Geometric construction of Yang-Mills fields in maximally symmetric spacetimes

In this thesis, we explore results from geometric construction methods of Yang--Mills fields in maximally symmetric spacetimes, including cosmological applications and abelian reductions in different settings. Such efforts were inspired by the development of a new construction method for electromagnetic knots in Minkowski space from an abelian reduction of an \SU(2) Yang--Mills theory in a Lorentzian cylinder over the three-sphere S^3 \simeq \SU(2), which is then conformally mapped into \Rds^{1,3}. My investigations and results will be presented in three parts, each with partially independent motivations, conclusions, and prospects for advancement.The first set of results concerns the previously constructed basis of electromagnetic knots in Minkowski space. We investigated conserved charges associated with the conformal invariance of the solutions, which helped us better understand some of their physical properties and to compare them with other known electromagnetic knots constructed from different established methods. In particular, we found specific parameter choices that allowed us to reproduce generalisations of the Rañada-Hopf knot, such as rotated Hopfions, and torus knots from Bateman's construction. Moreover, as studies on the reproduction of such knotted solutions in the lab advance, we performed a numerical investigation and documented the effects of the electromagnetic knots from our basis on charged particles, which is relevant data for possible experimental applications in the future.Motivated by the aforementioned results from the \SU(2) Yang--Mills field on \dS4, we explored what further results can be obtained from Yang--Mills theories constructed with similar use of the geometric structure of other maximally symmetric spacetimes. We first used a piecewise foliation of the Minkowski spacetime by orbits of the Lorentz group to construct an \SO(1,3) gauge theory in the different regions, which, after a regularisation on the lightcone, produced a Yang--Mills field whose energy-momentum tensor is global in \Rds^{1,3} and is given by an improvement term. Then, we explored an \SU(1,1) gauge theory on \AdS4 foliated with \AdS3 sheets and conformally mapped it into \Rds^{1,3}. The abelian reduction of the non-abelian gauge theory generated the hyperbolic equivalent of the Rañada-Hopf knot. We showed that one of its projections into a plane was a known magnetic vortex in

1{+}2

dimensions. We finalised by discussing the prospects of a method to generate an uncountable basis of hyperbolic equivalents of the previously discussed electromagnetic knots.The last set of results concerns the coupling of the Yang--Mills theory to General Relativity in Friedmann universes. We showed that, in any spacetime dimension, the equivariant Ansatz of any Yang--Mills theory on a cylinder over a symmetric space has its dynamics reduced to that of a Newtonian scalar degree of freedom subjected to a quartic potential. The effect of such gauge fields on the spacetime dynamics is simply reduced to the Wheeler-DeWitt constraint, which relates the initial conditions in the Einstein--Yang--Mills system. Moreover, we showed that, in spacetime dimensions other than four, a dynamical scale factor induces a Hubble friction-like term in conformal time in the reduced equation of motion for the gauge sector. Lastly, we investigated a model where an \SO(4)-symmetric \SU(2) Yang--Mills field thermodynamically stabilises the symmetric Higgs vacuum in a closed Friedmann universe in the electroweak epoch. We probed generic fluctuations of the gauge field and showed that it is unstable under any perturbation other than the singlet, which we solved exactly and found is marginally stable.

Yang-Mills theory
FLRW universes
Electromagnetic knots

References(56)

Figures(0)

[1]

Conservation of Isotopic Spin and Isotopic Gauge Invariance

Chen-Ning Yang
(
- Brookhaven
)
,
Robert L. Mills
(
- Brookhaven
)

- Phys.Rev. 96 (1954) 191-195,
- Also in *Lai, C. H. (Ed.): Gauge Theory Of Weak and Electromagnetic Interactions*, 29-33, and in *Yang, C.N.: Selected Papers 1945-1980*, 172-176,
•
DOI:
- 10.1103/PhysRev.96.191

[2]

A Unified Field Theory of Mesons and Baryons

T.H.R. Skyrme
(
- Harwell, AERE
)

- Nucl.Phys. 31 (1962) 556-569,
- Also in *Rebbi, C. ( Ed.), Soliani, G. ( Ed.): Solitons and Particles*, 499-512, and in *Lewes 1984, Proceedings, Solitons In Nuclear and Elementary Particle Physics*, 13-26
•
DOI:
- 10.1016/0029-5582(62)90775-7

[3]

Classical Yang-Mills theory in condensed matter physics

- Eur.J.Phys. 34 (2013) 161-180
•
e-Print:
- 1210.0105
•
DOI:
- 10.1088/0143-0807/34/1/161

[4]

Coset space dimensional reduction of gauge theories

- Phys.Rept. 219 (1992) 4-76
•
DOI:
- 10.1016/0370-1573(92)90101-5

[5]

Gauge Fields and Inflation

- Phys.Rept. 528 (2013) 161-261
•
e-Print:
- 1212.2921
•
DOI:
- 10.1016/j.physrep.2013.03.003

[6]

Origin of cosmological temperature

Daniel Friedan
(
- Rutgers U., Piscataway and
- Iceland U.
)

e-Print:
- 2005.05349

[7]

Exact Solution to the Einstein {Yang-Mills} Equation

Yutaka Hosotani
(
- Pennsylvania U.
)

- Phys.Lett.B 147 (1984) 44-46
•
DOI:
- 10.1016/0370-2693(84)90588-4

[8]

Modern differential geometry for physicists

C.J. Isham

[9]

Geometry, topology and physics

M. Nakahara

[10]

Lie Groups, Lie Algebras, and Representations, An Elementary Introduction, 2nd edition, Graduate Texts in Mathematics

B.C. Hall

[11]

Mathematical Gauge Theory, With Applications to the Standard Model of Particle Physics, Universitext

M.J.D. Hamilton

[12]

Solutions of Yang-Mills theory in four-dimensional de Sitter space

K. Kumar

[13]

An Introduction to Quantum Field Theory

M.E. Peskin
,
D.V. Schroeder

[14]

Gauge Field Theories (Cambridge University Press,)

S. Pokorski

[15]

General Relativity (Chicago University Press,)

R.M. Wald

[16]

Spacetime and Geometry, An Introduction to General Relativity (Cambridge University Press,)

S.M. Carroll

[17]

Classical Covariant Fields, Cambridge Monographs on Mathematical Physics (Cambridge University Press,)

M. Burgess

[18]

GAUGE THEORY AND VARIATIONAL PRINCIPLES

D. Bleecker

[19]

On Spherically Symmetric Fields in Gauge Theories

- Nucl.Phys.B 130 (1977) 209-220
•
DOI:
- 10.1016/0550-3213(77)90104-3

[20]

GAUGE COVARIANT CONFORMAL TRANSFORMATIONS

R. Jackiw
(
- MIT, LNS
)

- Phys.Rev.Lett. 41 (1978) 1635
•
DOI:
- 10.1103/PhysRevLett.41.1635

[21]

Symmetries in Gauge Theories

P.G. Bergmann
(
- Syracuse U.
)
,
E.J. Flaherty
(
- Syracuse U.
)

- J.Math.Phys. 19 (1978) 212-214
•
DOI:
- 10.1063/1.523540

[22]

Space-Time Symmetries in Gauge Theories

- Commun.Math.Phys. 72 (1980) 15
•
DOI:
- 10.1007/BF01200108

[23]

Group Actions on Principal Bundles and Invariance Conditions for Gauge Fields

- J.Math.Phys. 21 (1980) 2719,
- In *Montreal 1979, Proceedings, Geometrical and Topological Methods In Gauge Theories*, 149.(See Conference Index)
•
DOI:
- 10.1063/1.524389

[24]

SYMMETRIES OF GAUGE FIELDS

M. Molelekoa
(
- Lesotho Natl. U.
)

- J.Math.Phys. 26 (1985) 192-197,
- J. Math. Phys. 26 ( 1985) 192-197
•
DOI:
- 10.1063/1.526783

[25]

The structure of the solution set for the Yang-Mills equations

J.M. Arms

- Math.Proc.Cambridge Phil.Soc. 90 (1981) 361-372

1-25 of 56
1
2
3
25 / page