On the quantization of mechanical systems

We present a canonical way of assigning to each magnitude of a classical mechanical system a differential operator in the configuration space, thus rigorously establishing the Correspondence Principle for such systems. Here we show how each classical state given in the whole system determine, for each classical magnitude, a wave equation, whose solutions are the possible quantum states for the given state and classical magnitude. Classical states and quantum states corresponding with the same system are reciprocally conditioned.


Contents
Physical magnitudes 1 Introducing a metric on M. Quantization 3 Wave equations 5 The problem of the time evolution 5 References 6 Physical magnitudes. Let M be a smooth manifold of dimension n, T M be its tangent bundle and T * M be its cotangent bundle; let C ∞ (M) denote the ring of complex valued (infinitely) differentiable functions in M.
On T * M it is defined the Liouvile form θ, by θ α = α, for each α ∈ T * M, and where the equality is understood assuming that α is lifted by pull-back from M to T * M. In local coordinates (x 1 , . . . , x n ) for M and the corresponding ones (x 1 , . . . , x n , p 1 , . . . , p n ) for T * M, the expression of the Liouville form is θ = p j dx j (summation with respect repeated indexes is assumed). The 2-form ω 2 := dθ is the symplectic form in T * M; its expression in local coordinates is ω 2 = dp j ∧ dx j .
The interior product with ω 2 establishes an isomorphism of C ∞ (T * M)-modules between tangent vector fields on T * M and differential 1-forms on T * M: D → i D ω 2 . In this isomorphism, the vertical tangent fields (those that, as derivations, annihilate the subring C ∞ (M) of C ∞ (T * M)) correspond to the horizontal 1-forms (those 1-forms annihilated, by interior product, by vertical fields).
In local coordinates, the vertical tangent field ∂/∂p j is applied on dx j . The isomorphism established by the symplectic form between the C ∞ (T * M)-modules of vertical fields and horizontal 1-forms, is naturally extended to an isomorphism between the tensor algebra of the "vertical" contravariant tensor and the "horizontal" covariant tensor algebras.
For our present purpose we are not interested in all this tensors, but only those corresponding to symmetrical covariant tensors on M: let A be the C ∞ (M)-algebra of symmetric covariant tensors on M, considered (after pull-back) as covariant tensors on T * M; the symplectic form made to A a C ∞ (M)-algebra A * of symmetric contravariant tensor fields on T * M. In local coordinates, the polynomial P (dx 1 , . . . , dx n ) with coefficients in C ∞ (M), corresponds to the polynomial with the same coefficients, P (∂/∂p 1 , . . . , ∂/∂p n ) ∈ A * .
In the tangent bundle the notions of vertical tangent field and horizontal 1-forms are analogous to those we just have considered in T * M. In local coordinates (x 1 , . . . , x n ,ẋ 1 , . . . ,ẋ n ), the horizontal 1-forms are the linear combinations of the dx j with coefficients in C ∞ (T M) and vertical tangent fields are linear combinations of the ∂/∂ẋ j . But here, if no additional structure to that of manifold is given, we have no symplectic form. The realization of the tensor algebra A that, in T * M, was done an algebra of vertical differential operators, in T M is given as an algebra of functions as follows: for each function f ∈ C ∞ (M) let us denote byḟ the function on T M defined by the ruleḟ (v) := v(f ), for each v ∈ T M considered as a derivation v : C ∞ (M) → R; the functionḟ is, essentially, df . In general, for each horizontal 1-form α on T M it is defined the functionα byα(v) := α, v (duality). By means of this rule, each tensor a ∈ A define a function a ∈ C ∞ (T M), that is polynomial in the fibres (a polynomial in theẋ' with coefficients in C ∞ (M)). The function a is, essentially, the same as the tensor a. In conservative mechanical systems, which are now our subject of study, all the magnitudes appearing in the space of position-velocity states are of this type. For that, we will call classical magnitudes to these functions a; the ring of the classical magnitudes is, then the ring A ≃ A obtained by associating to each tensor a ∈ A the function a. In local coordinates, a(ẋ 1 , . . . ,ẋ n ) is obtained if dx j is replaced byẋ j in the polynomial a(dx 1 , . . . , dx n ). As C ∞ (M)-algebras we have isomorphisms A ≃ A ≃ A * , in such a way that each classical magnitude a (function on T M) is, substantially, the same object that (is canonically identified to) the vertical differential operator a * on T * M.
It is convenient to establish this correspondence by a different way, by using the Fourier transform, as follows: The Liouville form can be interpreted as a function on the fibred product T M × M T * M, by assigning to each vector v x ∈ T x M and each 1-form α When the "dimensions" are introduced for the classical magnitudes, θ has dimension of "action". For this reason, in order to give full sense to the transcendent functions of θ and they have a meaning independent of the measures unities, it must be introduced a constant with dimension of "action" and take θ/ instead of θ. With this, it makes sense the function exp (iθ/ ) on To specify, let us take as base space for the Fourier transform S(T M), the set of complex valued functions defined on T M, that on each fibre T x M are C ∞ and rapidly decreasing they and all their derivatives. Analogous meaning for S(T * M).
In each fibre T x M ≃ R n , the usual measure is, up to a constant factor, the unique translation invariant (Haar measure of the group R n ). Once chosen on each fibre this Haar measure, we can define the Fourier transform fibre to fibre: being dµ the Haar measure fixed on T x M. By taking local coordinates (x 1 , . . . , x n ) on an open set U of M and the corresponding ones (x 1 , . . . , x n ,ẋ 1 , . . . ,ẋ n ) on T U, (x 1 , . . . , x n , p 1 , . . . , p n ) in T * M, we have θ(ẋ, p) = p jẋ j and then: where λ x is the constant that fixes the choice of the measure. By derivation under the integral sign its is obtained the classical formula: for any function a on T M that is polynomial in theẋ, that is to say, for each symmetric covariant tensor a on M.
We see that, by changing the symplectic form ω 2 by (i/ )ω 2 , the correspondence between classical magnitudes a ∈ A and vertical differential operators on T * M is the same that the given by the Fourier transform. By introducing already the factor i/ , let us denote by A the vertical differential operator on T * M that corresponds to the magnitude a: a and A are the expresions on T M, T * M of the same object, the tensor a.
Introducing a metric on M. Quantization. When the structure of smooth manifold is the only one given on M, there is no correspondence between points of T M and points of T * M, although the symplectic structure (or the Fourier transform fiberwise) has allowed us to establish the correspondence a → A between functions on T M which are polynomial on fibers and vertical differential operators on T * M.
Let T 2 be a riemannian metric (of arbitrary signature) given on M. The metric establishes an isomorphism of fiber bundles T M → T * M by assigning to each tangent vector v . By means of that isomorphism we can translate each structure from one to the other of those bundles; we can talk about the Liouville form, the symplectic form, etc., on T M. In other to simplify the notation, if there is no risk of confusion, we will use the same notation for each object on T M and its translation to T * M. In this way, if the local coordinated expression for the metric is T 2 = g ij dx i dx j , the function p k on T * M is, in the coordinates of T M, p k = g jkẋ j , and In the previously established correspondence a → A, to the functionẋ j there corresponds −i ∂/∂p j ; therefore, to the polynomial p ℓ = g ℓjẋ j it corresponds To the very metric tensor T 2 there corresponds on T M the function 2T = g jkẋ iẋj and, on T * M, the differential operator ∆ = − 2 g kℓ ∂/∂p k ∂/∂p ℓ , which, once introduced the metric on M becomes By taking local coordinates (x j ) on an neighborhood of x 0 in M and the corresponding ones (x j ,ẋ j ) in T M, the expression of exp is (where the Γ' are the Christoffel symbols of the metric), if we get the Taylor expansion till second order terms. That formula is easily derived from the differential equation defining the geodesics and, from it and the inverse function theorem it results that exp is a local differentiable isomorphim. For further details, see [3]. The exponential map allows us to assign to each function f ∈ C ∞ (M) a function f , defined on a neighborhood of the 0 section of T M, by means of the rule f (v x ) := f (exp v x ), for the v x ∈ T x M on which the exponential map is defined. The function f is the description of f done from each point of the configuration space (from "each observer"). If we denote by O(M) the ring of germs of C ∞ functions on neighborhoods of 0 section of T M, the assignation f → f determines an injection of C ∞ (M) into O(M) that we will call the riemannian injection.
The trivial injection C ∞ (M) → C ∞ (T M) given by the pull-back associated with the projection T M → M, translates all vertical differential operators in T M that without 0-order terms, to the identically 0 operator in C ∞ (M). But, thanks to the riemannian immersion, each differential operator A in T M gives on M a non trivial differential operator: And this seems to be the key of the quantification: Definition. Let a be a symmetric covariant tensor on M, a ∈ C ∞ (T M) its associated function ("classical magnitude") and A the vertical differential operator in T M corresponding to a (once identified T M and T * M by means of the metric). The differential operator The quantification a → a is C ∞ (M)-linear for the module structure in the set of differential operators (given by left multiplication by functions). It is also injective, since, if the vertical differential operator A annihilates all f coming from C ∞ (M), it holds A = 0, because on each fibre of T M, A is a polynomial in the ∂/∂x with constant coefficients. However, the multiplicative structure changes, so losing the commutativity: given two symmetric covariant tensors a, b in M, for each f ∈ C ∞ (M), the computation of b( a f ) is made by applying the differential operator B to the function af , that differs from A f ; this is why, in general, it does not hold b • a = ba. In general neither is true a 2 = a 2 , as we will see later in some particular instance.
For the time being, the Taylor expansion of second order of the exponential map, allows us to find the quantum operators corresponding to the functions that are polynomials of degree lower or equal than 2 in theẋ.
For each f ∈ C ∞ (M) we have In particular, for the metric tensor T 2 we get where ∆ is the laplacian operator associated with the metric.
For the computation of a when a has arbitrary order, the formulae in [3] can be used.
Remark. The quantization is a local operation in M. Global conditions must be imposed once are fixed the space of admissible solutions for the wave equations.
The correspondence a → a is established for the tensor as a whole and it cannot be "factorized" because, in the general case, is not even a 2 = a 2 .
Also it is generally false that the Poisson bracket {a, b} corresponds to the commutator [ a, b]: in fact {a, b} contains just first order derivatives of the coefficients of tensors a, b, while in [ a, b] appear, in general, derivatives whose order is the greatest order of a, b.
Wave equations. It is not our current topic the prolongation of operators a to spaces out of C ∞ (M).
The classical magnitudes a that we are considering, are functions in T M, which are not automatically operators on C ∞ (M) (except when a is a tensor of order 0, a function in M).
For each section u : M → T M (a vector field on M), the restriction a(u) of the function a to the section u, is transported as a function on M and, as such, operates by multiplication on C ∞ (M).
Functions Ψ in C ∞ (M), or in a prefixed space of functions or distributions, where operators a and a(u) coincide are the functions proper for the magnitude a in the classical state u. They are the solutions of the wave equations ( a − a(u))Ψ = 0.
As an example, for a conservative mechanical system with hamiltonian H = T + U, the quantization of H is and the wave equation is When u is a section (a classical state When one takes as space of admissible "pure" states for the quantum-mechanic system the Hilbert space (complex and separable), Stone theorem about unitary one-parametric groups of automorphism (see [6], Ch.XI, §13) allows us to invert the direction of the transition classical → quantum: it is postulated that the evolution of the quantum system is given by a oneparametric group of unitary automorphisms; the infinitesimal generator has necessarily the form i × (self adjoint operator), and the parallelism with Classical Mechanics forces the group to be the one generated by a constant multiple of the quantified Hamilton function (it remains the problem of find that quantified operator, a problem we take for solved). Passing the time evolution of the states to that of operators ("Heisenberg representation"), the evolution law would be [ H, a t ] = i d a t /dt, in analogy with the equation of the evolution for the classical magnitudes: da/dt = {H, a} (along each trajectory).
The problem is the non-existence of a function which could be called "time" in the formalism of the Classical Mechanics (see [4]), although there exist a time of travel along each trajectory of a given mechanical system. To introduce a "time" function, in general it is needed the addition of a dimension to the configuration space and then to impose a constraint by a non univocally defined procedure. However, in case of conservative systems, for each lagrangian manifold of a complete integral of the Hamilton-Jacobi there is a "time" function that parameterizes any particular solution curve within the lagrangian manifold; except for the systems free of forces (geodesic), this time is not proportional to the "action" function in the lagrangian manifold, so that front waves of the action does not move to a constant temporal rate; the "time" which measures the movement of the wave fronts is another one: the quotient of action by energy. In the foundational memory of Schrödinger [5] the first part is dedicated to the hamiltonian analogy between Mechanics and Optics; it seems that Schrödinger is carried away by the intuition of absolute time and he use the "t" with two different meanings: in the formulae (3),(9), t is the time that parameterizes the trajectories while in formulae (5), (6), t is the quotient action/energy. With this second "time" the wave fronts of the action move at a constant rate. The primitive Quantum Theory leads us to interpret the t in the formulae of Quantum Mechanics in this way. On each stationary solution Φ = e −iEt/ Ψ, the "classical" meaning of t must be action/energy in the lagrangian manifold from which the Schrödinger equation for Ψ was written.
In a subsequent paper, we will study the nature of "time" in Classical, Undulatory and Quantum Mechanics in conservative systems.