Renormalization the quantum field model of particle interaction
The model simulates the interaction of abstract entities distinguished in a physical experiment and denoted as particles. Empirical data results in the non-hermitian anti-symmetric matrix of particle relationship. The real and imaginary parts of the matrix correspond to symmetric and asymmetric coup...
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Odessa National Academy of Telecommunication n.a. A.S.Popov
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irk-123456789-1069822016-10-11T03:02:07Z Renormalization the quantum field model of particle interaction Tikhonov, V.I. Tykhonov, A.V. Section A. Quantum Field Theory The model simulates the interaction of abstract entities distinguished in a physical experiment and denoted as particles. Empirical data results in the non-hermitian anti-symmetric matrix of particle relationship. The real and imaginary parts of the matrix correspond to symmetric and asymmetric coupling of particles. The relationship matrix evolves to multiplication of pure defined hermitian metric tensor and curvature vector. The real spectrum of metric tensor extended into the complex space with invariant spectrum power results in renormalized non-singular quantum field model of particle interaction. Рассматривается модель взаимодействия абстрактных сущностей, различимых в физическом эксперименте и названных частицами. Эмпирические данные представлены в форме неэрмитовой антисимметрической матрицы взаимодействия частиц. Вещественные и мнимые элементы матрицы соответствуют симметрической и асимметрической составляющим взаимодействующей пары частиц. Матрица взаимодействия приведена к произведению слабо обусловленного эрмитового метрического тензора на вектор кривизны. Вещественный спектр метрического тензора расширен в комплексную область с условием инвариантности мощности спектра, в результате чего получена ренормализованная несингулярная квантово-полевая модель взаимодействия частиц. Розглянуто модель взаємодії абстрактних сутностей, помітних у фізичному експерименті і названих частками. Емпіричні дані представлені у вигляді неермітової антисиметричної матриці взаємодії часток. Дійсні та уявні елементи матриці відповідають симетричній та асиметричній складовим взаємодіючої пари часток. Матриця взаємодії приведена к добутку слабо обумовленого метричного тензора на вектор кривизни. Дійсний спектр метричного тензора розширено у комплексну область за умови інваріантності потужності спектру, в результаті чого отримано ренормалізовану несингулярну квантово-польову модель взаємодії часток. 2012 Article Renormalization the quantum field model of particle interaction / V.I. Tikhonov, A.V. Tykhonov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 55-58. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 11.10.Gh, 11.10.Nx http://dspace.nbuv.gov.ua/handle/123456789/106982 en Вопросы атомной науки и техники Odessa National Academy of Telecommunication n.a. A.S.Popov |
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Section A. Quantum Field Theory Section A. Quantum Field Theory |
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Section A. Quantum Field Theory Section A. Quantum Field Theory Tikhonov, V.I. Tykhonov, A.V. Renormalization the quantum field model of particle interaction Вопросы атомной науки и техники |
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The model simulates the interaction of abstract entities distinguished in a physical experiment and denoted as particles. Empirical data results in the non-hermitian anti-symmetric matrix of particle relationship. The real and imaginary parts of the matrix correspond to symmetric and asymmetric coupling of particles. The relationship matrix evolves to multiplication of pure defined hermitian metric tensor and curvature vector. The real spectrum of metric tensor extended into the complex space with invariant spectrum power results in renormalized non-singular quantum field model of particle interaction. |
| format |
Article |
| author |
Tikhonov, V.I. Tykhonov, A.V. |
| author_facet |
Tikhonov, V.I. Tykhonov, A.V. |
| author_sort |
Tikhonov, V.I. |
| title |
Renormalization the quantum field model of particle interaction |
| title_short |
Renormalization the quantum field model of particle interaction |
| title_full |
Renormalization the quantum field model of particle interaction |
| title_fullStr |
Renormalization the quantum field model of particle interaction |
| title_full_unstemmed |
Renormalization the quantum field model of particle interaction |
| title_sort |
renormalization the quantum field model of particle interaction |
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Odessa National Academy of Telecommunication n.a. A.S.Popov |
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2012 |
| topic_facet |
Section A. Quantum Field Theory |
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http://dspace.nbuv.gov.ua/handle/123456789/106982 |
| citation_txt |
Renormalization the quantum field model of particle interaction / V.I. Tikhonov, A.V. Tykhonov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 55-58. — Бібліогр.: 8 назв. — англ. |
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Вопросы атомной науки и техники |
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| fulltext |
RENORMALIZATION THE QUANTUM FIELD MODEL OF
PARTICLE INTERACTION
V.I. Tikhonov 1∗and A.V. Tykhonov 2
1Odessa National Academy of Telecommunication n.a. A.S.Popov, 65029, Odessa, Ukraine
2Odessa National Polytechnic University, 65044, Odessa, Ukraine
(Received October 31, 2011)
The model simulates the interaction of abstract entities distinguished in a physical experiment and denoted as
particles. Empirical data results in the non-hermitian anti-symmetric matrix of particle relationship. The real and
imaginary parts of the matrix correspond to symmetric and asymmetric coupling of particles. The relationship matrix
evolves to multiplication of pure defined hermitian metric tensor and curvature vector. The real spectrum of metric
tensor extended into the complex space with invariant spectrum power results in renormalized non-singular quantum
field model of particle interaction.
PACS: 11.10.Gh, 11.10.Nx
1. INTRODUCTION
The classic field theory (CFT) studies the real and
complex functions (scalar fields, vector fields, tensor
fields) presented in a priory pre-defined linear spaces
[1]. This approach is truly relevant to the certain
universe stratum available for the human perception.
The enhanced experiments extend the scope of analy-
sis into macro- and microspheres, though the cogni-
tive horizon remains infinitesimal part of the Uni-
verse. The macro-world violates naive insights of
plane Euclidian space. The micro-world drastically
changes the understanding of what is the space even-
tually; it appears multidimensional, non-orthonormal
and asymmetric.
In contrast to CFT, the quantum field theory
(QFT) studies the models of abstract spaces simu-
lating the physical experiments [2, 3]. In this work,
authors try a holistic approach to the modeling of
arbitrary physical entities emerged in stochastic ex-
periment (called particles). The model assimilates
the space curvature and asymmetry properties of the
empirical data.
2. THE RELATIONSHIP MATRIX OF
EMPIRICAL DATA
Let X be an arbitrary open set of N interacting el-
ements (x1, ..., xN ) ∈ X , distinguished in the physi-
cal experiment within a certain time interval. These
elements we denote as particles. Any xn ∈ X con-
sider open, e.g. it may interact with particles of X
and out of X (latent interaction). The interaction
of particles may have the asymmetry. Let Q be the
non-hermitian complex anti-symmetric matrix of par-
ticle’s relationship. We define a binary operation ⊗
to pick hermitian matrix H out of the non-hermitian
matrix Q:
Q = H ⊗ ψ,
ψ = {ψn},
(1)
where ψn = eiθ(n) is the vector of curvature for ma-
trix Q. The operator ⊗ multiplies any diagonal entry
Q(n, n) by the ψ(n). The main diagonal of Q with
complex numbers evolves to real numbers and non-
hermitian matrix Q results in hermitian matrix H :
H = Q⊗ ψ∗ = Q⊗ e−iθ(n), (2)
where “*” is the complex conjugation symbol. We
will utilize the irreducible representation of hermitian
matrix H in eigenvector basis Z [4]:
H = Z∗ · Λ · Z,
Λ = I ⊗ λ,
(3)
where Λ is the diagonal matrix of eigenvalues λn,
I is the unit diagonal matrix, and λ = {λn} is the
vector of eigenvalues λn (also called spectrum of H).
If some eigenvalues λn are not positive, the matrix
H is poorly defined; it plagues singularity and needs
renormalization [5, 6].
3. RENORMALIZATION OF METRIC
TENSOR
To renormalize matrix H we make three steps.
Step 1. Suppose eigenvectors zn and eigenvalues
λn are ordered by λn decrement (if not they are to
be re-indexed):
λ1 ≥ λ2 ≥ ...λn ≥ ...λN . (4)
∗Corresponding author E-mail address: victor.tykhonov@onat.edu.ua
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 55-58.
55
Step 2. Let α = |λn|max be the maximal module
of λn, n = 1, 2, ..., N . Evolve spectrum λn = {λn} to
the spectrum β = {βn} where |βn|max = 1:
βn = λn/α. (5)
Now, the spectrum β and matrix Q become dimen-
sionless.
Step 3. Consider spectrum β = {βn} being the
real part of the complex spectrum ρ = {ρn}, where
ρn = exp(i · ϕn):
βn = Re(ρn) = Re[exp(i · ϕn)] = cos(ϕn). (6)
Let ΩR = I ⊗ β and CR = f(H) = Z∗ · ΩR · Z being
the real part of an abstract complex C = f(H); f
symbolizes the functional dependence on the argu-
ment. The imaginary part of the complex spectrum
is a vector γ = {γn}, where
γn = Im(ρn) = Im[exp(i · ϕn)] = sin(ϕn). (7)
Let ΩI = I ⊗ γ. The imaginary part of C is
CI = Z∗ · ΩI · Z. Thus, we have
C = CR + i · CI = Z∗ · ΩR · Z + Z∗ · ΩI · Z
= (Z∗ · ΩR + Z∗ · ΩI) · Z = Z∗ · (ΩR + ΩI) · Z
= Z∗ · Ω · Z;
(8)
C = Z∗ · Ω · Z,
Ω = ΩR + i · ΩI = I ⊗ (β + i · γ)
= I ⊗ exp(i · ϕ) = I ⊗ φ,
ϕ = {ϕn}, φ = exp(ϕ).
(9)
The matrix C = f(H) we denote as a renormalized
complex metric tensor (RMT) of the particle interac-
tion for the empirical relationship matrixQ = H ⊗ ψ.
Matrix C is non-singular as the inverse matrix C−1
always exists: C−1 = C∗. We have:
C · C−1 = (Z∗ · Ω · Z) · (Z∗ · Ω · Z)∗
= (Z∗ · Ω · Z) · (Z∗ · Ω∗ · Z)
= (Z∗ · Ω · (Z · Z∗) · Ω∗ · Z)
= Z∗ · (Ω · I · Ω∗) · Z = Z∗ · I · Z = I.
(10)
4. RENORMALIZATION THE QUANTUM
FIELD MODEL
We will define the multiplication
G = f(Q) = C ⊗ ψ = (Z · I ⊗ φ · Z∗) ⊗ ψ. (11)
as renormalized complex presentation of the empir-
ical data in form of relationship matrix Q. Three
objects present G = f(Q):
G = f(Q) = {Z, φ, ψ}, (12)
Z is the 2-valence one-covariant and one contra-
variant unitary matrix operator, φ is the 1-valence co-
variant complex phase vector, and ψ is the 1-valence
covariant complex curvature vector.
Therefore, the multiplication (9) is 4-valence 3-
co-variant and 1-contra-variant tensor of metrics and
curvature. We denote the system G = {Z, φ, ψ},
in respect to the multiplication (9), as co-variant
renormalized quantum field model (RQM) of parti-
cle interaction for the empirical relationship matrix
Q = H ⊗ ψ. The system G = {Z∗, φ∗, ψ∗} we de-
note as contra-variant RQM-model towards G. Ob-
viously, G is 4-valence 1-co-variant and 3-contra-
variant tensor.
Consider the obvious properties of the operator ⊗
spoken above:
ψ ⊗ ψ∗ = ψ∗ ⊗ ψ = e = {e1, e2, ..., en, ..., eN}
= {1, 1, ..., 1};
C ⊗ ψ ≡ ψ ⊗ C;
C ⊗ e ≡ C;
ψ ⊗ I ⊗ ψ∗ = ψ∗ ⊗ I ⊗ ψ = I.
(13)
The invariant tensor of RQM-model we define as
tripled convolution
G ·G = (C ⊗ ψ) · (C∗ ⊗ ψ∗)
= ψ ⊗ (C · C∗) ⊗ ψ∗ = ψ ⊗ I ⊗ ψ∗ = I.
(14)
The matrix I is 2-valence one-co-variant and one-
contra-variant tensor (neutral operator). It is clear
that tensor multiplication (13) is commutative:
G×G ≡ G×G = I. (15)
5. THE QUANTUM FIELD MODEL
ANALYSIS
We will discuss some properties and special cases
of the renormalized quantum field model (RQM) of
particle interaction.
Property 1. The RQM has η = N · (N + 1) − 1
degrees of freedom.
In fact, any square complex matrix Q with com-
plex main diagonal and anti-symmetric entries
Q(n,m) = Q(m,n) has N ×N +N = N · (N + 1)
independent entries, e.g. freedom degrees. The renor-
malization procedure given above provides: 1 free-
dom degree limitation (|βn|max = |λn|max/α = 1); N
freedom degrees extension (ΩI ) and N freedom de-
grees limitation (ΩI · ΩR = I ⊗ exp(i · ϕ) = I ⊗ φ).
Eventually η = N · (N + 1) − 1. Consider nor-
malization factor α, the particular renormalized
quantum field model (PRM) Gα = α ·G results in
η = N · (N + 1) freedom degrees.
Property 2. In the 4-dimensional space (N = 4)
the PRM-model has 20 freedom degrees.
The same number μ = N2 · (N2 − 1)/12 = 20 of free-
dom degrees results in the 4-valence 3-covariant and
1-contra-variant Riemann curvature tensor [2]. That
means that the 4-dimensional PRM model is isomor-
phic to the 4-dimensional space presented by the Rie-
mann curvature tensor. These two presentations bi-
jectively map each other.
56
Case 1: ψ = e (no curvature). The RQM evolves
into the normalized spectral phase C-filter in Z-basis:
G→ C = Z∗ · Ω · Z = Z · (I ⊗ φ) · Z∗. (16)
The filtering procedure for any sample
y = {y1, y2, ..., yn} is yC = C · y. Manipulating the
factor α = |λn|max we obtain the general spectral
phase filter
Cα = α · (F ∗ · Ω · F ). (17)
The α > 1 amplifies the y-output; α < 1 suppresses
the y-output.
Property 3. The C-filter (16) is spinor [3].
In special case φ = {exp(i · π ·m)n} all the diagonal
entries of matrix Ω = I ⊗ φ turn into the alternating
units ±1. We denote the correspondent vector s as
signature and matrix S as signature filter (S-filter):
s = {sn} = {exp(i · π ·m)n},
S = I ⊗ s.
(18)
Case 2: ψ = e (no curvature); α = 1 ;
Z →MS = [x,y, z, t] are real Euclidian orts
of the (3+1)-dimensional orthonormal space;
s → sM = [−1 , 1 , 1 , 1 ]. The sM is the signature
of the Minkovski space MS [7]:
C →M∗
S · sM ·MS = I ⊗ sM. (19)
Case 3: ψ = e; α = 1; Z → F is the Fourier ba-
sis. Now, the C-filter (16) turns into the conventional
digital phase Fourier filter Φ [8]:
C → Φ = F ∗ · Ω · F. (20)
The Fourier filtering for any sample y = {y1, y2, ..., yn}
is yC = Φ · y.
Here F · y = yF is direct Fourier transform; yF
is Fourier image; the Ω · yF = yFΩ is spectral cor-
rection of Fourier image yF ; the F ∗ · yFΩ = yC is
reverse Fourier transform for the corrected Fourier
image yFΩ .
Case 4: ψ = e; α = 1; Z → F ; Ω → ΩR = Re(Ω).
Now, the Φ-filter (20) turns into the passive spec-
tral density Fourier filter: Φ → ΦR = (F · ΩR · F ∗).
Manipulating the factor α = |λn|max we obtain the
general spectral density Fourier filter
ΦRα = α · (F · ΩR · F ∗). (21)
Property 4. Take equation (8), next derive
G = C ⊗ ψ = (CR + i · CI) ⊗ ψ.
Therefore, we have
G = QM + i QF,
QM = CR ⊗ ψ,
QF = CI ⊗ ψ,
(22)
where QM denotes abstract “Quantum Matter”, and
QF is abstract “Quantum Field”. The composition
G = QM + i QF has invariant total abstract power
PG obtained from (15):
PG = f(G) = Tr(G×G) = I = Tr(I) = N, (23)
where Tr is the matrix trace symbol.
6. CONCLUSIONS
The work studies the interaction model for an ar-
bitrary set of abstract physical entities called par-
ticles. The simulation model assumes the statisti-
cal experiment output data in form of non-hermitian
anti-symmetric complex matrix with non-real main
diagonal entries. To dissimulate the complexity of
the matrix diagonal a special binary operation es-
tablished in the work. Due to this operation, the
non-hermitian data matrix evolves into the composi-
tion of the two components: hermitian metrical ma-
trix of linear complex space and the curvature vector.
Hence, these two components are treated individu-
ally.
To override the inherent singularity of hermitian
matrix, the new axiom is applied: the spectrum of a
unitary operator metrics is a real part of the complex
spectrum defined for extended non-unitary operator
metrics. The extended complex spectrum considered
has constant power and is invariant to the phase ro-
tations (non-singular). From that point of view, the
concept of quantum matter and quantum field com-
ponent’s composition is originated in the work.
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57
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