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Quasi-Classical Model in SU(N) Gauge Field Theory
A.V.KOSHELKIN
Moscow Institute for Physics and Engineering
CONTENTS
1. Introduction.
2. Statement of Problem and Main Goal.
3. Self-Consistent Solution.
4. Fermion and Gauge Field in Developed Model.
5. Application to QCD.
6. Conclusion.
PEOPLE
C.N.Yang, R.L.Mills, Rev. 51 461 (1954).
S.Coleman. Phys. Lett. B 70 59 (1977).
T.Eguchi. Phys. Rev. D 13 1561 (1976).
P.Sikivie, N.Weiss, Phys. Rev. 20 487 (1979).
R.Jackiw, L.Jacobs, C.Rebbi, Phys. Rev. D21 426 (1980 ).
R.Teh, W.K.Koo, C.H.Oh, Phys. Rev. D24 2305 (1981)
V.M.Vyas,T.S.Raju, T.Shreecharan. e-Print: arXiv:0912.3993[het-th].
D.D.Dietrich, Phys Rev. D 80 (2009) 067701.
A.Slavnov, L.Faddeev, Introduction to Quantum Theory of Gauage
Fields, 2nd enl. and rev. ed. Moscow, Nauka, 1988.
E.S.Fradkin, Nucl.Phys. 76 (1966) 588.
and so on, so on …
1. Introduction.
2.Statement of Problem and Main Goal.
)2(0}])([){(
0)(}])([{
mTxAgiix
xmTxAgii
a
a
a
a
�
;],[c
cabbaTfiTT
0)( xAa
)1()()()(
)()()()()(
)()()()(
xTxxJ
xAxAfgxAxAxF
xgJxFxAfgxF
aa
cb
bc
aaaa
ac
bc
aba
abbaTTTr
21},{
Key Approximations
.1)4
;0)()()3
;0)())(()2
));(()()1
YM
FYM
aa
n
x
xx
xAxA
The main goals are
1) to obtained such solutions that both the Yang-Mills and Dirac Equation would be satisfied together;
2) to quantize the fields;
3) to apply the obtained results to QCD .
The Yang-Mills equation
WE ASSUME
)()()()()()())(()()(2 2
ars
bsrc
cabc
bcabc
bcab gJAAAffgAAfkgxAAfkg
)()()( xTxJ aa
)(;;;0)(
)())(sin()())(cos()()(
)1()2()2()1()2()1()1()1(
)2()1(
xkeeeekekeee
xBxexeAA aaaa
The Dirac equation
Provided that
)(2
)()(
0)())(()()(2)( 222
xm
mTigAix
xTAkigTAigTAgm
aa
aa
aa
aa
YMFYM
dx
d ;1
)()()( ,, Fexx ipx
,,,,,0
3
,,,,,0
3
),()(ˆ),()(ˆ2
)(
),()(ˆ),()(ˆ2
)(
pxpbpxpap
pdx
pxpbpxpap
pdx
SOLUTION IS
(Koshelkin,Phys.Lett.,B683 (2010) 205)
)()2(),(),( 3,
*,
3 pppxpxxd
3. Self-Consistent Solution.
a) Gauge field
b) Fermion field
)(;;;0)(
)())(sin()())(cos()()(
)1()2()2()1()2()1()1()1(
)2()1(
xkeeeekekeee
xBxexeAA aaaa
,,,,,0
3
,,,,,0
3
),()(ˆ),()(ˆ2
)(
),()(ˆ),()(ˆ2
)(
pxpbpxpap
pdx
pxpbpxpap
pdx
c) Relation equations
The problem is solvable when the dimension of the gauge group . Thereat, the currents generated by fermions and gauge field exactly compensate each other.
3N
)()()()( xAxTxJ aaa
,,,,,0
322 )(ˆ)(ˆ)(ˆ)(ˆ
2)1(
)cos()cos()cos()sin(2
pbpbpapap
pdNCA
BN
ffff s
bsc
asrbrbsrc
cabba
cab
0)cos()cos( asrbsrc
cab ffC
0)(sin21
0,
2
2
ba
N
ba
CN
(Koshelkin,Phys.Lett.,B696 (2011) 539)
4. Fermion and Gauge Field in Developed Model.
In terms of the multi particle problem,
the solutions correspond to individual states of particles
the solutions correspond to collective states (Fermi liquid-like)
1;2)1(
23
0
2/1
30
2
TC
n
T
mT
TC
n
Ng
NA
1;)1(
2
)1(
2 3/20
2/1
2
2/1
2
T
TnNg
N
Ng
NA
Fermion effective mass.
IN EQUILIBRIUM
1)
2)
1;2)1(
23
0
2/1
30
2/1
2*
TC
n
T
mT
TC
n
Ng
Nm
1;1;3
03/20*
TC
nnm
. 5. Application to QCD
6.Final remarks and conclusion.
1.The self-consistent solutions of the non-homogeneous YM equation and the Dirac equation in the external YM field is derived in the quasi-classical model when the YM field is assumed to be in form of the eikonal wave.
2. The quantum theory of the considered model is developed in the quasi-classical approximation.
3. The considered model is solvable when the dimension of the gauge group and assumes that the fermion and gauge fields have to exist together .That is an alternative to Glasma model by L.D.McLerran and R.Venugopalan.
4. The relation of the developed model to the generally accepted point of view on the matter generated in collisions of heavy ions of high energies is considered.
5. The fermion and gauge fields derived in the explicit form allow to develop diagram technique beyond perturbative consideration.
3N