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We show that the non-linear semi-quantum Hamiltonians which may be expressed as (where is the set of generators of some Lie algebra and are the classical conjugated canonical variables) always close a partial semi Lie algebra under commutation and , because of this, it is always possible to integrate the mean values of the quantum degrees of freedom of the semi-quantum non-linear system in the fashion: (where is the Maximum Entropy Principle density operator) and, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom’s mean values. Those invariants are useful to characterize the kind of dynamics (regular or irregular) the system displays given that they can be fixed by means of the initial conditions imposed on the semi-quantum non-linear system.

There exists certain kind of semiquantum non-linear systems which can be represented by the following Hamiltonian [

where

is an interaction term where a classical interacts with a quantum one in the fashion

being the classical canonically conjugated variables of position and momentum, respectively, and

bitrary quantum operator. Many of the systems given by Equation (1) may be expressed as a linear superposition

of quantum operators

where the coefficients belonging to the linear superposition,

of freedom in the fashion

quantum Hamiltonian given by Equation (2),

generators of some Lie algebra, 3) the mean value of the semiquantum Hamiltonian,

with a Hamiltonian function [

ues,

study the dynamics of these systems, that generally display two kind of regimes: regular and irregular. Such regimes can be differentiated by means of the values that the invariants adopt (and these values are fixed by the initial conditions imposed on the system). The interested reader may consult [

The paper is organized as follows: Section 2 introduces the basic tools of MEP approach. In Section 3, we focus attention on the specific Hamiltonian representation of semiquantum non-linear systems. In Section 4, we apply MEP tools to our semiquantum non-linear systems and integrate the quantum degrees of freedom. In Section 5, we show how the general dynamics invariants emerge out of the algebra’s closure and derive the specific dynamic invariants associated to SU(2), Heisenberg, SO(2,1), and SU(1,1) Lie algebras. Finally, some conclusions are drawn in Section 6.

The description of the quantum state of a system is made by means of the density or statistical operator

According to Jayne’s Information Theory [

where the subindex 0 refers to the normalization condition

given that the identity operator

established by Alhassid & Levine [

which is expressed in terms of

The normalized statistical operator of maximal entropy given by Equation (6) enables one to obtain the en-

tropy

The statistical operator

where

Levine in [

“so that the equation of motion of the density operator has thus been converted to a set of coupled equations of motion for the Lagrange parameters. The number of coupled equations equals the number of constraints” [

“the boundary conditions of the equation of motion are determined by the requirement that the initial state

As a consequence of the fact that the statistical operator obeys Equation (9), the entropy (8) is a constant of the motion, i.e.

for any two times

Equation (13) is known as the generalized Eherenfest theorem. Finally, one can obtain the mean values of the relevant operators for all times as [

Semiquantum Hamiltonians are often found in the literature [

“one composed by a quantum part coupled to a classical part. The essential structure of all these models is a classical part acting directly on the quantum part, with the quantum part reacting back on the classical part through the expectation value of some observable [...] we refer to a system as semiquantum if one part is treated classically and the other part quantum mechanically” [

Thus, the Hamiltonian representing a semi-quantum system may be expressed in the form given by Equation

(1) [

is an interaction term coupling quantum and classical degree of freedom. This

to be a non-linear one. The semiquantum Hamiltonians we are interested in are those in which the

given by a linear superposition of, say,

bra. These quantum operators are the quantum degrees of freedom of the semiquantum system. The classical degrees of freedom are the canonical conjugate variables

cast as

some Lie algebra. Thus, Equation (1) (or its equivalent Equation (2)) may be re-written in the fashion [

where the first term includes the

coefficients

nians are used to model some nanotechnology devices (like molecular transistors, nanotubes, quantum dots and SQUIDS, for instance [

[

where

The Maximum Entropy Principle Approach (MEP) is able to generate a semiquantum formalism to deal with semi-quantum non-linear Hamiltonians like Equation (15) for which a set of relevant operators is invoked so as to fulfill the closure condition expressed in Equation (10) (the generators of some Lie algebra). This formalism was developed in [

Let us consider a mixed physical system represented by the semiquantum Hamiltonian represented by Equation (15) with a coupling term [

with

With

freedom. Since the identity operator commutes with the Hamiltonian, the classical term does not appear in the final result of the quantum commutation operation given by Equation (17). Accordingly, it is possible to generalize the prescription given by Alhassid & Levine in [

into the Equation (9) of motion. The time dependence of

where

Thus, the equation of motion for the density operator (9) has been converted into a set of coupled Equation (20). Nevertheless, there exist a difference with respect to the full quantum case of [

The normalization Equation (5) enables us to obtain the

where the

The integration of non-linear semiquantum differential Equation (21) can be accomplished in the fashion

exclusively on account of the fact it was possible to close the algebra by means of Equation (17) as we will show in the following [

Since in the Schrödinger representation the quantum operators do not depend on time explicitly, all the time

dependence is contained in the MEP density operator

rameters. Accordingly, from Equation (24) we obtain (see also Equation (9))

Take now into account the invariance of the trace under commutation operation [

Finally, taking into account the semi-quantum closure condition, Equation (17), we can replace the operator

which is the generalized Ehrenfest theorem given by Equation (21). In short, if we are able to close a semi Lie algebra under commutation with the semiquantum non-linear Hamiltonian (16), then we can integrate the equations of motion of the quantum degrees of freedom even though the Hamiltonian exhibits a nonlinearity via the

Equation (23) can also be written in the fashion [

The density operator of maximum entropy may be used to calculate the mean value of the Hamiltonian given by Equation (16)

and the entropy at the maximum acquires the form

and is a constant of the motion [

Summing up: as it is possible to close a semi Lie algebra under commutation with the non-linear semiquantum Hamiltonian, Equation (16), it is also possible to obtain the maximum entropy density operator

Concerning the system's classical degrees of freedom, the energy is taken to coincide with the quantum expectation value of the semiquantum Hamiltonian [

Thus, the semiquantum non-linear dynamics displayed by Hamiltonians of the type given by Equation (16) may be represented in a semiquantum phase space

whose dimension is

nifold of the system

Non-linear dynamics of semiquantum Hamiltonians of the type given by Equation (16), exhibits two kinds of invariants: 1) general dynamic invariants [

The density operator

they obey a special case of Equation (9) which is

and holds for any positive integer

Taking into account that

the invariant given by Equation (35) may be expressed as

where the

We are interested in the particular form which the former invariant acquires for the case

in terms of the so-called quantum correlation coefficients

quantum operators

Equation (21) enables us to obtain the non-linear evolution equation for the quantum correlation coefficients

The motion invariant given by Equation (38) is the so-called second order centered invariant of [

associated to an inner product (see [

Any non-linear semiquantum Hamiltonian of the type given by Equation (16) which fulfills the closure condi-

tion (17) with the generators of some Lie algebra exhibits the second order centered invariant (38)

dynamic invariant. This is of importance because, from it, it is possible to recover the generalized uncertainty principle [

The left hand side of Equation (41) is a principal minor of order 2 belonging to the correlation matrix

where

non-commuting operators, we can define the following expression, which is obtained as the summation over the

principal minors of order 2 belonging to the metric matrix

We call Equation (42) the generalized uncertainty principle [

The uncertainty principle given by Equation (42) imposes strong constraints on the system and avoids the making of wrong choices for the initial conditions in their semiquantum non-linear equations of motion.

The semiquantum closure condition (17) defines a

a particular kind of dynamic invariants that we will enumerate in the following.

Let

Equation (17) (i.e. the generators of some Lie algebra). We define the operator

On the other side, the closure Equation (17) enables us to obtain the following commutation relations [

where the

Equations (45) and (46) and (43) and (47) have terms that couple quantum and classical degrees of freedom. With the help of these equations it is possible to demonstrate that the anti-symmetry of matrix

· The

· The Bloch “hypersphere” invariant [

· The

· The

· The summation over the principal minors of order

The summation over the principal minors of order

the coefficients of the secular equation of the correlation matrix [

where

set composed only by

1) case

2) case

3) case

With

The methodology to demonstrate that the expressions given by Equations (53) to (55) are invariants is the same for all of them. We restrict ourselves to the case of invariant

(56)

From Equation (56) it can be seen that if

It is well-known that

tion rules [

where

Proposition 1: If a set of operators, which fulfills the commutation relation, Equation (57), closes a commutation algebra with a Hamiltonian of the type given by Equation (58), then the semiquantum matrix

Every Hamiltonian that closes an algebra with the SU(2) generators is accompanied by the invariants given by Equations (48) to (55). Some examples of these Hamiltonians (58) are:

· Spin 1/2 particle interacting with the classical harmonic oscillator [

· Spin 1/2 particle interacting with a biquadratic oscillator [

netic field's frequency,

stant between classical and quantum degrees of freedom. Equation (17) yields to the following anti-symmetric

matrix

· Spin 1/2 particle interacting with the double well [

Equation (17) yields to the following anti-symmetric matrix

Note that Equations (59), (61) and (63) give rise to anti-symmetric

with

principle (42), so

Lie algebra, that can be expressed in the guise

defining the celebrated Bloch sphere of the system. The quantum degrees of freedom’s mean values can be obtained from the density operator [

with

by means of

In virtue of Equations (70)-(72), we can obtain the relationship [

corresponding to the

The Heisenberg group

i.e. Hamiltonians that are quadratic in

from Equation (17) has the following form [

The correlation matrix's characteristic polynomial has two coefficients:

but only

miltonians of the type given by Equation (74) which close a partial Lie algebra with

· The Hamiltonian representing the production of charged meson pairs [

With

momentum respectively and

renced in literature as representative of the zeroth mode contribution of an strong external field to the production of charged meson pairs [

where

sical variable

· The generalized harmonic oscillator [

Let us consider the following generalized harmonic oscillator Hamiltonian [

where

responding

Proposition 2: Let us have

tion with a semiquantum Hamiltonian

matrix such that

Proof: The correlation matrix’s determinant is

If we take time derivative in Equation (83) and use Equation (40), we obtain

As

As matrices

The quantum degrees of freedom's mean values were integrated as

density operator is [

where:

tively (see [

We revisit now the Hamiltonian given by Equation (79) and the generators of the SO(2,1) Lie algebra,

Let us define

such that

Accordingly, the Hamiltonian from Equation (79) may be recast as

so that the commutation relations

lead to the following antisymmetric semi-quantum matrix

which enables us to obtain the semi-quantum non-linear equations of motion

Taking into account Equations (99) and/or Equation (103), we can easily see that

is an invariant of the motion when the Hamiltonian (79) is associated to the SO(2,1) Lie algebra. Further, as the matrix

With

hand side of the generalized uncertainty principle, Equation (42) which, for this particular case, remains as a constant of the motion.

Let’s consider again the Hamiltonian [

With

momentum respectively and

is referenced in literature as representative of the zeroth mode contribution of an strong external field to the production of charged meson pairs [

where

classical position variable

closes a partial Lie algebra either with the Heisenberg group

as relevant operators

we obtain the following semi-quantum matrix [

which is a block diagonal one, i.e. we see how the subspaces corresponding to the Heisenberg and SU(1.1) algebras are independent one from each other.

Making use of Equations (20), (21), (31) and (32) we obtain the equations of motion for the quantum (mean values and

Information Theory tells us that the statistical operator is [

The inclusion of the Hamiltonian into the relevant set does not modify the dynamics of the system but transforms it in a thermodynamic one (see [

with

If we replace Equations (119)-(124) into Equation (118) (the reader may find in [

with

and

with

system and write [

With

sub-matrix associated to the Heisenberg group

which is the same as that given by Equation (75) (a null trace one), so, the the dynamic invariant

this invariant was used in [

we are going to see that it corresponds to the SU(1,1) Lie algebra. In fact, lets consider first the quantum opera-

tors [

As in [

one has

On the other side, remembering that

we can make the identification

responding to the SU(1,1) Lie algebra [

may be expressed in terms of

It is easy to see that the Casimir operator given by Equation (145) commutes with the semiquantum Hamiltonian given by Equation (108). Accordingly, this Hamiltonian exhibits an SU(1,1) structure and the following dynamic invariant

It is also possible to demonstrate the invariance of Equation (146), making use of Equations (21) and (17) (within the MEP context).

We have discussed properties of non-linear semi-quantum Hamiltonians of the form

conjugated canonical variables. We saw that this Hamiltonian always closes a partial Lie algebra under commutation with the

seen that these Hamiltonians are always associated to dynamic invariants, which are expressed in terms of

the quantum degrees of freedom’s mean values

characterize the kind of dynamics that the system displays, as several examples have amply illustrated.