjunction of events α and β: ”α and β”;
-
p ∩
α β : disjunction of events α and β: ”α or/and β”;
e ∪
h α : event, complementary to event α: ”not α”;
:
v : generality quantifier: ”for every”;
i ∀
X : existenial quantifier: ”for some”;
∃
r e1, e2, e3 are the Cartesian basis vectors;
a
Def
x = (x e +x e +x e );
1 1 2 2 3 3
Def
x = t;
0
d3x D=ef dx dx dx ;
1 2 3
Def
∂Rk = ∂/∂Rxk; R R
Def Def
∂ = ∂ = ∂/∂t;
t 0
∂′ D=ef ∂/∂x′ ;
k k
∞ ∞ ∞
Def
= ;
Xk k1X=−∞k2X=−∞k3X=−∞
1
1 0 0 0
Def Def
1 = , 0 = ,
2 0 1 2 0 0
" # " #
1 0 1 0
β[0] D=ef 1 D=ef 2 2 , 1 D=ef 4 4 ,
− 4 −" 02 12 # 8 " 04 14 #
1 0 0 0
γ[5] D=ef 2 2 , 0 D=ef 2 2 ,
0 1 4 0 0
" 2 2 # " 2 2 #
−
the Pauli matrices:
0 1 0 i 1 0
σ = , σ = − , σ = ;
1 1 0 2 i 0 3 0 1
! ! !
−
2 Introduction
I call event, occured in single point of space-time, as a point event. And
event, formed by the point events, is called as a physics event.
Obviously, the double-slits experiment proves that an elementary particle
does not exist continuously, but such particle is a set of the point events,
bounded by probabilities [1], [2], [3], [4], [5].
These probabilities are expressed by the spinor functions and by opera-
tors of a probability creation and a probability annihilation. These operators
are similar to the field operators of QFT. The motion equations in form of
the Dirac equations with the additional fields are obtained for the spinor
functions. Some of these additional fields behave as the mass members, and
other ones behave as the gauge fields.
A 3+1 vector of a probability current and 3-vectors of a average velocity
and of a local velocity for the probability propagation are defined by these
motion equations.
A set C of complex n n matrices is called as Clifford’s set [10] if the
×
following conditions are fulfilled:
e
if α C and α C then α α +α α = 2δ ;
k r k r r k k,r
∈ ∈
if α α +α α = 2δ for all elements α of set C then α C.
k r r k k,r r k
e e ∈
If n = 4 then the Clifford’s set either contains 3 matrices (Clifford’s
e e
triplet) or contains 5 matrices (Clifford’s pentad).
2
6 Clifford’s pentads exists, only [10]:
one light pentad β:
σ 0 σ 0 σ 0
β[1] D=ef 1 2 , β[2] D=ef 2 2 , β[3] D=ef 3 2 , (1)
0 σ 0 σ 0 σ
" 2 1 # " 2 2 # " 2 3 #
− − −
0 1
γ[0] D=ef 2 2 , (2)
1 0
" 2 2 #
0 1
β[4] D=ef i 2 2 ; (3)
·" 12 02 #
−
three chromatic pentads (see Appendix): red, blue and green ones; and
two taste pentads: sweet and bitter pentads.
At first the motion equations, held the light pentad elements, only, are
considered. Such equations are called as equations for a leptonn motion.
The Dirac equation contains four elements of Clifford’s pentads, only.
Three of these elements (1) accord to three space coordinates, and fourth
element (2) either constitute the mass member or accords to the time coor-
dinate. But Clifford’s pentad holds five elements. Certainly fifth element (3)
of the pentad should be added to the motion equation. That is the Dirac
equationmasspartwillholdtwo members. Moreover, iftwoadditionalquasi-
space coordinates shall be put in accordance to these two Clifford’s pentads
mass elements ((2) and (3)) then the homogeheous Dirac equation will be
obtained. Allfiveelements ofClifford’spentads andallfive spacecoordinates
are held alike in this equation. All local velocities magnitudes equal to unit
(c) in such five- dimensional space.
The overdefinited by the such ways equation of motion is invariant for
a rotation in 2-space of fourth and fifth coordinates. This transformation
defines a field, similar to B-boson field.
Since values of probability are not defined absolutely exactly then the
forming mass members additional fields are expressible by sufficiently fine
stratums. And values of this two spatial coordinates, added for homogene-
ity of mass members, can be delimitted by a large number without loss of
generality. In this case a masses spectrum is got discrete, and for every 3+1
space-time points: either the single mass is put in this point or this point is
empty.
3
A mass is expressed by the 2-root from sum of two quadrates of integer
numbers from two mass members of the motion equation:
m = n2 +s2.
0 0 0
q
But this motion equation is invariant for rotations in 2-space of fourth
and fifth coordinates. That is a mss must be expressed by natural number,
and numbers of the mass members must remain integer in these rotations.
Hence a mass and the making this mass members numbers should be
Pythagorean triplet m ;n ,s [7]. Here m is called as a triplet father.
0 0 0 0
h i
One triplet substitutes another triplet with the same father at rotations.
Let( = α ,α ,...,α )beafinitesetofangles(positiverealnumbers)
0 1 n
{ }
such that
n
α = 2π
k
k=0
X
and
ε = max(α ,α ,...,α )
0 1 n
with ε as a tiny real positive number.
In that case is called as 2π coat with precision ε.
Let m be an natural number such that a Pythagorean triplet m;n,s
h i
exists for every element α of with
k
n = msinα ,
k
s = mcosα .
k
In that case m is called as a father of .
At far areas of the natural number array the 2π coats fathers exist for
any high precision.
I believe that these families of the Pythagorean triplets conform to the
families of elementary particles.
The particles creationandthe particles annihilation operatorsaredenoted
astheFouriertransformationsofcorrespondingoperatorsforprobability, and
antiparticles are denoted by the standard way.
Subsequently here are considered all unitary transformations on the two-
masses functions such that these transformations retain the probability 4-
vectors. The transformations, adequate to electroweak gauge fields, exist
4
among these unitary transformations. These electroweak unitary transfor-
mations are expressed by rotations in 2-space of fourth and fifth coordinates,
too. Particles, similar toneutrino(neutrinno), ariseinthesetransformations.
Neutrinnos prove to be connected essentially with their leptonns.
The motions equations is invariant for these transformations, and fields,
similar to the boson ones, arise as result. The massless field
g
2
W = ∂ W ∂ W i (W W W W )
µ,ν µ ν ν µ µ ν ν µ
− − 2 −
denoted by usual way, but the motion equations for fields W are similar
µ
to Klein-Gordon equation with nonzero mass.
Massless field A and the massive field Z are denoted by standard way by
fields B and W.
The leptonn motion equation is invariant for rotations in 3-space of first
usual spatial coordinates and for Lorentz’s rotations in 3+1 space-time. The
motion equations, maked by the chromatic pentads, are mixed between each
other. That is particles, conformed to pentads of different colors, is unsepa-
rable in space-time (confinement). Two sorts of three colors of such particles
are held in one family. Hence - six. I call these particles as quarrks.
3 Events
Let Ω be a finite set of N‘ point-events A t(ι),x(ι) (N‘ is natural number,
0 ι
ι is natural index number). (cid:16) (cid:17)
Let Ω be set of events for which:
1) Ω Ω;
0
⊆
2) if B Ω and C Ω then (B C) Ω and (B C) Ω;
∈ ∈ ∪ ∈ ∩ ∈
3) if B Ω then B Ω;
∈ ∈
4) if A t(ι),x(ι) Ω then
ι
∈
x(ι) :(cid:16)A t(ι),x(cid:17)(ι) Ω and x(ι) : A t(ι),x(ι) Ω.
ι ι
∀ ∈ ∃ ∈
(cid:16)I call eleme(cid:16)nts of Ω(cid:17)(cid:17)as a physic(cid:16)s events. (cid:16) (cid:17)(cid:17)
Let:
α t,x(ι1),x(ι2),...,x(ιs) D=ef
ι1,ι2,...,ιs
(cid:16) (cid:17)
A t,x(ι1) A t,x(ι2) A t,x(ιs)
D=ef ι1 ∩ ι2 ∩···∩ ιs ∩ .
(cid:16) (cid:16) N(cid:17)‘ (cid:16) x((cid:17)ι) : A (t,x(ι))(cid:16) (cid:17)(cid:17)
∩ι∈/{ι1,ι2,...,ιs} ∀ ι
(cid:16) (cid:16) (cid:17)(cid:17)
5
For example:
α (t) D=ef N‘ x(ι) : A (t,x(ι));
0 ∩ι=1∀ ι
α t,x(1) D=ef
1
D=ef A t,x(1) (cid:16) N‘ (cid:17)x(ι) : A (t,x(ι)) ;
1 ∩ ∩ι6=1∀ ι
(cid:16) (cid:17) (cid:16) (cid:17)
α t,x(1),x(2) D=ef
1,2
(cid:16) (cid:17)
D=ef A t,x(1) A t,x(2) N‘ x(ι) : A (t,x(ι)) .
1 ∩ 2 ∩ ∩ι∈/{1,2}∀ ι
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) (cid:16) (cid:17)
Let P be a probability function, defined on Ω.
Because A t(ι),x(ι) is a point-event then for m = n:
ι
6
(cid:16) (cid:17)
P A t(ι),x(ι) A t(ι),x(ι) = 0.
ι m m ∩ ι n n
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
By idempotence:
P A t(ι),x(ι) A t(ι),x(ι) = P A t(ι),x(ι)
ι ι ι
∩
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
Let H be a Hilbert space.
Let linear operators ψ(ι) x(ι) ( j 1,2,3,4 ) act on elements of H.
j ∈ { }
And these operators have go(cid:16)t the(cid:17)following properties:
1. H contains the element Φ for which:
0
Φ†Φ = 1
0 0
and
ψ(ι)Φ = 0, Φ†ψ(ι)† = 0;
j 0 0 j
2.
ψ(ι) x(ι) ψ(ι) x(ι) = 0
j j
(cid:16) (cid:17) (cid:16) (cid:17)
and
6
ψ(ι)† x(ι) ψ(ι)† x(ι) = 0;
j j
(cid:16) (cid:17) (cid:16) (cid:17)
3.
ψ(ι′)† y(ι′) ,ψ(ι) x(ι) D=ef
j′ j
D=ef ψ(ι′)† yn(ι′) ψ(cid:16)(ι) x(cid:17)(ι) +ψ(cid:16)(ι) (cid:17)xo(ι) ψ(ι′)† y(ι′) (4)
j′ j j j′
(cid:16) =(cid:17)δι′,ιδ(cid:16)y(ι′(cid:17)) x(ι) (cid:16)δj′,j(cid:17) (cid:16) (cid:17)
−
(cid:16) (cid:17)
I call operator ψ(ι)†(x) as a creation operator and ψ(ι)(x) -as an annihi-
lation operator of the event A probability at point x.
ι
4 An one event case
4.1 Hamiltonians
Let N‘ = 1.
Let real function ρ(t,x) be the probability density of the event α (t,x).
1
That is for each domain D (D R3):
⊆
d3x ρ(t,x) D=ef P( x D : α (t,x)).
1
· ∃ ∈
ZD
Complex functions ϕ (t,x) (j 1,2,3,4 ) exist such that
j
∈ { }
4
ρ(t,x) = ϕ∗(t,x)ϕ (t,x). (5)
j j
j=1
X
Let us denote Ψ(t,x) as the following:
4
Ψ(t,x) D=ef ϕ (t,x)ψ†(x)Φ (6)
j j 0
j=1
X
From (4):
4
Ψ†(t,x′)Ψ(t,x) = ϕ∗(t,x′)ϕ (t,x)δ(x′ x).
j j −
j=1
X
Iconsider theevents, fulfilledtothefollowingcondition, only: thereexists
a tiny real positive number h such that if x π (r 1,2,3 ) then
| r| ≥ h ∈ { }
7
ϕ (t,x) = 0.
j
Let (V) be denoted as the following: x (V) if and if, only, x π for
∈ | r| ≤ h
r 1,2,3 . That is:
∈ { }
π π π
d3x = h dx h dx h dx .
1 2 3
π π π
Z(V) Z−h Z−h Z−h
If
x(t) = d3x xρ(t,x)
·
Z
then x(t) is the average coordinate of the event α and:
1
4
x(t) = d3x x ϕ∗(t,x)ϕ (t,x),
· j j
Z j=1
X
and the average velocity of the event α probability is:
1
v(t) = ∂ x(t) =
t
∂ ϕ∗(t,x) ϕ (t,x) (7)
= d3x x 4 t j j
· j=1 +(cid:16)ϕ∗j (t,x)(∂(cid:17)tϕj(t,x)) !
R P
Because
[s]∗ [s]
β = β
j,k k,j
and ϕ (t, ) = 0 then from (7):
j
±∞
3 4 4
v(t) = d3x e ϕ∗β[α]ϕ .
− · α j j,k k
Z α=1 k=1j=1
X XX
If denote:
4 4
j D=ef ϕ∗β[α]ϕ (8)
α − j j,k k
k=1j=1
XX
then
8
j = j e +j e +j e
1 1 2 2 3 3
is the probability current for which:
v(t) = d3x j
·
Z
If
Def
j = ρu (9)
then
v(t) = d3x ρu,
·
Z
that is u is an event α local probability velocity.
1
Let j 1,2,3,4 , k 1,2,3,4
∈ { } ∈ { }
Let
ϕ (t,x) = c ς (t,x)
j j,w,p w,p
w,p
X
def
with ς (t,x) = exp(ih(wt px)) be the Fourier series for ϕ (t,x).
w,p j
−
def
Let ϕ (t,x) = c ς (t,x).
j,w,p j,w,p w,p
Let t,x be any space-time point.
h i
Denote value of function ϕ at this point as
k,w,p
ϕ = A
k,w,p ht,xi k
|
and value of function ∂ ϕ 4 3 β[α]∂ ϕ at this point as
t j,w,p− s=1 α=1 j,s α s,w,p
P P
4 3
[α]
∂ ϕ β ∂ ϕ = C .
t j,w,p− j,s α s,w,p!|ht,xi j
s=1α=1
X X
There A and C are complex numbers. Hence the following equations
k j
set:
4 z A = C ,
k=1 j,k,w,p k j (10)
z∗ = z
( P j,k,w,p − k,j,w,p (cid:12)(cid:12)
(cid:12)
(cid:12)
9 (cid:12)
is a set of 20 algebraic complex equations with 16 complex unknown num-
bers z . This set can be reformulated as the set of 8 linear real equations
k,j,w,p
with 16 real unknown numbers Re(z ) for j < k and Im(z ) for
j,k,w,p j,k,w,p
j k. This set has got solutions by the Kronecker-Capelli theorem. Hence
≤
at every point t,x such complex number z exists.
j,k,w,p ht,xi
h i |
Let κ be a linear operator on the linear space, spanned by functions
w,p
ς (t,x), and
w,p
κw,pςw′,p′ d=ef 0ς,wi′f,pw′, =if ww′=or/wa′n,dpp==p′p;′ .
( (cid:12)
6 6 (cid:12)
(cid:12)
(cid:12)
Let Q be a operator such that in every point t,x : (cid:12)
j,k
h i
def
Q = z κ
j,k ht,xi j,k,w,p ht,xi w,p
| |
Xw,p(cid:16) (cid:17)
ThereforeforeveryϕthereexistoperatorQ suchthattheϕdependence
j,k
upon t is characterized by the following differential equations1:
4
[1] [2] [3]
∂ ϕ = β ∂ +β ∂ +β ∂ +Q ϕ . (11)
t j j,k 1 j,k 2 j,k 3 j,k k
kX=1(cid:16) (cid:17)
and Q∗ = z∗ κ = Q .
j,k w,p j,k,w,p|ht,xi w,p − k,j
In that case if (cid:16) (cid:17)
P
def [1] [2] [3]
H = i β ∂ +β ∂ +β ∂ +Q
j,k j,k 1 j,k 2 j,k 3 j,k
(cid:16) (cid:17)
c
then H is called as the hamiltonian of the moving with equation (11).
If (t,x) is denoted as:
H c
4 4
(t,x) D=ef ψ†(x) H (t,x)ψ (x) (12)
H j j,k k
j=1 k=1
X X
c
then (τ,x) is called as the hamiltonian density.
H
From (6):
i d3x (t,x)Ψ(t,x ) = ∂ Ψ(t,x ).
0 t 0
− ·H
Z
1
This set of equations is similar to the Dirac equation with the mass matrix [11], [12],
[13]. I choose form of this set of equations in order to describe behaviour of ρ (t,x) by
℘
spinors and by the Clifford’s set elements.
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