Rob (uni)bam. - bab6,-2,,. (2.7b). We see .... 6abb,,+; m 9 - Rob (V~)ham. From (3.14) and (3.19), we .... J. Goldstone and R. Jackiw, Phys. Lett. m, 81 (1978). 15.
mSLAC - PUB June 1985
- 3700
(T)
GAUGE-FIXING THE SU(N) LATTICE GAUGE FIELD HAMILTONIAN*
BELAL E. BAAQUIE t Stanford Stanford
Linear Accelerator
University,
Stanford,
Center
California,
94305
ABSTRACT We exactly gauge-fix the Hamiltonian
for the SU(N)
lattice gauge field and
eliminate the redundant gauge degrees of freedom. The gauge-fixed lattice Hamiltonian, in particular
for the Coulomb gauge, has many new terms in addition to
the ones obtained in the continuum
Submitted
-
formulation.
to Physical Review D
* Work supported by the Department of Energy, contract DE - AC03 - 76SF00515. t Permanent Address: Department of Physics National University of Singapore, Kent Ridge, Singapore 0511
1. INTRODUCTION The Hamiltonian
for QCD (quantum chromodynamics)
ied using the lattice and continuum
formulations.
has been widely stud-
In a remarkable
paper by
Drell,’ a derivation was given of the running coupling constant of QCD using the continuum
Hamiltonian;
this calculation used weak field perturbation
the Coulomb gauge. The mathematical Hamiltonian
treatment
theory and
of gauge-fixing the Yang-Mills
goes back to Schwinger ;2 the more recent paper by Christ and Lee3
gives a clear and complete treatment
of gauge-fixing the continuum
gauge field
Hamiltonian. The continuum
Hamiltonian
preserves gauge invariance; and Lee,3 a momentum for two-loops
has until now been given no regulation
for the one-loop calculation
invariance and renders the theory non-renormalizable; it is known that dimensional ficient to renormalize dimensional
regularization
the action.
regularization
QCD Hamiltonian
carried out by Drelll
cut-off is sufficient to ensure renormalizability.
and higher it is known that a momentum
which
However,
cut-off violates gauge-
for the action formulation
of the Feynmann
For the Hamiltonian,
diagrams4 is suf-
there is no analog of
and hence it is not clear how to regulate continuum
to all orders.
The lattice Hamiltonian516 is regulated to all orders and could be used for two loops or higher.
involving
Hamiltonian
using weak coupling approximation,
for example the Coulomb gauge. -
If we want to analyze the lattice
calculations
Gauge-fixing
it is necessary to fix a gauge, the action of the lattice gauge
&eory has been solved,7 and in this paper we extend gauge-fixing to the lattice Hamiltonian.
Gauge fixing essentially involves only lattice gauge-field and the
quarks enter only through the quark color charge operator. .
2
So we will essentially
study only the gauge field and introduce the quark fields when necessary. Gauge-fixing
the lattice Hamiltonian
the continuum
Hamiltonian;
formulation.7l8
For the Hamiltonian
is very similar in spirit to gauge-fixing
this similarity
can be clearly seen in the action
we will basically follow the treatment
given
by Christ and Lee.3 There are, however, significant differences between the lattice and continuum
Hamiltonians
both for the kinetic operator and the potential term.
The lattice gauge field is defined using finite group elements of SU(N) fundamental
as the
degrees of freedom whereas the continuum uses only the infinitesimal Th’is d’1ff erence will introduce
elements of SU(N). Given appropriate
generalized interpretation
a lot of extra complications.
of the basic symbols, it will turn out
however that the form of the gauge-fixed continuum
and lattice Hamiltonians
are
very similar. In Sec. 2 we discuss the Hamiltonian electric field operator. we perform freedom.
and give a construction
of the chromo-
We then discuss Gauss’s Law for the system. In Sec. 3
a change of variable and eliminate
the redundant
gauge degrees of
In Sec. 4 we evaluate Gauss’s Law for the new variables and find that
the constrained
variables decouple exactly from the Gauss’s constraint.
3 we evaluate the gauge-fixed lattice Hamiltonian, introduce
the quark charge operator.
In Sec.
discuss operator ordering and
In Sec. 6 we discuss the main feature of
our results.
3
2. DEFINITIONS Consider a d-dimensional Euclidean spatial lattice with spacing a; let Uni, i = 192 , . . . . d, be the SU(N)
link degree of freedom from lattice site
the unit lattice vector in the ith direction) field. The Hamiltonian
for SU(N)
and let &,
n to
n + i (2 is
$,, be the lattice quark
lattice gauge field in the temporal axial gauge
is given by5s6
H =
HYM[U] + HF[&$3u]
- $
c
(2.la)
where H
YM=
v2 (Uni)
n,i
-
$
C
Tr
(U,i
U,+;,juz+;,iU,G)
n,ij
and HF is the quark-gauge field part. operator.
The Hamiltonian
Gauge transformation
acts only on gauge-invariant
Laplace-Beltrami wave-functionals
and the wave-functionals
+
Uni (P)
G PnUni
P;t+;
(2.2)
@ are invariant under (2.2), that is
P-3)
W I = wb)l
- By performing
@.
is given by
Uni
_-.
Note V2 is the SU(N)
an infinitesimal
gauge-transformation
4
(and introducing
the
quark field) we have from (2.3) Gauss’s Law’
{@(Uni)
2
’
E,L(“n-;S}
-
Pna
[ i
The operators
1
I@)
EF and E,” are first order hermetian
with the commutation
=
(24
O
differential
operators
equation6
(2.5a)
[Ef,
E;]
Et
(U),
E,R(U)=Rab(U)
= -icab,
E,R
Rob (U)=
1
Ef
E,R,
where Rab is the adjoint representation,
Tr(Xa
(2.5b)
UXb U+)
--0
(2.5~)
(2.5d)
Xa the generators and C&c the structure
constants of SU(N). The operator
pna ($3 +, U) is the lattice quark color charge operator6
and
satisfies [ha,
Ptnb]
=
icabc
(2.5e)
Pnc &a
From (2.4) and (2.5~) we have
0 =
C
{ Rab(U,i)Ef’(U,i)
-
Ef(u,-I,i)
Pna
1
i
=-
r cm,i
Dtni
@(urni)
-
5
IQ)
Pna
1IQ>
(2.6a)
(2.6b)
where D~“,i is the lattice covariant backward derivative. of lattice site n and nonabelian
Let In, a) be a ket vector
index a; then, from (2.6) we have the real matrix
Di given by
D&i
=
(n,
aIQlm,
=
Rob (uni)bam
(2.7a)
b)
-
(2.7b)
bab6,-2,,
We see from above that Di performs a finite rotation
Rab on the ket vector and
then displaces it in the backward direction. We write the Hamiltonian
as sum of the kinetic and potential H
= K(U)
+ qJ,
?A q
energy, that is (2.8)
where K = -c
C
(2.9)
V’(Uni)
n,i
and P is the rest of (2.la).
It is known that9 -V2(U)
= c
(2.10)
Ek(U)E,L(U) a
In light of Gauss’s Law and (2.10) we identify tric operator
of the gauge field corresponding
canonical coordinates
E,L(Uni)
as the chromoelec-
to the link variable Uni. Choose
B~i such that Uni =
eXp(iB$ Xa)
-
(2.11)
Then we have, suppressing the lattice and vector indices and summing -
Tepeated nonabelian
on
indices (2.12a)
.
6
(2.12b)
Note e:b(“)
Explicit
(2.13)
= eii(“)
expressions for cab L(R) are given in (3.8).
3. GAUGEFIXING We can see from Gauss’s Law that all the Uni’s are not required to describe the gauge-invariant
wave-functional
a. We gauge-transform
variables Vni which are constrained;
the constrained
Uni to a new set of
variables Vni will decouple
from Gauss’s Law. Consider the change of variables from { Uni} to {pm, Vni}, with {Vni} having one constraint
for each n. That is tin Uni
= (P&9 = Pn
4n Ki
P+
= GaPZ
(3.la)
(3.lb)
n+;
and choosing the Coulomb gauge for the lattice gives xi(Vni)
z Im C
TrXa (Vni - Vn-;,i) = 0
(3.k)
i
In canonical coordinates we have
pn =
Vni = eXp(i&Xa), -
--For small variation
exp(i+iXa)
(3.2)
A” + dAa, we have V(A
+ dA) = V(A)
I+
V+(A)sdA”]
(3.3)
(3=4
[l + iXajz(A)dAb]
= V(A)
where
Define &p)Aa
= j,$@)(A)dAb
V(A
+ dA) = V(A)(l
then + iXa6RA”)
(3.7a)
= (1 + iXaGLA”)V(A)
(3.7b)
It can be shown that L(R) L(R) = (Ijab
eaa
and hence matrix (3.1) from function
Uni
t0
f ab
P-8)
e can be determined
from (3.5). Under the charge of variables
Vni, the potential
energy P in (2.8) can be expressed as a
of only Vnim For the kinetic energy K we need the expression for E:(U).
Note, using the chain rule and formula
(3.8)
w-4 =C
f,R,(Ani)z mi
n,i
Therefore, -
-
(3.10)
efp(Ani)$-+... ni
from (2.12) and (3.10)
Ef(umj)=’
(3.11)
iSL B~j
We now evaluate the coefficient functions .
a
of above equation.
The constraint
Eq.
(3.1~)
is valid under variations
of A~i to Aa,i + Dali,
i.e. (3.12)
0 = x;(A) = x&i
(3.13)
+ dA)
Hence, from (3.12) and (3.13) C
IT&i(A)
6RAki
(3.14)
= 0
m,i
where, for constraint
(3.lb) we have I?ab nma. =
(n, alI’ilm, b)
(3.15a) (3.15b)
= ~X~I~LA~i =
W$6,m
-
WlL;
ibn-;
9
,
(3.15c)
m
where from (3.1~) W$
The constraint
=
Tr (xavnixb
(3.16)
+ xbv$xa)
(3.14) on A~i determines 6p/6B.
Consider from (3.lb),
the
following variation
Jk(A
+ dA) = CP:(~ + d4)Uni(B
+ dB)pn+;(4
+ d4) .
(3.17)
which yields from (3.7a) bRA:i
--
= 6R4:+2
-
=- c D
&i
Rab(VL)JR$f:
bR6fn
+ Rab
+ Rab(‘Pi+;)6RB%i
(3.18)
(3.19)
m
From (3.18) and (3.19), we have the lattice covariant forward derivative operator .
9
Di given
by ab Dnmr . ='(n,alDilm, b) = 6abb,,+; 9m -
(3.20) (3.21)
Rob (V~)ham
From (3.14) and (3.19), we have
c
(%alriDilm,b)6R&,,
-I- c
m,i,b
(n, alriRT[m,
b)6RBki
(3.22)
= 0
m,i,b
where T stands for transpose and
bvlRilm,b)
(3.23)
= barn Rab (r~,+;)
Hence, from (3.22) we have (3.24) where (I’ - D)- ’ is the inverse of operator
Ci
I'iDi. We
also have from (3.19) and
(3.24) ‘RAii
_
- -
‘LBRj
1
n,a (
II
PiI--D
I’jRT
(3.25)
- RT&j
Hence, from (3.11), (3.24) and (3.25)
(3.26)
-
-
Equation moelectric .
(3.26) provides the solution for expressing the unconstrained
operator
6/6~B
in terms of the new constrained 10
operator
6/6~A
chroand
S/6,4.
the gauge transformation
i
In essence, this solves the problem of guage-
fixing the lattice Hamiltonian. Note that from (3.25) we have the identity
C(L,cll?iln,a) $$$
=0
(3.27)
mi
n,i
as expected. We have from (3.14)
C(n,alWd$$-- = 0
(3.28)
mi
m,i
Hence, from (2.12) and (3.28) 6 SL Aa,i ’
Ainj
] =
(6mrnbijJac
-
(%
a ( r?
&
rj(
m, C))
efb (Amj)
(3.29)
4. GAUSS’S LAW _
We check that constrained variables
Vni
decouple from Gauss’s Law. Recall
from (2.7) and (3.26), we have
-
-
-
c( I
1 r&-D?
I) ec
6
3 3 3 ’ s,4;
n,am
n,ij
(4-l)
From the definitions .
of
Di
and
Pi given
in (2.7) and (3.21) respectively, we have 11
the crucial operator identity
(4.2) where h
alRIm,
b) = Jnm
(4.3)
Rab(%)
Hence, from (4.2) we see that the first term in (4.1) is zero and we have
(44 -=
We see that
Vni
(4.5)
6R4;
has decoupled from Gauss’s constraint,
and we have from
(2.6) and (4.5) 6 + Pna ibR4;
IQ> = O
(4.6)
1
Solving (4.6)) we have from (13.1)~ qih
$9 U) =
&
w-$4%)
e-E,pn.4=
(4.7)
cp(