On unified theoretical models, dynamical torsion and spin
Unified field theoretical models based on generalized affine geometries are proposed, described and analyzed from the physical and mathematical points of view. The relation between torsion and spin of the models is explicitly shown and some of their physical consequences, as the modification of the...
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irk-123456789-1070132016-10-12T03:02:13Z On unified theoretical models, dynamical torsion and spin Cirilo-Lombardo, Diego Julio Section C. Theory of Elementary Particles. Cosmology Unified field theoretical models based on generalized affine geometries are proposed, described and analyzed from the physical and mathematical points of view. The relation between torsion and spin of the models is explicitly shown and some of their physical consequences, as the modification of the anomalous momentum of the elementary particles, are discussed. Модели единой теории поля, основанные на обобщенных аффинных геометриях, предложены, описаны и проанализированы с физико-математической точки зрения. Явно показана связь между кручением и спином в этих моделях. Обсуждаются некоторые их физические следствия, например, модификация аномального импульса элементарных частиц. Моделі єдиної теорії поля, засновані на узагальнених афінних геометріях, запропоновані, описані й проаналізовані з фізико-математичної точки зору. Явно показаний зв'язок між крутінням і спином у цих моделях. Обговорюються деякі їхні фізичні наслідки, наприклад, модифікація аномального імпульсу елементарних часток. 2012 Article On unified theoretical models, dynamical torsion and spin / Diego Julio Cirilo-Lombardo // Вопросы атомной науки и техники. — 2012. — № 1. — С. 139-142. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk http://dspace.nbuv.gov.ua/handle/123456789/107013 en Вопросы атомной науки и техники International Institute of Physics, Capim Macio, Brazil |
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Section C. Theory of Elementary Particles. Cosmology Section C. Theory of Elementary Particles. Cosmology |
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Section C. Theory of Elementary Particles. Cosmology Section C. Theory of Elementary Particles. Cosmology Cirilo-Lombardo, Diego Julio On unified theoretical models, dynamical torsion and spin Вопросы атомной науки и техники |
| description |
Unified field theoretical models based on generalized affine geometries are proposed, described and analyzed from the physical and mathematical points of view. The relation between torsion and spin of the models is explicitly shown and some of their physical consequences, as the modification of the anomalous momentum of the elementary particles, are discussed. |
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Article |
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Cirilo-Lombardo, Diego Julio |
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Cirilo-Lombardo, Diego Julio |
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Cirilo-Lombardo, Diego Julio |
| title |
On unified theoretical models, dynamical torsion and spin |
| title_short |
On unified theoretical models, dynamical torsion and spin |
| title_full |
On unified theoretical models, dynamical torsion and spin |
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On unified theoretical models, dynamical torsion and spin |
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On unified theoretical models, dynamical torsion and spin |
| title_sort |
on unified theoretical models, dynamical torsion and spin |
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International Institute of Physics, Capim Macio, Brazil |
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2012 |
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Section C. Theory of Elementary Particles. Cosmology |
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http://dspace.nbuv.gov.ua/handle/123456789/107013 |
| citation_txt |
On unified theoretical models, dynamical torsion and spin / Diego Julio Cirilo-Lombardo // Вопросы атомной науки и техники. — 2012. — № 1. — С. 139-142. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT cirilolombardodiegojulio onunifiedtheoreticalmodelsdynamicaltorsionandspin |
| first_indexed |
2025-07-07T19:22:47Z |
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2025-07-07T19:22:47Z |
| _version_ |
1837017251502358528 |
| fulltext |
Section C. Theory of Elementary Particles. Cosmology
ON UNIFIED THEORETICAL MODELS, DYNAMICAL
TORSION AND SPIN
Diego Julio Cirilo-Lombardo 1,2∗
1International Institute of Physics, Capim Macio, 59078-400, Natal-RN, Brazil
2Bogoliubov Laboratory of Theoretical Physics, Joint Institute of Nuclear Research, 141980, Dubna, Russia
(Received October 13, 2011)
Unified field theoretical models based on generalized affine geometries are proposed, described and analyzed from the
physical and mathematical points of view. The relation between torsion and spin of the models is explicitly shown
and some of their physical consequences, as the modification of the anomalous momentum of the elementary particles,
are discussed.
PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk
1. THE THEORY
In this report the geometrical analysis of a new type
of Unified Field Theoretical (UFT) models intro-
duced previously in [1, 2] by the authors is presented.
These new unified theoretical models are character-
ized by an underlying hypercomplex structure, zero
non-metricity and the geometrical action is deter-
mined fundamentally by the curvature arising thanks
to the breaking of symmetry of a group manifold
in higher dimensions. This mechanism of Cartan-
MacDowell-Mansouri type, permits us to construct
geometrical actions of determinantal type. Such
mechanism also leads to a non topological physical
Lagrangian due to the splitting of a reductive geom-
etry. The starting point of the type of theory is a
space-time basis Manifold equipped with a metric,
e.g.
M, gμν ≡ eμ · eν , (1)
where for each point p ∈ M ∃ a local space affine A.
The connection over A Γ̃ defines a generalized affine
connection Γ on M specified by (∇, K) where K is
an invertible (1, 1) tensor over M . We will demand
that the connection is compatible and rectilinear
∇K = KT, ∇g = 0, (2)
where T is the torsion, and g (the space-time met-
ric used to raise and to low indices and determines
the geodesics) is preserved under parallel transport.
This generalized compatibility condition ensures that
the affine generalized connection Γ maps autoparal-
lels of Γ on M in straight lines over the affine space
A (locally). The first equation in (2) is equal to the
condition determining the connection in terms of the
fundamental field in the UFT non-symmetric. For in-
stance, K can be identified with the fundamental ten-
sor in the non-symmetric fundamental theory. This
fact gives us the possibility to restrict the connection
to an (anti) Hermitian theory.
The second important point is the following: let
us consider [1] the extended curvature
Rab
μν = Rab
μν + Σab
μν (3)
with
Rab
μν = ∂μωab
ν − ∂νωab
μ + ωac
μ ω b
νc − ωac
ν ω b
μc , (4)
Σab
μν = − (ea
μeb
ν − ea
νeb
μ
)
.
We assume here ωab
ν a SO (d − 1, 1) connection and
ea
μ is a vierbein field. The eqs. (3) and (4) can be ob-
tained, for example, using the formulation that was
pioneering introduced in seminal works by E. Cartan
long time ago [1]. It is well known that in such a
formalism the gravitational field is represented as a
connection of one form associated with some group
which contains the Lorentz group as subgroup. The
typical example is provided by the SO (d, 1) de Sit-
ter gauge theory of gravity. In this specific case, the
SO (d, 1) the gravitational gauge field ωAB
μ = −ωBA
μ
is broken into the SO (d − 1, 1) connection ωab
μ and
the ωda
μ = ea
μ vierbein field, with the dimension d
fixed. Then, the de Sitter (anti-de Sitter) curvature
RAB
μν = ∂μωAB
ν − ∂νωAB
μ + ωAC
μ ω B
νC −ωAC
ν ω B
μC (5)
splits in the curvature (3). At this point, our goal
is to enlarge the group structure of the space-time
Manifold of such manner that the curvature (5), ob-
viously after the breaking of symmetry, permits us to
∗E-mail address: diego@theor.jinr.ru
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 139-142.
139
define the geometrical Lagrangian of the theory as
Lg =
√
detRa
μRaν =
√
detGμν . (6)
Then the action is
S =
b2
4π
∫ √−gdx4
R,
R ≡
√
γ4 − γ2
2
G
2 − γ
3
G
3
+
1
8
(
G
2
)2
− 1
4
G
4
,
Gμν ≡ λ2
(
gμν + fa
μfaν
)
+ (7)
+2λ
(
R(μν) + fa
μR[aν]
)
+ Ra
μRaν ,
Gν
ν ≡ λ2 (d + fμνfμν) + (8)
+2λ (RS + RA) +
(
R2
S + R2
A
)
with (the upper bar on the tensorial quantities in-
dicates traceless condition, b2 in principle constant
homogenizing the units)
RS ≡ gμνR(μν), RA ≡ fμνR[μν],
G
ν
ρG
ρ
ν ≡ G
2
,
γ ≡ Gν
ν
d , Gμν ≡ Gμν − gμν
4 Gν
ν ,
G
ν
λG
λ
ρG
ρ
ν ≡ G
3
,
(
G
ν
ρG
ρ
ν
)2
≡
(
G
2
)2
,
G
ν
μG
μ
λG
λ
ρG
ρ
ν ≡ G
4
,
(9)
where the variation was made with respect to the as-
sumed for fρτ (electromagnetic) potential aτ as fol-
lows:
R(μν) =
◦
Rμν − T α
μρ T ρ
αν = −λgμν , (10)
R[μν] = ∇αT α
μν = −λfμν , (11)
δ
√
G
δaτ
= ∇ρ
(
∂
√
G
∂fρτ
)
≡ ∇ρF
ρτ =
= ∇ρ
[
λ2Nμν
(
δσ
μ fρ
ν + δσ
ν fρ
μ
)
2R
]
= 0. (12)
From this set, the link between the torsion T and f
will be determined. Notice that total antisymmetry
of the torsion is assumed due to the physical conse-
quences that it property for T carry up. Also, f is
not a priori potential for the torsion T and Nμν is
a tensor coming from the variational procedure and
defined in [1, 2]. Our goal is to take advantage of the
geometrical and topological properties of this theory
in order to determine the minimal group structure
of the resultant space-time Manifold able to support
a fermionic structure. From this fact, the relation
between antisymmetric torsion and Dirac structure
of the space-time is determined and the existence of
an important contribution of the torsion to the gyro-
magnetic factor of the fermions as follows [1, 2, 5]:[(
P̂μ − eÂμ
)2
− m2 − 1
2
eσμνFμν
]
uλ+ (13)
+
1
2
σμνRλ
ρ[μν]u
ρ − 1
2
eσμν
(
ÂμP̂ν − Âν P̂μ
)
uλ = 0.
It is interesting to see that
i) the above formula is absolutely general for the
type of geometrical Lagrangians involved containing
the generalized Ricci tensor inside,
ii) for instance, the variation of the action will
carry the symmetric contraction of components of
the torsion tensor, then the arising of terms as
hμ hν (hν is an axial vector dual of the torsion) .
iii) the only thing that changes is the mass (see
[2]) and the explicit form of the tensors involved as
Rλ
ρ[μν], Fμν , etc. is without variation of the Dirac gen-
eral structure of the equation under consideration,
iv) eq. (13) differs from the one obtained by Lan-
dau and Lifshitz by the appearance of last two terms:
the term involving the curvature tensor is due to the
spin interaction with the gravitational field (due to
the torsion term in Rλ
ρ[μν]) and the last term is the
spin interaction with the the electromagnetic and the
mechanical momenta,
v) expression (13) is valid for another vector vλ,
then is valid for a bispinor of the form Ψ = u + iv,
vi) the meaning for a quantum measurement of
the space-time curvature is mainly due to the term
in (13) involving explicitly the curvature tensor.
The important point here is that the spin-gravity
interaction term is so easily derived as the spinors
are represented as space-time vectors whose covariant
derivatives are defined in terms of the G-(affine) con-
nection. In their original form the Dirac equations
would have, in curved space-time, their momentum
operators replaced by covariant derivatives in terms
of “spin-connection” whose relation is not immedi-
ately apparent.
2. DIRAC STRUCTURE,
ELECTROMAGNETIC FIELD AND
ANOMALOUS GYROMAGNETIC
FACTOR
The interesting point now is based on the observation
that if we introduce expression (11) in (13) then:
[(
P̂μ − eÂμ
)2
− m2 − 1
2
eσμνFμν
]
uλ−
−λ
d
1
2
γ5σμνf[μν]u
λ − 1
2
eσμν
(
ÂμP̂ν − Âν P̂μ
)
uλ = 0,
(14)
[(
P̂μ − eÂμ
)2
− m2 − 1
2
σμν
(
eFμν + γ5 λ
d
fμν
)]
uλ−
(15)
−e
2
σμν
(
ÂμP̂ν − Âν P̂μ
)
uλ = 0.
We can see clearly that if Âμ = jaμ (with j being an
arbitrary constant), Fμν = jfμν the last expression
140
takes the suggestive form:[(
P̂μ − eÂμ
)2
− m2 − 1
2
(
ej + γ5 λ
d
)
σμνfμν
]
uλ−
(16)
−e
2
σμν
(
ÂμP̂ν − Âν P̂μ
)
uλ = 0
with the result that the term corresponding to the gy-
romagnetic factor has been modified to
(
j + γ5 λ
ed
)
/2.
Notice that in an Unified Theory, with the character-
istics introduced here, is reasonable the identification
introduced in the previous step (F ↔ f) in order
that the fields arise from the same geometrical struc-
ture (as is possible due to the U(2, C) fundamental
structure of the space-time, main ingredient of the
arising of matter from the geometrical structure of
the Manifold).
The concrete implications about this important
contribution of the torsion to the gyromagnetic fac-
tor will be given elsewhere with great detail on the
dynamical property of the torsion field. We remark
only the following:
i) there exists an important contribution of the
torsion to the gyromagnetic factor that can have im-
plicancies to the trouble of the anomalous momentum
of fermionic particles,
ii) this contribution appears (taking the second
equality of expression (11)), as a modification on the
vertex of interaction, almost from the effective point
of view;
iii) it is quite evident that this contribution will
probably justify the little appearance of the torsion at
great scale, because we can bound the torsion due to
the other well known contributions to the anomalous
momenta of the elementary particles (QED, weak,
hadronic contribution, etc),
iv) the form of the coupling spin-geometric struc-
ture coming from the first principles, as the Dirac
equation, not prescriptions,
v) then, from iii) how the covariant derivative
works in presence of torsion is totally determined by
the G structure of the space-time,
vi) the Dirac equation (14) (where the second part
of the equivalence (11) was introduced coming from
the equation of motion), said us that the vertex was
modified without a dynamical function of propaga-
tion. Then, other form to see the problem treated in
this paragraph is to introduce the propagator for the
torsion corresponding to the first part of the equiva-
lence (11) written as follows:
[(
P̂μ − eÂμ
)2
− m2 − 1
2
eσμνFμν
]
uλ−
− 1
2d
γ5σμν∇μhνuλ− (17)
−1
2
eσμν
(
ÂμP̂ν − Âν P̂μ
)
uλ = 0,
where the relation between the totally antisymmetric
torsion tensor and its dual hμ has been used. Ana-
lyzing the first two terms of expression (17) we can
guess the explicit form of the vertex and the effective
Dirac Lagrangian where the new modification comes
from:
Lfeff = Ψ
(
iγμ∂μ − m + γ5γμhμ + eγμAμ
)
Ψ.
Then, the formal Feynman propagator that we are
looking for is evidently such:
S (p) =
(
iγμpμγμ − m + γ5γμhμ + eγμAμ
) · Δ
Λ
,
where we have defined
Δ ≡ (p2 − m2 + h2 + e2A2
)
+
+ 2
[
γ5 (p · h + mγμhμ) + e (p · A + emγμAμ)
]
,
Λ ≡ (p2 − m2 + h2 + e2A2 + iε
)2 −
− 4
[(
p · (γ5h + eA
))2 − m2
(
h2 + e2A2
)]
.
This important possibility including several
processes of interest in modern physics (anomalous
momentum of the muon, velocity of the neutrino,
neutrino spin flip, etc) will be studied soon else-
where [5].
Acknowledgements
I am very grateful to the organizers of this extraor-
dinary meeting in memory of Alexander I. Akhiezer:
outstanding person from the scientific and the hu-
man points of view. Many thanks are given to Pro-
fessors Yu. P. Stepanovsky and A. Dorokhov for my
scientific formation and to my friend and collabora-
tor S. N. Shulga for their interest and discussions; and
particularly to Professor Yu Xin who introduced me
into the subject of the Unified Theories based only
on geometrical concepts and the Mach principle. This
work was partially supported by CNPQ-MEC Brazil-
ian funds.
References
1. D.J. Cirilo-Lombardo // Int. J. Theor. Phys.
2010, v. 49, p. 1288-1301 (and references
therein).
2. D.J. Cirilo-Lombardo // Int. J. Theor. Phys.
DOI 10.1007/s10773-011-0678-1 (and references
therein).
3. D.J. Cirilo-Lombardo // Class. Quantum
Grav. 2005, v. 22, p. 4987-5004 (and references
therein).
4. P. Singh // Class. Quantum Grav. 1990, v. 7,
p. 2125.
5. D.J. Cirilo-Lombardo and S.N. Shulga, in prepa-
ration.
141
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