Method of orbits of co-associated representation in thermodynamics of the Lie non-compact groups

07/11/2017
Auteurs : Vitaly Mikheev
Publication GSI2017
OAI : oai:www.see.asso.fr:17410:22350
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A method of the solution of the main problem of homogeneous spaces thermodynamics for non-compact Lie groups is presented in the work. The method originates from formalism of non-commutative Fourier analysis based on method of coadjoint orbits. A formula that allows eciently evaluate heat kernel and statistic sum on non-compact Lie group is obtained. The algorithm of construction of high temperature heat kernel expansion is also discussed.

Method of orbits of co-associated representation in thermodynamics of the Lie non-compact groups

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application/pdf Method of orbits of co-associated representation in thermodynamics of the Lie non-compact groups Vitaly Mikheev
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Method of orbits of co-associated representation in thermodynamics of the Lie non-compact groups

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Method of orbits of co-associated representation in thermodynamics of the Lie non-compact groups Vitaly Mikheev1 Omsk State Technical University, 644050, Prospekt Mira, 11/8, Omsk, Russian Federation, vvm125@mail.ru Abstract. A method of the solution of the main problem of homoge- neous spaces thermodynamics for non-compact Lie groups is presented in the work. The method originates from formalism of non-commutative Fourier analysis based on method of coadjoint orbits. A formula that allows efficiently evaluate heat kernel and statistic sum on non-compact Lie group is obtained. The algorithm of construction of high temperature heat kernel expansion is also discussed. Keywords: heat kernel, statistic sum, partition function, non-commutative integration, non-compact Lie groups, high-temperature expansion 1 Introduction The purpose of that work is to work out the method for solution of the main problem of homogeneous spaces thermodynamics which consists of evaluation of statistic sum (partition function) Zβ = X n dn exp(−βEn), (1) where dn is degeneration degree of corresponding En. It also may be found as a trace of density matrix (heat kernel) Zβ = Z ρβ(x, x)dµ(x), dµ(x) = p |g|dx. (2) That problem is important not only because statistic sum and heat kernel are important features of the space and can reveal thermodynamic properties of particles in that manifold [1]. The solution of main problem of homogeneous spaces thermodynamics for arbitrary manifold can move one step further to understanding the problem formulated by Kac ”Can we hear the shape of the drum?”. In other words we try to understand how can geometry and topology of the space influence spectral properties of Laplace operator on it ([2],[3],[4]). All existing results in that field were related to the compact manifolds or non-compact manifolds of finite volume. There is no algorithm of building heat kernel and statistic sum for arbitrary non-compact manifold because in this case series (1) and integral (2) are divergent since the volume of the manifold is infinite. Density matrix (heat kernel) is to be found from heat kernel equation (Bloch equation) on homogeneous space with special initial condition ∂ρβ(x, x0 ) ∂β + H(x)ρβ(x, x0 ) = 0, ρβ(x, x0 )|β=0 = δ(x, x0 ). (3) Solution of equation (3) has two problems which can hardly be overcome by existing methods of integration of PDEs, for instance by widely used separa- tion of variables. Firstly one must obtain global solution on entire manifold but separation of variables sufficiently connected with the coordinate system on the manifold and therefor can give only local solutions. Secondly we have to build the solution of Bloch equation (3) from the functions which form the solution basis which must satisfy special initial condition chosen as δ - function. That is also a complicated problem. 2 Integration of heat kernel equation on non-compact Lie groups Let’s consider equation (3) on n–dimensional real Lie group G with operator H being a quadratic function of left-invariant vector fields ξ on the group. That means that H is Laplace operator on a group space with left-invariant Riemann metric H(−i~ξ) = −~2 Gab ξaξb = −~2 ∆. (4) Solution of equation (3) on non-compact Lie group will be obtained using the formalism of non-commutative Fourier analysis on Lie groups based on method of orbits. The method originates from works by Kirillov [6], Souriau [7] and Kostant [8]. For that purpose we induce special irreducible representation of Lie algebra G (so called λ–representation) on Lagrange submanifold Q to a co-adjoint orbit Oλ ∈ G∗ [li(q, ∂q, λ), lj(q, ∂q, λ)] = Ck ijlk(q, ∂q, λ). (5) where Ck ij are structural constants of Lie algebra G, and li(q, ∂q, λ) - first order differential operators. It can be shown that any irreducible representation of Lie algebra can be acquired as a certain λ – representation determined by the choice of linear func- tional λ ∈ G∗ . Linear functional λ ≡ λ(j) where number of parameters j is equal to the number of Casimir functions - index of Lie algebra G. Since that measure dµ(λ) is a spectral measure of Casimir operators on Lie group. Let’s consider representation of the Lie group G in the functional space C∞ (Q) which acts on the functions from that space as follows Tλ g ψ(q) = Z Dλ qq0 (g)ψ(q0 )dµ(q0 ), (6) and appears to be the lift of λ–representation of Lie algebra to a group li(q, ∂q, λ) = ∂ ∂gi Tλ g |g=e. (7) Linear functional λ must be integer, i.e Z γ∈H2(Oλ) ωλ = 2πin, n ∈ Z, (8) where ωλ is well known Kirillov 2-form on the orbit [6]. Functions Dj qq0 (g) are matrix elements of representation (6) and can be found from equations [ξi(g) + li(q0, ∂0 q, j)]Dλ qq0 (g) = 0, Dλ qq0 (e) = δ(q, q0), (9) here e is identity element of the group. Functions Dλ qq0 (g) perform generalized Fourier transform on Lie group solving the main problem of harmonic analysis [9]. Here J if a manifold of parameters determining covector λ. ϕ(g) = Z Q×Q×J ϕ̂j(q, q0 )Dλ qq0 (g) dµ(q)dµ(q0 )dµ(λ). (10) So action of right-invariant and left-invariant vector fields on group goes into action of operators of λ–representation on Lagrange submanifold of the coadjoint orbit [5] ξiϕ(g) ⇐⇒ li(q0 , ∂0 q, λ)ϕ̂j(q, q0 ); ηiϕ(g) ⇐⇒ li(q, ∂q, λ)ϕ̂j(q, q0 ). (11) After transition from the group space to the Lagrange submanifold of the orbit Oλ we have heat kernel equation on coadjoint orbit with smaller number of variables [10] ∂Rβ(q, q̃, j) ∂β + H(−i~l)Rβ(q, q̃, j) = 0, Rβ(q, q̃, j)|β=0 = δ(q, q̃), (12) which appears to be ODE and is to be integrated in quadratures if following condition (dim G − ind G)/2 = 1 (13) is satisfied. In (12) heat kernel Rβ(q, q̃, j) is connected with ρβ(g, g0 ) on entire space by expression ρβ(g, g0 ) = Z Rβ(q, q̃, j)Dλ qq̃(g0−1 g)dµ(q)dµ(q̃)dµ(λ) (14) From solution of (12) we can obtain statistic sum on non-compact Lie group using properties of Dλ qq0 (g) Zβ = Z G dµ(x) Z Q×J Rβ(q, q, j)dµ(q)dµ(λ) = V olG Z Q×J Rβ(q, q, j)dµ(q)dµ(λ). (15) One can see that that integration in (15) over the volume of the manifold goes in- dependently from integration over measure dµ(q) on coadjoint orbit and spectral measure dµ(λ). So we have opportunity to factorize in statistic sum divergences connected with infinite volume of non-compact space and since that we have following expression for specific statistic sum zβ = Zβ/V olG = Z Rβ(q, q, j)dµ(q)dµ(λ), (16) which is sufficiently finite. So instead of solution of (2) with n independent variables we solve equation (12) with smaller number of variables and get specific statistic sum on non- compact group manifold. Application of statistic sum touches numerous fields of theoretical physics from quantum statistic mechanics and quantum field theory to information theory where traditional physical objects such as entropy find new and quite productive interpretation [11],[12]. 3 High temperature expansion of heat kernel on non-compact Lie groups Representation of the partition function and heat kernel itself as a power series (heat kernel expansion) is a significant problem. That expansion in the most general for the homogeneous case is to be written as Zβ = V ol(M) (4πβ)d/2 ∞ X n=0 anβn . (17) here β is an inverse thermodynamic temperature. In order to find the coefficients of heat kernel expansion on Lie group is proposed to express the heat kernel as Rβ(q, q̃, j) = exp( i ~ Sβ(q, q̃, j)), (18) where Sβ(q, q̃, j is a complex function. Using regular Fourier transform in respect to the variable q̃ φ(q, p) = Z φ(q, q̃) exp( ipq̃ ~ )dq̃, φ(q, q̃) = 1 (2π~) dimOλ 2 Z φ(q, p) exp(− ipq ~ )dp, it’s possible to pass to the function Rβ(q, p, j), which satisfies heat kernel equa- tion ∂Rβ(q, p, j) ∂β + Ĥ(−i~l(q, ∂q))Rβ(q, p, j) = 0. (19) The equation for the function Sβ(q, q̃, j) is i ~ ∂Sβ(q, p, j) ∂β + exp (− i ~ Sβ(q, p, j))Ĥ(−i~l(q, ∂q)) exp i ~ Sβ(q, p, j)) = 0 (20) with initial condition Sβ(q, p, j)|β=0 = pq. To be also represented as a power series Sβ(q, p, j) = ∞ X n=0 Sn(q, p, j)βn . (21) Ĥ(−i~l(q, ∂q)) is a second order differential operator and can be represented as Ĥ(−i~l(q, ∂q)) = −~2 Gab la(q, ∂q)lb(q, ∂q) = hab (q) ∂2 ∂qa∂qb + ha (q) ∂ ∂qa + h(q), (22) where coefficients hab , ha , h(q) can be easily obtained through the operators of λ — representation (7). Expression (22) using standard notation p̂a = i~ ∂ ∂qa can be rewritten as Ĥ(−i~l(q, ∂q)) = Hab (q)p̂ap̂a + Ha (q)p̂a + H(q). (23) So the equation (20) transforms at i h ∞ X k=0 kSkβ(k−1) (q, p, j) + ∞ X k=0 Θ(k) (q, p, j)βk = 0, (24) with notation Θ(k) (q, p, j) = −i~Hab Sk,ab(q, p, j)+Hab k X m=0 Sm,a(q, p, j)Sk−m,b(q, p, j)+Ha Sk,a(q, p, j)+Hδ0 k, Finally we get the recurrent expression to determine coefficients Sk+1(q, p, j) Sk+1(q, p, j) = i~ k + 1 Θ(k) (q, p, j). (25) It’s obvious that a coefficient corresponding to the first power β in (21) is H(q, p) - a qp-symbol of the hamiltonian Ĥ(−i~l(q, ∂q)). That allows to get the formula of the first order for the high temperature expansion of partition function zβ ≈ 1 (2π~) dimOλ 2 Z exp( i(q − q) h − H(q, p))dpdqdj. Power series of the heat kernel expansion on symplectic sheet to the coadjoint orbit Rβ(q, p, j) can be obtained through coefficients Sk(q, p, j) by expression Rβ(q, p, j) = ∞ X n=0 1 n! d dβn n+1 Y k=1 n+1−k X m=0 (( i ~ Sk(q, p, j))βk )m m! |β=0βn . (26) High temperature asymptotic of partition function (statistic sum) is to be found by the formula (16), which after inverse Fourier transformations is per- formed looks as follows zβ = 1 (2π~) dimOλ 2 Z Rβ(q, p, j) exp(− ipq ~ )dpdqdj = ∞ X n=0 znβn , so the coefficients zn of the partition sum expansion are zn = 1 (2π~) dimOλ 2 Z 1 n! d dβn n+1 Y k=1 n+1−k X m=0 (( i ~ Sk(q, p, j))βk )m m! |β=0e(− ipq ~ ) dµ(p)dµ(q)dµ(λ), (27) and for the expansion of the heat kernel itself Rn(q, q̃, j) = 1 (2π~) dimOλ 2 Z 1 n! d dβn n+1 Y k=1 n+1−k X m=0 (( i ~ Sk(q, p, j))βk )m m! |β=0e− ipq̃ ~ dµ(p). (28) The final result for the heat kernel expansion on the Lie group manifold G is obtained after substitution of coefficients (21) in the formula (14) ρn(x, x0 ) = Z Rn(q, q̃, j)Dλ qq̃(x0−1 x)dµ(q)dµ(q̃)dµ(λ). (29) Applications of heat kernel and partition sum are quite useful in many fields of theoretical physics. Among them are worth mentioning problems of quantum field theory and quantum thermodynamics as well as problems of theory of information being considered from a geometric point of view. As an application example of presented method must be mentioned result obtained by the author in [13] for heat kernel on group E(2). References 1. N.Hurt Geometric quantization in action, D.Reidel publishing company, 1983 2. S. Minakshisundaram and A. Pleijel Some Properties of the Eigen functions of the Laplace Operator on Riemannian Manifolds.// Can. J. Math. 1949. V. 1. P. 242-256. 3. S. R. S. Varadhan On the Behavior of the Fundamental Solution of the Heat Equa- tion. // Comm. Pure Appl. Math. 1967. V. 20. P. 431-455. 4. S.A.Molchanov Diffusion processes and Riemannian geometry. Uspekhi Mathem. Nauk, 1975, 30, No.1, 57 pp. 5. I.V. Shirokov Darboux coordinates on K-orbits and the spectra of Casimir operators on Lie groups.//Theoretical and Mathematical Physics, Vol. 123, No. 3, 2000. 6. A.A. Kirillov Elements of the Theory of Representations. Springer Verlag, Berlin- New York-Heidelberg, 1976. 7. Souriau J.-M, Structures des systmes dynamiques, Dunod, Paris, 1970 8. B.Kostant, Quantization and Unitary Representations, Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics, Vol. 170, Springer-Verlag. 9. A.Barut, R.Razcka Theory of Group Representations and Applications, World Sci- entific, 1986. 10. S.P. Baranovsky, V.V. Mikheyev, I.V. Shirokov Quantum Hamiltonian systems on K-orbits: Semiclassical spectrum of the asymmetric top. // Theoretical and Math- ematical Physics, Vol. 129, No. 1, 2001. 11. Nencka, H., Streater, R.F. Information Geometry for some Lie algebras. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1999, 2, 441460. 12. Barbaresco, F. Geometric Theory of Heat from Souriau Lie Groups Thermody- namics and Koszul Hessian Geometry, Entropy 2016, 18, 386. 13. V.V. Mikheyev, I.V. Shirokov Application of Coadjoint Orbits in the Thermody- namics of Non-Compact Manifolds. // Electronic Journal of Theoretical Physics, Vol.2, Issue 7, 2005, pp.1-10.