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,

RomaniaSSSCP - Nis 2009

Structure of the paperI. Basic facts on the symmetries of the gauge fields: I.1 Point-like symmetry I.2 From the local to the global (BRST) symmetryII. 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

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AbstractWe 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), δu=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:

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)t,q(q ,qqq iiiii

)t,q(t ,ttt

)t,q(Uuqq

ut

t

uu i

i

n

1i

ii

n

1i q)t,q(

t)t,q(U

iiii

i

n

1i

q,q

UU

iiiii

i

n

1i

qq2,q

U"U

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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n,1i ,0q

)t,q(Vq

ii

n,1i ,0)q

)t,q(Vq("U

iiL

0)q

)t,q(V(U

iLi

0qq

V

tq

Vqq2

ji

2

j

n

1ji

2

iii

jj

n

1jiijij

n

1jiLi

ii

n

1iL

qq)t(qbq2

1)t,q(

q)t()t()t,q(

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

...0 SS

sS 0; sF 0 #

sA 0 B s. t.A sB

sAB AsB 1 BsAB #

s , ; s2 0 #

:

s , 0 # * Master equation:

* Acyclicity :

* BRST Charge:

* Extended action:

* BRST operator:

* Right derivative:

* Extended Hamiltonian: H0 H H0 . . . ;sH H,S 0

I.2 Gauge symmetry: (2) The extended sp(3) symmetry

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A gauge theory = constrained dynamical system described by a set of irreducible constraints and by the canonical Hamiltonian

0},,...,2,1,{ HmG

The gauge algebra have the form : G ,G c G , H0 ,G V

G #

The sp(3) BRST symmetry: sT s1 s2 s3 , #

sasb sbsa 0, a,b 1,2,3. #

The extended phase space: QA q i,Q a , a , , a 1,2,3 #

PA pi,P a , a , , a 1,2,3 . #

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aP b abG . # daQ b adc db c 12

c Q Q . #

a , a 1,2,3 a

a a a a a a

a b abcP c #

da b ab 1

2c

bQ c ca 112

f f

bcdQ cQ dQ e ea #

a

, a 1

a ab b #

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

G , 1, ,m

P a ,Q a , a 1,2,3 P a Q a 1

The extended phase space will be generate by introduction, for each constraint

of three pairs of canonical conjugate ghost variables

and

da 12

c Q b ba 1

12f

f f

f cQ b baQ c. #

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a , b 0; a,b 1,2,3. #

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

a G Q b ba abcP c b 12

c P cQ cQ a a 1

2c

c cQ a

112

f f

bcd bQ cQ dQ e ea 12

c Q a

112

f f

f f

cQ b baQ c .

The extended Hamiltonian will be given by the BRST invariance requirement:

H, a 0 ; a 1,2,3. # H|P Q 0 H0 . #

H H0 V P aQ a a a a . #

c V

constant,

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II.1 The non-abelian gauge field (1) The BRST Approach

S0Am 1

4d4x F

m Fm # F

m Am A

m g nrm A

nAr . # where

The canonical analysis leads to the irreducible first class constraints:

Gm1x pm

0 x 0 # Gm2x ipm

i x gmnrA irxpinx 0 #

The gauge algebra is given by:

Gm1,Gn

1 0, Gm1,Gn

2 0, Gm2,Gn

2 fmnrG r

2, H0 ,Gm1 Gm

2, H0 ,Gm2 fmn

rA0nG r2. #

The gauge fixed action: SY d4x 14

F m Fm

p0m Am Pma

1 D nm Q2na

nmnm

namnma DD )2()1()2()1( ))(())((

12

fnrm abc ma

1 Q2ncD er Q2eb m

1D en 2eaQ2ra

2naD er Q2ea 1

6fne

m f rqe abc m

1Q2qaQ2nbD gr Q2gc. #

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:

)( rn

ranar

mnrn

mnrm PQgFAgF

,0)(2

2

qbncrbc

emq

rne QQ

g

,0)( namn QD

0))((

2)( ebr

encm

nrbcnm

n QDQg

D

,0)(2

))(()( rbn

nemreab

ebrenmnrabna

nm QD

gQDgPD

.0)( nnmD

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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The fields are expressed by a finite set of color factors.

Am0 0, jAm

i 0, Ami t 1

g Omi f mt, Om

i Oni mn #

Qmar, t hm tumar # jQmar, t hm t jumar mnqhm tO jnuqar. #

One obtains the system of "mechanical" equations:..

fm

f m f2 f m 2 0. # ..

hm

t 2hm t f nhq 0, m n q, m,n,q 1, ,d. #

f i,hi, i 1,2,3

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:

Fm gmnrA

nF r 0 # D nm Qna 0 #

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II.1 The non-abelian gauge field(3) A four dimensional dynamics

Let us use the notations: f 1 x, f 2 f 3 y,h1 u,h2 h3 v

By choosing arbitrary constant factors, the system becomes:

vyuxv

vyuxu

yxyy

yxxx

43

1

2

)42

11(

)2

123(

..

..

22..

22..

)(;)( )3()2()1()3()2()1(..

hhhfffwwwMw T

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, uder, v, vder}, pretty different behaviors:

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Figure 1: Initial conditions forx, 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), uder(t), vder(t) .

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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.

References:

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(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