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35-472: The term R-matrix is used in connection with the Yang–;Baxter equation , first introduced in the field of statistical mechanics in the works of J. B. McGuire in 1964 and C. N. Yang in 1967 and in the group algebra C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} of the symmetric group in the work of A. A. Jucys in 1966. The classical R-matrix arises in

70-668: A module of A {\displaystyle A} , and P i j = ϕ i j ( P ) {\displaystyle P_{ij}=\phi _{ij}(P)} . Let P : V ⊗ V → V ⊗ V {\displaystyle P:V\otimes V\to V\otimes V} be the linear map satisfying P ( x ⊗ y ) = y ⊗ x {\displaystyle P(x\otimes y)=y\otimes x} for all x , y ∈ V {\displaystyle x,y\in V} . The Yang–Baxter equation then has

105-557: A multiplicative parameter, the Yang–Baxter equation is for all values of u {\displaystyle u} and v {\displaystyle v} . The braided forms read as: In some cases, the determinant of R ( u ) {\displaystyle R(u)} can vanish at specific values of the spectral parameter u = u 0 {\displaystyle u=u_{0}} . Some R {\displaystyle R} matrices turn into

140-413: A one dimensional projector at u = u 0 {\displaystyle u=u_{0}} . In this case a quantum determinant can be defined . Then the parametrized Yang-Baxter equation (in braided form) with the multiplicative parameter is satisfied: There are broadly speaking three classes of solutions: rational, trigonometric and elliptic. These are related to quantum groups known as

175-564: A scalar function, then R ′ {\displaystyle R'} also satisfies the Yang–Baxter equation. The argument space itself may have symmetry. For example translation invariance enforces that the dependence on the arguments ( u , u ′ ) {\displaystyle (u,u')} must be dependent only on the translation-invariant difference u − u ′ {\displaystyle u-u'} , while scale invariance enforces that R {\displaystyle R}

210-510: Is R {\displaystyle R} followed by a swap of the two objects. In one-dimensional quantum systems, R {\displaystyle R} is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is integrable . The Yang–Baxter equation also shows up when discussing knot theory and the braid groups where R {\displaystyle R} corresponds to swapping two strands. Since one can swap three strands in two different ways,

245-456: Is a function of the scale-invariant ratio u / u ′ {\displaystyle u/u'} . A common ansatz for computing solutions is the difference property, R ( u , u ′ ) = R ( u − u ′ ) {\displaystyle R(u,u')=R(u-u')} , where R depends only on a single (additive) parameter. Equivalently, taking logarithms, we may choose

280-573: Is a professor of mathematics at Rikkyo University . He is a grandson of the linguist Kaku Jimbo  [ ja ] . After graduating from the University of Tokyo in 1974, he studied under Mikio Sato at the Research Institute for Mathematical Sciences in Kyoto University . He has made important contributions to mathematical physics , including (independently of Vladimir Drinfeld )

315-400: Is homogeneous in parameter dependence in the sense that if one defines R ′ ( u i , u j ) = f ( u i , u j ) R ( u i , u j ) {\displaystyle R'(u_{i},u_{j})=f(u_{i},u_{j})R(u_{i},u_{j})} , where f {\displaystyle f} is

350-583: Is then defined using the 'twisted' alternate form above, asserting ( i d × r ) ( r × i d ) ( i d × r ) = ( r × i d ) ( i d × r ) ( r × i d ) {\displaystyle (id\times r)(r\times id)(id\times r)=(r\times id)(id\times r)(r\times id)} as maps on X × X × X {\displaystyle X\times X\times X} . The equation can then be considered purely as an equation in

385-421: The ℏ 2 {\displaystyle \hbar ^{2}} coefficient of the quantum YBE (and the equation trivially holds at orders ℏ 0 , ℏ {\displaystyle \hbar ^{0},\hbar } ). Michio Jimbo Michio Jimbo ( 神保 道夫 , Jimbō Michio , born November 28, 1951) is a Japanese mathematician working in mathematical physics and

SECTION 10

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420-468: The Yangian , affine quantum groups and elliptic algebras respectively. Set-theoretic solutions were studied by Drinfeld . In this case, there is an R {\displaystyle R} -matrix invariant basis X {\displaystyle X} for the vector space V {\displaystyle V} in the sense that the R {\displaystyle R} -matrix maps

455-547: The Yang–Baxter equation (or star–triangle relation ) is a consistency equation which was first introduced in the field of statistical mechanics . It depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states. It states that a matrix R {\displaystyle R} , acting on two out of three objects, satisfies where R ˇ {\displaystyle {\check {R}}}

490-460: The category of sets . Solutions to the classical YBE were studied and to some extent classified by Belavin and Drinfeld. Given a 'classical r {\displaystyle r} -matrix' r : V ⊗ V → V ⊗ V {\displaystyle r:V\otimes V\rightarrow V\otimes V} , which may also depend on a pair of arguments ( u , v ) {\displaystyle (u,v)} ,

525-586: The symmetric group in the work of A. A. Jucys in 1966. Let A {\displaystyle A} be a unital associative algebra . In its most general form, the parameter-dependent Yang–Baxter equation is an equation for R ( u , u ′ ) {\displaystyle R(u,u')} , a parameter-dependent element of the tensor product A ⊗ A {\displaystyle A\otimes A} (here, u {\displaystyle u} and u ′ {\displaystyle u'} are

560-503: The Yang–Baxter equation are often constrained by requiring the R {\displaystyle R} matrix to be invariant under the action of a Lie group G {\displaystyle G} . For example, in the case G = G L ( V ) {\displaystyle G=GL(V)} and R ( u , u ′ ) ∈ End ( V ⊗ V ) {\displaystyle R(u,u')\in {\text{End}}(V\otimes V)} ,

595-469: The Yang–Baxter equation enforces that both paths are the same. According to Jimbo , the Yang–Baxter equation (YBE) manifested itself in the works of J. B. McGuire in 1964 and C. N. Yang in 1967. They considered a quantum mechanical many-body problem on a line having c ∑ i < j δ ( x i − x j ) {\displaystyle c\sum _{i<j}\delta (x_{i}-x_{j})} as

630-466: The Yang–Baxter equation is for all values of u 1 {\displaystyle u_{1}} , u 2 {\displaystyle u_{2}} and u 3 {\displaystyle u_{3}} . Let A {\displaystyle A} be a unital associative algebra. The parameter-independent Yang–Baxter equation is an equation for R {\displaystyle R} , an invertible element of

665-1040: The alternate form is In the parameter-independent special case where R ˇ {\displaystyle {\check {R}}} does not depend on parameters, the equation reduces to and (if R {\displaystyle R} is invertible) a representation of the braid group , B n {\displaystyle B_{n}} , can be constructed on V ⊗ n {\displaystyle V^{\otimes n}} by σ i = 1 ⊗ i − 1 ⊗ R ˇ ⊗ 1 ⊗ n − i − 1 {\displaystyle \sigma _{i}=1^{\otimes i-1}\otimes {\check {R}}\otimes 1^{\otimes n-i-1}} for i = 1 , … , n − 1 {\displaystyle i=1,\dots ,n-1} . This representation can be used to determine quasi-invariants of braids , knots and links . Solutions to

700-400: The classical YBE is (suppressing parameters) [ r 12 , r 13 ] + [ r 12 , r 23 ] + [ r 13 , r 23 ] = 0. {\displaystyle [r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0.} This is quadratic in the r {\displaystyle r} -matrix, unlike

735-421: The components of the matrices R ∈ End ( V ) ⊗ End ( V ) ≅ End ( V ⊗ V ) {\displaystyle R\in {\text{End}}(V)\otimes {\text{End}}(V)\cong {\text{End}}(V\otimes V)} are written R i j k l {\displaystyle R_{ij}^{kl}} , which is the component associated to

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770-457: The definition of the classical Yang–Baxter equation. In quasitriangular Hopf algebra , the R-matrix is a solution of the Yang–Baxter equation . The numerical modeling of diffraction gratings in optical science can be performed using the R-matrix propagation algorithm. There is a method in computational quantum mechanics for studying scattering known as the R-matrix. This method

805-507: The eight vertex model in 1972. Another line of development was the theory of factorized S-matrix in two dimensional quantum field theory. Zamolodchikov pointed out that the algebraic mechanics working here is the same as that in the Baxter's and others' works. The YBE has also manifested itself in a study of Young operators in the group algebra C [ S n ] {\displaystyle \mathbb {C} [S_{n}]} of

840-721: The following alternate form in terms of R ˇ ( u , u ′ ) = P ∘ R ( u , u ′ ) {\displaystyle {\check {R}}(u,u')=P\circ R(u,u')} on V ⊗ V {\displaystyle V\otimes V} . Alternatively, we can express it in the same notation as above, defining R ˇ i j ( u , u ′ ) = ϕ i j ( R ˇ ( u , u ′ ) ) {\displaystyle {\check {R}}_{ij}(u,u')=\phi _{ij}({\check {R}}(u,u'))} , in which case

875-403: The induced basis on V ⊗ V {\displaystyle V\otimes V} to itself. This then induces a map r : X × X → X × X {\displaystyle r:X\times X\rightarrow X\times X} given by restriction of the R {\displaystyle R} -matrix to the basis. The set-theoretic Yang–Baxter equation

910-563: The initial development of the study of quantum groups , the development of the theory of τ {\displaystyle \tau } -functions for the KP ( Kadomtsev–Petviashvili ) integrable hierarchy, and other related integrable hierarchies , and development of the theory of isomonodromic deformation systems for rational covariant derivative operators. In 1993 he won the Japan Academy Prize for this work. In 2010 he received

945-642: The map e i ⊗ e j ↦ e k ⊗ e l {\displaystyle e_{i}\otimes e_{j}\mapsto e_{k}\otimes e_{l}} . Omitting parameter dependence, the component of the Yang–Baxter equation associated to the map e a ⊗ e b ⊗ e c ↦ e d ⊗ e e ⊗ e f {\displaystyle e_{a}\otimes e_{b}\otimes e_{c}\mapsto e_{d}\otimes e_{e}\otimes e_{f}} reads Let V {\displaystyle V} be

980-762: The only G {\displaystyle G} -invariant maps in End ( V ⊗ V ) {\displaystyle {\text{End}}(V\otimes V)} are the identity I {\displaystyle I} and the permutation map P {\displaystyle P} . The general form of the R {\displaystyle R} -matrix is then R ( u , u ′ ) = A ( u , u ′ ) I + B ( u , u ′ ) P {\displaystyle R(u,u')=A(u,u')I+B(u,u')P} for scalar functions A , B {\displaystyle A,B} . The Yang–Baxter equation

1015-791: The parameters, which usually range over the real numbers ℝ in the case of an additive parameter, or over positive real numbers ℝ in the case of a multiplicative parameter). Let R i j ( u , u ′ ) = ϕ i j ( R ( u , u ′ ) ) {\displaystyle R_{ij}(u,u')=\phi _{ij}(R(u,u'))} for 1 ≤ i < j ≤ 3 {\displaystyle 1\leq i<j\leq 3} , with algebra homomorphisms ϕ i j : A ⊗ A → A ⊗ A ⊗ A {\displaystyle \phi _{ij}:A\otimes A\to A\otimes A\otimes A} determined by The general form of

1050-511: The parametrization R ( u , u ′ ) = R ( u / u ′ ) {\displaystyle R(u,u')=R(u/u')} , in which case R is said to depend on a multiplicative parameter. In those cases, we may reduce the YBE to two free parameters in a form that facilitates computations: for all values of u {\displaystyle u} and v {\displaystyle v} . For

1085-572: The potential. Using Bethe's Ansatz techniques, they found that the scattering matrix factorized to that of the two-body problem, and determined it exactly. Here YBE arises as the consistency condition for the factorization. In statistical mechanics , the source of YBE probably goes back to Onsager's star-triangle relation, briefly mentioned in the introduction to his solution of the Ising model in 1944. The hunt for solvable lattice models has been actively pursued since then, culminating in Baxter's solution of

R-matrix - Misplaced Pages Continue

1120-522: The tensor product A ⊗ A {\displaystyle A\otimes A} . The Yang–Baxter equation is where R 12 = ϕ 12 ( R ) {\displaystyle R_{12}=\phi _{12}(R)} , R 13 = ϕ 13 ( R ) {\displaystyle R_{13}=\phi _{13}(R)} , and R 23 = ϕ 23 ( R ) {\displaystyle R_{23}=\phi _{23}(R)} . Often

1155-448: The unital associative algebra is the algebra of endomorphisms of a vector space V {\displaystyle V} over a field k {\displaystyle k} , that is, A = End ( V ) {\displaystyle A={\text{End}}(V)} . With respect to a basis { e i } {\displaystyle \{e_{i}\}} of V {\displaystyle V} ,

1190-620: The usual quantum YBE which is cubic in R {\displaystyle R} . This equation emerges from so called quasi-classical solutions to the quantum YBE, in which the R {\displaystyle R} -matrix admits an asymptotic expansion in terms of an expansion parameter ℏ , {\displaystyle \hbar ,} R ℏ = I + ℏ r + O ( ℏ 2 ) . {\displaystyle R_{\hbar }=I+\hbar r+{\mathcal {O}}(\hbar ^{2}).} The classical YBE then comes from reading off

1225-688: Was originally formulated for studying resonances in nuclear scattering by Wigner and Eisenbud . Using that work as a basis, an R-matrix method was developed for electron , positron and photon scattering by atoms . This approach was later adapted for electron, positron and photon scattering by molecules . R-matrix method is used in UKRmol and UKRmol+ code suits. The user-friendly software Quantemol Electron Collisions (Quantemol-EC) and its predecessor Quantemol-N are based on UKRmol/UKRmol+ and employ MOLPRO package for electron configuration calculations. Yang%E2%80%93Baxter equation In physics ,

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