LECTURE7—10/14/2020WEYL QUANTIZATION:EXAMPLES1.WEYLQUANTIZATIONOFPOLYNOMIAL-TYPEFUNCTIONSToday wefocus onthe Weyl quantization /()asaW(0)(r) =2We shall compute the operator aw for some simple classes of functions. A formulathat we will use several times is the Fourier inversion formulaLeif(y)dydsf(r) = (2h)n Je"ormoreprecisely,itsvariation1eip f(r, y)dyds,f(r,r) =(2元h)nJRwhich can be obtained from the following identity by setting u = r:.Lei--ps f(u, y)dyde.f(u,a) =(2元h）nJRnI Weyl quantization of a(r).We start with the case a = a(r)/ ea()0()dyde =a(a)0(a),aw()(a) =(2元h)nJRm2So, as one can expect (which holds for all t-quantizations)=“multiplication by a(r)".a(r)T Weyl quantization of a($)Next let's consider the casea = a(s)()()()= (Fr-)s→ [a(S)(Fnp)(S) (r)So we get (again the same formula holds (trivially) for all t-quantizations):a()" = Frl oa(E)o Fh
LECTURE 7 — 10/14/2020 WEYL QUANTIZATION: EXAMPLES 1. Weyl quantization of polynomial-type functions Today we focus on the Weyl quantization a❜ W (ϕ)(x) = 1 (2π~) n ❩ Rn ❩ Rn e i (x−y)·ξ ~ a( x + y 2 , ξ)ϕ(y)dydξ. We shall compute the operator a❜W for some simple classes of functions. A formula that we will use several times is the Fourier inversion formula, f(x) = 1 (2π~) n ❩ Rn ❩ Rn e i (x−y)·ξ ~ f(y)dydξ, or more precisely, its variation f(x, x) = 1 (2π~) n ❩ Rn ❩ Rn e i (x−y)·ξ ~ f(x, y)dydξ, which can be obtained from the following identity by setting u = x:. f(u, x) = 1 (2π~) n ❩ Rn ❩ Rn e i (x−y)·ξ ~ f(u, y)dydξ. ¶ Weyl quantization of a(x). We start with the case a = a(x) : a❜ W (ϕ)(x) = 1 (2π~) n ❩ Rn ❩ Rn e i (x−y)·ξ ~ a( x + y 2 )ϕ(y)dydξ = a(x)ϕ(x). So, as one can expect (which holds for all t-quantizations) aÕ(x) W = “multiplication by a(x)”. ¶ Weyl quantization of a(ξ). Next let’s consider the case a = a(ξ) : a❜ W (ϕ)(x) = 1 (2π~) n ❩ Rn ❩ Rn e i (x−y)·ξ ~ a(ξ)ϕ(y)dydξ = (F −1 ~ )ξ→x [a(ξ)(F~ϕ)(ξ)] (x). So we get (again the same formula holds (trivially) for all t-quantizations): aÔ(ξ) W = F −1 ~ ◦ a(ξ) ◦ F~. 1
2LECTURE7—10/14/2020 WEYLQUANTIZATION:EXAMPLESSuch operators are known as Fourier multipliers. Note that in particular we getaW = Pa, and thus (lsP/2 + V(r)" =-h2/2+ V.As an application, we calculate the Weyl quantization of the quadratic expo-nential a(s) = esTQs , where Q is a non-singular symmetric n x n matrix. DenotePQ(S) = $TQE.Proposition 1.1. For a(s) = e"Qs we haveawo(z) = [det Q-1/2.-eisgnoLe-hyTQ-yp(r+y)dy(1)(2元h)n/2Proof. In Lecture 4 we showedF(eQr) = (2)n/2eigsgn(Q)e-1TQ-'gIdetQ|?It follows1n etesTosds = Fr (esro(0)(a) - detol1e-eifsgnQe-PQ-1(a)(2元h)n/2(2元h)nJThuset(r-)esTQsp(y)dydeawp(r)=(2元h)nIdet Q|-1/2 L Idet Q-1/2-eisnQ [e-PQ-1()p(r +y)dy.(2元h)n/2口Theformula (1)will beused next timeto computethesymbol ofthecompositionof twoWeyloperators.Weyl quantization of polynomials in both and .By linearity, to compute the Weyl quantization of a polynomial in both r andS, it is enough to compute the Weyl quantization of monomialsa(r,s) = ragL etp("+)aPp(y)dydea(0)(z)= (2元h)n J".inti-KNTTa-EB(0)(r)21a7r(hDa)(ra-(r)2lal
2 LECTURE 7 — 10/14/2020 WEYL QUANTIZATION: EXAMPLES Such operators are known as Fourier multipliers. Note that in particular we get ξ❝α W = P α , and thus (|ξ|Û2/2 + V (x)) W = −~ 2∆/2 + V . As an application, we calculate the Weyl quantization of the quadratic exponential a(ξ) = e i 2~ ξ T Qξ , where Q is a non-singular symmetric n × n matrix. Denote pQ(ξ) = ξ TQξ. Proposition 1.1. For a(ξ) = e i 2~ ξ T Qξ we have (1) a❜ W ϕ(x) = | det Q| −1/2 (2π~) n/2 e i π 4 sgnQ ❩ Rn e − i 2~ y T Q−1yϕ(x + y)dy. Proof. In Lecture 4 we showed F(e i 2 x T Qx) = (2π) n/2 e i π 4 sgn(Q) | det Q| 1 2 e − i 2 ξ T Q−1 ξ . It follows 1 (2π~) n ❩ Rn e i ~ x·ξ e i 2~ ξ T Qξdξ = F −1 ~ (e i 2~ pQ(ξ) )(x) = | det Q| −1/2 (2π~) n/2 e i π 4 sgnQe − i 2~ pQ−1 (x) . Thus a❜ W ϕ(x) = 1 (2π~) n ❩ Rn ❩ Rn e i ~ (x−y)·ξ e i 2~ ξ T Qξϕ(y)dydξ = | det Q| −1/2 (2π~) n/2 e i π 4 sgnQ ❩ Rn e − i 2~ pQ−1 (x−y)ϕ(y)dy = | det Q| −1/2 (2π~) n/2 e i π 4 sgnQ ❩ Rn e − i 2~ pQ−1 (y)ϕ(x + y)dy. The formula (1) will be used next time to compute the symbol of the composition of two Weyl operators. ¶ Weyl quantization of polynomials in both x and ξ. By linearity, to compute the Weyl quantization of a polynomial in both x and ξ, it is enough to compute the Weyl quantization of monomials a(x, ξ) = x α ξ β : a❜ W (ϕ)(x) = 1 (2π~) n ❩ Rn ❩ Rn e i (x−y)·ξ ~ ( x + y 2 ) α ξ βϕ(y)dydξ = ❳ γ≤α 1 2 |α| ✥ α γ ✦ x γxØα−γξ β anti−KN (ϕ)(x) = ❳ γ≤α 1 2 |α| ✥ α γ ✦ x γ (~Dx) β (x α−γϕ(x)).
3LECTURE7—10/14/2020WEYLQUANTIZATION:EXAMPLESIn other words, we get the following McCoy's formula:TERW=PBQ-S021alwhere ≤a means ≤aj for all j, andQ!(a- Weyl quantization of polynomials in .Next let's compute the Weyl quantization of a(r,E) = Zja/≤x aa(r)EUsingthe fact(-hDy)cei-ps= sei-psand the Fourier inversion formula we get+y(eips a()sp(y)dydeaW(p)(r) =(2元h)nJ2lal<kT+y((-hDy)eip)p(y)dydelalk (2元h)n /Rn P(hD,) (aa(“)() dyde0lal/≤x (2h)n /Rn1= Z(D,) [aa(“)0()]2lol<k= 2-hi() [(hD)aa(r)] (D)α-(r),[al<≤Sowe getaw= 2-mi(a) [(hD)aa(α)] (D)a-la<≤aAs a consequenceCorollary 1.2. If a(r,E) = jal<h aa(r)sa is a polynomial of degree k in E, thenaw is a semiclassical differential operator of order k of the formaW = Z aa(r)(hD)°+ "terms of order <k -1".[α]=kNote that the same result holds for the Kohn-Nirenberg quantization and theanti-Kohn-Nirenberg quantization, and in fact for all semiclassical t-quantizations
LECTURE 7 — 10/14/2020 WEYL QUANTIZATION: EXAMPLES 3 In other words, we get the following McCoy’s formula: xÕαξ β W = ❳ γ≤α 1 2 |α| ✥ α γ ✦ Q γP βQ α−γ , where γ ≤ α means γj ≤ αj for all j, and ✥ α γ ✦ := α! γ!(α − γ)! = ✥ α1 γ1 ✦ · · · ✥ αn γn ✦ . ¶ Weyl quantization of polynomials in ξ. Next let’s compute the Weyl quantization of a(x, ξ) = P |α|≤k aα(x)ξ α . Using the fact (−~Dy) α e i (x−y)·ξ ~ = ξ α e i (x−y)·ξ ~ and the Fourier inversion formula we get a❜ W (ϕ)(x) = 1 (2π~) n ❩ Rn ❩ Rn e i (x−y)·ξ ~ ❳ |α|≤k aα( x + y 2 )ξ αϕ(y)dydξ = ❳ |α|≤k 1 (2π~) n ❩ Rn ❩ Rn ⑩ (−~Dy) α e i (x−y)·ξ ~ ❿ aα( x + y 2 )ϕ(y)dydξ = ❳ |α|≤k 1 (2π~) n ❩ Rn ❩ Rn e i (x−y)·ξ ~ (~Dy) α ⑩ aα( x + y 2 )ϕ(y) ❿ dydξ = ❳ |α|≤k (~Dy) α ➉ aα( x + y 2 )ϕ(y) ➌ y=x = ❳ |α|≤k ❳ γ≤α 2 −|γ| ✥ α γ ✦ [(~D) γ aα(x)] · (~D) α−γϕ(x). So we get a❜ W = ❳ |α|≤k ❳ γ≤α 2 −|γ| ✥ α γ ✦ [(~D) γ aα(x)] · (~D) α−γ . As a consequence Corollary 1.2. If a(x, ξ) = P |α|≤k aα(x)ξ α is a polynomial of degree k in ξ, then a❜W is a semiclassical differential operator of order k of the form a❜ W = ❳ |α|=k aα(x)(~D) α + “terms of order ≤ k − 1”. Note that the same result holds for the Kohn-Nirenberg quantization and the anti-Kohn-Nirenberg quantization, and in fact for all semiclassical t-quantizations
4LECTURE7—10/14/2020WEYLQUANTIZATION:EXAMPLES2.SYMPLECTICINVARIANCEAND APPLICATIONSSymplectic invariance of Weyl quantization.According to the computations above.it seems that Wevl guantization ismuchmore complicated than the Kohn-Nirenberg or the anti-Kohn-Nirenberg quantiza-tions. A natural question is: what is the advantage of the Weyl quantization? Wehave seen the first big advantage: Weyl quantization will quantize real-valued func-tions to formally self-adjoint operators. Here we explain the second big advantage:the (unitary) invariance under linear symplectomorphisms '(this conception wil beexplained later).Theorem 2.1 (Symplectic invariance of Weyl quantization). Given any“linearsymplectomorphism" : T*Rn → T*R", there is a metaplectic operator U (whichis an isomorphism on(R")and on(Rn),and is unitary on L?(Rn))such that(2)ao"=U-loaw。Ua.Remark.Moreover,it can be shown that such“symplectic invariance"characterizethe Weyl quantization:If there is a“quantization process"Q:'(Rn × Rn)-→C((Rn), (Rn)) which is sequentially continuous, quantizes any bounded func-tion a()to the operator“multiplication by a(r)"and satisfies the symplectic in-variance property above, then it is the Weyl quantization!Here are three special cases of this theorem for which we can easily check (2) bydirect computations:(A) The linear symplectomorphism is the map: Rn ×R"-→R"× R", (C,s)-(S,-r)which“intertwines" r and with a twisting.In this case U=Fh-(So FncanberegardedasthequantizationofJ!)(B) The linear symplectomorphism is the map: R" × R"→Rn ×R", (r,s)-(c,S+Cr),where C is a symmetric n × n matrix. In this case U is the map “multipli-cation by thefunction eiaTCr/2h"(C)The linear symplectomorphism is themapΦ: R" ×Rn →R" × Rn, (c,S) → (Ac, (AT)-1),where A is an invertible matrix. In this case Ua is the map U is given by(Ue)(r) = (Ar)1From the classical-quantum correspondence point of view, a nice quantization should preservesymplecticproperties,twosymplecticallyequivalentclassicalobjectsshouldcorrespondstounitar-ilyequivalent quantumobjects.InLectureI wehavementioned theEgorov theorem,which canbeexplained astheunitaryinvarianceundergeneral symplectomorphisms (which onlyhold inthesemiclassical limit).Here,for linear symplectomorphisms, the invariance is an exact relation
4 LECTURE 7 — 10/14/2020 WEYL QUANTIZATION: EXAMPLES 2. Symplectic invariance and applications ¶ Symplectic invariance of Weyl quantization. According to the computations above, it seems that Weyl quantization is much more complicated than the Kohn-Nirenberg or the anti-Kohn-Nirenberg quantizations. A natural question is: what is the advantage of the Weyl quantization? We have seen the first big advantage: Weyl quantization will quantize real-valued functions to formally self-adjoint operators. Here we explain the second big advantage: the (unitary) invariance under linear symplectomorphisms 1 (this conception will be explained later). Theorem 2.1 (Symplectic invariance of Weyl quantization). Given any “linear symplectomorphism” Φ : T ∗R n → T ∗Rn , there is a metaplectic operator UΦ (which is an isomorphism on S 0 (R n ) and on S (R n ), and is unitary on L 2 (R n )) such that (2) aÖ◦ Φ W = U −1 Φ ◦ a❜ W ◦ UΦ. Remark. Moreover, it can be shown that such “symplectic invariance” characterize the Weyl quantization: If there is a “quantization process” Q : S 0 (R n × R n ) → L(S (R n ), S 0 (R n )) which is sequentially continuous, quantizes any bounded function a(x) to the operator “multiplication by a(x)” and satisfies the symplectic invariance property above, then it is the Weyl quantization! Here are three special cases of this theorem for which we can easily check (2) by direct computations: (A) The linear symplectomorphism is the map Φ : R n × R n → R n × R n , (x, ξ) 7→ (ξ, −x) which “intertwines” x and ξ with a twisting. In this case UΦ = F~. (So F~ can be regarded as the quantization of J!) (B) The linear symplectomorphism is the map Φ : R n × R n → R n × R n , (x, ξ) 7→ (x, ξ + Cx), where C is a symmetric n × n matrix. In this case UΦ is the map “multiplication by the function e ixT Cx/2~ ”. (C) The linear symplectomorphism is the map Φ : R n × R n → R n × R n , (x, ξ) 7→ (Ax,(A T ) −1 ξ), where A is an invertible matrix. In this case UΦ is the map UΦ is given by (UΦϕ)(x) = ϕ(Ax). 1From the classical-quantum correspondence point of view, a nice quantization should preserve symplectic properties, two symplectically equivalent classical objects should corresponds to unitarily equivalent quantum objects. In Lecture 1 we have mentioned the Egorov theorem, which can be explained as the unitary invariance under general symplectomorphisms (which only hold in the semiclassical limit). Here, for linear symplectomorphisms, the invariance is an exact relation
LECTURE7—10/14/2020WEYLQUANTIZATION:EXAMPLES5Remark.Infact,one can prove that anylinear symplectomorphism canbewrittenas a composition of the three classes of linear symplectomorphisms above.As aresult, the general theorem is proven as long as we can check the three cases (A)(B) and (C).Wewill not prove thefull theorem here2.Instead, in what follows wewill prove case (A) and a special case of (B), and give two applications. We willleave the proof of the general cases of (B) and (C) as an exercise. Case (A): Conjugation by Fourier transform.Weprove case (A)by direct computation:Theorem 2.2 (Conjugation by Fourier transform). Let b(r,) = a(E,-r), thenF-loawoFh=6w(3)Proof.We compute[(F-l)m→2(aW)y→n(Fn)z→yl (α)[e-t*p(2)dz]=(F-")n→r(aW)y→n[[a(()e)a(,)p(2)dy(2元h)n(2元h)n1+()a()()(2元h)"（2元h）2+a()d(d(2元h)n(2元h)n1[ek(r+y+2)net(-25-29)-Sa(C,E)dcdedn(y)dy(2元h）m2元h）(2h) / ete(-y)Sa(, ))drdn ) dc(y)dy(2元h）nJB2Using theFourier inversionformula/ etr f(r)drde,f(0) = [FFr"f](0) = 7(2元h）nJ起the expression in (-..) above can be simplified toet(a-)a(c,-) = et(r-)<b(“+,.1口and the conclusion follows.2For a proof, c.f. Folland, Harmonic analysis in phase space, Chapter 4
LECTURE 7 — 10/14/2020 WEYL QUANTIZATION: EXAMPLES 5 Remark. In fact, one can prove that any linear symplectomorphism can be written as a composition of the three classes of linear symplectomorphisms above. As a result, the general theorem is proven as long as we can check the three cases (A), (B) and (C). We will not prove the full theorem here 2 . Instead, in what follows we will prove case (A) and a special case of (B), and give two applications. We will leave the proof of the general cases of (B) and (C) as an exercise. ¶ Case (A): Conjugation by Fourier transform. We prove case (A) by direct computation: Theorem 2.2 (Conjugation by Fourier transform). Let b(x, ξ) = a(ξ, −x), then (3) F −1 ~ ◦ a❜ W ◦ F~ = ❜b W . Proof. We compute ➈ (F −1 ~ )η→x(a❜ W )y→η(F~)z→yϕ ➋ (x) =(F −1 ~ )η→x(a❜ W )y→η ➉❩ Rn e − i ~ z·yϕ(z)dz➌ =(F −1 ~ )η→x ➊ 1 (2π~) n ❩ Rn ❩ Rn ❩ Rn e i ~ (η−y)·ξ a( η + y 2 , ξ)e − i ~ z·yϕ(z)dzdydξ➍ = 1 (2π~) n 1 (2π~) n ❩ Rn ❩ Rn ❩ Rn ❩ Rn e i ~ x·η e i ~ (η−y)·ξ a( η + y 2 , ξ)e − i ~ z·yϕ(z)dzdydξdη = 1 (2π~) n ❩ Rn ➊ 1 (2π~) n ❩ Rn ❩ Rn ❩ Rn e i ~ [x·η+(η−z)·ξ−y·z] a( η + z 2 , ξ)dzdξdη➍ ϕ(y)dy = 1 (2π~) n ❩ Rn ➊ 2 n (2π~) n ❩ Rn ❩ Rn ❩ Rn e i ~ [x·η+2(η−ζ)·ξ−y·(2ζ−η)]a(ζ, ξ)dζdξdη➍ ϕ(y)dy = 1 (2π~) n ❩ Rn ➊ 2 n (2π~) n ❩ Rn ❩ Rn ❩ Rn e i ~ (x+y+2ξ)·η e i ~ (−2ξ−2y)·ζ a(ζ, ξ)dζdξdη➍ ϕ(y)dy = 1 (2π~) n ❩ Rn ➊❩ Rn ❶ 1 (2π~) n ❩ Rn ❩ Rn e i ~ τ·η e i ~ (x−y−τ)·ζ a(ζ, τ − x − y 2 )dτ dη➀ dζ➍ ϕ(y)dy Using the Fourier inversion formula f(0) = [F~F −1 ~ f](0) = 1 (2π~) n ❩ Rn ❩ Rn e i ~ x·ξ f(x)dxdξ, the expression in (· · ·) above can be simplified to e i ~ (x−y)·ζ a(ζ, −x − y 2 ) = e i ~ (x−y)·ζ b( x + y 2 , ζ) and the conclusion follows. 2For a proof, c.f. Folland, Harmonic analysis in phase space, Chapter 4.