# Quantum Hamiltonian Mechanics on Symplectic Manifolds

## Abstract

In the previous chapter we presented the general theory of quantum deformations of classical Poisson algebras. In the following chapter we develop a deformation procedure applied to classical statistical Hamiltonian mechanics (described in Sect. 3.3) in order to construct its quantum analogue on the phase space. First, we define quantum states as appropriate deformations of classical states and their time development through the respective deformation of the classical Liouville equation. Then we introduce quantum Hamiltonian equations of motion being a deformation of classical Hamiltonian equations and time development of quantum observables. With particular care we present the theory of quantum flow and quantum trajectories on a phase space together with a wide range of examples which illustrate the presented formalism. Such constructed quantum theories (each related with an appropriate quantum algebra) reduce to a common classical counterpart as deformation parameter \(\hslash \) tends to zero: \(\hslash \rightarrow 0.\)

In the previous chapter we presented the general theory of quantum deformations of classical Poisson algebras. In the following chapter we develop a deformation procedure applied to classical statistical Hamiltonian mechanics (described in Sect. 3.3) in order to construct its quantum analogue on the phase space. First, we define quantum states as appropriate deformations of classical states and their time development through the respective deformation of the classical Liouville equation. Then we introduce quantum Hamiltonian equations of motion being a deformation of classical Hamiltonian equations and time development of quantum observables. With particular care we present the theory of quantum flow and quantum trajectories on a phase space together with a wide range of examples which illustrate the presented formalism. Such constructed quantum theories (each related with an appropriate quantum algebra) reduce to a common classical counterpart as deformation parameter \(\hslash \) tends to zero: \(\hslash \rightarrow 0.\)

## 7.1 General Theory of Quantization

### 7.1.1 Quantum States

*quantum states*related to quantum Poisson algebra Open image in new window are those functions

*ρ*∈

*L*

^{2}(

*M*,

*d*Ω

_{ħ}) which satisfy the following conditions

- 1.
*ρ*=*ρ*^{∗}(self-conjugation), - 2.
Open image in new window (normalization),

- 3.
Open image in new window for \(f\in C_{0}^{\infty }(M)\) (positive-definiteness).

*L*

^{2}(

*M*,

*d*Ω

_{ħ}).

*Pure states*are defined as extreme points of the set of states, i.e. as those states which cannot be written as convex linear combinations of some other states. Thus

*ρ*

_{pure}is a pure state if and only if there do not exist two different states

*ρ*

_{1}and

*ρ*

_{2}such that

*ρ*

_{pure}=

*pρ*

_{1}+ (1 −

*p*)

*ρ*

_{2}for some

*p*∈ (0, 1). A state which is not pure is further called a

*mixed state*.

*M*pure states can be alternatively characterized as functions

*ρ*

_{pure}∈

*L*

^{2}(

*M*,

*d*Ω

_{ħ}) which are self-conjugated, normalized, and idempotent (cf. Sect. 3.3.1):

*ρ*

_{mix}∈

*L*

^{2}(

*M*,

*d*Ω

_{ħ}) can be characterized as convex linear combinations, possibly infinite, of pure states \(\rho _{\text{pure}}^{(\lambda )}\)

*p*

_{λ}≥ 0 and ∑

_{λ}

*p*

_{λ}= 1.

*ρ*are the analogue of the classical distribution functions representing states of the classical Hamiltonian system and considered in Sect. 3.3.1. The difference between classical and quantum distribution functions is that the latter do not have to be non-negative everywhere. Thus,

*ρ*(

*x*,

*p*) cannot be interpreted as a probability density of finding a particle in a point (

*x*,

*p*) of the phase space. This is a consequence of the fact that

*x*and

*p*coordinates do not commute with respect to the ⋆ -multiplication, which yield, from the Heisenberg uncertainty principle, that it is impossible to measure simultaneously the position and momentum of a particle like in classical mechanics. Hence, the point position of a particle in the phase space does not make sense anymore. On the other hand it is possible to introduce the so called

*marginal distributions*

*x*or momentum

*p*(see Sect. 8.1.2). In (7.1.3)

*S*-operator links a given ⋆ -product with the Moyal one. The result is not surprising as each marginal distribution depends on commuting coordinates only. Note, however, that only in the case of the ⋆

_{M}-product the marginal distributions are received by simple integration of a distribution function with respect to

*x*or

*p*variable. In general, the distribution function first has to be transformed with the isomorphism

*S*.

*ρ*the expectation value of the observable

*A*in the state

*ρ*is defined by

### 7.1.2 Time Evolution of Quantum Systems

*H*

_{C}and self-conjugated with respect to involution ∗ of respective ⋆ -algebra. Like in the classical theory, there are two equivalent points of view on the time evolution: quantum Schrödinger picture and quantum Heisenberg picture. In the Schrödinger picture states undergo time development while observables do not. An equation of motion for states which is the analogue of classical Liouville equation ( 3.3.11) takes the form

*unitary function*as

*H*is self-conjugated

In consequence, the time evolution of states can be expressed in terms of the one-parameter group of unitary functions *U*(*t*). Notice that *quantum Liouville equation* (*von Neumann equation*) on the symplectic manifold is represented like its classical analogue by a linear PDE.

*ρ*which do not depend explicitly on time: \( \frac {\partial \rho }{\partial t}=0\), are called stationary states and hence fulfill the relation

*ρ*is a pure state, then ( 7.1.9) is equivalent to a pair of ⋆ -genvalue problems

*E*in (7.1.10) is equal to the expectation value of Hamiltonian

*H*in a pure stationary state

*ρ*, i.e. is equal to energy of a system in that state

*ρ*(

*t*), i.a. \(\left \langle A\right \rangle _{\rho (t)}\), fulfills the following equation of motion

*U*(

*t*) from (7.1.7) on

*A*:

*U*(

*t*) is given by (7.1.7).

*t*results in the evolution equation for

*A*:

*quantum Hamiltonian equations*of motion are of the form

*quantum Hamiltonian transport*. What is important, the system of PDE’s (7.1.14) is equivalent to the system of ordinary differential equations but in the space of ⋆ -functions. Indeed, any function \(A\in \mathcal {A}_{Q}\) can be expanded in a ⋆ -power series ( 6.1.47). In particular, any monomial

*x*

^{n}

*p*

^{m}can be expressed as an ⋆ -polynomial ( 6.2.16)

*x*,

*p*) coordinates and Weyl ordering, with transformed Hamiltonian

*H*(

*ħ*) =

*S*(

*ħ*)

*H*.

### Example 7.1

*x*,

*p*)

*x*,

*p*) chart takes the Moyal form, then quantum equations of motion, according to (7.1.16), take the form where

*Q*(0) =

*x*,

*P*(0) =

*p*and \(\star \equiv \star _{M}^{(x,p)}.\) The solution of classical and quantum dynamics will be considered in the next section.

### Observation 14

*Comparing the results of Sects.* *3.3.2* *and*7.1.2*we observe that the linear aspect of classical and quantum Hamiltonian mechanics is represented by time evolution of states, described on both levels by linear PDE (the so called Schrödinger picture). On the other hand, the nonlinear aspect of both theories is represented by time evolution of observables, described on both levels by nonlinear ODE (the so called Heisenberg picture) defined on an appropriate space of ordinary-functions and star-functions, respectively. Contrary to a classical case, on a quantum level Hamiltonian equations of motion belong only to the Heisenberg picture as pure coherent classical states* ( *3.3.16* ) *are not admissible as quantum states.*

## 7.2 Quantum Trajectories in Phase Space

The time evolution of a classical Hamiltonian system is fully determined by trajectories (a flow) in a phase space (see Sect. 3.3.2 ). Once we calculate a classical flow Φ_{t} for the given system a time evolution of states and observables can be received by simply composing them with Φ_{t}. A classical flow is defined as a map Φ_{t}: *M* → *M* on the phase space *M*, which at every point *ξ*_{0} ∈ *M* gives a trajectory (curve) *γ*(*t*) = Φ_{t}(*ξ*_{0}) on *M* passing through the point *ξ*_{0} and being a solution of the Hamilton’s equations ( 3.2.35). Moreover, any trajectory Φ_{t}(*ξ*_{0}) has the property of being a classical canonical transformation for every *t*, and the set \(\{\Phi _{t}\}_{t\in \mathbb {R}}\) have a structure of a group with multiplication being a composition of maps.

From the very beginning of quantum physics, a lot of efforts have been taken to formulate some kind of an analogue of phase space trajectories in quantum mechanics [95]. The most common approaches to quantum dynamics are the de Broglie-Bohm approach [50, 51, 154], the average value approach [181, 266], and the Moyal trajectories approach (see [93, 174] and references therein).

In the following section we develop the theory of Moyal trajectories resulting from quantum Hamiltonian equations (7.1.14). In consequence, the time evolution of observables cannot be given as a simple composition of observables with a quantum flow. For this reason in papers [93] and [174] observables were considered to be ⋆ -functions. Then the action of a flow on observables was given as a ⋆ -composition.

In the approach presented in this section we treat observables as ordinary functions on a classical phase space. We also present in an explicit form a quantum action of a flow on observables, which is a deformation of the respective classical action. The resulting time dependence of observables gives an appropriate solution of a quantum time evolution equation for observables (7.1.13). Then, we show that a set of quantum symplectomorphisms (quantum flow) has a structure of a group with multiplication (quantum composition) being a deformation of the ordinary composition regarded as a multiplication in a group of classical symplectomorphisms (classical flow) [34]. Such an approach to quantum trajectories has a benefit in that it is not needed to calculate the form of observables as star-functions, but only a quantum action of a given trajectory needs to be found.

### 7.2.1 Quantum Flow

*M*,

*π*,

*H*), where \(M=\mathbb {R}^{2N}\), \(\pi =\partial _{x^{i}}\wedge \partial _{p_{i}}\), and

*H*∈

*C*

^{∞}(

*M*) is an arbitrary real function. Then the solution of quantum Hamiltonian equations

*Q*

^{i}(

*x*,

*p*, 0) =

*x*

^{i}and

*P*

_{j}(

*x*,

*p*, 0) =

*p*

_{j}, i.e., the Heisenberg representation for observables of position and momentum, generates a quantum flow Φ

_{t}in a phase space according to an equation

*t*the map Φ

_{t}is a quantum canonical transformation ( quantum symplectomorphism) from coordinates (

*x*,

*p*) to new coordinates

*x*

^{′}=

*Q*(

*x*,

*p*,

*t*;

*ħ*),

*p*

^{′}=

*P*(

*x*,

*p*,

*t*;

*ħ*). It means that Φ

_{t}preserves the quantum Poisson bracket Open image in new window , which can be seen from (7.2.3) and the fact that Open image in new window Open image in new window .

_{t}, treated as a quantum canonical transformation, can act on observables and states as simple composition of maps. Such a classical action can also be used to transform the algebraic structure of the quantum Poisson algebra so that the action will be an isomorphism of the initial algebra and its transformation. So, a star-product ⋆

_{t}being the Moyal product transformed by \(\Phi _{t}^{-1}\) is defined by the formula As we know from our previous considerations, the ⋆

_{t}- product takes the form where vector fields \(D_{x^{i}}\), \(D_{p_{i}}\) are transformations of coordinate vector fields \(\partial _{x^{i}}\), \(\partial _{p_{i}}\):

_{t}-product is gauge equivalent to the Moyal product. In other words, to a quantum flow Φ

_{t}there corresponds a unique isomorphism

*S*

_{t}satisfying Observe, that for the ⋆

_{t}-algebra the involution is also the complex-conjugation.

*P*

_{j}(

*x*,

*p*, 0) =

*p*

_{0j}represents a particular quantum trajectory.

_{t}on

*A*. The composition of Φ

_{t}with observables, i.e. the classical action of Φ

_{t}on observables, does not result in a proper time evolution of observables and thus it is necessary to deform this classical action. It will be proved that a proper action of the quantum flow Φ

_{t}on functions from \(\mathcal {A}_{Q}\) (a pull-back of Φ

_{t}) is given by the new formula

*S*

_{t}is an isomorphism associated to the quantum canonical transformation \(\Phi _{t}^{-1}\).

*S*

_{t}

*x*

^{i}=

*x*

^{i}and

*S*

_{t}

*p*

_{j}=

*p*

_{j}was used, being on the other hand a consequence of the quantum canonicity of Φ

_{t}. Secondly, \(\Phi _{t}^{\ast }\) given by (7.2.4) is an automorphism of \(\mathcal {A}_{Q}\) as

_{t}denotes a star-product transformed by \(\Phi _{t}^{-1}\). Thus

_{t}is a flow of the quantum Hamiltonian vector field

*ζ*

_{H}. Moreover, in an analogy with classical mechanics, { Φ

_{t}} is a one-parameter group of quantum canonical transformations with respect to a new multiplication defined by

_{t}} is a group. Further on we will call it a quantum composition. The quantum composition rule given by (7.2.6) is properly defined since it respects the quantum pull-back of flows:

*x*⋆

*p*. Using the fact that

*S*

_{t}

*x*=

*x*and

*S*

_{t}

*p*=

*p*for every

*t*one calculates that

*T*

_{1},

*T*

_{2}defined on the whole phase space (\( S_{T_{1}\circ T_{2}}\) is an isomorphism intertwining star-products ⋆ and \(\star _{T_{1}\circ T_{2}}\), \(S_{T_{1},T_{2}}\) intertwines \(\star _{T_{1}}\) with \(\star _{T_{1}\circ T_{2}}\), and \(S_{T_{1}}\) intertwines ⋆ with \(\star _{t_{1}}\), where \(\star _{t_{1}}\) and \(\star _{T_{1}\circ T_{2}}\) are the Moyal products transformed, respectively, by transformations

*T*

_{1}and

*T*

_{1}∘

*T*

_{2}), one finds that

### Observation 15

*Quantum trajectories of the linear Hamiltonian systems coincide with classical trajectories. It follows from the fact that for the Hamiltonian functions being quadratic polynomials of phase space coordinates:*\(\left [ \left \vert H,\cdot \right \vert \right ] =\{H,\cdot \}\)*(*7.2.8)*. Besides, because solutions Q*(*t*) *and P*(*t*) *are linear in x*^{i}, *p*_{i}*, so S*_{t} = 1*. It means that the quantum group multiplication* (7.2.6) *(quantum composition) coincides with the classical composition* (7.2.10) *and in consequence, the quantum time evolution of any observable A is the same as the classical time evolution of A. For such systems, the only difference between the classical and quantum dynamics relies on different admissible states in which the evolution takes place. On the other hand, even in such simplest cases, classical and quantum systems differ fundamentally on the level of stationary states.*

*ħ*→ 0, (7.2.3) reduces to classical phase space trajectories

*ħ*→ 0 limit of the quantum flow (7.2.2) (quantum symplectomorphism). An action of the classical flow Φ

_{t}on functions from \(\mathcal {A}_{C}\) (a pull-back of Φ

_{t}) is just a simple composition of functions with Φ

_{t}, being an

*ħ*→ 0 limit of (7.2.4)

_{t}} forms a one-parameter group of canonical transformations, preserving a classical Poisson bracket: \(\{Q^{i}(t),P_{j}(t)\}=\delta _{j}^{i}\), with multiplication being an ordinary composition of maps

*ħ*→ 0 limit of quantum composition (7.2.6).

### 7.2.2 Quantum Dynamics with Classical Trajectories

*m*= 1. This Hamiltonian describes only the kinetic energy of the particle. It does not contain any terms describing the potential energy, i.e. there are no forces acting on the particle (the particle is free).

_{t}(

*x*,

*p*;

*ħ*) = Φ

_{t}(

*x*,

*p*) = (

*Q*(

*t*),

*P*(

*t*)), where

*S*

_{t}= 1 for any linear transformation, so the time evolution of any classical and quantum observable

*A*(

*x*,

*p*) is given by

*ρ*. For the classical system, Eqs. (7.2.11) and (7.2.12) represent simultaneously the dynamics of expectation values of observables (position and momentum in particular) in the pure coherent classical state

*ρ*of the form

*ρ*

_{C}=

*δ*(

*x*

^{′}−

*x*)

*δ*(

*p*

^{′}−

*p*). Nevertheless, contrary to the classical case, for the quantum system we get

*ΔP*(

*t*) of the free particle described by the state (7.2.15) does not change in time and is equal to its initial value

*Δp*, whereas the uncertainty of the position

*ΔQ*(

*t*) initially equal

*Δx*increases in time. Note also that the uncertainties of the position and momentum satisfy the Heisenberg uncertainty principle, i.e. \(\varDelta {Q(t)} \varDelta P{(t)}\geq \tfrac {\hbar }{2}\). Moreover, initially the free particle is in a state which minimizes the Heisenberg uncertainty principle since \( \varDelta Q(0)\varDelta P(0)=\varDelta x\varDelta p=\tfrac {\hbar }{2}\). It is also worth noting that the expectation value of the momentum \(\langle {P(t)} \rangle _{\rho _{Q}}\) is constant and equal

*p*, whereas the expectation value of the position \(\langle {Q(t)}\rangle _{\rho _{Q}}\) is equal

*x*+

*pt*. Hence, the time evolution of the free particle described by the state (7.2.15) can be interpreted as the movement of the particle along a straight line with the constant momentum equal

*p*, similarly as in the classical case. The difference between the classical and quantum case is that in the quantum case there is some uncertainty of the position and momentum, in contrast to the classical case where the position and momentum are known precisely. Observe also that for any admissible value of

*γ*the coherence is not preserved during time evolution.

*ħ*→ 0

^{+}. The limit has to be calculated in the distributional sense, i.e. one has to calculate the limit \(\lim _{\hbar \rightarrow 0^{+}}\langle {\rho }_{Q}{,\phi }\rangle =\lim _{\hbar \rightarrow 0^{+}}Tr(\rho _{Q}(\hbar )\phi )\) for every test function

*ϕ*. One easily calculates that

*ρ*

_{Q}( 7.2.15), describing a quantum free particle, converges in the limit

*ħ*→ 0

^{+}to the classical pure state describing a classical free particle moving along a straight line with the constant momentum equal

*p*.

_{t}(

*x*,

*p*;

*ħ*) = Φ

_{t}(

*x*,

*p*) = (

*Q*(

*t*),

*P*(

*t*)), where

*S*

_{t}= 1, so the time evolution of any classical and quantum observable

*A*(

*x*,

*p*) is given by (7.2.12). For the classical system, like in the previous example, equations (7.2.20) represent simultaneously the dynamics of expectation values of position and momentum in the pure coherent classical state (7.2.13), for which the minimal classical uncertainty relation (7.2.14) is fulfilled. As the state (7.2.13) is not an admissible quantum state we again consider a one-parameter family of pure and coherent quantum states

*ρ*of the form (7.2.15).

*ρ*

_{C}=

*δ*(

*x*

^{′}−

*x*)

*δ*(

*p*

^{′}−

*p*) . Nevertheless, contrary to the classical case, for the quantum system we get

*γ*=

*ω*, which remains coherent for arbitrary value of

*t*. Indeed, as

As was proved earlier, in the limit *ħ* → 0^{+}, pure coherent quantum states (7.2.15) converge to the pure coherent classical state ( 7.2.13).

*holomorphic coordinates*( 6.2.2)

*a*and \(\bar {a}\) are called the

*annihilation*and

*creation*functions since they decrease and increase the number of excitations of the vibrational mode with frequency

*ω*(annihilate and create the quanta of vibrations). Note, that \(a\star =(\bar {a}\star )^{\dagger }\), \(\bar {a}\star =(a\star )^{\dagger }\) and

*H*takes the form

*m*,

*n*are numbering the ⋆ -genvalues of

*H*. It can be shown that

*m*,

*n*are non-negative integer numbers. The energy levels

*E*

_{n}of the harmonic oscillator are equal

*n*= 0, 1, 2, … and ⋆ -genfunctions are the ⋆ -genfunctions

*ρ*

_{mn}of

*H*, i.e.

*ω*.

*ρ*

_{00}

*ρ*

_{00}takes the form

*ρ*

_{mn}can be now calculated giving

*r*,

*θ*)

*ρ*

_{n}in the limit

*ħ*→ 0

^{+}converge. Again it has to be calculated in a distributional sense, hence the limit \(\lim _{\hbar \rightarrow 0^{+}}\langle {\rho _{n},\phi }\rangle \) has to be calculated for every test function

*ϕ*. One finds that for fixed

*n*

i.e. all quantum stationary pure states *ρ*_{n} of the harmonic oscillator converge, in the limit *ħ* → 0^{+}, to the single classical state (*x* = 0, *p* = 0) describing a particle with the position and momentum equal 0. This result is not surprising as the state (*x* = 0, *p* = 0) is the only classical stationary pure state of the harmonic oscillator.

The reader can find other interesting examples of ⋆ -genvalue problems and their solutions in [75] and [76]. Moreover, the reader can find the general solution of ⋆ -genvalue problem for Hamiltonians quadratic in phase space coordinates in [92].

Previous results of this subsection give us an ambiguous answer to the question of time development of the initial coherent state. We investigated it indirectly, calculating time development of uncertainty relation *ΔQ*(*t*)*ΔP*(*t*), with minimal initial value \(\varDelta {Q(0)}\varDelta P{(0)=} \tfrac {\hbar }{2}\). For a free particle we found that the coherence is not preserved during time evolution (7.2.2) while for harmonic oscillator it is preserved for a distinguished initial coherent state (7.2.15) for *γ* = *ω*. Thus, let us investigate more systematically that problem for arbitrary linear Hamiltonian system in \(\mathbb {R}^{2}\).

*x*and

*p*variables is of the above form for some values of constants

*ω*,

*α*and

*β*. It should be noted that this type of Hamiltonian is very often found in quantum optics where admissible coherent and squeezed states of the light are investigated [132, 133, 144, 243, 257].

*Q*(

*t*) and momentum

*P*(

*t*) take the common form

*ω*

^{2}>

*α*

^{2}+

*β*

^{2}reads

*ω*

^{2}<

*α*

^{2}+

*β*

^{2}we get

*ω*

^{2}=

*α*

^{2}+

*β*

^{2}

*ω*

^{2}>

*α*

^{2}+

*β*

^{2}. In an initial coherent state

*ΔQ*)

^{2}and (

*ΔP*)

^{2}

*ξ*=

*α*+

*iβ*=

*re*

^{iθ}and writing it in polar variables (

*r*,

*θ*) the uncertainties (

*ΔQ*)

^{2}and (

*ΔP*)

^{2}take the form

*ω*,

*α*,

*β*are arbitrary.

*γ*= 1 (

*γ*=

*ω*for old

*x*and

*p*)

*Rt*=

*kπ*and \(Rt=\arctan (\tfrac {R}{ \omega }\tan \theta )+k\pi \), \(k\in \mathbb {Z}\). Thus, we will consider a further reduction. First, let us take

*β*= 0 and

*α*> 0. Then

*r*=

*α*and

*θ*= 0. In this case we receive

*α*= 0, but this is exactly the case of the harmonic oscillator (7.2.24).

*ω*

^{2}<

*α*

^{2}+

*β*

^{2}by taking

*ω*= 0. Then

*R*=

*r*and

*γ*= 1 we get

*t*we have to take

*β*= 0. Then

*θ*= 0,

*r*=

*α*, equations (7.2.30) reduce to

*ω*

^{2}=

*α*

^{2}+

*β*

^{2}. In this case

*ΔQ*)

^{2}and momentum (

*ΔP*)

^{2}increase quadratically with time, so the minimal uncertainty is not preserved for any

*t*except

*t*= 0. In the particular case, when \(\beta =\omega =\tfrac {1}{2}\) and

*α*= 0, formulas (7.2.33) and (7.2.34) reduce to these for a free particle, considered at the beginning of this subsection.

*ω*

^{2}>

*α*

^{2}+

*β*

^{2}the following one-parameter family of linear canonical transformations of coordinates

*ω*

^{2}<

*α*

^{2}+

*β*

^{2}another one-parameter family of linear canonical transformations of coordinates

*H*(

*q*

^{′},

*p*

^{′}) the initial coherence is preserved during time evolution.

### Observation 16

*From the above considerations it follows that the conservation of coherence property during time development of quantum state is rather rare phenomenon. Even for a three-parameter family of linear quantum Hamiltonian equations* (7.2.26) *in*\(\mathbb {R}^{2}\)*, initial coherence is preserved during time evolution only for two cases: β *=* α *= 0 *and β *=* ω *= 0*. So why should we expect such a property for nonlinear quantum Hamiltonian equations? On the other hand, when ω*^{2} > *α*^{2} + *β*^{2}*we can always reduce the dynamics to the harmonic oscillator (β *=* α *= 0*), and when ω*^{2} < *α*^{2} + *β*^{2}*we can reduce the dynamics to the case ω *=* β *= 0*, provided that we will be working with new variables q*^{′}, *p*^{′}*. In the frame of original variables q*, *p it means that for the considered class of systems there always exist canonically conjugated observables q*^{′} =* q*^{′}(*q*, *p*), *p*^{′} =* p*^{′}(*q*, *p*) *for which the minimal uncertainty is preserved during time evolution.*

*m*

_{1},

*m*

_{2}are masses of particles and

*k*is a coupling constant. Quantum equations of motion (7.1.16) for observables of position and momentum are of the form

*x*

^{2}is a cyclic coordinate, so

*P*

_{2}is a constant of motion equal to its initial value

*P*

_{2}=

*p*

_{2}and in consequence, equations (7.2.36) reduce to their classical counterparts

*T*(

*x*

^{1},

*x*

^{2},

*p*

_{1},

*p*

_{2}) = (

*Q*

^{1},

*Q*

^{2},

*P*

_{1},

*P*

_{2}) in a four-dimensional phase space \(\mathbb {R}^{4}\) with the following generating function

*F*

_{4}from Sect. 6.1.4. In accordance with (7.2.1) the received quantum flow Φ

_{t}transforms the Moyal product to the following product

*S*

_{t}associated with Φ

_{t}and intertwining the Moyal product at

*t*= 0 with the ⋆

_{t}-product takes the form

*S*

_{t}is an isomorphism (unitary operator) of the Hilbert space \(L^{2}(\mathbb {R}^{4})\) onto itself.

As in this case \(S_{t_{2}}\Phi _{t_{1}}=\Phi _{t_{1}}\), the group multiplication for { Φ_{t}} is just a composition of maps, as one could expect since Φ_{t} is simultaneously the classical and quantum trajectory. However, the action of Φ_{t} on observables and states does not reduce in general to a composition of maps (7.2.1) like in the classical case as now *S*_{t} ≠ 1, which is a direct consequence of the fact that the Hamiltonian is a cubic function of phase space coordinates. As the result, the time evolution of quantum observables is governed by (7.2.4). This shows that for the considered case the time evolution of quantum observables differs in general from the time evolution of classical observables.

_{t}on an observable

*A*, given by (7.2.4), indeed describes the quantum time evolution of

*A*. As the illustration of that fact let us take \(A(x,p)=x_{1}x_{2}^{2}\). Then

### 7.2.3 Pure Quantum Trajectories

*t*, we have

*Q*⋆

*P*is a constant of motion

*A*is a symmetric, nonsingular, 2

*n*× 2

*n*matrix. In a particular case, for factorization

*ω*is a symplectic matrix ( 6.1.2) and \(a\in \mathbb {C}\), the noncommutative exponential is given by In our case (7.2.40)

*H*=

*xp*and

*ħ*→ 0

_{t}is an example of a flow for which Φ

_{t}, for every \(t\neq \tfrac {k\pi }{\kappa \hbar }\), is not a classical symplectomorphism, since

_{t}transforms the Moyal product to the following product

*S*

_{t}associated with Φ

_{t}and intertwining the Moyal product with the ⋆

_{t}-product, up to the second order in

*ħ*, takes the form

*ħ*one can show that

*S*

_{t}in the above form satisfies these relations up to

*O*(

*ħ*

^{2}).

_{t}is a purely quantum trajectory, we deal with the quantum group multiplication (7.2.6) for { Φ

_{t}} as well as the quantum action (7.2.4) of Φ

_{t}on observables and states. Indeed, expanding (7.2.41) with respect to

*ħ*:

*S*

_{t}(7.2.42), the quantum composition law

*O*(

*ħ*

^{2}). Note also that the flow Φ

_{t}is not defined for all \(t\in \mathbb {R}\) as it is singular for \(t=\tfrac {2k+1}{2} \tfrac {\pi }{\kappa \hbar }\), contrary to classical flows which are globally defined. This is an interesting result showing that in general the quantum time evolution do not have to be defined for all instances of time

*t*.

### Observation 17

*Singularities of* *classical trajectories are not admissible as each classical trajectory represents measurable quantities, actually expectation values of position and momentum of a system in a pure coherent classical state* (7.2.13) *for all*\(t\in \mathbb {R}\)*. On the contrary, pure quantum trajectories themselves are not “physical” objects as states* (7.2.13) *are not admissible so, singularities of pure quantum trajectories are acceptable.*

*Q*and

*P*from (7.2.41) in the coherent state (7.2.15). The result after introducing

*Q*〉

_{ρ}and 〈

*P*〉

_{ρ}are well defined only on intervals \((-\tfrac {1}{4}+n)\tfrac {\pi }{\kappa \hbar }<t<( \tfrac {1}{4}+n)\tfrac {\pi }{\kappa \hbar }\), \(n\in \mathbb {Z}\). This once again shows that time evolution of the considered system is not defined for all values of the evolution parameter

*t*and even time development of expectation values of position and momentum is only well defined on certain intervals of

*t*.

### Observation 18

*We have found that for the considered quantum trajectories, expectation values of observables of position and momentum in the coherent state* (7.2.15) *were well defined only on certain intervals of t, which raises problems and questions of interpretation of such a kind of time evolution. If we assume, like in the classical case, that the expectation values of position and momentum have to be smooth functions for any*\(t\in \mathbb {R}\)*, then we have two options. Either, for a chosen quantization, there exist quantum states for which our assumption is fulfilled (the state* (7.2.15) *does not belong to that class) or, if there are no such states, our quantization is not ‘physical’ and we have to chose another quantization which fulfills the imposed assumption.*

The above observation was made from the mathematical point of view. Let’s have a look on the problem from the physical side. In other words, let us asses the length of the interval on which time evolution of the system is well defined. Notice that dimension of *κ* in joule-seconds is *J*^{−1}*s*^{−2} so we put \(\kappa =\left |\kappa \right |J^{-1}s^{-2}\) and moreover *ħ* ≃ 10^{−34}*Js*. So, the length of the time interval \(\tfrac {\pi }{4\hbar \kappa }\) will be approximately equal \(\left |\kappa \right |{ }^{-1}0.785\times 10^{34}s\). Notice that the age of Universe is 0.437 × 10^{18}*s*. So, for a large range of *κ* singularities appearing in time evolution are nonphysical.

After reading this chapter the reader might be disappointed with a small number of examples of stationary problems of known quantum systems, presented in deformation quantization formalism. The exception was made for the case of quantum harmonic oscillator. The reason is that such calculations directly in a Hilbert space over the phase space are very cumbersome and complex. Fortunately, at least for the canonical quantization, we can simplify that problem passing to so called position representation of quantum systems. This is the subject of the next chapter in which we present many known and new examples of separable eigenvalue quantum problems defined in an appropriate Hilbert spaces over Riemannian configuration spaces.

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