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Quantum Localisation on the Circle7th International Workshop on New Challenges in Quantum
Mechanics: Integrability and Supersymmetry
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil)
Benasque, 3rd of September of 2019
J. Math. Phys., 59(5), 52105 (2018), in collaboration with J.P. Gazeau(Univ. Paris-Diderot) and D. Noguera (CBPF, Rio de Janeiro)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 2: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/2.jpg)
Introduction and Motivation
If ψ(α) is the 2π-periodic wave function on the circle, thequantum angle α cannot be a multiplication operator,αψ(α) = αψ(α) without breaking periodicity.
Except if α stands for the 2π-periodic discontinuous anglefunction,
(αψ)(α) :=(α− 2π
⌊ α2π
⌋)ψ(α) . (1)
However, for α Self-Adjoint (SA), spec(α) ⊂ [0, 2π], the CCR[α, pα] = i~I does not hold for SA quantum angularmomentum pα = −i~ ∂
∂α .
Instead, one has
[α, pα] = i~I
[1− 2π
∑n
δ(α− 2nπ)
]. (2)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 3: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/3.jpg)
Introduction and Motivation
If ψ(α) is the 2π-periodic wave function on the circle, thequantum angle α cannot be a multiplication operator,αψ(α) = αψ(α) without breaking periodicity.
Except if α stands for the 2π-periodic discontinuous anglefunction,
(αψ)(α) :=(α− 2π
⌊ α2π
⌋)ψ(α) . (1)
However, for α Self-Adjoint (SA), spec(α) ⊂ [0, 2π], the CCR[α, pα] = i~I does not hold for SA quantum angularmomentum pα = −i~ ∂
∂α .
Instead, one has
[α, pα] = i~I
[1− 2π
∑n
δ(α− 2nπ)
]. (2)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 4: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/4.jpg)
Introduction and Motivation
If ψ(α) is the 2π-periodic wave function on the circle, thequantum angle α cannot be a multiplication operator,αψ(α) = αψ(α) without breaking periodicity.
Except if α stands for the 2π-periodic discontinuous anglefunction,
(αψ)(α) :=(α− 2π
⌊ α2π
⌋)ψ(α) . (1)
However, for α Self-Adjoint (SA), spec(α) ⊂ [0, 2π], the CCR[α, pα] = i~I does not hold for SA quantum angularmomentum pα = −i~ ∂
∂α .
Instead, one has
[α, pα] = i~I
[1− 2π
∑n
δ(α− 2nπ)
]. (2)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 5: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/5.jpg)
Introduction and Motivation
If ψ(α) is the 2π-periodic wave function on the circle, thequantum angle α cannot be a multiplication operator,αψ(α) = αψ(α) without breaking periodicity.
Except if α stands for the 2π-periodic discontinuous anglefunction,
(αψ)(α) :=(α− 2π
⌊ α2π
⌋)ψ(α) . (1)
However, for α Self-Adjoint (SA), spec(α) ⊂ [0, 2π], the CCR[α, pα] = i~I does not hold for SA quantum angularmomentum pα = −i~ ∂
∂α .
Instead, one has
[α, pα] = i~I
[1− 2π
∑n
δ(α− 2nπ)
]. (2)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 6: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/6.jpg)
This is an old problem (dating back to Dirac ”The QuantumTheory of the Emission and Absorption of Radiation”).
Most approaches rely on replacing the angle operator by aquantum version of a smooth periodic function of the classicalangle at the cost of losing localisation.
We revisit the problem of the quantum angle throughcoherent state (CS) quantisation, which is a particular methodbelonging to covariant integral quantisation (S.T. Ali, J.-P.Antoine, and J.-P. Gazeau, Coherent States, Wavelets andtheir Generalizations).
Our approach is group theoretical, based on the unitaryirreducible representations of the (special) Euclidean groupE(2) = R2oSO(2) (see also S. De Bievre, Coherent statesover symplectic homogeneous spaces).
One of our aims is to build acceptable angle operators fromthe classical angle function through a consistent andmanageable quantisation procedure.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 7: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/7.jpg)
This is an old problem (dating back to Dirac ”The QuantumTheory of the Emission and Absorption of Radiation”).
Most approaches rely on replacing the angle operator by aquantum version of a smooth periodic function of the classicalangle at the cost of losing localisation.
We revisit the problem of the quantum angle throughcoherent state (CS) quantisation, which is a particular methodbelonging to covariant integral quantisation (S.T. Ali, J.-P.Antoine, and J.-P. Gazeau, Coherent States, Wavelets andtheir Generalizations).
Our approach is group theoretical, based on the unitaryirreducible representations of the (special) Euclidean groupE(2) = R2oSO(2) (see also S. De Bievre, Coherent statesover symplectic homogeneous spaces).
One of our aims is to build acceptable angle operators fromthe classical angle function through a consistent andmanageable quantisation procedure.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 8: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/8.jpg)
This is an old problem (dating back to Dirac ”The QuantumTheory of the Emission and Absorption of Radiation”).
Most approaches rely on replacing the angle operator by aquantum version of a smooth periodic function of the classicalangle at the cost of losing localisation.
We revisit the problem of the quantum angle throughcoherent state (CS) quantisation, which is a particular methodbelonging to covariant integral quantisation (S.T. Ali, J.-P.Antoine, and J.-P. Gazeau, Coherent States, Wavelets andtheir Generalizations).
Our approach is group theoretical, based on the unitaryirreducible representations of the (special) Euclidean groupE(2) = R2oSO(2) (see also S. De Bievre, Coherent statesover symplectic homogeneous spaces).
One of our aims is to build acceptable angle operators fromthe classical angle function through a consistent andmanageable quantisation procedure.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 9: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/9.jpg)
This is an old problem (dating back to Dirac ”The QuantumTheory of the Emission and Absorption of Radiation”).
Most approaches rely on replacing the angle operator by aquantum version of a smooth periodic function of the classicalangle at the cost of losing localisation.
We revisit the problem of the quantum angle throughcoherent state (CS) quantisation, which is a particular methodbelonging to covariant integral quantisation (S.T. Ali, J.-P.Antoine, and J.-P. Gazeau, Coherent States, Wavelets andtheir Generalizations).
Our approach is group theoretical, based on the unitaryirreducible representations of the (special) Euclidean groupE(2) = R2oSO(2) (see also S. De Bievre, Coherent statesover symplectic homogeneous spaces).
One of our aims is to build acceptable angle operators fromthe classical angle function through a consistent andmanageable quantisation procedure.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 10: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/10.jpg)
This is an old problem (dating back to Dirac ”The QuantumTheory of the Emission and Absorption of Radiation”).
Most approaches rely on replacing the angle operator by aquantum version of a smooth periodic function of the classicalangle at the cost of losing localisation.
We revisit the problem of the quantum angle throughcoherent state (CS) quantisation, which is a particular methodbelonging to covariant integral quantisation (S.T. Ali, J.-P.Antoine, and J.-P. Gazeau, Coherent States, Wavelets andtheir Generalizations).
Our approach is group theoretical, based on the unitaryirreducible representations of the (special) Euclidean groupE(2) = R2oSO(2) (see also S. De Bievre, Coherent statesover symplectic homogeneous spaces).
One of our aims is to build acceptable angle operators fromthe classical angle function through a consistent andmanageable quantisation procedure.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 11: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/11.jpg)
Covariant integral quantisation - general scheme
Let G be a Lie group with left Haar measure dµ(g) and g 7→ U(g) aUIR of G in H. For ρ ∈ B(H) , suppose the following operator isdefined in a weak sense:
R :=
∫Gρ (g)dµ (g) , ρ (g) := U (g) ρU† (g)
Then , R = cρI , since U (g0)RU† (g0) =∫G ρ (g0g)dµ (g) = R
That is, the family of operators ρ(g) provides a resolution of theidentity ∫
Gρ (g)
dµ (g)
cρ= I , cρ =
∫Gtr(ρρ(g))dµ (g) <∞
This allows an integral quantisation of complex-valued functions onthe group
f 7→ Af =
∫Gρ(g) f (g)
dµ (g)
cρ,
which is covariant in the sense that
U(g)Af U†(g) = AU(g)f , (U(g)f )(g ′) = f (g−1g ′)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 12: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/12.jpg)
Covariant integral quantisation - general scheme
Let G be a Lie group with left Haar measure dµ(g) and g 7→ U(g) aUIR of G in H. For ρ ∈ B(H) , suppose the following operator isdefined in a weak sense:
R :=
∫Gρ (g)dµ (g) , ρ (g) := U (g) ρU† (g)
Then , R = cρI , since U (g0)RU† (g0) =∫G ρ (g0g)dµ (g) = R
That is, the family of operators ρ(g) provides a resolution of theidentity ∫
Gρ (g)
dµ (g)
cρ= I , cρ =
∫Gtr(ρρ(g))dµ (g) <∞
This allows an integral quantisation of complex-valued functions onthe group
f 7→ Af =
∫Gρ(g) f (g)
dµ (g)
cρ,
which is covariant in the sense that
U(g)Af U†(g) = AU(g)f , (U(g)f )(g ′) = f (g−1g ′)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 13: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/13.jpg)
Covariant integral quantisation - general scheme
Let G be a Lie group with left Haar measure dµ(g) and g 7→ U(g) aUIR of G in H. For ρ ∈ B(H) , suppose the following operator isdefined in a weak sense:
R :=
∫Gρ (g)dµ (g) , ρ (g) := U (g) ρU† (g)
Then , R = cρI , since U (g0)RU† (g0) =∫G ρ (g0g)dµ (g) = R
That is, the family of operators ρ(g) provides a resolution of theidentity ∫
Gρ (g)
dµ (g)
cρ= I , cρ =
∫Gtr(ρρ(g))dµ (g) <∞
This allows an integral quantisation of complex-valued functions onthe group
f 7→ Af =
∫Gρ(g) f (g)
dµ (g)
cρ,
which is covariant in the sense that
U(g)Af U†(g) = AU(g)f , (U(g)f )(g ′) = f (g−1g ′)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 14: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/14.jpg)
Covariant integral quantisation - general scheme
Let G be a Lie group with left Haar measure dµ(g) and g 7→ U(g) aUIR of G in H. For ρ ∈ B(H) , suppose the following operator isdefined in a weak sense:
R :=
∫Gρ (g)dµ (g) , ρ (g) := U (g) ρU† (g)
Then , R = cρI , since U (g0)RU† (g0) =∫G ρ (g0g)dµ (g) = R
That is, the family of operators ρ(g) provides a resolution of theidentity ∫
Gρ (g)
dµ (g)
cρ= I , cρ =
∫Gtr(ρρ(g))dµ (g) <∞
This allows an integral quantisation of complex-valued functions onthe group
f 7→ Af =
∫Gρ(g) f (g)
dµ (g)
cρ,
which is covariant in the sense that
U(g)Af U†(g) = AU(g)f , (U(g)f )(g ′) = f (g−1g ′)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 15: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/15.jpg)
Covariant integral quantisation - general scheme
Let G be a Lie group with left Haar measure dµ(g) and g 7→ U(g) aUIR of G in H. For ρ ∈ B(H) , suppose the following operator isdefined in a weak sense:
R :=
∫Gρ (g)dµ (g) , ρ (g) := U (g) ρU† (g)
Then , R = cρI , since U (g0)RU† (g0) =∫G ρ (g0g)dµ (g) = R
That is, the family of operators ρ(g) provides a resolution of theidentity ∫
Gρ (g)
dµ (g)
cρ= I , cρ =
∫Gtr(ρρ(g))dµ (g) <∞
This allows an integral quantisation of complex-valued functions onthe group
f 7→ Af =
∫Gρ(g) f (g)
dµ (g)
cρ,
which is covariant in the sense that
U(g)Af U†(g) = AU(g)f , (U(g)f )(g ′) = f (g−1g ′)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 16: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/16.jpg)
Covariant integral quantisation - homogeneous spaces
We consider the quantisation of functions on a homogeneousspace X , the left coset manifold X ∼ G/H for the action of aLie Group G , where the closed subgroup H is the stabilizer ofsome point of X .
The interesting case is when X is a symplectic manifold (e.g.,co-adjoint orbit of G ) and can be viewed as the phase spacefor the dynamics.
Given a quasi-invariant measure ν on X , one has for a globalBorel section σ : X → G a unique quasi-invariant measureνσ(x).
Let U be a square-integrable UIR, and ρ a density operatorsuch that cρ :=
∫X tr (ρ ρσ(x)) dνσ(x) <∞ with
ρσ(x) := U(σ(x))ρU(σ(x))†.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 17: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/17.jpg)
Covariant integral quantisation - homogeneous spaces
We consider the quantisation of functions on a homogeneousspace X , the left coset manifold X ∼ G/H for the action of aLie Group G , where the closed subgroup H is the stabilizer ofsome point of X .
The interesting case is when X is a symplectic manifold (e.g.,co-adjoint orbit of G ) and can be viewed as the phase spacefor the dynamics.
Given a quasi-invariant measure ν on X , one has for a globalBorel section σ : X → G a unique quasi-invariant measureνσ(x).
Let U be a square-integrable UIR, and ρ a density operatorsuch that cρ :=
∫X tr (ρ ρσ(x)) dνσ(x) <∞ with
ρσ(x) := U(σ(x))ρU(σ(x))†.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 18: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/18.jpg)
Covariant integral quantisation - homogeneous spaces
We consider the quantisation of functions on a homogeneousspace X , the left coset manifold X ∼ G/H for the action of aLie Group G , where the closed subgroup H is the stabilizer ofsome point of X .
The interesting case is when X is a symplectic manifold (e.g.,co-adjoint orbit of G ) and can be viewed as the phase spacefor the dynamics.
Given a quasi-invariant measure ν on X , one has for a globalBorel section σ : X → G a unique quasi-invariant measureνσ(x).
Let U be a square-integrable UIR, and ρ a density operatorsuch that cρ :=
∫X tr (ρ ρσ(x)) dνσ(x) <∞ with
ρσ(x) := U(σ(x))ρU(σ(x))†.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 19: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/19.jpg)
Covariant integral quantisation - homogeneous spaces
We consider the quantisation of functions on a homogeneousspace X , the left coset manifold X ∼ G/H for the action of aLie Group G , where the closed subgroup H is the stabilizer ofsome point of X .
The interesting case is when X is a symplectic manifold (e.g.,co-adjoint orbit of G ) and can be viewed as the phase spacefor the dynamics.
Given a quasi-invariant measure ν on X , one has for a globalBorel section σ : X → G a unique quasi-invariant measureνσ(x).
Let U be a square-integrable UIR, and ρ a density operatorsuch that cρ :=
∫X tr (ρ ρσ(x)) dνσ(x) <∞ with
ρσ(x) := U(σ(x))ρU(σ(x))†.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 20: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/20.jpg)
One has the resolution of the identity
I =1
cρ
∫Xρσ(x) dνσ(x) .
We then define the quantisation of functions on X as thelinear map
f 7→ Aσf =1
cρ
∫X
f (x) ρσ(x)dνσ(x) .
Covariance holds in the sense U(g)Aσf U(g)† = AσgUl (g)f , where
σg (x) = gσ(g−1x) with Ul(g)f (x) = f(g−1x
).
For ρ = |η〉〈η|, we are working with CS quantisation, wherethe CS’s are defined as |ηx〉 := |U(σg (x))η〉.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 21: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/21.jpg)
One has the resolution of the identity
I =1
cρ
∫Xρσ(x) dνσ(x) .
We then define the quantisation of functions on X as thelinear map
f 7→ Aσf =1
cρ
∫X
f (x) ρσ(x)dνσ(x) .
Covariance holds in the sense U(g)Aσf U(g)† = AσgUl (g)f , where
σg (x) = gσ(g−1x) with Ul(g)f (x) = f(g−1x
).
For ρ = |η〉〈η|, we are working with CS quantisation, wherethe CS’s are defined as |ηx〉 := |U(σg (x))η〉.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 22: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/22.jpg)
One has the resolution of the identity
I =1
cρ
∫Xρσ(x) dνσ(x) .
We then define the quantisation of functions on X as thelinear map
f 7→ Aσf =1
cρ
∫X
f (x) ρσ(x)dνσ(x) .
Covariance holds in the sense U(g)Aσf U(g)† = AσgUl (g)f , where
σg (x) = gσ(g−1x) with Ul(g)f (x) = f(g−1x
).
For ρ = |η〉〈η|, we are working with CS quantisation, wherethe CS’s are defined as |ηx〉 := |U(σg (x))η〉.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 23: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/23.jpg)
Coherent states for semi-direct product groups
Let V , dimV = n, S ≤ GL(V ) and G = V o S
Given k0 ∈ V ∗ , one can show that
H0 = {g ∈ G |(k0, 0) = Ad#g (k0, 0)} = N0 o S0
for (k0, 0) ∈ g∗ and N0 = {x ∈ V :< p; x >= 0 ∀p ∈ T ∗k0O∗}
Furthermore, X = G/H0 ' T ∗O∗ admits a symplecticstructure.
Finally, given a UIR χ of V and a UIR L of S , one canconstruct an irreducible representation (v , s) 7→ UχL (v , s) ofG induced by the representation χ⊗ L of V o S0.
Given η ∈ H = L2(O∗, dν), one constructs a family ηp,q:ηp,q(k) =
(UχL (σ(p,q))η
)(k)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Coherent states for semi-direct product groups
Let V , dimV = n, S ≤ GL(V ) and G = V o S
Given k0 ∈ V ∗ , one can show that
H0 = {g ∈ G |(k0, 0) = Ad#g (k0, 0)} = N0 o S0
for (k0, 0) ∈ g∗ and N0 = {x ∈ V :< p; x >= 0 ∀p ∈ T ∗k0O∗}
Furthermore, X = G/H0 ' T ∗O∗ admits a symplecticstructure.
Finally, given a UIR χ of V and a UIR L of S , one canconstruct an irreducible representation (v , s) 7→ UχL (v , s) ofG induced by the representation χ⊗ L of V o S0.
Given η ∈ H = L2(O∗, dν), one constructs a family ηp,q:ηp,q(k) =
(UχL (σ(p,q))η
)(k)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 25: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/25.jpg)
Coherent states for semi-direct product groups
Let V , dimV = n, S ≤ GL(V ) and G = V o S
Given k0 ∈ V ∗ , one can show that
H0 = {g ∈ G |(k0, 0) = Ad#g (k0, 0)} = N0 o S0
for (k0, 0) ∈ g∗ and N0 = {x ∈ V :< p; x >= 0 ∀p ∈ T ∗k0O∗}
Furthermore, X = G/H0 ' T ∗O∗ admits a symplecticstructure.
Finally, given a UIR χ of V and a UIR L of S , one canconstruct an irreducible representation (v , s) 7→ UχL (v , s) ofG induced by the representation χ⊗ L of V o S0.
Given η ∈ H = L2(O∗, dν), one constructs a family ηp,q:ηp,q(k) =
(UχL (σ(p,q))η
)(k)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Coherent states for semi-direct product groups
Let V , dimV = n, S ≤ GL(V ) and G = V o S
Given k0 ∈ V ∗ , one can show that
H0 = {g ∈ G |(k0, 0) = Ad#g (k0, 0)} = N0 o S0
for (k0, 0) ∈ g∗ and N0 = {x ∈ V :< p; x >= 0 ∀p ∈ T ∗k0O∗}
Furthermore, X = G/H0 ' T ∗O∗ admits a symplecticstructure.
Finally, given a UIR χ of V and a UIR L of S , one canconstruct an irreducible representation (v , s) 7→ UχL (v , s) ofG induced by the representation χ⊗ L of V o S0.
Given η ∈ H = L2(O∗, dν), one constructs a family ηp,q:ηp,q(k) =
(UχL (σ(p,q))η
)(k)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 27: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/27.jpg)
Coherent states for semi-direct product groups
Let V , dimV = n, S ≤ GL(V ) and G = V o S
Given k0 ∈ V ∗ , one can show that
H0 = {g ∈ G |(k0, 0) = Ad#g (k0, 0)} = N0 o S0
for (k0, 0) ∈ g∗ and N0 = {x ∈ V :< p; x >= 0 ∀p ∈ T ∗k0O∗}
Furthermore, X = G/H0 ' T ∗O∗ admits a symplecticstructure.
Finally, given a UIR χ of V and a UIR L of S , one canconstruct an irreducible representation (v , s) 7→ UχL (v , s) ofG induced by the representation χ⊗ L of V o S0.
Given η ∈ H = L2(O∗, dν), one constructs a family ηp,q:ηp,q(k) =
(UχL (σ(p,q))η
)(k)
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 28: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/28.jpg)
Coherent-state quantisation for semi-simple Lie groups
If one can prove∫X dµ(p,q)〈φ |ηp,q 〉H〈ηp,q |ψ 〉H = cη〈φ |ψ 〉 whereφ, ψ : O∗ → C and 0 < cη <∞
we obtain the resolution of the identity1
cη
∫X dµ(p,q)|ηp,q 〉〈ηp,q | = I
CS quantisation maps the classical function f (p,q) ∈ X to
the operator on H, Af =1
cη
∫X dµ(p,q) |ηp,q〉 〈ηp,q | f (p,q)
The quantisation is covariant UχL (g)Af UχL (g)† =
AσgUl (g)f , A
σgf :=
1
cη
∫X dµ(p,q)
∣∣ησgp,q⟩ ⟨ησgp,q∣∣ f (p,q) , with
|ησgp,q 〉 = UχL (gσ(g−1(p,q)))|η 〉The semiclassical portrait of the operator Af is defined as
f (p,q) =1
cη
∫X dµ(p′,q ′)f (p′,q ′)
∣∣⟨ηp′,q′ |ηp,q⟩∣∣2 .
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Coherent-state quantisation for semi-simple Lie groups
If one can prove∫X dµ(p,q)〈φ |ηp,q 〉H〈ηp,q |ψ 〉H = cη〈φ |ψ 〉 whereφ, ψ : O∗ → C and 0 < cη <∞we obtain the resolution of the identity1
cη
∫X dµ(p,q)|ηp,q 〉〈ηp,q | = I
CS quantisation maps the classical function f (p,q) ∈ X to
the operator on H, Af =1
cη
∫X dµ(p,q) |ηp,q〉 〈ηp,q | f (p,q)
The quantisation is covariant UχL (g)Af UχL (g)† =
AσgUl (g)f , A
σgf :=
1
cη
∫X dµ(p,q)
∣∣ησgp,q⟩ ⟨ησgp,q∣∣ f (p,q) , with
|ησgp,q 〉 = UχL (gσ(g−1(p,q)))|η 〉The semiclassical portrait of the operator Af is defined as
f (p,q) =1
cη
∫X dµ(p′,q ′)f (p′,q ′)
∣∣⟨ηp′,q′ |ηp,q⟩∣∣2 .
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Coherent-state quantisation for semi-simple Lie groups
If one can prove∫X dµ(p,q)〈φ |ηp,q 〉H〈ηp,q |ψ 〉H = cη〈φ |ψ 〉 whereφ, ψ : O∗ → C and 0 < cη <∞we obtain the resolution of the identity1
cη
∫X dµ(p,q)|ηp,q 〉〈ηp,q | = I
CS quantisation maps the classical function f (p,q) ∈ X to
the operator on H, Af =1
cη
∫X dµ(p,q) |ηp,q〉 〈ηp,q | f (p,q)
The quantisation is covariant UχL (g)Af UχL (g)† =
AσgUl (g)f , A
σgf :=
1
cη
∫X dµ(p,q)
∣∣ησgp,q⟩ ⟨ησgp,q∣∣ f (p,q) , with
|ησgp,q 〉 = UχL (gσ(g−1(p,q)))|η 〉The semiclassical portrait of the operator Af is defined as
f (p,q) =1
cη
∫X dµ(p′,q ′)f (p′,q ′)
∣∣⟨ηp′,q′ |ηp,q⟩∣∣2 .
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Coherent-state quantisation for semi-simple Lie groups
If one can prove∫X dµ(p,q)〈φ |ηp,q 〉H〈ηp,q |ψ 〉H = cη〈φ |ψ 〉 whereφ, ψ : O∗ → C and 0 < cη <∞we obtain the resolution of the identity1
cη
∫X dµ(p,q)|ηp,q 〉〈ηp,q | = I
CS quantisation maps the classical function f (p,q) ∈ X to
the operator on H, Af =1
cη
∫X dµ(p,q) |ηp,q〉 〈ηp,q | f (p,q)
The quantisation is covariant UχL (g)Af UχL (g)† =
AσgUl (g)f , A
σgf :=
1
cη
∫X dµ(p,q)
∣∣ησgp,q⟩ ⟨ησgp,q∣∣ f (p,q) , with
|ησgp,q 〉 = UχL (gσ(g−1(p,q)))|η 〉
The semiclassical portrait of the operator Af is defined as
f (p,q) =1
cη
∫X dµ(p′,q ′)f (p′,q ′)
∣∣⟨ηp′,q′ |ηp,q⟩∣∣2 .
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 32: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/32.jpg)
Coherent-state quantisation for semi-simple Lie groups
If one can prove∫X dµ(p,q)〈φ |ηp,q 〉H〈ηp,q |ψ 〉H = cη〈φ |ψ 〉 whereφ, ψ : O∗ → C and 0 < cη <∞we obtain the resolution of the identity1
cη
∫X dµ(p,q)|ηp,q 〉〈ηp,q | = I
CS quantisation maps the classical function f (p,q) ∈ X to
the operator on H, Af =1
cη
∫X dµ(p,q) |ηp,q〉 〈ηp,q | f (p,q)
The quantisation is covariant UχL (g)Af UχL (g)† =
AσgUl (g)f , A
σgf :=
1
cη
∫X dµ(p,q)
∣∣ησgp,q⟩ ⟨ησgp,q∣∣ f (p,q) , with
|ησgp,q 〉 = UχL (gσ(g−1(p,q)))|η 〉The semiclassical portrait of the operator Af is defined as
f (p,q) =1
cη
∫X dµ(p′,q ′)f (p′,q ′)
∣∣⟨ηp′,q′ |ηp,q⟩∣∣2 .
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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The Euclidean group E(2)
Now G = E(2), where V = R2 and S = SO(2), soE(2) = R2 o SO(2) = {(r , θ) , r ∈ R2 , θ ∈ [0, 2π)} , withcomposition (r , θ)(r ′, θ′) = (r +R(θ)r ′, θ + θ′).
V ∗ = R2, O∗ = {k = R(θ)k0 ∈ R2 |R(θ) ∈ SO(2)} ' S1
The stabilizer under the coadjoint action Ad#E(2) is
H0 ={
(x , 0) ∈ E(2) | c · x = 0, c ∈ R2, ‖c‖ = 1, fixed}
.
The classical phase space X ≡ T ∗S1 ' R× S1 carriescoordinates (p, q) and has symplectic measure dp ∧ dq.
The UIR of E(2) are L2(S1,dα) 3 ψ(α) 7→ (U(r , θ)ψ) (α) =e i(r1 cosα+r2 sinα)ψ(α− θ).
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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The Euclidean group E(2)
Now G = E(2), where V = R2 and S = SO(2), soE(2) = R2 o SO(2) = {(r , θ) , r ∈ R2 , θ ∈ [0, 2π)} , withcomposition (r , θ)(r ′, θ′) = (r +R(θ)r ′, θ + θ′).
V ∗ = R2, O∗ = {k = R(θ)k0 ∈ R2 |R(θ) ∈ SO(2)} ' S1
The stabilizer under the coadjoint action Ad#E(2) is
H0 ={
(x , 0) ∈ E(2) | c · x = 0, c ∈ R2, ‖c‖ = 1, fixed}
.
The classical phase space X ≡ T ∗S1 ' R× S1 carriescoordinates (p, q) and has symplectic measure dp ∧ dq.
The UIR of E(2) are L2(S1,dα) 3 ψ(α) 7→ (U(r , θ)ψ) (α) =e i(r1 cosα+r2 sinα)ψ(α− θ).
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 35: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/35.jpg)
The Euclidean group E(2)
Now G = E(2), where V = R2 and S = SO(2), soE(2) = R2 o SO(2) = {(r , θ) , r ∈ R2 , θ ∈ [0, 2π)} , withcomposition (r , θ)(r ′, θ′) = (r +R(θ)r ′, θ + θ′).
V ∗ = R2, O∗ = {k = R(θ)k0 ∈ R2 |R(θ) ∈ SO(2)} ' S1
The stabilizer under the coadjoint action Ad#E(2) is
H0 ={
(x , 0) ∈ E(2) | c · x = 0, c ∈ R2, ‖c‖ = 1, fixed}
.
The classical phase space X ≡ T ∗S1 ' R× S1 carriescoordinates (p, q) and has symplectic measure dp ∧ dq.
The UIR of E(2) are L2(S1,dα) 3 ψ(α) 7→ (U(r , θ)ψ) (α) =e i(r1 cosα+r2 sinα)ψ(α− θ).
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 36: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/36.jpg)
The Euclidean group E(2)
Now G = E(2), where V = R2 and S = SO(2), soE(2) = R2 o SO(2) = {(r , θ) , r ∈ R2 , θ ∈ [0, 2π)} , withcomposition (r , θ)(r ′, θ′) = (r +R(θ)r ′, θ + θ′).
V ∗ = R2, O∗ = {k = R(θ)k0 ∈ R2 |R(θ) ∈ SO(2)} ' S1
The stabilizer under the coadjoint action Ad#E(2) is
H0 ={
(x , 0) ∈ E(2) | c · x = 0, c ∈ R2, ‖c‖ = 1, fixed}
.
The classical phase space X ≡ T ∗S1 ' R× S1 carriescoordinates (p, q) and has symplectic measure dp ∧ dq.
The UIR of E(2) are L2(S1,dα) 3 ψ(α) 7→ (U(r , θ)ψ) (α) =e i(r1 cosα+r2 sinα)ψ(α− θ).
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 37: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/37.jpg)
The Euclidean group E(2)
Now G = E(2), where V = R2 and S = SO(2), soE(2) = R2 o SO(2) = {(r , θ) , r ∈ R2 , θ ∈ [0, 2π)} , withcomposition (r , θ)(r ′, θ′) = (r +R(θ)r ′, θ + θ′).
V ∗ = R2, O∗ = {k = R(θ)k0 ∈ R2 |R(θ) ∈ SO(2)} ' S1
The stabilizer under the coadjoint action Ad#E(2) is
H0 ={
(x , 0) ∈ E(2) | c · x = 0, c ∈ R2, ‖c‖ = 1, fixed}
.
The classical phase space X ≡ T ∗S1 ' R× S1 carriescoordinates (p, q) and has symplectic measure dp ∧ dq.
The UIR of E(2) are L2(S1,dα) 3 ψ(α) 7→ (U(r , θ)ψ) (α) =e i(r1 cosα+r2 sinα)ψ(α− θ).
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 38: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/38.jpg)
Coherent states for E(2)
Theorem
Given the unit vector c ∈ R2 and the corresponding subgroup H0,there exists a family of affine sections σ : R× S1 → E(2) definedas σ(p, q) = (R(q)(κp + λ), q) , where κ,λ ∈ R2 are constantvectors, and c · κ 6= 0.
From the section σ(p, q), the representation U(r, θ), and a vectorη ∈ L2(S1, dα), we define the family of states|ηp,q〉 = U(σ(p, q))|η〉.Theorem
The vectors ηp,q form a family of coherent states for E(2) which
resolves the identity on L2(S1,dα), I =∫R×S1
dp dq
cη|ηp,q 〉〈ηp,q | ,
if η(α) is admissible in the sense that supp η ∈ (γ − π, γ)mod 2π,
and 0 < cη :=2π
κ
∫S1|η(q)|2
sin(γ − q)dq <∞.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Coherent states for E(2)
Theorem
Given the unit vector c ∈ R2 and the corresponding subgroup H0,there exists a family of affine sections σ : R× S1 → E(2) definedas σ(p, q) = (R(q)(κp + λ), q) , where κ,λ ∈ R2 are constantvectors, and c · κ 6= 0.
From the section σ(p, q), the representation U(r, θ), and a vectorη ∈ L2(S1, dα), we define the family of states|ηp,q〉 = U(σ(p, q))|η〉.
Theorem
The vectors ηp,q form a family of coherent states for E(2) which
resolves the identity on L2(S1,dα), I =∫R×S1
dp dq
cη|ηp,q 〉〈ηp,q | ,
if η(α) is admissible in the sense that supp η ∈ (γ − π, γ)mod 2π,
and 0 < cη :=2π
κ
∫S1|η(q)|2
sin(γ − q)dq <∞.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 40: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/40.jpg)
Coherent states for E(2)
Theorem
Given the unit vector c ∈ R2 and the corresponding subgroup H0,there exists a family of affine sections σ : R× S1 → E(2) definedas σ(p, q) = (R(q)(κp + λ), q) , where κ,λ ∈ R2 are constantvectors, and c · κ 6= 0.
From the section σ(p, q), the representation U(r, θ), and a vectorη ∈ L2(S1, dα), we define the family of states|ηp,q〉 = U(σ(p, q))|η〉.Theorem
The vectors ηp,q form a family of coherent states for E(2) which
resolves the identity on L2(S1,dα), I =∫R×S1
dp dq
cη|ηp,q 〉〈ηp,q | ,
if η(α) is admissible in the sense that supp η ∈ (γ − π, γ)mod 2π,
and 0 < cη :=2π
κ
∫S1|η(q)|2
sin(γ − q)dq <∞.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Quantisation of classical variables
Given the family of coherent states |ηp,q〉, we apply the linear
map f 7→ Aσf =∫R×S1
dp dq
cηf (p, q) |ηp,q 〉〈ηp,q | to classical
observables f (p, q).
For f (p, q) = u(q) with u(q + 2π) = u(q), Au is themultiplication operator (Auψ)(α) = (Eη;γ ∗ u) (α)ψ(α) where
Eη;γ(α) := 2πκcη
|η(α)|2sin(γ−α) , suppEη;γ ⊂ (γ − π, γ) is a probability
distribution on the interval [−π, π].
For the momentum f (p, q) = p,
(Apψ) (α) =
(−i ∂∂α− λa
)ψ (α) for real η.
For a general polynomial f (q, p) =∑N
k=0 uk(q) pk one gets∑Nk=0 ak(α)(−i∂α)k .
The classical limit is obtained by considering the large κ limitwith the condition |η(α)|2 → δ(α− γ + π/2). Thenf (q, p) = f (q, p) + o(1/κ).
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Quantisation of classical variables
Given the family of coherent states |ηp,q〉, we apply the linear
map f 7→ Aσf =∫R×S1
dp dq
cηf (p, q) |ηp,q 〉〈ηp,q | to classical
observables f (p, q).
For f (p, q) = u(q) with u(q + 2π) = u(q), Au is themultiplication operator (Auψ)(α) = (Eη;γ ∗ u) (α)ψ(α) where
Eη;γ(α) := 2πκcη
|η(α)|2sin(γ−α) , suppEη;γ ⊂ (γ − π, γ) is a probability
distribution on the interval [−π, π].
For the momentum f (p, q) = p,
(Apψ) (α) =
(−i ∂∂α− λa
)ψ (α) for real η.
For a general polynomial f (q, p) =∑N
k=0 uk(q) pk one gets∑Nk=0 ak(α)(−i∂α)k .
The classical limit is obtained by considering the large κ limitwith the condition |η(α)|2 → δ(α− γ + π/2). Thenf (q, p) = f (q, p) + o(1/κ).
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Quantisation of classical variables
Given the family of coherent states |ηp,q〉, we apply the linear
map f 7→ Aσf =∫R×S1
dp dq
cηf (p, q) |ηp,q 〉〈ηp,q | to classical
observables f (p, q).
For f (p, q) = u(q) with u(q + 2π) = u(q), Au is themultiplication operator (Auψ)(α) = (Eη;γ ∗ u) (α)ψ(α) where
Eη;γ(α) := 2πκcη
|η(α)|2sin(γ−α) , suppEη;γ ⊂ (γ − π, γ) is a probability
distribution on the interval [−π, π].
For the momentum f (p, q) = p,
(Apψ) (α) =
(−i ∂∂α− λa
)ψ (α) for real η.
For a general polynomial f (q, p) =∑N
k=0 uk(q) pk one gets∑Nk=0 ak(α)(−i∂α)k .
The classical limit is obtained by considering the large κ limitwith the condition |η(α)|2 → δ(α− γ + π/2). Thenf (q, p) = f (q, p) + o(1/κ).
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Quantisation of classical variables
Given the family of coherent states |ηp,q〉, we apply the linear
map f 7→ Aσf =∫R×S1
dp dq
cηf (p, q) |ηp,q 〉〈ηp,q | to classical
observables f (p, q).
For f (p, q) = u(q) with u(q + 2π) = u(q), Au is themultiplication operator (Auψ)(α) = (Eη;γ ∗ u) (α)ψ(α) where
Eη;γ(α) := 2πκcη
|η(α)|2sin(γ−α) , suppEη;γ ⊂ (γ − π, γ) is a probability
distribution on the interval [−π, π].
For the momentum f (p, q) = p,
(Apψ) (α) =
(−i ∂∂α− λa
)ψ (α) for real η.
For a general polynomial f (q, p) =∑N
k=0 uk(q) pk one gets∑Nk=0 ak(α)(−i∂α)k .
The classical limit is obtained by considering the large κ limitwith the condition |η(α)|2 → δ(α− γ + π/2). Thenf (q, p) = f (q, p) + o(1/κ).
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 45: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/45.jpg)
Quantisation of classical variables
Given the family of coherent states |ηp,q〉, we apply the linear
map f 7→ Aσf =∫R×S1
dp dq
cηf (p, q) |ηp,q 〉〈ηp,q | to classical
observables f (p, q).
For f (p, q) = u(q) with u(q + 2π) = u(q), Au is themultiplication operator (Auψ)(α) = (Eη;γ ∗ u) (α)ψ(α) where
Eη;γ(α) := 2πκcη
|η(α)|2sin(γ−α) , suppEη;γ ⊂ (γ − π, γ) is a probability
distribution on the interval [−π, π].
For the momentum f (p, q) = p,
(Apψ) (α) =
(−i ∂∂α− λa
)ψ (α) for real η.
For a general polynomial f (q, p) =∑N
k=0 uk(q) pk one gets∑Nk=0 ak(α)(−i∂α)k .
The classical limit is obtained by considering the large κ limitwith the condition |η(α)|2 → δ(α− γ + π/2). Thenf (q, p) = f (q, p) + o(1/κ).
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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the Angle operator: analytic and numerical results
For the 2π-periodic and discontinuous angle functiona(α) = α for α ∈ [0, 2π), we get the multiplication operator(Eη,γ ∗a)(α) = α+2π(1−
∫ α−π Eη;γ(q)dq)−
∫ γγ−π q Eη,γ(q)dq .
We choose a specific section with λ = 0, γ = π/2 and asfiducial vectors the family η(s,ε)(α) of periodic smooth evenfunctions, suppη = [−ε, ε] mod 2π, parametrized by s > 0 and0 < ε < π/2,
η(s,ε)(α) =1√εe2s
ωs
(αε
)where es :=
∫ 1
−1dx ωs(x) .
and
ωs(x) =
exp
(− s
1− x2
)0 ≤ |x | < 1 ,
0 |x | ≥ 1 ,
are smooth and compactly supported test functions.(η(s,ε)
)2(α)→ δ(α) as ε→ 0 or as s →∞.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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the Angle operator: analytic and numerical results
For the 2π-periodic and discontinuous angle functiona(α) = α for α ∈ [0, 2π), we get the multiplication operator(Eη,γ ∗a)(α) = α+2π(1−
∫ α−π Eη;γ(q)dq)−
∫ γγ−π q Eη,γ(q)dq .
We choose a specific section with λ = 0, γ = π/2 and asfiducial vectors the family η(s,ε)(α) of periodic smooth evenfunctions, suppη = [−ε, ε] mod 2π, parametrized by s > 0 and0 < ε < π/2,
η(s,ε)(α) =1√εe2s
ωs
(αε
)where es :=
∫ 1
−1dx ωs(x) .
and
ωs(x) =
exp
(− s
1− x2
)0 ≤ |x | < 1 ,
0 |x | ≥ 1 ,
are smooth and compactly supported test functions.
(η(s,ε)
)2(α)→ δ(α) as ε→ 0 or as s →∞.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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the Angle operator: analytic and numerical results
For the 2π-periodic and discontinuous angle functiona(α) = α for α ∈ [0, 2π), we get the multiplication operator(Eη,γ ∗a)(α) = α+2π(1−
∫ α−π Eη;γ(q)dq)−
∫ γγ−π q Eη,γ(q)dq .
We choose a specific section with λ = 0, γ = π/2 and asfiducial vectors the family η(s,ε)(α) of periodic smooth evenfunctions, suppη = [−ε, ε] mod 2π, parametrized by s > 0 and0 < ε < π/2,
η(s,ε)(α) =1√εe2s
ωs
(αε
)where es :=
∫ 1
−1dx ωs(x) .
and
ωs(x) =
exp
(− s
1− x2
)0 ≤ |x | < 1 ,
0 |x | ≥ 1 ,
are smooth and compactly supported test functions.(η(s,ε)
)2(α)→ δ(α) as ε→ 0 or as s →∞.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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α−0.6 −0.3 0 0.3 0.6
η(s,ε)
0
2.5
5
7.5
10
ε = 0.3s = 2
ε = 0.4s = 3.5555
(a) τ = 22.22
α−0.6 −0.3 0 0.3 0.6
η(s,ε)
0
2.5
5
7.5
10
ε = 0.45s = 16.5306
ε = 0.35s = 10
(b) τ = 81.63
Figure: Plots of η(s,ε) for various values of τ =s
ε2.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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α0 π
23π2
2ππ
Eη(s,ε);π2∗ a
0
π2
π
3π2
2π
ε = 0.3s = 2
ε = 0.4s = 3.5555
(a) τ = 22.22
α0 π
23π2
2ππ
Eη(s,ε);π2∗ a
0
π2
π
3π2
2π
ε = 0.35s = 10
ε = 0.45s = 16.5306
(b) τ = 81.63
Figure: Plots of(Eη(s,ε);π2 ∗ a
)(α) for various values of τ =
s
ε2.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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q0 π
23π2
2ππ
q
0
π2
π
3π2
2π
ε = 0.3s = 2
ε = 0.4s = 3.5555
(a) τ = 22.22
q0 π
23π2
2ππ
q
0
π2
π
3π2
2π
ε = 0.35s = 10
ε = 0.45s = 16.5306
(b) τ = 81.63
Figure: Plots of the lower symbol q(q) of the angle operator Aa for
various values of τ =s
ε2.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Angle-angular momentum: commutation relations andHeisenberg inequality
For λ = 0 and ψ(α) ∈ L2(S1, dα), we find the non-canonicalCR ([Ap,Aa]ψ) (α) = −i (1− 2πEη;γ(α))ψ(α)
Since limε→0 Eη;γ(α) = δ(α), one has, in the limit ε→ 0, forα ∈ [0, 2π) mod 2π, ([Ap,Aa]ψ) (α) = (−i + i2πδ(α))ψ(α).
The uncertainty relation for Ap and Aa, with the coherentstates ηp,q, is ∆Ap ∆Aa > 1
2 |〈ηp,q |[Ap,Aa]|ηp,q 〉|.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Angle-angular momentum: commutation relations andHeisenberg inequality
For λ = 0 and ψ(α) ∈ L2(S1, dα), we find the non-canonicalCR ([Ap,Aa]ψ) (α) = −i (1− 2πEη;γ(α))ψ(α)
Since limε→0 Eη;γ(α) = δ(α), one has, in the limit ε→ 0, forα ∈ [0, 2π) mod 2π, ([Ap,Aa]ψ) (α) = (−i + i2πδ(α))ψ(α).
The uncertainty relation for Ap and Aa, with the coherentstates ηp,q, is ∆Ap ∆Aa > 1
2 |〈ηp,q |[Ap,Aa]|ηp,q 〉|.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Angle-angular momentum: commutation relations andHeisenberg inequality
For λ = 0 and ψ(α) ∈ L2(S1, dα), we find the non-canonicalCR ([Ap,Aa]ψ) (α) = −i (1− 2πEη;γ(α))ψ(α)
Since limε→0 Eη;γ(α) = δ(α), one has, in the limit ε→ 0, forα ∈ [0, 2π) mod 2π, ([Ap,Aa]ψ) (α) = (−i + i2πδ(α))ψ(α).
The uncertainty relation for Ap and Aa, with the coherentstates ηp,q, is ∆Ap ∆Aa > 1
2 |〈ηp,q |[Ap,Aa]|ηp,q 〉|.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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q0 π
23π2
2ππ
∆Aa
0
0.5
1
1.5
2
2.5
3
3.5
ε = 0.3
s = 2
ε = 0.4
τ = 3.5555
(a) τ = 22.22
q0 π
23π2
2ππ
∆Aa
0
0.5
1
1.5
2
2.5
3
3.5
ε = 0.35
s = 10
ε = 0.45
s = 16.5306
(b) τ = 81.63
p0 5 10 15 20 25
∆Ap
0
15
30
45
60
75
90
105
120
ε = 0.3
s = 2
ε = 0.4
s = 3.5555
(c) τ = 22.22
p0 5 10 15 20 25
∆Ap
0
15
30
45
60
75
90
105
120
ε = 0.35
s = 10
ε = 0.45
s = 16.5306
(d) τ = 81.63
Figure: Plots of the dispersions ∆Aa and ∆Ap with respect to the
coherent state |η(s,ε)p,q 〉 for various values of τ =s
ε2.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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L.H.S.-R.H.S.
q0 π
23π2
2ππ0
0.5
1
1.5
2
s = 2; ε = 0.3
s = 3.5555; ε = 0.4
(a) τ = 22.22
L.H.S.-R.H.S.
q0 π
23π2
2ππ0
0.5
1
1.5
2
s = 10; ε = 0.35
s = 16.5306; ε = 0.45
(b) τ = 81.63
L.H.S.-R.H.S.
τ5 30 60 90 110
0
0.05
0.1
0.15
0.2
ε = 0.3
ε = 0.4
(c) q = 1
L.H.S.-R.H.S.
τ5 30 60 90 110
0
0.05
0.1
0.15
0.2
ε = 0.35
ε = 0.45
(d) q = 1
Figure: Plots of the difference L.H.S.-R.H.S. of the uncertainty relation
with respect to the coherent state |η(s,ε)p,q 〉 for various values of τ = sε2 .
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Conclusions
We have presented a picture of a quantisation based on the resolutionof the identity provided by coherent states for the special Euclideangroup E(2).
The cylinder R× S1 depicts the classical phase space of the motion ofa particle on a circle, and is mathematically realized as the left cosetE(2)/H, where H is a stabilizer subgroup under the coadjoint actionof E(2).The coherent states for E(2) are constructed from a UIR ofE(2) = R2 o SO(2) restricted to an affine sectionR× S1 3 (p, q) 7→ σ(p, q) ∈ E(2).For functions on the cylindric phase space, the correspondingoperators and lower symbols are determined . For periodic functionsf (q) of the angular coordinate q, the operators Af are multiplicationoperators whose spectra are given by periodic functions.The angle function a(α) = α is mapped to a SA multiplication angleoperator Aa with continuous spectrum.The translation covariance mod 2π U(θ)Aσf U
†(θ) = Aσf (·−θ) is aquantum expression of the transition map between different charts onthe circle.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Conclusions
We have presented a picture of a quantisation based on the resolutionof the identity provided by coherent states for the special Euclideangroup E(2).The cylinder R× S1 depicts the classical phase space of the motion ofa particle on a circle, and is mathematically realized as the left cosetE(2)/H, where H is a stabilizer subgroup under the coadjoint actionof E(2).
The coherent states for E(2) are constructed from a UIR ofE(2) = R2 o SO(2) restricted to an affine sectionR× S1 3 (p, q) 7→ σ(p, q) ∈ E(2).For functions on the cylindric phase space, the correspondingoperators and lower symbols are determined . For periodic functionsf (q) of the angular coordinate q, the operators Af are multiplicationoperators whose spectra are given by periodic functions.The angle function a(α) = α is mapped to a SA multiplication angleoperator Aa with continuous spectrum.The translation covariance mod 2π U(θ)Aσf U
†(θ) = Aσf (·−θ) is aquantum expression of the transition map between different charts onthe circle.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Conclusions
We have presented a picture of a quantisation based on the resolutionof the identity provided by coherent states for the special Euclideangroup E(2).The cylinder R× S1 depicts the classical phase space of the motion ofa particle on a circle, and is mathematically realized as the left cosetE(2)/H, where H is a stabilizer subgroup under the coadjoint actionof E(2).The coherent states for E(2) are constructed from a UIR ofE(2) = R2 o SO(2) restricted to an affine sectionR× S1 3 (p, q) 7→ σ(p, q) ∈ E(2).
For functions on the cylindric phase space, the correspondingoperators and lower symbols are determined . For periodic functionsf (q) of the angular coordinate q, the operators Af are multiplicationoperators whose spectra are given by periodic functions.The angle function a(α) = α is mapped to a SA multiplication angleoperator Aa with continuous spectrum.The translation covariance mod 2π U(θ)Aσf U
†(θ) = Aσf (·−θ) is aquantum expression of the transition map between different charts onthe circle.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
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Conclusions
We have presented a picture of a quantisation based on the resolutionof the identity provided by coherent states for the special Euclideangroup E(2).The cylinder R× S1 depicts the classical phase space of the motion ofa particle on a circle, and is mathematically realized as the left cosetE(2)/H, where H is a stabilizer subgroup under the coadjoint actionof E(2).The coherent states for E(2) are constructed from a UIR ofE(2) = R2 o SO(2) restricted to an affine sectionR× S1 3 (p, q) 7→ σ(p, q) ∈ E(2).For functions on the cylindric phase space, the correspondingoperators and lower symbols are determined . For periodic functionsf (q) of the angular coordinate q, the operators Af are multiplicationoperators whose spectra are given by periodic functions.
The angle function a(α) = α is mapped to a SA multiplication angleoperator Aa with continuous spectrum.The translation covariance mod 2π U(θ)Aσf U
†(θ) = Aσf (·−θ) is aquantum expression of the transition map between different charts onthe circle.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 61: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/61.jpg)
Conclusions
We have presented a picture of a quantisation based on the resolutionof the identity provided by coherent states for the special Euclideangroup E(2).The cylinder R× S1 depicts the classical phase space of the motion ofa particle on a circle, and is mathematically realized as the left cosetE(2)/H, where H is a stabilizer subgroup under the coadjoint actionof E(2).The coherent states for E(2) are constructed from a UIR ofE(2) = R2 o SO(2) restricted to an affine sectionR× S1 3 (p, q) 7→ σ(p, q) ∈ E(2).For functions on the cylindric phase space, the correspondingoperators and lower symbols are determined . For periodic functionsf (q) of the angular coordinate q, the operators Af are multiplicationoperators whose spectra are given by periodic functions.The angle function a(α) = α is mapped to a SA multiplication angleoperator Aa with continuous spectrum.
The translation covariance mod 2π U(θ)Aσf U†(θ) = Aσf (·−θ) is a
quantum expression of the transition map between different charts onthe circle.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 62: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/62.jpg)
Conclusions
We have presented a picture of a quantisation based on the resolutionof the identity provided by coherent states for the special Euclideangroup E(2).The cylinder R× S1 depicts the classical phase space of the motion ofa particle on a circle, and is mathematically realized as the left cosetE(2)/H, where H is a stabilizer subgroup under the coadjoint actionof E(2).The coherent states for E(2) are constructed from a UIR ofE(2) = R2 o SO(2) restricted to an affine sectionR× S1 3 (p, q) 7→ σ(p, q) ∈ E(2).For functions on the cylindric phase space, the correspondingoperators and lower symbols are determined . For periodic functionsf (q) of the angular coordinate q, the operators Af are multiplicationoperators whose spectra are given by periodic functions.The angle function a(α) = α is mapped to a SA multiplication angleoperator Aa with continuous spectrum.The translation covariance mod 2π U(θ)Aσf U
†(θ) = Aσf (·−θ) is aquantum expression of the transition map between different charts onthe circle.
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle
![Page 63: Quantum Localisation on the Circlebenasque.org/2019qmis/talks_contr/036_RodrigoFresneda.pdf · Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations). Our](https://reader033.fdocuments.net/reader033/viewer/2022060217/5f065e9d7e708231d417a6ee/html5/thumbnails/63.jpg)
obrigado!
Figure: UFABC Campus in Santo Andre, Sao Paulo
Rodrigo Fresneda (UFABC - Sao Paulo, Brasil) Quantum Localisation on the Circle