Symmetric multiplets in Quantum Algebras

We consider a modified version of the coproduct for U(suq(2)) and show that in the limit when q → 1, there exists an essentially non-cocommutative coproduct. We study the implications of this non-cocommutativity for a system of two spin-1/2 particles. Here it is shown that, unlike the usual case, this non-trivial coproduct allows for symmetric and antisymmetric states to be present in the multiplet. We surmise that our analysis could be related to the ferromagnetic and antiferromagnetic cases of the Heisenberg magnets. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected] In the standard Drinfeld-Jimbo[1] q-deformation of su(2) algebra which is defined by the relations [J+, J−] = [2J0] [J0, J±] = ±J± (1) where [x] = q − q q − q−1 and the coproduct is given by ∆(J±) = J± ⊗ q0 + q0 ⊗ J± (2a) ∆(J0) = J0 ⊗ l + l ⊗ J0, (2b) which is non-cocommutative for generic q, i.e. ∆(J±) 6= σ◦∆(J±) and σ(a⊗b) = b⊗a is the flip automorphism. The coproduct dictates the tensor multiplications of two representations. For instance in the case of two spin-1/2 representations the above coproduct leads to the following states: |0, 0 > = 1 √ 2 (q 1 2 |+ > ⊗|− > −q 12 |− > ⊗|+ >) (3a) |1, 1 > = |+ > ⊗|+ > (3b) |1, 0 > = 1 √ 2 (q 1 2 |+ > ⊗|− > +q 1 2 |− > ⊗|+ >) (3c) |1,−1 > = |− > ⊗|− > . (3d) In ref[2], Zachos considered the q → −1 limit of this suq(2) algebra and and showed that it has some interesting consequences for the wavefunctions. Specifically, he showed that for a system of two spin-1/2 particles, the singlet state which is ordinarily antisymmetric transforms into a symmetric state while one of the triplet states becomes antisymmetric. This odd behaviour can be traced to the fact that the coproduct under the q → −1 limit remains noncocommutative while the half-integer spin representations of suq(2) reduce to those of su(2). So on the surface, it appears that one has an unconventional composition law for the usual su(2) algebra. There is a caveat in this argument however, the limit is singular in at least two respects. Firstly, the coproduct obtained by taking the q → −1 limit In the case of integer-spin representations, the q → −1 limit reduces to those of su(1,1). 2 in (2a) is not an algebra homomorphism in the case of half integral spin representations. Secondly, as pointed out in ref[2] the Casmir becomes divergent when one considers half integral spin representations: [j][j + 1] q→−1 −→ 4 ǫ + 1 2 + j(j + 1) + {− 1 32 + 1 24 (j(j + 1)(2j + 2j − 1)}ǫ + o(ǫ) (4) where ǫ = q − q. In this letter, we look at the admissability of such states and ask whether a non-trivial coproduct exists for the su(2) algebra. The latter would essentially lead to a universal enveloping algebra (UEA) of ordinary su(2) Lie algebra with a non-cocommutative Hopf structure. We start by modifying the suq(2) algebra coproduct as: ∆(J+) = J+ ⊗ q0e0 + e0q0 ⊗ J+ (5a) ∆(J−) = J− ⊗ q0e0 + e0q0 ⊗ J− (5b) ∆(J0) = J0 ⊗ 1 + 1 ⊗ J0 (5c) ǫ(J+) = ǫ(J−) = ǫ(J0) = 0 (5d) S(J±) = −eqJ±, S(J0) = −J0 (5e) where n is any integer. In the limit when q → 1, the coproduct, counit and antipode become ∆(J±) = J± ⊗ e0 + e0 ⊗ J± (6a) ∆(J0) = J0 ⊗ l + l ⊗ J0 (6b) ǫ(J±) = ǫ(J0) = 0 (6c) S(J±) = −eJ± S(J0) = −J0. (6d) It is important to observe that for odd n, the coproduct eq(6 a) remains non-cocommutative. To elucidate this non-triviality, it is instructive to evaluate the coproduct and its opposite One can ascertain this by computing explicitly the matrices of ∆J± and ∆J0 in the spin 1/2 ⊗ 1/2 representation of suq(2). By taking the limit q → −1 of these matrices one then finds that the commutator between ∆J+ and ∆J− yields −2∆J0 instead of the requisite 2∆J0. 3 (∆ ≡ σ ◦∆) in a tensor product representation. To this end we consider the tensor product of two representations labelled by j1 and j2 i.e. (j1 ⊗ j2). Since the commutation relations are just the standard ones, we have for each representation J0|j,m > = m|j,m > (7a) J±|j,m > = √ (j ∓m)(j ±m+ 1)|j,m± 1 > (7b) where −j ≤ m ≤ j. The value j characterizes the representation: J|j,m >≡ (J+J− + J 0 − J0)|j,m >= j(j + 1)|j,m > (8) where J is the quadratic casmir and j takes the values 0, 1/2, 1, 3/2, · · ·. For an arbitrary vector |j1, m1 > ⊗|j2, m2 > belonging to the representation space of (j1 ⊗ j2) and using eq(6a), we have ∆(J±)|j1, m1 > ⊗|j2, m2 > = f±(j1, m1)e |j1, m1 + 1 > ⊗|j2, m2 > +f±(j2, m2)e ∓iπnm1 |j1, m1 > ⊗|j2, m2 + 1 > (9) where f±(ji, mi) = √ (ji ∓mi)(ji ±mi + 1) and −ji ≤ mi ≤ ji (i = 1, 2). For the opposite coproduct one obtains ∆(J±)|j1, m1 > ⊗|j2, m2 > = f±(j1, m1)e |j1, m1 + 1 > ⊗|j2, m2 > +f±(j2, m2)e ±iπnm1 |j1, m1 > ⊗|j2, m2 + 1 > . (10) Now if the coproduct is cocommutative at the algebraic level it is necessary that this property be reflected by any tensor product representation. In other words the rhs. of eq(9) and eq(10) should be equal for any allowed values of j1 and j2 and their corresponding m’s. On the other hand, the existence of a tensor product representation in which the two do not agree is sufficient proof for non-cocommutativity of ∆. When one of the two representations that appears in the tensor product is characterized by a half-integer value (i.e.ji = 1/2, 3/2, 5/2, ...), then for odd integer n, we see that ∆(J±) 6= ∆(J±). So we can surmise that ∆(J±) 6= ∆(J±) in general. 4 We stress that in the q → 1 limit of eq(5a,b), the coproduct in (6a) differs from the usual one for suq(2) for q = e . By identifying e ≡ q in (6a) one has ∆(J±) = J± ⊗ q0 + q0 ⊗ J± (11) which clearly differs from eq(2a) by a sign in J0 for ∆(J−). Note further that this is a Hopf ∗-algebra in the sense that the canonical conjugation[3] (J0) + = J0 (J±) † = J∓ (12) is compatible with the Hopf structure. For generic q, the tensor product of two representations closely imitates the results in eq(9), with the expressions f±(ji, mi) replaced by its q-deformed form, f ′ ±, where f ′ ±(ji, mi) = √ [ji ∓mi][ji ±mi + 1]. (13) We next consider the implications of these results for a system of two spin-1/2 particles. It is convenient to introduce a new set of generators in which the coproduct assumes an equivalent form. We define J ′ + = e J+, J ′ − = J−e −iπnJ0 J ′ 0 = J0 (14) with the primed generators satisfying the same commutation relations as the unprimed ones so that we can regard these operators as the generators of original algebra. The coproduct in eq(5a -c) now reads ∆(J ′ +) = J ′ + ⊗ q ′ 0e ′ 0 + q ′ 0 ⊗ J ′ + (15a) ∆(J ′ −) = J ′ − ⊗ q ′ 0e ′ 0 + q ′ 0 ⊗ J ′ − (15b) ∆(J ′ 0) = J ′ 0 ⊗ 1 + 1 ⊗ J ′ 0. (15c) With j = 1 2 for spin-1/2 representation, we have J ′ ±|± >= 0 J ′ ±|∓ >= |± > J ′ 0|± >= ± 1 2 |± > (16) 5 where |± > denotes the q-deformed states | 2 ,± 2 > respectively. Using the same techniques for evaluating the normal Clebsch-Gordon coefficients based on the coproduct in eq(15), the explicit expressions for the singlet and triplet states can be written as: |0, 0 > = 1 √ 2 (q 1 2 |+ > ⊗|− > −(−1)q 12 |− > ⊗|+ >) (17a) |1, 1 > = |+ > ⊗|+ > (17b) |1, 0 > = 1 √ 2 (q 1 2 |+ > ⊗|− > +(−1)q 1 2 |− > ⊗|+ >) (17c) |1,−1 > = (−1)|− > ⊗|− > . (17d) The main difference from those resulting from the coproduct (2a, b) is the appearance of the factor (−1)n. For even n, we get the usual q-deformed multiplets and in the limit q → 1, we retrieve the usual antisymmetric singlet state and the usual symmetric |1, 0 > state in the triplet. For odd n, in the limit q → 1, we see that the singlet state becomes symmetric while the |1, 0 > state in the triplet becomes antisymmetric. The states here are similar to those obtained through the q → −1 limit shown in ref[2]. In the spin-1/2 representation eq (17a d), if we let ∆0 be the coproduct for even n and ∆1 be the coproduct for odd n, then the two coproducts are related by U∆1(a)U † = ∆0(a), where a is J ′ ± or J ′ 0 and U is given by U = 