Tensors, Fields Pt. 1 and the Lie Bracket Pt. 1

Transcription

1 Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 PHYS Southern Illinois University September 8, 2016 PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

2 Tensor Prouct Spaces Definition Let U an V be two vector spaces with arbitrary bases {u j } U j=1 an {v k } V k=1 respectively. Their tensor prouct W = U V is the vector space with basis elements {u j v k } U, V j,k=1. The elements of W are linear combinations of the basis vectors w = α j,k u j v k an are calle tensors. Tensor aition satisfies α(u j v k ) + β(u j v l ) = u j (αv k + βv l ) α(u j v k ) + β(u l v k ) = (αu j + βu l ) v k. Quantum mechanics; C n. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

3 Tensor Proucts on the Manifol For a manifol M an point p M, we introuce the tensor prouct space M N T p (M,N) := Tp (M) T p (M), which is M copies of T p (M) an N copies of T p (M). Elements of T (M,N) p are calle ( M N) tensors. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

4 Tensor Proucts on the Manifol For a manifol M an point p M, we introuce the tensor prouct space M N T p (M,N) := Tp (M) T p (M), which is M copies of T p (M) an N copies of T p (M). Elements of T (M,N) p are calle ( M N) tensors. For local coorinates {x k }, basis vectors of T (M,N) p take the form e j 1,,j N i 1, i M = x i 1 x i M x j 1 x j N. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

5 Tensor Proucts on the Manifol For a manifol M an point p M, we introuce the tensor prouct space M N T p (M,N) := Tp (M) T p (M), which is M copies of T p (M) an N copies of T p (M). Elements of T (M,N) p are calle ( M N) tensors. For local coorinates {x k }, basis vectors of T (M,N) p take the form e j 1,,j N i 1, i M = x i 1 x i M x j 1 x j N. An arbitrary ( ) M N tensor T takes the form T = t i 1,,i M j 1, j N e j 1,,j N i 1, i M. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

6 Tensor Proucts on the Manifol Contraction of inices An ( ) M (N,M) N tensor is a functional on the space T p. Let T = t i 1,,i M j 1, j N e j 1,,j N i 1, i M Let S = s j 1, j N i 1,,i M e i 1, i M j 1,,j N be an ( M N) tensor. be an ( N M) tensor. Then: T (S) = S(T ) = t i 1,,i M j 1, j N s j 1, j N i 1,,i M. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

7 Tensor Proucts on the Manifol Contraction of inices An ( ) M (N,M) N tensor is a functional on the space T p. Let T = t i 1,,i M j 1, j N e j 1,,j N i 1, i M Let S = s j 1, j N i 1,,i M e i 1, i M j 1,,j N be an ( M N) tensor. be an ( N M) tensor. Then: T (S) = S(T ) = t i 1,,i M j 1, j N s j 1, j N i 1,,i M. Therefore: T (N,M) p T (M,N) p = T (M,N) p = T (N,M) p. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

8 Tensor Proucts on the Manifol Vectors an one-forms as ( 1 0) an ( 0 1) tensors, respectively. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

9 Tensor Proucts on the Manifol Vectors an one-forms as ( 1 0) an ( 0 1) tensors, respectively. Matrices as ( 1 1) tensors. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

10 Tensor Proucts on the Manifol Vectors an one-forms as ( 1 0) an ( 0 1) tensors, respectively. Matrices as ( 1 1) tensors. Multipartite quantum states as ( N 0) tensors. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

11 Tensor Proucts on the Manifol Vectors an one-forms as ( 1 0) an ( 0 1) tensors, respectively. Matrices as ( 1 1) tensors. Multipartite quantum states as ( N 0) tensors. More examples in future lecture (metric tensor, Maxwell stress tensor, etc.). PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

12 Tensor Proucts on the Manifol Partial Contractions An ( ) ( N M tensor can be converte into an N K ) M tensor by acting on K one-forms: T = t i 1,,i M j 1, j N e j 1,,j N i 1, i M T = T( τ σ(1),, τ σ(k), ) = t i 1,,i M j 1, j N ω iσ(1) ω iσ(k) e j σ(k+1),,j σ(n) i 1, i M. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

13 Tensor Proucts on the Manifol Partial Contractions An ( ) ( N M tensor can be converte into an N K ) M tensor by acting on K one-forms: T = t i 1,,i M j 1, j N e j 1,,j N i 1, i M T = T( τ σ(1),, τ σ(k), ) = t i 1,,i M j 1, j N ω iσ(1) ω iσ(k) e j σ(k+1),,j σ(n) i 1, i M. An ( ) ( N M tensor can be converte into an N M K) tensor by acting on K tangent vectors: T = t i 1,,i M j 1, j N e j 1,,j N i 1, i M T = T(v σ(1),, v σ(k), ) = t i 1,,i M j 1, j N v i σ(1) v i σ(k) e j 1,,j M i σ(k+1), i σ(n). PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

14 Vector an Tensor Fiels Definition A vector fiel V on a manifol M is mapping V : M T p (M). That is, for every point p M, V assigns a vector v T p (M). Definition +- If {x k } is a local coorinate system for a neighborhoo of p, then we can express V (p) = v(p) = v k (p) x k. The components v k (p) are thus functions on M. The vector fiel V is smooth if the functions v k (p) = v k (x 1,, x n ) are smooth. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

15 Vector an Tensor Fiels Every curve on M has a tangent vector at every point along the curve. But is the converse true? Given a vector fiel V, is it possible to start at one point p an fin a curve whose tangent vectors are specifie by V? PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

16 Vector an Tensor Fiels Every curve on M has a tangent vector at every point along the curve. But is the converse true? Given a vector fiel V, is it possible to start at one point p an fin a curve whose tangent vectors are specifie by V? Definition The question is whether there exists a curve γ(λ) = (x 1 (λ),, x n (λ)) such that x k λ = v k (x 1,, x n ) k = 1,, n. (1) It the v k are smooth functions, then a (unique) infinite family of such curves can always be foun. They are calle the integral curves of the vector fiel. We enote V = λ for vector fiels with integral curves. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

17 Vector an Tensor Fiels λ = x 1 x 2. x 2 x 1 PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

18 Exponentiation of Vector Fiel Definition Let λ be a vector fiel, an x k (λ) coorinate values of the integral curves. For some parameter values λ 0 an λ 0 + ɛ, consier the Taylor series evaluate at ɛ = 0: x k (λ 0 + ɛ) = x k (λ 0 ) + ɛ λ x k (λ 0 ) ɛ2 λ 2 x k (λ 0 ) ) 12 2 = (1 + ɛ + ɛ2 λ 2 x k ( λ0 := exp ɛ λ x k). (2) λ 0 The operator exp ( ɛ ) λ = e ɛ λ is calle the exponentiation of the vector fiel. When acting on the x k, it moves by an ɛ along the integral curves (exactly). PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

19 Lie Bracket For a local coorinate system {x k }, each of the efine a vector fiel. x k They are linearly inepenent an provie a basis for T p (M) for each point p. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

20 Lie Bracket For a local coorinate system {x k }, each of the efine a vector fiel. x k They are linearly inepenent an provie a basis for T p (M) for each point p. Notice that the x k commute: [ x j, x k ] = 0. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

21 Lie Bracket For a local coorinate system {x k }, each of the efine a vector fiel. x k They are linearly inepenent an provie a basis for T p (M) for each point p. Notice that the Definition x k commute: [ x j, x k ] = 0. For an n-imensional manifol M, a collection of vector fiels {V k } n k=1 is calle a coorinate basis if V k = for some set of local coorinates {y k }. y k PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

22 Lie Bracket For a local coorinate system {x k }, each of the efine a vector fiel. x k They are linearly inepenent an provie a basis for T p (M) for each point p. Notice that the Definition x k commute: [ x j, x k ] = 0. For an n-imensional manifol M, a collection of vector fiels {V k } n k=1 is calle a coorinate basis if V k = for some set of local coorinates {y k }. y k For a coorinate basis, the local coorinates y k parametrize the integral curves themselves! PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

23 Lie Bracket Uner what conitions o the V k = λ k form a coorinate basis? PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

24 Lie Bracket Uner what conitions o the V k = λ k form a coorinate basis? A necessary conition is that commute: λ j λ k λ k λ j = v j an x j λ k = w k x k = (v j w k λ j x j w j v k ) x j x k. must PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

25 Lie Bracket Uner what conitions o the V k = λ k form a coorinate basis? A necessary conition is that commute: Definition λ j λ k λ k λ j = v j an x j λ k = w k x k = (v j w k λ j x j w j v k ) x j x k. must The commutator [ λ j, λ k ] is calle the Lie bracket of the two vector fiels. The Lie bracket is a vector fiel itself. PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

26 Lie Bracket Consier polar coorinates x = r cos θ, y = r sin θ an the vector fiels obtaine by the rotation: ( ) r = τ ( cos θ ) ( sin θ sin θ cos θ x y ). PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13

27 Lie Bracket Consier polar coorinates x = r cos θ, y = r sin θ an the vector fiels obtaine by the rotation: ( ) r = τ Picture of the commutator. ( cos θ ) ( sin θ sin θ cos θ x y ). PHYS Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8, / 13