Some Concepts used in the Study of Harish-Chandra Algebras of Matrices

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1 Intl J Engg Sci Adv Research 2015 Mar;1(1): Harish-Chandra Algebras Some Concepts used in the Study of Harish-Chandra Algebras of Matrices Vinod Kumar Yadav Department of Mathematics Rama University Kanpur India vinodkumaryadav3@gmailcom Abstract The main purpose of the work in this paper is to form base to know and explore the concepts involved in the papers [5] and [6] of Prof Harish-Chandra ( ) We have realized that it is very difficult to understand his work which certainly requires a very vast thinking and wide brain but we feel that our attempt is certainly pious Keywords Jacobian; metric; Hamiltonian; wave equation; representation We know that the set I INTRODUCTION where stands for the set of all real numbers forms a vector space over the field The dimension of this vector space is and the set is the standard basis for the vector space If we now assume the universe or the space as the collection of points in it and consider independent properties related to each such point these properties are expressed as real variables ; having different real values for different points We may think that in this way we are associating to the geometrical structure (the universe) an algebraic structure (the vector space ) The variables are called the coordinates of the related points Thus the coordinates are independent real valued functions depending on the points of the universe Any other independent functions may also serve the purpose of coordinates thus if are such that the functional determinant ie the Jacobian then can also be assumed to be the coordinates of the points We choose the coordinate system suitable to our particular geometric structure The universe related in this way will be denoted as When the coordinates are restricted to be between certain specified limits the totality of such points of is called a region of The set of all those points of whose coordinates may be expressed as function of single parameter is called a curve in It means that the equations define a curve in A congruence of curves is such a family of curves one of which passes through each point of The set of all those points of whose coordinates are expressible as functions of two independent parameters forms a surface in Thus the equations defining a surface are given by The totality of points whose coordinates are expressible as functions of independent parameters is called a variety or subspace of of dimension and we denote it as Such subspace is said to immersed in if is called a hyper surface of We can solve for independent parameters from any relations out of relations expressing coordinates in these parameters and substitute the values of these parameters in the remaining one relation to get the equation of the hyper surface in the form If and are two adjacent points of a and if we define the infinitesimal distance between them by the differential quadratic form (1) where the coefficients are functions of the coordinates then the attached with in this way is called a Riemannian space and is called Riemannian metric Geometry based upon a Riemannian metric is known as Riemannian geometry The equation (1) is also known as the fundamental quadratic form [2] [8] [18] In the equation (1) we have used Einstein summation convention ie both and are repeating in the single term hence are dummy indices and the summation is intended with respect each value of and In fact right hand side is the sum of terms and thus obtained is a scalar invariant are components of a contravariant vector and is open product of this vector with itself hence is a contravariant tensor of second order is a covariant tensor of second order which may be assumed to be symmetric

2 Vinod Kumar Yadav (2015) Riemannian metric is said to be Cartesian if all are constants The Cartesian system of coordinates is called orthogonal if If the equation (1) can be reduced to the form = 0 [ ] = -1 [ ] In the matrix form these conditions can be written as by suitable transformation of coordinates to then such coordinate system is called Euclidean coordinates system the metric and the space are said to be Euclidean Thus Euclidean coordinates are a particular case of Orthogonal Cartesian coordinates In case of Euclidean metric the fundamental metric tensor is defined by where is Kronecker tensor defined by or II A PARTICULAR RIEMANNIAN SPACE USED IN RELATIVITY THEORY Now we consider and define corresponding to points and of 4 In this case 00=1 and If we transform the coordinates to st each new coordinate is a linear function of i e and st then each such linear transformation is called a Lorentz transformation The set of all Lorentz transformations of forms a group under the multiplication of transformations and this group is known as the Lorentz Group [8] [13] The matrix represents a Lorentz transformation if where is the summation index Equating the coefficients for all values of ; we get the following 10 equations in 16 unknowns : = 1 [coefficients of ] = 0 [ ] = 0 [ ] = 0 [ ] = -1 [ ] = 0 [ ] = 0 [ ] or where and are defined as above In special theory of relativity the Lorentz transformation where is velocity of light and is the relative velocity of each coordinates system (reference frame) with respect to another is used in some important problems In this case the matrix of this transformation is We can verify that this special Lorentz transformation leaves the required quadratic form invariant ie It is also known and can be verified that under this Lorentz transformation the D alembertian operator is also invariant ie here is well known Laplacian operator Let be components of the linear momentum of a particle moving freely ie no external force is acting on it and let be the relativistic energy of this particle ie where is Planck s constant is the frequency and If we consider the function ; = -1 [ ] 135

3 Intl J Engg Sci Adv Research 2015 Mar;1(1): (2) Harish-Chandra Algebras In Special Relativity the quantity defined by is invariant under Lorentz transformation hence here and this gives get from which we ie (here ) and also Again from equation (3) we get from these calculation we get (3) using this relation we say that the vector four vector and its components are is a We note here that the square of the last component minus the sum of the squares of the first three is equal to 1 verifying the above required condition By multiplying each component of by using the formula Thus we have seen that the function given by equation (2) is a solution of the equation (4) which is known as the Klein-Gordon equation [11] [13] for the relativistic free motion of a particle of rest mass It is known that if rest mass is then the quantity is called the Lorentz mass of the particle Similarly the rest energy is related to the relativistic energy E by the formula The energy defined by this relation is the energy of free particle if the particle moves in a field of potential the total energy is Above in the derivation of the Klein-Gordon equation (4) we have denoted for the relativistic energy of the free particle If we retain the notation also for the relativistic energy of the free particle and assume that the velocity is small in comparison with then this vector by we get that ie So that we have the relation is a four vector hence get that ie In this formula the energy is expressed in terms of momentum We know that the energy expressed is terms of coordinates and corresponding momentum is known as Hamiltonian [12] Here we also want to point out that in quantum mechanics we assume the dynamical variables such as the coordinates components of momentum and angular momentum of the particles and functions of these quantities correspond to linear operators on a vector space which are subjected to the noncommutative algebra Hence the Hamiltonian in quantum mechanics is treated as a linear operator This is a main difference between classical and quantum mechanics The momentum components when represented by the linear operators is called the Schrödinger representation [3] [12] [14] III SCHRÖDINGER WAVE EQUATION For the non-relativistic consideration the energy for the free particle is just the kinetic energy hence let us consider the function ie Kinetic energy for non-relativistic case In the general case also we define the Kinetic energy of the particle by where In this case we get (5)

4 Vinod Kumar Yadav (2015) and Thus we get the equation (6) which is called non-relativistic wave equation for a free particle [10] [11] From the relation (5) we have which gives From the wave equation (6) may be written as ie (8) Here is non-relativistic Hamiltonian for free motion of the particle ie where are differential operators defined by the equation (7) If we now suppose that the particle is not free but external force applied on it is given by potential function ie external force then in the equation (8) will be replaced by and we get the wave equation in the form (7) Since we know that this equation becomes This function ie the solution of wave equation is known as wave function [1] [3] [10] [12] [14] [15] [17] IV THE RELATIVISTIC WAVE EQUATION OF ELECTRON The principle of relativity states that the laws of physics are the same for two observers moving with constant velocity relative to each other A physical law is expressed by a mathematical equation This equation should be of the same form in all the Lorentz frames ie it should be invariant under the Lorentz transformation An electron is an elementary particle ie a quanta of electromagnetic field having a spin of magnitude From the article II we know that the relativistic Hamiltonian for the free particle with rest mass is given by the formula (10) In quantum mechanical treatment is represented by a linear operator and due to the properties discussed in the article III the components of the momentum are represented by the operators is The Schrödinger wave equation obtained in the last article (11) with the Hamiltonian (10) equation (11) becomes (9) where is the total energy ie the sum of kinetic and potential energies This equation (9) is known as Schrödinger s wave equation for non-relativistic motion of the particle This equation has finite solutions only for special values of the energy For each such value of the system is said to be in corresponding energy state These values of are called energy levels We can write the Schrödinger s wave equation (9) in alternative forms In this case the Hamiltonian of the particle can be written as Hence the Schrödinger s wave equation becomes where we have written and Now writing we get (12) 137

5 Intl J Engg Sci Adv Research 2015 Mar;1(1): This wave equation as stated by Prof PAM Dirac [3] [4] is unsatisfactory from the point of view of relativistic theory and he obtained that the equation (13) is the desired wave equation In equation (13) stands for a vector and the quantity before is a linear operator ie are also linear operators are defined as above and are independent of s They are such that if then ( ) (14) here is the well known Kronecker delta function ie if and if Thus four s all anticommute with one another and the square of each is identity operator We should note here that we are adding and multiplying the operators and getting an operator ie we are concern here with the algebra of operators We know that if V is a vector space over a field F then the set A(V) of linear operators on V forms an algebra [7] [8] [16] with unit element We should also have in mind that A(V) plays the universal role in the sense that every algebra with unit element over F is isomorphic to a subalgebra of A(V) for some vector space V over F [7] Since our operator ( ) are real in the sense that they have real eigenvalues 1 and 1 because Hence if we represent concerning algebra by selecting V(F) C 4 (C) ie represent by operators on C 4 then following Dirac [3] [4] we get (15) Here We may verify that the s and s are all Hermitian hence s are also Hermitian The Hermitian matrices have real eigenvalues ie real operators correspond to the Hermitian matrices The wave equation (12) can be written as (16) Harish-Chandra Algebras where and This is field free wave equation for the electron [1] [10] [12] [14] [15] If an electromagnetic field is present then the required wave equation is (17) where is scalar potential and is vector potential of the field To prove that wave equation (17) is relativistic we require to show that it is invariant under the infinitesimal Lorentz transformation If is a 4-vector in one Lorentz frame and is a 4-vector in another Lorentz frame such that where the are small numbers of the first order then it is required that Such transformation from to is called an infinitesimal Lorentz transformation We can see that this condition is satisfied if the quantities are antisymmetric ie or by raising the indices by fundamental metric tensor we can write It has been proved by Dirac [3] that the wave equation (17) is invariant under such transformation V GROUP REPRESENTATION We know that if V is a vector space over a field F then the field F is contained in A(V) in the sense that each element α of F can be treated as a linear operator on V defined by for each Now if is an arbitrary set and we define a mapping on into A(V) st ie is a linear operator on V Under these circumstances we say that the pair (VP) is an S-module on F Let where is a group At this stage it is useful to remind the well known Cayley s theorem [7] in the theory of groups This theorem states that every group is isomorphic to a subgroup of A(S) for suitable S Here A(S) stands for all oneone onto functions defined on S We know that A(S) is a group under the product ie composition of two such functions Being one-one onto each function f is an invertible Now we have to see that which subgroup of A(S) is isomorphic G We take and to each we define the one-one onto function such that We prove that is a group with the product of any two elements of namely and by the formula Since Inverse of is It is well known that other properties of the group are easily verified and where φ is isomorphism of onto defined by

6 Vinod Kumar Yadav (2015) Thus every group can be isomorphically be represented by the group of selected one-one onto functions ie transformations on onto itself Now we will discuss representations [9] which are more useful for practical purposes The groups are represented by subgroups of A(V) where V is a vector space over a field F and A(V) is an algebra of linear operators defined on V Formally a representation of a group G is an homomorphism P on G into A(V) ie for each we have Here and the S-module (VP) is called representation space Clearly where e is identity in is the identity operator on V The operator is inverse operator of We know that any linear operator on a vector space V(F) of finite dimension is represented by a square matrix of order We select an ordered basis and obtain the image of each basic vector which will clearly be a linear combination of all basic vectors Then if is j th basic vector and ie ( are the coordinates of varying from 1 to we get the matrix of the linear operator as ie the coordinates of form the j th column of In the case of group representation if we represent the elements of the groups by linear operators then we say it as abstract representation and when we replace linear operators by corresponding matrices then we call the representation as matricial representation The change of basis gives equivalent representations ie two matricial representations of G are said to be equivalent if there exits an invertible matrix such that similarly two abstract representations are equivalent if their representation spaces are isomorphic We know that a vector subspace W of a vector space V is said to be invariant under an operator if If is a representation of G then V is said to be invariant with respect to if for every ie if W is invariant under each operator An S-module (VP) is said to be simple if it is of dimension >0 and if the only invariant subspaces of V are {0} and V If an S-module (VP) is representation space of a group and it is simple then the representation is also called simple or irreducible representation of Similarly an S- module is called semi-simple if it can be represented as a sum of simple sub-module where the term sub-module stands for ; being invariant subspace of V At the last we want to write the matricial representation of the group which is clearly of order 16 and it is supposed that We can say that this group is generated by the elements i where the composition is defined by above relations and as is well known that There are only two complex numbers namely 1 and -1 whose square is 1 and also product of complex numbers is commutative Here we need quantities other than 1 and -1 and whose product is not commutative hence we seek the representation of elements of as linear operators on C 2 (C) We can verify that the required matricial representation is given by with other elements of written using their properties stated above This representation of is clearly irreducible [9] [13] VI CONCLUSION The above mathematics helps in study of the whole papers [5] and [6] of a Kanpur born distinguished Mathematician of the world Prof Harish-Chandra ( ) a second FRS of India after Srinivasa Ramanujan ACKNOWLEDGMENT I am grateful to my supervisor Dr T N Trivedi for guidance and encouragement during the study of this work REFERENCES [1] M Alonso and H Valk Quantum Mechanics Principles and Applications Addison Wesley Publishing Company Limited 1973 [2] EM Corson Introduction to Tensors Spinors and Relativistic Wave- Equations Blackie & Son Limited Glasgow 1953 [3] PAM Dirac The Principles of Quantum Mechanics Clarendon Press Fourth edition Oxford 1958 [4] PAM Dirac The Quantum Theory of the Electron Proceedings of the Royal Society vol A117 No(778) pp [5] Harish-Chandra Algebra of the Dirac-Matrices Proceedings of the Indian Academy of Sciences vol 22 pp [6] Harish-Chandra Motion of an Electron in the Field of a Magnetic Pole Physical Review vol74 No8 pp [7] IN Herstein Topics in Algebra John Wiley & Sons New York 1975 [8] K Hoffman and R Kunze Linear Algebra Prentice-Hall Inc USA 1971 [9] N Jacobson Basic Algebra II WH Freeman and Company USA 1980 [10] PM Mathews and K Venkatesan A Text book of Quantum Mechanics Tata Mc Graw-Hill Publishing Company Limited New Delhi 1976 [11] J McConnell Quantum Particle Dynamics North-Holland Publishing Company Amsterdam 1960 [12] V Rojansky Introductory Quantum Mechanics Prentice-Hall Inc USA 1938 [13] P Roman Theory of Elementary Particles North-Holland Publishing Company Amsterdam 1964 [14] P Roman Advanced Quantum Theory Addison-Wesley Publishing Company Inc New York 1965 [15] ME Rose Relativistic Electron Theory John Wiley & Sons Inc New York 1961 [16] GF Simmons "Introduction to Topology and Modern Analysis Mc Graw-Hill Book Company Singapore 1963 [17] GL Trigg Quantum Mechanics D Van Nostrand Company Inc Princeton New Jersey 1964 [18] CE Weatherburn An Introduction to Riemannian Geometry and the Tensor Calculus Cambridge University Press New York

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msqm 2011/8/14 21:35 page 189 #197 Bibliography Dirac, P. A. M., The Principles of Quantum Mechanics, 4th Edition, (Oxford University Press, London, 1958). Feynman, R. P. and A. P. Hibbs, Quantum Mechanics