Introduction. Lattice gauge theories may well be one of the satis- factory candidates for a theory of the strong inter- actions. Wilson [1] originally proposed them in ...
Ze,tsch~ for Physik G Partic
Z. Phys, C - Particles and Fields 22, 77-81 (1984)
_
and Fields @ Springer-Verlag 1984
Residual Gauge Invariance of Hamiltonian Lattice Gauge Theories S. Ryang 1, T. Saito 2 and K. Shigemoto a 1 Department of Physics, Osaka University, Toyonaka, Osaka 560a, Japan 2 Department of Physics, Kyoto Prefectural University of Medicine, Kyoto 603, Japan 3 Department of Liberal Arts, Tezukayama University, Tezukayama 4, Nara 631, Japan Received 16 July 1983
Abstract. The time-independent residual gauge invariance of Hamiltonian lattice gauge theories is considered. Eigenvalues and eigenfunctions of the unperturbed Hamiltonian are found in terms of Gegenbauer's polynomials. Physical states which satisfy the subsidiary condition corresponding to Gauss' law are constructed systematically.
1. Introduction Lattice gauge theories may well be one of the satisfactory candidates for a theory of the strong interactions. Wilson [1] originally proposed them in the space-time discrete Lagrangian formalism, while Kogut and Susskind [2] reformulated them on a three-dimensional space lattice in the time-continuous Minkowski Hamiltonian formalism9 The later authors noticed that in a temporal gauge the dynamics is that of a collection of coupled rigid rotators in the strong coupling region. The latter approach is related with the former approach by the transfer matrix of statistical mechanics, or another methods [3]. In a previous paper [4] we have considered the residual-gauge invariance of the Hamiltonian lattice gauge theory and discussed a role of the longitudinal gluon in both Lagrangian and Hamiltonian lattice gauge theories. In the present paper we further investigate this residual-gauge invariant Hamiltonian lattice gauge theory, and find eigenvalues and eigenfunctions of the unperturbed Hamiltonian in terms of Gegenbauer polynomials. We clarify the relations between our eigenfunctions and the usual link variables. This makes it possible to construct physical states which satisfy the subsidiary condition corresponding to Gauss' law. These results suggest us various approximation methods in the Hamiltonian lattice
gauge theory by making use of these physical states, for instance, perturbative methods in strong coupling region, variational methods [5] in any coupling region, etc. This paper is organized as follows: In Sect. 2 we derive the residual gauge invariant Hamiltonian and discuss the algebra among the generators of right- and left-handed gauge transformations 9 In Sect. 3 the eigenfunctions for the kinetic part of our Hamiltonian are found9 In Sect. 4 we derive the relations between the eigenfunctions and the link variable and construct the physical states9 The final section is devoted to discussions and summary.
2. Residual Gauge Invariant Hamiltonian We summarize in this section the results of our previous paper. We consider the S U(2) gauge model with time continuous but space discrete lattice. In order to preserve the discrete space-dependent gauge symmetry, we take the following Lagrangian xoith the temporal gauge A~ = 0: L=~
.
o
Tr
\
ot
ot
9{Tr(U,(x, t) Uj(x + fa, t) U + (x
E
(1)
i~ j,x
+]a, t) U2 (x, t)) + h.c.]
where
0~(x, t) =
agAT(x, t),
a being the lattice spacing and g the coupling constant. In the continuum limit this Lagrangian is reduced to the conventional one with the temporal gauge. This Lagrangian is invariant under the following time-
S. Ryang et al. : Residual Gauge Invariance
78
we see the generator of this is given by
independent residual gauge transformation:
10~1 J0il'~/o.c
U,(x, 0 ~ V(x) G(x, OV+ (x + ta), U~+(x,t) ~ V(x + fa)U+(x,t)V+(x).
If instead of the kinetic term in (1) we take the following on e:
0'~
--
+ e abe ~ t- , n c, ; +.
a 3 0a(x, 0 ~ 0a(x, t) 2g 2 ,., , c3t (~t
(no sum on 0
this is not invariant under the above gauge transformation for finite lattice size. In the following it may be convenient to choose 0a(x, t) as a dynamical variable rather than U~(x, t) itself, because U~(x,t) is constrained by U + Ui = 1, whereas 0~(x, t) is not. In order to define the canonical m o m e n t u m rc~(x,t) conjugate to 07(x,t), we then express the kinetic part in (1) in term of 0a(x, t) as follows:
U,(O.,) ---,d ~~
a
olo7"~
(2) (10)
Similarly, for the left-handed gauge transformation
"a
T(kinetic) = ~ ~ O~(x, 0 M'~b(Oi(x, t)) O~(x, O, zg i,x where
/sinlO, I / 2 \ 2 / ,b M'~b(Oi(x'O)= t 1 0 ~ ) t 6
6~0~'~ 0~0~ 10i[2]+ [0,[2,
(3)
(4)
U,(O),
(11)
the generator of this is G~(0) = 89 a ac -
9/.c
(
0~0~/
- io,i 2 .
1 - Tco~T) ~abc
-
0bi
c
- S ' ;+ "
(12)
(no sum on 0 After somewhat tedious but straightforward calculation, we see that the generators G~ and G~ satisfy the following algebraic relations: [U,(x, t), G~(O,(x, t))] ---
.Ca
-
U,(x, t)~-, .Ca
and
[U~(x, t), G[(0,(x, t))] = - ~- U,(x, 0,
det(MO~) = (sin I0l/2 ,]4 \
(5)
1o~72 )
The conjugate m o m e n t u m ha(x, 0 is therefore defined by 8L _ a n~(x, t) - 801(x, t) g2 Mab(o')O~( x' t),
(6)
(14)
and E G O ) , G ( o ? ] - le "~ GR(O)6u, [GL(0), G~(0.,)] = - ze " " c GL(Oi)6U, "
(15) (16)
[G~(O), G[(0i) ] = 0.
(17)
a
--
Furthermore, we have for the Casimir operators (G~) 2 = (G~) 2
hence the Hamiltonian becomes 92 H = 2 a ~ n~'(x, O(M- lybn~(X, 0 + V(potential),
(7)
= rc~ -t
L2/4
1
sin2101/2
4'
(18)
where
where O"Ob'~
L ~ - C"b~obrc~,
0~ 0 b
L 2 = (La)2,
(19) (20)
"
(21)
10~ J + -101 ~-' 1
~0-=~
2gZ a i y',
V(potential) -
9{Tr(U,(x, 0 Ui(x + fa, 0 U+ (x + ]a) U + (x, 0 + h.c. }. The ordering of operators here will be defined later in such a way that the Hamiltonian becomes residual gauge invariant. For this purpose we put the commutation relations [0~(x, t), n~(y, t)]
(13)
=
i(~aO(~ij(~x,y etc.
(8)
For the right-handed gauge transformation
Ui(O.) ~ Ui(O)e i~~176 (e": infinitesimal)
(9)
(0a z i~+101
and [ 101,no] = i, [L ~, 101] = [L~, no] = O, EL~, L b] = ie"b~LL
(22) (23) (24)
Note that L~ corresponds to the angular momentum operator in the 0"-space. It may be easily seen that the Casimir operator a 2 --(GD a 2 is equal to l~a(M-1)ablr, b in the Hamil(GR) tonian (7) if the operator ordering is neglected. Therefore, we now fix the ordering of the operators in the
79
S. Ryang et al. : Residual Gauge Invariance
Hamiltonian in such a way that the kinetic part in (7) becomes residual-gauge invariant. Namely, we take the Casimir operator (G]) 2 itself instead of the kinetic term in (7), i.e.,*
Since the normalization of the wave function is defined by the S U(2) invariant measure [d#(O)] as
. [d30]/sin 210[/2\ 2 1 = j l ~ n 2 ~ ~[]~ ) q~+(O")q~(O")
92 = 2aa ~ [G~(0i(x))]2
H~
= 5 4~2 sin 2 IOI/2R+(IOI)R(IOI),
9z - ~a ~ E=a(M'- ~)=~ - 88 __ 92a ~ I r c ~ , +
we should represent rc0 as
1 d in~ - sin 10[/2 dlOI sin 10[/2,
L~/4
Hence the Schr6dinger equation is given by
H gJ = E 7"
(26)
(35)
in order to make s o Hermitian. From this we have the Schr6dinger equation for R(]0I)
d2 R
~-n 1(l + 1) d0~2 2 c o t e - ~ + ~ R
with H = Ho(kinetic) + V(potential).
(27)
Note that, starting with this Hamiltonian, we obtain the Lagrangian formalism with the correct S U(2) invariant measure [d#(0)], i.e., [d#(0)] = I~ [d0~'(x, t)] [det(M~b)] 1/z
{ sin 10,1/2'~ 2 7
Let us solve the free Schr6dinger equation (29)
where H o is given by (25). Since H o is composed of an ensemble of the non-interacting fields, the wave function 7"o can be solved as a direct product of the function of independent points, i.e.,
7"0 = l~ 4~(O~'(x)),
(36)
R(e) = sift ~ C(e),
(37)
then Eq. (36) is reduced to
dC 1)cot~-l(l
+ 2)C= - 42C,
(38)
dZC dC (z 2 - 1)~-z2 + ( 2 / + 3 ) - d ~ z - ( 4 2 - 1 2 - 2 / ) c = o ,
(39)
or
3. 'Free' Eigenfunetions Ho 7"0 = E0 7'0,
=42R,
where e = [0[/2 and 2 =(2a/92)8. If we put
d 2 Cs + 2(l
i,X
V......
(34)
where z=cos0c This equation is nothing but Gegenbauer's differential equation, and the nonsingular solution of this is given by the Gegenbauer polynomial
C(z) = C', +- ~(cos 101/2)
(40)
with eigenvalues
(30)
i,x
where 4~(O~(x)) satisfies
2t
and
)
g 2 _ 88 ~aa re~ -~ sin 2 IOi[/2 -~b(O~'(x)) -= e,,~~(Oa(X))
1 = 0, 1,2 ..... n. (31)
and Go = E/3i,x"
(32)
i,x
(42)
To sum up, the eigenvalues and the eigenfunctions of (31) are
92 n[n_ 1), 8,,x = Ta "~l, ~ + q~,t=(0~'(x)) = N,, sin z]Oi]/2Ct,+_~(cos i01i/2) Y~m(f2.), n = 0 , 1 , 2 .... ; l = 0 , 1 , 2 , . . . , n ; m = - - l ..... l; (43)
In order to solve this equation we put
9 (0 ~) -- R(101) Ytm(Q),
= 2t+l[(n + 1)(n -- /) !(1!)2 ]1/2
where 0 denotes angle variables in the three dimensional O%pace and
N,t
L 2 Y,m(Q) = l(l + 1) Ytm((2).
C"+-l(c~
(33)
9 We understand that T. Maskawa has given the similar Hamiltonianand obtainedthe invariantmeasurein the Lagrangian formalism.
L_
(n+l+l)!
rcj
,
"-t F(I + 1 + k)F(n + 1 - k)
k=OZ~ n - - - / - S k~.V~ 0
Iol .cos(2k - n + / ) T .
+ 1)] y
8O
S. Ryang et al. : Residual Gauge Invariance
We see that the total degeneracy of the energy level n is (n + 1)2.
4,2 =(4,111 -- 4,11,- 1)/iN/~,
4. Physical States
Hence substituting the compact form (51) into (48), we can express the gauge invariant state in terms of our eigenfunctions with n = 1. Next, in order to construct the physical state with n = 2, we consider the tensor product U~p(U +)ya, which is reduced to the following form:
By using the right and left-handed gauge generators G~ and G~, the generator of the residual gauge transformation (2) is given by 3 G(x) ~ Z [GaR(Oi(x))ga(X) -- GaR(Oi(X))ga( X § fa)] i=1
4'3 = 4'11o,
(52)
1 Uafl(U+)v3 = ~ 6 ( ~ 1 3 y
(44) with the property
..~ 1
ab
~U(3)("c
b
)~a(za)~,~,
(53)
where U{'3b, = 89Tr ( U + z b U r
Ta
.Ca
[ U,(x), G(x)] = - ~- e"(x) U~(x) + U,(x) ~ ~~ +
ta). (45)
By taking the linear combination of G(x, t) with respect to the position and eliminating the gauge parameters g~ the generator of the residual gauge transformation, which corresponds to the 'Gauss operator' D'ZbE} in the continuum limit, is given by
(54)
Since U(~) can be decomposed as
.b
(
0"0"'~ 0"0 b
v 3, = cos t01.a ~ - 10?-) +
10f
t3abc sin 101 1-74- 0
(55)
we can express 4,2z= in terms of UI'}) as follows-
3
Q"(x) = ~, {G[(0~(x)) - G,~(0~(x - fa))}.
(46)
-
+ i ( u g , + u2
]
i=1
The subsidiary condition, which corresponds to Gauss' law and eliminates the longitudinal mode of the gluon, should be imposed on the state T, i.e., Q"(x) T = 0
(47)
4,=,0 = @ [2 v s -
for any x. A few physical states which satisfy the condition (47) are shown below in terms of our eigenfunctions (43). As usual we can construct the gauge invariant states by constructing the link variables U/(x) along a closed path on the lattice, i.e.,
4,21,0 - N2~ EU(3)21 U~],
T(C) = Tr (Ui(x) U j ( x -oF fa) ..... Uv(x - fia)),
4,20,0 = Tr U(3 ).
(48)
where C is a space-like closed loop. Let us write U,~ as
/{ U=a=\c~
101
0. i__zasin[O[~.
101
2 ~/=a
(49)
Then the eigenfunctions 4,,~,, with n = 1 can be expressed in terms of U,a as follows:
- vs
4,2,,_+ 1 -- ~'~3 F U ~ - U~z~ § 2 -
(3)~a,
(55)
Conversely, U("3b) is expressed by the above 4,% then we can construct the gauge invariant state in terms of the eigenfunctions with n = 2 as I//(n = 2)(C) : Tr(U(3)i(x ) U(3)j( x _}_fa)... U(3)p(X -- fia)),
(57)
4,111 = -- i N / 2 U 2 1 ,
The gauge invariance of ~"=2)(C) may be apparent because under the gauge transformation (2), each Ut'3b) transforms as
(/)11,-1 = - - i N / / 2 U 1 2 ,
ab U(3)i(X ) = W - 1 (X)ac ucd(3)i(X) W db (x + fa),
4,1oo = Tr(U),
4,a 1o = --
i(U11
-- U22)"
(50)
where W "b is the adjoint representation of V in the sense of
(51)
v + T a v = w a b T b.
Conversely U=p is expressed by the above 4,'s:
U~p = 89
§ Ta4,a)ctl3,
where 4,0 = r
4,' = ( 4 , ~ + ~l 1,- ~)/,,f2,
(58)
5. Discussion and Conclusion We have constructed the lattice Hamiltonian so as to preserve the residual-gauge invariance. The kinetic
81
S. Ryang et al. : Residual Gauge Invariance part of this is given by the Casimir o p e r a t o r of the residual-gauge transformation. We found eigenvalues and eigenfunctions of this kinetic part in terms of Gegenbauer's polynomials. Owing to the requirement of residual-gauge invariance, our lattice H a m i l t o n i a n has a slightly different form from the K o g u t - S u s s k i n d Hamiltonian, but the eigenvalues of the kinetic part are the same to theirs. We represented the usual link variables in terms of our eigenfunctions, i.e., Gegenbauer's polynomials. F r o m this it became possible to construct systematically physical state which satisfy the subsidiary condition corresponding to Gauss' law. As examples, we have constructed explicitly the physical states with n = 1 and n = 2, which are given by (48) and (57), respectively. We suppose that they correspond to bare glue ball-like states with even parity. In general the physical state with 'principal q u a n t u m n u m b e r ' n m a y be represented by ~(,)(C) = Tr [U(n)I(X) U(n)j(X + i'a)... U(,)v(x - - / ) a ) ] ,
where U~,) = exp(iTaO a) and T" is spin n/2-representation of colour S U(2). O u r formulation suggests us various a p p r o x i m a t i o n m e t h o d s in the H a m i l t o n i a n lattice gauge theory by making use of these physical states, for instance, perturbative calculations in strong coupling region, variational calculations in any coupling region, etc. [5]. These are left for future studies.
References 1. K. Wilson: Phys. Rev. DI0, 2445 (1974) 2. J. Kogut, L. Susskind: Phys. Rev. Dll, 395 (1975) 3. G.P. Canning, D. Frrster: Nucl. Phys. B9I, 151 (1975); V. Baluni, J.F. Willemsen: Phys. Rev. D13, 3342 (1976); M. Creutz: Phys. Rev. D15, 1128 (1977) 4. S. Ryang, T. Saito, K. Shigemoto: Hamiltonian lattice gauge theories and a role of longitudinal modes. Kyoto Prefectural University Prepfint 5. J.P. Greensite: Nucl. Phys. B166, 113 (1980); D. Horn, M. Weinstein: Phys. Rev. D25, 3331 (1982); A. Patkbs: Phys. Lett. ll0B, 391 (1982); T. Hofsiiss, R. Horsley: Phys. Lett. 123B, 65 (1982)
