Gauge theories attractive and repulsive force

1
Gauge theoriesattractive and repulsive force

• Venue USM physics school
• Time11.00a.m.-11.30a.m.
• Presented by Koh Pin Wai
• Date19-3-2008

2
Gauge theoriesattractive and repulsive forces

• that close to theirs properties is an
  asymptotic values, except the bounded region
  space.
• These linear dimension in these region is .
  
• m is denoted as the smallest mass generated by
  the Higgs effect.
• These region is separated by a distance , the
  field generally close to a spherically symmetric
  fundamental solution.
• The fundamental solution for d2,GU(1) is a
  vortex for
• arbitrarily G, d3, fundamental solutions is
  monopoles.

3
Solitons considered as particles

• Solitons as a waves also arise as a particles.
• The basic particles has the topological field N1
  which considered as stable and it will not decay
  to the trivial field.
• It is convenient to say the solitons is Newtonian
  particles because
• i) It can be calculated out by the classical
  mechanics method, i.e. Lagrangian method and
  Hamiltonian method.
• ii) It obey some of the nature properties in
  macroscopic level such as the attractive and
  repulsion forces inside the system.

4

• The graph electric potential of solitons (V)
  against the separation between solitons (r) is
  almost same like the empirical potential energy
  function (potential V against the separation r)
• The most obvious example is the monopoles with
  the potential formula 3

5

• In the quantum interpretations,
• The quantum interpretation for soliton-soliton
  collisions can produce elementary particles
• 4

6
Gauge theory Homotopy

• Homotopy is used in topology algebraic to
  classify the topological spaces.
• Topology is a study of shape and continuous map
  between them.
• Example A topologist cannot find a differences
  between the donuts and a cups.
• In all the Yang-Mills Higgs group, these group
• are either the integer
• This explain why the addition and subtraction for
  the homotopy classes is defined.

7

• There exists a solitons with the inverse class in
  the given homotopy class and this is correspond
  to one-to-one.
• Fundamental solutions for the positive and
  negative elements can be defined if all the
  integer from the homotopy group is sum up
  together.
• The group playing important to determine
  the particle and the antiparticles of the
  systems.
• The positive elements of is called the particles
  whereas the negative elements of is called
  the antiparticles.
• i.e vortex/antivortex and monopoles/antimonopoles.
  
• The reaction between the soliton-antisoliton
  pairs is believed by the physicist is just like
  the electron positron pair annihilation.

8

• The soliton-antisoliton pair annihilation
• Is look like the electron positron pair
  annihilation like the Fenyman diagram below
• Fig 1 electron-positron pair
  annihilation
• The difference between solitons and the
  elementary particles is the elementary particles
  do not have its own topological structure.

9
Interparticle forces

• What are the interparticle forces?
• Interparticles forces is zero in order for a
  static solution.
• When particles attract to each other, the static
  solution for the configurations of all the
  particles is found in the same point.
• If particles repel, boundary conditions on the
  walls of the finite domain is needed to stabilize
  the static configurations of the particles in the
  interior.
• In fact, there are always exists solutions in the
  interior of the bounded domain for all the values
  of the Higgs coupling constant if and only if
  the net forces is not repulsive.

10

• There is no any multiparticles solutions to the
  equation if the net forces is repulsive.
• If all the particles located in the same point .
• Solution exists as a minimum action only.
• Discussed in more briefly ways
• Formula is a solution that
  contain a distant pair of a fundamental solitons.
• Potential energy function is defined as
• V is a function of the separation, whereas the
  gradient interpreted as a interparticle forces.

11
Scalar and vector field

• Vector fields is defined as a construction in
  calculus which associates with vector in every
  point (locally in) Euclidean space. i.e. A (gauge
  field)
• Scalar fields associates with the scalar values,
  which can be either in physical or mathematical
  definition in every point of space. i.e.
  (Higgs field)
• In electromagnetism, Gauge field and Higgs field
  are used to mediate the force.
• The gauge potentials creates attractive force
  between two opposite charges fundamental solitons
  and repulsive forces between two like charges
  fundamental solitons .

12
Interaction energy and coupling constant

• Force balancing only will occurs for the like
  charge with the condition coupling constant must
  be critical value.
• .
• For the case , net force are attractive
  force for like change.
• For the case , net force are repulsive
  force for like change.
• The forces between two opposite charge in the
  system are always attractive forces.
• The paper Interaction energy of superconducting
  vortices, Physical review B, volume 19, number
  9, by Laurence Jacobs and Claudio Rebbi had shown
  the

13

• critical coupling constant for the vortices in
  theirs paper.
• Their prove the coupling constant for vortices
• is the critical values in the system which the
  vortices do not interact.
• When , energy functional can be written as
• When , energy functional can be written as
• The final result is

14

• This is the table 1 that constructed by using the
  term
• and the graph is like below
• The second table is the energies for a single
  vortex and two superimpose vortices as functions
  of
• Comparison for both graph
• Table 1 energies as functions of
  Table 2 energies as functions of 2
• (prediction)

15

• Comparison of two graphs

16

• From the both graph that shown at the slide
  above, the two lines that represented as the
  energy of two superimposed vortices and twice
  energy single vortex is crossover at .
• This show that is a critical coupling
  constant for the vortices.
• By using the variational method, the constrained
  minimun of with two vortices for kept at
  a fixed separation can be determined.
• Evaluation of as functions of the and
  minimizes the expression is introduces by a set
  of field configurations depending on a
  variational parameters , i1,N.
• The step of minimization is expressed as the
  following equation

17

• and also
• The numerical result generated by computer for
  the energy of two vortices at a separation d for
  
• is shown at table below
• Table 3 Energy of two vortices at a
  separation d
• for
• 2

18

• Fig 3 Energy of a two vortex field configuration
  for
• as a function of the inter- vortex
  separation. The black lines correspond to
  asymptotic values.2

19

• The critical value for vortices and the
  critical value for monopoles . In this
  state, the interparticles forces eliminates and
  solitons do not interact.
• These solutions contain only particles and
  antiparticles, phenomenon time dependent bound
  state may happen in the system is conjectured if
  the particles and antiparticles exist in the
  static solution.

20

• References
• 1 Vortices and Monopoles, Arthur Jaffe and
  Clifford Taubes, 1980,
• edited by A.Jaffe and D.Ruelle Birkhauser
  Boston.
• 2 Interacting energy of superconducting
  vortices, L.Jacobs and C.Rebbi, Phys. Rev. B 19,
  4486 1979.
• 3Introduction to electrodynamics, David
  J.Griffiths and Reed College, 1999, Prentice
  Hall.
• 4Topological soliton, Nicholas Manton and Paul
  Schuliffe, 2004, Cmabridge University Press.

21

• Thank you