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Gauge theory of quantum groups
0083-6656/93 $24.00
Vistas in Astronomy, VoL 37, pp. I 11-114,1993
@1993Perpmon PressLtd
Printedin C,reat Britain.Allrigbmm~ea'ved.
GAUGE THEORY OF QUANTUM GROUPS MinoruHirayama Toyama University, Toyama 930, Japan
The gauge field theory of the quantum group SUq(2) is formulated. The parallelism to the case of SU(2) is maintained with the help of the careful definition of SUq(2) gauge transformations preserving some group-like properties. The Lagrangian invariant under local SUq(2) is obtained.
Nearly forty years ago, Yang and ~Xlills (1954) succeeded in formulating the SU(2) gauge theory. On the other hand, the notion of the Lie group has been generalized recently to some noncommutative and noncocommutative Hopf algebras, which are now called quantum groups. The most fundamental example of quantum group is SUq(2), containing a real parameter v. It is a generalization of SU(2) in the sense that it reduces to SU(2) when v equals 1. We here investigate how the SU(2) gauge theory of Yang-Mills can be generalized to the case of SUq(2). The matrix entries of the fundamental representation of SUq(2) are noncommutative operators and SUq(2) is not closed for the conventional product. So a special care must be taken in the definition of the product of gauge transformations. The interesting fact is that the product denoted by in Woronowicz (1987) and by @ in Reshetikhin et al. (1990) inherits some group-like properties. The product ~ is intimately related to the coproduc t of SUq(2). It was shown by Masuda et al. (1991) that the operator entry I[qj of the irreducible unitary representation of SUq(2) can Z .'#¢ be chosen so as to satisfy ~h,l ¢rlct~IkiI|lj ~" #i.i [. Here, I is the uhit operator and or a diagonal numerical matrix with ~kl = u~(k-1)6kt- If each Wi, i = 1, 2,-- -, m satisfies the above relation, the product W - W , , ~ I I ~ _ ~ • -. ~ W , (a) satisfies
~aw~,~I]~ = ~ j . ,
(2)
k.I
with Im = I ® I ® . " ® I (m times). We hereafter regard the local version IV(z) = W,,(z) (2)~ - 1 ( z ) ~ - - - ~ [ I ~ ( z ) of W,,,(~II%_I ~ ) . . . (~Ilr~ as the matrix specifying the SUv(2) gauge, where x denotes the coordinate of the flat spacetime. The differential calculus on SUq(2) has been investigated by some authors. We consider a function g[z] - g (z, t~(x), 7(x), t~*(z), 7*(x)) where t~(x), etc., are the SUq(2) coordinates at z and are noncommutative operators. With the held of the differential calculus of Woronowicz (1987. 1989~.
M. Hirayama
112
the partial derivative D~,g[z] with respect to z ~ can be defined so that the Leibniz rule and the commutativity are maintained:
D.(g[x]h[z]) = (Dug[zl)h[z]+g[z]Duh[x],
(3)
D~,D.g[x] = D,.Dt,g[z].
(4) t
$.
- W ~
We are to define the gauge potential A~V(x) in the gauge IV and its gauge transform (A~T (x)) by W'(x) = W~'(z)(DI, V~_I(x ) ~ . - -
x TI"e
A. (~))
= AT,~I,~(~),
(5)
corresponding to the viewpoint stated in the previous paragraph. We define the gauge field by F ;IT" , ( z ) = I V W, , V .W] ,
V .W = 0 , +
igA~V(z).
(6)
It is desirable that A~V(x) and F~, IT."(x) are hermitian fields satisfying IV
(7)
It turns out that the appropriate definitions o f . C ( ~ ) and /ha.IT" (~)),, Tl,'e are
(s)
k=O
1
--- ( D . W ( x ) ) W(x) -I, ig
At...IT-,= (~)) (w'(~) ® r,~) (~. ® .a',T'(~)) (w'(~)-~ ® ~.) _ 1 ( P . W ' ( z ) ) W ' ( x ) -1 ® Ira, mg where ah,t,(x), k = 0, 1, 2 are operator-valued component fields and ical matrices satisfying
(9)
Xk(W),
k = 0, 1, 2 are numer-
x ~ ( w ) x , ( w ) - c,~X,(W)X~(W) = ~ dm~,x,,,(w).
(lO)
2
rtt-~O
For algebraic properties ofak.~(z), see Hirayama (1992). In (10), the numerical factors c~h and d,nht are defined by Ckk = 1, Cuo = (Cou)-t = v u, CUt = CI0 = (OU)-t = (C0,)-~ = v a, d,20 = v -*, d, ou = - v , dulu = doo~ = -vU(1 + vU), duut = d0,o = v-2(1 + v u) and d,~kt = 0 otherwise. We notice that A~I'(z) and (A WCx)) n', belong to different spaces:
w
r#~,
w'
F#(,,,+n),
(11)
where I' is some operator ring. The transformation law of the gauge field is induced from (6) and
(9) as W "TT'~ F;.(x)) =
(W'(z)®
In)(In® F::'(z))(W'(z)-1®In).
(12)
Gauge Theory of Quantum Groups
113
To keep the parallelism to the Yang-Mills case, the Lagrangian density of the SUq(2) gauge theory should be a bilinear combination of -Fwt~ "~ It should be independent of the choice of W(z), the /z~, ~,--I" dimensionality N of W(z) and the integer m. W e begin with defining Ski by
Skt= tr (o-l Xh ( Wt )~-2 X,( Wt ) ) , k, I = O, 1, 2
(13)
where I tq is a representation of SUq(2). We find that Sht vanishes unless (k, I) = (0, 2), (2, 0), (1, 1). The important property of Sht is that the ratios of its non-vanishing elements do not depend on IVy: S0~: S~0 : Stt = - 1 : - u 2 : v ( 1 + v2). (14) We note that the ratios of the non-vanishing elements of any simpler expressions, e.g., tr (Xh(WL) Xt(Hq)) and tr (oXk(1$q)Xt(Wt)) depend on the IVL adopted. We define L(x) by 2
L(~) = ~ ~ . s~,f~..~(~)f~"(~).
(15)
k,l=O
It can be written as
L(~) = g {f0,.~(~) f~ (~) + (1 + v~)fl,.~(z)*f~(z) + f2,.~(z)*f~'~(z)} •
(15a)
In (15), t¢ and h,gv(x) are given by
=
-8 ($20) -~ ,
(16) 2
(17) pjq=O
We see that L(x) is independent of Wt. Turning back to the case of the N × N matrix W(x) = W,.(z) ~IV,,,_t(z) (D"" ~W~(z), we define Lw(x) by LWCz) = d r (oG~V Cx)GrL"~Cx)) ,
(18)
a.~(x) " = ~(W)F'~C~)rCW),,
(19)
rCW) : ( W ~ ( ~ ) ¢ . . . e W ~ ( ~ ) ~ - ' ~ ) W C x )
-~.
(20)
We then find that LW(x) is related to L(x) by
This result shows that LW(z) depends on W(z) only through the integer m. Reminding that we have defined the SUq(2)gauge transformations by the ~ product, it is not unnatural to introduce the equivalence relation I ® a ~ a. (22) Now LW(x) is equivalent to L(x):
:(~)
~ L(~).
(23)
We interpret (23) as the gauge invariance of LW(z). We regard L(z) as the Lagrangian of the SUq(2) gauge theory since it becomes the Yang-Mills Lagrangian for u = 1. We have thus seen that the SUq(2) gauge theory can be formulated in a parallel way to the case of SU(2). The
M. Hirayama
114
coupling of the matter field with the gauge potential can be introduced through V~VffW(z). Here, the matter field ~w(x) in the gauge IV is defined by =
®
(24)
~(x) being a column vector independent of W(x). The gauge invariant Lagrangian can be constructed in a usual way. We finally note the crucial difference of the SUq(2) gauge theory from the SU(2) one. Let U and V be L- and N- dimensional representations of SUq(2), respectively. In accordance with Woronowicz(1987), the tensor product of U and V is denoted by U ¢DV. U ¢DV is also a representation of SUq(2), the dimension of which being LN. In the case of gauge theory of Lie groups, we have A~©V = A~, v ~E~v + EL ¢~Au v (Yang-Mills), (25) where Eu is the N-dimensional unit matrix. On the other hand, for the Sgq(2) ease, we have
._,:+'"= A,"+E,, + (V +E,,) (E,.®AV) (u-'+E,,,). W e see that .4~(Dv and
(26)
A~"(D[:are not related simply.
Hirayama, M. (1992) Gauge field theory of the quantum group SUq(2). Progr.Theor.Phys. 88, No. 1. Masuda, T., Mimachi, K., Nakagami, U., Noumi, M., and Ueno, K. (1991) Representations of the quantum group SUq(2) and the little q-Jacobi polynomials. J.Funct.Anal. 99, 357. Reshetikhin, N. Yu., Takhtadzhyan, L. A., and Faddeev, L. D., (1990) Quantizations of Lie groups and Lie algebras. LeningradMath.Z 1,193. V¢oronowicz, S. L. (1987) Twisted SU(2) group. An example of a non-commutative differential calculus. PubI.RIMS, KyotoUniv. 23, 117. Woronowicz, S. L. (1989). Differential calculus on compact matrix pseudogroups (quantum groups), Commun.Math.Phys. 122 125. Yang, C. N. and Mills, R. L. (1954) Conservation of isotopic spin and isotopic gauge invariance. Phys.Rev. 96, 191.
Vistas in Astronomy, VoL 37, pp. I 11-114,1993
@1993Perpmon PressLtd
Printedin C,reat Britain.Allrigbmm~ea'ved.
GAUGE THEORY OF QUANTUM GROUPS MinoruHirayama Toyama University, Toyama 930, Japan
The gauge field theory of the quantum group SUq(2) is formulated. The parallelism to the case of SU(2) is maintained with the help of the careful definition of SUq(2) gauge transformations preserving some group-like properties. The Lagrangian invariant under local SUq(2) is obtained.
Nearly forty years ago, Yang and ~Xlills (1954) succeeded in formulating the SU(2) gauge theory. On the other hand, the notion of the Lie group has been generalized recently to some noncommutative and noncocommutative Hopf algebras, which are now called quantum groups. The most fundamental example of quantum group is SUq(2), containing a real parameter v. It is a generalization of SU(2) in the sense that it reduces to SU(2) when v equals 1. We here investigate how the SU(2) gauge theory of Yang-Mills can be generalized to the case of SUq(2). The matrix entries of the fundamental representation of SUq(2) are noncommutative operators and SUq(2) is not closed for the conventional product. So a special care must be taken in the definition of the product of gauge transformations. The interesting fact is that the product denoted by in Woronowicz (1987) and by @ in Reshetikhin et al. (1990) inherits some group-like properties. The product ~ is intimately related to the coproduc t of SUq(2). It was shown by Masuda et al. (1991) that the operator entry I[qj of the irreducible unitary representation of SUq(2) can Z .'#¢ be chosen so as to satisfy ~h,l ¢rlct~IkiI|lj ~" #i.i [. Here, I is the uhit operator and or a diagonal numerical matrix with ~kl = u~(k-1)6kt- If each Wi, i = 1, 2,-- -, m satisfies the above relation, the product W - W , , ~ I I ~ _ ~ • -. ~ W , (a) satisfies
~aw~,~I]~ = ~ j . ,
(2)
k.I
with Im = I ® I ® . " ® I (m times). We hereafter regard the local version IV(z) = W,,(z) (2)~ - 1 ( z ) ~ - - - ~ [ I ~ ( z ) of W,,,(~II%_I ~ ) . . . (~Ilr~ as the matrix specifying the SUv(2) gauge, where x denotes the coordinate of the flat spacetime. The differential calculus on SUq(2) has been investigated by some authors. We consider a function g[z] - g (z, t~(x), 7(x), t~*(z), 7*(x)) where t~(x), etc., are the SUq(2) coordinates at z and are noncommutative operators. With the held of the differential calculus of Woronowicz (1987. 1989~.
M. Hirayama
112
the partial derivative D~,g[z] with respect to z ~ can be defined so that the Leibniz rule and the commutativity are maintained:
D.(g[x]h[z]) = (Dug[zl)h[z]+g[z]Duh[x],
(3)
D~,D.g[x] = D,.Dt,g[z].
(4) t
$.
- W ~
We are to define the gauge potential A~V(x) in the gauge IV and its gauge transform (A~T (x)) by W'(x) = W~'(z)(DI, V~_I(x ) ~ . - -
x TI"e
A. (~))
= AT,~I,~(~),
(5)
corresponding to the viewpoint stated in the previous paragraph. We define the gauge field by F ;IT" , ( z ) = I V W, , V .W] ,
V .W = 0 , +
igA~V(z).
(6)
It is desirable that A~V(x) and F~, IT."(x) are hermitian fields satisfying IV
(7)
It turns out that the appropriate definitions o f . C ( ~ ) and /ha.IT" (~)),, Tl,'e are
(s)
k=O
1
--- ( D . W ( x ) ) W(x) -I, ig
At...IT-,= (~)) (w'(~) ® r,~) (~. ® .a',T'(~)) (w'(~)-~ ® ~.) _ 1 ( P . W ' ( z ) ) W ' ( x ) -1 ® Ira, mg where ah,t,(x), k = 0, 1, 2 are operator-valued component fields and ical matrices satisfying
(9)
Xk(W),
k = 0, 1, 2 are numer-
x ~ ( w ) x , ( w ) - c,~X,(W)X~(W) = ~ dm~,x,,,(w).
(lO)
2
rtt-~O
For algebraic properties ofak.~(z), see Hirayama (1992). In (10), the numerical factors c~h and d,nht are defined by Ckk = 1, Cuo = (Cou)-t = v u, CUt = CI0 = (OU)-t = (C0,)-~ = v a, d,20 = v -*, d, ou = - v , dulu = doo~ = -vU(1 + vU), duut = d0,o = v-2(1 + v u) and d,~kt = 0 otherwise. We notice that A~I'(z) and (A WCx)) n', belong to different spaces:
w
r#~,
w'
F#(,,,+n),
(11)
where I' is some operator ring. The transformation law of the gauge field is induced from (6) and
(9) as W "TT'~ F;.(x)) =
(W'(z)®
In)(In® F::'(z))(W'(z)-1®In).
(12)
Gauge Theory of Quantum Groups
113
To keep the parallelism to the Yang-Mills case, the Lagrangian density of the SUq(2) gauge theory should be a bilinear combination of -Fwt~ "~ It should be independent of the choice of W(z), the /z~, ~,--I" dimensionality N of W(z) and the integer m. W e begin with defining Ski by
Skt= tr (o-l Xh ( Wt )~-2 X,( Wt ) ) , k, I = O, 1, 2
(13)
where I tq is a representation of SUq(2). We find that Sht vanishes unless (k, I) = (0, 2), (2, 0), (1, 1). The important property of Sht is that the ratios of its non-vanishing elements do not depend on IVy: S0~: S~0 : Stt = - 1 : - u 2 : v ( 1 + v2). (14) We note that the ratios of the non-vanishing elements of any simpler expressions, e.g., tr (Xh(WL) Xt(Hq)) and tr (oXk(1$q)Xt(Wt)) depend on the IVL adopted. We define L(x) by 2
L(~) = ~ ~ . s~,f~..~(~)f~"(~).
(15)
k,l=O
It can be written as
L(~) = g {f0,.~(~) f~ (~) + (1 + v~)fl,.~(z)*f~(z) + f2,.~(z)*f~'~(z)} •
(15a)
In (15), t¢ and h,gv(x) are given by
=
-8 ($20) -~ ,
(16) 2
(17) pjq=O
We see that L(x) is independent of Wt. Turning back to the case of the N × N matrix W(x) = W,.(z) ~IV,,,_t(z) (D"" ~W~(z), we define Lw(x) by LWCz) = d r (oG~V Cx)GrL"~Cx)) ,
(18)
a.~(x) " = ~(W)F'~C~)rCW),,
(19)
rCW) : ( W ~ ( ~ ) ¢ . . . e W ~ ( ~ ) ~ - ' ~ ) W C x )
-~.
(20)
We then find that LW(x) is related to L(x) by
This result shows that LW(z) depends on W(z) only through the integer m. Reminding that we have defined the SUq(2)gauge transformations by the ~ product, it is not unnatural to introduce the equivalence relation I ® a ~ a. (22) Now LW(x) is equivalent to L(x):
:(~)
~ L(~).
(23)
We interpret (23) as the gauge invariance of LW(z). We regard L(z) as the Lagrangian of the SUq(2) gauge theory since it becomes the Yang-Mills Lagrangian for u = 1. We have thus seen that the SUq(2) gauge theory can be formulated in a parallel way to the case of SU(2). The
M. Hirayama
114
coupling of the matter field with the gauge potential can be introduced through V~VffW(z). Here, the matter field ~w(x) in the gauge IV is defined by =
®
(24)
~(x) being a column vector independent of W(x). The gauge invariant Lagrangian can be constructed in a usual way. We finally note the crucial difference of the SUq(2) gauge theory from the SU(2) one. Let U and V be L- and N- dimensional representations of SUq(2), respectively. In accordance with Woronowicz(1987), the tensor product of U and V is denoted by U ¢DV. U ¢DV is also a representation of SUq(2), the dimension of which being LN. In the case of gauge theory of Lie groups, we have A~©V = A~, v ~E~v + EL ¢~Au v (Yang-Mills), (25) where Eu is the N-dimensional unit matrix. On the other hand, for the Sgq(2) ease, we have
._,:+'"= A,"+E,, + (V +E,,) (E,.®AV) (u-'+E,,,). W e see that .4~(Dv and
(26)
A~"(D[:are not related simply.
Hirayama, M. (1992) Gauge field theory of the quantum group SUq(2). Progr.Theor.Phys. 88, No. 1. Masuda, T., Mimachi, K., Nakagami, U., Noumi, M., and Ueno, K. (1991) Representations of the quantum group SUq(2) and the little q-Jacobi polynomials. J.Funct.Anal. 99, 357. Reshetikhin, N. Yu., Takhtadzhyan, L. A., and Faddeev, L. D., (1990) Quantizations of Lie groups and Lie algebras. LeningradMath.Z 1,193. V¢oronowicz, S. L. (1987) Twisted SU(2) group. An example of a non-commutative differential calculus. PubI.RIMS, KyotoUniv. 23, 117. Woronowicz, S. L. (1989). Differential calculus on compact matrix pseudogroups (quantum groups), Commun.Math.Phys. 122 125. Yang, C. N. and Mills, R. L. (1954) Conservation of isotopic spin and isotopic gauge invariance. Phys.Rev. 96, 191.