Post-Modern Topics in Discrete Clifford Nelson Faustino Analysis - PowerPoint PPT Presentation

Post-Modern Topics in Discrete Clifford Analysis Post-Modern Topics in Discrete Clifford Nelson Faustino Analysis What I’ve learned from Frank The radial algebra approach Beyond Landau-Weyl Calculus Nelson Faustino Lie-algebraic discretization Weyl-Heisenberg symmetries Center of Mathematics, Computation and Cognition, UFABC Appell Set Formulation su ( 1 , 1 ) nelson.faustino@ufabc.edu.br symmetries Ongoing Research Past and Future Directions in Hypercomplex and Harmonic Analysis Towards Dirac-K¨ ahler – Celebrating Frank Sommen’s 60th birthday formalism Discrete Quantum Mechanics 1 / 23

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank 1 The radial algebra approach What I’ve learned from Frank Beyond Landau-Weyl Calculus The radial algebra approach Beyond Landau-Weyl Calculus Lie-algebraic discretization 2 Lie-algebraic discretization Weyl-Heisenberg symmetries Weyl-Heisenberg symmetries Appell Set Formulation Appell Set Formulation su ( 1 , 1 ) symmetries su ( 1 , 1 ) symmetries Ongoing Research Ongoing Research 3 Towards Dirac-K¨ ahler Towards Dirac-K¨ ahler formalism formalism Discrete Quantum Discrete Quantum Mechanics Mechanics 2 / 23

Radial Algebra Formulation of Clifford Analysis F. Sommen, An Algebra of Abstract Vector Variables , (1997), Portugalia Math. Post-Modern Clifford Analysis: Study of operators belonging to the algebra Topics in Discrete Clifford Analysis � � Alg x j , ∂ x j , e j : j = 1 , . . . , n , Nelson Faustino What I’ve learned from Frank x j and ∂ x j satisfy the Weyl-Heisenberg graded commuting 1 The radial algebra approach relations Beyond Landau-Weyl Calculus � � � � ∂ x j , ∂ x k = [ x j , x k ] = 0 and ∂ x j , x k = δ jk I . Lie-algebraic discretization Weyl-Heisenberg e 1 , e 2 , . . . , e n are the generators of the Clifford algebra C ℓ 0 , n . The 2 symmetries Appell Set remainder graded anti-commuting relations are given by Formulation su ( 1 , 1 ) symmetries e j e k + e k e j = − 2 δ jk . Ongoing Research Multivector derivative: D = � n j = 1 e j ∂ x j is the standard Dirac Towards Dirac-K¨ ahler formalism operator (embedding of the gradient derivative on C ℓ 0 , n ). Discrete Quantum Mechanics Multivector multiplication: X : f ( x ) �→ � n j = 1 e j x j f ( x ) is the standard left multiplication of f ( x ) by the Clifford vector x = � n j = 1 x j e j . 3 / 23

Radial Algebra Formulation of Clifford Analysis F. Sommen, An Algebra of Abstract Vector Variables , (1997), Portugalia Math. Post-Modern Clifford Analysis: Study of operators belonging to the algebra Topics in Discrete Clifford Analysis � � Alg x j , ∂ x j , e j : j = 1 , . . . , n , Nelson Faustino What I’ve learned from Frank x j and ∂ x j satisfy the Weyl-Heisenberg graded commuting 1 The radial algebra approach relations Beyond Landau-Weyl Calculus � � � � ∂ x j , ∂ x k = [ x j , x k ] = 0 and ∂ x j , x k = δ jk I . Lie-algebraic discretization Weyl-Heisenberg e 1 , e 2 , . . . , e n are the generators of the Clifford algebra C ℓ 0 , n . The 2 symmetries Appell Set remainder graded anti-commuting relations are given by Formulation su ( 1 , 1 ) symmetries e j e k + e k e j = − 2 δ jk . Ongoing Research Multivector derivative: D = � n j = 1 e j ∂ x j is the standard Dirac Towards Dirac-K¨ ahler formalism operator (embedding of the gradient derivative on C ℓ 0 , n ). Discrete Quantum Mechanics Multivector multiplication: X : f ( x ) �→ � n j = 1 e j x j f ( x ) is the standard left multiplication of f ( x ) by the Clifford vector x = � n j = 1 x j e j . 3 / 23

Radial Algebra Formulation of Clifford Analysis F. Sommen, An Algebra of Abstract Vector Variables , (1997), Portugalia Math. Post-Modern Clifford Analysis: Study of operators belonging to the algebra Topics in Discrete Clifford Analysis � � Alg x j , ∂ x j , e j : j = 1 , . . . , n , Nelson Faustino What I’ve learned from Frank x j and ∂ x j satisfy the Weyl-Heisenberg graded commuting 1 The radial algebra approach relations Beyond Landau-Weyl Calculus � � � � ∂ x j , ∂ x k = [ x j , x k ] = 0 and ∂ x j , x k = δ jk I . Lie-algebraic discretization Weyl-Heisenberg e 1 , e 2 , . . . , e n are the generators of the Clifford algebra C ℓ 0 , n . The 2 symmetries Appell Set remainder graded anti-commuting relations are given by Formulation su ( 1 , 1 ) symmetries e j e k + e k e j = − 2 δ jk . Ongoing Research Multivector derivative: D = � n j = 1 e j ∂ x j is the standard Dirac Towards Dirac-K¨ ahler formalism operator (embedding of the gradient derivative on C ℓ 0 , n ). Discrete Quantum Mechanics Multivector multiplication: X : f ( x ) �→ � n j = 1 e j x j f ( x ) is the standard left multiplication of f ( x ) by the Clifford vector x = � n j = 1 x j e j . 3 / 23

Radial Algebra Formulation of Clifford Analysis F. Sommen, An Algebra of Abstract Vector Variables , (1997), Portugalia Math. Post-Modern Clifford Analysis: Study of operators belonging to the algebra Topics in Discrete Clifford Analysis � � Alg x j , ∂ x j , e j : j = 1 , . . . , n , Nelson Faustino What I’ve learned from Frank x j and ∂ x j satisfy the Weyl-Heisenberg graded commuting 1 The radial algebra approach relations Beyond Landau-Weyl Calculus � � � � ∂ x j , ∂ x k = [ x j , x k ] = 0 and ∂ x j , x k = δ jk I . Lie-algebraic discretization Weyl-Heisenberg e 1 , e 2 , . . . , e n are the generators of the Clifford algebra C ℓ 0 , n . The 2 symmetries Appell Set remainder graded anti-commuting relations are given by Formulation su ( 1 , 1 ) symmetries e j e k + e k e j = − 2 δ jk . Ongoing Research Multivector derivative: D = � n j = 1 e j ∂ x j is the standard Dirac Towards Dirac-K¨ ahler formalism operator (embedding of the gradient derivative on C ℓ 0 , n ). Discrete Quantum Mechanics Multivector multiplication: X : f ( x ) �→ � n j = 1 e j x j f ( x ) is the standard left multiplication of f ( x ) by the Clifford vector x = � n j = 1 x j e j . 3 / 23

Basic operators and relations in Clifford Analysis Post-Modern Topics in Discrete Clifford Analysis Euler operator: E = � n j = 1 x j ∂ x j Nelson Faustino Physical meaning: Hamiltonian of a field of free 1 What I’ve learned from Frank non-interacting bosons (useful in the the study of the spectra The radial algebra approach of the Harmonic Oscillator ). Beyond Landau-Weyl Calculus Basic properties: Lie-algebraic discretization Laplacian splitting: ∆ := � n j = 1 ∂ 2 x j = − D 2 1 Weyl-Heisenberg symmetries Invariant properties: [ E , X ] = X and [ E , D ] = − D . 2 Appell Set Formulation Euler operator splitting: XD + DX = − 2 ( E + n 2 id ) . 3 su ( 1 , 1 ) symmetries 2 ∆ , p † = 1 2 X 2 and Harmonic Analysis representation: p = − 1 Ongoing q = E + n Research 2 are the canonical generators of the Lie algebra sl 2 ( R ) . Towards Dirac-K¨ ahler formalism � p , p † � � q , p † � = p † , = q , [ q , p ] = − p . Discrete Quantum Mechanics 4 / 23

Basic operators and relations in Clifford Analysis Post-Modern Topics in Discrete Clifford Analysis Euler operator: E = � n j = 1 x j ∂ x j Nelson Faustino Physical meaning: Hamiltonian of a field of free 1 What I’ve learned from Frank non-interacting bosons (useful in the the study of the spectra The radial algebra approach of the Harmonic Oscillator ). Beyond Landau-Weyl Calculus Basic properties: Lie-algebraic discretization Laplacian splitting: ∆ := � n j = 1 ∂ 2 x j = − D 2 1 Weyl-Heisenberg symmetries Invariant properties: [ E , X ] = X and [ E , D ] = − D . 2 Appell Set Formulation Euler operator splitting: XD + DX = − 2 ( E + n 2 id ) . 3 su ( 1 , 1 ) symmetries 2 ∆ , p † = 1 2 X 2 and Harmonic Analysis representation: p = − 1 Ongoing q = E + n Research 2 are the canonical generators of the Lie algebra sl 2 ( R ) . Towards Dirac-K¨ ahler formalism � p , p † � � q , p † � = p † , = q , [ q , p ] = − p . Discrete Quantum Mechanics 4 / 23

Basic operators and relations in Clifford Analysis Post-Modern Topics in Discrete Clifford Analysis Euler operator: E = � n j = 1 x j ∂ x j Nelson Faustino Physical meaning: Hamiltonian of a field of free 1 What I’ve learned from Frank non-interacting bosons (useful in the the study of the spectra The radial algebra approach of the Harmonic Oscillator ). Beyond Landau-Weyl Calculus Basic properties: Lie-algebraic discretization Laplacian splitting: ∆ := � n j = 1 ∂ 2 x j = − D 2 1 Weyl-Heisenberg symmetries Invariant properties: [ E , X ] = X and [ E , D ] = − D . 2 Appell Set Formulation Euler operator splitting: XD + DX = − 2 ( E + n 2 id ) . 3 su ( 1 , 1 ) symmetries 2 ∆ , p † = 1 2 X 2 and Harmonic Analysis representation: p = − 1 Ongoing q = E + n Research 2 are the canonical generators of the Lie algebra sl 2 ( R ) . Towards Dirac-K¨ ahler formalism � p , p † � � q , p † � = p † , = q , [ q , p ] = − p . Discrete Quantum Mechanics 4 / 23