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1 УДК 539.192 (043.3) Klimko G.T. Donetsk National Technical University (DonNTU), Ukraine Ukraine, 83001, c. Donetsk, Artema st., 58, [email protected] Recovery of the basis of group π2n representation on its subgroup πn×n and Harriman’s theorem Климко Г.Т. Донецкий национальный технический университет, Украина Украина, 83001, г. Донецк, ул. Артема, 58, [email protected] Восстановление базиса представления группы π2n на её подгруппе πn×n и теорема Гарримана Климко Г.Т. Донецький національний технічний університет, Україна Україна, 83001, м. Донецьк, вул. Артема, 58, [email protected] Відновлення базису, що дає представлення групи π2n на базисі її підгрупи πn×n , і теорема Гаррімана The proof of Garriman’s theorem [1] is given for arbitrary order reduced density matrix of both the clear, and the mixed states of fermions at once. Essential its parts are a Pauli's exclusion principle, rotation group symmetry of spin functions and new commutation relations. Теорема Гарримана [1] доказана для редуцированных матриц плотности чистого и смешанного состояний фермионов из принципа Паули, симметрии спиновых функций, новых следствий из перестановок штрихованных и не штрихованных переменных. Теорема Гаррімана [1] доведена для редуцироних матриць густини чистого і змішаного станів ферміонів з використанням принципу Паулі, симетрії спінових функцій і нових наслідків, пов'язаних з переміщеннями штрихованих і нештрихованих змінних 1 Reduced density matrix Reduced density matrix of arbitrary order p (RDM-p) [2, 3], including transitional RDM-p, for fermions states with spin projection values M and M can be written in form [4] Γ ( p) x , x ,..., x 1 2 p x1, x2 ,..., xp () R((p))μ Tˆ((p))μ . Its spin and “conjugated” space components look like (1) 2 ( p) μ T(γ)ω σ pit t i t i t pit μ σ (2) pit it μ pγ ωμ , (p)μ R 1ε i t ε i t μ p γ ωμ pit it μ γ t i i t tμ p Lμt i t * jp t it μ* jp t μ There are conjugated unitary transformation coefficients p γ ωμ pit it μ . p i t it μ pγ ωμ (3) and are appeared in Eqs (2), (3). The contravariant spinors σ p it α(i1 ) α(i2 ) α(it ) β(j1` ) β(j2 ) β(j p t ) , its conjugated covariant spinors and hermit spin tensors Tˆ((p)μ γ)ω (4) of a rank ω are ortho- normalized. Antisymmetry of wave functions of states is concluded in Eq. (1). Therefore for 2 antisimmetrizer Aˆ Aˆ is true Rp Tˆ p Aˆ Rp Tˆ p Aˆ . Both spin and space coordinates of particles are designated by its numbers it j p t 1, 2,... p . 2. Recovery of the basis of group π2n representation on its subgroup πn×n Unlike the one-particle contravariant spinors α and β, which connected to spin space irreducible representation (IR) D ½ of rotation group SU 2 [5, 6], covariant spinors α+(i) and β +(i) belong to the equivalent IR D½* = V ½∙D½∙(V ½) + with unitary matrix [5, page 165]: V ½ α 0 1 ½ + = . Then the spinors (V ) ∙ β = 1 0 β are transforming on space of IR D ½, as α 3 α well as spin functions . Therefore conjugated to Eq. (4) covariant spinors are equivalent β to contravariant ones. After replacement every α + (i) on a spinor –β(i) and β + (j) respectively on α(j) its generate the basis of spinors σ p i t . It’s belong the same space of a direct product p i 1 1 Di 2 of p identical IR D 1 2 of group SU 2 , as contravariant spinors (see Eq. (4)), σ p i t 1 β(i1 ) β(i2 ) β(i pt ) α(j1 ) α(j2 ) α(j pt ) . t (5) As result, the basis from products of kontra- and of covariant spinors, as the basis of spinors of a 2p-rank from products of only contravariant spinors σ p it σ p it , be- long to the same representation of rotation group. It allows properties of one basis, established on the ground of its group-theoretical reviewing, to transfer on properties of equivalent basis. From theory of IR functions p , p for permutation group 2 p it follows, that the u Yˆu Qˆuv and function v Yˆv , where permutation Qˆ vu Qˆuv1 substitutes numbers of coordinates in Young table of u on numbers from the Young table similate one IR basis. Instead the permutation Q̂uv connects functions v [5, 6], as- u and v [6] Qˆuvv QˆuvYˆv QˆuvYˆvQˆuv1Qˆuv YˆuQˆuv u . (6) Call to mind, that both in lines and in columns of standard Young tables [5, 6] the number of coordinates are ranking from left to right and from top to down according to increase of their place number in the initial list. For list 1, 2, …, p-, p-+1,…,p, 1, 2, p-, p-+1′,…,p′ such Young table will be obtained, when the coulombs of Young scheme [p+, p-], where ω = 0, 1, …, р, are filling by numbers 1, 1, 2, 2,…, p-, p-, p-+1,…,p, p-+1′,…,p′ 1 2 … im … p- 1′ 2′ … i′m … (p-)′ p-+1 Fig. 1. … p (p-+1)′ … p′ 4 The number of standard Young tables [5, 6] equals to dimension of IR p , p 2p 2p 2 p ! 2 1 f2 p , p p 1 p 1! p ! (7) A permutation may be represented by product not intersected cyclical permutations. Its do not leave invariant variable places from their list. Cycles are representing by product of transposition, which are not commute. Such transpositions commutate according to a rule [6] i, k k , l i , l i , k k , l i , l , (8) We prove now main assertion according to which it is possible to receive any function u simmetrized on the standard Young table u for IR p , p from function v permuting its arguments only inside each of sets on p variable with the primed or not primed numbers separately, if in the second line of the “standard” table of Young v the variables of one type set only remain, as in the Young table v in a Fig. 1. Clearly, if transition from v to u demands rearrangement of primed variables with not primed, then in Eq. (6) Qˆ uv 2 p . ˆ Really, it is possible to separate exactly luv transpositions Q l uv p 1 i, i luv i 1 ˆ , conserving after their operation on the table v correct regularity of not primed in Q uv 2p numbers concerning primed both in table lines and in table columns. The number pairs luv of permuted primed and not primed variables is restricted an inequality luv min p , p 2 , where p 2 – the whole part from p/2. The factorization of permutations Qˆ uv Qˆ uv Qˆ uv Qˆ l uv reduces our assertion to the proof of a possibility of replacement of each transposition imik in Qˆ operation on Yˆv of permutations belonging to only a subgroup Yˆ by an identical l uv v p p : ik , im Yˆv Yˆik ,im v ik , im 2 Aˆik im Aˆik im Yˆv . In equality (9) we always have k < m and as in Eq. (6) Young projectors are equal (9) 5 p Yˆv c Aˆi i Sˆ i i p 1 Yˆik ,im ·v c There p Sˆi p ˆ A ˆ A 1 k ,m i i Sˆ Aˆ ik im ˆ ... p S... p i i p , (10) ˆ Sˆ ˆ A ...p \ im ,i S ...p \i ,im i im k p k k p . (11) is product of antisimmetrizers and of simmetrizers re- spectively, acting on the corresponding sets of variables, c is constant to ensure that Yˆv 2 = Yˆv . ˆ in (6) may be changed on equal operator but only from Due equality (9) any operator Q uv permutations belonging to subgroup p p . It constructs group 2 p the subgroup operators. If ik , im ... p , ˆ Sˆ =A ik im ... Sˆ... p p k p 1 Aˆik im Aˆik im 2 1 ik , im ... p , then Aˆik im Sˆ... Sˆ... p p 0 and in a right-hand part of Eq. (9) instead of Aˆi i Aˆimim · Yˆv the kk i , i i transpositions basis { u } using k m , im 4 will stay, when the definition (10) for Yˆv is used: Aˆi i ik ik imim Sˆ... p Sˆ... p . (12) k , m Here ... p and ... p are the same sets of variables, as in Eq. (10). Take in mind the equality: Sˆ... p Sˆ... p imik Sˆ .... p \im ,ik p ... p \ik ,im p m k Sˆ i i . , (13) and rules (8) we transform expression (12) into an operator Aˆ i k i m Aˆ i k i m 1 2 p-ω Aˆi i ik im ik ik ik im imim Sˆ... 1 ν k , m Sˆ... p \ i k , i m p \ i m , i k p (14) p imik , In Eq. (14) first transpositions are retracted into suitable antisimmetrizers with a change of sign, and second are retracted into simmetrizers. Transposition imik will be commutated to 6 left with all operators transforming its. As a result we obtain the left-hand part of identity (9). For the function M n M ! n M ! to give the IR n s, n s basis of group u Here 1 2 2n ... n M ... nM [6, 7] we always can in the following form Cuv Qˆ1Qˆ 2 Qˆ1Qˆ 2 Yˆv M , ˆ s M s. Q1ˆ n M Qˆ 2ˆ n M (15) Cuv Qˆ1Qˆ 2 are simple numerical coefficients and u is a standard Young table. 3 The proof of the Garriman’s theorem For the given p and = M – M= t – t ´ in Eq. (3) the number of "independent" space components RDM is determined both impossibility of build-up of tensors of a higher rank, than ω = p, in p–partial spin space, and Garriman’s theorem, proved for some cases [1]. It defines a possibility to receive all components R((p))μ in Eqs. (1), (2) from "standard" component with the same and , which obtained according to a "standard addition schema of the spin moments (o)” [5, 6] , that is equivalent Young table v in a Fig. 1. The obtained above results (see Eq. 15)) allow us to give the common proof of the Garriman’s theorem. To build the irreducible tensors Eq. (2) routinely the products of the conjugated spinors , , and must be changed by one-particle operators [8], Iˆ , Sˆz tˆ1(0) 12 ( ), (16a) S 2 t1( 1) , S 2 t1( 1) , (16b) Spσ Iˆ2 2, Spσtˆ1(μ) tˆ1(μ) μμ 2 , Spσtˆ1μ Iˆ 0. (16c) (μ ) Here Iˆ is a unit operator, tˆ1` are components of one-particle tensor of the first rank. Then the complete set of addition schemes {()} may be realized by serial adding the operator Iˆ or tˆ1`(μ ) to already built irreducible tensor of smaller number of particles. As result, we can define the union list from p one-particle operators and from the intermediate values of tensor rank for addition schema (). 7 The operators σ p it σ p i t , as in Eq. (5), and Tˆ γ ω spinors σ p it (p)μ Tˆ(γ)ω can be transformed on the p it σ and the tensor (2p)μ depending out of 2p of spin variables. Then last save the same addition schema [] = (), the moment and its projection . It is true, “as from the permutation operators of spin coordinates of particles commute with the spin rotation operators [6] it follows that the dimension Eq. (7) of groups 2p p , p IR for and the number of addition schemes {[]} of , spins with same complete val- ues and are equal.” Each such addition schema correspond the line of IR of permutation group 2 p . Therefore 2p-particle functions Tˆ γ(2p)μ ω one set {[]}. They form basis for IR can be numbered by addition schemes of λω = p , p of symmetric group 2 p and ac- 2 p cording to Eq. (15) they can be obtained from a function Tˆ 0 . Let's find mutual conformi 2 p ˆ ty between spinor T 0 p ˆ and tensor T 0 . It is evident the tensor T(1(p)μ with a special p ω ...)ω addition schema (o) = (1p-…)ω gives this appropriation. It looks like Tˆ (pp)ωω (1 ...)ω 2 p ω 2 Iˆ(1)Iˆ(2) Iˆ(p ω)Sˆ (p ω 1) Sˆ (p). (17) It has the predicted values of ranks of the all intermediate tensors for any p and after using Eqs. (16), (5) it is equivalent to a function-spinor of a rank Tˆ (2p p)( ) 2 [1 ...]ω p 2 p ( (i ) (i) (i) (i)) i 1 p ( j ) ( j ) . j p 1 (18) Function (18) is really simmetrized under the Young table [6], which columns are completed by numbers 1, 1, 2, 2, …, p-, p-, …, as in a Fig. 1, and from Eq. (15) follows T[(2]ωp) p C Qˆ ,Qˆ p p ˆ ˆ T (2 p) ˆ p T (2 p) . Q,QQQ [1 p ..]ω [1 p ...]ω (19) After we return to tensors Eq. (2), using Eqs. (16) in Eqs. (18), (19), the same expan(p) sion remains valid both for the highest component T̂(γ)ω and for all components of the tensor 8 (p)μ (p) Tˆ(γ)ω too, as they may be obtained from T̂(γ)ω by spin projection decreasing operators Ŝ Sˆ , Tˆ p 1 1 Tˆ p 1 . (20) p As operators Sˆ Sˆ i commute with all permutation operators of particles and with i 1 Qˆ Qˆ p p too, we have general proof of “Garriman’s theorem” in spin space Tˆ ( p )( ) ω p C Q,Q p p ˆˆ Q,QQT ( p )( ) ˆ ˆ Tˆ Q p 1 p ... ω ( p )( ) 1 p ... ω (21) . These tensors may be orthonormalized, as in (1), without change its obtained structure (21). In this case applying expansion (2), we make up the identity (21) into linear relationships between coefficients p it it μ p γ ωμ inside of Eqs. (2), (3). They contain same factors, as in Eq. (21), p it it μ p γ ωμ p C Q, Q pQˆ 1it Qˆ 1it μ p 1p-ω ... ωμ . Q, Q p p (22) Taken into account that the number sets in Eq. (22) and ones at sign factor in Eq. (3) are same we shall receive the connection of space components RDM-p R(pp)μ R(p)(ω ) (1 ...) R((p))μ with standard component , which is similar to Eq. (21): 1 Qˆ , Qˆ ˆ p ˆ p q q p ˆ ( p )( ) Qˆ ˆ p Rˆ ( p )( ) C Q, Q QR p p 1 ...ω 1 ...ω .(23) , p where ˆ p p , (-1)q+q' is sign factor for permutation Q·Q'. Moreover, it is true that R(p)μ Sp Γ p x1 ,..., x p x1,..., xp Tˆ p , γ (24) if spin tensors (21) are orthonormalized. This simple process always can be applied to obtain exact result type Eq. (23) for any RDM. It completes generally proof of Garriman’s theorem. 9 References 1 Harriman J.E. Geometry of Density Matrices. III. Spin Components // Int. J. Quantum Chem. – 1979. – v. 15, № 6. – p. 611 – 643. 2 Husimi К. Some Formal Properties of the Density Matrix // Proc. Phys. – Math. Soc. Japan – 1940. – v. 22, № 24. – p. 264 – 314. 3 Löwdin P.-O. Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method. of Configurational Interaction, - Phys. Rev., 1955, v.97, № 6, p. 1474 - 1489. 4 McWeeny R., Mizuno Y. The Density Matrix in Many–Electron Quantum Mechanics. II. Separation of Space and Spin Variables, Spin Coupling Problems // Proc. Roy. Soc. (London). – 1961. – v. A 259, № 1299, р.554 –569. 5 Petrashen М. I., Trifinov Е. D. Applications of group theory in quantum chemistry. – М.: Nauka, 1967. – 307 с. 6 Hamermesh М. Group theory and its applications to physical problems.– М:, Мir, 1966.– 587 с. 7 Mestechkin M.M., Klimko G.T. Spin-dependent operators in the spin-free quantum chemistry // Intern. J. Quantum Chem. – 1978. – V. 13, № 5. – P.579-596. 8 Klimko G.T., Mestechkin M.M., Whyman G.E. Fock coordinate function method for separation of spin variables in transition density matrices // Intern. J. Quantum Chem.-1980. V. 17, № 3. - P. 415 - 428.