Spin, confinement, localization and supersymmetry
New mechanism of localization of fields in a particular sector of a special supermanifold is proposed and described, based on previous works of the authors. Different aspects of this new mechanism are discussed as an alternative to confinement in elementary particle physics and to the Randall-Sundru...
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International Institute of Physics, Capim Macio, Brazil
2012
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| Цитувати: | Spin, confinement, localization and supersymmetry / Diego Julio Cirilo-Lombardo // Вопросы атомной науки и техники. — 2012. — № 1. — С. 39-42. — Бібліогр.: 12 назв. — англ. |
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Cirilo-Lombardo, Diego Julio 2016-10-10T15:46:20Z 2016-10-10T15:46:20Z 2012 Spin, confinement, localization and supersymmetry / Diego Julio Cirilo-Lombardo // Вопросы атомной науки и техники. — 2012. — № 1. — С. 39-42. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk https://nasplib.isofts.kiev.ua/handle/123456789/106978 New mechanism of localization of fields in a particular sector of a special supermanifold is proposed and described, based on previous works of the authors. Different aspects of this new mechanism are discussed as an alternative to confinement in elementary particle physics and to the Randall-Sundrum scenarios in warped manifolds. Предложен и описан новый возможный механизм локализации поля в определенном секторе специального супермногообразия. Различные аспекты этого нового механизма рассматриваются в качестве альтернативы конфайнменту в физике элементарных частиц и сценариям Рэндалл-Сандрама в искривленных многообразиях. Запропоновано та описано новий можливий механізм локалізації поля у певному секторі спеціальної супермногостатності. Різні аспекти цього нового механізму розглядаються в якості альтернативи конфайнменту в фізиці елементарних частинок і сценаріям Рендалл-Сандрама у викривлених многостатностях. This work is in memory of Alexander I. Akhiezer: outstanding person from the scientific and the human point of view and of Professor Vyacheslav Soroka pioneering in the development of the supergeometry and supergravity. Many thanks are given, however, to Professors Yu. P. Stepanovsky and A. Dorokhov for my scientific formation and to my friend and collaborator S. N. Shulga for their interest and discussions; This work was partially supported by CNPQ-PNPD Brazilian funds. en International Institute of Physics, Capim Macio, Brazil Вопросы атомной науки и техники Section A. Quantum Field Theory Spin, confinement, localization and supersymmetry Спин‚ конфайнмент‚ локализация и суперсимметрия Спін‚ конфайнмент‚ локалізація і суперсиметрія Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Spin, confinement, localization and supersymmetry |
| spellingShingle |
Spin, confinement, localization and supersymmetry Cirilo-Lombardo, Diego Julio Section A. Quantum Field Theory |
| title_short |
Spin, confinement, localization and supersymmetry |
| title_full |
Spin, confinement, localization and supersymmetry |
| title_fullStr |
Spin, confinement, localization and supersymmetry |
| title_full_unstemmed |
Spin, confinement, localization and supersymmetry |
| title_sort |
spin, confinement, localization and supersymmetry |
| author |
Cirilo-Lombardo, Diego Julio |
| author_facet |
Cirilo-Lombardo, Diego Julio |
| topic |
Section A. Quantum Field Theory |
| topic_facet |
Section A. Quantum Field Theory |
| publishDate |
2012 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
International Institute of Physics, Capim Macio, Brazil |
| format |
Article |
| title_alt |
Спин‚ конфайнмент‚ локализация и суперсимметрия Спін‚ конфайнмент‚ локалізація і суперсиметрія |
| description |
New mechanism of localization of fields in a particular sector of a special supermanifold is proposed and described, based on previous works of the authors. Different aspects of this new mechanism are discussed as an alternative to confinement in elementary particle physics and to the Randall-Sundrum scenarios in warped manifolds.
Предложен и описан новый возможный механизм локализации поля в определенном секторе специального супермногообразия. Различные аспекты этого нового механизма рассматриваются в качестве альтернативы конфайнменту в физике элементарных частиц и сценариям Рэндалл-Сандрама в искривленных многообразиях.
Запропоновано та описано новий можливий механізм локалізації поля у певному секторі спеціальної супермногостатності. Різні аспекти цього нового механізму розглядаються в якості альтернативи конфайнменту в фізиці елементарних частинок і сценаріям Рендалл-Сандрама у викривлених многостатностях.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/106978 |
| citation_txt |
Spin, confinement, localization and supersymmetry / Diego Julio Cirilo-Lombardo // Вопросы атомной науки и техники. — 2012. — № 1. — С. 39-42. — Бібліогр.: 12 назв. — англ. |
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| first_indexed |
2025-11-26T08:07:17Z |
| last_indexed |
2025-11-26T08:07:17Z |
| _version_ |
1850615132708667392 |
| fulltext |
SPIN, CONFINEMENT, LOCALIZATION AND
SUPERSYMMETRY
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 30, 2011)
New mechanism of localization of fields in a particular sector of a special supermanifold is proposed and described,
based on previous works of the authors. Different aspects of this new mechanism are discussed as an alternative to
confinement in elementary particle physics and to the Randall-Sundrum scenarios in warped manifolds.
PACS: 03.65.Pm, 03.65.Ge, 61.80.Mk
1. INTRODUCTION AND MOTIVATION
The study of symmetries plays a fundamental role
in modern physics. The geometrical interpretation
of the physical phenomena takes as basic object the
action, where all the dynamics of the theory is de-
rived. The idea of associating an underlying geomet-
rical structure to these physical phenomena coming
from a fundamental idea of unification of all the inter-
actions into the natural world and not from an heuris-
tic thought. The interrelation between physical and
mathematical definitions and concepts (i.e. geome-
try, groups, topology↔space-time, internal structure,
fields) turns more and more concrete and basic in the
physics of the XX and XXI centuries. If there are
well elegant formulations of the physical problems of
interest from the mathematical point of view, there
exists a lack of uniqueness in the geometrical defini-
tion of the Lagrangian density.
Great difficulties appear (or almost are evidently
explicit) at the quantum level where the geometri-
cal objects playing the role of Lagrangian or Hamil-
tonian pass to play the role of (super) operators.
These troubles carry inexorably to the utilization of
diverse methods or prescriptions that change the orig-
inal form of the action (or Hamiltonian). This distor-
tion of the original form of these fundamental opera-
tors at the classical level generally does not produce
changes into the dynamical equations of the theory
but quantically introduces several changes, because
the spectrum of physical states is closely related with
the form of the Hamiltonian. This fact was pointed
out by the author in the previous paper [1]. Clearly,
in order to construct the Lagrangian and other fun-
damental invariants of the theory, the introduction
of a manifold as the important ingredient is the rel-
evant thing. In particular it can be very interesting
to introduce a super-manifold (in the sense of [1] and
references therein) in order to include the fermionic
fields in a natural manner.
It is therefore of interest to study the geometry
not only of the simplest superspaces, but also the
more unusual or non-standard ones and elucidate all
the gauge degrees of freedom that they possess. This
fact will clarify and expand the possibilities to con-
struct more realistic physical models and new math-
ematically consistent theories of supergravity. On
the other hand, the appearance of supergroups must
draw attention to the study of the geometries of the
homogeneous superspaces whose groups of motions
they are. Another motivation of the study of these
Riemannian superspaces is the establishment of some
degree of uniqueness in the obtained supersymmetric
solutions.
Motivated by the above, we comment and discuss
our previous paper [1] reviewing, studying and ana-
lyzing from the point of view of the possible vacuum
solutions, the simplest non trivial supermetric given
by Volkov and Pashnev in [2] that was the “starting
point” toy model of the first part of this work:
ds2 = ωμωμ + aωαωα − a∗ω
.
αω .
α. (1)
This particular non-degenerate supermetric contains
the complex parameters a and a∗ that make it differ-
ent from other more standard supermetrics. As we
showed in [1, 3], the degenerate supermetrics are not
consistent into a well theoretically formulated super-
geometry. Then, our main task is to find the meaning
and the role played by these complex parameters from
the geometrical and physical points of view.
Our goal is to show that, from the point of view
of the obtained solutions, the complex parameters a
localize the fields in a specific region of the bosonic
part of this special superspace, that they explicitly
∗E-mail addresses: diego@theor.jinr.ru and diego777jcl@gmail.com
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 39-42.
39
break down the chiral symmetry when some condi-
tions are required and that all these very important
properties remain although the supersymmetry of the
model was completely broken. Also, besides all these
highlights, we also show that the obtained vacuum
states from the extended supermetric are very well
defined in any Hilbert space.
2. “WARPED” GRAVITY MODELS,
CONFINEMENT AND THE
SUPERMETRIC
It is well known that large extra dimensions offer an
opportunity for a new solution to the hierarchy prob-
lem [4]. Field theoretical localization mechanisms for
scalar and fermions [5] as well as for gauge bosons [6]
were found. The crucial ingredient of this scenario is
a brane on which standard model particles are local-
ized. In string theory, fields can naturally be local-
ized on D-branes due to the open strings ending on
them [7]. Up until recently, extra dimensions had to
be compactified, since the localization mechanism for
gravity was not known. It was suggested in Ref. [8]
that gravitational interactions between particles on a
brane in uncompactified five dimensional space could
have the correct four dimensional Newtonian behav-
ior, provided that the bulk cosmological constant and
the brane tension are related. Recently, it was found
by Randall and Sundrum that gravitons can be local-
ized on a brane which separates two patches of AdS5
space-time [9]. The necessary requirement for the
four-dimensional brane Universe to be static is that
the tension of the brane is fine-tuned to the bulk cos-
mological constant [8, 9]. On the other hand, recent
papers present an interesting model in which the ex-
tra dimensions are used only as a mathematical tool
taking advantage of the AdS/CFT correspondence
that claims that the 5D warped dimension is related
with a strongly coupled 4D theory [10].
A remarkable property of the vacuum solution of
(1) given by the expression
gab (t) = e−( m
|a| )
2
t2+c1t+c2eξ�(t)gab (0) (2)
with
� (t) =
◦
φα
[(
αeiωt/2 + βe−iωt/2
)
− (
σ0
)α
.
α
(
αeiωt/2 − βe−iωt/2
)]
+
2i
ω
[(
σ0
) .
β
α
Z .
β
+
(
σ0
)α
.
α
Zα
]
and
gab (0) = 〈Ψ (0)|Lab |Ψ (0)〉
is that the physical state gab (x) is localized in a par-
ticular position of the space-time. The supermetric
coefficients a and a∗ play the important role of local-
izing the fields in the bosonic part of the superspace in
similar and suggestive form as the well known “warp
factors” in multidimensional gravity [11] for a pos-
itive (or negative) tension brane. But the essential
difference is, because the C-constants a and a∗ com-
ing from the BL,0 (even) fermionic part of the su-
perspace under consideration, no additional and/or
topological structures that break the symmetries of
the model (i.e. reflection Z2-symmetry) are required:
the natural structure of the superspace produces this
effect.
Also it is interesting to remark here that the
Gaussian type solution (2) is a very well defined
physical state in a Hilbert space [10, 12] from the
mathematical point of view, contrarily to the case
u (y) = ce−H|y| given in [11] that, although it was
possible to find a manner to include it in any Hilbert
space, it is strongly needed to take special mathe-
matical and physical particular assumptions whose
meaning is obscure. The comparison with the case of
5-dimensional gravity plus cosmological constant [11]
is given in the following table:
Comparison of Superspace (1, d | 1) with the case of 5-dimensional gravity plus cosmological constant [11]
Space-time 5D gravity + Λ Superspace (1, d | 1)
Interval ds2 = A (y) dx2
3+1 − dy ds2 = ωμωμ + aωαωα − a∗ω
.
αω .
α
Equation
[−∂2
y − m2eH|y| + H2
[
|a|2 (
∂2
0 − ∂2
i
)
+ 1
4 (∂η − ∂ξ + i ∂μ (σμ) ξ)2 −
−2Hδ (y)] u (y) = 0 − 1
4 (∂η + ∂ξ + i ∂μ (σμ) ξ)2 + m2
]ab
cd
gab = 0
Solution u (y) = ce−H|y|, H ≡
√
− 2Λ
3 = |T |
M3 gab (x) = e−( m
|a| )
2
x2+c′1x+c′2eξ�(x) |f (ξ)|2
(
α
α∗
)
ab
Here, in order to make our comparison consistent,
the proposed superspace has d = n + 4 bosonic co-
ordinates and the extended superspace solution for
n = 0 can depend, in principle, on any or all the 4-
dimensional coordinates: x ≡ (t, x), c′1x ≡ c′1μxμ and
c′2 scalar (e.g.: the t coordinate in expression (2)); for
n �= 0 it depends on the n-additional coordinates.
Notice the following important observations:
i) The solution in the 5-dimensional gravity plus Λ
case, the explicit presence of the cosmological term is
40
necessary for the consistency of the model: the “fine-
tuning” H ≡
√
− 2Λ
3 = |T |
M3 , where T is the tension
of the brane and M3 is the constant of the Einstein-
Hilbert +Λ action.
ii) about the localization of the fields given by the
particular superspace treated here, the Z2 symmetry
is non-compatible with the solution that clearly is
not chiral or antichiral. This fact is consistent with
the analysis given for a similar superspace that the
considered here in Ref. [1,12] where the solutions are
superprojected in a sector of the physical states that
is not chiral or antichiral.
iii) because for n = 0 our solution (2) is attached
to the 3+1 space-time but the localization occurs on
the time coordinate (in any of the remanent 3 space
coordinates) the physics seems to be very different
with respect to the warped gravity model where the
field equation in final form for the 5-dimensional grav-
ity depends on the extra dimension1. This n = 0 case
can give some hints for the theoretical treatment of
the confinement mechanism with natural breaking of
the chiral symmetry in high energy physics (e.g., in-
stanton liquid models, etc);
iv) for n = 1, the situation in our model with the
solution depending on the extra coordinate changes
favorably: the localization of the field is in the addi-
tional bosonic coordinate (as the graviton in the RS
type model) but with all the good properties of the
solution (2) already mentioned in the beginning of
this paragraph.
From the points discussed above and the “state of
the art” of the problem, we have seen the importance
of the proposition of new mechanisms and alterna-
tive models that can help us to understand and to
handle the problem. Also it is clearly important that
the supermetric (1), cornerstone of this simple super-
model, is non-degenerate in order to solve in a simul-
taneous manner the localization-confinement of the
fields involved and the breaking of the chiral symme-
try. Then, it is not difficult to think to promote the
particular supermetric under study towards to build
a strongly coupled 4D model, using this particular
N=1 toy superspace.
3. DISCUSSION
The proposal for the choice of a model with underly-
ing basic structure starts from the very early times.
Today, a large effective group given by the standard
model (multiplicity in the representations and the dif-
ferent coupling constants) stimulates from time ago
the search of such models. As we saw in the first part
of this work and other references, starting from the
most simplest non-degenerate supermetric where the
supercoordinates are the fields of the theory and re-
taining the original form of the fundamental geomet-
rical operators (namely Lagrangian or Hamiltonian),
the physical states obtained are constructed from the
basic ones by means of operators that characterize
the most fundamental symmetries of the space-time.
The situation is more or less clear: although the
supersymmetry is broken, the physical states are lo-
calized in the “even” part of the manifold due to the
metric coefficients of a non-degenerate supermetric.
The physical states are composed by most fundamen-
tal (non-observable) basic states. Operators belong-
ing to the metaplectic group (the most fundamental
covering group of the SL(2C)) lead, due to a map pro-
duced by the basic CS, to the observable spectrum of
physical states. This fact is clearly important as the
“cornerstone” of a new realistic composite model of
particles based on coherent states where the space-
time symmetry is directly connected with the physi-
cal spectrum.
Acknowledgements
This work is in memory of Alexander I. Akhiezer:
outstanding person from the scientific and the hu-
man point of view and of Professor Vyacheslav Soroka
pioneering in the development of the supergeometry
and supergravity. Many thanks are given, however,
to Professors Yu. P. Stepanovsky and A. Dorokhov for
my scientific formation and to my friend and collab-
orator S. N. Shulga for their interest and discussions;
This work was partially supported by CNPQ-PNPD
Brazilian funds.
References
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