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 SSSCP - Nis 2009

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 SSSCP - Nis 2009 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 .

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 ε by: The variation of an arbitrary analytical function u(q,t), δ u=u(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: SSSCP - Nis 2009

A point-like transformation in the (q,t) space-time may be defined through an infinitesimal parameter ε by:

The variation of an arbitrary analytical function u(q,t), δ u=u(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:

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: SSSCP - Nis 2009

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:

SSSCP - Nis 2009 I.2 Gauge symmetry: (1) The global BRST symmetry : * Master equation: * Acyclicity : * BRST Charge: * Extended action: * BRST operator: * Right derivative: * Extended Hamiltonian:

I.2 Gauge symmetry: (2) The extended sp(3) symmetry SSSCP - Nis 2009 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:

SSSCP - Nis 2009 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, with and 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

SSSCP - Nis 2009 The master equations allow to determine the BRST charges, using the homological perturbation theory : The extended Hamiltonian will be given by the BRST invariance requirement: constant ,

SSSCP - Nis 2009 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:

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.

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

SSSCP - Nis 2009 The fields are expressed by a finite set of color factors . One obtains the system of "mechanical" equations: In the case d=3, 6 equations with 6 unknown color factors: II.1 The non-abelian gauge field (3) Towards a mechanical model The evolution of the “real” and “ghost” fields are given by:

SSSCP - Nis 2009 II.1 The non-abelian gauge field (3) A four dimensional dynamics Let us use the notations : By choosing arbitrary constant factors, the system becomes: The previous system has the general form:

The system has usually chaotic behavior, as the pictures from below show for y=y(x) and v=v(u). Although, some periodical solutions could be found.

Maintaining the initial conditions for x(t) and y(t), we obtain, for three different choices of the initial values of {u, u der , v, v der }, pretty different behaviors: SSSCP - Nis 2009 Figure 1: Initial conditions for x, y and their derivatives Figure 2: u,v,vder, vanish at the initial moment

Figure 3 and Figure 4 describe the behavior of the same system for slight different initial conditions for u(t), v(t), u der (t), v der (t) . SSSCP - Nis 2009 Figure 3 Figure 4

SSSCP - Nis 2009 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.

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.

References: A. Babalean, R. Constantinescu , C.Ionescu, J. Phys. A: Math. Gen. 31 (1998) 8653 M. Henneaux, C. Teitelboim , Quantization of Gauge Systems, Princeton Univ. Press (1992) Struckmeier J., Riedel C., Phys.Lett. 85, (2000), 3830 J.Struckmeier and C. Riedel, Phys.Rev.E 66, (2002) 066605 P.J.Olver, Springer-Verlag, New York, (1986) D.J.Arrigo, J.R.Beckham, J . Math. Anal. Appl. 289 (2004) 55 S. G. Matincan, G. K. Savvidi, N. G. Ter-Arutyunyan-Savvidi , Sov. Phys. JETP 53 (3) (1981) 421 T. S. Biro, S. G. Matinyan, B. Muller , Chaos and Gauge Theory, World Scientific Lecture Notes in Physics, Vol. 56 (1994) R.Constantinescu, C.Ionescu , Cent. Eur. J. Phys. (2009), D08-106

A. Babalean, R. Constantinescu , C.Ionescu, J. Phys. A: Math. Gen. 31 (1998) 8653

M. Henneaux, C. Teitelboim , Quantization of Gauge Systems, Princeton Univ. Press (1992)

Struckmeier J., Riedel C., Phys.Lett. 85, (2000), 3830

J.Struckmeier and C. Riedel, Phys.Rev.E 66, (2002) 066605

P.J.Olver, Springer-Verlag, New York, (1986)

D.J.Arrigo, J.R.Beckham, J . Math. Anal. Appl. 289 (2004) 55

S. G. Matincan, G. K. Savvidi, N. G. Ter-Arutyunyan-Savvidi , Sov. Phys. JETP 53 (3) (1981) 421

T. S. Biro, S. G. Matinyan, B. Muller , Chaos and Gauge Theory, World Scientific Lecture Notes in Physics, Vol. 56 (1994)

R.Constantinescu, C.Ionescu , Cent. Eur. J. Phys. (2009), D08-106