A connection between the Uncertainty Principles on the real line - PowerPoint PPT Presentation

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 ∈ L 1 ( R ) (or f ∈ C ∞ c ( R ) or f ∈ S ( R ), ...) : � ∞ 1 f ( x ) e − ixy dx , � √ f ( y ) = ( y ∈ R ) . 2 π −∞ Theorem (Heisenberg–Pauli–Weyl) Let f ∈ L 2 ( R ), and let a , b ∈ R . Then �� ∞ � 2 � ∞ � ∞ f ( y ) | 2 dy ≥ 1 ( x − a ) 2 | f ( x ) | 2 dx | f ( x ) | 2 dx ( y − b ) 2 | � . 4 −∞ −∞ −∞ Equality holds for Gaussian functions of the form f ( x ) = ce ibt e − dt 2 , 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 ∈ L 1 ( R ) (or f ∈ C ∞ c ( R ) or f ∈ S ( R ), ...) : � ∞ 1 f ( x ) e − ixy dx , � √ f ( y ) = ( y ∈ R ) . 2 π −∞ Theorem (Heisenberg–Pauli–Weyl) Let f ∈ L 2 ( R ), and let a , b ∈ R . Then �� ∞ � 2 � ∞ � ∞ f ( y ) | 2 dy ≥ 1 ( x − a ) 2 | f ( x ) | 2 dx | f ( x ) | 2 dx ( y − b ) 2 | � . 4 −∞ −∞ −∞ Equality holds for Gaussian functions of the form f ( x ) = ce ibt e − dt 2 , 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 ∈ L 2 ( R ), � f � 2 = � � f � 2 , (Plancherel), and, for f ∈ S ( R ), � df dx ( y ) = iy � f ( y ) . Then we can also write � � d � � � � ≥ 1 � � 2 � f � 2 � ( x − a ) f � 2 dx − b f 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 ∈ L 2 ( R ), � f � 2 = � � f � 2 , (Plancherel), and, for f ∈ S ( R ), � df dx ( y ) = iy � f ( y ) . Then we can also write � � d � � � � ≥ 1 � � 2 � f � 2 � ( x − a ) f � 2 dx − b f 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 2 π -periodic function. We will consider f as a function on the interval ] − π, π ] or on the circle S 1 . Theorem Let f be a ”nice” function, and a , b ∈ R . Then � � d � � � � � π � � � � � � ≥ 1 � � � ( e i θ − a ) f e i θ | f ( θ ) | 2 d θ � � � � � d θ − b � . f � � � 4 π 2 − π 2 The RHS could be zero! (when f ( θ ) = ce ik θ , 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 2 π -periodic function. We will consider f as a function on the interval ] − π, π ] or on the circle S 1 . Theorem Let f be a ”nice” function, and a , b ∈ R . Then � � d � � � � � π � � � � � � ≥ 1 � � � ( e i θ − a ) f e i θ | f ( θ ) | 2 d θ � � � � � d θ − b � . f � � � 4 π 2 − π 2 The RHS could be zero! (when f ( θ ) = ce ik θ , 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 2 π -periodic function. We will consider f as a function on the interval ] − π, π ] or on the circle S 1 . Theorem Let f be a ”nice” function, and a , b ∈ R . Then � � d � � � � � π � � � � � � ≥ 1 � � � ( e i θ − a ) f e i θ | f ( θ ) | 2 d θ � � � � � d θ − b � . f � � � 4 π 2 − π 2 The RHS could be zero! (when f ( θ ) = ce ik θ , 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