《半经典微局部分析 Semiclassical Microlocal Analysis》课程教学资源（英文讲义）Lec03 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT.pdf

2LECTURE 3-09/27/2020 QUANTIZATION V.S.SEMI-CLASSICAL LIMIT1.QUANTIZATIONThe word “quantization"refers to any systematical way of constructing a quan-tum mechanical description of a system from its classical mechanical description. Inthe language of mathematics,it should be a“functor"that givesSymplectic space (M,w)Hilbert space (H,(, ),(C(M, R), {, ) (self-adjoint operators on H, At theverybeginning of quantummechanics,Dirac suggested several natural axiomsthat a quantization procedure should satisfy:Dirac's Axioms. A quantization procedure assigns to any classical observable a ECo(M, R) a self-adjoint operator Q(a) on some Hilbert space H, such that(D1) (linearity) Q(μa+vb) =μQ(a)+vQ(b) for any μ,VER.(D2)(normalization) Q(1)=Id.(quantizaiton condition) Q({a,b)) = [Q(a), Q(b)].(D3)(minimality) Any complete family of functions on M passes to a complete(D4)family of self-adjoint operators on H.-A family of functions on M is complete if they separate points almosteverywhere on M.-A family of operators on H is complete if they act irreducibly on H, i.e.no nonzero proper closed subspace of H is invariant under the family ofoperators.Unfortunately, such a quantization does not exists!Theorem 1.1 (Groenewold-van Hove no-go theorem). A quantization procedurethat satisfies (D1)-(D4) for all f E Co(M) does not erist.In fact, one can not even quantize all polynomials of degree ≤ 4 in Coo(T*R),:To see this one need to observe the following facts:. The first key ingredient is a Schur-type lemma:Lemma 1.2. Suppose Y is an self-adjoint operator on H such that[Y,Q(r)] = [Y,Q()] = 0,thenY =c·Id for some constant c.Proof. We will need some facts from functional analysis:-By a theorem in functional analysis (c.f. K. Yosida, Functional analysis,6th ed., page 339), if A is a self-adjoint operator and B is a boundedself-adjoint operator, and A, B commutes, then for any Baire functionf, f(A) commutes with B.Inparticular,B commutes with any spectralprojection xE(A) of A onto an interval finite E

2 LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT 1. Quantization The word “quantization” refers to any systematical way of constructing a quantum mechanical description of a system from its classical mechanical description. In the language of mathematics, it should be a “functor” that gives Symplectic space (M, ω) Hilbert space (H,h·, ·i), (C ∞(M, R), {·, ·}) (self-adjoint operators on H, i ~ [·, ·]). At the very beginning of quantum mechanics, Dirac suggested several natural axioms that a quantization procedure should satisfy: Dirac’s Axioms. A quantization procedure assigns to any classical observable a ∈ C ∞(M, R) a self-adjoint operator Q(a) on some Hilbert space H, such that (D1) (linearity) Q(µa + νb) = µQ(a) + νQ(b) for any µ, ν ∈ R. (D2) (normalization) Q(1) = Id. (D3) (quantizaiton condition) Q({a, b}) = i ~ [Q(a), Q(b)]. (D4) (minimality) Any complete family of functions on M passes to a complete family of self-adjoint operators on H. – A family of functions on M is complete if they separate points almost everywhere on M. – A family of operators on H is complete if they act irreducibly on H, i.e. no nonzero proper closed subspace of H is invariant under the family of operators. Unfortunately, such a quantization does not exists! Theorem 1.1 (Groenewold-van Hove no-go theorem). A quantization procedure that satisfies (D1)-(D4) for all f ∈ C ∞(M) does not exist. In fact, one can not even quantize all polynomials of degree ≤ 4 in C ∞(T ∗R),. To see this one need to observe the following facts: • The first key ingredient is a Schur-type lemma: Lemma 1.2. Suppose Y is an self-adjoint operator on H such that [Y, Q(x)] = [Y, Q(ξ)] = 0, then Y = c · Id for some constant c. Proof. We will need some facts from functional analysis: – By a theorem in functional analysis (c.f. K. Yosida, Functional analysis, 6th ed., page 339), if A is a self-adjoint operator and B is a bounded self-adjoint operator, and A, B commutes, then for any Baire function f, f(A) commutes with B. In particular, B commutes with any spectral projection χE(A) of A onto an interval finite E

3LECTURE3—09/27/2020QUANTIZATION V.S.SEMI-CLASSICAL LIMIT-Conversely, if B commutes with any spectral projection Xe(A) of A,then by spectral theorem, B commutes with A.Note that as a consequence of thefirst fact,for any finite interval F. Xe(B)commutes with Xe(A).Then as a consequence of the second fact, XF(B)commutes with A.Now we start to prove the theorem. First assume Y is a bounded selfadjoint operator. Then as we just argued, for any finite interval F C R,the spectral projection XF(Y) commutes with Q(r) and Q(),i.e.the rangeof the operator XF(Y) is a subspace of H which is invariant under bothQ(r) and Q(s). But since the two functions and from a complete set offunctions on T*R, by (D4) the two operatorsQ(r)and Q($)act irreduciblyon H. As a result, the range of XF(Y) is either O or H for any F. So thespectrum ofY containsonly onepoint.The case where Y is unbounded is more subtle, because all operatorsinvolved areonlydenselydefined.Under some extraassumptionsthat allcomputations involved make sense in a dense subspace,we can concludethat y?, and thus the operator y2+Id, commutes with Q(r) and Q()Since Y? + Id is positive and self-adjoint, it is invertible.Moreover, itsinverse (y2 +Id)-1 is a bounded self-adjoint operator which commutes withbothQ(r)and Q(s).Nowwecan applywhat we justprovedto claim that口(y2 + Id)-1 has the form cld, and the conclusion follows.. Using the above lemma one can show Q(r) = Q(r)Q() + cId for someconstant c,and then inductivelyproveQ(rm) =Q(r)m and Q(sm) =Q(s)mThe second key observation is that we have the following two different ways towrite a degree 4 polynomial as Poisson brackets of two degree 3polynomials:[3,53] =-9r22=3[r25,r2]A contradiction will appearif calculate Q(r2) and Q(r2) explicitly.Wewill leave the details as an exercise.This is both a bad news and a good news for mathematicians: on one hand, thereis no “perfect theory"describing every aspect of the classical-quantum correspon-dence. On the other hand side, by focusing on different aspects of the correspon-dence, mathematicians and physicists has developed several different theories underthe title "quantization".Among these different quantization schemes, three of themare most popular and are widely studied by (different groups of) mathematicians:(A)Weyl quantization.This is one of the first quantization theory in mathemat-ics (proposed by H.Weyl in 1927).It can be viewed as analysts'quantizationmethod sinceitmainlyuses thelanguage of (Fourier)analysis,butit isalsoveryclosely related to representation theory. In Weyl quantization we only quantizecotangent bundles. One start point to understand the theory is the canonical

LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT 3 – Conversely, if B commutes with any spectral projection χE(A) of A, then by spectral theorem, B commutes with A. Note that as a consequence of the first fact, for any finite interval F, χF (B) commutes with χE(A). Then as a consequence of the second fact, χF (B) commutes with A. Now we start to prove the theorem. First assume Y is a bounded selfadjoint operator. Then as we just argued, for any finite interval F ⊂ R, the spectral projection χF (Y ) commutes with Q(x) and Q(ξ), i.e. the range of the operator χF (Y ) is a subspace of H which is invariant under both Q(x) and Q(ξ). But since the two functions x and ξ from a complete set of functions on T ∗R, by (D4) the two operators Q(x) and Q(ξ) act irreducibly on H. As a result, the range of χF (Y ) is either 0 or H for any F. So the spectrum of Y contains only one point. The case where Y is unbounded is more subtle, because all operators involved are only densely defined. Under some extra assumptions that all computations involved make sense in a dense subspace, we can conclude that Y 2 , and thus the operator Y 2 + Id, commutes with Q(x) and Q(ξ). Since Y 2 + Id is positive and self-adjoint, it is invertible. Moreover, its inverse (Y 2 + Id)−1 is a bounded self-adjoint operator which commutes with both Q(x) and Q(ξ). Now we can apply what we just proved to claim that (Y 2 + Id)−1 has the form cId, and the conclusion follows. • Using the above lemma one can show Q(xξ) = Q(x)Q(ξ) + cId for some constant c, and then inductively prove Q(x m) = Q(x) m and Q(ξ m) = Q(ξ) m. The second key observation is that we have the following two different ways to write a degree 4 polynomial as Poisson brackets of two degree 3 polynomials: {x 3 , ξ3 } = −9x 2 ξ 2 = 3{x 2 ξ, xξ2 }. A contradiction will appear if calculate Q(x 2 ξ) and Q(xξ2 ) explicitly. We will leave the details as an exercise. This is both a bad news and a good news for mathematicians: on one hand, there is no “perfect theory” describing every aspect of the classical-quantum correspondence. On the other hand side, by focusing on different aspects of the correspondence, mathematicians and physicists has developed several different theories under the title “quantization”. Among these different quantization schemes, three of them are most popular and are widely studied by (different groups of) mathematicians: (A) Weyl quantization. This is one of the first quantization theory in mathematics (proposed by H. Weyl in 1927). It can be viewed as analysts’ quantization method since it mainly uses the language of (Fourier) analysis, but it is also very closely related to representation theory. In Weyl quantization we only quantize cotangent bundles. One start point to understand the theory is the canonical

4 LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT quantization introduced by P. Dirac in his doctoral thesis in 1926, in which the position functions xk’s and the momentum functions ξk’s are quantized to the position operators Qk’s and the momentum operators Pk’s. The central relation between these operators is the canonical commutative relations1 [Qk, Qk] = 0, [Pk, Pj ] = 0, [Qk, Pj ] = ~ i δjk · Id, which is of course the quantum analogue of the classical facts {xk, xj} = 0, {ξk, ξj} = 0, {xk, ξj} = δkj . As we mentioned in lecture 1, in Weyl quantization we take xk Qk = multiplicaton by xk ξk Pk = ~ i ∂ ∂xk , and as a result we can quantize H = |ξ| 2 2 + V (x) Hˆ = − ~ 2 2 ∆ + V (x). However, it is not clear how to quantize functions in mixed variables like x1ξ1 = ξ1x1 = x1ξ1+ξ1x1 2 , since the multiplications of functions are commutative, while the compositions of operators are not commutative. [In some sense it is the noncommutativity in the quantum side that makes the whole theory interesting.]H. Weyl had the idea of using Fourier transform to solve this problem: for any nice function a(x, ξ) ∈ C ∞(T ∗R n ), we can quantize it to the operator A : L 2 (R n ) → L 2 (R n ) defined via Fourier transform, by (Af)(x) = (a W f)(x) := 1 (2π) n ❩ e i (x−y)]·ξ ~ a( x + y 2 , ξ)ψ(y)dydξ. Such operators are called semi-classical pseudodifferential operators. We will see that the Weyl quantization does not satisfy the quantization condition [a W , bW ] 6= ~ i {a, b} W , but it does satisfy a weakened version of quantization condition: [a W , bW ] = ~ i {a, b} W + O(~ 2 ). We will also quantize canonical relations (like the Hamiltonian flow) between cotangent bundles. The resulting operator are called Fourier integral operators. Pseudodifferential operators and Fourier integral operators are the major objects of this course. 1 In the language of representation, we really get a representation of the Heisenberg Lie group

LECTURE3—09/27/2020QUANTIZATIONV.S.SEMI-CLASSICAL LIMIT5(B) Geometric quantization. The theory was introduced by Kostant and Souriauin 1970s,and ismainly studied bygeometers.In afuture course on symplecticgeometry, I will describe in detail how to quantize in the general symplecticsetting. In geometric quantization, the Hilbert space will be taken to be the setof smooth sections of some “pre-quantum line bundle" satisfying some extraconditions.(C) Deformation quantization. The theory was introduced by Flato, Lichnerowiczand Sternheimer in 1976, and has been studied extensively since then mainly byalgebraists. In this theory, the main focus is trying to understand quantizationas a deformation of the (Poisson) structure of the algebra of classical observ-ables.For example,people want to construct a new noncommutativeproduct* on Co(M)[] (whose elements are formal power series in h with coefficientsin C(M), where M is a symplectic or Poisson manifold) such that(f,g) + O(h2).f*g-g*f=We will see that in Weyl quantization, such a star product can be defined forcotangent bundles.2. SEMICLASSICAL ANALYSISRoughly speaking, semiclassical analysis is the reverse of quantization: We wouldlike to get the information about the classical system from the information aboutthe corresponding quantum system. As we mentioned in Lecture 1, the guideline ofthe subject is the so-called Bohr's correspondence principle:Bohr's Correspondence Principle. The classical mechanics can be realized astheformal h-→o limit of the corresponding quantum mechanics.Remark. When we say h -→ O limit, we don't really evaluate the limit, but insteadstudy the asymptotic behavior of our objects as h -→ 0For example, the three theorems we mentioned in Lecture I are all in this spirit:(1) The Egorov's theoremeitQ/naWe-itQ/n = (pta)W+O(h)is a rigorous mathematical theorem justifying our claim “the propagatorU(t) = e-itQ/n is the quantum analogue of the classical geodesic flow ptassociated to the Hamiltonian q(r,s).(2) The Weyl law(2h) (Vol(r,5) | H(z,5) ≤) + (h)#tells us that asymptotically, the number of quantum energy below coincideswith the measure of the classical region with classical energy below >

LECTURE 3 — 09/27/2020 QUANTIZATION V.S. SEMI-CLASSICAL LIMIT 5 (B) Geometric quantization. The theory was introduced by Kostant and Souriau in 1970s, and is mainly studied by geometers. In a future course on symplectic geometry, I will describe in detail how to quantize in the general symplectic setting. In geometric quantization, the Hilbert space will be taken to be the set of smooth sections of some “pre-quantum line bundle” satisfying some extra conditions. (C) Deformation quantization. The theory was introduced by Flato, Lichnerowicz and Sternheimer in 1976, and has been studied extensively since then mainly by algebraists. In this theory, the main focus is trying to understand quantization as a deformation of the (Poisson) structure of the algebra of classical observables. For example, people want to construct a new noncommutative product ? on C ∞(M)[~] (whose elements are formal power series in ~ with coefficients in C ∞(M), where M is a symplectic or Poisson manifold) such that f ? g − g ? f = ~ i {f, g} + O(~ 2 ). We will see that in Weyl quantization, such a star product can be defined for cotangent bundles. 2. Semiclassical Analysis Roughly speaking, semiclassical analysis is the reverse of quantization: We would like to get the information about the classical system from the information about the corresponding quantum system. As we mentioned in Lecture 1, the guideline of the subject is the so-called Bohr’s correspondence principle: Bohr’s Correspondence Principle. The classical mechanics can be realized as the formal ~ → 0 limit of the corresponding quantum mechanics. Remark. When we say ~ → 0 limit, we don’t really evaluate the limit, but instead study the asymptotic behavior of our objects as ~ → 0. For example, the three theorems we mentioned in Lecture 1 are all in this spirit: (1) The Egorov’s theorem e itQ/~ a W e −itQ/~ = (ρ ∗ t a) W + O(~) is a rigorous mathematical theorem justifying our claim “the propagator U(t) = e −itQ/~ is the quantum analogue of the classical geodesic flow ρt associated to the Hamiltonian q(x, ξ)”. (2) The Weyl law #{j | λj ≤ λ} = 1 (2π~) n (Vol({(x, ξ) | H(x, ξ) ≤ λ}) + O(~)) tells us that asymptotically, the number of quantum energy below λ coincides with the measure of the classical region with classical energy below λ