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A connection between the Uncertainty Principles on the real line (Heisenberg) and on the circle (Breitenberger)

Nils Byrial Andersen Aarhus University Alba, 18 June, 2013

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 1 / 14

Heisenberg’s Uncertainty Principle

The Fourier transform for f ∈ L1(R) (or f ∈ C ∞

c (R) or f ∈ S(R), ...) :

• f (y) =

1 √ 2π ∞

−∞

f (x)e−ixy dx, (y ∈ R).

Theorem (Heisenberg–Pauli–Weyl)

Let f ∈ L2(R), and let a, b ∈ R. Then ∞

−∞

(x − a)2|f (x)|2 dx ∞

−∞

(y − b)2| f (y)|2 dy ≥ 1 4 ∞

−∞

|f (x)|2 dx 2 . Equality holds for Gaussian functions of the form f (x) = ceibte−dt2, where c, d ∈ R.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 2 / 14

Heisenberg’s Uncertainty Principle

The Fourier transform for f ∈ L1(R) (or f ∈ C ∞

c (R) or f ∈ S(R), ...) :

• f (y) =

1 √ 2π ∞

−∞

f (x)e−ixy dx, (y ∈ R).

Theorem (Heisenberg–Pauli–Weyl)

Let f ∈ L2(R), and let a, b ∈ R. Then ∞

−∞

(x − a)2|f (x)|2 dx ∞

−∞

(y − b)2| f (y)|2 dy ≥ 1 4 ∞

−∞

|f (x)|2 dx 2 . Equality holds for Gaussian functions of the form f (x) = ceibte−dt2, where c, d ∈ R.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 2 / 14

Recall that, for f ∈ L2(R), f 2 = f 2, (Plancherel), and, for f ∈ S(R),

• df

dx (y) = iy f (y). Then we can also write (x − a)f 2

• d

dx − b

• f
• 2

≥ 1 2f 2

2.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 3 / 14

Recall that, for f ∈ L2(R), f 2 = f 2, (Plancherel), and, for f ∈ S(R),

• df

dx (y) = iy f (y). Then we can also write (x − a)f 2

• d

dx − b

• f
• 2

≥ 1 2f 2

2.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 3 / 14

Many proofs and generalizations...

1 Basic Undergraduate Maths. 2 Operator inequalities (Physics...). 3 Heat equation / semigroup approach (Ciatti–Ricci–Sundari). 4 Expansion by Hermite functions (De Bruijn). 5 Other...

Uncertainty Principle generalized to various integral transforms using 2–5... Today: Use item 2 on difference operators on Bernstein functions (Paley–Wiener functions) to find a new connection between the Heisenberg Uncertainty Principle and the Breitenburger Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 4 / 14

Many proofs and generalizations...

1 Basic Undergraduate Maths. 2 Operator inequalities (Physics...). 3 Heat equation / semigroup approach (Ciatti–Ricci–Sundari). 4 Expansion by Hermite functions (De Bruijn). 5 Other...

Uncertainty Principle generalized to various integral transforms using 2–5... Today: Use item 2 on difference operators on Bernstein functions (Paley–Wiener functions) to find a new connection between the Heisenberg Uncertainty Principle and the Breitenburger Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 4 / 14

Many proofs and generalizations...

1 Basic Undergraduate Maths. 2 Operator inequalities (Physics...). 3 Heat equation / semigroup approach (Ciatti–Ricci–Sundari). 4 Expansion by Hermite functions (De Bruijn). 5 Other...

Uncertainty Principle generalized to various integral transforms using 2–5... Today: Use item 2 on difference operators on Bernstein functions (Paley–Wiener functions) to find a new connection between the Heisenberg Uncertainty Principle and the Breitenburger Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 4 / 14

Many proofs and generalizations...

1 Basic Undergraduate Maths. 2 Operator inequalities (Physics...). 3 Heat equation / semigroup approach (Ciatti–Ricci–Sundari). 4 Expansion by Hermite functions (De Bruijn). 5 Other...

Uncertainty Principle generalized to various integral transforms using 2–5... Today: Use item 2 on difference operators on Bernstein functions (Paley–Wiener functions) to find a new connection between the Heisenberg Uncertainty Principle and the Breitenburger Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 4 / 14

Many proofs and generalizations...

1 Basic Undergraduate Maths. 2 Operator inequalities (Physics...). 3 Heat equation / semigroup approach (Ciatti–Ricci–Sundari). 4 Expansion by Hermite functions (De Bruijn). 5 Other...

Uncertainty Principle generalized to various integral transforms using 2–5... Today: Use item 2 on difference operators on Bernstein functions (Paley–Wiener functions) to find a new connection between the Heisenberg Uncertainty Principle and the Breitenburger Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 4 / 14

Many proofs and generalizations...

1 Basic Undergraduate Maths. 2 Operator inequalities (Physics...). 3 Heat equation / semigroup approach (Ciatti–Ricci–Sundari). 4 Expansion by Hermite functions (De Bruijn). 5 Other...

Uncertainty Principle generalized to various integral transforms using 2–5... Today: Use item 2 on difference operators on Bernstein functions (Paley–Wiener functions) to find a new connection between the Heisenberg Uncertainty Principle and the Breitenburger Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 4 / 14

Many proofs and generalizations...

1 Basic Undergraduate Maths. 2 Operator inequalities (Physics...). 3 Heat equation / semigroup approach (Ciatti–Ricci–Sundari). 4 Expansion by Hermite functions (De Bruijn). 5 Other...

Uncertainty Principle generalized to various integral transforms using 2–5... Today: Use item 2 on difference operators on Bernstein functions (Paley–Wiener functions) to find a new connection between the Heisenberg Uncertainty Principle and the Breitenburger Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 4 / 14

Breitenberger’s Uncertainty Principle

Let f be a 2π-periodic function. We will consider f as a function on the interval ] − π, π] or on the circle S1.

Theorem

Let f be a ”nice” function, and a, b ∈ R. Then

• (eiθ − a)f
• 2
• d

dθ − b

• f
• 2

≥ 1 4π

• π

−π

eiθ|f (θ)|2 dθ

• .

The RHS could be zero! (when f (θ) = ceikθ, k ∈ Z). LHS could also be zero...

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 5 / 14

Breitenberger’s Uncertainty Principle

Let f be a 2π-periodic function. We will consider f as a function on the interval ] − π, π] or on the circle S1.

Theorem

Let f be a ”nice” function, and a, b ∈ R. Then

• (eiθ − a)f
• 2
• d

dθ − b

• f
• 2

≥ 1 4π

• π

−π

eiθ|f (θ)|2 dθ

• .

The RHS could be zero! (when f (θ) = ceikθ, k ∈ Z). LHS could also be zero...

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 5 / 14

Breitenberger’s Uncertainty Principle

Let f be a 2π-periodic function. We will consider f as a function on the interval ] − π, π] or on the circle S1.

Theorem

Let f be a ”nice” function, and a, b ∈ R. Then

• (eiθ − a)f
• 2
• d

dθ − b

• f
• 2

≥ 1 4π

• π

−π

eiθ|f (θ)|2 dθ

• .

The RHS could be zero! (when f (θ) = ceikθ, k ∈ Z). LHS could also be zero...

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 5 / 14

Breitenberger’s Uncertainty Principle

Let f be a 2π-periodic function. We will consider f as a function on the interval ] − π, π] or on the circle S1.

Theorem

Let f be a ”nice” function, and a, b ∈ R. Then

• (eiθ − a)f
• 2
• d

dθ − b

• f
• 2

≥ 1 4π

• π

−π

eiθ|f (θ)|2 dθ

• .

The RHS could be zero! (when f (θ) = ceikθ, k ∈ Z). LHS could also be zero...

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 5 / 14

Abstract Set-up

H: Hilbert space with inner product ·, · and norm · = ·, ·1/2. A, B: linear operators with domains D(A), D(B). [A, B] := AB − BA, with domain D(AB) ∩ D(BA). (can be small!) Expectation Value of A w.r.t. f ∈ D(A): τA(f ) := Af , f f , f , Standard deviation, or variance, of A w.r.t. f ∈ D(A): σA(f ) := Af − τA(f )f = min

a∈C (A − a)f .

Notice that τA(f )f is the orthogonal projection of Af on f . Notice that Af , f ∈ R for self-adjoint A. (quantum mechanics

• bservables correspond to self-adjoint operators on Hilbert spaces)

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 6 / 14

Abstract Set-up

H: Hilbert space with inner product ·, · and norm · = ·, ·1/2. A, B: linear operators with domains D(A), D(B). [A, B] := AB − BA, with domain D(AB) ∩ D(BA). (can be small!) Expectation Value of A w.r.t. f ∈ D(A): τA(f ) := Af , f f , f , Standard deviation, or variance, of A w.r.t. f ∈ D(A): σA(f ) := Af − τA(f )f = min

a∈C (A − a)f .

Notice that τA(f )f is the orthogonal projection of Af on f . Notice that Af , f ∈ R for self-adjoint A. (quantum mechanics

• bservables correspond to self-adjoint operators on Hilbert spaces)

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 6 / 14

Abstract Set-up

H: Hilbert space with inner product ·, · and norm · = ·, ·1/2. A, B: linear operators with domains D(A), D(B). [A, B] := AB − BA, with domain D(AB) ∩ D(BA). (can be small!) Expectation Value of A w.r.t. f ∈ D(A): τA(f ) := Af , f f , f , Standard deviation, or variance, of A w.r.t. f ∈ D(A): σA(f ) := Af − τA(f )f = min

a∈C (A − a)f .

Notice that τA(f )f is the orthogonal projection of Af on f . Notice that Af , f ∈ R for self-adjoint A. (quantum mechanics

• bservables correspond to self-adjoint operators on Hilbert spaces)

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 6 / 14

Abstract Set-up

H: Hilbert space with inner product ·, · and norm · = ·, ·1/2. A, B: linear operators with domains D(A), D(B). [A, B] := AB − BA, with domain D(AB) ∩ D(BA). (can be small!) Expectation Value of A w.r.t. f ∈ D(A): τA(f ) := Af , f f , f , Standard deviation, or variance, of A w.r.t. f ∈ D(A): σA(f ) := Af − τA(f )f = min

a∈C (A − a)f .

Notice that τA(f )f is the orthogonal projection of Af on f . Notice that Af , f ∈ R for self-adjoint A. (quantum mechanics

• bservables correspond to self-adjoint operators on Hilbert spaces)

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 6 / 14

Abstract Set-up

H: Hilbert space with inner product ·, · and norm · = ·, ·1/2. A, B: linear operators with domains D(A), D(B). [A, B] := AB − BA, with domain D(AB) ∩ D(BA). (can be small!) Expectation Value of A w.r.t. f ∈ D(A): τA(f ) := Af , f f , f , Standard deviation, or variance, of A w.r.t. f ∈ D(A): σA(f ) := Af − τA(f )f = min

a∈C (A − a)f .

Notice that τA(f )f is the orthogonal projection of Af on f . Notice that Af , f ∈ R for self-adjoint A. (quantum mechanics

• bservables correspond to self-adjoint operators on Hilbert spaces)

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 6 / 14

Abstract Set-up

H: Hilbert space with inner product ·, · and norm · = ·, ·1/2. A, B: linear operators with domains D(A), D(B). [A, B] := AB − BA, with domain D(AB) ∩ D(BA). (can be small!) Expectation Value of A w.r.t. f ∈ D(A): τA(f ) := Af , f f , f , Standard deviation, or variance, of A w.r.t. f ∈ D(A): σA(f ) := Af − τA(f )f = min

a∈C (A − a)f .

Notice that τA(f )f is the orthogonal projection of Af on f . Notice that Af , f ∈ R for self-adjoint A. (quantum mechanics

• bservables correspond to self-adjoint operators on Hilbert spaces)

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 6 / 14

An operator inequality

Theorem

Assume that A, B are symmetric or normal operators on H, then (A − a)f (B − b)f ≥ σA(f )σB(f ) ≥ 1 2|[A, B]f , f |, for all 0 = f ∈ D(AB) ∩ D(BA), and all a, b ∈ C. There is equality if, and only if, (B − b)f = λ(A − a)f = −λ(A∗ − a)f , for some λ ∈ C.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 7 / 14

An operator inequality

Theorem

Assume that A, B are symmetric or normal operators on H, then (A − a)f (B − b)f ≥ σA(f )σB(f ) ≥ 1 2|[A, B]f , f |, for all 0 = f ∈ D(AB) ∩ D(BA), and all a, b ∈ C. There is equality if, and only if, (B − b)f = λ(A − a)f = −λ(A∗ − a)f , for some λ ∈ C.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 7 / 14

Smart choices of H and operators A, B give the classical results:

Example (Heisenberg’s Uncertainty Principle)

H = L2(R), Af (x) = xf (x) og Bf (x) = i d

dx f (x).

Note : A and B are self-adjoint and [A, B]f (x) = if (x).

Exemple (Breitenberger’s Uncertainty Principle)

H = L2(−π, π), Af (x) = eixf (x) og Bf (x) = i d

dx f (x).

Note : A is unitary, B is self-adjoint and [A, B]f (x) = eixf (x).

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 8 / 14

Smart choices of H and operators A, B give the classical results:

Example (Heisenberg’s Uncertainty Principle)

H = L2(R), Af (x) = xf (x) og Bf (x) = i d

dx f (x).

Note : A and B are self-adjoint and [A, B]f (x) = if (x).

Exemple (Breitenberger’s Uncertainty Principle)

H = L2(−π, π), Af (x) = eixf (x) og Bf (x) = i d

dx f (x).

Note : A is unitary, B is self-adjoint and [A, B]f (x) = eixf (x).

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 8 / 14

Smart choices of H and operators A, B give the classical results:

Example (Heisenberg’s Uncertainty Principle)

H = L2(R), Af (x) = xf (x) og Bf (x) = i d

dx f (x).

Note : A and B are self-adjoint and [A, B]f (x) = if (x).

Exemple (Breitenberger’s Uncertainty Principle)

H = L2(−π, π), Af (x) = eixf (x) og Bf (x) = i d

dx f (x).

Note : A is unitary, B is self-adjoint and [A, B]f (x) = eixf (x).

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 8 / 14

Proof I

Assume (w.l.o.g.) a = b = 0. Assume first A∗ = A and B∗ = B. Then |[A, B]f , f | = |(AB − BA)f , f | = |ABf , f − BAf , f | = |Bf , A∗f − Af , B∗f | = |Bf , Af − Af , Bf | = |Af , Bf − Af , Bf | = 2|Im Af , Bf | (|Im z| ≤ |z|) ≤ 2|Af , Bf | (Cauchy–Schwartz) ≤ 2Af Bf . Equality when Af = λBf (Cauchy–Schwartz) for an imaginary skalar λ.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 9 / 14

Proof II

Assume A and B normal. Then |[A, B]f , f | = |(AB − BA)f , f | = |ABf , f − BAf , f | = |Bf , A∗f − Af , B∗f | (triangle inequality) ≤ |Bf , A∗f | + |Af , B∗f | (Cauchy–Schwartz) ≤ Bf A∗f + Af B∗f = 2Af Bf .

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 10 / 14

The Bernstein space B2

R

(or, Paley–Wiener space) B2

R = {f ∈ L2(R) | f ′2 ≤ Rf 2}

= {f ∈ L2(R) | supp f ⊂ [−R, R]}. B2

π and l2(Z) are isomorphic, with proportional norms.

Isomorphism by the W–K–S Sampling Formula f (z) =

• n∈Z

ak sinc (z − n), ({ak}k∈Z ∈ l2(Z)). Inverse: B2

π ∋ f → {f (k)}k∈Z ∈ l2(Z).

˙ B2

R = {f ∈ B2 R : xf (x) ∈ L2(R)} (Note : f ∈ ˙

B2

R ⇒ xf (x) ∈ B2 R)

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 11 / 14

The Bernstein space B2

R

(or, Paley–Wiener space) B2

R = {f ∈ L2(R) | f ′2 ≤ Rf 2}

= {f ∈ L2(R) | supp f ⊂ [−R, R]}. B2

π and l2(Z) are isomorphic, with proportional norms.

Isomorphism by the W–K–S Sampling Formula f (z) =

• n∈Z

ak sinc (z − n), ({ak}k∈Z ∈ l2(Z)). Inverse: B2

π ∋ f → {f (k)}k∈Z ∈ l2(Z).

˙ B2

R = {f ∈ B2 R : xf (x) ∈ L2(R)} (Note : f ∈ ˙

B2

R ⇒ xf (x) ∈ B2 R)

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 11 / 14

The Bernstein space B2

R

(or, Paley–Wiener space) B2

R = {f ∈ L2(R) | f ′2 ≤ Rf 2}

= {f ∈ L2(R) | supp f ⊂ [−R, R]}. B2

π and l2(Z) are isomorphic, with proportional norms.

Isomorphism by the W–K–S Sampling Formula f (z) =

• n∈Z

ak sinc (z − n), ({ak}k∈Z ∈ l2(Z)). Inverse: B2

π ∋ f → {f (k)}k∈Z ∈ l2(Z).

˙ B2

R = {f ∈ B2 R : xf (x) ∈ L2(R)} (Note : f ∈ ˙

B2

R ⇒ xf (x) ∈ B2 R)

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 11 / 14

Connection between the two uncertainty principles

Let Aδf (x) = f (x) − f (x − δ) δ (f ∈ B2

R, x ∈ R),

where δ ∈ (0, 1], and Bf (x) = xf (x) (f ∈ ˙ B2

R, x ∈ R).

Since [Aδ, B]f (x) = f (x − δ), the operator inequality gives (Aδ − a)f 2 (B − b)f 2 ≥ 1 2|f (· − δ), f |, for 0 = f ∈ ˙ B2

R, and all a, b.

δ → 0 yields Heisenberg’s Uncertainty Principle (Aδ → d

dx )

δ = 1 and R = π yields Breitenberger’s Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 12 / 14

Connection between the two uncertainty principles

Let Aδf (x) = f (x) − f (x − δ) δ (f ∈ B2

R, x ∈ R),

where δ ∈ (0, 1], and Bf (x) = xf (x) (f ∈ ˙ B2

R, x ∈ R).

Since [Aδ, B]f (x) = f (x − δ), the operator inequality gives (Aδ − a)f 2 (B − b)f 2 ≥ 1 2|f (· − δ), f |, for 0 = f ∈ ˙ B2

R, and all a, b.

δ → 0 yields Heisenberg’s Uncertainty Principle (Aδ → d

dx )

δ = 1 and R = π yields Breitenberger’s Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 12 / 14

Connection between the two uncertainty principles

Let Aδf (x) = f (x) − f (x − δ) δ (f ∈ B2

R, x ∈ R),

where δ ∈ (0, 1], and Bf (x) = xf (x) (f ∈ ˙ B2

R, x ∈ R).

Since [Aδ, B]f (x) = f (x − δ), the operator inequality gives (Aδ − a)f 2 (B − b)f 2 ≥ 1 2|f (· − δ), f |, for 0 = f ∈ ˙ B2

R, and all a, b.

δ → 0 yields Heisenberg’s Uncertainty Principle (Aδ → d

dx )

δ = 1 and R = π yields Breitenberger’s Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 12 / 14

Connection between the two uncertainty principles

Let Aδf (x) = f (x) − f (x − δ) δ (f ∈ B2

R, x ∈ R),

where δ ∈ (0, 1], and Bf (x) = xf (x) (f ∈ ˙ B2

R, x ∈ R).

Since [Aδ, B]f (x) = f (x − δ), the operator inequality gives (Aδ − a)f 2 (B − b)f 2 ≥ 1 2|f (· − δ), f |, for 0 = f ∈ ˙ B2

R, and all a, b.

δ → 0 yields Heisenberg’s Uncertainty Principle (Aδ → d

dx )

δ = 1 and R = π yields Breitenberger’s Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 12 / 14

Connection between the two uncertainty principles

Let Aδf (x) = f (x) − f (x − δ) δ (f ∈ B2

R, x ∈ R),

where δ ∈ (0, 1], and Bf (x) = xf (x) (f ∈ ˙ B2

R, x ∈ R).

Since [Aδ, B]f (x) = f (x − δ), the operator inequality gives (Aδ − a)f 2 (B − b)f 2 ≥ 1 2|f (· − δ), f |, for 0 = f ∈ ˙ B2

R, and all a, b.

δ → 0 yields Heisenberg’s Uncertainty Principle (Aδ → d

dx )

δ = 1 and R = π yields Breitenberger’s Uncertainty Principle.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 12 / 14

Proof : In terms of the Fourier coefficients {f (n)},

• n∈Z

|f (n − 1) − af (n)|2 1

2

n∈Z

|(n − b)f (n)|2 1

2

≥ 1 2

• n∈Z

f (n − 1)f (n)

• ,

for all {f (n)}n∈Z ∈ l2(Z). ˇ f ∈ L2(−π, π) Fourier inverse of {f (n)}. Then eiθˇ f ∼ {f (n − 1)} and i d

dθˇ

f ∼ {nf (n)}. Transferring from l2(Z) to L2(−π, π) via Fourier inverse

• (eiθ − a)ˇ

f

• 2
• d

dθ − b

• ˇ

f

• 2

≥ 1 2

• eiθˇ

f , ˇ f

• .

Q.E.D.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 13 / 14

Proof : In terms of the Fourier coefficients {f (n)},

• n∈Z

|f (n − 1) − af (n)|2 1

2

n∈Z

|(n − b)f (n)|2 1

2

≥ 1 2

• n∈Z

f (n − 1)f (n)

• ,

for all {f (n)}n∈Z ∈ l2(Z). ˇ f ∈ L2(−π, π) Fourier inverse of {f (n)}. Then eiθˇ f ∼ {f (n − 1)} and i d

dθˇ

f ∼ {nf (n)}. Transferring from l2(Z) to L2(−π, π) via Fourier inverse

• (eiθ − a)ˇ

f

• 2
• d

dθ − b

• ˇ

f

• 2

≥ 1 2

• eiθˇ

f , ˇ f

• .

Q.E.D.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 13 / 14

Proof : In terms of the Fourier coefficients {f (n)},

• n∈Z

|f (n − 1) − af (n)|2 1

2

n∈Z

|(n − b)f (n)|2 1

2

≥ 1 2

• n∈Z

f (n − 1)f (n)

• ,

for all {f (n)}n∈Z ∈ l2(Z). ˇ f ∈ L2(−π, π) Fourier inverse of {f (n)}. Then eiθˇ f ∼ {f (n − 1)} and i d

dθˇ

f ∼ {nf (n)}. Transferring from l2(Z) to L2(−π, π) via Fourier inverse

• (eiθ − a)ˇ

f

• 2
• d

dθ − b

• ˇ

f

• 2

≥ 1 2

• eiθˇ

f , ˇ f

• .

Q.E.D.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 13 / 14

Proof : In terms of the Fourier coefficients {f (n)},

• n∈Z

|f (n − 1) − af (n)|2 1

2

n∈Z

|(n − b)f (n)|2 1

2

≥ 1 2

• n∈Z

f (n − 1)f (n)

• ,

for all {f (n)}n∈Z ∈ l2(Z). ˇ f ∈ L2(−π, π) Fourier inverse of {f (n)}. Then eiθˇ f ∼ {f (n − 1)} and i d

dθˇ

f ∼ {nf (n)}. Transferring from l2(Z) to L2(−π, π) via Fourier inverse

• (eiθ − a)ˇ

f

• 2
• d

dθ − b

• ˇ

f

• 2

≥ 1 2

• eiθˇ

f , ˇ f

• .

Q.E.D.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 13 / 14

Proof : In terms of the Fourier coefficients {f (n)},

• n∈Z

|f (n − 1) − af (n)|2 1

2

n∈Z

|(n − b)f (n)|2 1

2

≥ 1 2

• n∈Z

f (n − 1)f (n)

• ,

for all {f (n)}n∈Z ∈ l2(Z). ˇ f ∈ L2(−π, π) Fourier inverse of {f (n)}. Then eiθˇ f ∼ {f (n − 1)} and i d

dθˇ

f ∼ {nf (n)}. Transferring from l2(Z) to L2(−π, π) via Fourier inverse

• (eiθ − a)ˇ

f

• 2
• d

dθ − b

• ˇ

f

• 2

≥ 1 2

• eiθˇ

f , ˇ f

• .

Q.E.D.

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 13 / 14

See also: Prestin, Quak, Rauhut, Selig, ”On the connection of uncertainty principles for functions on the circle and on the real line”, J. Fourier Anal.

• Appl. 9 (2003), 387-409.

Other difference operator: Cδf (z) = f (z + δ) − f (z − δ) 2δ (f ∈ B2

R, z ∈ C),

yields Heisenberg as before, and

• (sin(θ) − a)ˇ

f

• 2
• d

dθ − b

• ˇ

f

• 2

≥ 1 2

• cos(θ)ˇ

f , ˇ f

• .

Why look at B2

R? Localized frequencies...

Observation: Let f ∈ B2

R, i.e., frequencies are in the interval [−R, R].

Then the Bernstein inequality f ′2 ≤ Rf 2 gives (x − a)f 2 ≥ f 2 2R (f ∈ B2

R).

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 14 / 14

See also: Prestin, Quak, Rauhut, Selig, ”On the connection of uncertainty principles for functions on the circle and on the real line”, J. Fourier Anal.

• Appl. 9 (2003), 387-409.

Other difference operator: Cδf (z) = f (z + δ) − f (z − δ) 2δ (f ∈ B2

R, z ∈ C),

yields Heisenberg as before, and

• (sin(θ) − a)ˇ

f

• 2
• d

dθ − b

• ˇ

f

• 2

≥ 1 2

• cos(θ)ˇ

f , ˇ f

• .

Why look at B2

R? Localized frequencies...

Observation: Let f ∈ B2

R, i.e., frequencies are in the interval [−R, R].

Then the Bernstein inequality f ′2 ≤ Rf 2 gives (x − a)f 2 ≥ f 2 2R (f ∈ B2

R).

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 14 / 14

See also: Prestin, Quak, Rauhut, Selig, ”On the connection of uncertainty principles for functions on the circle and on the real line”, J. Fourier Anal.

• Appl. 9 (2003), 387-409.

Other difference operator: Cδf (z) = f (z + δ) − f (z − δ) 2δ (f ∈ B2

R, z ∈ C),

yields Heisenberg as before, and

• (sin(θ) − a)ˇ

f

• 2
• d

dθ − b

• ˇ

f

• 2

≥ 1 2

• cos(θ)ˇ

f , ˇ f

• .

Why look at B2

R? Localized frequencies...

Observation: Let f ∈ B2

R, i.e., frequencies are in the interval [−R, R].

Then the Bernstein inequality f ′2 ≤ Rf 2 gives (x − a)f 2 ≥ f 2 2R (f ∈ B2

R).

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 14 / 14

See also: Prestin, Quak, Rauhut, Selig, ”On the connection of uncertainty principles for functions on the circle and on the real line”, J. Fourier Anal.

• Appl. 9 (2003), 387-409.

Other difference operator: Cδf (z) = f (z + δ) − f (z − δ) 2δ (f ∈ B2

R, z ∈ C),

yields Heisenberg as before, and

• (sin(θ) − a)ˇ

f

• 2
• d

dθ − b

• ˇ

f

• 2

≥ 1 2

• cos(θ)ˇ

f , ˇ f

• .

Why look at B2

R? Localized frequencies...

Observation: Let f ∈ B2

R, i.e., frequencies are in the interval [−R, R].

Then the Bernstein inequality f ′2 ≤ Rf 2 gives (x − a)f 2 ≥ f 2 2R (f ∈ B2

R).

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 14 / 14

See also: Prestin, Quak, Rauhut, Selig, ”On the connection of uncertainty principles for functions on the circle and on the real line”, J. Fourier Anal.

• Appl. 9 (2003), 387-409.

Other difference operator: Cδf (z) = f (z + δ) − f (z − δ) 2δ (f ∈ B2

R, z ∈ C),

yields Heisenberg as before, and

• (sin(θ) − a)ˇ

f

• 2
• d

dθ − b

• ˇ

f

• 2

≥ 1 2

• cos(θ)ˇ

f , ˇ f

• .

Why look at B2

R? Localized frequencies...

Observation: Let f ∈ B2

R, i.e., frequencies are in the interval [−R, R].

Then the Bernstein inequality f ′2 ≤ Rf 2 gives (x − a)f 2 ≥ f 2 2R (f ∈ B2

R).

Nils Byrial Andersen Aarhus University A connection between the Uncertainty Principles on the real line (Heisenberg) and on the Alba, 18 June, 2013 14 / 14