Global symmetries in non-linear models of gauge theories

1
The nonabelian gauge fields and their dynamics in
the finite space of color factors
Radu Constantinescu, Carmen Ionescu
University of Craiova, 13 A. I. Cuza Str.,
Craiova 200585, Romania
2
Structure of the paper I. Basic facts on the
symmetries of the gauge fields I.1
Point-like symmetry I.2 From the local to
the global (BRST) symmetry II. Passage to the
mechanical model II.1 The general
(non-abelian) electromagnetic field II.2 The
attached mechanical model II.3 The combined
dynamics of the gauge and ghost fields

• Abstract
• We shall focus on the possibility of reducing the
  study of the nonabelian gauge field (with an
  infinite number of degrees of freedom) to a
  simpler mechanical one (with finite number of
  degrees of freedom). We shall express the whole
  set of generators of the extended BRST space in
  terms of a finite number of color factors.

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I.1 Point-like symmetry (1) Lie operators

• A point-like transformation in the (q,t)
  space-time may be defined through an
  infinitesimal parameter e by  
• The variation of an arbitrary analytical function
  u(q,t), duu(q',t')-u(q,t)
• The operator U denotes the generator of the
  infinitesimal point-like transformation and is
  called Lie operator. Its concrete form is
• The first extension
• The second extension

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I.1 Point-like symmetry (2) the example of a
non-autonomous systems

• For a dynamical system described by the equations
  of motion
• The Lie symmetries leave invariant these
  equations
• When the physical system does not involve
  velocity terms
• It is equivalent with
• The solutions for the 2 unknown functions are

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I.2 Gauge symmetry (1) The global BRST symmetry

Extended action
BRST operator
Right derivative
Acyclicity
BRST Charge
Master equation
Extended Hamiltonian
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I.2 Gauge symmetry (2) The extended sp(3)
symmetry
A gauge theory constrained dynamical system
described by a set of irreducible constraints and
by the canonical Hamiltonian

The gauge algebra have the form  
The sp(3) BRST symmetry
The extended phase space
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Construction of the extended phase-space
The extended phase space will be generate by
introduction, for each constraint
of three pairs of canonical conjugate ghost
variables
and
For assuring the crucial property of the Koszul
differentials,
namely the acyclicity of the positive
resolution numbers, it is necessary to introduce
the new generators,
with
and their conjugates
with
so that
The same property, the acyclicity of
imposes the introduction of new generators,
and
with
8

The master equations allow to determine the BRST
charges, using the homological perturbation
theory


constant
,

The extended Hamiltonian will be given by the
BRST invariance requirement
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II.1 The non-abelian gauge field (1) The BRST
Approach
where
The canonical analysis leads to the irreducible
first class constraints
The gauge algebra is given by
The gauge fixed action
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II.1 The non-abelian gauge field (2) Ghosts as
real variables

• By considering that ghosts are, them too, real
  variables, one can write down their equations of
  motion

On the basis of these equations part of the terms
containing ghosts can be eliminated from the
gauge fixed action. The vertex ghost-ghost-gauge
field is remaining.
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II.1 The non-abelian gauge field (3) Towards a
mechanical model
The evolution of the real and ghost fields
are given by
The fields are expressed by a finite set of color
factors.
One obtains the system of "mechanical" equations
In the case d3, 6 equations with 6 unknown color
factors
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II.1 The non-abelian gauge field (3) A four
dimensional dynamics
The previous system has the general form
Let us use the notations
By choosing arbitrary constant factors, the
system becomes
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The system has usually chaotic behavior, as the
pictures from below show for yy(x) and vv(u).
Although, some periodical solutions could be
found.
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Maintaining the initial conditions for x(t) and
y(t), we obtain, for three different choices of
the initial values of u, uder, v, vder, pretty
different behaviors
Figure 1 Initial conditions for x, y and their
derivatives
Figure 2 u,v,vder, vanish at the initial moment
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Figure 3 and Figure 4 describe the behavior of
the same system for slight different initial
conditions for u(t), v(t), uder(t), vder(t) .
Figure 3
Figure 4
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Conclusions

• The main objective of the paper how the study of
  a gauge field could be reduced to the study of a
  system defined in terms of a finite number of
  parameters, the color factors
• The starting point was the fact that field obeys
  to a global symmetry (the BRST symmetry) which
  includes all gauge invariances.
• The ghost fields can be seen as real fields and
  can be expressed, them too, in terms of some
  color factors.
• The dynamical evolution of the color factors
  attached to the ghosts is sensitive dependent on
  the evolution of the real fields.

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