Particles in the early universe : high-energy limit of the standard model from the contraction of its gauge group

172 Pages·2020·2.932 MB·English

by  Gromov, Nikolaĭ Alekseevich

PARTICLES IN THE
EARLY UNIVERSE
High-Energy Limit of the Standard Model from
the Contraction of Its Gauge Group
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PARTICLES IN THE
EARLY UNIVERSE
High-Energy Limit of the Standard Model from
the Contraction of Its Gauge Group
Nikolai A Gromov
Russian Academy of Sciences, Russia
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
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PARTICLES IN THE EARLY UNIVERSE
High-Energy Limit of the Standard Model from the Contraction of Its Gauge Group
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Preface
Group-TheoreticalMethodsareanessentialpartofmoderntheoreticaland
mathematical physics. The most advanced theory of fundamental inter-
actions, namely Standard Model, is a gauge theory with gauge group
SU(3)×SU(2)×U(1).Alltypesofclassicalgroupsofinfiniteseries:orthog-
onal, unitary and symplectic as well as inhomogeneous groups, which are
semidirectproductsoftheirsubgroups,areusedindifferentareasofphysics.
Euclidean,Lobachevsky,Galilean,Lorentz,Poincar´e,(anti)deSittergroups
arethebasesforspaceandspace-timesymmetries.Supergroupsandsuper-
symmetric models in the field theory predict the existence of new super-
symmetric partners of knownelementary particles. Quantumdeformations
of Lie groupsandLie algebrasleadto non-commutativespace-timemodels
(or kinematics).
Contractions of Lie groups is the method for receiving new Lie groups
from the initial ones. In the standard E. Wigner and E. Ino¨nu¨ approach,
[In¨onu¨ and Wigner (1953)] continuous parameter (cid:2) is introduced in such
a way that in the limit (cid:2) → 0 group operation is changed but Lie group
structure and its dimension are conserved. In general, a contracted group
is a semidirect product of its subgroups. In particular, a contraction of
semisimple groups gives non-semisimple ones. Therefore, the contraction
method is atoolforstudying non-semisimplegroupsstartingfromthe well
known semisimple (or simple) Lie groups.
The method of contractions (limit transitions) was extended later to
other types of groups and algebras.Gradedcontractions [de Montigny and
Patera (1991); Moody and Patera (1991)] additionally conserve grading
of Lie algebra. Lie bialgebra contractions [Ballesteros, Gromov, Herranz,
del Olmo and Santander (1995)] conserve both Lie algebra structure and
cocommutator. Contractions of Hopf algebras (or quantum groups) are
v
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vi Particlesin the Early Universe
introduced in such a way [Celeghini, Giachetti, Sorace and Tarlini (1990,
1992)] that in the limit (cid:2) → 0 new expressions for coproduct, counit and
antipode appear which satisfies Hopf algebra axioms. All these give rise to
the following generalization of the notion of group contraction on contrac-
tion of algebraic structures [Gromov (2004)].
Definition 0.1. Contraction of algebraic structure (M,∗) is the map φ(cid:2)
dependent on parameter (cid:2)
φ(cid:2) :(M,∗)→(N,∗(cid:2)), (0.1)
where(N,∗(cid:2))isanalgebraicstructureofthesametype,whichisisomorphic
in (M,∗) when (cid:2)(cid:4)=0 and non-isomorphic when (cid:2)=0.
There is another approach [Gromov (2012)] to the description of non-
semisimpleLiegroups(algebras)basedontheirconsiderationoverPimenov
algebraPn(ι) with nilpotent commutative generators.In this approachthe
motion groups of constant curvature spaces (or Cayley–Klein groups) are
realized as matrix groups of special form over Pn(ι) and can be obtained
from the simple classical orthogonal group by substitution of its matrix
elements for Pimenov algebraelements.It turns outthat such substitution
coincides with the introduction of Wigner–Ino¨nu¨ contraction parameter
(cid:2). So our approach demonstrates that the existence of the corresponding
structures over algebra Pn(ι) is the mathematical base of the contraction
method.
It should be noted that both approaches supplement each other and
in the final analysis give the same results. Nilpotent generators are more
suitableinthemathematicalconsiderationofcontractionswhereasthecon-
traction parameter continuously tending to zero corresponds more to the
physical intuition where a physical system continuously changes its state
and smoothly goes into its limit state. In chapter 5, devoted to the appli-
cations of the contraction method to the modern theory of the elementary
particles, the traditional approach is used.
It is well known in geometry (see, for example, review [Yaglom, Rosen-
feldandYasinskaya(1964)])thatthereare3ndifferentgeometriesofdimen-
sion n, which admit the motion group of maximal order. R.I. Pimenov
suggested [Pimenov (1959, 1965)] a unified axiomatic description of all 3n
geometriesofconstantcurvature(orCayley–Kleingeometries)anddemon-
strated that all these geometries can be locally simulated in some region
of n-dimension sphericalspace with named coordinates,which can be real,
imaginary and nilpotent ones. According to Erlanger program by F. Klein
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Preface vii
the main content in geometry is its motion group whereas the properties
of transforming objects are secondary. The motion group of n-dimensional
sphericalspaceisisomorphictotheorthogonalgroupSO(n+1).Thegroups
obtained from SO(n+1) by contractions and analytical continuations are
isomorphic to the motion groups of Cayley–Klein spaces. This correspon-
dence provides the geometrical interpretation of Cayley–Klein contraction
scheme.By analogythis interpretationis transferredto the contractionsof
other algebraic structures.
TheaimofthisbookistodevelopcontractionmethodforCayley–Klein
orthogonaland unitary groups (algebras) and apply it to the investigation
of physical structures. The contraction method apart from being of inter-
est to group theory itself is of interest to theoretical physics too. If there
is a group-theoretical description of a physical system, then the contrac-
tion of its symmetry group corresponds to some limit case of the system
underconsideration.Sothereformulationofthesystemdescriptioninterms
of this method and the subsequent physical interpretations of contraction
parameters give an opportunity to study different limit behaviours of the
physical system. Examples of such an approach are given in chapter 2 for
the space–time models, in chapter 3 for the Jordan–Schwinger representa-
tions of groups which are closely connected to the properties of stationary
quantumsystemswhoseHamiltoniansarequadraticincreationandannihi-
lationoperators.Inchapter5themodifiedStandardModelwithcontracted
gauge group is considered. The contraction parameter tending to zero is
associated with the inverse average energy (temperature) of the Universe
which makes it possible to re-establish the evolution of particles and their
interactions in the early Universe.
Itislikelythatthedevelopedformalismisessentialinconstructing“gen-
eral theory of physical systems” where it will necessarily turn the group-
invariant study of a single physical theory in Klein’s understanding (i.e.
characterizedby symmetrygroup)toa simultaneousstudy ofa setoflimit
theories. Then some physical and geometricalproperties will be the invari-
ant properties of all sets of theories and they should be considered in the
first place. Other properties will be relevant only for particular represen-
tatives and will change under limit transition from one theory to another
[Zaitsev (1974)].
In chapter 1, definitions of Cayley–Klein orthogonal and unitary
groupsaregiven;their generators,commutatorsandCasimiroperatorsare
obtained by transformations of the corresponding quantities of classical
groups.
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viii Particles inthe Early Universe
In chapter 4, contractions of irreducible representations of unitary
and orthogonal algebras in Gel’fand–Tsetlin basis [Gromov (1991, 1992)],
which are especially convenient for applications in quantum physics, are
considered. Possible contractions, which give different representations of
contracted algebras, are found and general contractions leading to repre-
sentations with nonzero eigenvalues of Casimir operators are studied in
detail. For multi-parametric contractions, when a contracted algebra is a
semidirect sum of a nilpotent radical and a semisimple subalgebra, our
method ensures the construction of irreducible representations for such
algebras starting from well known irreducible representations of classical
algebras.Whenalgebrascontractedondifferentparametersareisomorphic,
the irreducible representations in different bases (discrete and continuous)
are obtained.
The list of references does not pretend to be complete and reflects the
author’s interests.
N. A. Gromov
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Contents
Preface v
1. The Cayley–Kleingroups and algebras 1
1.1 Dual numbers and the Pimenov algebra . . . . . . . . . . 1
1.1.1 Dual numbers . . . . . . . . . . . . . . . . . . . . 1
1.1.2 The Pimenov algebra . . . . . . . . . . . . . . . . 3
1.2 The Cayley–Klein orthogonalgroups and algebras . . . . 4
1.2.1 Three fundamental geometries on a line . . . . . . 4
1.2.2 Nine Cayley–Kleingroups . . . . . . . . . . . . . 8
1.2.3 Extension to higher dimensions . . . . . . . . . . 14
1.3 The Cayley–Klein unitary groups and algebras . . . . . . 17
1.3.1 Definitions, generators,commutators . . . . . . . 17
1.3.2 The unitary group SU(2;j1) . . . . . . . . . . . . 20
1.3.3 Representations of the group SU(2;j1) . . . . . . 22
1.3.4 The unitary group SU(3;j). . . . . . . . . . . . . 28
1.3.5 Invariant operators . . . . . . . . . . . . . . . . . 31
1.4 Classification of transitions between the Cayley–Klein
spaces and groups . . . . . . . . . . . . . . . . . . . . . . 32
2. Space–time models 35
2.1 Kinematics groups . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Carroll kinematics . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Non-relativistic kinematics. . . . . . . . . . . . . . . . . . 43
3. The Jordan–Schwingerrepresentations of Cayley–Klein groups 47
3.1 The second quantization method and matrix
elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
ix