Open access peer-reviewed chapter - ONLINE FIRST

Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos

By Paul Bracken

Submitted: July 27th 2020Reviewed: October 14th 2020Published: November 17th 2020

DOI: 10.5772/intechopen.94491

Downloaded: 28

Abstract

The concept of integrability of a quantum system is developed and studied. By formulating the concepts of quantum degree of freedom and quantum phase space, a realization of the dynamics is achieved. For a quantum system with a dynamical group G in one of its unitary irreducible representative carrier spaces, the quantum phase space is a finite topological space. It is isomorphic to a coset space G/R by means of the unitary exponential mapping, where R is the maximal stability subgroup of a fixed state in the carrier space. This approach has the distinct advantage of exhibiting consistency between classical and quantum integrability. The formalism will be illustrated by studying several quantum systems in detail after this development.

Keywords

  • classical
  • quantum
  • chaos
  • integrability
  • conservation law
  • algebra

1. Introduction

In classical mechanics, a Hamiltonian system with Ndegrees of freedom is defined to be integrable if a set of Nconstants of the motion Uiwhich are in involution exist, so their Poisson bracket satisfies UiUj=0, i,j=1,,N. For an integrable system, the motion is confined to an invariant two-dimensional torus in 2N-dimensional phase space. If the system is perturbed by a small nonintegrable term, the KAM theorem states that its motion may still be confined to the N-torus but deformed in some way [1, 2, 3]. The first computer simulation of nonequilibrium dynamics for a finite classical system was carried out by Fermi and his group. They considered a one-dimensional classical chain of anharmonic oscillators and found it did not equilibrate.

Classically, chaotic motion is longtime local exponential divergence with global confinement, a form of instability. Confinement with any kind of divergence is produced by repeatedly folding, a type of mixing that can only be analyzed by using probability theory. The motion of a Hamiltonian system is usually neither completely regular nor properly described by statistical mechanics, but shows both regular and chaotic motion for different sets of initial conditions. There exists generally a transition between the two types of motion as initial conditions are changed which may exhibit complicated behavior. As entropy or the phase space area quantifies the amount of decoherence, the rate of change of the phase space area quantifies the decoherence rate. In other words, the decoherence rate is the rate at which the phase space area changes.

It is important to extend the study of chaos into the quantum domain to better understand concepts such as equilibration and decoherence. Both integrable as well as nonintegrable finite quantum systems can equilibrate [4, 5]. Integrability does not seem to play a crucial role in the structure of the quasi-stationary state. This is in spite of the fact that integrable and nonintegrable quantum systems display different level-spacing statistics and react differently to external perturbation. Although integrable systems can equilibrate, the main difference from nonintegrable systems may be longer equilibration times. This kind of behavior is contrary to integrable classical finite systems that do not equilibrate at all. Nonintegrable classical systems can equilibrate provided they are chaotic.

The properties of a quantum system are governed by its Hamiltonian spectrum. Its form should be important for equilibration of a quantum system. The equilibration of a classical system depends on whether the system is integrable or not. Integrable classical systems do not tend to equilibrate, they have to be nonintegrable. Quantum integrability in ndimensions may be defined in an analogous way requiring the existence of nmutually commuting operators, but there is no corresponding theorem like the Liouville theorem. An integrable system in quantum mechanics is one in which the spectral problem can be solved exactly, and such systems are few in number [6, 7].

In closed classical systems, equilibration is usually accompanied by the appearance of chaos. Defining quantum chaos is somewhat of an active area of study now. The correspondence principle might suggest we conjecture quantum chaos exists provided the corresponding classical system is chaotic and the latter requires the system to be nonintegrable. Classical chaos does not necessarily imply quantum chaos, which seems to be more related to the properties of the energy spectrum.

It was proposed that the spectrum of integrable and nonintegrable quantum systems ought to be qualitatively different. This would be seen in the qualitative difference of the density of states. At a deeper level, one may suspect that changes in the energy spectrum as a whole may be connected to the breaking of some symmetry or dynamical symmetry. This is the direction taken here [8, 9, 10].

It is the objective to see how algebraic and geometric approaches to quantization can be used to give a precise definition of quantum degrees of freedom and quantum phase space. Thus a criterion can be formulated that permits the integrability of a given system to be defined in a mathematical way. It will appear that if the quantum system possesses dynamical symmetry, it is integrable. This suggests that dynamical symmetry breaking should be linked to nonintegrability and chaotic dynamics at the quantum level [11, 12, 13].

Algebraic methods first appeared in the context of the new matrix mechanics in 1925. The importance of the concept of angular momentum in quantum mechanics was soon appreciated and worked out by Wigner, Weyl and Racah [14, 15, 16]. The close relationship of the angular momentum and the SO3algebra goes back to the prequantum era. The realization that SO4is the symmetry group of the Kepler problem was first demonstrated by Fock. A summary of the investigation is as follows. To familiarize those who are not familiar with algebraic methods in solving quantum problems, an introduction to the algebraic solution of the hydrogen atom is presented as opposed to the Schrödinger picture. This approach provides a platform for which a definition of quantum integrability of quantum systems can be established. Thus, at least one approach is possible in which a definition of concepts such as quantum phase space, degrees of freedom as well as how an idea of quantum integrability and so forth can be formulated [17, 18, 19, 20, 21]. After these issues are addressed, a number of quantum models will be discussed in detail to show how the formalism is to be used [22, 23, 24].

1.1 The hydrogen atom

The hydrogen atom is a unique system. In this system, almost every quantity of physical interest can be computed analytically as it is a completely degenerate system. The classical trajectories are closed and the quantum energy levels only depend on the principle quantum number. This is a direct consequence of the symmetry properties of the Coulomb interaction. Moreover, the properties of the hydrogen atom in an external field can be understood using these symmetry properties. They allow a parallel treatment in the classical and quantum formalisms.

The Hamiltonian of the hydrogen atom in atomic units is

H0=p221r.E1

The corresponding quantum operator is found by replacement of pby i. Due to the spherical symmetry of the system, the angular momentum components are constants of the motion,

L=r×p,H0L=0.E2

So H0L2Lzis a complete set of commuting operators classically, so three quantities in mutual involution, which implies integrability of the system.

The Coulomb interaction has another constant of the motion associated with the Runge-Lenz vector R. This has the symmetrized quantum definition

R=12p×LL×prr,H0R=0.E3

If the Rdirection is chosen as the reference axis of a polar coordinate system in the plane perpendicular to L, one deduces the equation of the trajectory as

r=L21+Rcosϑ.E4

The modulus determines whether the trajectory is an ellipse, a parabola or a hyperbola.

There are then 7constants of the motion LRH0are not independent and satisfy

RL=0,L2R22H0=12H01.E5

The minus one on the right in (5) is not present in classical mechanics. The mutual commutation relations are given in terms of εijk, the fully antisymmetric tensor as follows,

LiLj=iεijkLk,LiRj=iεijkRk,RiRj=iεijk2H0Lk,E6

Let us look at the symmetry group of the hydrogen atom. The symmetry group is the set of phase space transformations which preserve the Hamiltonian and the equations of motion. It can be identified from the commutation relations between constants of motion. For hydrogen, for negative energies, the group of rotations in 4-dimensional space is called SO4.

The generators of the rotation group in an n-dimensional space are the n1n/2components of the n-dimensional angular momentum

Lij=xipjxjpi,1i,jn.E7

In (7), Lijis the generator of the rotations in the ij-plane and has the following commutation relations

LijLkl=0,LijLik=iLjk.E8

The first bracket in (8) holds if all four indices are different.

Define the reduced Runge-Lenz vector to be

R=R2H0.E9

The commutation relations (6) are those of a four-dimensional angular momentum with the identification

L12=LzL23=LzL31=LyL14=RzL24=RyL34=Rz,E10

and Casimir operator

L2=i<jLij2=L2+R2=12H01.E11

The classical trajectory is thus uniquely defined with the 6 components of Land LR=0. Any trajectory can be transformed into any other one having the same energy by a 4-dimensional rotation. An explicit realization of this four-dimensional invariance is to use a stereographic projection from the momentum space onto the 4-dimensional sphere with radius p0=2H0. On this sphere, the solutions to Schrödinger’s equation as well as the classical equations of motion are those of the free motion. Schrödinger’s equation on the four-dimensional sphere can be separated into six different types of coordinates each associated with a set of commuting operators.

Spherical coordinates correspond to the most natural set, and choosing the quantization axis in the 4 direction and inside the (1,2,3) subspace, the z-axis or usual 3-axis as reference axis, the three operators can be simultaneously diagonalized,

L2=12H01,L2=L122+L312+L232=Lx2+Ly2+Lz2,Lz=L12,E12

The respective eigenvalues of these operators are n21, ll+1and Msuch that Mln1, so the total degeneracy is n2. It corresponds to a particular subgroup chain given by

SO4nSO3lSO2M.E13

Other choices are possible, such as other spherical coordinates obtained from the previous by interchanging the role of the 3 and 4 axes. This simultaneously diagonalizes the three operators

L2=12H01,
λ2=L122+L142+L242=Rx'2+R`y2+Lz2,Lz=L12.E14

The respective eigenvalues of these operators are n21, λλ+1and Msuch that Mλn1. The subgroup chain for this situation is

SO4nSO3λSO2M.E15

Another relevant case is the adoption of cylindrical coordinates on the 4-dimensional sphere associated with the following set of commuting operators

L2=12H01,L12=Lz,L34=Rz.E16

This set has the following associated subgroup chain,

SO4SO2SO2.E17

In configuration space, this is associated with separability in parabolic coordinates. This is a specific system but it exhibits many of the mathematical and physical properties that will appear here.

2. Quantum degrees of freedom

The time evolution of a system in classical mechanics in time is usually represented by a trajectory in phase space and the dynamical variables are functions defined on this space. The dimension of phase space is twice the number of degrees of freedom, and a point represents a physical state. The space is even-dimensional and it is endowed with a symplectic Poisson bracket structure. Dynamical properties of the system are described completely by Hamilton’s equations within this space.

For a quantum system, on the other hand, the dynamical properties are discussed in the setting of a Hilbert space. Dynamical observables are self-adjoint operators acting on elements of this space. A physical state is represented by a ray of the space, so the Hilbert space plays a role similar to phase space for a classical system. The Hilbert space cannot play the role of a quantum phase space since its dimension does not in general relate directly to degrees of freedom. Nor can it be directly reduced to classical phase space in the classical limit. Let us define first the quantum degrees of freedom as well as giving a suitable meaning to quantum phase space.

Suppose His a Hilbert space of a system characterized completely by a complete set of observables denoted C. Set Cis composed of the basic physical observables, such as coordinates, momenta, spin and so forth, but excludes the Hamiltonian. The basis vectors of the space can be completely specified by a set of quantum numbers which are related to the eigenvalues of what are usually referred to as the fully non-degenerate commuting observables Cof C. A fully degenerate operator or observable OChas for some constant λthe action

Oψi=λψi,ψiH.E18

Definition 1: (Quantum Dynamical Degrees of Freedom) Let C:OjOiOj=0ij=1Nbe a complete set of commuting observables of a quantum system. A basis set of its Hilbert space Hcan be labeled completely by Mnumbers αi:i=1Mcalled quantum numbers which are related to the eigenvalues of the non-fully degenerate observables Oi:i=1MMN, a subset of C. Then the number Mis defined to be the number of quantum dynamical degrees of freedom.

Since the members of Care provided by the system, not including the Hamiltonian, it depends only on the structure of the system’s dynamical group G. Thus the number of quantum dynamical degrees of freedom based on this definition is unique for a given system with a specific Hilbert space H.

The physical and mathematical considerations for defining the dimension of the nonfully degenerate operator subset Cof C, not the dimension of Citself, as the number of quantum dynamical degrees of freedom is as follows. In a given H, all fully degenerate operators in Care equivalent to a constant multiple of the identity operator guaranteeing the irreducibility of H. The expectation values of any fully degenerate operator is a constant and contains no dynamical information.

A given quantum system generally has associated with it a well-defined dynamical group structure due to the fact that the mathematical image of a quantum system is an operator algebra gin a linear Hilbert space. This was seen in the case of hydrogen. It comes about from the mathematical structure of quantum mechanics. The dynamical group Gwith algebra gis generated out of the basic physical variables, with the corresponding algebraic structure defined by the commutation relations.

The Hamiltonian Hand all transition operators Ocan be expressed as functions of a closed set of operators

H=HTi,O=OTi,TiTj=kCijkTk.E19

The Cijkin (19) are called the structure constants of algebra g. The Hilbert space is decomposed into a direct sum of the carrier spaces of unitary irreducible (irrep) representations of the group. Consequently, the dynamical symmetry properties of the system can be restricted to an irreducible Hilbert space which acts as one of the irrep carrier spaces of G.

From group representation theory, it will be given that a total of σsubgroup chains exist for a given group

Gα=GsααGsα1αG1α,α=1,,σ.E20

For each subgroup chain Gαof G, there is a complete set of commuting operators Cwhich specifies a basis set of its irreducible basis carrier space H, so the dimension of Cfor all subgroup chains of Gis the same. A subgroup chain of dynamical group Gserves to determine the Mquantum dynamical degrees of freedom for a given quantum system with Hilbert space Han irrep carrier space of G.

Definition 2: For a quantum system with Mindependent quantum dynamical degrees of freedom the quantum phase space is defined to be a 2M-dimensional topological space. The space is isomorphic to the coset space G/Rwith explicit symplectic structure. Here Gis the dynamical group of the system and RGis the maximal stability subgroup of the Hilbert space.

3. Quantum integrability and dynamical symmetry

Quantum phase space defined here can be compact or noncompact depending on the finite or infinite nature of the Hilbert space. A consequence of this development is that the classical definition of integrability can in general be directly transferred to the quantum case.

Definition 3: (Quantum Integrability) A quantum system with Mindependent dynamical degrees of freedom, hence a 2M-dimensional quantum phase space, is integrable if and only if there are Mquantum constraints of motion, or good quantum numbers, which are related to the eigenvalues of Mnon-fully degenerate observables: O1,O2,,On.

Any set of variables that commute may be put in the form of a complete set of commuting observables Cby including certain additional observables with it. The definition then says that if the system is integrable, a complete set of commuting variables Ccan be found so that the Hamiltonian is always diagonal in the basis referred to by C. In the reverse sense, the definition implies that if the system is integrable, simultaneous accurate measurements of Mnon-fully degenerate observables in the energy eigenvalues can be carried out.

The link with the dynamical group structure can be developed. This specifies exactly the integrability of a quantum system. To this end the definition of dynamical symmetry is needed.

Definition 4: (Dynamical Symmetry) A quantum system with dynamical group Gpossesses a dynamical symmetry if and only if the Hamiltonian operator of the system can be written and presented in terms of the Casimir operators of any specific chain with αfixed

H=FCkjαE21

The index of a particular subgroup chain Ckjαthe i-th Casimir operator of subgroup Gkα, k=sα,,1, i=1,,lkαand lkαdenotes that the rank of subgroup Gkαis l. It is now possible to state a theorem which gives a condition for integrability to apply.

Proposition 1: (Quantum Integrability) A quantum system with dynamical group Gis said to be integrable if it possesses a dynamical symmetry of G.

To prove this, note that it can be broken down into two cases or subgroup classes for a given dynamical group Gand are referred to as canonical and noncanonical.

First consider the case in which Gαis a canonical subgroup chain of G. The Casimir operators of G, CGiand all Casimir operators Ckiαcorresponding to the subgroups in chain Gαform a complete set of commuting operators Cαof any carrier irrep space Hof Gαso for fixed α,

Cα:CGiCkiαQjj=1N.E22

When Gαis the dynamical symmetry of the system, all operators in Cαare constants of motion

HQj=0.E23

There are always Mnonfully dynamical operators in Cα. By the third definition, the system is integrable.

For a non-canonical subgroup chain Gαthe number of Casimir operators CGiof Gand all Casimir operators of Ckiαof Gαis less than the number of the complete set of commuting operators Cof any irrep carrier space of G. By definition, of any complete set of commuting operators, there must exist other commuting operators Ojthat commute with CGiand Ckiα. These have to be included in the union as well when putting together Cα

Cα:CGiCkiαOjQj:j=1N.E24

When the system is characterized by the dynamical symmetry of Gα, the operators in (24) satisfy relation (23) as well. In this case as well, there must exist Mnon-fully degenerate operators of constants of motion as in the previous case.

Based on this proposition, it can be stated that nonintegrability of a quantum system involves the breaking of the dynamical symmetry of the system. It may be concluded that dynamical symmetry breaking can be said to be a property which characterizes quantum nonintegrability.

Let us summarize what has been found as to what quantum mechanics tells us. In a given quantum system with dynamical Lie group Gwhich is of rank land dimension n, the dimension of a complete set of commuting operators Cof Gwith any particular subgroup chain is d=l+nl/2in which the loperators are Casimirs of Gand are fully degenerate for any given irrep of G. The number Mof the non-fully degenerate operators in Cfor a given irrep of Gcannot exceed Mnl/2. When dynamical symmetry is broken such that any of the Mconstants of the motion for the system is destroyed the system becomes nonintegrable.

4. Quantum phase space

It is of interest then to develop a model for phase space for quantum mechanics which may be regarded as an analogue to classical physics. By what has been said so far, the Hilbert space Hof the system can be broken up into a direct sum of the unitary irreps carrier spaces of G,

H=ΛYΛHΛ.E25

In (25), the subscript Λlabels a particular irrep of Lie group G, Λis the largest weight of the irrep and YΛthe degeneracy of Λin Hwith no correlations existing between various HΛ. The study of the dynamical properties of the system can be located on one particular irreducible subspace HΛof H. For a quantum system with MΛindependent quantum dynamical degrees of freedom, the corresponding quantum phase space should be a 2MΛ-dimensional, topological phase space without additional constraints.

To construct the quantum phase space from the quantum dynamical degrees of freedom for an arbitrary quantum system, the elementary excitation operators can be obtained from the structures of Gand HΛ. Let aibe a subset of generators of Gsuch that any states Ψof the system are generated for all ΨHΛby means of

Ψ=Fai0.E26

Moreover F(ai) is a polynomial in the operators aiand 0HΛis the reference state. The requirement placed on state 0is that one can use a minimum subset of gto generate the entire subspace HΛfrom 0. In this event, the collection aiis called the set of elementary excitation operators of the quantum dynamical degrees of freedom. If Gis compact, 0is the lowest Λ,Λor highest weight Λ,Λstate of HΛ. If Gis noncompact, it is merely the lowest state. The number of aiis the same as the number of quantum dynamical degrees of freedom. Physically this has to be the case since that is how the operators are defined. Thus the set aiand Hermitian conjugate aiin gΛform a dynamical variable subspace μof gso we can write

μ:aiaii=1MΛ.E27

With respect to μthere exists a manifold whose dimension is twice that of the quantum dynamical degrees. It can be realized by means of a unitary exponential mapping of the dynamical variable operator subspace μ

Ω=expi=1MΛηiaiηiai.E28

The ηiare complex parameters and i=1,,MΛ. In fact, Ωis a unitary coset representation of G/R, where RGis generated by the subalgebra κ=gμ. Thus (28) shows that qis isomorphic to the 2MΛ-dimensional coset space G/R, and will be denoted this way from now on. The discussion will apply just to semi-simple Lie groups whose gsatisfies the usual Cartan decomposition g=κ+μand κκκ, κμμand μμκ. Thus G/Rwill be a complex homogeneous space with topology and a group transformation acting on G/Ris a homomorphic mapping of G/Rinto itself.

The homogeneous space G/Rhas a Riemannian structure with metric

gij=2logKzz¯ziz¯jE29

The function Kzz¯is called the Bergmann kernel of G/Rand can be represented as

Kzz¯=λfλzfλz¯.E30

The functions fλzin (30) constitute an orthogonal basis for a closed linear subspace L2G/Rof L2G/Rsuch that

G/Rfλzfλz¯K1zz¯zz¯=δλλ,E31

and zz¯is the group invariant measure on the space G/R. It will be written

zz¯=ζdetgiji=1Mλdzidz¯iπ.E32

In (32)ζis a normalization factor given by the condition that (32) integrated over the space G/Ris equal to one. There is also a closed, nondegenerate two-form on G/Rwhich is expressed as,

ω=ii,jgijdzidz¯j.E33

Corresponding to this two form there is a Poisson bracket which is given by

fh=1ii,jgijfzihz¯jfz¯jhzi.E34

In (34)fand hare functions defined on G/R. By introducing canonical coordinates qpthese quantities can be rewritten in terms of these coordinates.

4.1 Phase space quantum dynamics

Based on what has been stated about G/R, it would be useful to describe the quantum phase space. This means for a given quantum system a phase space representation must exist. Such a representation can be found if there exists an explicit mapping such that

OTiUqp,Ψρq+ip.E35

Here Ois given by (19), and ρqpL2. For a quantum system with a quantum phase space G/R, this mapping can be realized by coherent states. To construct coherent states of Gand HΛdefined on G/R, the fixed state 0is chosen as the initial state

g0=Ωr0=Λ,Ωer,gG,rR,ΩG/R.E36

Then Ris the maximal stability subgroup of 0so any rRacting on 0will leave 0invariant up to a phase factor

r0=er0.E37

The Λ,Ωare the coherent states which are isomorphic to G/R. Therefore,

Λ,ΩΩ0=expi=1MΛηiaiηiai0=K1/2zz¯expi=1MΛziai0=K1/2zz¯Λ,z.E38
Kzz¯=0expi=1MΛz¯iaiexpi=1MΛziai0=ΛzΛz=0Λ02=λfΛλzfΛλz.

The Bargmann kernel was introduced in (30), and for a semisimple Lie group, the parameters ziare given by

z=ηtanηη1/2ηη1/2,Gcompact,ηtanhηη1/2ηη1/2,Gnoncompact.E39

Here ηrepresents the nonzero k×pblock matrix of the operator i=1MΛηiaiηiai. The state Λ,zin (38) is an unnormalized form of Λ,Ωand fΛ,λzis the orthogonal basis of L2G/Rthe function space

fΛ,λz=ΛλΛλ,E40

where Λ,λis a basis for HΛ, a particular irreducible subspace of the Hilbert space. The coherent states of (38) are over-complete

G/RΛ,ΩΛ,Ωz=I.E41

A classical-like framework or analogy has been established in the form of a quantum phase space specified by Gand HΛ. Variables which reside in this classical analogy are denoted thus c˜. The 2MΛ-dimensional quantum phase space G/Rhas all the required structures of a classical mechanical system. It is always possible a classical dynamical theory can be established in G/Rwhose motion is confined to G/Rand is determined by the following equations of motion

dU˜dt=U˜qpH˜qp,q,pG/R.E42

This equation can be replaced by Hamilton’s equations

dqidt=H˜qppi,dpidt=H˜qpqi.E43

In (42) and (43), H˜qpis the Hamiltonian of the system, and U˜qpis a physical observable. A correspondence principle is implied here and requires that suitable conditions can be found such that the quantum dynamical Heisenberg equations can be written this way.

Clearly, if suitable conditions hold the phase space representation of the commutator of any two operators is equal to the Poisson bracket of the phase space representation of these two operators so that

1iΛΩAHBHΛΩ=A˜B˜.E44

Then the phase space representation of the Heisenberg equation

dAHdt=1iAHHH,E45

given by (42) is therefore equivalent to (43). In (45), AHis the Heisenberg operator

AH=UAU1,U=eiHt/,E46

and Ais time-independent in the Schrödinger picture. The coherent state on the left of (44) is time-independent. Observables on the right side are the expectation values of the Schrödinger operators in the time-dependent coherent state. The quantum phase space maintains many of the quantum properties which are important, such as internal degrees of freedom, the Pauli principle, statistical properties and dynamical symmetry. Formally the equation of motion is classical. The phase space representation is based on the whole quantum structure of the coset space G/R.

Let us discuss integrability and dynamical symmetry. A quantum system with MΛindependent degrees of freedom is integrable if and only if the MΛnon-fully degenerate observables can simultaneously be measured in the energy representation. There exist non-fully degenerate observables Ci:i=1MΛ1which commute with each other and H

CiCj=0,CiH=0.E47

It follows that in the classical limit which has been formulated,

C˜iC˜j=0,C˜iH˜=0.E48

Together with the Hamilton equations, (47) also formally defines classical integrability, so quantum integrability is completely consistent with the classical theory. In the classical analogy, the group structure of the system is defined by Poisson brackets. The concept of dynamical symmetry is naturally preserved in the classical analogy, so the theorem on dynamical symmetry and integrability is also meaningful for the classical analogy. If the Hamiltonian has the symmetry S, then its phase space picture representation has the same symmetry. To see this, if

SHS1=H,E49

in the phase space representation, it holds that

ΛΩHΛΩ=ΛΩSHS1ΛΩ=ΛΩHΛΩ.E50

To put this concisely, we write

H˜qp=H˜qp,E51

where S1Λ,Ω=S1Ω0=Λ,Ωeh.

5. Applications to physical systems

5.1 Harmonic oscillator

The harmonic oscillator has dynamical group H4and is a single-degree of freedom system [13, 14, 15, 23]. To the dynamical group corresponds the algebra h4defined by the set aaaaIwith Hilbert space the Fock space VF:nn=12, so the fixed state is the ground state 0, and elementary excitation operator a. The quantum phase space is constructed from the unitary exponential mapping of the subspace μ:aaof h4,

Ωz=expzaz¯aH4/U1U1.E52

With generators aaand I, U1U1in (52) is the maximal stability subgroup of 0. As H4/U1U1is isomorphic to the one-dimensional complex plane, the quantum phase space has metric gij=δijand z=dzdz¯/π. It is noncompact due to the infiniteness of the Fock space. There is a well-known symplectic structure on the complex plane with Poisson bracket of two functions F˜1,F˜2defined by

F˜1F˜2=1iF˜1zF˜2z¯F˜1z¯F˜2z.E53

It is useful to introduce the standard canonical position and momentum coordinates

z=12q+ip,z¯=12qip.E54

The Glauber coherent states can be realized by the states zwith the set of these states isomorphic to H4/U1U1and given as

zΩz0=expzaz¯a0=ez2/2expza0.E55

The normalization constant in (55) is the Bargmann kernel

Kzz¯=ez2.E56

The phase space representation of the wavefunction ΨVFis

fz=Ψz=n=0fnznn!.E57

By Wick’s Theorem, it is always possible to write an operator Ain normal product form

A=Aaa=k,lAk,lnakal.E58

The phase space representation of Ais just

U˜zz¯=zAz=k,lAk,lnz¯kzl.E59

In the case Ais simply a generator of H4, we can write (59) as

a˜=zaz,a˜=zaz,a˜a˜=zaaz=z2,I˜=zIz=I.E60

The corresponding algebraic structure of H4in the phase-space representation is

ica˜a˜=I˜,ica˜a˜=a˜,ica˜a˜a˜=a˜.E61

Here cis used in the classical analogy. The algebraic structure of the H4generators is preserved when commutators are replaced by Poisson brackets in phase space. Using (54) the Dirac quantization condition and H˜are given by

qp=icqp,H˜qp=zHz.E62

For the forced harmonic oscillator, the classical analogy of the Hamiltonian is given by

H˜qp=ω2p2+q2+i2λtq2λtp=ω2p2+q2+12λt+λ¯tq)+12iλtλ¯tp.E63

Hamilton’s equations in (44) can be used to evaluate the tderivatives of qand p:

dqdt=ωp+12iλtλ¯t,dpdt=ωq12λt+λ¯t.E64

Hence combining these two derivatives, we obtain

ddtq+ip=q+ip2iλt.E65

Multiplying both sides by the integrating factor eiωtand then integrating with respect to t, the solution is

qt+ipt=eiωtq0+ip0i2eiωt0tλτeiωτ=zt2c.E66

If the initial state is 0or a coherent state z0, then the exact quantum solution is

ψt=ztetE67

and ztis given by (66). The phase φis a quantum effect obtained from zt

φt=12ωt0tλτzτ.E68

This seems to imply the classical analogy provides an exact quantum solution if the Hamiltonian is a linear function of the generators of G.

5.2 SU2spin system

The phase space structure of a spin system will be constructed and as well the phase-space distribution and classical analogy.

Since the dynamical group of the spin system is SU2and the Hilbert space is described by the states V2j+1=jmwhere m=j,j+1,,jand jis an integer or half-integer, the fixed state is j,j. This is the lowest weight state of V2j+1. Thus the elementary excitation operator of the spin system is J+and the explicit form of j,mis

j,m=1j+m!2jj+m1/2J+j+mj,j.E69

Any state Ψ=m=jjfmj,mV2j+1can be generated by a polynomial of J+acting on j,j. This means the number of quantum dynamical degrees of freedom is equal to the number of elementary excitation operators. The quantum phase space can be found by mapping μ:J+Jto the coset space SU2/U1by means of ηJ+η¯JexpηJ+η¯Jwhere η=ϑ/2e, 0ϑπ, 0φ2π. The coset space SU2/U1is isomorphic to a two-dimensional sphere. The coherent states of SU2/U1are well known

jΩ=expηJ+η¯Jj,j=1+z2jexpzJ+j,j=1+z2jjz,z=tanϑ2e.E70

The generalized Bargmann kernel on S2is Kzz¯=1+z22j. Then the metric gijand measure are given by

gij=δij2j1+z22,=1π2j+1dzdz¯1+z2)2.E71

Given the canonical coordinates

14jq+ip=z1+z2=sinϑ2e,E72

there obtains the bracket

F˜1F˜2=F˜1qF˜2pF˜1pF˜2q,E73

where q2+p24j, which implies the phase space of a spin system is compact.

The phase space representation of the state ΨV2j+1is for fL2S2,

fz=ΨjΩ=n=0fn2jj+m1/2zj+m,E74

The phase space representation of an operator B=BJiis

B˜zz¯=jΩBjΩ.E75

When the operator Bin (75) is chosen to be one of the three operators J+, Jor J0, the results are

J˜+=jΩJ+jΩ=2jz¯1+z2,J˜=jΩJjΩ=2jz1+z2,J˜0=jΩJ0jΩ=jz211+z2.E76

These can also be given in terms of q,pby using (72). The algebraic structure of SU2in the phase space representation is given by the Poisson bracket

icJ˜J˜+=2J˜0,icJ˜0J˜±=±J˜±.E77

The classical analogy of an observable BJiis given by the following expression

B˜qp=jΩBJijΩ.E78

The classical limit is found by taking jand the classical Hamiltonian function is

H˜Cqp=HjΩJijΩ=HJ˜+J˜J˜0.E79

5.3 SU11quantum systems and a two-level atom

A two-level atom is considered which interacts with two coupled quantum systems that can be represented in terms of a su11Lie algebra. When for example mixed four-waves are injected into a cavity containing a single two level-atom an interaction occurs between the four waves and the atom that is electromagnetic radiation and matter. The Hamiltonian has the form

1H=i=12ωiaiai+12+12ω0σz+λa12a22σ++a12a22σ.E80

It is similar to 5.1, so we sketch the physical situation. The σ±, σzare raising lowering and inversion operators which satisfy the commutation relations σzσ±=2σ±, σ+σ=σz, whereas the ai, aiare basic creation and annihilation operators with ajaj=δij. The interaction term in (80) can be thought of as the interaction between two different second harmonic modes. This can be cast in terms of three su11Lie algebra generators K+, Kand Kzwhich satisfy the commutation relations,

KzK±=±K±,KK+=2Kz.E81

The corresponding Casimir Kwhich has eigenvalue kk1given by

K2=Kz212K+K+KK+.E82

Given that this is the Lie algebra, it can be said that the Fock space is spanned by the set of vectors VF:mkand the operators in (81) act on these states as follows,

Kzm;k=m+km;k,K2m;k=kk1m;k,K+m;k=m+1m+2km+1;k,Km;k=mm+2k1m1;k.E83

It is the case that K0;m=0so this is the lowest level state. The su11Lie algebra can be realized in terms of boson annihilation and creation operators and it is isomorphic to the Lie algebra of the non-compact SU11group. For the Hamiltonian (80) define operators K±iand Kzias

K+i=12ai2,Ki=12ai2,Kzi=12aiai+12,i=1,2,E84

where the Bargmann index kis either 1/4for the even parity states while 3/4applies to the odd-parity states.

Using these operators, (80) is written in terms of su2and su11operators such that it has the form,

1H=i=12ηiKzi+ω2σz+λK+1K+2σ+K1K2σ+.E85

The Heisenberg equations of motion obtained from (85) gives

iddtKz1=λK+1K+2σK1K2σ+,iddtKz2=λK+1K+1σK1K2σ+,iddtσz=λK1K2σ+K+1K+1σ.E86

The following two operators N1and N2are constants of the motion

N1=Kz1+σz,N2=Kz2+σz.E87

Hamiltonian (80) can now be put in the equivalent form

1H=N+C+I,E88

where Iis the identity operator and Nand Care the operators

N=i=12ηiNi,C=Δσz+λK+1K+2σ+K1K2σ+.E89

The constant Δis the detuning parameter defined as

Δ=ω2η1η2.E90

As Nand Ccommute, each commutes with the Hamiltonian Hso Nand Care constants of the motion. The time evolution operator Utis given by

Ut=expiHtexpiNtexpiCt.E91

In the space of the two-level eigenstates

eiNt=eiW1t00eiW2t.E92

The operators Wi, i=1,2are defined by W1=η1Kz1+η2Kz2+1and W2=η1Kz1+η2Kz21. The second exponential on the right of (91) takes the form,

expiCt=cosτitiΔτsinτ1tsinτ1tτ1K1K2K+1K+2sinτ1tτ1cosτ2tiΔτ2sinτ2tE93

where τj2=Δ2+νj, j=1,2and

τ1=λ2K1K+1K2K+2,τ2=λ2K+1K1K+2K2E94

The coherent atomic state ϑ,φis considered to be the initial state that contains both excited and ground states and has the structure,

ϑ,φ=cosϑ2e+sinϑ2eg.E95

where ϑis the coherence angle, φthe relative phase of the two atomic states. The excited state is attained by taking ϑ0, while the ground state of the atom is derived from the limit ϑπ. The initial state of the system that describes the two su11Lie algebras is assumed to be prepared in the pair correlated state ξ,qdefined by

K1K2ξ,q=ξξ,q,Kz1Kz1ξ,q=qξ,q.E96

Since the operators K1K2and Kz1Kz2commute, ξ,qcan be introduced which is simultaneously an eigenstate of both operators,

ξ,q=n=0Cnq+n+k2k1;k1;n,k2.E97

Then applying K2and then K1we obtain,

K1K2ξ,q=n=0Cnnn+2k1q+n+k2k1(q+n+k2k1+2k11q+n+k2k11,k1;n1,k2,=n=0Cn+1n+1n+2kq+n+k2k1+1(q+n+k2k1+2k1q+n+k2k1,k1;n,k2.E98

This calculation implies that the normalization constant Cncan be obtained by solving

n+1n+2kq+n+k2k1+1n+q+k1+k2Cn+1=ξC0.E99

The new state is of the form,

ξ,q=Nqn=0Cnq+n+k2k1,k;n,k2,Nq2=n=0Cn2.E100

If it is assumed that at t=0the wave function of the system is ψ0=ϑ,φξ,q, using (91) on ψ0, the state can be calculated for t>0can be determined

ψt=eiW1tcosτ1tiΔτ1sinτ1t)cosϑ2iλτ1sinτ1tK1K2esinϑ2eξ,q+eiW2tcosτ2t+iΔτ2sinτ2tesinϑ2iλτ2sinτ2tK+1K+2cosϑ2gξ,q.E101

The reduced density matrix is constructed from this

ρft=Tratomψtψt.E102

6. Summary and conclusions

Explicit structures for quantum phase space have been examined. Quantum phase space provides an inherent geometric structure for an arbitrary quantum system. It is naturally endowed with sympletic and quantum structures. The number of quantum dynamical degrees of freedom has a great effect on determining the quantum phase space. Inherent properties of quantum theory, the Pauli principle, quantum internal degrees of freedom and quantum statistical properties are included. A procedure can be stated for constructing this quantum phase space and canonical coordinates should be derivable for all semi-simple dynamical Lie groups with Cartan decomposition. The coset space G/Rprovides a way to define coherent states which link physical Hilbert space and quantum phase space. This motivates the study of the algebraic structure of the phase space representation of observables. The algebraic structure of operators is preserved in phase space if the operators are those of the dynamical group G. Through this approach, this property results in an explicit realization of the classical limit of quantum systems. A classical analogy was developed and seen in the examples as well for an arbitrary quantum system independently of the existence of the classical counterpart, so the classical limit of the quantum system can be obtained explicitly if it exists. The classical analogy will contain the first-order quantum correlation. A theorem which pertains to the relationship between dynamical symmetry and integrability has been proved, and is also valid in classical mechanics. It is then possible to construct a way to look for the quantum manifestation of chaos. Finally, it is then consistent with Berry’s definition, the study of semi-classical but nonclassical, behavior characteristic of systems whose classical motion exhibits chaos.

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Paul Bracken (November 17th 2020). Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos [Online First], IntechOpen, DOI: 10.5772/intechopen.94491. Available from:

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