Wigner Functions for the Canonical Pair Angle and Orbital...

Wigner Functions for the Canonical PairAngle and Orbital Angular Momentum

Hans Kastrup

Theory Group

DESY Hamburg

Institute for Theoretical Physics

RWTH Aachen

2. Intern. Wigner Workshop, June 5, 2017

HKa, (PRA 94, 062113 (2016) {arXiv:1601.02520}; PRA 95, 052111 (2017){arXiv:1702.05615}) IW2, June 5, 2017 1 / 30

Outline

Wigner functions for S1 × R: angle and orbital angular momentum

Previous difficulties with Wigner functions on Pθ,p ∼= S1 × RSolution of the quantum angle problemEuclidean group E(2) of the planeWigner operator on Pθ,pWigner function of wave functions ψ(ϕ)ExamplesConclusions and OutlookA personal tribute to Eugene Wigner


Wigner functions for Pθ,p

Wigner functions for S1 × R:angle and orbital angular momentum

Typical example: Rotator around a fixed axis

• L(θ, θ) = m2 (x2 + y2)− V (θ) =

m r20

2 θ2 − V (θ),

• x = r0 cos θ, y = r0 sin θ; V (θ + 2π) = V (θ);

• (in case of pendulum: V (θ) = mgr0(1− cos θ) )• pθ = ∂L/∂θ = mr2

0 θ: orbital angular momentum,

• H(θ,pθ) = 12mr2

0p2θ + V (θ),

• V (θ) = 0 in the following.• pθ = −∂H/∂θ = 0⇒ pθ = const . ∈ R.


Wigner functions for Pθ,p Previous difficulties with Wigner functions on Pθ,p ∼= S1 × R

The previous two main obstacles of goingfrom Wigner functions on R× R to those on S1 × R

Construction of Wigner functions on S1 × Rin analogy to those on R× R:First papers by Berry (1977) and Mukunda (1979)- and then many more - met two basic obstacles:• Quantizing the angle θ!

Only formal, mathematically unsatisfactory “solutions”!• Compatibility of continuous classical p and

discontinuous quantum l = ~m, m ∈ Z. (~ = 1 in the following)Attempted way out: make classical p discontinuous, too:S1 × R→ S1 × Z:no longer a classical phase space (cotangent bundle)

Removel of both obstacles in what follows!



The difficulties of quantizing the angle

• canonical coordinates:q = θ ∈ [0,2π) ≡ R mod 2π, pθ ≡ p = mr2

0 θ ∈ R (rotator),

• phase space: (θ,p) ∈ Pθ,p ∼= S1 × R,

• θ is discontinuous at the borders 0 and 2π!

• (local) Poisson brackets:{f1, f2}θ,p = ∂θf1(θ,p)∂p − ∂pf1(θ,p)∂θf2(θ,p), θ ∈ (0,2π),{θ,p}θ,p = 1,

• Quantum mechanics: θ → Θ, p → L, [Θ,L] ≡ Θ · L− L ·Θ = i ;



• angular momentum operator L has eigenvaluesn ∈ Z , i.e. n = 0,±1,±2, ..., L|n〉 = n|n〉,

• 〈n|Θ · L− L ·Θ|n〉 = i〈n|n〉,⇒ (n − n)〈n|Θ|n〉 = i ,

• ⇒ 0 = i~: contradiction!! Operator Θ cannot exist!

• Many attempts since 1926 to circumvent the problem, at leastformally!


Wigner functions for Pθ,p Solution of the quantum angle problem

Solution of the problem in terms of sin θ and cos θAngle, geometrically: point of intersection of a ray from the origin withthe unit circle S1 around the origin;point is uniquely determined by ~n = (cos θ, sin θ) or χ~n, χ > 0.

~n = (cos θ, sin θ) ∈ S1

r =1

θcos θ

sinθ


Wigner functions for Pθ,p Solution of the quantum angle problem

Use the pair (cos θ, sin θ) instead of θ itself for quantization (1963:Mackey and Louisell independently; HKa, PRA 73, 052104 (2006)),• basis functions (observables):

h1 = cos θ, h2 = sin θ, h3 = pθ ≡ p.

• on classical phase space

Pθ,p = {(θ,p); θ ∈ R mod 2π, p ∈ R }

• hj obey Poisson brackets

{h3, h1}θ,p = h2, {h3, h2}θ,p = −h1, {h1, h2}θ,p = 0 :

• Lie algebra of the 3-parametric Euclidean group E(2) of the plane.• Unitary representations of E(2) provide QM on Pθ,p.


Wigner functions for Pθ,p Euclidean group E(2) of the plane

Euclidean group E(2)

Action of E(2) = {g(ϑ,~a)} on plane: ~x → R(ϑ) · ~x + ~a,•

g(ϑ,~a) ◦ ~x =

(cosϑ − sinϑsinϑ cosϑ

)(x1x2

)+

(a1a2

),

or•

g(ϑ,~a) ◦

x1x21

=

cosϑ − sinϑ a1sinϑ cosϑ a2

0 0 1

x1x21

;



• generators of infinitesimal transformations:

L =

0 −1 01 0 00 0 0

, K1 =

0 0 10 0 00 0 0

, K2 =

0 0 00 0 10 0 0

,• Lie algebra commutators:

[L, K1] = K2, [L, K2] = −K1, [K1, K2] = 0 ;

isomorphic to Poisson/Lie algebra of hj !

• QM: hj become self-adjoint operators p → L and(cos θ, sin θ)→ ~K = (K1,K2).



Hilbert space

• Hilbert space Hϕ for unitary representations of Euclidean group:

(ψ2, ψ1) =

∫ π

−π

dϕ2π

ψ∗2(ϕ)ψ1(ϕ),

with basis

en(ϕ) = einϕ, (em,en) = δmn, m,n ∈ Z,

• ψ(ϕ+ 2π) = ψ(ϕ):

ψ(ϕ) =∑n∈Z

cn en(ϕ), cn = (en, ψ),



Action of operators

•~Kψ(ϕ) = χ (cosϕ, sinϕ)ψ(ϕ), χ > 0,

Lψ(ϕ) =1i∂ϕψ(ϕ),

• ~K : multiplication operator,• en(ϕ) are eigenfunctions of angular momentum operator:

L en(ϕ) = n en(ϕ),

• If A is operator in Hϕ its matrix elements Amn are given byAmn = (em,Aen)


Wigner functions for Pθ,p Wigner operator on Pθ,p

Wigner operator on the cylindrical Pθ,pConstruction of Wigner operator on Pθ,p, partially following B. Wolf etal.:• Start with the group element

g0(ϑ,~b) = exp(b1K1 + b2K2 + ϑL)(R(ϑ) sinc(ϑ/2) R(ϑ/2)~b0 0 1

),

sinc x ≡ sin xx

,

• Translations now parametrized in an angle-dependent way:

~a = sinc(ϑ/2) R(ϑ/2)~b,



• It follows that

g0(ϑ,~b) = e(ϑ/2)L ◦ esinc(ϑ/2)~b·~K ◦ e(ϑ/2)L,

or, withR(−ϑ/2)~a = sinc(ϑ/2)~b,

thatg0(ϑ,~a) = e(ϑ/2)L ◦ e[R(−ϑ/2)~a]·~K ◦ e(ϑ/2)L.

Kind of Weyl symmetrization!



• Given the framework of unitary representations of E(2)appropriate Wigner operator can be constructedfrom unitary operator

U0(ϑ,~a) = ei(ϑ/2)L ◦ ei~a(−ϑ/2)·~K ◦ ei(ϑ/2)L,

~xϑ ≡ R(ϑ) · ~x ,• by kind of group averaging (similar to the planar case):

V [~χ(θ),p] =1

(2π)3

∫ π

−πdϑ∫ ∞−∞

da1da2 U0,

U0 = ei(L−p)(ϑ/2) ◦ ei(~K−~χ(θ))·~a(−ϑ/2)

◦ei(L−p)(ϑ/2).

• ~χ(θ) = χ~n, χ > 0, ~n = (cos θ, sin θ).



Wigner matrixMatrix elements (em,V [~n(θ),p]en) lead to following Wigner matrixV (θ,p) = (Vmn):•

Vmn(θ,p) =1

(2π)2 ei(n−m)θ

∫ π

−πdϑei[(n+m)/2−p]ϑ

=1

2πei(n−m)θ sincπ[p − (m + n)/2],

• wheresinc x =

sin xx

, sinc(x = 0) = 1,

−3 −2 −1 1 2 30

1

p−m

2πVm(θ, p) = sincπ(p−m)


Wigner functions for Pθ,p Wigner function of wave functions ψ(ϕ)

Wigner function of a wave function ψ(ϕ)

• ψ(ϕ) =∑

n∈Z cnen(ϕ),•

Vψ(θ,p) =1

(2π)2

∫ π

−πdϑe−ipϑψ∗(θ − ϑ/2)ψ(θ + ϑ/2)

=∑

m,n∈Zc∗mVmn(θ,p)cn

= (ψ,V (θ,p)ψ),

• Vψ real because V = (Vmn) is Hermitean ( V ∗mn = Vnm).



Some general Properties of Vψ(θ, p)• Marginal distributions in the planar case:∫

Rdp Wφ(q,p) = |φ(q)|2,

∫R

dq Wφ(q,p) = |φ(p)|2.

• Marginal distributions in the (θ,p) case are: |ψ(θ)|2and |cm|2,m ∈ Z; but

• ∫ ∞−∞

dp Vψ(θ,p) =1

2π|ψ(θ)|2,

• ∫ π

−πdθVψ(θ,p) =

∑n∈Z|cn|2 sincπ(p − n) ≡ ωψ(p) :

Whittaker’s cardinal interpolation function (1915)!Important tool in sampling and signal processing theories!



• As ∫ ∞−∞

dp sincπ(m − p) sincπ(n − p) = δmn,

• one has ∫ ∞−∞

dp ωψ(p) sincπ(p −m) = |cm|2.

• Transition probabilities∫ ∞−∞

dp∫ π

−πdθVψ2(θ,p) Vψ1(θ,p) =

12π|(ψ2, ψ1)|2.

•tr(A · B) = 2π

∫ ∞−∞

dp∫ π

−πdθ tr[A · V (θ,p)] tr[B · V (θ,p)].

• expectation value of operator A for a given density matrix ρ:

〈A〉ρ = tr(ρ · A)

= 2π∫ ∞−∞

dp∫ π

−πdθ tr[ρ · V (θ,p)] tr[A · V (θ,p)],



•A (θ,p) ≡ tr[A · V (θ,p)]

analogue to Weyl symbol in planar case!• For the product A · B one has

AB(θ,p) = A(θ,p) e~2i Λ(θ,p)B(θ,p)

≡ A(θ,p) ? B(θ,p)

A2(θ,p) = A2(θ,p).

Λ(θ,p) =←∂ p→∂ θ −

←∂ θ→∂ p,

• Representation of Hilbert space operator A in terms of its symbol:

A = (Amn) =

∫ ∞−∞

dp∫ π

−πdθV (θ,p) tr[A · V (θ,p)]


Wigner functions for Pθ,p Examples

Examples for (θ, p) Wigner functions

1. Wigner function for the basis function em(ϕ)

• ψ(ϕ) = em(ϕ) : cm = 1, cn = 0 for n 6= m,•

Vm(θ,p) = (1/2π) sincπ(p −m),

−3 −2 −1 1 2 30

1

p−m

2πVm(θ, p) = sincπ(p−m)

• Vm(θ,p) is negative for certain p values!



2. Wigner function for a simple “cat” state

• e1(ϕ) = eiϕ and e−1(ϕ) = e−iϕ different eigenfunctions ofHamiltonian H = εL2, both with same eigenvalue ε;Le1 = (+1)e1; Le−1 = (−1)e−1;

• Here: +1: “cat alive”, -1: “cat dead”!• superposition f+(ϕ) = 1√

2[eiϕ + e−iϕ] =

√2 cosϕ, (f+, f+) = 1,

• density |f+(ϕ)|2 = 1 + cos 2ϕ = 2 cos2 ϕ,• cos 2ϕ: interference/entanglement term.• operators: L = (1/i)∂ϕ, C = cosϕ, S = sinϕ.

(f+,Lf+) = 0, (f+,L2f+) = 1; (f+,Cf+) = 0, (f+,C2f+) = 3/4;

(f+,Sf+) = 0, (f+,S2f+) = 1/4; (f+,C · Lf+) = 0, (f+,L ·Cf+) = 0.



Wigner function of f+(ϕ) = 1√2

[eiϕ + e−iϕ] :

2πVf+(θ,p) = cos 2θ sincπp +12

[sincπ(p + 1) + sincπ(p − 1)].

−3 −2 −1 1 2 3

−1

0

1

p

2πVf+(θ, p)

θ = 0,±π θ = ±π8,± 7π

8θ = ±π

4,± 3π

4

θ = ± 3π8,± 5π

8θ = ±π

2



cos(2θ)sinc(πp) +0.5[sinc(π(p-1))+sinc(π(p+1))]0.5

0-0.5

−π

0

π

θ

−3−2

−10

12

3

p

−1−0.5

00.5

1



3. Wigner function of a qubit

Consider e1(ϕ) and e−1(ϕ) as basis of qubit!:

χα,β(ϕ) = cosβe1(ϕ) + sinβeiαe−1(ϕ).

Properties:

(χα,β, χα,β) = 1, Lχα,β = χα+π,β, L2χα,β = χα,β.

Wigner function:

Vχα,β (θ,p) =1

2π{sin 2β cos(2θ − α) sincπp

+ cos2 β sincπ(p − 1) + sin2 β sincπ(p + 1)}.

Previous examples special cases: α = 0, β = 0 and α = 0, β = π/4.



Marginal distributionsθ marginal distribution:∫ ∞

−∞dp Vχα,β (θ,p) =

12π

[sin 2β cos(2θ − α) + 1] =1

2π|χα,β(θ)|2.

Whittaker cardinal function for p and m = ±1:∫ π

−πdθVχα,β (θ,p) = cos2 β sincπ(p−1)+sin2 β sincπ(p+1) ≡ ωχα,β (p).

Marginal probability for p = 1:∫ ∞−∞

dp ωχα,β (p) sincπ(p − 1) = cos2 β.

Marginal probability for p = −1:∫ ∞−∞

dp ωχα,β (p) sincπ(p + 1) = sin2 β.


Wigner functions for Pθ,p Conclusions and Outlook

Conclusions and Outlook

• Wigner functions for cylindrical phase spaces aremathematically and physically consistent,structurally very similar to those of planar phase spaces.

• Many more properties,e.g. ? products or quantum Liouville equations etc.,can be found in my last two PRA papers.

• The simple example of a cat state (more generally:angular momentum qubit) suggests a possible rolefor quantum information science etc.

• Two more typical examples in PRA 94:Minimal uncertainty states (von Mises distribution)and thermal states.


Wigner functions for Pθ,p A personal tribute to Eugene Wigner

A personal tribute to Eugene Wigner

• In 1962 I got my PhD from the University of Munich;Thesis: Conformal symmetries of field theories.

• In 1963 Wigner received the Nobel prize.• In his Nobel Lecture he quoted a publication of mine

[Phys.Lett.3 (1962),78], in which I proposedconformal transformations as asymptotic symmetriesat very high energies.

• This led to an invitation by Wigner to cometo Princeton for a year!



• I was in Princeton in the academic year 1965/66;(in 1964/65 in Berkeley).

• Wigner and I had different ideasabout possible applications of the conformal group!

• So there was no joint publication!• After a while the interpretation of conformal transformations

as an asymptotic symmetry became generally acceptedand an elaborate analysis of conformally invariant field theoriestook off!



• At that time in Princeton there wasn’t any mentionof the Wigner function between Wigner and myself!

• I learnt about it much later (and initially was sceptical).• But I should have known earlier, because one of my

Thesis advisers, Fritz Bopp, had written about it in 1956 and 1961!• So now my last two papers are my late tribute to Eugene Wigner,

who played an important rolein my education as a theoretical physicist!


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