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Text of the page (random words):
ion of möbius transformations the möbius transformations can be represented by matrices α β γ δ α δ β γ 1 displaystyle begin pmatrix alpha beta gamma delta end pmatrix qquad alpha delta beta gamma 1 since a common factor of α β γ δ cancels for the same reason the matrix is not uniquely defined since multiplication by i has no effect on either the determinant or the möbius transformation the composition law of möbius transformations follow that of the corresponding matrices the conclusion is that each möbius transformation corresponds to two matrices g g sl 2 c using this correspondence one may write π u g ϕ π u cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 e i ϕ 2 0 0 e i ϕ 2 π u g θ π u 1 0 0 0 cos θ sin θ 0 sin θ cos θ cos θ 2 i sin θ 2 i sin θ 2 cos θ 2 displaystyle begin aligned pi _ u g_ phi pi _ u left begin pmatrix cos phi sin phi 0 sin phi cos phi 0 0 0 1 end pmatrix right pm begin pmatrix e i frac phi 2 0 0 e i frac phi 2 end pmatrix pi _ u g_ theta pi _ u left begin pmatrix 1 0 0 0 cos theta sin theta 0 sin theta cos theta end pmatrix right pm begin pmatrix cos frac theta 2 i sin frac theta 2 i sin frac theta 2 cos frac theta 2 end pmatrix end aligned these matrices are unitary and thus π u so 3 su 2 sl 2 c in terms of euler angles nb 1 one finds for a general rotation g ϕ θ ψ g ϕ g θ g ψ cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 1 0 0 0 cos θ sin θ 0 sin θ cos θ cos ψ sin ψ 0 sin ψ cos ψ 0 0 0 1 cos ϕ cos ψ cos θ sin ϕ sin ψ cos ϕ sin ψ cos θ sin ϕ cos ψ sin ϕ sin θ sin ϕ cos ψ cos θ cos ϕ sin ψ sin ϕ sin ψ cos θ cos ϕ cos ψ cos ϕ sin θ sin ψ sin θ cos ψ sin θ cos θ displaystyle begin aligned g phi theta psi g_ phi g_ theta g_ psi begin pmatrix cos phi sin phi 0 sin phi cos phi 0 0 0 1 end pmatrix begin pmatrix 1 0 0 0 cos theta sin theta 0 sin theta cos theta end pmatrix begin pmatrix cos psi sin psi 0 sin psi cos psi 0 0 0 1 end pmatrix begin pmatrix cos phi cos psi cos theta sin phi sin psi cos phi sin psi cos theta sin phi cos psi sin phi sin theta sin phi cos psi cos theta cos phi sin psi sin phi sin psi cos theta cos phi cos psi cos phi sin theta sin psi sin theta cos psi sin theta cos theta end pmatrix end aligned 1 one has 5 π u g ϕ θ ψ e i ϕ 2 0 0 e i ϕ 2 cos θ 2 i sin θ 2 i sin θ 2 cos θ 2 e i ψ 2 0 0 e i ψ 2 cos θ 2 e i ϕ ψ 2 i sin θ 2 e i ϕ ψ 2 i sin θ 2 e i ϕ ψ 2 cos θ 2 e i ϕ ψ 2 displaystyle begin aligned pi _ u g phi theta psi pm begin pmatrix e i frac phi 2 0 0 e i frac phi 2 end pmatrix begin pmatrix cos frac theta 2 i sin frac theta 2 i sin frac theta 2 cos frac theta 2 end pmatrix begin pmatrix e i frac psi 2 0 0 e i frac psi 2 end pmatrix pm begin pmatrix cos frac theta 2 e i frac phi psi 2 i sin frac theta 2 e i frac phi psi 2 i sin frac theta 2 e i frac phi psi 2 cos frac theta 2 e i frac phi psi 2 end pmatrix end aligned 2 for the converse consider a general matrix π u g α β α β β α su 2 displaystyle pm pi _ u g_ alpha beta pm begin pmatrix alpha beta overline beta overline alpha end pmatrix in operatorname su 2 make the substitutions cos θ 2 α sin θ 2 β 0 θ π 1 2 ϕ ψ arg α 1 2 ψ ϕ arg β displaystyle begin aligned cos tfrac theta 2 alpha sin tfrac theta 2 beta 0 leq theta leq pi tfrac 1 2 phi psi arg alpha tfrac 1 2 psi phi arg beta end aligned with the substitutions π g α β assumes the form of the right hand side rhs of 2 which corresponds under π u to a matrix on the form of the rhs of 1 with the same φ θ ψ in terms of the complex parameters α β g α β 1 2 α 2 β 2 α 2 β 2 i 2 α 2 β 2 α 2 β 2 α β α β i 2 α 2 β 2 α 2 β 2 1 2 α 2 β 2 α 2 β 2 i α β α β α β α β i α β α β α α β β displaystyle g_ alpha beta begin pmatrix frac 1 2 left alpha 2 beta 2 overline alpha 2 overline beta 2 right frac i 2 left alpha 2 beta 2 overline alpha 2 overline beta 2 right alpha beta overline alpha overline beta frac i 2 left alpha 2 beta 2 overline alpha 2 overline beta 2 right frac 1 2 left alpha 2 beta 2 overline alpha 2 overline beta 2 right i left alpha beta overline alpha overline beta right alpha overline beta overline alpha beta i left alpha overline beta overline alpha beta right alpha overline alpha beta overline beta end pmatrix to verify this substitute for α β the elements of the matrix on the rhs of 2 after some manipulation the matrix assumes the form of the rhs of 1 it is clear from the explicit form in terms of euler angles that the map p su 2 so 3 π u g α β g α β displaystyle begin cases p operatorname su 2 to operatorname so 3 pm pi _ u g_ alpha beta mapsto g_ alpha beta end cases just described is a smooth 2 1 and surjective group homomorphism it is hence an explicit description of the universal covering space of so 3 from the universal covering group su 2 lie algebra edit associated with every lie group is its lie algebra a linear space of the same dimension as the lie group closed under a bilinear alternating product called the lie bracket the lie algebra of so 3 displaystyle operatorname so 3 is denoted by s o 3 displaystyle mathfrak so 3 and consists of all skew symmetric 3 3 matrices 6 this may be seen by differentiating the orthogonality condition a t a i a so 3 displaystyle a t a i a in operatorname so 3 nb 2 the lie bracket of two elements of s o 3 displaystyle mathfrak so 3 is as for the lie algebra of every matrix group given by the matrix commutator a 1 a 2 a 1 a 2 a 2 a 1 displaystyle a_ 1 a_ 2 a_ 1 a_ 2 a_ 2 a_ 1 which is again a skew symmetric matrix the lie algebra bracket captures the essence of the lie group product in a sense made precise by the baker campbell hausdorff formula the elements of s o 3 displaystyle mathfrak so 3 are the infinitesimal generators of rotations i e they are the elements of the tangent space of the manifold so 3 displaystyle operatorname so 3 at the identity element if r ϕ n displaystyle r phi boldsymbol n denotes a counterclockwise rotation with angle ϕ displaystyle phi about the axis specified by the unit vector n displaystyle boldsymbol n then u r 3 d d ϕ ϕ 0 r ϕ n u n u displaystyle forall boldsymbol u in mathbb r 3 qquad left frac operatorname d operatorname d phi right _ phi 0 r phi boldsymbol n boldsymbol u boldsymbol n times boldsymbol u this can be used to show that the lie algebra s o 3 displaystyle mathfrak so 3 with commutator is isomorphic to the lie algebra r 3 displaystyle mathbb r 3 with cross product under this isomorphism an euler vector ω r 3 displaystyle boldsymbol omega in mathbb r 3 corresponds to the linear map ω displaystyle widetilde boldsymbol omega defined by ω u ω u displaystyle widetilde boldsymbol omega boldsymbol u boldsymbol omega times boldsymbol u in more detail most often a suitable basis for s o 3 displaystyle mathfrak so 3 as a 3 dimensional vector space is l x 0 0 0 0 0 1 0 1 0 l y 0 0 1 0 0 0 1 0 0 l z 0 1 0 1 0 0 0 0 0 displaystyle boldsymbol l _ x begin bmatrix 0 0 0 0 0 1 0 1 0 end bmatrix quad boldsymbol l _ y begin bmatrix 0 0 1 0 0 0 1 0 0 end bmatrix quad boldsymbol l _ z begin bmatrix 0 1 0 1 0 0 0 0 0 end bmatrix the commutation relations of these basis elements are l x l y l z l z l x l y l y l z l x displaystyle boldsymbol l _ x boldsymbol l _ y boldsymbol l _ z quad boldsymbol l _ z boldsymbol l _ x boldsymbol l _ y quad boldsymbol l _ y boldsymbol l _ z boldsymbol l _ x which agree with the relations of the three standard unit vectors of r 3 displaystyle mathbb r 3 under the cross product as announced above one can identify any matrix in this lie algebra with an euler vector ω x y z r 3 displaystyle boldsymbol omega x y z in mathbb r 3 7 ω ω l x l x y l y z l z 0 z y z 0 x y x 0 s o 3 displaystyle widehat boldsymbol omega boldsymbol omega cdot boldsymbol l x boldsymbol l _ x y boldsymbol l _ y z boldsymbol l _ z begin bmatrix 0 z y z 0 x y x 0 end bmatrix in mathfrak so 3 this identification is sometimes called the hat map 8 under this identification the s o 3 displaystyle mathfrak so 3 bracket corresponds in r 3 displaystyle mathbb r 3 to the cross product u v u v displaystyle left widehat boldsymbol u widehat boldsymbol v right widehat boldsymbol u times boldsymbol v the matrix identified with a vector u displaystyle boldsymbol u has the property that u v u v displaystyle widehat boldsymbol u boldsymbol v boldsymbol u times boldsymbol v where the left hand side we have ordinary matrix multiplication this implies u displaystyle boldsymbol u is in the null space of the skew symmetric matrix with which it is identified because u u 0 displaystyle boldsymbol u times boldsymbol u boldsymbol 0 a note on lie algebras edit main article angular momentum operator see also representation theory of su 2 and jordan map in lie algebra representations the group so 3 is compact and simple of rank 1 and so it has a single independent casimir element a quadratic invariant function of the three generators which commutes with all of them the killing form for the rotation group is just the kronecker delta and so this casimir invariant is simply the sum of the squares of the generators j x j y j z displaystyle boldsymbol j _ x boldsymbol j _ y boldsymbol j _ z of the algebra j x j y j z j z j x j y j y j z j x displaystyle boldsymbol j _ x boldsymbol j _ y boldsymbol j _ z quad boldsymbol j _ z boldsymbol j _ x boldsymbol j _ y quad boldsymbol j _ y boldsymbol j _ z boldsymbol j _ x that is the casimir invariant is given by j 2 j j j x 2 j y 2 j z 2 i displaystyle boldsymbol j 2 equiv boldsymbol j cdot boldsymbol j boldsymbol j _ x 2 boldsymbol j _ y 2 boldsymbol j _ z 2 propto boldsymbol i for unitary irreducible representations d j the eigenvalues of this invariant are real and discrete and characterize each representation which is finite dimensional of dimensionality 2 j 1 displaystyle 2j 1 that is the eigenvalues of this casimir operator are j 2 j j 1 i 2 j 1 displaystyle boldsymbol j 2 j j 1 boldsymbol i _ 2j 1 where j is integer or half integer and referred to as the spin or angular momentum so the 3 3 generators l displayed above act on the triplet spin 1 representation while the 2 2 generators below t act on the doublet spin 1 2 representation by taking kronecker products of d 1 2 with itself repeatedly one may construct all higher irreducible representations d j that is the resulting generators for higher spin systems in three spatial dimensions for arbitrarily large j can be calculated using these spin operators and ladder operators for every unitary irreducible representations d j there is an equivalent one d j 1 all infinite dimensional irreducible representations must be non unitary since the group is compact in quantum mechanics the casimir invariant is the angular momentum squared operator integer values of spin j characterize bosonic representations while half integer values fermionic representations the antihermitian matrices used above are utilized as spin operators after they are multiplied by i so they are now hermitian like the pauli matrices thus in this language j x j y i j z j z j x i j y j y j z i j x displaystyle boldsymbol j _ x boldsymbol j _ y i boldsymbol j _ z quad boldsymbol j _ z boldsymbol j _ x i boldsymbol j _ y quad boldsymbol j _ y boldsymbol j _ z i boldsymbol j _ x and hence j 2 j j 1 i 2 j 1 displaystyle boldsymbol j 2 j j 1 boldsymbol i _ 2j 1 explicit expressions for these d j are j z j b a j 1 a δ b a j x j b a 1 2 δ b a 1 δ b 1 a j 1 a b 1 a b j y j b a 1 2 i δ b a 1 δ b 1 a j 1 a b 1 a b displaystyle begin aligned left boldsymbol j _ z j right _ ba j 1 a delta _ b a left boldsymbol j _ x j right _ ba frac 1 2 left delta _ b a 1 delta _ b 1 a right sqrt j 1 a b 1 ab left boldsymbol j _ y j right _ ba frac 1 2i left delta _ b a 1 delta _ b 1 a right sqrt j 1 a b 1 ab end aligned where j is arbitrary and 1 a b 2 j 1 displaystyle 1 leq a b leq 2j 1 for example the resulting spin matrices for spin 1 j 1 displaystyle j 1 are j x 1 2 0 1 0 1 0 1 0 1 0 j y 1 2 0 i 0 i 0 i 0 i 0 j z 1 0 0 0 0 0 0 0 1 displaystyle begin aligned boldsymbol j _ x frac 1 sqrt 2 begin pmatrix 0 1 0 1 0 1 0 1 0 end pmatrix boldsymbol j _ y frac 1 sqrt 2 begin pmatrix 0 i 0 i 0 i 0 i 0 end pmatrix boldsymbol j _ z begin pmatrix 1 0 0 0 0 0 0 0 1 end pmatrix end aligned note however how these are in an equivalent but different basis the spherical basis than the above i l in the cartesian basis nb 3 for higher spins such as spin 3 2 displaystyle tfrac 3 2 j 3 2 displaystyle j tfrac 3 2 j x 1 2 0 3 0 0 3 0 2 0 0 2 0 3 0 0 3 0 j y 1 2 0 i 3 0 0 i 3 0 2 i 0 0 2 i 0 i 3 0 0 i 3 0 j z 1 2 3 0 0 0 0 1 0 0 0 0 1 0 0 0 0 3 displaystyle begin aligned boldsymbol j _ x frac 1 2 begin pmatrix 0 sqrt 3 0 0 sqrt 3 0 2 0 0 2 0 sqrt 3 0 0 sqrt 3 0 end pmatrix boldsymbol j _ y frac 1 2 begin pmatrix 0 i sqrt 3 0 0 i sqrt 3 0 2i 0 0 2i 0 i sqrt 3 0 0 i sqrt 3 0 end pmatrix boldsymbol j _ z frac 1 2 begin pmatrix 3 0 0 0 0 1 0 0 0 0 1 0 0 0 0 3 end pmatrix end aligned for spin 5 2 displaystyle tfrac 5 2 j 5 2 displaystyle j tfrac 5 2 j x 1 2 0 5 0 0 0 0 5 0 2 2 0 0 0 0 2 2 0 3 0 0 0 0 3 0 2 2 0 0 0 0 2 2 0 5 0 0 0 0 5 0 j y 1 2 0 i 5 0 0 0 0 i 5 0 2 i 2 0 0 0 0 2 i 2 0 3 i 0 0 0 0 3 i 0 2 i 2 0 0 0 0 2 i 2 0 i 5 0 0 0 0 i 5 0 j z 1 2 5 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 5 displaystyle begin aligned boldsymbol j _ x frac 1 2 begin pmatrix 0 sqrt 5 0 0 0 0 sqrt 5 0 2 sqrt 2 0 0 0 0 2 sqrt 2 0 3 0 0 0 0 3 0 2 sqrt 2 0 0 0 0 2 sqrt 2 0 sqrt 5 0 0 0 0 sqrt 5 0 end pmatrix boldsymbol j _ y frac 1 2 begin pmatrix 0 i sqrt 5 0 0 0 0 i sqrt 5 0 2i sqrt 2 0 0 0 0 2i sqrt 2 0 3i 0 0 0 0 3i 0 2i sqrt 2 0 0 0 0 2i sqrt 2 0 i sqrt 5 0 0 0 0 i sqrt 5 0 end pmatrix boldsymbol j _ z frac 1 2 begin pmatrix 5 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 0 5 end pmatrix end aligned main article spin physics higher spins isomorphism with 𝖘𝖚 2 edit the lie algebras s o 3 displaystyle mathfrak so 3 and s u 2 displaystyle mathfrak su 2 are isomorphic one basis for s u 2 displaystyle mathfrak su 2 is given by 9 t 1 1 2 0 i i 0 t 2 1 2 0 1 1 0 t 3 1 2 i 0 0 i displaystyle boldsymbol t _ 1 frac 1 2 begin bmatrix 0 i i 0 end bmatrix quad boldsymbol t _ 2 frac 1 2 begin bmatrix 0 1 1 0 end bmatrix quad boldsymbol t _ 3 frac 1 2 begin bmatrix i 0 0 i end bmatrix these are related to the pauli matrices by t i 1 2 i σ i displaystyle boldsymbol t _ i longleftrightarrow frac 1 2i sigma _ i the pauli matrices abide by the physicists convention for lie algebras in that convention lie algebra elements are multiplied by i the exponential map below is defined with an extra factor of i in the exponent and the structure constants remain the same but the definition of them acquires a factor of i likewise commutation relations acquire a factor of i the commutation relations for the t i displaystyle boldsymbol t _ i are t i t j ε i j k t k displaystyle boldsymbol t _ i boldsymbol t _ j varepsilon _ ijk boldsymbol t _ k where ε ijk is the totally anti symmetric symbol with ε 123 1 the isomorphism between s o 3 displaystyle mathfrak so 3 and s u 2 displaystyle mathfrak su 2 can be set up in several ways for later convenience s ...