Beyond coherent state quantization

Beyond Coherent State Quantization Laure Gouba1 1 The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, I-34151 Trieste Italy. Email: [email protected] (Dated: September 11, 2019) We present an original approach to quantization based on operator-valued measure that general- izes the so-called Berezin-Klauder-Toeplitz quantization, and more generally coherent state quanti- zation approches. I. INTRODUCTION In Physics, quantization is generally understood as a correspondence between a classical and a arXiv:1909.03928v1 [math-ph] 9 Sep 2019 quantum theory and dequantization is the opposite process by which one starts with a quantum theory and arrives back at its classical counterpart. The processes of quantization and dequantiza- tion have evolved into mathematical theories, affecting the areas of group representation theory and symplectic geometry. Quantization is not a method for deriving quantum mechanics but a way to understand the deeper physical reality which underlies the structure of both classical and quantum mechanics and which unifies the two from a geometrical point of view. There is a certain mathemat- ical richness in the various theories of quantization where the procedure does make sense. However, not every quantum system has a classical counterpart and then for such a system a quantization method does not make sense, moreover different quantum systems may reduce to the same classical theory. Originally P. A. M. Dirac introduced the canonical quantization in his 1926 doctoral thesis, the method of classical analogy for quantization [1]. The canonical quantization or correspondence principle is an attempt to take a classical theory described by the phase space variables, let’s say p and q, and a Hamiltonian H(q, p) to define or construct its corresponding quantum theory. The following simple technique for quantizing a classical system is used. Let q i , pi , i = 1, 2 . . . n, be the canonical positions and momenta for a classical system with n degrees of freedom. Their quantized counterparts q̂ i , p̂i are to be realized as operators on the Hilbert space H = L2 (Rn , dx) by the prescription ∂ (q̂ i ψ)(x) = xi ψ(x); (p̂i ψ)(x) = −i~ ψ(x); i = 1, 2, . . . n, x ∈ Rn . (1) ∂xi This procedure is known as canonical quantization and is the basic procedure of quantization of a classical mechanics model [2–4]. More general quantities, such as the Hamiltonians, become operators according to the rule H(p, q) → H = H(p̂, q̂), (2) an expression that may have ordering ambiguities [5, 6]. In which canonical coordinates system does such a quantization procedure work? 1. According to Dirac replacing classical canonical coordinates by corresponding operators is found in practice to be successful only when applied with the dynamical coordinates and mo- menta referring to a cartesian system of axes and not to more general curvilinear coordinates. 2. Cartesian coordinates can only exist on a flat space. 3. The canonical quantization seems to depend on the choice of coordinates.  2 4. Beyond the ordering problem, one should keep in mind that [Q, P ] = i~ Id holds true with self adjoint operators Q, P , only if both have continuous spectrum (−∞, +∞), and there is uniqueness of the solution, up to unitary equivalence (von Neumann). There are two attitudes that may be taken towards this apparent dependence of the procedure of the canonical quantization procedure on the choice of coordinates. The first view would be to acknowl- edge the cartesian character that is seemingly part of the procedure. The second view would be to regard it as provisional and seek to find a quantization formulation that eliminates this apparently unphysical feature of the current approaches. The aim of eliminating the dependence on cartesian coodinates in the standard approaches is no doubt one of the motivations for several procedures such as the geometric quantization [7–10], the path integral quantization [11], the deformation quanti- zation [12–14, 16, 17], the Klauder-Berezin-Toeplitz quantization [18–20]. However, the canonical quantization is of the most famous and simplest procedure that is mostly accepted due to its ex- perimental validations. In a recent review [21] by S. T. Ali and M. Engliš various techniques of quantization conceived to make the transition from classical mechanics to quantum mechanics in the last five decades are presented. The point of departure is always an analysis of the geometrical structure of either the classical phase space or the classical configuration space. There is no general theory of quantization presently available which is applicable in all cases, and indeed, often the techniques used to quantize has to be tailored to the problem in question. As we have already said, canonical quantization generally requires the use of cartesian coordinates and not more general coordinates. So for a dynamical system without any constraints, its phase space is assumed to be flat and admits a standard quantization of its canonical variables. In presence of constraints, all the variables of the system are not physical and the unphysical variables that may cause some little concerns at a classical level are highly unwelcome at the quantum level. It is then necessary to eliminate the unphysical variables and keep only the true physical degrees of freedom. The quantization of the physical degrees of freedom follows as in standard quantization. The study of constrained systems for the purpose of quantization was initiated by Dirac [22–24], where he developed a method for treating singular systems and constraints. While Dirac’s method and subsequent developments can cope with most models of interest, there are some problems in the sense that the quantization of the remaining degrees of freedom after one eliminates the unphysical variables may not be straightforward because the reduced (physical) phase space is non-euclidean in the sense that an obstruction has arisen where none existed before and that obstruction precludes the existence of self-adjoint canonical operators satisfying the canonical commutation relations. Other approaches for quantizing constrained systems of classical theories have been subsequently developed [25–29]. Recently we have been interested in applying the Dirac’s method in the study of the two-dimensional damped harmonic oscillator in the extended phase space [30] as some years ago, we have used the method to study nonperturbatively scalar and spinor abelian gauge theories in 1 + 1 dimensions [31]. Inspired by Klauder’s paper on coherent state quantization of constrained systems [32], we are interested in understanding a procedure of quantization beyond the coherent states quantization, generically called integral quantization mostly developed by J. P. Gazeau and al. [33–37], in order to better study some of the constrained systems where the method is applicable. The paper is organized as follows. In section II basic notions on coherent states are given. In section III, starting from the Klauder-Berezin-Toeplitz quantization and the prime quantization, we reach the coherent state or frame quantization. In section IV, we start by introducing first the sea star algebra and then we introduce the integral quantization. The perspectives are included in the later section. II. BASIC NOTIONS ON COHERENT STATES Are coherent states the natural language of quantum theory? According to J. R. Klauder [38] the answer to the question is Yes! In his paper he displayed various fundamental aspects of quantum  3 theory from the perspective of a coherent state formulation. The study of coherent states affect almost all branches of quantum physics: quantum optics, nuclear physics, atomic physics, solid -state physics, quantum electrodynamics (the infrared problem), quantization and dequantization problems, path integrals, quantum gravity, quantum information. Coherent-state methods have been proved convenient in the analysis of dual models of strings [39], and likewise they have been useful in the interacting boson model in nuclear physics, semiclassical calculations in chemistry have also made effective use of coherent state methods. The first example of what is now called coherent states was discored by Schrödinger [43] as he was interested in studying quantum states which restore the classical behavior of the position operator of a quantum system in the Heisenberg picture, i i Q(t) = e ~ Ht Qe− ~ Ht , (3) 2 P where H = 2m + V (Q) is the quantum Hamiltonian system. Schrödinger understood by classical behavior that the expected value or average q̄(t) of the position operator Q(t) in the desired state would obey the classical equation of motion ∂ V̄ mq̄¨(t) + = 0. (4) ∂q Schrödinger discovered the first example of coherent states pertaining to the harmonic oscillator V (q) = 21 m2 ω 2 q 2 , known universally to physicists, and the prototype of every integrable model. These states are parametrized by the complex number z, satisfying hz|Q(t)|zi = 2Q0 |z| cos(ωt − ϕ), (5) 1 z = |z|eiϕ and Q0 = (~/2mω) 2 is the fundamental quantum length, ~ the universal constant. Using the generating function found in the classic book of Courant and Hilbert, Schrödinger realized that a Gaussian wave-function could be constructed from a particular superposition of the wave functions corresponding to the discrete eigenvalues of the harmonic oscillator. Further these new states followed the classical motion. At this time the probability-amplitude nature of the wave function was not yet known, so the complex nature of the wave function bothered Schrödinger. He wondered if, perhaps, it was only the real part of the wave function that is physical. At that time the uncertainty relations had yet to be discovered, but from the point of view that most closely resembles the modern minimum uncertainty method, Schrödinger had discovered the coherent states. The notion of coherent states is rooted in quantum physics and its relation to classical physics and the term coherent was introduced by R. J. Glauber in 1963 in the field of quantum optics, and it is for a special property in quantum optics that the name coherent states was originally chosen, for instance from the current language of quantum optic there are expressions like coherent radiation, sources emitting coherently, . . . The importance of coherent states became widely recognized during the 1960’s due to the works of Glauber, Klauder and Sudarshan [40–42, 44–46]. Although the name coherent states is applied to a wide class of objects, in every case the set of states referes to vectors in a Hilbert space H (finite or countably infinite dimensional). We denote the states in question, in Dirac notation, by |li, where l is an element (in general multidimensional) of an appropriate label space D endowed with a notation of continuity (topology). More specifically, an appropriate label space is one for which every finite dimensional subspace is locally euclidean. There are in essence just two properties that all coherent states share in common: 1. Continuity: the vector |li is a strongly continuous function of the label l. 2. Resolution of unity (completeness): there exists a positive R measure δl on D such that the unit operator Id admits the ”resolution of unity” Id = |lihl|δl when integrated over D. The Schrödinger-Klauder-Glauber-Sudarshan coherent states, denoted here |zi, also called canon- ical coherent states or standard coherent states satisfy the following properties:  4 1. the states |zi saturate the Heisenberg inequality h∆Qiz h∆P iz = 21 ~; 2. the states |zi are eigenvectors of the annihilation operator, with eigenvalue z, a|zi = z|zi, z ∈ C; 3. the states |zi are obtained from the ground state |0i of the harmonic oscillator by a unitary action of the Weyl Heisenberg group, † |zi = eza −z̄a |0i; (6) 4. the coherent state {|zi} constitute an overcomplete family of vectors in the Hilbert space of the states of the Harmonic oscillator. This property is encoded in the following resolution of the identity 1 Z Id = |zihz|dRe(z)dImz. (7) π C These four properties are, to various extents, the basis of the many generalizations of the canonical notion of coherent state, illustrated by the family {|zi}. The unique qualification of the coher- ent states to allow for a classical reading in a host of quantum situations results from these four properties. III. COHERENT STATES QUANTIZATION Quantization procedure is an important aspect of coherent states, for instance standard coherent states offer a classical-like representation of the evolution of quantum observables. In coherent states quantization, we are interested in quantization of sets, more precisely measure spaces through coherent states. For the normalized coherent states |zi, the resolution of the unity 1 Z |zihz|d2 z = Id (8) π C is a crucial property for our purpose in setting the bridge between the classical and the quantum worlds. A. Klauder-Berezin-Toeplitz quantization The Klauder-Berezin coherent state quantization also named anti-Wick quantization or Toeplitz quantization by many authors, consists in associating with any classical operator f that is a function of phase space variable (q, p) or equivalently of (z, z̄), the operator valued integral 1 Z f (z, z̄)|zihz|d2 z = Af . (9) π C The function f is usually supposed to be smooth, but we will not retain in the sequel this too restrictive attribute. The resulting operator Af , if it exists, at least in a weak sense, acts on the Hilbert space H of quantum states for which the set of Fock (or number ) states |ni is an orthogonal basis. It is worthwhile being more explicit about what we mean by weak sense: the integral d2 z Z hψ|Af |ψi = f (z, z̄)|hψ|zi|2 (10) C π should be finite for any |ψi in some dense subset in H. One should note that if |ψi is normalized then the integral (10) represents the mean value of the function f with respect to the ψ -dependent  5 probability distribution z → |hψ|zi|2 on the phase space. More mathematical rigor is necessary here, and we will adopt the following acceptance criteria for a function (or distribution) to belong to the class of quantizable classical observables. A function f : C → C, z 7→ f (z, z̄) (more generally a distribution T ∈ D′R(R2 ) ) is a coherent state quantizable classical observable along the map f 7→ Af defined by Af = π1 C f (z, z̄)|zihz|d2 z ( more generally T 7→ AT ) : 1. if the map z 7→ hz|Af |zi (or z 7→ hz|AT |zi), z = √12 (q + ip) ≡ (q, p), is a smooth ( ∈ C ∞ ) function with respect to the (q, p) coordinates of the phase space, 2. and if we restore the dependence on ~ through z → √z , we must get the right semiclassical ~ limit, which mean that h √z~ |Af | √z~ i → f ( √z~ , √z̄~ ), the same asymptotic behavior must hold in a distributional sense if we are quantizing distributions. B. Ordering problem in quantum mechanics: prime quantization Let assume we have a classical observable, precisely a real-valued function f of the position variable q and the momentum variable p. We assume that the configuration space of the system is R1 means P that it moves on the phase space T ⋆ (R1 ) ∼ R2 . If f admits a Taylor expansion, one has ∞ f (q, p) = m,n=0 Cmn q m pn , where Cmn are appropriate coefficients which could possibly be zero for m, n > N for some integer N . In a quantized theory, the variables q, p are replaced by essentially self-adjoint unbounded operators Q and P , respectively, on a (separable) Hilbert space H, satisfying the canonical commutation relations (over a common dense domain D, [Q, P ] = i, (~ = 1). What the self-adjoint operator F on H, corresponding P∞ to the classical observable f should be? It seems plausible to quantize f via F = m,n=0 Cmn Qm P n . Although the infinite sum of unbounded P∞ m n operators appearing in F = m,n=0 Cmn Q P has a meaning and even if F was a self-adjoint operator on a dense domain DF ⊂ H, one has to decide on aP certain ordering of the noncommutating ∞ operators Q and P , when their products appear in F = m,n=0 Cmn Qm P n . For instance, it is a matter of some arbitrariness as to whether the classical observable qp2 is to be replaced by the quantum operator 12 [QP 2 +P 2 Q] or 14 [QP 2 +2P QP +P 2 Q] or some other self-adjoint combinations. Some of the combinations are mathematically equivalent, however, in our opinion, not enough physical justification is given for choosing one ordering over another. We introduce the prime quantization, that is a quantization technique developped by H. D. Doebner [47–49]. This technique sheds light on the time honoured ordering problem of quantum mechanics and has connections to coherent state quantization and Klauder-Berezin-Toeplitz quantization. The method starts out on the phase space Γ of the classical system and exploits the functional analytical rather than geometric structures of the system. One starts with the symplectic manifold (Γ, Ω) and the Hilbert space H = L2 (Γ, Ω). The next step is to look for reproducing kernel subspaces of H. Let HK a subspace of H, on which the evaluation maps Eγ : HK → C, Eγ (ψ) = ψ(γ), γ ∈ Γ, ψ ∈ HK (11) are continuous. Then, K(γ, γ ′ ) = Eγ Eγ⋆′ is a reproducing kernel. For any Borel set ∆ ⊂ Γ, the operator Z aK (∆) = Eγ⋆′ Eγ dµ(γ), (12) ∆ on HK is positive. A classical observable f : Γ → R is then quantized by the prescription, Z Q(f ) = f (γ)Eγ⋆′ Eγ dµ(γ). (13) Γ  6 One has to prove the asymptotic validity of [Q(f ), Q(g)] ∼ i~Q({f, g}). (14) The challenge in this method is to find the appropriate reproducing kernel Hilbert space HK , which is reflective of the physical problem. C. Coherent states or frame quantization We consider the Hilbert space L2K (X, µ)(K = R or K = C) of square-integrable real or complex functions f (x) on the observation set X : X |f (x)|2 µ(dx) < ∞. A classical observable is a function R f (x) on X having specific properties with respect to some supplementary structure allocated to X, such as topology, geometry, or something else. Given a frame[52] {|xi, x ∈ X}, a function f : x(∈ X) 7→ f (x)(∈ K), possibly undertood in a Rdistribution sense, is a coherent state quantizable classical observable along the map f 7→ Af = X f (x)|xihx|ν(dx), if the map x(∈ X) 7→ hx|Af |xi is a regular function with respect to some additional structures allocated to X. We must get the right classical limit, which means that hx|Af |xi = f (x) as a certain parameter goes to 0. A quantization R of the observation set X is in one-to-one correspondence with the choice of a frame in the sense of X |xihx|ν(dx) = Id where ν(dx) = N (x)µ(dx). This term of frame is more appropri- ate for designating the total family {|xi}x∈X . This frame can be discrete or continuous, depending on the topology additionnaly allocated to the set X, and it can be overcompleted. The validity of a precise frame choice is determined by comparing spectral characteristics of quantum observables Af with experimental data. A quantization method associated with a specific R frame is intrinsi- cally limited to all those classical observables for which the expansion Af = X f (x)|xihx|ν(dx) is mathematically justified within the theory of operators in Hilbert space (weak convergence). The coherent state quantization in this context requires a minimal significant structure on X, namely, the existence of a measure µ(dx), together with a σ-algebra of measurable subsets, and some additional structure to be defined depending on the context. The construction of the Hilbert space H is equivalent to the choice of a class of eligible quantum states, together with a technical condition of continuity. A correspondence between classical and quantum observables is then provided through a suitable generalization of the standard coherent states. IV. BEYOND COHERENT STATE QUANTIZATION: INTEGRAL QUANTIZATION Let us brieffly recall the following. We consider an orthonormal basis (or frame) of the Euclidean plane R2 defined by the two vectors (in Dirac ket notations)     ~i ≡ |0i ≡ 1 ~ π 0 and j ≡ | i ≡ , (15) 0 2 1 and |θi denotes the unit vector with polar angle θ ∈ [0, 2π). This basis or frame is such that h0|0i = 1 = h π2 | π2 i and h0| π2 i = 0. The sum of their corresponding orthogonal projectors resolves the unit Id = |0ih0| + | π2 ih π2 | , that is equivalent to       1 0 1 0 0 0 = + . (16) 0 1 0 0 0 1  7 To the unit vector |θi = cos θ|0i + sin θ| π2 i corresponds the orthogonal projector Pθ given by Pθ = |θihθ| and cos2 θ cos θ sin θ     cos θ |θihθ| = (cos θ sin θ) = (17) sin θ cos θ sin θ sin2 θ Pθ = R(θ)|θihθ|R(−θ), where   cos θ − sin θ R(θ) = ∈ SO(2). (18) sin θ cos θ A. Sea star algebra The five-fold symmetry of sea star (starfish) motivated the development in a comprehensive way, the quantization of functions on a set with five elements which yields in particular the notion of a quantum angle. The purpose of this introductory model is to enhance our intuition that quantization can be viewed as the analysis of a set from the point of view of a family of coherent states. FIG. 1: A sea star as a five-fold frame for the plane 1. sea stars have five arms, or rays, connected to a small round body; 2. sea stars detect light with five purple eyespots at the end of each arm; 3. sea stars typically show pentameral symmetry. How the ocean floor is ”five-fold orientationally ’ explored by a sea star? A sea star possibly do this in a noncommutative way through the pentagonal set of unit vectors (the arms) forming a five-fold frame in the plane. The unit vectors (arms) are determined by | 2nπ 2nπ 5 i = R( 5 )|0i ≡ ”coherent” state n = 0, 1, 2, 3, 4 mod 5. We get the resolution of the identity: 4   2 X 2πn 2πn 1 0 | ih |= ≡ Id . (19) 5 n=0 5 5 0 1 The property holds for any regular N-fold polygon in the plane N −1   2 X 2πn 2πn 1 0 | ih |= . (20) N n=0 N N 0 1  8 Let’s make more precise the set X explored by the sea star, determine a probabilistic construction of the frame by using the Hilbert space structure of the plane and an exact way of exploring X through functions on this finite set. The sea star senses its possible orientations via the five angles 2πn 5 and X = {0, 1, 2, 3, 4} the set of orientations. It is equipped with discrete measure with uniform weight 4 2X Z f (x)dµ(x) = f (n). (21) x 5 n=0 Let us choose two orthonormal elements, φ0 (n) = cos 2πn 2πn 5 and φ1 (n) = sin 5 in the Hilbert space 2 L (X, µ), and build the five unit vectors in the real two-dimensional Hilbert space, that is the Euclidean plane R2 = H, with the usual orthonormal basis |0i, | π2 i: 2πn π n ∈ X, n 7→ |ni ≡ | i = φ0 (n)|0i + φ1 (n)| i. (22) 5 2 The operators Pn = | 2πn 2πn 2 5 ih 5 | acts on H = R . Given a n0 ∈ {0, 1, 2, 3, 4}, one derives the probability distribution on X = {0, 1, 2, 3, 4} 2π(n0 − n)   2πn0 2πn 2 tr(Pn0 Pn ) = |h | i| = cos2 . (23) 5 5 5 The five unit vectors |ni ≡ | 2πn 5 i resolve the identity in H. They form a finite unit frame for analysing the complex-valued functions n 7→ f (n) on X through what we call a coherent state (CS) quantization: 4 2X 2nπ 2nπ Z f (n) 7→ |xihx|f (x)dµ(x) = f (n)| ih | ≡ Af . (24) X 5 n=0 5 5 If we had chosen instead as a finite frame the orthonormal basis |0i, | π2 i, in R2 , we would have obtained the trivial commutative quantization: (f (0), f (1)) 7→ Af = diag(f (0), f (1)). (25) This map should be analysed more precisely through spectral values of the 2 × 2 symmetric matrix Af , and also through coherent states mean values of Af : 4 2nπ 2nπ 2 X 2(n − m)π n 7→ fˇ(n) := h |Af | i= f (m) cos2 , n = 0, 1, 2, 3, 4. (26) 5 5 5 m=0 5 B. Integral quantization An original approach to quantization based on operator valued measures is named generically integral quantization. The approach generalize the coherent state quantization. The probabilistic aspects appearing is highlighted at each stage of the quantization procedure. To be more mathe- matically precise and still remaining at a basic level, quantization is a linear map Q : C(X) → A(H) from a vector space C(X) of complex valued function f (x) on a set X to a vector space A(H) of linear operator Q(f ) ≡ Af in some complex Hilbert space H, such that 1. to the function f = 1 there corresponds the identity operator Id on H; 2. to a real function f ∈ C(X) there corresponds a (an essentially) self adjoint operator Af in H.  9 The above conditions may be easily fulfilled if one uses the resources offered by the pair (measure, integration). Let (X, ν) be a measure space. Let H be a complex Hilbert space and x ∈ X, x 7→ M (x) ∈ L(H) an X-labelled family of bounded operators on H resolving the identity: Z M (x)dν(x) = Id , (27) X provided that the equality is valid in a weak sense, which implies ν integrability for the family M (x). If the operators M (x) are positive and have unit trace, they will be preferentially denoted by M (x) = ρ(x) in order to comply with the usual notation for a density matrix in quantum mechanics. The corresponding quantization of complex-valued functions f (x) on X is then defined by the linear map: Z f 7→ Af = M (x)f (x)dν(x). (28) X This operator-valued integral is again understood in the weak sense, as the sesquilinear form, Z Bf (ψ1 , ψ2 ) = hψ1 |M (x)|ψ2 if (x)dν(x). (29) X The form Bf is assumed to be defined on a dense subspace of H. If f is real and at least semi- bounded, the Friedrichs extension of Bf univocally defines a self-adjoint operator. 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