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

SLIDE 1

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Post-Modern Topics in Discrete Clifford Analysis

Nelson Faustino

Center of Mathematics, Computation and Cognition, UFABC nelson.faustino@ufabc.edu.br

Past and Future Directions in Hypercomplex and Harmonic Analysis – Celebrating Frank Sommen’s 60th birthday

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SLIDE 2

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

1

What I’ve learned from Frank The radial algebra approach Beyond Landau-Weyl Calculus

2

Lie-algebraic discretization Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

3

Ongoing Research Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

2 / 23

SLIDE 3

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Radial Algebra Formulation of Clifford Analysis

F. Sommen, An Algebra of Abstract Vector Variables, (1997), Portugalia Math.

Clifford Analysis: Study of operators belonging to the algebra Alg

xj, ∂xj , ej : j = 1, . . . , n
,

1

xj and ∂xj satisfy the Weyl-Heisenberg graded commuting relations

∂xj , ∂xk
= [xj, xk] = 0 and
∂xj , xk
= δjkI.

2

e1, e2, . . . , en are the generators of the Clifford algebra Cℓ0,n. The remainder graded anti-commuting relations are given by ejek + ekej = −2δjk. Multivector derivative: D = n

j=1 ej∂xj is the standard Dirac

perator (embedding of the gradient derivative on Cℓ0,n).

Multivector multiplication: X : f(x) → n

j=1 ejxjf(x) is the

standard left multiplication of f(x) by the Clifford vector x = n

j=1 xjej.

3 / 23

SLIDE 4

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Radial Algebra Formulation of Clifford Analysis

F. Sommen, An Algebra of Abstract Vector Variables, (1997), Portugalia Math.

Clifford Analysis: Study of operators belonging to the algebra Alg

xj, ∂xj , ej : j = 1, . . . , n
,

1

xj and ∂xj satisfy the Weyl-Heisenberg graded commuting relations

∂xj , ∂xk
= [xj, xk] = 0 and
∂xj , xk
= δjkI.

2

e1, e2, . . . , en are the generators of the Clifford algebra Cℓ0,n. The remainder graded anti-commuting relations are given by ejek + ekej = −2δjk. Multivector derivative: D = n

j=1 ej∂xj is the standard Dirac

perator (embedding of the gradient derivative on Cℓ0,n).

Multivector multiplication: X : f(x) → n

j=1 ejxjf(x) is the

standard left multiplication of f(x) by the Clifford vector x = n

j=1 xjej.

3 / 23

SLIDE 5

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Radial Algebra Formulation of Clifford Analysis

F. Sommen, An Algebra of Abstract Vector Variables, (1997), Portugalia Math.

Clifford Analysis: Study of operators belonging to the algebra Alg

xj, ∂xj , ej : j = 1, . . . , n
,

1

xj and ∂xj satisfy the Weyl-Heisenberg graded commuting relations

∂xj , ∂xk
= [xj, xk] = 0 and
∂xj , xk
= δjkI.

2

e1, e2, . . . , en are the generators of the Clifford algebra Cℓ0,n. The remainder graded anti-commuting relations are given by ejek + ekej = −2δjk. Multivector derivative: D = n

j=1 ej∂xj is the standard Dirac

perator (embedding of the gradient derivative on Cℓ0,n).

Multivector multiplication: X : f(x) → n

j=1 ejxjf(x) is the

standard left multiplication of f(x) by the Clifford vector x = n

j=1 xjej.

3 / 23

SLIDE 6

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Radial Algebra Formulation of Clifford Analysis

F. Sommen, An Algebra of Abstract Vector Variables, (1997), Portugalia Math.

Clifford Analysis: Study of operators belonging to the algebra Alg

xj, ∂xj , ej : j = 1, . . . , n
,

1

xj and ∂xj satisfy the Weyl-Heisenberg graded commuting relations

∂xj , ∂xk
= [xj, xk] = 0 and
∂xj , xk
= δjkI.

2

e1, e2, . . . , en are the generators of the Clifford algebra Cℓ0,n. The remainder graded anti-commuting relations are given by ejek + ekej = −2δjk. Multivector derivative: D = n

j=1 ej∂xj is the standard Dirac

perator (embedding of the gradient derivative on Cℓ0,n).

Multivector multiplication: X : f(x) → n

j=1 ejxjf(x) is the

standard left multiplication of f(x) by the Clifford vector x = n

j=1 xjej.

3 / 23

SLIDE 7

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Basic operators and relations in Clifford Analysis

Euler operator: E = n

j=1 xj∂xj

1

Physical meaning: Hamiltonian of a field of free non-interacting bosons (useful in the the study of the spectra

f the Harmonic Oscillator).

Basic properties:

1

Laplacian splitting: ∆ := n

j=1 ∂2 xj = −D2

2

Invariant properties: [E, X] = X and [E, D] = −D.

3

Euler operator splitting: XD + DX = −2(E + n

2id).

Harmonic Analysis representation: p = − 1

2∆, p† = 1 2X 2 and

q = E + n

2 are the canonical generators of the Lie algebra sl2(R).

p, p†

= q,

q, p†

= p†, [q, p] = −p.

4 / 23

SLIDE 8

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Basic operators and relations in Clifford Analysis

Euler operator: E = n

j=1 xj∂xj

1

Physical meaning: Hamiltonian of a field of free non-interacting bosons (useful in the the study of the spectra

f the Harmonic Oscillator).

Basic properties:

1

Laplacian splitting: ∆ := n

j=1 ∂2 xj = −D2

2

Invariant properties: [E, X] = X and [E, D] = −D.

3

Euler operator splitting: XD + DX = −2(E + n

2id).

Harmonic Analysis representation: p = − 1

2∆, p† = 1 2X 2 and

q = E + n

2 are the canonical generators of the Lie algebra sl2(R).

p, p†

= q,

q, p†

= p†, [q, p] = −p.

4 / 23

SLIDE 9

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Basic operators and relations in Clifford Analysis

Euler operator: E = n

j=1 xj∂xj

1

Physical meaning: Hamiltonian of a field of free non-interacting bosons (useful in the the study of the spectra

f the Harmonic Oscillator).

Basic properties:

1

Laplacian splitting: ∆ := n

j=1 ∂2 xj = −D2

2

Invariant properties: [E, X] = X and [E, D] = −D.

3

Euler operator splitting: XD + DX = −2(E + n

2id).

Harmonic Analysis representation: p = − 1

2∆, p† = 1 2X 2 and

q = E + n

2 are the canonical generators of the Lie algebra sl2(R).

p, p†

= q,

q, p†

= p†, [q, p] = −p.

4 / 23

SLIDE 10

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Basic operators and relations in Clifford Analysis

Euler operator: E = n

j=1 xj∂xj

1

Physical meaning: Hamiltonian of a field of free non-interacting bosons (useful in the the study of the spectra

f the Harmonic Oscillator).

Basic properties:

1

Laplacian splitting: ∆ := n

j=1 ∂2 xj = −D2

2

Invariant properties: [E, X] = X and [E, D] = −D.

3

Euler operator splitting: XD + DX = −2(E + n

2id).

Harmonic Analysis representation: p = − 1

2∆, p† = 1 2X 2 and

q = E + n

2 are the canonical generators of the Lie algebra sl2(R).

p, p†

= q,

q, p†

= p†, [q, p] = −p.

4 / 23

SLIDE 11

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Basic operators and relations in Clifford Analysis

Euler operator: E = n

j=1 xj∂xj

1

Physical meaning: Hamiltonian of a field of free non-interacting bosons (useful in the the study of the spectra

f the Harmonic Oscillator).

Basic properties:

1

Laplacian splitting: ∆ := n

j=1 ∂2 xj = −D2

2

Invariant properties: [E, X] = X and [E, D] = −D.

3

Euler operator splitting: XD + DX = −2(E + n

2id).

Harmonic Analysis representation: p = − 1

2∆, p† = 1 2X 2 and

q = E + n

2 are the canonical generators of the Lie algebra sl2(R).

p, p†

= q,

q, p†

= p†, [q, p] = −p.

4 / 23

SLIDE 12

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Basic operators and relations in Clifford Analysis

Euler operator: E = n

j=1 xj∂xj

1

Physical meaning: Hamiltonian of a field of free non-interacting bosons (useful in the the study of the spectra

f the Harmonic Oscillator).

Basic properties:

1

Laplacian splitting: ∆ := n

j=1 ∂2 xj = −D2

2

Invariant properties: [E, X] = X and [E, D] = −D.

3

Euler operator splitting: XD + DX = −2(E + n

2id).

Harmonic Analysis representation: p = − 1

2∆, p† = 1 2X 2 and

q = E + n

2 are the canonical generators of the Lie algebra sl2(R).

p, p†

= q,

q, p†

= p†, [q, p] = −p.

4 / 23

SLIDE 13

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Basic operators and relations in Clifford Analysis

Euler operator: E = n

j=1 xj∂xj

1

Physical meaning: Hamiltonian of a field of free non-interacting bosons (useful in the the study of the spectra

f the Harmonic Oscillator).

Basic properties:

1

Laplacian splitting: ∆ := n

j=1 ∂2 xj = −D2

2

Invariant properties: [E, X] = X and [E, D] = −D.

3

Euler operator splitting: XD + DX = −2(E + n

2id).

Harmonic Analysis representation: p = − 1

2∆, p† = 1 2X 2 and

q = E + n

2 are the canonical generators of the Lie algebra sl2(R).

p, p†

= q,

q, p†

= p†, [q, p] = −p.

4 / 23

SLIDE 14

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Beyond Landau-Weyl Calculus

D. Constales, N.F

. & R.S. Kraußhar, J. Phys. A: Math. Theor. 44 135303 (2011)

Classical harmonic oscillator: H = 1

2(−∆ + |x|2) = 1 2(D2 − X 2)

Ladder operator splitting: D± =

1 √ 2(X ∓ D) give rise to

D+D− + D−D+ = −2H. Clifford operators as Classes of Wigner Quantal Systems: span 1

2D2 −, 1 2D2 +, H

⊕ span {D−, D+} equipped with the standard

graded commutator [·, ·] is a isomorphic to a Lie superalgebra

sp(1|2).

Spherical Dirac operator: Γ := −XD − E is radially independent since [Γ, X 2] = 0. Displacement Operator vs. Intertwining Properties: exp λ n (D − X)

D =
D + 2λ

n Γ − λI

exp

λ n (D − X)

,

exp λ n (D − X)

X =
X + 2λ

n Γ − λI

exp

λ n (D − X)

.

5 / 23

SLIDE 15

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Beyond Landau-Weyl Calculus

D. Constales, N.F

. & R.S. Kraußhar, J. Phys. A: Math. Theor. 44 135303 (2011)

Classical harmonic oscillator: H = 1

2(−∆ + |x|2) = 1 2(D2 − X 2)

Ladder operator splitting: D± =

1 √ 2(X ∓ D) give rise to

D+D− + D−D+ = −2H. Clifford operators as Classes of Wigner Quantal Systems: span 1

2D2 −, 1 2D2 +, H

⊕ span {D−, D+} equipped with the standard

graded commutator [·, ·] is a isomorphic to a Lie superalgebra

sp(1|2).

Spherical Dirac operator: Γ := −XD − E is radially independent since [Γ, X 2] = 0. Displacement Operator vs. Intertwining Properties: exp λ n (D − X)

D =
D + 2λ

n Γ − λI

exp

λ n (D − X)

,

exp λ n (D − X)

X =
X + 2λ

n Γ − λI

exp

λ n (D − X)

.

5 / 23

SLIDE 16

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Beyond Landau-Weyl Calculus

D. Constales, N.F

. & R.S. Kraußhar, J. Phys. A: Math. Theor. 44 135303 (2011)

Classical harmonic oscillator: H = 1

2(−∆ + |x|2) = 1 2(D2 − X 2)

Ladder operator splitting: D± =

1 √ 2(X ∓ D) give rise to

D+D− + D−D+ = −2H. Clifford operators as Classes of Wigner Quantal Systems: span 1

2D2 −, 1 2D2 +, H

⊕ span {D−, D+} equipped with the standard

graded commutator [·, ·] is a isomorphic to a Lie superalgebra

sp(1|2).

Spherical Dirac operator: Γ := −XD − E is radially independent since [Γ, X 2] = 0. Displacement Operator vs. Intertwining Properties: exp λ n (D − X)

D =
D + 2λ

n Γ − λI

exp

λ n (D − X)

,

exp λ n (D − X)

X =
X + 2λ

n Γ − λI

exp

λ n (D − X)

.

5 / 23

SLIDE 17

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Beyond Landau-Weyl Calculus

D. Constales, N.F

. & R.S. Kraußhar, J. Phys. A: Math. Theor. 44 135303 (2011)

Classical harmonic oscillator: H = 1

2(−∆ + |x|2) = 1 2(D2 − X 2)

Ladder operator splitting: D± =

1 √ 2(X ∓ D) give rise to

D+D− + D−D+ = −2H. Clifford operators as Classes of Wigner Quantal Systems: span 1

2D2 −, 1 2D2 +, H

⊕ span {D−, D+} equipped with the standard

graded commutator [·, ·] is a isomorphic to a Lie superalgebra

sp(1|2).

Spherical Dirac operator: Γ := −XD − E is radially independent since [Γ, X 2] = 0. Displacement Operator vs. Intertwining Properties: exp λ n (D − X)

D =
D + 2λ

n Γ − λI

exp

λ n (D − X)

,

exp λ n (D − X)

X =
X + 2λ

n Γ − λI

exp

λ n (D − X)

.

5 / 23

SLIDE 18

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Beyond Landau-Weyl Calculus

D. Constales, N.F

. & R.S. Kraußhar, J. Phys. A: Math. Theor. 44 135303 (2011)

Classical harmonic oscillator: H = 1

2(−∆ + |x|2) = 1 2(D2 − X 2)

Ladder operator splitting: D± =

1 √ 2(X ∓ D) give rise to

D+D− + D−D+ = −2H. Clifford operators as Classes of Wigner Quantal Systems: span 1

2D2 −, 1 2D2 +, H

⊕ span {D−, D+} equipped with the standard

graded commutator [·, ·] is a isomorphic to a Lie superalgebra

sp(1|2).

Spherical Dirac operator: Γ := −XD − E is radially independent since [Γ, X 2] = 0. Displacement Operator vs. Intertwining Properties: exp λ n (D − X)

D =
D + 2λ

n Γ − λI

exp

λ n (D − X)

,

exp λ n (D − X)

X =
X + 2λ

n Γ − λI

exp

λ n (D − X)

.

5 / 23

SLIDE 19

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Beyond Landau-Weyl Calculus

D. Constales, N.F

. & R.S. Kraußhar, J. Phys. A: Math. Theor. 44 135303 (2011)

Solutions of the time-harmonic Maxwell equation with angular part For each Ps ∈ Ps ∩ ker D, Pλ,s(x) = exp λ

n (D − X)

exp
− ∆

2

Ps(x)

equals to Pλ,s(x) = cosh √ 2s + n λ

n

Ps(x) −

1 √ 2s+n sinh

√ 2s + n λ

n

x Ps(x)

solves the coupled system of equations:    (D − λI)f(x) = − 2λ

n Γ (f(x))

E(f(x)) + n

2 − 2s

f(x) = − 2λ

n xf(x)

Moreover, Pλ,s(x) is harmonic.

1

The polynomial solutions for the time-harmonic Maxwell equations correspond to Pλ,s(x) = Pλ,s(|x|), establishing the correspondence with Z. Xu, (1991).

2

Contrary to the approach of I. Cac ¸ ˜ ao , D. Constales & R.S. Kraußhar (2009) , this approach takes into account linear combinations of radial and angular wave functions.

6 / 23

SLIDE 20

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Beyond Landau-Weyl Calculus

D. Constales, N.F

. & R.S. Kraußhar, J. Phys. A: Math. Theor. 44 135303 (2011)

Solutions of the time-harmonic Maxwell equation with angular part For each Ps ∈ Ps ∩ ker D, Pλ,s(x) = exp λ

n (D − X)

exp
− ∆

2

Ps(x)

equals to Pλ,s(x) = cosh √ 2s + n λ

n

Ps(x) −

1 √ 2s+n sinh

√ 2s + n λ

n

x Ps(x)

solves the coupled system of equations:    (D − λI)f(x) = − 2λ

n Γ (f(x))

E(f(x)) + n

2 − 2s

f(x) = − 2λ

n xf(x)

Moreover, Pλ,s(x) is harmonic.

1

The polynomial solutions for the time-harmonic Maxwell equations correspond to Pλ,s(x) = Pλ,s(|x|), establishing the correspondence with Z. Xu, (1991).

2

Contrary to the approach of I. Cac ¸ ˜ ao , D. Constales & R.S. Kraußhar (2009) , this approach takes into account linear combinations of radial and angular wave functions.

6 / 23

SLIDE 21

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Beyond Landau-Weyl Calculus

D. Constales, N.F

. & R.S. Kraußhar, J. Phys. A: Math. Theor. 44 135303 (2011)

Solutions of the time-harmonic Maxwell equation with angular part For each Ps ∈ Ps ∩ ker D, Pλ,s(x) = exp λ

n (D − X)

exp
− ∆

2

Ps(x)

equals to Pλ,s(x) = cosh √ 2s + n λ

n

Ps(x) −

1 √ 2s+n sinh

√ 2s + n λ

n

x Ps(x)

solves the coupled system of equations:    (D − λI)f(x) = − 2λ

n Γ (f(x))

E(f(x)) + n

2 − 2s

f(x) = − 2λ

n xf(x)

Moreover, Pλ,s(x) is harmonic.

1

The polynomial solutions for the time-harmonic Maxwell equations correspond to Pλ,s(x) = Pλ,s(|x|), establishing the correspondence with Z. Xu, (1991).

2

Contrary to the approach of I. Cac ¸ ˜ ao , D. Constales & R.S. Kraußhar (2009) , this approach takes into account linear combinations of radial and angular wave functions.

6 / 23

SLIDE 22

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Finite difference toolbox

1

Equidistant lattice with mesh width h > 0: hZn =

x = (x1, . . . , xn) ∈ Rn : x

h ∈ Zn

2

Forward/backward finite difference operators (∂+j

h f)(x) = f(x + hej) − f(x)

h , (∂−j

h f)(x) = f(x) − f(x − hej)

h .

3

Translation property: ∂+j

h and ∂−j h

are interrelated by (T ±j

h f)(x) = f(x ± hej) i.e.

T −j

h (∂+j h f)(x) = (∂−j h f)(x)

and T +j

h (∂−j h f)(x) = (∂+j h f)(x).

4

Product rules for finite difference operators: ∂+j

h (g(x)f(x))

= (∂+j

h g)(x)f(x + hej) + g(x)(∂+j h f)(x)

∂−j

h (g(x)f(x))

= (∂−j

h g)(x)f(x − hej) + g(x)(∂−j h f)(x).

7 / 23

SLIDE 23

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Finite difference toolbox

1

Equidistant lattice with mesh width h > 0: hZn =

x = (x1, . . . , xn) ∈ Rn : x

h ∈ Zn

2

Forward/backward finite difference operators (∂+j

h f)(x) = f(x + hej) − f(x)

h , (∂−j

h f)(x) = f(x) − f(x − hej)

h .

3

Translation property: ∂+j

h and ∂−j h

are interrelated by (T ±j

h f)(x) = f(x ± hej) i.e.

T −j

h (∂+j h f)(x) = (∂−j h f)(x)

and T +j

h (∂−j h f)(x) = (∂+j h f)(x).

4

Product rules for finite difference operators: ∂+j

h (g(x)f(x))

= (∂+j

h g)(x)f(x + hej) + g(x)(∂+j h f)(x)

∂−j

h (g(x)f(x))

= (∂−j

h g)(x)f(x − hej) + g(x)(∂−j h f)(x).

7 / 23

SLIDE 24

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Finite difference toolbox

1

Equidistant lattice with mesh width h > 0: hZn =

x = (x1, . . . , xn) ∈ Rn : x

h ∈ Zn

2

Forward/backward finite difference operators (∂+j

h f)(x) = f(x + hej) − f(x)

h , (∂−j

h f)(x) = f(x) − f(x − hej)

h .

3

Translation property: ∂+j

h and ∂−j h

are interrelated by (T ±j

h f)(x) = f(x ± hej) i.e.

T −j

h (∂+j h f)(x) = (∂−j h f)(x)

and T +j

h (∂−j h f)(x) = (∂+j h f)(x).

4

Product rules for finite difference operators: ∂+j

h (g(x)f(x))

= (∂+j

h g)(x)f(x + hej) + g(x)(∂+j h f)(x)

∂−j

h (g(x)f(x))

= (∂−j

h g)(x)f(x − hej) + g(x)(∂−j h f)(x).

7 / 23

SLIDE 25

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Finite difference toolbox

1

Equidistant lattice with mesh width h > 0: hZn =

x = (x1, . . . , xn) ∈ Rn : x

h ∈ Zn

2

Forward/backward finite difference operators (∂+j

h f)(x) = f(x + hej) − f(x)

h , (∂−j

h f)(x) = f(x) − f(x − hej)

h .

3

Translation property: ∂+j

h and ∂−j h

are interrelated by (T ±j

h f)(x) = f(x ± hej) i.e.

T −j

h (∂+j h f)(x) = (∂−j h f)(x)

and T +j

h (∂−j h f)(x) = (∂+j h f)(x).

4

Product rules for finite difference operators: ∂+j

h (g(x)f(x))

= (∂+j

h g)(x)f(x + hej) + g(x)(∂+j h f)(x)

∂−j

h (g(x)f(x))

= (∂−j

h g)(x)f(x − hej) + g(x)(∂−j h f)(x).

7 / 23

SLIDE 26

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Radial-type discretization

Lie-algebraic formulation

Radial-type approach: Study of finite difference operators belonging to the algebra Alg {Lj, Mj, ej : j = 1, . . . , n},

1

Lj and Mj are position and momentum operators, respectively, satisfying the set of Weyl-Heisenberg algebra relations [Lj, Lk] = [Mj, Mk] = 0 and [Lj, Mk] = δjkI

2

e1, e2, . . . , en are the generators of the Clifford algebra of signature (0, n). Multivector operators: Basic left endomorphisms acting that act on functions with values on Cℓ0,n. Multivector derivative: L = n

j=1 ejLj stands the Lie-algebraic

counterpart of the Dirac operator D = n

j=1 ej∂xj .

Multivector multiplication: M = n

j=1 ejMj stands the

Lie-algebraic counterpart for the left multiplication of f(x) by a Clifford vector X = n

j=1 xjej.

8 / 23

SLIDE 27

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Radial-type discretization

Lie-algebraic formulation

Radial-type approach: Study of finite difference operators belonging to the algebra Alg {Lj, Mj, ej : j = 1, . . . , n},

1

Lj and Mj are position and momentum operators, respectively, satisfying the set of Weyl-Heisenberg algebra relations [Lj, Lk] = [Mj, Mk] = 0 and [Lj, Mk] = δjkI

2

e1, e2, . . . , en are the generators of the Clifford algebra of signature (0, n). Multivector operators: Basic left endomorphisms acting that act on functions with values on Cℓ0,n. Multivector derivative: L = n

j=1 ejLj stands the Lie-algebraic

counterpart of the Dirac operator D = n

j=1 ej∂xj .

Multivector multiplication: M = n

j=1 ejMj stands the

Lie-algebraic counterpart for the left multiplication of f(x) by a Clifford vector X = n

j=1 xjej.

8 / 23

SLIDE 28

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Radial-type discretization

Lie-algebraic formulation

Radial-type approach: Study of finite difference operators belonging to the algebra Alg {Lj, Mj, ej : j = 1, . . . , n},

1

Lj and Mj are position and momentum operators, respectively, satisfying the set of Weyl-Heisenberg algebra relations [Lj, Lk] = [Mj, Mk] = 0 and [Lj, Mk] = δjkI

2

e1, e2, . . . , en are the generators of the Clifford algebra of signature (0, n). Multivector operators: Basic left endomorphisms acting that act on functions with values on Cℓ0,n. Multivector derivative: L = n

j=1 ejLj stands the Lie-algebraic

counterpart of the Dirac operator D = n

j=1 ej∂xj .

Multivector multiplication: M = n

j=1 ejMj stands the

Lie-algebraic counterpart for the left multiplication of f(x) by a Clifford vector X = n

j=1 xjej.

8 / 23

SLIDE 29

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Radial-type discretization

Lie-algebraic formulation

Radial-type approach: Study of finite difference operators belonging to the algebra Alg {Lj, Mj, ej : j = 1, . . . , n},

1

Lj and Mj are position and momentum operators, respectively, satisfying the set of Weyl-Heisenberg algebra relations [Lj, Lk] = [Mj, Mk] = 0 and [Lj, Mk] = δjkI

2

e1, e2, . . . , en are the generators of the Clifford algebra of signature (0, n). Multivector operators: Basic left endomorphisms acting that act on functions with values on Cℓ0,n. Multivector derivative: L = n

j=1 ejLj stands the Lie-algebraic

counterpart of the Dirac operator D = n

j=1 ej∂xj .

Multivector multiplication: M = n

j=1 ejMj stands the

Lie-algebraic counterpart for the left multiplication of f(x) by a Clifford vector X = n

j=1 xjej.

8 / 23

SLIDE 30

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Radial-type discretization

Examples

1

Forward finite differences: The set of operators ∂+j

h and

xjT −j

h

: f(x) → xjf(x − hej) span the Weyl-Heisenberg algebra of dimension 2n + 1. Moreover D+

h = n j=1 ej∂+j h and

Xh = n

j=1 ejxjT −j h

are the corresponding multivector ladder

perators on the lattice hZn.

2

Backward finite differences: ∂−j

h

and xjT +j

h

: f(x) → xjf(x + hej) also span the Weyl-Heisenberg algebra of dimension 2n + 1. This turns out D−

h = n j=1 ej∂−j h

and X−h = n

j=1 ejxjT +j h

as the corresponding multivector ladder operators on the lattice hZn.

3

Discretization of the Hermite operator: D+

h and Xh − D− h are

btained from the set of ladder operators Lj = ∂+j

h and

Lj = xjT −j

h

− ∂−j

h . Moreover Xh − D− h generate hypercomplex

extensions of the Poisson-Charlier polynomials (cf. N.F., Appl.

Math. Comp., Vol. 247, pp. 607-622, 2014).

9 / 23

SLIDE 31

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Radial-type discretization

Examples

1

Forward finite differences: The set of operators ∂+j

h and

xjT −j

h

: f(x) → xjf(x − hej) span the Weyl-Heisenberg algebra of dimension 2n + 1. Moreover D+

h = n j=1 ej∂+j h and

Xh = n

j=1 ejxjT −j h

are the corresponding multivector ladder

perators on the lattice hZn.

2

Backward finite differences: ∂−j

h

and xjT +j

h

: f(x) → xjf(x + hej) also span the Weyl-Heisenberg algebra of dimension 2n + 1. This turns out D−

h = n j=1 ej∂−j h

and X−h = n

j=1 ejxjT +j h

as the corresponding multivector ladder operators on the lattice hZn.

3

Discretization of the Hermite operator: D+

h and Xh − D− h are

btained from the set of ladder operators Lj = ∂+j

h and

Lj = xjT −j

h

− ∂−j

h . Moreover Xh − D− h generate hypercomplex

extensions of the Poisson-Charlier polynomials (cf. N.F., Appl.

Math. Comp., Vol. 247, pp. 607-622, 2014).

9 / 23

SLIDE 32

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Radial-type discretization

Examples

1

Forward finite differences: The set of operators ∂+j

h and

xjT −j

h

: f(x) → xjf(x − hej) span the Weyl-Heisenberg algebra of dimension 2n + 1. Moreover D+

h = n j=1 ej∂+j h and

Xh = n

j=1 ejxjT −j h

are the corresponding multivector ladder

perators on the lattice hZn.

2

Backward finite differences: ∂−j

h

and xjT +j

h

: f(x) → xjf(x + hej) also span the Weyl-Heisenberg algebra of dimension 2n + 1. This turns out D−

h = n j=1 ej∂−j h

and X−h = n

j=1 ejxjT +j h

as the corresponding multivector ladder operators on the lattice hZn.

3

Discretization of the Hermite operator: D+

h and Xh − D− h are

btained from the set of ladder operators Lj = ∂+j

h and

Lj = xjT −j

h

− ∂−j

h . Moreover Xh − D− h generate hypercomplex

extensions of the Poisson-Charlier polynomials (cf. N.F., Appl.

Math. Comp., Vol. 247, pp. 607-622, 2014).

9 / 23

SLIDE 33

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Quasi-Monomiality Approach

The Exponential Generating Function (EGF) approach

Many degrees of freedom for choose discretization operators: (cf. N.F., SIGMA 9 (2013), 065) The set of operators

xj + h

2

T +j

h

: f(x) →

xj + h

2

f(x + hej) and
xj − h

2

T −j

h

: f(x) →

xj − h

2

f(x − hej) satisfy
∂−j

h ,

xk + h

2

T +k

h

=
∂+j

h ,

xk − h

2

T −k

h

= δjkI
cf. N. F. Appl. Math. Comp., 2014

The EGF of the form Gh(x, y; κ) = n

j=1

1 κ 1

h log (1 + hyj)

(1 + hyj)

xj h

yield the set of operators Lj = ∂+j

h and Mj =

xj − κ′(∂xj )κ
∂xj

−1 T −j

h

as generators of the Weyl-Heisenberg algebra of dimension 2n + 1. Moreover, they are unique.

10 / 23

SLIDE 34

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Quasi-Monomiality Approach

The Exponential Generating Function (EGF) approach

Proposition (N.F ., Appl. Math. Comp., 2014) Let κ(t) defined as above and Xh the multiplication operator. If there is a multi-variable function λ(y) (y ∈ Rn) such that λ D+

h exp(x · y)

exp(x · y)

=

n

j=1

κ(yj) then the Fourier dual Λh of D+

h is given by

Λh = Xh −

log λ
D+

h

, x
.

Quasi-Monomiality formulation: Based on Fock space formalism one can construct each Clifford-vector-valued polynomial wk(x; h; λ) of order k by means of the operational rule wk(x; h; λ) = µk (Λh)k a, a ∈ Cℓ0,n.

11 / 23

SLIDE 35

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Quasi-Monomiality Approach

The Exponential Generating Function (EGF) approach

Appell set definition: {wk(x; h; λ) : k ∈ N0} is an Appell set carrying D+

h if w0(x; h; λ) = a is a Clifford number and D+ h wk(x; h; λ)

is a Clifford-vector-valued polynomial of degree k − 1 satisfying D+

h wk(x; h; λ) = kwk−1(x; h; λ).

Appell set equivalent formulation: Find for each (x, t) ∈ hZn × R a EGF Gh(x, t; λ) satisfying the set of equations    D+

h Gh(x, t; λ) = tGh(x, t; λ)

for (x, t) ∈ hZn × R \ {0} Gh(x, 0; λ) = a for x ∈ hZn. Bessel type hypergeometric functions: Gh(x, t; λ) =

0F1

n 2; −t2 4 (Λh)2

a + tΛh 0F1

n 2 + 1; −t2 4 (Λh)2

a

= Γ n 2 tΛh 2 − n

2 +1

J n

2 −1(tΛh)a + n J n 2 (tΛh)

a

12 / 23

SLIDE 36

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Quasi-Monomiality Approach

The Exponential Generating Function (EGF) approach

Appell set definition: {wk(x; h; λ) : k ∈ N0} is an Appell set carrying D+

h if w0(x; h; λ) = a is a Clifford number and D+ h wk(x; h; λ)

is a Clifford-vector-valued polynomial of degree k − 1 satisfying D+

h wk(x; h; λ) = kwk−1(x; h; λ).

Appell set equivalent formulation: Find for each (x, t) ∈ hZn × R a EGF Gh(x, t; λ) satisfying the set of equations    D+

h Gh(x, t; λ) = tGh(x, t; λ)

for (x, t) ∈ hZn × R \ {0} Gh(x, 0; λ) = a for x ∈ hZn. Bessel type hypergeometric functions: Gh(x, t; λ) =

0F1

n 2; −t2 4 (Λh)2

a + tΛh 0F1

n 2 + 1; −t2 4 (Λh)2

a

= Γ n 2 tΛh 2 − n

2 +1

J n

2 −1(tΛh)a + n J n 2 (tΛh)

a

12 / 23

SLIDE 37

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Quasi-Monomiality Approach

The Exponential Generating Function (EGF) approach

Appell set definition: {wk(x; h; λ) : k ∈ N0} is an Appell set carrying D+

h if w0(x; h; λ) = a is a Clifford number and D+ h wk(x; h; λ)

is a Clifford-vector-valued polynomial of degree k − 1 satisfying D+

h wk(x; h; λ) = kwk−1(x; h; λ).

Appell set equivalent formulation: Find for each (x, t) ∈ hZn × R a EGF Gh(x, t; λ) satisfying the set of equations    D+

h Gh(x, t; λ) = tGh(x, t; λ)

for (x, t) ∈ hZn × R \ {0} Gh(x, 0; λ) = a for x ∈ hZn. Bessel type hypergeometric functions: Gh(x, t; λ) =

0F1

n 2; −t2 4 (Λh)2

a + tΛh 0F1

n 2 + 1; −t2 4 (Λh)2

a

= Γ n 2 tΛh 2 − n

2 +1

J n

2 −1(tΛh)a + n J n 2 (tΛh)

a

12 / 23

SLIDE 38

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Why one needs su(1, 1) based symmetries?

The Weyl-Heisenberg symmetry breaking

Main Goal: For a given polynomial w(t) of degree 1, with µ = ∂+j

h w(xj) = ∂−j h w(xj),

study the spectra of the coupled eigenvalue problem E+

h f(x) = E− h f(x) = εf(x)

carrying E±

h = n j=1 µ−1w

xj ± h

2

∂±j

h .

Drawback: The set of operators ∂+j

h , ∂−j h , W −j h

= µ−1w

xj + h

2

T −j

h , W +j h

= µ−1w

xj + h

2

T +j

h

and I, with j = 1, 2, . . . , n, do not endow a canonical realization of an Weyl-Heisenberg type algebra of dimension 4n + 1. Fill the Weyl-Heisenberg gap: The set of operators W −j

h

= µ−1w

xj + h

2

T −j

h , W +j h

= µ−1w

xj + h

2

T +j

h

and Wj = µ−1w (xj) I generate a Lie algebra isomorphic to sl(2n, R) (N.F., SIGMA 9 (2013), 065– Lemma 1).

13 / 23

SLIDE 39

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Why one needs su(1, 1) based symmetries?

The Weyl-Heisenberg symmetry breaking

Main Goal: For a given polynomial w(t) of degree 1, with µ = ∂+j

h w(xj) = ∂−j h w(xj),

study the spectra of the coupled eigenvalue problem E+

h f(x) = E− h f(x) = εf(x)

carrying E±

h = n j=1 µ−1w

xj ± h

2

∂±j

h .

Drawback: The set of operators ∂+j

h , ∂−j h , W −j h

= µ−1w

xj + h

2

T −j

h , W +j h

= µ−1w

xj + h

2

T +j

h

and I, with j = 1, 2, . . . , n, do not endow a canonical realization of an Weyl-Heisenberg type algebra of dimension 4n + 1. Fill the Weyl-Heisenberg gap: The set of operators W −j

h

= µ−1w

xj + h

2

T −j

h , W +j h

= µ−1w

xj + h

2

T +j

h

and Wj = µ−1w (xj) I generate a Lie algebra isomorphic to sl(2n, R) (N.F., SIGMA 9 (2013), 065– Lemma 1).

13 / 23

SLIDE 40

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Why one needs su(1, 1) based symmetries?

The Weyl-Heisenberg symmetry breaking

Main Goal: For a given polynomial w(t) of degree 1, with µ = ∂+j

h w(xj) = ∂−j h w(xj),

study the spectra of the coupled eigenvalue problem E+

h f(x) = E− h f(x) = εf(x)

carrying E±

h = n j=1 µ−1w

xj ± h

2

∂±j

h .

Drawback: The set of operators ∂+j

h , ∂−j h , W −j h

= µ−1w

xj + h

2

T −j

h , W +j h

= µ−1w

xj + h

2

T +j

h

and I, with j = 1, 2, . . . , n, do not endow a canonical realization of an Weyl-Heisenberg type algebra of dimension 4n + 1. Fill the Weyl-Heisenberg gap: The set of operators W −j

h

= µ−1w

xj + h

2

T −j

h , W +j h

= µ−1w

xj + h

2

T +j

h

and Wj = µ−1w (xj) I generate a Lie algebra isomorphic to sl(2n, R) (N.F., SIGMA 9 (2013), 065– Lemma 1).

13 / 23

SLIDE 41

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Discrete series representations of SU(1, 1)

The construction

Ladder operators on hZn: W +

h = n j=1 W +j h , W − h = n j=1 W −j h

and W = n

j=1 Wj generate a Lie algebra isomorphic to

su(1, 1) ∼ = sl(2, R). The remaining commuting relations are given by W +

h

h , W h

=

W +

h

W −

h

h , W h

= −

W −

h

W +

h

h , W −

h

= 2

hW.

Casimir operator: The operator of the form Kh = W

h

2 − 1

2

W +

h

h W −

h

+

W −

h

h W +

h

determines an irreducible

unitary representation πλ of SU(1, 1) on the enveloping algebra U(su(1, 1)). This representation is labeled by the eigenvalues λ of Kh.

14 / 23

SLIDE 42

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Discrete series representations of SU(1, 1)

The construction

Ladder operators on hZn: W +

h = n j=1 W +j h , W − h = n j=1 W −j h

and W = n

j=1 Wj generate a Lie algebra isomorphic to

su(1, 1) ∼ = sl(2, R). The remaining commuting relations are given by W +

h

h , W h

=

W +

h

W −

h

h , W h

= −

W −

h

W +

h

h , W −

h

= 2

hW.

Casimir operator: The operator of the form Kh = W

h

2 − 1

2

W +

h

h W −

h

+

W −

h

h W +

h

determines an irreducible

unitary representation πλ of SU(1, 1) on the enveloping algebra U(su(1, 1)). This representation is labeled by the eigenvalues λ of Kh.

14 / 23

SLIDE 43

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Discrete series representations of SU(1, 1)

Positive/Negative series representations

1

Positive series representation of SU(1,1): π+

λ is thus determined

by the set of ladder operators              π+

λ

W −

h

=

E+

h − E− h

π+

λ

W +

h

=

W +

h

π+

λ

W

h

=

E+

h + n 2I

π+

λ (Kh)

=

E+

h + n 2I

E+

h +

n

2 − 1

I
−

W +

h

h (E+ h − E− h )

2

Negative series representation of SU(1,1): π−

λ is thus determined

by the set of ladder operators                π−

λ

W −

h

=

W −

h

π−

λ

W +

h

= E+

h − E− h

π−

λ

W

h

= −E−

h − n 2I

π−

λ (Kh) =

E−

h + n 2I

E−

h +

n

2 − 1

I
−

W −

h

h (E+ h − E− h )

15 / 23

SLIDE 44

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Discrete series representations of SU(1, 1)

Positive/Negative series representations

1

Positive series representation of SU(1,1): π+

λ is thus determined

by the set of ladder operators              π+

λ

W −

h

=

E+

h − E− h

π+

λ

W +

h

=

W +

h

π+

λ

W

h

=

E+

h + n 2I

π+

λ (Kh)

=

E+

h + n 2I

E+

h +

n

2 − 1

I
−

W +

h

h (E+ h − E− h )

2

Negative series representation of SU(1,1): π−

λ is thus determined

by the set of ladder operators                π−

λ

W −

h

=

W −

h

π−

λ

W +

h

= E+

h − E− h

π−

λ

W

h

= −E−

h − n 2I

π−

λ (Kh) =

E−

h + n 2I

E−

h +

n

2 − 1

I
−

W −

h

h (E+ h − E− h )

15 / 23

SLIDE 45

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

The Howe dual pair technique

N.F . SIGMA 9 (2013), 065

Invariant subspaces Hs;h : Spaces with basic polynomials of the form ws(x; h) = W +

h

s m0(x; h), with E+

h m0(x; h) = E− h m0(x; h) = 0.

Invariant subspaces Hs;−h: Spaces with basic polynomials of the form ws(x; −h) =

W −

h

s m0(x; h), with E+

h m0(x; h) = E− h m0(x; h) = 0.

Irreducible subspaces: The SO(n)−invariant subspaces of the form W +

h

r (Hs−r;h ∩ Hs−r;−h) resp.

W −

h

r (Hs−r;h ∩ Hs−r;−h). Fourier decomposition of Hs;±h: is determined from the Howe dual pair (SO(1, 1), su(1, 1)). as a direct sum of the (s + 1)− irreducible pieces

W ±

h

r (Hs−r;h ∩ Hs−r;−h).

16 / 23

SLIDE 46

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

The Howe dual pair technique

N.F . SIGMA 9 (2013), 065

Invariant subspaces Hs;h : Spaces with basic polynomials of the form ws(x; h) = W +

h

s m0(x; h), with E+

h m0(x; h) = E− h m0(x; h) = 0.

Invariant subspaces Hs;−h: Spaces with basic polynomials of the form ws(x; −h) =

W −

h

s m0(x; h), with E+

h m0(x; h) = E− h m0(x; h) = 0.

Irreducible subspaces: The SO(n)−invariant subspaces of the form W +

h

r (Hs−r;h ∩ Hs−r;−h) resp.

W −

h

r (Hs−r;h ∩ Hs−r;−h). Fourier decomposition of Hs;±h: is determined from the Howe dual pair (SO(1, 1), su(1, 1)). as a direct sum of the (s + 1)− irreducible pieces

W ±

h

r (Hs−r;h ∩ Hs−r;−h).

16 / 23

SLIDE 47

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

The Howe dual pair technique

N.F . SIGMA 9 (2013), 065

Invariant subspaces Hs;h : Spaces with basic polynomials of the form ws(x; h) = W +

h

s m0(x; h), with E+

h m0(x; h) = E− h m0(x; h) = 0.

Invariant subspaces Hs;−h: Spaces with basic polynomials of the form ws(x; −h) =

W −

h

s m0(x; h), with E+

h m0(x; h) = E− h m0(x; h) = 0.

Irreducible subspaces: The SO(n)−invariant subspaces of the form W +

h

r (Hs−r;h ∩ Hs−r;−h) resp.

W −

h

r (Hs−r;h ∩ Hs−r;−h). Fourier decomposition of Hs;±h: is determined from the Howe dual pair (SO(1, 1), su(1, 1)). as a direct sum of the (s + 1)− irreducible pieces

W ±

h

r (Hs−r;h ∩ Hs−r;−h).

16 / 23

SLIDE 48

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

The Howe dual pair technique

N.F . SIGMA 9 (2013), 065

Invariant subspaces Hs;h : Spaces with basic polynomials of the form ws(x; h) = W +

h

s m0(x; h), with E+

h m0(x; h) = E− h m0(x; h) = 0.

Invariant subspaces Hs;−h: Spaces with basic polynomials of the form ws(x; −h) =

W −

h

s m0(x; h), with E+

h m0(x; h) = E− h m0(x; h) = 0.

Irreducible subspaces: The SO(n)−invariant subspaces of the form W +

h

r (Hs−r;h ∩ Hs−r;−h) resp.

W −

h

r (Hs−r;h ∩ Hs−r;−h). Fourier decomposition of Hs;±h: is determined from the Howe dual pair (SO(1, 1), su(1, 1)). as a direct sum of the (s + 1)− irreducible pieces

W ±

h

r (Hs−r;h ∩ Hs−r;−h).

16 / 23

SLIDE 49

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Application to Cauchy problems

N.F . SIGMA, 2013

Homogeneous Cauchy problem in [0, ∞) × hZn:    ∂tg(t, x) + E+

h g(t, x) − E− h g(t, x) = 0

, t > 0 g(0, x) = f(x) , t = 0 E+

h g(t, x) = E− h g(t, x)

, t ≥ 0. Semigroup action: The one-parameter representation Eh(t) = exp(tE−

h − tE+ h ) of the Lie group SU(1, 1) yields

g(t, x) = Eh(t)f(x) as a solution of the above homogeneous Cauchy

problem. Hereby f(x) ∈ ∞

s=0 Hs;h ∩ Hs;−h.

Discrete series connection: Since g(0, x) = f(x) ∈ ∞

s=0 Hs;h and

g(1, x) = Eh(1)f(x) ∈ ∞

s=0 Hs;−h, one can see that the semigroup

(Eh(t))t≥0 gives, in particular, a direct link between the positive series representation of SU(1, 1) with the negative ones.

17 / 23

SLIDE 50

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Application to Cauchy problems

N.F . SIGMA, 2013

Homogeneous Cauchy problem in [0, ∞) × hZn:    ∂tg(t, x) + E+

h g(t, x) − E− h g(t, x) = 0

, t > 0 g(0, x) = f(x) , t = 0 E+

h g(t, x) = E− h g(t, x)

, t ≥ 0. Semigroup action: The one-parameter representation Eh(t) = exp(tE−

h − tE+ h ) of the Lie group SU(1, 1) yields

g(t, x) = Eh(t)f(x) as a solution of the above homogeneous Cauchy

problem. Hereby f(x) ∈ ∞

s=0 Hs;h ∩ Hs;−h.

Discrete series connection: Since g(0, x) = f(x) ∈ ∞

s=0 Hs;h and

g(1, x) = Eh(1)f(x) ∈ ∞

s=0 Hs;−h, one can see that the semigroup

(Eh(t))t≥0 gives, in particular, a direct link between the positive series representation of SU(1, 1) with the negative ones.

17 / 23

SLIDE 51

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Application to Cauchy problems

N.F . SIGMA, 2013

Homogeneous Cauchy problem in [0, ∞) × hZn:    ∂tg(t, x) + E+

h g(t, x) − E− h g(t, x) = 0

, t > 0 g(0, x) = f(x) , t = 0 E+

h g(t, x) = E− h g(t, x)

, t ≥ 0. Semigroup action: The one-parameter representation Eh(t) = exp(tE−

h − tE+ h ) of the Lie group SU(1, 1) yields

g(t, x) = Eh(t)f(x) as a solution of the above homogeneous Cauchy

problem. Hereby f(x) ∈ ∞

s=0 Hs;h ∩ Hs;−h.

Discrete series connection: Since g(0, x) = f(x) ∈ ∞

s=0 Hs;h and

g(1, x) = Eh(1)f(x) ∈ ∞

s=0 Hs;−h, one can see that the semigroup

(Eh(t))t≥0 gives, in particular, a direct link between the positive series representation of SU(1, 1) with the negative ones.

17 / 23

SLIDE 52

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Overview

Why must we use Lie-algebraic based discretizations?

Lie-algebraic based discretizations:

1

Preserve canonical symmetries: Get exact representation formulae for the polynomial solutions from methods already known in continuum;

2

Deep understanding of physical models: Provides a general scheme to construct sequences of polynomials as eigenfunctions of a discrete Hamiltonian operator.

3

Application to Cauchy problems: The 1-parameter representation

f SU(1, 1) produces solutions of homogeneous Cauchy-problems

as hypergeometric series representations (cf. N.F., SIGMA 065, 2013– Section 4).

4

Provides an operational framework: The construction polynomials on the lattice based on the knowledge of the EGF makes intuitive and fully rigorous the study special functions and integral transforms on the lattice (cf. N.F., Appl. Math. Comp., 2014, Subsection 3.3).

18 / 23

SLIDE 53

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Overview

Why must we use Lie-algebraic based discretizations?

Lie-algebraic based discretizations:

1

Preserve canonical symmetries: Get exact representation formulae for the polynomial solutions from methods already known in continuum;

2

Deep understanding of physical models: Provides a general scheme to construct sequences of polynomials as eigenfunctions of a discrete Hamiltonian operator.

3

Application to Cauchy problems: The 1-parameter representation

f SU(1, 1) produces solutions of homogeneous Cauchy-problems

as hypergeometric series representations (cf. N.F., SIGMA 065, 2013– Section 4).

4

Provides an operational framework: The construction polynomials on the lattice based on the knowledge of the EGF makes intuitive and fully rigorous the study special functions and integral transforms on the lattice (cf. N.F., Appl. Math. Comp., 2014, Subsection 3.3).

18 / 23

SLIDE 54

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Overview

Why must we use Lie-algebraic based discretizations?

Lie-algebraic based discretizations:

1

Preserve canonical symmetries: Get exact representation formulae for the polynomial solutions from methods already known in continuum;

2

Deep understanding of physical models: Provides a general scheme to construct sequences of polynomials as eigenfunctions of a discrete Hamiltonian operator.

3

Application to Cauchy problems: The 1-parameter representation

f SU(1, 1) produces solutions of homogeneous Cauchy-problems

as hypergeometric series representations (cf. N.F., SIGMA 065, 2013– Section 4).

4

Provides an operational framework: The construction polynomials on the lattice based on the knowledge of the EGF makes intuitive and fully rigorous the study special functions and integral transforms on the lattice (cf. N.F., Appl. Math. Comp., 2014, Subsection 3.3).

18 / 23

SLIDE 55

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Overview

Why must we use Lie-algebraic based discretizations?

Lie-algebraic based discretizations:

1

Preserve canonical symmetries: Get exact representation formulae for the polynomial solutions from methods already known in continuum;

2

Deep understanding of physical models: Provides a general scheme to construct sequences of polynomials as eigenfunctions of a discrete Hamiltonian operator.

3

Application to Cauchy problems: The 1-parameter representation

f SU(1, 1) produces solutions of homogeneous Cauchy-problems

as hypergeometric series representations (cf. N.F., SIGMA 065, 2013– Section 4).

4

Provides an operational framework: The construction polynomials on the lattice based on the knowledge of the EGF makes intuitive and fully rigorous the study special functions and integral transforms on the lattice (cf. N.F., Appl. Math. Comp., 2014, Subsection 3.3).

18 / 23

SLIDE 56

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Star-Laplacian factorization

Several possibilities

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Finite difference representation: ∆h = n

j=1 ∂+j h ∂−j h

= n

j=1 1 h

∂+j

h − ∂−j h

.

2

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

3

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

4

An alternative factorization: Obtained in my joint work with U. K¨ ahler &

F. Sommen (AACA, Volume 17, Issue 1,

pp 37-58 (2007)). Figure: The star laplacian in hZ3

19 / 23

SLIDE 57

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Star-Laplacian factorization

Several possibilities

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Finite difference representation: ∆h = n

j=1 ∂+j h ∂−j h

= n

j=1 1 h

∂+j

h − ∂−j h

.

2

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

3

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

4

An alternative factorization: Obtained in my joint work with U. K¨ ahler &

F. Sommen (AACA, Volume 17, Issue 1,

pp 37-58 (2007)). Figure: The star laplacian in hZ3

19 / 23

SLIDE 58

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Star-Laplacian factorization

Several possibilities

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Finite difference representation: ∆h = n

j=1 ∂+j h ∂−j h

= n

j=1 1 h

∂+j

h − ∂−j h

.

2

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

3

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

4

An alternative factorization: Obtained in my joint work with U. K¨ ahler &

F. Sommen (AACA, Volume 17, Issue 1,

pp 37-58 (2007)). Figure: The star laplacian in hZ3

19 / 23

SLIDE 59

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Star-Laplacian factorization

Several possibilities

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Finite difference representation: ∆h = n

j=1 ∂+j h ∂−j h

= n

j=1 1 h

∂+j

h − ∂−j h

.

2

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

3

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

4

An alternative factorization: Obtained in my joint work with U. K¨ ahler &

F. Sommen (AACA, Volume 17, Issue 1,

pp 37-58 (2007)). Figure: The star laplacian in hZ3

19 / 23

SLIDE 60

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Star-Laplacian factorization

Several possibilities

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Finite difference representation: ∆h = n

j=1 ∂+j h ∂−j h

= n

j=1 1 h

∂+j

h − ∂−j h

.

2

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

3

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

4

An alternative factorization: Obtained in my joint work with U. K¨ ahler &

F. Sommen (AACA, Volume 17, Issue 1,

pp 37-58 (2007)). Figure: The star laplacian in hZ3

19 / 23

SLIDE 61

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Star-Laplacian factorization

Several possibilities

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Finite difference representation: ∆h = n

j=1 ∂+j h ∂−j h

= n

j=1 1 h

∂+j

h − ∂−j h

.

2

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

3

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

4

An alternative factorization: Obtained in my joint work with U. K¨ ahler &

F. Sommen (AACA, Volume 17, Issue 1,

pp 37-58 (2007)). Figure: The star laplacian in hZ3

19 / 23

SLIDE 62

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Star-Laplacian factorization

Several possibilities

Star Laplacian: ∆hf(x) =

n

j=1

f(x + hej) + f(x − hej) − 2f(x) h2

1

Finite difference representation: ∆h = n

j=1 ∂+j h ∂−j h

= n

j=1 1 h

∂+j

h − ∂−j h

.

2

Using forward and backward Dirac

perators: ∆h = − 1

2

D+

h D− h + D− h D+ h

3

Using a central difference Dirac

perator: ∆h = − 1

4

D+

h/2 + D− h/2

2

4

An alternative factorization: Obtained in my joint work with U. K¨ ahler &

F. Sommen (AACA, Volume 17, Issue 1,

pp 37-58 (2007)). Figure: The star laplacian in hZ3

19 / 23

SLIDE 63

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Some attempts

What De Ridder have tried to downplay through his talk: The Howe dual pair technique with respect to the pair (G, h), where G stands a Lie group and h a Lie (super)algebra. The choice of G = SO(n, C) corresponds to the group of symmetries that Paul A.

M. Dirac have considered on his former paper, entitled The Electron

Wave Equation in De-Sitter Space (Annals Math. 36 (1935) 657-669). Up to now, the choice of h (h = sl(2|1) (?), as conjectured

n her Ph.D dissertation (2013)) seems to be an open problem to

be solved. Possible direction of research, towards Langlands program: In my recent preprint on arXiv (arXiv:1602.02252) the null solution

f the massless [discrete] Dirac operator Dh − mχh(x) was derived,

in a natural way, from the compactification of Rn−1,n at infinity, described in terms of the Cayley transform ϕ(w) = 1 + w 1 − w. The appearance of the resulting vacuum vector suggests that it should be possible to obtain plane-wave solutions for Dh − mχh(x) in terms of automorphic forms, lying to the universal cover of Rn−1,n– the so-called Einstein static universe.

20 / 23

SLIDE 64

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Some attempts

What De Ridder have tried to downplay through his talk: The Howe dual pair technique with respect to the pair (G, h), where G stands a Lie group and h a Lie (super)algebra. The choice of G = SO(n, C) corresponds to the group of symmetries that Paul A.

M. Dirac have considered on his former paper, entitled The Electron

Wave Equation in De-Sitter Space (Annals Math. 36 (1935) 657-669). Up to now, the choice of h (h = sl(2|1) (?), as conjectured

n her Ph.D dissertation (2013)) seems to be an open problem to

be solved. Possible direction of research, towards Langlands program: In my recent preprint on arXiv (arXiv:1602.02252) the null solution

f the massless [discrete] Dirac operator Dh − mχh(x) was derived,

in a natural way, from the compactification of Rn−1,n at infinity, described in terms of the Cayley transform ϕ(w) = 1 + w 1 − w. The appearance of the resulting vacuum vector suggests that it should be possible to obtain plane-wave solutions for Dh − mχh(x) in terms of automorphic forms, lying to the universal cover of Rn−1,n– the so-called Einstein static universe.

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SLIDE 65

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Discrete Quantum Mechanics

The SUSY approach

Discrete Electromagnetic Schr¨

dinger Operator:

Lhf(x) = 1 2µ

n

j=1

1 qh2 f(x) − 1 hah(xj)f(x + hej) − 1 hah(xj − h)f(x − hej)

+

q Φh(x)f(x). An Inverse Problem towards SUSY QM: Given a pair of Clifford-vector-valued operators (A+

h , A− h ) satisfying

Lh = 1 2

A+

h A− h + A− h A+ h

,

can we recover the discrete electric and magnetic potentials of Lh, Φh(x) and ah(x) respectively, from the knowledge of its k−bound states ψk(x; h) (k ∈ N0)?

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SLIDE 66

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Ongoing Research

N.F . arXiv:1505.05926

Bayesian Probability Formulation: For a given ground state ψ0(x; h) satisfying ψ0, ψ0h = 1, the quantity Pr  

n

j=1

ejXj = x   = hnψ0(x; h)†ψ0(x; h) may be regarded as a discrete quasi-probability law on hZn, carrying a set of independent and identically distributed (i.i.d.) random variables X1, X2, . . . , Xn. Some recovery formulae: For a given vacuum vector of the form ψ0(x; h) = φ(x; h)s (s ∈ Pin(n)) the discrete electric and magnetic potentials, Φh(x) and ah(x) respectively, are uniquely determined from the formulae Φh(x) = 1 4µ

n

j=1

1 q2h2

φ(x; h)2

φ(x + hej; h)2 + φ(x − hej; h)2 φ(x; h)2

ah(x)

=

n

j=1

ej φ(x; h) qh φ(x + hej; h).

22 / 23

SLIDE 67

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Ongoing Research

N.F . arXiv:1505.05926

Bayesian Probability Formulation: For a given ground state ψ0(x; h) satisfying ψ0, ψ0h = 1, the quantity Pr  

n

j=1

ejXj = x   = hnψ0(x; h)†ψ0(x; h) may be regarded as a discrete quasi-probability law on hZn, carrying a set of independent and identically distributed (i.i.d.) random variables X1, X2, . . . , Xn. Some recovery formulae: For a given vacuum vector of the form ψ0(x; h) = φ(x; h)s (s ∈ Pin(n)) the discrete electric and magnetic potentials, Φh(x) and ah(x) respectively, are uniquely determined from the formulae Φh(x) = 1 4µ

n

j=1

1 q2h2

φ(x; h)2

φ(x + hej; h)2 + φ(x − hej; h)2 φ(x; h)2

ah(x)

=

n

j=1

ej φ(x; h) qh φ(x + hej; h).

22 / 23

SLIDE 68

Post-Modern Topics in Discrete Clifford Analysis Nelson Faustino What I’ve learned from Frank

The radial algebra approach Beyond Landau-Weyl Calculus

Lie-algebraic discretization

Weyl-Heisenberg symmetries Appell Set Formulation su(1, 1) symmetries

Ongoing Research

Towards Dirac-K¨ ahler formalism Discrete Quantum Mechanics

Last but not least...

Happy Birthday Franciscus by your 22(24 − 1)th birthday!

www.cbpf.br/group31/ June 19 - June 24: 31st International Colloquium on Group Theoretical Methods in Physics (GROUP 31), Rio de Janeiro,

Brazil. There will be a Clifford-oriented parallel session (M6. Lie

algebras and Groups, Clifford algebras, Representation Theory, Special Functions) organized by Jean-Pierre Gazeau (U Paris Diderot, Paris) and Rold˜ ao da Rocha (UFABC, S˜ ao Paulo).

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