Can the zero-point energy of the quantized harmonic oscillator be lower? Possible implications for the physics of "dark energy" and "dark matter"
CCan the zero-point energy of the quantized harmonic oscillator be lower?Possible implications for the physics of “dark energy” and “dark matter”
H. A. Kastrup ∗ DESY Hamburg, Theory Group, Notkestrasse 85, D-22607 Hamburg, Germany
Replacing the canonical pair q and p of the classical harmonic oscillator (HO) by the locally andsymplectically equivalent pair angle ϕ and action variable I implies a qualitative change of the globaltopological structure of the associated phase spaces: the pair ( q, p ) is an element of a topologicallytrivial plane R whereas the pair ( ϕ, I > ∈ S × R + is an element of a topologically non-trivial, infinitely connected, punctured plane R −{ } , which has the orthochronous “Lorentz” group SO ↑ (1 ,
2) (or its two-fold covering, the symplectic group Sp (2 , R )) as its “canonical” group. Due toits infinitely many covering groups the resulting (“symplectic”) spectrum of the associated quantumHamiltonian H = ω ˆ I is given by { (cid:126) ω ( n + b ) , n = 0 , , . . . ; b ∈ (0 . , e.g. b = 1 /s, s ∈ N and large } ,in contrast to the ( q, p ) version, where the Hamiltonian has the “orthodox” spectrum { (cid:126) ω ( n +1 / } .The deeper reason for the difference is that for the description of the periodic orbit { p = p ( q ) } onecovering of S suffices, whereas one generally needs many coverings for the time evolution ϕ ( t ). Andthis, in turn, can lead to a lowering of the zero-point energies.Several theoretical and possible experimental implications of the “symplectic” spectra of the HOare discussed: The potentially most important ones concern the vibrations of diatomic moleculesin the infrared, e.g. those of molecular hydrogen H . Those symplectic spectra of the HO mayprovide a simultaneous key to two outstanding astrophysical puzzles, namely the nature of dark(vacuum) energy and that of dark matter: To the former because the zero-point energy b (cid:126) ω offree electromagnetic wave oscillator modes can be extremely small > b ≈ exp ( −
35) for themeasured dark energy density). And a key to the dark matter problem because the quantum zero-point energies of the electronic Born-Oppenheimer potentials in which the two nuclei of H or thenuclei of other primordial diatomic molecules vibrate can be lower, too, and, therefore, may leadto spectrally detuned “dark” H molecules during the “Dark Ages” of the universe and formingWIMPs in the hypothesized sense! All results appear to be in surprisingly good agreement with theΛCDM model of the universe.Besides laboratory experiments the search for 21-cm radio signals from the Dark Ages of theuniverse and other astrophysical observations can help to explore those hypothetical implications. CONTENTS
I. Introduction 2II. Motions on the classicalphase spaces S ϕ,I and S q,p S ϕ,I
41. Global coordinates 42. Orbits on S ϕ,I
53. Time evolution 7B. Orbits on S q,p ω as an external field 8III. Quantum mechanics of the phase space S ϕ,I K j ofthe symplectic group Sp (2 , R ) 8B. Time evolution 91. Heisenberg picture 92. Schr¨odinger Picture 10C. Relationship between the operators ˜ K j and the conventional operators Q and P H = (cid:126) ω ( ˜ K + g ˜ K ) 11 ∗ [email protected]
1. Transition matrix elements with respect tothe number states in 1st order 112. Exact eigenvalues of ˜ C g ( K ) 11E. Explicit Hilbert spaces for ˜ K j , j = 0 , , C g ( K ) = ˜ K + g ˜ K , spectra andeigenfunctions 111. Hardy space on the unit circle 122. Hardy space related Hilbert spaces forgeneral b ∈ (0 ,
1] 123. A unitary transformation by a change ofbasis 144. Aharonov-Bohm-, (fractional) quantumHall-effects, anyons, Berry’s phase,Bloch waves etc. 155. The operator ˜ C g ( K ) = ˜ K + g ˜ K on H b, + K j K + ˜ K − and˜ K − ˜ K + a r X i v : . [ phy s i c s . g e n - ph ] D ec C. Vibrations of diatomic moleculeswith different isotopic atoms 20D. Interferences of time dependent energyeigenstates 21E. Transitions associated with the Hamiltonian H ( K ) = (cid:126) ω ˜ C g ( K ) 21F. Traps for neutral molecules andoptomechanics 21G. Perelomov coherent states 21H. (Dispersive) van der Waals forces 22V. Possible astrophysical implications 23A. Dark energy and the cosmological constant 23B. “Dark” b-H and other primordial moleculesas dark matter? 24Acknowledgments 26A. Hilbert space for ˜ K j and ˜ C g ( K ) on R +0 K j on L ( du, R +0 ) 262. Eigenfunctions and spectrum of˜ C g = ˜ K + g ˜ K I. INTRODUCTION
It very probably appears presumptuous and provoca-tive to question the well-known quantum properties ofthe primeval prototype of quantum mechanical systems:the harmonic oscillator (HO in the following)!The motive for daring a new look at the physical sys-tem HO arise from its well-known locally - but not globally- equivalent canonical descriptions: either in terms of theCartesian coordinates ( q, p ) ∈ R or in terms of the an-gle and action variables ( ϕ, I ) ∈ { R mod 2 π × R + } ∼ = S × R + ∼ = R − { } , where x ∈ R + iff x ∈ R and x > q ( ϕ, I ) = (cid:114) Im ω cos ϕ , p ( ϕ, I ) = −√ m ω I sin ϕ . (1)This mapping is locally symplectic: dq ∧ dp = dϕ ∧ dI , or ∂ ( q, p ) ∂ ( ϕ, I ) = 1 . (2)The canonically equivalent Hamiltonians are given by H ( q, p ) = 12 m p + 12 m ω q = H ( ϕ, I ) = ω I , (3) with their respective canonical Eqs. of motion˙ q = ∂H∂p = p/m, ˙ p = − ∂H∂q = − mω q ; (4)˙ ϕ = ∂H∂I = ω, ˙ I = − ∂H∂ϕ = 0 , (5)the latter with the obvious solutions ϕ ( t ) = ω t + ϕ , I = const. = I = E/ω > . (6)The solutions of the Eqs. (4) and (5) describe - as func-tions of time t - orbits in the repective phase spaces S q,p = { ( q, p ) ∈ R } ∼ = R (7)and S ϕ,I = { ( ϕ, I ) , ϕ ∈ R mod 2 π , I > } (8) ∼ = S × R + ∼ = R − { } . The crucial point - for all what follows in the presentpaper - is this: the two phase spaces (7) and (8) are globally (topologically) qualitatively different! Whereasthe ( q, p )-space is a simply connected and topologicallytrivial plane R , the ( ϕ, I )-space is topologically a “punc-tured” plane, i.e. a R with the origin { (0 , } deleted!This is so for several reasons: the variables ϕ and I can be considered as polar coordinates of a plane, with- obviously - ϕ the angle and I > I = 0 has to be excluded because otherwise theangle ϕ becomes undefined at that point. In addition thevalue I = 0 describes a branch point for the transforma-tion functions (1). More arguments can be found in Ref.[2].The topology of the phase space (8) may equivalentlybe characterized as that of a simple cone with the tipdeleted or as that of a semi-cylinder without the pointsof the finite circular surface at I = 0.If ϕ ( t ) and I ( t ) describe the moving points of a peri-odic orbit on S ϕ,I then those points may loop around theorigin arbitrarily many times (the first homotopy group π of S ϕ,I consists of the integers ∈ Z ), because the or-bit coordinate ϕ ( t ) can circle the origin of S arbitrarilymany times in the course of time t ! Thus, the config-uration space of ϕ corresponds to one of the infinitelymany covering spaces of the circle S , the universal cov-ering being the real line R . A physical example for ahigh number of coverings is provided by the oscillationsof electromagnetic vibrations.That missing point in the phase space R ≡ S q,p −{ , } , or in S ϕ,I , has dramatic consequences for the as-sociated quantum theory which will be discussed in moredetail below.The crucial result is the following:The quantum operator version ˆ I = K of the classicalaction variable I of the HO has the possible spectrum K | b, n (cid:105) = (cid:126) ( n + b ) | b, n (cid:105) , n = 0 , , , . . . ; 1 > b > . (9)Here K is the self-adjoint Lie algebra generator of thecompact subgroup O(2) in an irreducible unitary repre-sentation of the 3-dimensionsl “orthochronous Lorentz”group SO ↑ (1 , ∼ = Sp (2 , R ) /Z or of one of its in-finitely many covering groups, where the double cover-ing Sp (2 , R ) is the symplectic group of the plane. (seebelow). This means that the HO Hamilton operator H ϕ,I = ω K , (10)associated with the phase space S ϕ,I , can have a groundstate ( n = 0) with eigenvalue (zero-point energy) (cid:126) ω b, > b >
0, especially with 1 / > b > R is thesymplectic group Sp (2 , R ) which acts transitively on R (for any two points on R there is an element of Sp (2 , R )which connects the two but leaves the origin invariant )!The group Sp (2 , R ) is a twofold covering of the “or-thochronous Lorentz” group SO ↑ (1 ,
2) in “one time andtwo space dimensions” which acts correspondingly on thephase space S ϕ,I , i.e. acting transitively and leaving theorigin invariant! The number b, > b > SO ↑ (1 ,
2) [3].The main mathematical aspects of the present paperhave been presented previously in Refs. [2, 4–7]. A brief -introductory but probably helpful - summary of them isgiven in Ref. [8]. Essential mathematical references are[9–14].The present paper tries to draw attention to possibleexperimental and observational implications of the ( ϕ, I )-framework for the HO. Hopefully, appropriate laboratoryexperiments and astrophysical obervations will be able tofind out whether nature has “made use of the availablemathematical possibilities”(Dirac) or not!There are - at least - two immediate crucial questions:i) Why should the classical canonical pair ( ϕ, I ) be a“better” - or at least equivalent - basis for the quantumdescription of a system like the HO, compared to theconventional pair ( q, p )?ii) If the predictions of the quantized ( ϕ, I ) frameworkare richer than the ( q, p ) framework - but not contradic-tory -, why haven’t they been observed yet?Ad i): A crucial obstacle for using the “observable”angle ϕ itself for the quantum description of a physicalsystem has been that there exist no corresponding self-adjoint operator ˆ ϕ [15, 16]! This shortcoming can, how-ever, be remedied by the following observation [15, 16]:Geometrically an angle ϕ can be defined by two ori-ented rays (vectors) both originating from the same givenpoint. The two rays then span a plane. In order to de-scribe the angle uniquely analytically one chooses a thirdray which “emanates” from the same point and which isorthogonal to one of the two original rays. Projectingthe second original ray onto the two orthogonal ones bymeans of a circle, with radius a , around the origin yields apair a (cos ϕ, sin ϕ ) , a > , which determines ϕ uniquely,after chosing a clockwise- or counter-clockwise orienta-tion. It is convenient, but not necessary, to put a = 1. Quantizing the system then means: quantizing thecomponents a cos ϕ and a sin ϕ which combined representone “observable”, the angle ϕ ! The details depend on thechoice of a and possibly other elements of the associatedPoisson algebra.We now come to a crucial physical point:For periodic motions - like that of a HO - the an-gle ϕ ( t ) of Eq. (5) does not stop at ϕ ( T ) ≡ ω T = 2 π but “runs” around the origin, say at least s times, i.e. ϕ ( t ) ≥ ϕ ( t s ) = s π, s ∈ Z , , and in this way generates an(s+1)-fold covering of the unit circle S . In this way theconfiguration space S s +1] of the angle becomes (s+1)-fold connected, and, as s can be an arbitrary integer,infinitely connected.So, for dynamical (time-dependent) periodic systemsthe “observable” angle consists of 2 parts: the number s of completed coverings S s ] of the unit circle and a “rest” χ ∈ [0 , π ): ϕ = (2 π s + χ ) ∈ S s +1] , s ∈ Z , χ ∈ [0 , π ) . (11)The crucial point for periodic motions is that s = 0 is suf-ficient to describe the orbit p ( q ) of Eq. (1), but in orderto describe the time evolution ϕ ( t ) one needs to know thepair ( χ ; s ) of Eq. (11). This is a consequence of the non-trivial topology of the phase space (8). In many casesthe angle ϕ ( t ) appears in the form ϕ ( t ) = ωt ≡ ˜ t , i.e. itis essentially a time variable. Thus, the pair ( q, p ) ∈ S q,p of a periodic orbit is independent of the number s ofcoverings. But this number of coverings is essential inconnection with the phase space S ϕ,I . This importantdifference leads to corresponding different quantum me-chanical properties of the two phase spaces, e.g. for S ϕ,I to the set of spectra (9), containing the “orthodox” case b = 1 / s -foldcovering ( s > b = 1 /s .The introduction of the canonical pair angle and actionvariables is conventionally motivated by the aim to makethe action variable I a constant of motion, i.e. to have an“integrable” system [17–19]. But this is not necessary:one can try to describe systems in the phase space (8)in terms of the local coordinate pair ( ϕ, I ) or the globalones h = I, . h = I cos ϕ, h − I sin ϕ which will beillustrated by an example in the next chapter.Phases play an important role in many physical sys-tems with periodic properties like vibrations, waves etc.,e.g. in optics, atomic and molecular spectroscopy, con-densed matter physics etc. Thus, it is important to un-derstand the corresponding quantum theories in terms ofthe canonical pair angle and action variable properly andconsistently and look for experimental consequences.Ad ii): One reason might be that nobody up to nowhas been looking for the newly predicted physical phe-nomena! Another reason could be that the associatedsignals are very weak and obscured by the “orthodox”spectrum (cid:126) ω ( n + 1 / h j , j = 0 , ,
2, provided interms of the group SO ↑ (1 , S ϕ,I ofthe HO. These “classical” considerations are intended toprovide an intuitive background for the discussions of theassociated quantum mechanics in Ch. III.In Ch. III several aspects of quantum mechanical sys-tems are discussed the basic “observables” of which aregiven by the Lie algebra elements K , K and K of thecanonical group Sp (2 , R ) of the punctured plane. Thoughthis Lie algebra is also that of the isomorphic groups SU (1 ,
1) or SL (2 , R ), of the group SO ↑ (1 ,
2) and all cov-ering groups as well, the “symplectic” variant appears tobe preferable because of possible generalizations to higherdimensional phase spaces [2, 6].One of the main topics in this chapter consists of thediscussion of the spectrum (9) and related explicit Hilbertspaces for the representation of the self-adjoint operators K , K , K . Extended use is made of mathematical re-sults contained in Refs. [2, 6]. Hardy spaces (i.e. Hilbertspaces which have non-vanishing Fourier components for n = 0 , , , . . . only) on the circle play a prominent rolefor constructing those explicit Hilbert spaces.Ch. IV contains a number of suggestions to find con-crete physical systems to which the theoretical frameworkmay apply.Possibly the most important application concerns thevibrations of diatomic molecules which are harmonicin the neighbourhood of the minima of their Born-Oppenheimer (BO) potentials and where “symplectic”spectra (9) may lead to a lower ( b < /
2) ground stateenergy compared to the “orthodox” value b = 1 / . As this moleculehas no permanent electric dipole moment, its “orthodox”infrared emission and absorption signals are already veryweak. Therefore very probably even more so the “non-orthodox” ones. Other diatomic molecules of the lightestelements with an electric dipole element (like, e.g. LiH)may be more appropriate for laboratory infrared experi-ments.For H itself Raman scattering or atomic and molec-ular collisions may induce transient electric dipole mo-ments leading to characteristic emissions or absorptionsassociated with vibrations (and rotations) [20].Extremely important are possible astrophysical appli-cations, discussed in Ch. V, especially concerning theproblems of dark energy and dark matter; here the spec-trum (9) may provide the key to the simultaneous under-standing of both problems:As the index b may be arbitrarily small >
0, the associ-ated estimate of the cosmological constant Λ - or the vac-uum “dark” energy density - can be compatible with theexperimentally observed value, leading to b ≈ exp( −
35) .In addition, the possible lowering of the vibrational zero-point energies of electronic Born - Oppenheimerpotentials for diatomic molecules suggests to look atmolecular hydrogen b-H and other primordial diatomicmolecules as candidates for dark matter.Altogether one finds that the consequences of a sym-plectic spectrum (0 < b < /
2) of the HO are sur-prisingly well compatible with the cosmological ΛCDMmodel, with - mainly - b -H molecules as WIMPs !There is, however, one important caveat: the dynamicsof the transitions (rates) to and from the new additionalenergy levels has still to be worked out !Experimentally, 21-cm radio telescopes directed to-wards the Dark Ages of the universe are of special impor-tance (see, e.g. Ref. [21]). Recent observations [22] indi-cate – unexpected for the present interpretations of darkmatter – non-gravitational (electromagnetic?) interac-tions between atomic hydrogen and dark matter [23, 24] !This appears to be compatible with b -H molecules asdark matter Similarly the recently observed discrepancybetween computer simulated dark matter models andgravitational lensing [25] is of interest in this context.If the observed cosmic dark matter indeed consists of- infrared detuned - primordial diatomic molecules thenthere is no need for the introduction of any kind of “new”matter, a point which has also been emphasized in therecent discussions of dark matter as being formed by pri-mordial black holes (for a recent review see, e.g. Ref.[26]). II. MOTIONS ON THE CLASSICALPHASE SPACES S ϕ,I AND S q,p The present chapter discusses a simple classical modelon the phase space S ϕ,I as a preparation for the discus-sion of the corresponding quantum mechanical one later. A. Coordinates and orbits on S ϕ,I
1. Global coordinates
It was already indicated above that the angle ϕ itselfis not a “good” global coordinate on S ϕ,I . The situa-tion is even worse for the corresponding quantum theory[15]. As described above, a way out is to characterize thegeometrical quantity “angle” by the pair (cos ϕ, sin ϕ ).However the triple I, C ( ϕ ) = cos ϕ, S ( ϕ ) = sin ϕ is stillnot appropriate for our present purpose:Consider the Poisson brackets { f , f } ϕ,I ≡ ∂ ϕ f ∂ I f − ∂ I f ∂ ϕ f (12)for locally smooth functions on S ϕ,I . The 3 functions I, C ( ϕ ) and S ( ϕ ) obey the Poisson Lie algebra { I, C } ϕ,I = S , { I, S } ϕ,I = − C , { C, S } ϕ,I = 0 , (13)which constitutes the Lie algebra of the Euclidean group E (2) of the plane: rotations (generated by I ) and 2 in-dependent translations (generated by C and S ). Theyare the proper coordinates for a phase space with thetopology of an infinite cylinder S × R , like that of thecanonical system angle and orbital angular momentum[15, 16].It can be justified systematically [27] that the appro-priate global coordinates on S ϕ,I are the functions h ( ϕ, I ) = I > , (14) h ( ϕ, I ) = I cos ϕ , h ( ϕ, I ) = − I sin ϕ , which obey (cid:126)h ≡ h + h = h , h > . (15)and, therefore, describe a simple (“light”) cone, with thetip deleted. The functions h j ( ϕ, I ) obey the Poisson Liealgebra { h , h } ϕ,I = − h , { h , h } ϕ,I = h , (16) { h , h } ϕ,I = h , which constitutes - as mentioned above - the Lie algebraof the 3-dimensional group SO ↑ (1 ,
2) or of the symplecticgroup Sp (2 , R ) of a ( x, y )-plane, the transformations ofwhich leave the skew-symmetric form dx ∧ dy invariant.The triple ( h , h , h ) transforms as a 3-vector withrespect to the group SO ↑ (1 , q, p ) transformsas a vector with respect to the symplectic group [2]!As the symplectic group Sp (2 , R ) is isomorphic to thegroups SL (2 , R ) and SU (1 ,
1) [28], one may use thosehere, too. But the identification as the symplectic groupappears to be more appropriate in the framework of clas-sical mechanics and, above all, it can be generalized tohigher dimensions [2].Justification of the global “canonical” coordinates (14)in a nutshell [8]: The three 1-dimensional subgroups ofthe (transitive) group SO ↑ (1 ,
2) = Sp (2 , R ) /Z (1 ro-tation, 2 “Lorentz boosts”) generate global orbits on S ϕ,I . The generators of these orbits are global Hamil-tonian vector fields the associated Hamiltonian functionsof which are the “coordinates” (14). This is in completeanalogy to the usual phase space S q,p the global coordi-nates q and p of which are the Hamiltonian functions ofthe vector fields which generate the global translations in p - and q -directions on S q,p , endowed with a symplecticstructure in terms of the Poisson bracket {· , ·} q,p .
2. Orbits on S ϕ,I The graph of the motion (6) in S ϕ,I is utterly simple:a circle of radius I > t is given by the angle ϕ ( t ) = ωt . We assume that ϕ ( t =0) = 0 and that ϕ ( t ) starts clockwise off a given rayemanating from the point I = 0. That ray also definesan horizontal abscissa of an orthogonal coordinate system with an ordinate of pointing upwards (see Fig. 1). Theclockwise orientation of the angle is induced by the choiceof h and h in Eqs. (14). Note that h is the projection of I on the positive ab-scissa and h the one on the negative ordinate (see Fig.1).Note also that these two projections do not commute(see the last of the Eqs. (16)), again a consequence of thefact that the point (0 ,
0) does not belong to the phasespace!Things become more interesting if we “disturb” the HOby introducing new interactions. Note that on S ϕ,I func-tions have to be expressed in terms of the basic variables h j ( ϕ, I ) , j = 0 , ,
2. This means for the HO: H ( ϕ, I ) = ω h . (17)A simple but interesting modification is [29] H ( ϕ, I ) = ω ( h + g h ) = ω I (1 + g cos ϕ ) , g ≥ , (18)with the Eqs. of motion˙ ϕ = ∂ I H ( ϕ, I ) = ω (1 + g cos ϕ ) , (19)˙ I = − ∂ ϕ H ( ϕ, I ) = g ωI sin ϕ. (20)Now I ( t ) is no longer a constant.The present discussion is an extension of the usual onefor (completely) integrable systems [17–19, 30] in whichthe action variables are constants of motion as functionson the original ( q, p )-phase space and where the original“tori” (determined by I = const. and ϕ ∈ S in our veryspecial case) are rather stable against small perturbations(KAM theory [31]).Here the global phase space formed by angle and actionvariables is being considered and the action variable I may be a function of time t , like the angle variable ϕ .Recall that an action variable is originally defined asa global variable - like the energy - on the phase space(7), namely as a closed path integral along the border ofa volume determined by the energy and the potential ofthe system [17–19, 30]: I ( E ) = 12 π (cid:73) C ↑ ( E ) dq p ( q, E ) , p ( q, E ) = ± (cid:112) m ( E − V ( q ) , (21)where the clockwise oriented closed path C ↑ ( E ) is de-termined by the energy equation p / m + V ( q ) = E .According to Stokes’ theorem the path integral is equalto the volume with the border C ↑ ( E ). Here I ( E ) is aconstant of motion because E is a constant along theorbits { [ q ( t ) , p ( t )] } .The more general case I ( t ) can be obtained from thelocal relation (2): Integrating both sides simultaneouslyat time t gives∆ ϕ ( t ) · ∆ I ( t ) = (cid:90) ∆ G q,p ( t ) dqdp = ∆ V q,p ( t ) , (22)where ∆ V q,p ( t ) is the volume of the region ∆ G q,p ( t ) ⊂S q,p , with ∆ V q,p ( t ) = 0 for ∆ ϕ ( t ) = 0 or ∆ I = 0. Puttingthe lower value I = 0 in ∆ I = I − I the special case(21) is obtained for ∆ ϕ ( t ) = 2 π .With I = 0 it follows from Eq. (22) that I ( t ) = ∂V q,p ( t ) /∂ϕ, (23)where the pair ( q, p ) is assumed to be a function of ϕ likein Eqs. (1). Thus, I ( t ) may be interpreted as the differ-ential change with ϕ of the phase space volume V q,p ( t ) attime t .The additional term g h in the Hamiltonian (18)breaks several related symmetries: rotation invariancein the ( h , h )-plane (which can be remedied by usingthe combination cos α h + sin α h instead of h ), specialLorentz “boosts” in the directions “1” or “2” and reflec-tion parity ( ϕ → ϕ ± π ). Time reversal ( ϕ → − ϕ ) isfulfilled.The Eqs. of motion (19) and (20) can be integratedimmediately:As the energy is still conserved, ω I (1 + g cos ϕ ) = E = const. , (24)we have the orbit equation I ( ϕ ) = I g cos ϕ , I = E/ω . (25)This equation describes a conical section with a givenfocus as the origin for the polar coordinates ϕ (“trueanomaly”), distance I ( ϕ ) from that focus , “semi-latusrectum” I and “numerical eccentricity” (cid:15) = g .For g < g = 1 a parabola andfor g > ϕ increases clockwisefrom the fixed ray which starts from the focus nearest tothe orbit point I − = I / (1 + g ), the “perihelion”, andfurther passes through that latter point (see Fig. 1). Ellipse, g < • semi-latus rectum: I , • numerical eccentricity: g , • perihelion: I − = I / (1 + g ), • aphelion: I + = I / (1 − g ) • major semi-axis: a = [ I / (1 + g ) + I / (1 − g )] = I / (1 − g ), • linear eccentricity: e = g a = gI / (1 − g )(2 e is the distance of the two foci), • minor semi-axis: b = √ a − e = I / (1 − g ) / , • area of ellipse: abπ = I π (1 − g ) / .Thus the shape of the ellipse is completely determinedby the coupling constant g and the integration constant I . g = g = 0 b e a I − I + I I Iϕh h FIG. 1. Phase space S ϕ,I : 1. Graph of the circular orbit { ( ϕ ∈ [0 , π ) , I = I > } generated by the periodic motion(6) of the ”orthodox” HO (in black). 2. Graph of the ellipticalorbit I ( ϕ ) = I / (1 + g cos ϕ ) associated with the periodicmotions (31) and (36) (in red, with g=1/2). Origin of thepolar coordinates ( ϕ, I ) here is the right focus. The meaningof the different letters with their related quantities and theirhistorical names are described in the list (26) The same holds for the hyperbola with the focus of theleft branch as the origin for the polar coordinates:
Hyperbola, g > : (27) • If g > I / (1 + g ) where ϕ = 0. • semi-latus rectum: I , • numerical eccentricity: g , • linear eccentricity: e = gI / ( g − e is the distance of the 2 foci), • major and minor semi-axis: a = I / ( g − , b = I / ( g − / , • The two angles ϕ ∞ ( i ) , i = 1 , ϕ ∞ ( i ) = − /g, ϕ ∞ (1) ∈ ( π/ , π ) and ϕ ∞ (2) ∈ ( π, π +3 π/ Parabola, g = 1 : (28)This simple case can be treated in the same way as thetwo others above.Obviously, the orbits of the last two cases extend toinfinity.
3. Time evolution
Ellipse
The time evolution follows from Eq. (19): (cid:90) ϕϕ dϑ g cos ϑ = ω ( t − t ) . (29)For g < ϕ = 0 for t = 0: (cid:112) − g ω t = 2 arctan (cid:20)(cid:114) − g g tan( ϕ/ (cid:21) , (30)or tan[ ϕ ( t ) /
2] = (cid:114) g − g tan( (cid:112) − g ω t/ . (31)Thus, the interaction g h ( ϕ, I ) leads to an effective red-shifted angular frequency ω g = (cid:112) − g ω, (32)with a branch point for g → t ( π/
2) neededto pass from ϕ = 0 to ϕ = π/ ω g t ( π/
2) = 2 arctan (cid:114) − g g = arccos g. (33)At that time I [ t ( π/ I (see Eq. (25)).The time needed to pass from ϕ = 0 to ϕ = π is ω g t ( π ) = π. (34)Here we have I [ t ( π )] = I / (1 − g ) (“aphelion”).For reasons of symmetry of the ellipse we get from Eq.(34) for one period ω g T g ;2 π = 2 π, T g ;2 π = 2 πω (cid:112) − g . (35)Thus, 1 / (cid:112) − g is a kind of “refractive index”.Once the time evolution ϕ ( t ) is known that of I ( t )can be obtained from the orbit equation (25). Using therelation cos ϕ = 1 − tan ( ϕ/ / [1+tan ( ϕ/ I ( t ) = I − g (1 − g cos[ ω g t ]) , (36)which again gives the relations (33), (34) and (35).The above results may be looked at as follows: Forvanishing g we have on S ϕ,I a clockwise periodic motionwith frequency ω on a circle of radius I . Adding theinteraction g h ( ϕ, I ) , ≤ g < , deformes the circle intoan ellipse with semi-latus rectum I and numerical ec-centricity g . In addition the original angular frequency ω of the periodic motion is reduced to ω g = (cid:112) − g ω .Pictorially speaking we start with a “circularly po-larized” motion ( g = 0) which encounters a medium (0 < g <
1) which induces an “elliptical polarization”and reduces the original angular frequency ω ! Hyperbola
As before the time evolution can be calculated from Eq.(19), the integration of which now gives [32]tan[ ϕ ( t ) /
2] = (cid:114) g + 1 g − (cid:112) g − ω t/ . (37) Parabola
Finally, the time evolution for the parabola ( g = 1) isgiven by [33] tan[ ϕ ( t ) /
2] = ω t . (38)
B. Orbits on S q,p Using the mappings (1) the orbit equation (25) in S ϕ,I can be mapped onto S q,p , where it has the parametriza-tion q g ( ϕ ) = (cid:115) I mω (1 + g cos ϕ ) cos ϕ (39) p g ( ϕ ) = − (cid:115) mωI g cos ϕ sin ϕ. (40)It implies g = 0 φg = q + q − qp tan φ = − p g q g = mω tan ϕ FIG. 2. Phase space S q,p : 1. Graph of the “orthodox”elliptical orbit generated by the usual HO (in black) 2. Graphof the image of the (red) ellipse in Fig. 1 as described by Eqs.(39) and (40) (in red). Here φ is the polar angle on S q,p and ϕ the polar angle on S ϕ,I . H ( q g , p g ) = p g m + 12 mω q g (41)= ω I = ω I g cos ϕ , − p g q g ≡ tan φ = m ω tan ϕ. (42)andcos ϕ = (cid:112) m/ ω q g (cid:112) H ( q g , p g ) = ± (cid:113) V ( q g , p g ) /H ( q g , p g ) . (43)Inserting the last expression into Eq. (41) yields a 4thorder equation in q g and p g for the orbit described byEqs. (39) and (40): H ( q g , p g ) ± g (cid:113) V ( q g , p g ) H ( q g , p g ) = ω I , (44)[ H ( q g , p g ) − ω I ] = g V ( q g , p g ) H ( q g , p g ) . For g = 0 this is reduced to the usual orbit ellipse of theHO on S q,p (See Fig. 2): p g =0 m + 12 mω q g =0 = E = ωI = const. (45)The functions (39) and (40) have the special values q g (0) ≡ q − = (cid:115) I mω (1 + g ) , p g (0) = 0 , (46) q g ( π/
2) = 0 , p g ( π/
2) = − (cid:112) mωI ,q g ( π ) ≡ q + = − (cid:115) I mω (1 − g ) , p g ( π ) = 0 ,q g (3 π/
2) = 0 , p g (3 π/
2) = (cid:112) mωI . Note that the 4th order figure defined by Eq. (44) is sym-metric with respect to the q -axis, but no longer symmet-ric with respect to the p -axis (see Fig. 2). Maximum andminimum of p g ( ϕ ) are given by the angles ϕ ± , whichobey cos ϕ ± = − (1 − (cid:112) − g ) /g. (47) C. The frequency ω as an external field According to Eq. (3) the Hamiltonian of the HO on S ϕ,I has the simple form H ( ϕ, I ) = ω I, (48)where the frequency ω appears as a parameter multiply-ing the basic action variable I .That parameter ω may also be considered as an ex-ternal “field” which can be “manipulated” from outside,e.g., as a function ω ( t ) of time t . The solution of Eq. (4)then is ϕ ( t ) − ϕ ( t ) = (cid:90) tt dτ ω ( τ ) . (49)Note that the Hamiltonian (48) is independent of ϕ andtherefore ˙ I = − ∂ ϕ H ( ϕ, I ) = 0, even if ω = ω ( t )! Thus,the action variable I is still conserved on S ϕ,I , but the energy E = ω ( t ) I is not! A possible interesting examplefor applications is a time dependent angular frequency ofthe form ω ( t ) = ω + λ sin( ρt ) , | λ | < ω . (50)In the present context it is also appropriate to brieflyrecall the so-called “adiabatic invariance” [17, 19, 34–36]of the action variable I: If energy E ( λ ) and frequency ω ( λ ) of a periodic motion with period T depend on aslowly varying parameter λ ( T dλ/dt (cid:28) λ ), then I = E ( λ ) /ω ( λ ) remains constant if λ varies.In the following discussions on the “non-orthodox”quantum mechanics of the HO it is essential to differ-entiate between the quantum counterparts of the actionvariable I and the Hamilton function H , the generatorof time evolution.For a given “binding” potential V ( q ) the angular fre-quency ω is generally defined as one half of the 2ndderivative of V ( q ) at its (local) minimum q = q : ω = (1 / d V ( q ) /dq | q = q . III. QUANTUM MECHANICS OF THE PHASESPACE S ϕ,I A. Basics: self-adjoint representations of thethree Lie algebra generators K j ofthe symplectic group Sp (2 , R ) The quantization of the global “coordinates” h j fromEq. (14) is implemented by reinterpreting them as self-adjoint operators in a given Hilbert space [37], h j → K j = (cid:126) ˜ K j (51)which obey the associated Lie algebra (16):[ ˜ K , ˜ K ] = i ˜ K , [ ˜ K , ˜ K ] = − i ˜ K , (52)[ ˜ K , ˜ K ] = − i ˜ K . (53)(Quantities ˜ A with a ”tilde”, here and in the following,are considered to be dimensionless).The three self-adjoint operators ˜ K j can be obtainedas Lie algbra generators of irreducible unitary represen-tations of the corresponding groups SO ↑ (1 , Sp (2 , R )(the latter being isomorphic to the groups SL (2 , R ) and SU (1 , K is the generator of the maximal compact abeliansubgroup U (1), its eigenstates may be used as a Hilbertspace basis (here formally in Dirac’s notation, explicitexamples will be discussed later):˜ K | b, n (cid:105) = ( n + b ) | b, n (cid:105) , where b is some real number ∈ (0 ,
1) (“Bargmann index”[9]) and n = 0 , , , . . . . This central result can be derivedas followsThe operators˜ K ± = ˜ K ± i ˜ K , (54) K = 12 ( ˜ K + + ˜ K − ) , K = 12 i ( ˜ K + − ˜ K − ) , (55)obey the relations[ ˜ K , ˜ K + ] = ˜ K + , [ ˜ K , ˜ K − ] = − ˜ K − , (56)[ ˜ K + , ˜ K − ] = − K . (57)They are raising and lowering operators:˜ K + | b, n (cid:105) = [(2 b + n )( n + 1)] / | b, n + 1 (cid:105) , (58)˜ K − | b, n (cid:105) = [(2 b + n − n ] / | b, n − (cid:105) . (59)The relations (58) and (59) are derived under the as-sumptions that there exists a state | b, (cid:105) such that˜ K | b, (cid:105) = b | b, (cid:105) , ˜ K − | b, (cid:105) = 0 , b ∈ [1 , . (60)Eq. (58) implies | b, n (cid:105) = 1 (cid:112) (2 b ) n n ! ( ˜ K + ) n | b, (cid:105) , (61)(2 b ) n = 2 b (2 b + 1) · · · (2 b + n − b + n ) / Γ(2 b ) . It then follows that˜ K | b, n (cid:105) = ( n + b ) | b, n (cid:105) , n = 0 , , . . . ; 1 > b > . (62)This is the so-called “positive discrete series” D (+) b among the different types of possible irreducible unitaryrepresentations of Sp (2 , R ) [9, 28] The Bargmann index b - called “B-index” in the following - characterizes anirreducible unitary representation (IUR) D (+) b .The Casimir operator C = ˜ K + ˜ K − ˜ K (63)= ˜ K + ˜ K − − ˜ K ( ˜ K − K − ˜ K + − ˜ K ( ˜ K + 1)of the IUR D (+) b has the value C = b (1 − b ) . (64)This means that the “classical Pythagoras” (15) is vi-olated quantum mechanically for b (cid:54) = 1, e.g. in the caseof the usual HO with b = 1 / SO ↑ (1 ,
2) has infinitely many coveringgroups because its compact subgroup O (2) ∼ = S ∼ = U (1)is infinitely connected!Let us denote the s -fold covering by SO ↑ [ s ] (1 , , s = 1 , , . . . . (65) Its irreducible unitary representations D (+) b have the in-dices b = rs , r = 1 , , . . . , s − . (66)This means that b min = 1 /s can be arbitrarily small > s is large enough!The 2-fold coverings Sp (2 , R ) = SL (2 , R ) ∼ = SU (1 ,
1) (67)have b = 1 / ϕ, I )-Hamiltonian H ( ϕ, I ) → H osc ( K ) = ω K , K = (cid:126) ( ˜ K , ˜ K , ˜ K ) (68)can have the b -dependent spectra E b, n ( ϕ, I ) = (cid:126) ω ( n + b ) , n = 0 , , . . . ; 1 > b > , (69)As this result is due to the properties of the symplecticgroup Sp (2 , R ), especially its compact subgroup U (1) ∼ = O (2), we call it the “symplectic spectrum” of the HO, andthe conventional special case b = 1 / (cid:104) b, n | ˜ K j | b, n (cid:105) = 0 , j = 1 , , (70)implying for the mean square deviations(∆ ˜ K j ) b,n = 12 ( n + 2 nb + b ) , j = 1 , , (71)so that (∆ ˜ K ) b,n (∆ ˜ K ) b,n = 12 ( n + 2 bn + b ) , (72)(∆ ˜ K ) b,n =0 (∆ ˜ K ) b,n =0 = b . (73)The last relation shows that b → B. Time evolution
1. Heisenberg picture
The appearence of covering groups (65) has a naturalphysical background: Take the time dependence of theangle ϕ ( t ) = ω t in Eq. (6) (with t = 0 , ϕ = 0): Ingeneral the system will not stop after covering a circlejust once, ω T = 2 π , but will circle the origin, say, atleast s times, ω T s = 2 π s , where s ∈ N can be arbitrarilylarge.In the following the dimensionless time variable˜ t = ω t (74)0will be used. It is an angle variable.The unitary time evolution operator is given by U (˜ t ) = e − i ˜ K ˜ t , ˜ K = N + b , (75)where the number operator N can be considered a func-tion of the operators ˜ K j , j = 0 , ,
2, as will be shownbelow.The unitary operator (75) implies the usual HeisenbergEqs. of motion: U ( − ˜ t ) ˜ K + U (˜ t ) = e − i ˜ t ˜ K + , (76) U ( − ˜ t ) ˜ K − U (˜ t ) = e + i ˜ t ˜ K − , (77) U ( − ˜ t ) ˜ K U (˜ t ) = cos ˜ t ˜ K + sin ˜ t ˜ K , (78) U ( − ˜ t ) ˜ K U (˜ t ) = − sin ˜ t ˜ K + cos ˜ t ˜ K . (79)For ˜ t = 2 π the operator (75) becomes U (˜ t = 2 π ) = e − πib . (80)If b = r/s , r, s ∈ N , r < s and divisor free , (81)this implies for SO ↑ [ s ] (1 , U (˜ t = s π ) = . (82)The ground state | b, (cid:105) has the time evolution U (˜ t ) | b, (cid:105) = e − i b ˜ t | b, (cid:105) , (83)with the associated time period T π,b = 2 πω b , ω b ≡ b ω , (84)which can become arbitrarily large for b = 1 /s, s → ∞ .Symbolically speaking: 1 /b is a kind of refraction index n . Whereas ˜ K generates rotations and time evolutionsby performing many phase rotations, the operators˜ K and ˜ K generate special “Lorentz” transformations(“boosts”) in directions 1 and 2, respectively [38]:With U ( w ) = e ( w/
2) ˜ K + − ( w ∗ /
2) ˜ K − = e i w ˜ K + i w ˜ K , (85) w = w + i w = | w | e iθ one obtains U ( − w ) (cid:126) ˜ KU ( w ) = (cid:126) ˜ K (cosh | w | − (cid:126)n · (cid:126) ˜ K ) · (cid:126)n (86)+ sinh | w | (cid:126)n ˜ K ,U ( − w ) ˜ K U ( w ) = cosh | w | ˜ K + sinh | w | ( (cid:126)n · (cid:126) ˜ K ) , (87) (cid:126) ˜ K = ( ˜ K , ˜ K ) , (cid:126)n = (cos θ, − sin θ ) . Eqs. (86) and (87) describe a Lorentz “boost” in direction (cid:126)n .
2. Schr¨odinger Picture If | b ; t (cid:105) is a state vector of the Hilbert space associatedwith the B-index b at time t , then - according to theunitary time evolution (75) - we have at time t : | b ; t (cid:105) = U ( t − t ) | b ; t (cid:105) = e − iωK ( t − t ) / (cid:126) | b ; t (cid:105) , (88)which implies (cid:126) i ∂ t | b ; t (cid:105) = ωK | b ; t (cid:105) , K = (cid:126) ˜ K . (89)For a representation | b ; t (cid:105) = ∞ (cid:88) n =0 c n ( t ) | b, n (cid:105) (90)we get from Eq. (88), with t = 0, c n ( t ) = e − iω ( n + b ) t , (91)so that | b ; t (cid:105) = e − ib ωt ∞ (cid:88) n =0 e − i n ωt | b, n (cid:105) . (92) C. Relationship between the operators ˜ K j and the conventional operators Q and P The relations (1) expressed in terms of the functions h j ( ϕ, I ) from Eqs. (14) take the form q ( ϕ, I ) = (cid:114) m ω h ( ϕ, I ) (cid:112) h ( ϕ, I ) , (93) p ( ϕ, I ) = √ m ω h ( ϕ, I ) (cid:112) h ( ϕ, I ) . (94)There is a corresponding relationship at the operatorlevel: Define the operators A ( K ) = ( ˜ K + b ) − / ˜ K − , (95) A † ( K ) = ˜ K + ( ˜ K + b ) − / . (96)According to Eqs. (62), (58) and (59) they act on thenumber states | b, n (cid:105) as A † | b, n (cid:105) = √ n + 1 | b, n + 1 (cid:105) , (97) A | b, n (cid:105) = √ n | b, n − (cid:105) , (98) n = 0 , , . . . . This means [
A, A † ] = ∀ D (+) b , (99) independent of the value of b ! A ( K ) and A † ( K ) arethe usual Fock space annihilation and creation operatorsfor all b and independent of b !Note that the denominator in Eqs. (95) and (96) iswell-defined, because ˜ K is a positive definite operatorand b a positive number for each representation of theseries D (+) b .The quantum mechanical position and momentum op-erators Q and P can now be defined as usual: Q = Q ( K ) = λ √ A † ( K ) + A ( K )] , (100) P = P ( K ) = i (cid:126) √ λ [ A † ( K ) − A ( K ] , (101) λ = (cid:112) (cid:126) / ( m ω ) , (102)where λ has the dimension of a length.The ( q, p )-Hamilton operator H ( Q, P ) = 12 m P ( K ) + 12 m ω Q ( K ) (103)= (cid:126) ω ( N ( K ) + 12 ) ,N ( K ) = A † ( K ) A ( K ) (104)= ˜ K + ( ˜ K + b ) − ˜ K − has the usual “orthodox” spectrum E n = (cid:126) ω ( n + 1 /
2) !Thus, it turns out that the quantum mechanics associ-ated with the phase space S ϕ,I is rather more subtle thanthe one associated with S q,p and that those subtleties getlost if one passes from the ( ϕ, I )-case to the ( q, p )-case!As the creation and annihilation operators A † A withtheir ( b -independent) defining properties (97), (98) and(99) are essential building blocks for many quantum sys-tems, that loss of b -dependent subtleties may in turn leadto a corresponding loss of physical insights. The bigquestion is: Did nature implement those subtleties?
The time evolution of the composite operators (100)and (101) is the usual one. It follows from the relations(76) and (79): U ( − ˜ t ) ˜ Q U (˜ t ) = cos ˜ t ˜ Q + sin ˜ t ˜ P , (105) U ( − ˜ t ) ˜ P U (˜ t ) = − sin ˜ t ˜ Q + cos ˜ t ˜ P . (106)
D. The model H = (cid:126) ω ( ˜ K + g ˜ K )
1. Transition matrix elements with respect to the numberstates in 1st order
The quantum mechanical counterpart of the classicalHamiltonian (18) is the operator H ( K ) = (cid:126) ω ˜ C g ( K ) , ˜ C g ( K ) = ˜ K + g ˜ K , g ≥ . (107)Before discussing a special explicit choice for the op-erators ˜ K j , their associated Hilbert space and the exact eigenfunctions of the Hamiltonian (107) we mention thevalues of the (formal) 1st order matrix elements (cid:104) b, m | ˜ C g ( K ) | b, n (cid:105) , m, n = 0 , , , . . . (108)From the relations (55), (58) and (59) we get for m (cid:54) = n (the case m = n appears trivial, but that is only so in 1storder. It follows from the exact solution - discussed below- that the 2nd order g and higher ones contribute): (cid:104) b, m | ˜ C g ( K ) | b, n (cid:105) = g (cid:104) b, m | ( ˜ K + + ˜ K − ) | b, n (cid:105) (109)= g b + n )( n + 1)] / δ m ( n +1) + g b + n − n ] / δ m ( n − . Thus, we have the selection rule∆ n = ± , (110)for the Hamiltonian (107), the same as, e.g., for vibra-tional (electric dipole) transitions of diatomic molecules[39]!Examples: (cid:104) b, m = 0 | ˜ C g ( K ) | b, n = 1 (cid:105) = g √ b, (111) (cid:104) b, m = 2 | ˜ C g ( K ) | b.n = 1 (cid:105) = g (cid:112) b + 1) , (112) (cid:104) b, m = n − | ˜ C g ( K ) | b, n (cid:105) = g (cid:112) n (2 b + n − , (113) (cid:104) b, m = n + 1 | ˜ C g ( K ) | b, n (cid:105) = g (cid:112) ( n + 1)(2 b + n )(114)Eq. (111) shows that the associated transition probabilityfor 0 ↔ b .
2. Exact eigenvalues of ˜ C g ( K ) It follows from the explicit Hilbert space calculationsin Ch. III.E and in Appendix A that the operator ˜ C g ( K )has the exact eigenvalues˜ c g,b ; n = ( n + b ) (cid:112) − g , n = 0 , , . . . , (115)so that the Hamiltonian (107) has the the eigenvalues˜ E g,b ; n = (cid:126) ω g ( n + b ) , n = 0 , , . . . , ω g = (cid:112) − g ω, (116)which reflects the (”redshifted”) frequency reduction (32)of the classical motions in Ch. II. E. Explicit Hilbert spaces for ˜ K j , j = 0 , , and ˜ C g ( K ) = ˜ K + g ˜ K , spectra and eigenfunctions Several explicit Hilbert spaces for concrete irreducibleunitary representations of the group SO ↑ (1 , Sp (2 , R ) (or theisomorphic ones SU (1 ,
1) and SL (2 , R ) and of all othercovering groups as well have been discussed in the liter-ature [9, 11–14, 40].The associated self-adjoint Lie algebra generators ˜ K j all obey the same commutation relations (10). The rep-resentation spaces include Hardy spaces on the unit circle S , Hilbert spaces of holomorphic functions on the unitdisc D = { λ ∈ C , | λ | < } and also Hilbert spaces on thepositive real line R +0 = { x ∈ R , x ≥ } . We shall presentHardy space related Hilbert spaces for b ∈ (0 ,
1) here anddiscuss corresponding Hilbert spaces on R +0 in AppendixA: Hardy spaces on the unit circle are closely related tothe variable angle, whereas Hardy Hilbert spaces on R +0 are associated with the action variable I .The following discussion follows closely those of Secs.7.1 and 7.2 of Ref. [2]. Mathematical details like, e.g.questions concerning the convergence of series or inte-grals, will be ignored in the following! The associatedjustification can be found in the mathematical literaturequoted above.
1. Hardy space on the unit circle
A “Hardy space” H ( S , dϑ ) is a closed subspace ofthe usual Hilbert space L ( S , dϑ ) on the unit circle S with the scalar product( f , f ) = 12 π (cid:90) S dϑ f ∗ ( ϑ ) f ( ϑ ) , (117)and the orthonormal basis e n ( ϑ ) = e i n ϑ , n ∈ Z . (118)The associated Hardy subspace H ( S , dϑ ) is spannedby the basis consisting of the elements with non-negative n , namely e n ( ϑ ) = e i n ϑ , n = 0 , , , · · · . (119)If we have two Fourier series ∈ H ( S , dϑ ), f ( ϑ ) = ∞ (cid:88) n =0 c n, e i n ϑ , f ( ϑ ) = ∞ (cid:88) n =0 c n, e i n ϑ , (120)they have the scalar product( f , f ) + = 12 π (cid:90) S dϑ f ∗ ( ϑ ) f ( ϑ ) = ∞ (cid:88) n =0 c ∗ n, c n, (121)and obey the boundary condition f j ( ϑ + 2 π ) = f j ( ϑ ) , j = 1 , . (122)The coefficients c n,j are given by c n.j = ( e n , f j ) + . (123) Sp (2 , R ) Lie algebra generators are˜ K = 1 i ∂ ϑ + 12 , (124)˜ K + = e i ϑ ( 1 i ∂ ϑ + 1) , (125)˜ K − = e − i ϑ i ∂ ϑ . (126)Thus, the Hardy space with the basis (119) provides aHilbert space for the conventional HO with the spectrum { n + 1 / } and the operators (124)-(126) act on the basis(119) as ˜ K e n ( ϑ ) = ( n + 12 ) e n ( ϑ ) , (127)˜ K + e n ( ϑ ) = ( n + 1) e n +1 ( ϑ ) , (128)˜ K − e n ( ϑ ) = n e n − ( ϑ ) , (129)which are special cases of the relations (62), (58) and (59)with b = 1 /
2. Note that the ground state here is givenby e n =0 ( ϑ ) = 1.A possible generalization of the case b = 1 / b ∈ (0 ,
1) within the same Hilbert space as above canbe obtained [5, 41] by inspection of the relations (62),(58) and (59):˜ K = N + b, N = 1 i ∂ ϑ , (130)˜ K + = e iϑ [( N + 2 b )( N + 1)] / , (131)˜ K − = [( N + 2 b )( N + 1)] / e − iϑ . (132)Applied to the basis (119) these operators have the cor-rect properties.Defining the self-adjoint operator M b ( N ) = +[( N + 2 b )( N + 1)] / = M † b , (133)= +[( ˜ K o + b )( N + 1)] / , with M b ( N ) e inϑ = +[( n + 2 b )( n + 1)] / e inϑ , (134)the operators (131) and (132) can be written as˜ K + = ( e iϑ M b ) , ˜ K − = ( M b e − iϑ ) . (135)
2. Hardy space related Hilbert spaces for general b ∈ (0 , Another possible representation for the more generalcase b ∈ (0 ,
1) can be obtained by a generalization of thethe scalar product (121):Introducing on H ( S , dϑ ) the positive definite (self-adjoint) operator A b by the action A b e n ( ϑ ) = n !(2 b ) n e n ( ϑ ) , n = 0 , , . . . , b > , (136)( a ) n = a ( a + 1) ( a + 2) . . . ( a + n − a + n ) / Γ( a ) , ( a ) n =0 = 1 , f j ( ϑ ) = ∞ (cid:88) n =0 c n,j e n ( ϑ ) , j = 1 , , (137)by( f , f ) b, + ≡ ( f , A b f ) + = ∞ (cid:88) n =0 n !(2 b ) n c ∗ n, c n, , (138)so that ( e n , f ) b, + = n !(2 b ) n c n, . (139)We denote the (Hardy space associated) Hilbert spacewith the scalar product (138) by H b, + ( S , dϑ ).An orthonormal basis in this space is given byˆ e b,n ( ϑ ) = (cid:114) (2 b ) n n ! e n ( ϑ ) , (140)(ˆ e b,n , ˆ e b,n ) b, + = δ n n . Two series f j ( ϑ ) = ∞ (cid:88) n =0 a n,j ˆ e b,n ( ϑ ) , j = 1 , , (141)have the obvious scalar product( f , f ) b, + = ∞ (cid:88) n =0 a ∗ n, a n, , (142) a n, =(ˆ e b,n , f ) b, + (143)It follows that for a given function f ( ϑ ) its expansioncoefficients c n or a n with respect to e n or e b,n are relatedas follows c n = ( e n , f ) + = (cid:115) n !(2 b ) n (ˆ e b,n , f ) + = (cid:115) n !(2 b ) n a n (144)As in general ( f, f ) + (cid:54) = ( f, f ) b, + , (145)one has to be careful in the case of quantum mechanicalapplications:In a Hilbert space with scalar product ( f , f ) one gen-erally needs the normalization ( f, f ) = 1 for the usualprobability interpretations. If initially ( f, f ) (cid:54) = 1 one hasto renormalize the state f : f → f / (cid:112) ( f, f ). So, if, e.g.( f, f ) + = 1 in inequality (145), one has to renormalize f if one wants to determine transition probabilities andexpectation values etc. with respect to (138).In H b, + ( S , dϑ ) the generators ˜ K j have the form [37,40] ˜ K = 1 i ∂ ϑ + b, (146)˜ K + = e i ϑ ( 1 i ∂ ϑ + 2 b ) , (147)˜ K − = e − i ϑ i ∂ ϑ , (148) so that˜ K = 12 ( ˜ K + + ˜ K − ) = cos ϑ i ∂ ϑ + b e iϑ , (149)˜ K = 12 i ( ˜ K + − ˜ K − ) = sin ϑ i ∂ ϑ − ib e iϑ (150)= ˜ K ( ϑ − π/ . The operators (147) and (148) have the correct actions(58) and (59) on the basis (140):˜ K ˆ e b,n = ( n + b ) ˆ e b,n , (151)˜ K + ˆ e b,n = (cid:112) (2 b + n )( n + 1) ˆ e b,n +1 , (152)˜ K − ˆ e b,n = (cid:112) (2 b + n − n ˆ e b,n − . (153)The operators (147) and (148) are adjoint to each otheronly with respect to the scalar product (138), not withrespect to (121). Their adjointness with respect to thescalar product (138) can be verified by taking two series(141) and showing that( ˜ K − f , f ) b, + = ( f , ˜ K + f ) b, + . (154)This relation implies the self-adjointness of the operators(149) and (150). Note that( e n , ˆ e b,n ) + = (ˆ e b,n , e n ) + = (cid:115) (2 b ) n n ! δ n n , (155)( e n , ˆ e b,n ) b, + = (ˆ e b,n , e n ) b, + = (cid:115) n !(2 b ) n δ n n , (156)(ˆ e b,n , ˆ e b,n ) + = (2 b ) n n ! δ n n , (157)( e n , e n ) b, + = n !(2 b ) n δ n n . (158)The Fock space ladder operators A and A † associatedwith the Lie algebra generators (146)-(148) are given ac-cording to Eqs. (95).The so-called “reproducing kernel” on H b, + is given bythe “completeness” relationˆ A b ( ϑ − ϑ ) = ∞ (cid:88) n =0 ˆ e ∗ b,n ( ϑ ) ˆ e b,n ( ϑ ) (159)= [1 − e i ( ϑ − ϑ ) ] − b = ˆ A ∗ b ( ϑ − ϑ ) , (160)where the identity( a ) n n ! = ( − n (cid:18) − an (cid:19) (161)has been used. According to the relations (155) – (158)the kernel (159) has the properties12 π (cid:90) S dϑ ˆ A b ( ϑ − ϑ )ˆ e b,m ( ϑ ) = ˆ e b,m ( ϑ ) , (162)4or, written more formally in terms of the scalar product(138)( ˆ A b (1 , , ˆ e b,m (2)) b, + = ˆ e b,m ( ϑ ) , (163)( ˆ A b (1 , , ˆ e b,m (2)) + = (2 b ) m m ! ˆ e b,m ( ϑ ) , (164)( ˆ A b (1 , , e m (2)) b, + = (cid:115) m !(2 b ) m ˆ e b,m ( ϑ ) = e m ( ϑ ) , (165)( ˆ A b (1 , , e m (2)) + = (cid:114) (2 b ) m m ! ˆ e b,m ( ϑ ) (166)= (2 b ) m m ! e m ( ϑ ) . The numbers 1 and 2 mean the variables ϑ and ϑ , thelatter being an integration variable.The scalar product (138) itself may - according to Eq.(166) - be written as1(2 π ) (cid:90) S dϑ (cid:90) S dϑ f ∗ ( ϑ ) A b ( ϑ − ϑ ) f ( ϑ ) , (167)where the functions f j ( ϑ ) are as in Eq. (141). If a func-tion f ( ϑ ) = ∞ (cid:88) n =0 a n ˆ e b,n ( ϑ ) (168)is an element of H b, + then it follows from (163) that( ˆ A b (1 , , f (2)) b, + = f ( ϑ ) . (169)Thus, the “reproducing kernel” ˆ A b (2 ,
1) has - formally -similar properties as the usual δ -function.The property (169) has the following calculational ad-vantage: If one has two functions (141) considered aselements of H b, + , then their scalar product (138) can becalculated as( f , f ) b, + = 12 π (cid:90) S dϑ f ∗ ( ϑ ) f ( ϑ ) . (170) Space reflection and time reversal
According to Eq. (1) space reflections P can be imple-mented by the substitution P : ϑ → ϑ + π (171)and time reversal T by T : ϑ → − ϑ. (172)Quantum mechanically T is anti-unitary, i.e. accompa-nied by complex conjugation. Thus we get for the basis(140) and the operators (146), (149) and (150) P : e b,n ( ϑ ) → e b.n ( ϑ + π ) = ( − n e b,n ( ϑ ) , (173)˜ K ( ϑ ) → ˜ K ( ϑ + π ) = ˜ K ( ϑ ) , (174)˜ K ( ϑ ) → ˜ K ( ϑ + π ) = − ˜ K ( ϑ ) , (175)˜ K ( ϑ ) → ˜ K ( ϑ + π ) = − ˜ K ( ϑ ) , (176) and T : e b,n ( ϑ ) → [ e b.n ( − ϑ )] ∗ = e b,n ( ϑ ) , (177)˜ K ( ϑ ) → [ ˜ K ( − ϑ )] ∗ = ˜ K ( ϑ ) , (178)˜ K ( ϑ ) → [ ˜ K ( − ϑ )] ∗ = ˜ K ( ϑ ) , (179)˜ K ( ϑ ) → [ ˜ K ( − ϑ )] ∗ = − ˜ K ( ϑ ) , (180)
3. A unitary transformation by a change of basis
In the above discussion the b -dependence of the repre-sentation on H b, + is contained in the Lie operators (146)-(148) and in the metrical operator A b of Eq. (136), butnot in the basis e n ( ϑ ) of H we started from. Thus,all non-equivalent irreducible representations for differ-ent b are implemented by starting from the Hardy space H with the b -independent basis (119). By the unitarytransformation e n ( ϑ ) = e i n ϑ → e b,n ( ϑ ) = e i ( n + b ) ϑ , (181)one can pass to b -dependent Hilbert spaces ˆ H ( S , dϑ ; b )for functions with the boundary condition e b,n ( ϑ + 2 π ) = e i b π e b,n ( ϑ ) . (182)Now each irreducible unitary representation character-ized by the number b has its own Hilbert space, eachwith the scalar product (138) and with the basisˆ e b,n ( ϑ ) = (cid:114) (2 b ) n n ! e b,n ( ϑ ) . (183)The “reproducing kernel” here is A ( b ) ( ϑ − ϑ ) = ∞ (cid:88) n =0 e ∗ b,n ( ϑ ) e b,n ( ϑ ) (184)= e ib ( ϑ − ϑ [1 − e i ( ϑ − ϑ ) ] − b = e ib ( ϑ − ϑ ) ˆ A b ( ϑ − ϑ ) , The generators (146)-(148) now take the form˜ K = 1 i ∂ ϑ , (185)˜ K + = e i ϑ ( 1 i ∂ ϑ + b ) , (186)˜ K − = e − i ϑ ( 1 i ∂ ϑ − b ) (187)˜ K = cos ϑ i ∂ ϑ + ib sin ϑ, (188)˜ K = sin ϑ i ∂ ϑ − ib cos ϑ (189)= ˜ K ( ϑ − π/ K here, too, is obtained from ˜ K by replacing ϑ with ϑ − π/ P ( ϑ → ϑ + π ) and T ( ϑ →− ϑ ) applied to the basis (183) and the operators (185),(188) and (189) compared to the properties (173)-(180)there is only a change for the basis (183) for P : P : ˆ e b,n ( ϑ ) → ˆ e b,n ( ϑ + π ) = ( − n e iπb ˆ e b,n ( ϑ ) . (190)The global constant phase factor exp( iπb ) can be inter-preted as representing a new type of “fractional” statis-tics in 2 dimensions [42], of particles called “anyons” (seereferences below).
4. Aharonov-Bohm-, (fractional) quantum Hall-effects,anyons, Berry’s phase,Bloch waves etc.
The property of the “naive” planar rotation operator(185) to have a 1-parametric set of possible spectra -parametrized by the number b - is a mathematical conse-quence of the fact that the operator has a 1-parametricset of self-adjoint extensions [43].For physical systems topologically related to a punc-tured plane, the parameter b can have different physicalmeanings [44]:In the description of Aharonov-Bohm effect [44–46] (historically more appropriate: “Ehrenberg-Siday-Aharonov-Bohm effect” [47]) the index b is proportionalto the magnetic flux Φ crossing the plane.The magnetic flux model can also help to understandthe quantum Hall effect [48–50]. It can also do so for thefractional quantum Hall effect [50–52], especially in theframework of anyons [53–55] and related Chern-Simonstheories [56, 57]. As special Chern-Simons theories havethe structure group SO ↑ (1 ,
2) they may help to find theappropriate theoretical framework for the dynamics as-sociated with with the symplectic spectra Eq. (62).Closely related are the properties of Berry’s phase [49,58–60].In the case of Bloch waves b represents the momentuminside the first Brillouin zone [61].
5. The operator ˜ C g ( K ) = ˜ K + g ˜ K on H b, + According to Eq. (149) the operator ˜ C g ( K ) = ˜ K + g ˜ K here has the form˜ C g ( K ) = (1 + g cos ϑ ) 1 i ∂ ϑ + b (1 + ge iϑ ) (191)The eigenvalue differential equation˜ C g ( K ) f g,b ( ϑ ) = ˜ c g,b f g,b ( ϑ ) (192) has the general solution [62], with 0 ≤ g < f g,b ( ϑ ) = C (1 + g cos ϑ ) − b e i ˜ c g,b (1 − g ) − / χ ( ϑ ) − ibϑ , (193) χ ( ϑ ) = 2 arctan (cid:20)(cid:114) − g g tan( ϑ/ (cid:21) , (194) C = const . The boundary condition f g,b ( ϑ + 2 π ) = f g,b ( ϑ ) (195)implies˜ c g,b = ( n + b ) (cid:112) − g ≡ ˜ c g,b ; n , n = 0 , , . . . , (196)The implementation of the boundary condition (195) in-cludes the transformation χ → χ + 2 π .The result (196) coincides with Eq. (A24) in Ap-pendix A and corresponds to the classical result (32).Thus, we have for ˜ C g ( K ) the eigenfunctions f g,b ; n ( ϑ ) = C (1 + g cos ϑ ) − b e i ( n + b ) χ ( ϑ ) − ibϑ , (197) n = 0 , , . . . , ˜ C g ( K ) f g,b ; n ( ϑ ) =( n + b ) (cid:112) − g f g,b ; n ( ϑ ) . (198)The constant C in the solution (193) can be determinedfrom the normalization condition ( f g,b , f g,b ) = 1. Usingthe relation (170) leads to the integral [63]1 = | C g,b | (cid:90) π dϑ π (1 + g cos ϑ ) − b (199)= | C g,b | (1 − g ) − b P b − [(1 − g ) − / ] , from which the normalization constant C g,b = | C g,b | can be determined. It is independent of n . Here P ν ( x ) , ν ∈ R , x ≥
0, is the Legendre function of thefirst kind [64, 65]. It has - among others - the properties P ν ( x ) = P − ν − ( x ) , P ν (1) = 1 , P ν =0 ( x ) = 1. F. Nonlinear interactions in termsof the operators ˜ K j The HO model plays an important role in molecularphysics (see the next chapter): For example, the nucleiof diatomic molecules can oscillate relative to each otheralong their connecting axis. As long as the associatedenergy levels are small compared to the dissociation en-ergy V one can approximate those vibrations by a one-dimensional HO the potential of which is centered at theequilibrium point q = r >
1. The Morse potential for molecular vibrations
In order to take dissociation into account Morse sug-gested the potential [66] V Mo ( q ) = V ( e − aq − , V , a > , (200)where q is the distance of the atomic nuclei from theirpoint of equlibrium q . For aq (cid:28) V Mo ( q ) ≈ mω q , ω = a (cid:112) V /m, (201)where m is the reduced mass of the two nuclei.In addition V Mo ( q → ∞ ) = V , V Mo ( q → −∞ ) = + ∞ . (202)For aq (cid:28) − q = r ≥ E < V the classical motions are bounded and pe-riodic. The system is also integrable, i.e. there existcanonical angle and (constant) action variables in orderto describe the sytem [66]. The relationship between con-stant energy E < V and action variable I turns out tobe (see Eq. (21)) ω I = 2 V (1 − (cid:112) − E/V , E = ω I (cid:18) − ω I V (cid:19) . (203)This gives the Hamilton function H Mo ( ϕ, I ) = ω I (cid:18) − ω I V (cid:19) , (204)with the associated Eqs. of motion˙ ϕ = ∂ I H = ω (1 − ω I V ) ≡ ω I , (205)˙ I = − ∂ ϕ H = 0 , ⇒ ω I = const. (206)Note that here E ( I ) = 12 ( ω + ω I ) I. (207)Replacing the action variable I in the Hamilton function(204) by the operator (cid:126) ˜ K leads to the spectrum E b,n = (cid:126) ω ( n + b ) (cid:20) − (cid:126) ω V o ( n + b ) (cid:21) , (208) E b,n =0 = (cid:126) ω b (cid:18) − (cid:126) ω V o b (cid:19) . (209)For the bracket [. . . ] in Eq. (208) to be positive onlythose n are allowed which imply this property.This system is a simple but instructive example howthe use of the canonical pair angle and action variablesinstead of the canonical position and momentum can sim-plify the description of the dynamics of the system, atthe expense of making it intuitively less accessible! Thatmight be especially so if the system is not completely in-tegrable and the action variable a function of time, too,as in the model of Ch. II above.
2. Potentials involving the terms ˜ K + ˜ K − and ˜ K − ˜ K + Due to the Casimir operator relations (63) the eigen-value equations of the (dimensionless) Hamiltonians (upto a factor (cid:126) ω ) ˜ F − = ˜ K + g − ˜ K + ˜ K − (210)and ˜ F + = ˜ K + g + ˜ K − ˜ K + (211)can be solved immediately: Eigenvectors are still thoseof ˜ K (see Eq. (62)) and the eigenvalues are˜ f g − ; b,n = n + b + g − ( n + 2 b − n ; ˜ f g − ; b,n =0 = b, (212)and˜ f g + ; b,n = n + b + g + ( n +2 b )( n +1) ; ˜ f g + ; b,n =0 = b (1+2 g + ) . (213)The models describe the annihilation and creation ofquanta (Eq. (210)) and vice versa (Eq. (211)). The cou-plings g − and g + may depend on external parameters. IV. REFLECTIONS ON POSSIBLEEXPERIMENTSAND OBSERVATIONS
Replacing the ingrained and very successful habit ofdescribing the quantum HO by the “canonical” pair po-sition and momentum operators (or the associated cre-ation and annihilation operators) by the quantum versionof its classical - locally - equivalent canonical pair angleand action variables may appear unnecessarily artificialand even unnatural:Compared to position and momentum variables thepair angle and action variables is less familiar as far asvisualization and perception are concerned:Whereas the angle can be illustrated well as a fractionof the unit circle and its s-fold coverings by the corre-sponding number of rotations of the hand of a clock,a visualization of the action variable is not so obvious.True, all quantum mechanical action variables must - inprinciple - be proportional to Planck’s constant (cid:126) and forintegrable systems it appears to be closely related to theconserved quantity “energy”. But we have seen in Ch.II that the action variable I may be quite useful as a co-ordinate even if it is not a constant of motion. For suchtime–dependent I ( t ) the relation (23) may be a helpfultool for an intuitive interpretation.An important lesson from Ch. II for the discussionsbelow is that the energy E may be conserved even if theaction variable I ( t ) is not!Perhaps we have to go beyond the use of position andmomentum as the basic observables in a part of the quan-tum world where other “canonical” observables are moreappropriate! This is, of course, a larger challenge for7a reformulation of (perturbative) quantum field theoriesetc., for which the orthodox description of the HO is afundamental building block!In view of the qualitative differences between the globalphase spaces (7) and (8) and their possible physical impli-cations - especially for the associated quantum theory - itis obviously important to make experimental and obser-vational attempts to look for corresponding phenomenain nature!All the following considerations apply, of course, only, if the mathematical models from the previous chaptershave counterparts in nature! For this reason all possibleapplications discussed in the following are hypothetical!The good news is that the relevance of the model canbe tested in the laboratory and by astrophysical observa-tions! The (hopefully preliminary) bad news is that theassociated theoretical framework for the dynamics gov-erning transition rates etc. involving the new spectra stillhas to be worked out! A. Generalities
In view of their possible far-reaching implications theabove theoretical results should, of course, be subject tocritical reviews and be probed experimentally! In thefollowing - as a kind of “tour d’horizon” - ideas and sug-gestions for such experiments and observations are dis-cussed, in the hope that a few experimentalists will bemotivated and inclined to meet the challenge and thatexperts - experimentalists and theoreticians - in the ar-eas of physics mentioned below, will point out possiblemisunderstandings and will suggest improvements andconsequences!Harmonically oscillating quantum systems can befound in many areas of physics, at least approximatelyclose to the corresponding (local) minima of classical“binding” potentials with periodic motions.It is important to note that the “symplectic” or “frac-tional” spectrum (9) is tied to the groups U (1) or O (2)and their infinitely many coverung groups, but not tothe rotation group SO (3) and its single 2-fold covering SU (2). Accordingly one has to look for 2-dimensional(sub)systems with “effective” phase spaces (8). Such sys-tems may be found in molecular spectroscopy (e.g. di-atomic molecules), quantum optics, optomechanics and- possibly - in astrophysics (“dark” energy and “dark”matter, see below).One obvious question is: Why haven’t we seen thosesymplectic b -dependent spectra yet? Several answers arepossible:0. They just don’t exist in nature!1. One possible reason is that no one has lookedfor them. This is quite plausible if the “visibility” ofthose symplectic spectra is very weak, as, e.g. for in-frared emission or absorption lines of homonuclear di-atomic molecules like H , because they have no electricdipole moment or because their Stokes or Anti-Stokes lines in inelastic Raman scattering off vibrating and ro-tating molecules (see below) are very weak.2. As discussed in Section III.C the impact of the (com-posite) ”orthodox” Fock space annihilation and creationoperators (95) and (96) with the usual properties (97),(98) and (99) may dominate and obscure the symplecticspectra (69), except for the value b = 1 /
2. Thus, one hasto find means in order to discover other (fractional) partsof the spectra (69), if they exist at all! In any case, theirobservability appears to be rather weak.3. Transitions - radiative, non-radiative, collisional,Raman-type etc. - between different levels of the spectra(69) require appropiate kinds of electromagnetic interac-tions, the dynamics of which has not yet been workedout!Consider two generally different levels of the spectra(69): E b j ,n bj = (cid:126) ω j ( n b j + b j ) , j = 1 , n b j = 0 , , . . . . (214)For a fixed b = b = b ; and ω = ω = ω the observ-able energy difference between an upper level character-ized by n (cid:48) b and a lower level characterized by n (cid:48)(cid:48) b < n (cid:48) b . E b,n (cid:48) b − E b,n (cid:48)(cid:48) b = (cid:126) ω ( n (cid:48) b − n (cid:48)(cid:48) b ) (215)cannot be distinguished from the corresponding differ-ence for a b (cid:54) = b .More interesting is a transition with a change of theB-index ( b ↔ b ): E b ,n (cid:48) b − E b ,n (cid:48)(cid:48) b = (cid:126) ω [( n (cid:48) b − n (cid:48)(cid:48) b ) + ( b − b )] . (216)If such transitions are possible, e.g. for n (cid:48)(cid:48) b = n (cid:48) b = 0and as - up to now - the only condition on b (and b ) isthe inequality b > b → b → · · · b m → b > , (217)accompanied by the emission of m − ω ) quanta. Even a continuum between b and b appears possible. All this depends on the stillto be established associated dynamics, which determinesrates and selection rules! If the initial quanta cascadedown the “fluorescence” sequence (217) they can end upin the microwave or even radiowave region, without lossof the total energy!If ω (cid:54) = ω (occurs for diatomic molecules with differ-ent isotopic atoms and for electronic transitions betweenlocal minima of different Born-Oppenheimer potentialsfor the nuclei; see below), one has( E b ,n (cid:48) b − E b ,n (cid:48)(cid:48) b ) / (cid:126) = ω [ n (cid:48) b − σ n (cid:48)(cid:48) b + b − σb ] , σ = ω ω . (218)4. As discussed in Section III.B above, for a given ω one needs the time t s , ω t s = 2 πs , s ∈ Z , in order to“run” through an s -fold covering S s ] of the circle S . Theprefactor exp( − ib ωt ) of the general state (92) shows that8 orthodoxHO spectrum n / n / = 11 n / = 0 ∼∼∼ ∼∼∼ symplecticHO spectra n b b b b + 1 b + 1 FIG. 3. Comparison of the orthodox and the symplecticspectra of the HO, with indications of possible transitionsbetween levels (ignoring any selection rules) here the time “angle” ˜ t = ωt can be reduced by a small b (cid:28) / s of coverings is obviouslyvery large for a finite time interval ∆ t (cid:29) T = 1 / (2 πω ),where ω ≈ Hz in the near infrared.5. In Section III.D above we discussed the n → n ± g to dependexplicitly on time or via other external parameters orfields [68].6. For experimental tests it is essential to find quanti-ties (“observables”) which are especially sensitive to val-ues of the B(argmann)-index b . The following is an -incomplete - list of possible theoretically interesting ex-periments (without proper knowledge of their feasibilityin the laboratory or of their observability in the sky)! B. Vibrating diatomic molecules
Among the most important and interesting oscillatorsthe above results may apply to are vibrating diatomicmolecules (for introductions to their physics see, e.g. thetextbooks [69–74]). They have one vibrational degree offreedom: oscillations about the point of equilibrium alongthe line connecting the two nuclei (“internuclear axis” =INA). Near that equilibrium point the potential may beconsidered to be harmonic. In the Born-Oppenheimer(BO) approximation the effective potentials for the vi-brating nuclei are provided by energy configurations ofthe electron “cloud” the dynamics of which depends only “adiabatically” on the state of the nuclei, especially ontheir distance R (see Fig.4).The (classical) angular frequency ω = 2 πν for the mu-tual harmonic vibrations of the nuclei is given by ω = 2 π ν = (cid:112) k/µ, (219)where k is the “force constant”, determined - in the BOapproximation - solely by the actual electronic configu-ration and µ = m m / ( m + m ) is the “reduced” massof the two vibrating atoms.Spectroscopists denote the vibrational level numbers n of the HO by v and give the frequencies ν [s − ] in termsof the “wave number” ˜ ν = ν/c [cm − ]. One then has the(approximate) equivalences1 eV ∼ = 8066 cm − ∼ = 11605 ◦ K . (220)Spectroscopically the differences between homonuclear(equal nuclei like molecular hydrogen H or oxygen O )and heteronuclear (different nuclei like carbon monoxyde C O) diatomic molecules are important: because ofspace reflection symmetry the homonuclear molecules donot have a permanent electric dipole moment and, there-fore, no corresponding infrared emissions or absorptions[75]. If, however, their polarizability is nonvanishing,they can have induced electric dipole moments, e.g. incase of elastic and inelastic Raman-type scattering or bycollisions.In addition to the vibrational energy levels character-ized by the numbers v = 0 , , . . . the diatomic moleculeshave rotational levels J = 0 . . , . . . due to the rotationsof the molecule around an axis which lies in a plane per-pendicular to the INA and passing through the centreof mass on that axis. So in general one has the com-bined vibration - rotation (“rovibrational”) transitions( v (cid:48) , J (cid:48) ) → ( v (cid:48)(cid:48) , J (cid:48)(cid:48) ). The frequencies of the vibrationaltransitions are generally in the “near-infrared” (frequen-cies around ν ≈ s − ) and those of the rotational onesare at least one order of magnitude smaller and are in the“far-infrared” or microwave region.Example: Molecular hydrogen H Here are some essential properties of the molecule H which are importent for our present discussion: As ahomonuclear diatomic molecule H has no permanentelectric dipole moment (this property is frequently men-tioned in the literature, but very rarely proven; for aproof see Ref. [75]). Because of this missing electricdipole moment their is no corresponding infrared emis-sion or absorption.There is, however, (weak) magnetic dipole and electricquadrupole infrared radiation [77].Due to that missing electric dipole moment there areno direct vibrational transitions v ↔ v ± Σ + g (see Fig. 4).As a consequence, in order to experimentally analysethe ladder of vibrational states of, e.g. the BO electronicground state potential X Σ + g , an “ultraviolet detour” has9 internuclear distance n = 1: 0 .
52 eVX Σ + g D = 4 .
48 eVE Σ + g F Σ + g n = 0: 12 . n = 0: 11 . Σ + u C Π u ˚AeV FIG. 4. Some of the lowest ”binding” electronic BO po-tentials for the vibrations of the two H nuclei as a functionof their distance R . The zero energy on the ordinate coin-cides with the zero-point energy of the electronic potential X Σ + g . The numerical values are taken from Ref. [76]. For theinterpretation of the terms denoting the different curves seeAppendix B. to be taken: one first initiates an ultraviolet allowed (1-or 2- γ ) absorptive transition from the electronic groundstate to a vibrational level of a higher BO electronic po-tential (Fig. 4), from where the photons cascade down(in 1 or more steps) to a vibrational level of X Σ + g which is different of the one the photons originally startedfrom. The difference of the observed ultraviolet frequen-cies then provides information about the vibrational lev-els of the selected BO potential [76, 78–88].The “ultraviolet detour” also plays an essential role inthe so-called “Solomon process” which leads to photodis-sociation of H [89–92].Another possibility to observe vibrational and rovibra-tional levels of H electronic BO potentials is providedby the polarizability of the molecule, which allows forRaman-type transitions associated with induced electricdipole moments, induced by by external light beams orbe collisions[93–97].The two nuclei (protons) oscillating in the binding elec-tronic BO potentials may have antiparallel spins (para-H ) or parallel ones (ortho-H ). For recent summeriesand reviews of the role of H in different areas of physicssee, e.g. [76, 89, 98–101]. More references will be quoted in the course of the discussions below. (Numerical val-ues of quantities mentined below are rounded up/downfrom their impressively determined accurate theoreticaland experimental values).For the nuclear vibrations of the diatomic homonuclearmolecules H in the electronic ground state X Σ + g BOpotential (Fig. 4) one has for the “transition” (”groundtone”) [86, 102](∆ v, ∆ J ) : ( v (cid:48)(cid:48) = 0 , J (cid:48)(cid:48) = 0) ↔ ( v (cid:48) = 1 , J (cid:48) = 0) (221) ≈ ( ± [˜ ν ↔ ≈ − ] , Σ + g isan effective potential for the vibrations of the two nuclei,depending on their distance R (Fig. 4).The vibrational transition value (221) correponds toabout 0 .
516 eV, a wave length λ = 1 / ˜ ν ≈ , µ m and atemperature of ≈ ◦ K.In comparison the rotational transition (0 , ↔ (0 , ν ≈
118 cm − ∼ = 0 . λ ≈ µ m.If the vibrating H molecule were an ideal HO, its “or-thodox” zero-point energy, according to Eq. (221), wouldbe E ( X Σ + g ) = 12 ˜ ν ↔ ( X Σ + g ) (222) ≈ − ∼ = 0 .
258 eV ∼ = 2994 ◦ K . The vibrating molecule H is, of course, no ideal HObecause it dissociates at a finite energy D > E . TheMorse potential (200) takes this qualitatively into ac-count, as can be seen from the relations (202). The“anharmonic” modifications of energy (204) and angu-lar frequency (205) are small as long as ω I (cid:28) V .The quantum mechanical energy (208) can be writtenas E b,n = (cid:126) ω ( n + b ) − ( (cid:126) ω ) V o ( n + b ) , (223)which may be considered as a polynomial in ( n + b ). Asthe Morse potential still is only a rough approximation,one has taken - for the orthodox value b = 1 / E v,J = (cid:88) i ≥ ,j ≥ Y i,j ( v + 1 / i J j ( J + 1) j , (224)where the coefficients Y i,j are determined (mainly) ex-perimentally. The ground state (“zero-point”) energy isgiven by E , = (cid:88) i ≥ Y i, (1 / i . (225)For the Morse potential one has Y , = (cid:126) ω , Y , = − (cid:126) ω / (4 V ), all other Y i, vanishing.0The approximation ansatz (224) gives for H insteadof (222) the value [104] E [H ) ≈ − . (226)Thus, by passing from the orthodox HO spectrum ( b =1 / ), usually associated with H infrared vibrations, tothe symplectic one [ b ∈ (0 , ] one can lower the zero-point energy of the BO potential X Σ + g maximally bythe (approximate) amount E ( H ; b → − E ( H ; b = 1 / ≡ ˆ E ( H ) (227) ≈ − ∼ = 0 .
26 eV ∼ = 3000 ◦ K . Similarly, the known dissociation energy of H [98, 101,105, 106] D ≈ − ∼ = 4 .
48 eV (228)- theoretically - increases maximally by the the amount(227).Thus, the orthodox H vibrational spectrum (b = 1/2)can be considerably “detuned” for 1 / > b > moleculeimplies an additional effecive Boltzmann factor e − ˆ E ( H ) / ( k B T ) = e − ◦ /T . (229)Preliminarily ignoring all dynamical mechanisms the lastEq. says that for T < ◦ K the symplectic ground statebecomes preferred statistically . This will play a role inour astrophysical discussion below. It also indicates thatthe symplectic HO spectra may be observed better atvery low temperatures.As mentioned above, in the BO approximation theelectronic ground state X Σ + g (which includes the actionof the nuclear Coulomb potentials on the electrons) pro-vides a potential for the two vibrating nuclei as a functionof their distance R . The potential has a minimum aroundwhich the oscillations are approximately harmonic. Thesame applies to the next higher electronic (metastable)states B Σ + u , E F Σ + g and C Π u . They have localminima in appropriate neighbourhoods of which the vi-brations are harmonic, too (see Fig. 4).In the following list one can find the measured en-ergy differences between the ground states of the differ-ent electronic levels relative to X Σ + g ( v = 0 , J = 0)and the energies E ↔ of the first vibrational excitations( v = 0 , J = 0) → ( v = 1 , J = 0) above those groundstates [76, 78, 80–83, 85, 87, 107]. The data here arefrom Ref. [83]: ≥ X Σ + g [eV] E ↔ [eV] (230)X Σ + g . Σ + u .
19 0 . Σ + g .
30 0 . Σ + g .
32 0 . Π u .
30 0 .
29 The second E ↔ -column shows that the first vibrationalexcitations are generally quite different for the differentelectronic levels, reflecting the curvature differences atthe minima of the potential curves. If the five BO po-tentials are approximately harmonic near their minima,the above numerical values of E ↔ are twice the valuesof their zero-point energies.Note that the transitions from (to) the listed higher electronic levels to (from) the electronic ground state X Σ + g ( v = 0 , J = 0) are in the vacuum UV ( ≥ α tran-sition of atomic hydrogen (10.20 eV). This is importantfor a gas mixture of H and H : The relative energy dif-ferences (230) are all larger than the Lyman α transitionand they become even larger for the symplectic spectra.This is important for astrophysical applications (see be-low).As mentioned above, in order to determine the vibra-tional transition energies E ↔ in the list (230) experi-mentally one has to take a “detour” of determining re-lated (electronic) UV transitions first and then subtractthe corresponding energies [86, 88, 98, 101, 105].The set of UV transitions from the states of the po-tential X Σ + g to those of B Σ + u or vice versa is calledthe “Lyman-band”, and those of X Σ + g to C Π u the“Werner-band” [78, 80–83]. C. Vibrations of diatomic moleculeswith different isotopic atoms
Such systems played an important but nowadaysmostly forgotten role in the early history of quantummechanics:Even before Heisenberg derived the now well-established spectrum of the HO in his famous paper fromJuly 1925 [108], Mullikan had concluded from his in-vestigations of diatomic molecules that their vibrationalspectra should be described by half-integers, not inte-gers as the Bohr-Sommerfeld quantization prescriptionhad suggested [30, 34]. Mullikan compared the vibra-tional spectra of diatomic molecules in which one atomwas replaced by an isotope (B O and B O; AgCl andAgCl ) [109].Classically the vibrating atoms have angular fre-quences ω i = (cid:112) k/µ i , where the µ i , i = 1 , µ for one and µ forthe other molecule containing one or two isotopic atoms.The (electronic) oscillator strength k is assumed to bethe same in both cases (BO approximation).Let E ( i ) = γ (cid:126) ω i be the two slightly different oscilla-tor ground state energy levels for the two “isotopic” oscil-lators. Let further E a and E b be two known electronic en-ergy levels (they may be equal) from which transitions tothe ground states with energies E ( i ) are possible. Thenthe difference ω a, − ω b, = ( E a − E b ) / (cid:126) − γ ( ω − ω ) (231)1of the frequencies ω a, = [ E a − E (1)] / (cid:126) , ω b, = [ E b − E (2)] / (cid:126) (232)can be used in order to determine γ . Mullikan concludedthat γ ≈ /
2. A good review of the method can be foundin Ref. [110].Due to the tremendous experimental and technolog-ical advances since those experiments from almost 100years ago it appears possible to perform similar more re-fined experiments [86] in order to find fractional valuesof the B-index b other than 1 /
2. However, one first hasto account for the deficits of the BO approximation andfor the corrections due to rotational, electronic and QEDeffects [106, 111]!
D. Interferences of time dependent energyeigenstates
Applying the unitary time evolution operator (75) tothe energy eigenstates | b, n (cid:105) yields ( (cid:126) = 1 in the follow-ing) U (˜ t ) | n, b (cid:105) = e − i ( n + b )˜ t | n, b (cid:105) , ˜ t = ω t. (233)Reccall that ˜ t is a (dimensionless) angle variable. Let˜ t increase by an amount δ ˜ t which may be implementedby either a change of ω or of t or of both. Consider thesuperposition | n, b ; ˜ t, δ ˜ t (cid:105) = (1 + e − i ( n + b ) δ ˜ t ) e − i ( n + b )˜ t | n, b (cid:105) . (234)Then the oscillations of the “intensity” |(cid:104) n, b ; ˜ t, δ ˜ t | n, b ; ˜ t, δ ˜ t (cid:105)| = 4 cos [( n + b ) δ ˜ t/ . (235)are sensitive to the value of bδ ˜ t , especially for n = 0. Foran analogous approach in a recent experiment see Ref.[112].An alternative to generate such interferences by achange δ ˜ t one may also use - at least theoretically - achange δb . The question, how to generate states like | n, b (cid:105) experimentally has, unfortunately, to be left openhere. E. Transitions associated with the Hamiltonian H ( K ) = (cid:126) ω ˜ C g ( K ) In case the model Hamiltonian (107) with its ”effec-tive” electric dipole moment can somehow be imple-mented experimentally, either by heteronuclear moleculeslike. e.g. LiH (it has the rather large electric dipolemoment 5.9 D[ebeye]) or by Raman-type induced elec-tric dipole moments of homonuclear diatomic molecules,then especially the transitions (111) depend sensitivelyon the value of b : The probability for the transition n = 0 ↔ n = 1 is given by |(cid:104) b, m = 0 | ˜ C g ( K ) | b, n = 1 (cid:105)| = b g / . (236) So, if the index b is very small > { n + b } of ˜ K is rescaled for ˜ C g ( K ) by an overall ”redshifting”factor (cid:112) − g (see Eq. (197)). F. Traps for neutral molecules and optomechanics
A speculatively optimal experimental situation wouldbe a diatomic neutral molecule in a cooled down trapwhich allows the vibrational emission and absorptionproperties of the molecule to be observed, especiallythose of its different electronic potential ground states.As already stressed above, the conditions are differentfor heteronuclear and homonuclear molecules, the formerhaving an electric dipole moment, the latter not, whichrequires some Raman-type excitations. In view of thevery impressive developments of experimental possibili-ties involving such traps [113–115], it appears possible toachieve at least a few of the required aims. Closely re-lated are optical devices coupled to mechanical oscillators[112, 116–118]
G. Perelomov coherent states
Among the three different types of coherent states[119] associated with the Lie algebra (52), the so-called“Perelomov” coherent states appear to be the mostpromising ones in order to detect traces of HO spec-tra with b (cid:54) = 1 /
2: Their matrix elements contain theBargmann index b quite explicitly and they can be gen-erated experimentally [119].The states | b, λ (cid:105) can either be defined as eigenstates ofa composite “annihilation” operator, E b, − | b, λ (cid:105) = λ | b, λ (cid:105) , E b, − = ( ˜ K + b ) − ˜ K − , (237) λ = | λ | e − iθ ∈ D = { λ ∈ C , | λ | < } , or by generating them from the ground state | b, (cid:105) bymeans of the unitary operator U ( λ ) P = e ( w/
2) ˜ K + − ( w ∗ /
2) ˜ K − = e λ ˜ K + e ln(1 −| λ | ) ˜ K e − λ ∗ ˜ K − , (238) w = | w | e − iθ ∈ C , λ = tanh( | w | / e − iθ , | w | = ln (cid:18) | λ | − | λ | (cid:19) , so that | b, λ (cid:105) = U ( λ ) P | b, (cid:105) . (239)2In terms of number states we have the expansion | b, λ (cid:105) = (1 − | λ | ) b ∞ (cid:88) (cid:18) (2 b ) n n ! (cid:19) / λ n | b, n (cid:105) , (240)(2 b ) n = 2 b (2 b + 1) . . . (2 b + n −
1) = Γ(2 b + n ) / Γ(2 b ) , (2 b ) n =0 = 1 . Note that the coefficient of λ n in this expansion is thesame as that of e n ( ϑ ) in Eq. (140).Important expectation values with respect to | b, λ (cid:105) are (cid:104) b, λ | ˜ K | b, λ (cid:105) ≡ ˜ K b,λ = b cosh | w | , (241) (cid:104) b, λ | ˜ K | b, λ (cid:105) ≡ ˜ K b,λ = b sinh | w | cos θ, , (242) (cid:104) b, λ | ˜ K | b, λ (cid:105) ≡ ˜ K b,λ = − b sinh | w | sin θ, (243) (cid:104) b, λ | N | b, λ (cid:105) ≡ N b,λ = b (cosh | w | − , (244)(∆ N ) b,λ ≡ N b,λ − N b,λ = 12 b sinh | w | (245)It follows that most quantities can be expressed in termsof the “observables” N b,λ and (∆ N ) b,λ : As b sinh | w | = √ b (∆ N ) b,λ (246)and cosh | w | = (sinh | w | + 1) / we have, e.g.,˜ K b,λ =[2 b (∆) b,λ + b ] / , (247) N b,λ [1 + N b,λ / (2 b )] =(∆ N ) b,λ . (248)It follows from the last equation that Paul’s parameterR [120] here has the value R b,λ ≡ (∆ N ) b,λ − N b,λ N b,λ = 12 b . (249) R is a measure for deviations from a Poisson distributionfor which (∆ n ) − ¯ n = 0.In addition we have | λ | = tanh( | w | /
2) = cosh | w | − | w | = N b,λ √ b (∆ N ) b,λ (250) | λ | = cosh | w | − | w | + 1 = N b,λ N b,λ + 2 b (251)and | w | = ln (cid:18) | λ | − | λ | (cid:19) = ln (cid:20) b/N b,λ ) / (1 + 2 b/N b,λ ) / − (cid:21) . (252)For N b,λ / b (cid:29) | w | ≈ ln(2 N b,λ /b ) . (253)The transition probability p b ( n ↔ λ ) = |(cid:104) b, n | b, λ (cid:105)| = (1 − | λ | ) b (2 b ) n n ! | λ | n (254) can be expressed as (see Eq. (251)) p b ( n ↔ λ ) = (2 b ) n (2 b ) b N nb,λ n ! ( N b,λ + 2 b ) n +2 b , (255) p b (0 ↔ λ ) = (2 b ) b ( N b,λ + 2 b ) b . (256)Comparing the classical quantities (14) with the relations(73) and (241) - (243) one infers the classical limit I (cid:16) lim ( (cid:126) b ) → , | w |→∞ [( (cid:126) b ) cosh | w | (cid:16) ( (cid:126) b ) sinh | w | ] (257) H. (Dispersive) van der Waals forces
F. London was the first one to associate the attractivevan der Waals forces between neutral atoms or moleculeswith the nonvanishing zero-point energy of the HO [121–124]. For atoms or molecules of the same type and with-out retardation he derived - using a HO model - the po-tential (see also Refs. [125, 126]) V ( R ) = − γ ( 12 (cid:126) ω ) α R , (258)where γ > α the (static)polarizability of the two atoms or molecules [93, 127], R ≤ λ = (2 πc ) /ω the distance of their nuclei and ω the angular frequency of an oscillating electric field modewhich acts, e.g., either on the permanent electric dipolemoments of two molecules or on the their induced electricdipole moments. If (cid:126)d is the electric dipole moment gen-erated at its position by the effective electric field (cid:126)E eff ,then α in an isotropic situation is defined by (cid:126)d = α (cid:126)E eff .It has the dimension [ L ] ( (cid:126)E eff includes a charge factor).The relation (258) holds for vanishing temperature T (for T > (cid:126) ω/ , both in their ground states.The corresponding values are α [ H ( S )] = 0 . · − m , α [ H ( Σ + g )] = 0 . · − m (259)[129]. As the polarizability α is closely related to thedispersion properties of an optical medium [130], Lon-don called the forces associated with the potential (258)“dispersion” van der Waals forces [124].If one applies London’s [122–124] and later heuristicarguments [125, 131] for the derivation of the van derWaals potential to the symplectic spectrum of the HO,one obtains instead of Eq. (258): V ( R ) = − γ ( b (cid:126) ω ) α R . (260)3If b < / V. POSSIBLE ASTROPHYSICALIMPLICATIONS
In case the above theoretically possible “symplectic”- or “fractional” - spectra (69) of the HO are - at leastpartially - realized in nature they could shed new lighton some unsolved basic astrophysical problems of whichI shall mention the two most important ones [146]:
Dark energy [147–150] and dark matter [151, 152]. Herethe symplectic spectra (69) may be a (the) key to thesolutions of both problems simultaneously!For reasons mentioned above those spectra have notyet been seen in the laboratory. But, surprisingly,physical implications of those spectra are supported bythe observationally favoured cosmological ΛCDM model[148, 153–156] and by the associated WIMP hypothesis[151, 157–159].
It probably sounds provocative, but the observed darkenergy and dark matter properties may provide the firstempirical support for the existence of the spectra (69) innature!
The following discussions and arguments are mostlyqualitative. The obviously necessary and crucial quanti-tative arguments will still have to be reviewed and workedout in detail!
A. Dark energy and the cosmological constant
Describing the existing astrophysical observations interms of the Einstein-Friedmann-Lemaˆıtre cosmologi-cal model [148, 160] leads to the conclusion that the(“vacuum”) energy density c ρ Λ , associated with theso-called “Lambda”-term in the Einstein-Friedmann-Lemaˆıtre equations, has the same order of magnitude asthe critical energy density [161] c ρ crit = 3 c H / (8 πG N ) ≈ − h GeV cm − , (261)where the scale factor h for the present Hubble expansionrate H has the approximate value h ≈ . h is not to be confused with Planck’s constantin the following):Taking into account that the observed “dark” energydensity c ρ Λ is about 0 . c ρ Λ with the vacuum energy density of thequantized free electromagnetic field [162], u em ;0 (ˆ ω, b ) = b (cid:126) π c ˆ ω , (262) where ˆ ω is an appropriate cutoff for the correspondingdivergent frequency integral u em ;0 ( b ) = b (cid:126) π c (cid:90) ω ≥ dω ω , (263)allows to make a numerical estimate of b :Introducing the cutoff length (cid:96) = 2 πc ˆ ω (264)leads to the approximate equality u em ;0 ( (cid:96), b ) = b π (cid:126) c(cid:96) ≈ . · c ρ crit . (265)Taking for (cid:96) the (reduced) Compton wave length of theelectron, (cid:96) ≈ λ = (cid:126) / ( m e c ) ≈ . · − cm (266)and inserting h ≈ . b the approximate value b ≈ − . (267)This is an extremely small b – value, but it is theoreticallyallowed in the present framework! This is in contrast tothe conventional theoretical estimates of the dark energywith b = 1 / b . This simplification is, of course, not neces-sary. b can depend on the frequency ω : b = b ( ω ) ∈ (0 , (cid:96) : if we. e.g., replace the factor 10 − by10 − the estimate in Eq. (267) is reduced to b ≈ − .In addition all non-electromagnetic effects were neglected(they would lead to an even smaller value of b than that inEq. (267)!). This will be justified by the discussion belowconcerning the nature of dark matter as being essentiallymolecular hydrogen the dynamics of which is essentiallydetermined by electromagnetic forces.Let me make another very crude estimate related tothe “cosmic” order of magnitude (267) of the B-index b : Consider the relation (84) between the angular fre-quency ω = 2 πν , the time period T π ; b and the index b :Most of the very first molecules and molecular ions af-ter the beginning of the recombination epoch in the veryearly universe were diatomic, with the vibrating nuclei lo-cally emitting infrared light with frequencies ν = ω/ (2 π )around 10 s − . Even though homonuclear elements like H do not have a permanent electric dipole element, theystill radiate in the infrared [77] and especially can emitRaman radiation by induced dipole momennts (see Ch.IV.B above).4Taking for T π ; b the extreme value 10 yr ≈ · sand ignoring cosmic red shifts z (i.e. being in the restframe of the molecule) we get the crude estimate b = 1 / ( ν T π ; b ) ≈ − . (268)An important open question is whether the vacuum(“dark”) energy has changed with cosmic time whichwould imply a corresponding time dependence of b ! B. “Dark” b-H and other primordial moleculesas dark matter? The following most intriguing but perhaps also dan-gerously seductive or even deceptive attempt intends tointerpret the cosmic “dark matter” in the “symplectic”framework of the HO. The central hypothetical role hereis being played by “symplectic” molecular hydrogen b-H as the main candidate for dark matter. The possibilitythat molecular hydrogen H may play a role for the un-derstanding of dark matter has been tentatively discussedbefore [166–176] without, it seems, having a lasting im-pact. But the possible existence of a (weak or hidden)“detuned” symplectic spectrum of the vibrating b-H al-lows for a new and probably more promising approach! Inaddition to this ”symplectic” detuning there is, of course,the usual cosmic redshift z due to the expansion of theuniverse.All the directly obtained experimental and oberva-tional data like those of the “Planck” Kollaboration etc.are, of course, not affected, but all the calculated par-ticle and cosmic standard model dependent dynamical -vibration related - electromagnetic properties (transitionprobabilities of emissions, absorptions, dissociations, ion-izations and other rates etc.) have to be re-evaluated.The same applies to the Big Bang Nucleosynthesis andthe primordial photon-baryon ratio [177, 178]!Molecular hydrogen plays already an important rolein the present standard (“orthodox”) cosmologicalparadigm [89, 92, 99, 100, 153, 170, 179–183]. Due tothe missing electric dipole moment it is difficult to detectastrophysically. For searches in the intergalactic medium(IGM) one uses a plausible correlation between the den-sities of H and of carbon monoxyde CO [184], which hasa permanent electric dipole moment and is more visible.But the atoms C and O are not primordial ones and haveto be bred in (first) stars etc..Presently, however, we are primarily interested in theepoch of the universe which is called its “Dark Ages” [21,185–188], i.e. the cosmic time period which started whenthe photons decoupled from matter and primordial atoms(mainly He and H) could form (“recombination epoch”[185, 189–195]) at about 400000 years after the big bang(at redshift z = z (cid:63) ≈ ≈ ◦ K ≈ .
26 eV [196]).And which ended just before (around z ≈
30, i.e. about80 Myr after the big bang) density fluctuations of theprimordial gases led to the first gravitational “clumps”as seeds for the first stars [92, 185, 197, 198] and the first galaxies [92, 199]. The heat and radiation associatedwhith this gravitational process reionized the primordialneutral gases of the dark ages [92, 200], a cosmic periodcalled the “Dawn” of the universe [21].During those “dark ages” there were no “dust grains”,no “metals” (like O or C etc.) and no X- and nocosmic rays. There were only gases of primordial pho-tons, neutrinos, electrons, protons, deuterons and atomsHe, H, D, Li, their ions and first primordial diatomicmolecules and their ions [180, 181, 183] like H and,e.g. (HeH) + [183, 201–204] most of them at least par-tially in local thermodynamical equilibrium. Ions maybe “thrown” out of equilibrium by primordial magneticfields.As the main observable signal from that epoch the21 cm line from the electron spin flip in the field of themagnetic moment of the proton in atomic hydrogen hasintensively been discussed more recently [21, 185, 205–210] and experimentally implemented or planned [21, 22,211]. Molecular hydrogen is also assumed to play an es-sential role during the dark ages [185, 212–214] becauseof its cooling properties [92, 207].In the following a number of possibly important prop-erties of “symplectic” molecular hydrogen, called b-H ,are listed which support the hypothesis of essentiallyidentifying dark matter with b-H . The b -detuned spec-tra of other primordial diatomic molecules (containing D,He and Li [180, 181, 183]) probably play an essential role,too, but that of b-H is the most important one and hereserves as a prototype for a symplectically detuned molec-ular HO! The following discussion relies on properties ofb-H pointed out in Ch. IV.B above.What is still missing in the discussion presented hereis a necessary critical quantitative evaluation! This is, ofcourse, due to the fact that up to now no correspondingexperimental laboratory data are available and - aboveall - that the associated theory for the dynamics of the“symplectic” spectroscopy still has to be worked out! Butin principle it should be possible to decide the empiricalexistence of those symplectic spectra in the laboratory,keeping in mind that the index b might be very small(see Eq. (267)) and, therefore, optical transitions may bevery weak (slow)!Here is a list of properties and problems which showshow the b-H (plus other primordial molecules) darkmatter hypothesis compares with what essentially isknown about dark matter experimentally:1. It is quite surprising that the measured (relative)cosmic baryonic density Ω b ≈ .
05 and the darkmatter density Ω d ≈ .
25 are roughly of the sameorder of magnitude. And that the total matterdensity Ω m ≈ . Λ ≈ .
7! But this is no longer surprising if darkmatter consists essentially of “b-”H and other “b”-detuned primordial molecules, i.e. it is baryonic,too ! In addition there may be some kind of dy-namical relationships between vacuum energy and5matter.2. Whereas there appears to be - on average - 5 timesmore dark than baryonic matter in the universe,not a single non-standard particle has been de-tected, despite years-long tremendous and inge-nious efforts by experimentalists and theoreticians[151, 157–159, 215, 216]. That failure is no surpriseif dark matter essentially consists of b-H and otherb-detuned primordial diatomic molecules and ions!3. If indeed dark matter consists of such molecules andions, this implies that there is no dark matter inthe universe before the recombination/decouplingera when the formation of primordial atoms (He,H, D, Li), their molecules and ions sets in [92, 180,181, 183]. And there is then no annihilation of darkmatter!4. For the formation of b-H etc. as dark matter about5 times more primordial baryons and electrons arerequired. This needs, of course, a re-evaluation ofthe orthodox big bang nucleosynthesis (BBN) andthe related baryon to photon ratio [148, 177, 178,190, 192, 194].5. The dark matter approach discussed here may alsoshed some new light on the “Lithium problem” ofthe current BBN interpretation (a disturbing dis-crepancy between the observed Li/H ratio und theBBN predicted one) [217].6. If there are at the beginning of the recombina-tion phase - ignoring all other light primordial nu-clei, their atoms, molecules and ions - protons andelectrons, why should they form dominantly b-H molecules and H atoms so that on average we havea ratio of approximately 5:1? A preliminary ar-gument is that the b-H molecules are energeti-cally favoured compared to 2 free H atoms becauseof their binding energies (4.48 + 0.26 = 4,74) eV(see Eqs. (228) and (227)). In order to understandthe observed ratio 5:1 of “dark” matter to bary-onic matter one probably has to take the dynam-ics of the different atomic and molecular reactionsinto account! Note that the formation of primor-dial hydrogen molecules can take different routes[180, 181, 183].7. It also is worth mentioning that the “partial” Boltz-mann factor (229) becomes smaller than of order 1just at the beginning of the dark ages. This impliesthat as lower temperatures statistically the lower b-zero-point energy levels are correspondingly morepopulated than the “orthodox” ones!8. Even if the transitions between the detuned b-levels of the vibrating molecules are very weak thereshould be enough time (60-100 Myr) available dur-ing the dark ages to “settle down” to the b-zero-point energy levels. 9. Galaxies (first small ones which then becamelarger) are assumed to have grown from (Jeans)density perturbations/fluctuations leading to insta-bilities inside halos of dark matter [92, 216, 218–220]. That appears quite “natural” if those halosare essentially cold remains of primordial diatomicmolecular (mainly hydrogen) clouds from the darkages, in which most of the molecules are not yet dis-sociated. Consequently one has to expect dynam-ical relations between the ionized core of galaxies(including their stars, black holes and gases) andtheir dark cold molecular halos.10. If dark matter consists essentially of b-H and otherprimordial diatomic molecules and ions then therewas a rich amount of matter for the formation ofbaryonic supermassive black holes in the center ofevolving galaxies [220, 221].11. As H is considered to be an important coolant forthe primordial gases before the formation of cosmicstructures sets in [92, 222, 223] the cooling by b-H has to be analyzed anew.12. Molecular hydrogen has a mass of about 2 GeV.Comparing this with the reconstructed tempera-ture T γ ≈ .
26 eV of the photon gas at the begin-ning of the recombination period and assuming thematter gases to have roughly the same tempera-ture about that time implies that the molecules arehighly nonrelativistic. The same applies, of course,to the other primordial molecules and ions contain-ing He, D, Li etc.13. Neutral H molecules interact weakly among them-selves and with other neutral atoms or moleculesby van der Waals forces [224, 225]. In addition, therelation (260) shows that the strength of van derWaals forces becomes weaker than the “orthodox”ones for b < /
2. For the interactions of “orthodox”H molecules with atomic hydrogen and helium see,e.g., the Refs. [226–229].14. A theoretical analysis of the data obtained by the21-cm radio wave detector EDGES [22] concluded[23, 24, 230, 231] that most likely there are in-teractions beyond the gravitational ones betweenthe primordial atomic hydrogen and the dark mat-ter particles, the latter having a mass of about afew GeV! All this fits - at least qualitatively - theabove b-H interpretation of dark matter very well.In addition it supports the hypothesis that darkmatter consists of (w)eakly (i)nteracting (m)assive(p)articles (“WIMPs”).15. The recently observed discrepancy between gravi-tational lensing [25] and computer-simulated (stan-dard) dark matter models may also find an explana-tion within the framework discussed in the presentpaper!616. Of special interest in the context of the near-infrared “symplectic” spectra of H etc. moleculesdiscussed above are the recent unaccounted cosmicoptical background observations in that spectral re-gion by the CIBER and New Horizons collabora-tions [232, 233].All the properties listed above are - qualitatively - sur-prisingly compatible with the current ΛCDM model ofthe universe and the WIMP hypothesis for dark matter!All this suggests that the hypothesis: “the dark mat-ter observed in the universe consists essentially of b-H and of a smaller amount of other primordial diatomicb-detuned molecules and their ions” should be taken se-riously and analyzed more quantitatively and experimen-tally as well. This requires joint efforts of the physisistsinvolved in the field!One particular lesson to be learnt from the discuss-sions above is that we do not know yet enough about thephysics of the cosmic quantum“vacuum”! ACKNOWLEDGMENTS
I very much thank the DESY Theory Group for itsenduring kind and supportive hospitality after my retire-ment from the Institute for Theoretical Physics of theRWTH Aachen. I am grateful to David Kastrup for pro-viding the figures and for further technical support.
Appendix A: Hilbert space for ˜ K j and ˜ C g ( K ) on R +0
1. Representation of the Lie algebra generators ˜ K j on L ( du, R +0 ) The Hilbert spaces of square-integrable functions f ( u ) = u α/ e − u/ g ( u ) , u ≥ , α > − , (A1)with the scalar product( f , f ) ≡ (cid:90) ∞ du f ∗ ( u ) f ( u ) (A2)= (cid:90) ∞ du u α e − u g ∗ ( u ) g ( u ) . also provide Hilbert spaces for irreducible unitary repre-sentations of the group SO ↑ (1 , Sp (2 , R ) and all other covering groups[14, 40].The usual orthonormal basis for a fixed α is given bythe associated Laguerre functionsˆ e α ; n ( u ) = (cid:115) n !Γ( n + α + 1) u α/ e − u/ L αn ( u ) , n = 0 , , , . . . (A3)(ˆ e α ; m , ˆ e α ; n ) = δ mn , where the associated Laguerre polynomials L αn ( u ) are de-fined as .L αn ( u ) = m = n (cid:88) m =0 (cid:18) n + αn − m (cid:19) ( − u ) m m ! , L αn (0) = ( α + 1) n n ! , (A4)( a ) n = a ( a + 1) ( a + 2) . . . ( a + n − . Examples: L α ( u ) = 1 , L α ( u ) = α + 1 − u. (A5)The polynomials L αn ( u ) obey the differential equation u d L αn du + ( α − u + 1) dL αn du + n L αn = 0 . (A6)The operators ˜ K j here have the explicit form [2, 4], with α = 2 b − K = − u d du − ddu + (2 b − u + u , (A7)˜ K = − u d du − ddu + (2 b − u − u , (A8)˜ K = 1 i ( u ddu + 12 ) . (A9)They obey the Lie algebra (10). The inequality α > − b >
0, as desired!Eigenfunctions f b ; n ( u ) of ˜ K are the basis functions(A3) with α = 2 b − K f b ; n ( u ) =( n + b ) f b ; n ( u ) , n = 0 , , , . . . (A10) f b ; n ( u ) =ˆ e b − n ( u )= (cid:115) n !Γ(2 b + n ) u b − / e − u/ L b − n ( u ) . The ground state function is f b ;0 ( u ) = 1 (cid:112) Γ(2 b ) u b − / e − u/ . (A11)It obeys K − f b ;0 ( u ) = ( K − iK ) f b ;0 ( u ) = 0 . (A12)The expressions (A7) and (A8) allow for an interpretationof the variable u in terms of the classical variables ϕ and I : We have ˜ K − ˜ K = u . (A13)Comparing this with the difference h − h = v ≡ I (1 − cos ϕ ) = 2 I sin ( ϕ/
2) (A14)we note the correspondence u ↔ v = 4 I sin ( ϕ/ . (A15)7The relation (A14) can be interpreted geometrically asfollows: I > ϕ ∈ [0 , π ) can be considered as polar co-ordinates of a plane where ϕ = 0 describes the positivepart of the abscissa. According to Eq. (A14) the vari-able v can be considered as the difference between thedistance I of the point ( ϕ = 0 , I ) on the abscissa and theprojection of the position vector ( ϕ, I ) on the abscissa.From Eq. (A15) we have v ∈ [0 , I ] for ϕ ∈ [0 , π )] , v ∈ R +0 (A16)and v ∈ [2 I,
0] for ϕ ∈ [ π, π ] , v ∈ R +0 . (A17)Time reflection T : ϕ → − ϕ plus complex conjugation , (A18)can be implemented as expected: T : u → u, ˜ K → ˜ K , ˜ K → ˜ K , ˜ K → − ˜ K (A19)The homeomorphic doublings (A16) plus (A17) may beimportant for the implementation of space reflections (seeCh. III.E above) P : ϕ → ϕ + π, I → I, v ↔ v , (A20)˜ K → ˜ K , ˜ K → − ˜ K , ˜ K → − ˜ K . So, in order to implement space reflections, it appearsone has to double the Hilbert space defined by Eqs. (A1)and (A2).
2. Eigenfunctions and spectrum of ˜ C g = ˜ K + g ˜ K The eigenfunctions f g,b ; n of the operator ˜ C g = ˜ K + g ˜ K can be obtained from the eigenvalue Eq. (A10):We have˜ C g ( u ) f g,b ; n ( u ) = ( ˜ K + g ˜ K )( u ) f g,b ; n ( u ) (A21)= (1 + g ) (cid:20) − u d du − ddu + (2 b − u (cid:21) f g,b ; n ( u )+ (1 − g ) u f g,b ; n ( u ) = ˜ c g,b ; n f g,b ; n ( u ) . With u = (cid:114) g − g v, (A22)we get ˜ C g ( u ) = (cid:112) − g ˜ K ( v ) . (A23)Therefore ˜ C g ( u ) has the eigenvalues˜ c g,b ; n = ( n + b ) (cid:112) − g , n = 0 , , , . . . (A24) and the eigenfunctions f g,b ; n ( u ) = f b ; n [ v ( g, u )] . (A25)Thus, the Hamiltonian H = (cid:126) ω ˜ C g (A26)has the eigenvalues (cid:126) ω g ( n + b ) , ω g = (cid:112) − g ω, (A27)where ω g is the effective frequency (32) of the classicalsystem and the result (A27) coincides with the one ob-tained in Ch. III.E.5 above. Appendix B: Notational conventions for the statesof diatomic molecules
Here the essential features of the conventional nota-tions for the states of diatomic molecules are briefly re-called [70, 234, 235], for a better understanding of thediscussion in Ch. IV.B above:Starting point for the description employed is the BOapproximation [236] which makes use of the small ratioof electron and nucleon masses: This leads to a separa-tion of the original Schr¨odinger equation into two sim-pler ones: one for the electrons in their mutual Coulombpotentials and in those of the nuclei held fixed, and sec-ond one for the two nuclei moving in potentials providedby solutions E el ( R.a ) of the electronic Schr¨odinger equa-tion which depend “adiabatically” on the internucleardistance R = | (cid:126)R − (cid:126)R | and possibly on other param-eters a of the nuclei etc. Qualitatively it is important todifferentiate between homonuclear molecules (same nu-clei) and heteronuclear ones (different nuclei). In thehomonuclear case there is no permanent electric dipolemoment and, therefore, no corresponding light emissionor absorption!The main coordinate refererence is the internuclearaxis (INA), i.e. the straight line passing through the twonuclei. It is essential for the description of electronic mo-tions and the nuclear ones as well. The following remarksapply to diatomic molecules only, including their ions.
1. Electronic motions
Like in the case of atoms one starts by ignoring spin-orbit couplings ( (cid:126) = 1 in the following): the absolutevalues Λ = | M L | = 0 , , . . . L of the projections of thetotal electronic orbital angular momentum (cid:126)L on the INAare denoted by Σ (for Λ = 0) , Π (for Λ = 1) , ∆ (for Λ =2) , . . . (in analogy to the atomic s (for l = 0) , p (for l =1) , d (for l = 2) , . . . ), where the state Σ is non-degenerateand the states Π , ∆ , . . . are 2-fold degenerate.The following symmetries further specify a state:8Reflections on any plane containing the INA form asymmetry of diatomic molecules. For non-degeneratestates like Σ this means that its state vector stays in-variant or changes sign. Thus one can have Σ + or Σ − .For degenerate states like Π , ∆ , . . . a 2-dimensional statevector may be mapped into another one and no definite“parity” can be assigned. This holds for homo- and het-eronuclear diatomic molecules alike.For homonuclear diatomic molecules - like H - thereis still another reflection symmetry [237]: reflections onthe midpoint of the INA between the two nuclei trans-forms the molecule onto itself. Correspondingly the wavefunction may be even (= g from German “(g)erade”)or odd (= u from German “(u)ngerade”). This holdsalso for Λ (cid:54) = 0. So one can have the electronic statesΣ ± g , Σ ± u , Π g , Π u , . . . .For electric dipole transitions between different elec-tronic levels the following selection rules hold+ ↔ − , + (cid:54)↔ + , − (cid:54)↔ − ; g ↔ u, g (cid:54)↔ g, u (cid:54)↔ u. Denoting the projection of the total electron spin onthe INA by S one can have 2 S + 1 multiplets, i.e. singletsand triplets for diatomic molecules. The correspondingstates are denoted as usual, e.g. S +1 Σ + g etc.Different electronic levels may have the same S +1 Λ ± g,u .If they are singlets ( S = 0) one differentiates betweenthem by the capital letters X (ground state), A, B, . . . . If they are non-singlets one writes a,b, . . . . For the elec-tronic ground state (potential) of H one has X Σ + g (see Fig. 4). The choice of the capital letter for differentelectronic states is not stringent, but can have historicalbackgrounds.
2. Nuclear motions
As mentioned before, the electronic energy states E ( R, a ) just discussed depend on the distance R of thetwo nuclei and serve as potentials for their motions, asindicated in Fig. 4 and discussed in more detail in theliterature [238]. Again, the vibrational energy levels inthose potentials are denoted by v = 0 , , . . . , where thelowest levels are approximately “harmonic”. In additionthe molecule can rotate around any axis passing throughthe center of mass on the INA and being orthogonal tothe latter. The associated rotational quantum number isgenerally denoted by J .If electronic configurations E ( R, a ) have (local) min-ima, these generally lead to several vibrational statesand, therefore, one can have transitions (emissions andabsorptions) ( v (cid:48)(cid:48) ↔ v (cid:48) ) between vibrational levels of dif-ferent electronic states. This leads to so-called “bands”[239]. In addition, each vibrational state ( v ” and v ’) canbe associated with several rotational states ( J ” and J ’)leading to different “branches” [240]. [1] Another frequent choice is q = (cid:112) I/m ω sin ϕ, p = √ m ω I cos φ .[2] H. A. Kastrup, “A new look at the quantum mechan-ics of the harmonic oscillator,” Ann. Phys. (Leipzig) , 439 (2007), arXiv:quant-ph/0612032; this paper andRef. [6] contain many references to the associated liter-ature.[3] In most of the mathematical literature and in previ-ous papers by the author the number b is denoted by k . In order to avoid confusion with the wave number k = | (cid:126)k | = ω/c from optical spectroscopy the nota-tional change k → b has been made. It also is in-tended to honor the mathematical physicist Valentine”B”argmann (1908-1989) who was among the first toprovide a complete list of the irreducible unitary repre-sentations of the noncompact group SO ↑ (1 ,
2) and itscovering groups [9].[4] M. Bojowald, H. A. Kastrup, F. Schramm, andT. Strobl, “Group theoretical quantization of a phasespace S × R + and the mass spectrum of Schwarzschildblack holes in D space-time dimensions,” Phys. Rev. D , 044026 (2000).[5] M. Bojowald and T. Strobl, “Group theoretical quanti-zation and the example of a phase space S × R + ,” J.Math. Phys. , 2537 (2000).[6] H. A. Kastrup, “Quantization of the optical phase space S = { ϕ mod 2 π, I > } in terms of the group SO ↑ (1 , , 975 (2003), the e-printarXiv:quant-ph/0307069 is an expanded and corrected version of the printed article.[7] M. Bojowald and T. Strobl, “Symplectic cuts and pro-jection quantization for non-holonomic constraints,” In-tern. J. Mod. Phys. D , 713 (2003).[8] H. A. Kastrup, “Quantization in terms of symplecticgroups: the harmonic oscillator as a generic example,”J. Phys.: Conf. Series , 012036 (2011),doi:10.1088/1742-6596/284/1/012036.[9] V. Bargmann, “Irreducible unitary representations ofthe Lorentz group,” Annals Math. , 568 (1947).[10] L. Puk´anszky, “The Plancherel formula for the univer-sal covering group of sl ( R , , 96(1964).[11] N. Ja. Vilenkin, Special Functions and the Theoryof Group Representations , Transl. Math. Monographs,Vol. 22 (Amer. Math. Soc., Providence, R.I., USA, 1968)Ch. VII.[12] Jr. P. J. Sally,
Analytic Continuation of the Irre-ducible Unitary Representations of the Universal Cover-ing Group of SL(2, R ) , Mem. Amer. Math. Soc., Vol. 69(Amer. Math. Soc., Providence, R.I., USA, 1967).[13] Jr. P. J. Sally, “Intertwining operators and the repre-sentations of SL(2, R ),” J. Funct. Anal. , 441 (1970).[14] C. P. Boyer and K. B. Wolf, “Canononical transforms.III. Configuration and phase desciptions of quantumsystems possessing an sl (2 , R ) dynamical algebra,” J.Math. Phys. , 1493 (1975).[15] H. A. Kastrup, “Quantization of the canonically conju-gate pair angle and orbital angular momentum,” Phys. Rev. A , 052104 (2006).[16] H. A. Kastrup, “Wigner functions for the pair angle andorbital angular momentum,” Phys. Rev. A , 062113(2016).[17] L. D. Landau and E. M. Lifshitz, Mechanics , 2nd ed.,Course of Theoretical Physics, Vol. 1 (Pergamon PressLtd., Oxford, 1969) §
49 and § Classical Mathematical Physics: Dynam-ical Systems and Field Theories , 3rd ed. (Springer-Verlag, New York, 1997, pb: 2003) Ch. 3.3.[19] V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt,
Math-ematical Aspects of Classical and Celestial Mechanics ,3rd ed., Encyclopaedia of Mathematical Sciences, Vol. 3(Springer-Verlag, Berlin, Heidelberg, 2006) Chs. 5,2, 5.3and 6.1.[20] See Ch. III,2 of Ref. [69].[21] L. Koopmans et al.,
Peering into the dark (ages) withlow-frequency space interferometers, using the 21-cmsignal of neutral hydrogen from the infant universeto probe fundamental (astro)physics , arXiv:1908.04296;ESA Voyage 2050 – White Paper.[22] J. D. Bowman et al., “An absorption profile centred at78 megahertz in the sky-averaged spectrum,” Nature , 67 (2018).[23] R. Barkana, “Possible interaction between baryons anddark-matter particles revealed by the first stars,” Nature , 71 (2018).[24] A. Fialkov, R. Barkana, and A. Cohen, “Constrainingbaryon–dark–matter scattering with the cosmic dawn21–cm signal,” Phys. Rev. Lett. , 011101 (2018).[25] M. Meneghetti et al., “An excess of small-scale gravita-tional lenses observed in galaxy clusters,” Science ,1347 (2020).[26] G. Hasinger, “Illuminating the dark ages: Cosmic back-grounds from accretion onto primordial black hole darkmatter,” Journ. Cosm. Astropart. Phys. , 022 (2020),arXiv:2003.05150.[27] Detailed arguments can be found in Appendix A of Ref.[6], Ch. 4 of Ref. [2] and Ch. 2 of Ref. [8].[28] See, e.g., Appendix B of Ref. [6].[29] The model has been mentioned in Refs. [2] and [8].[30] M. Born, unter Mitwirkung von F. Hund, Vorlesungen¨uber Atommechanik , Struktur der Materie in Einzel-darstellungen II, Vol. 1 (Julius Springer, Berlin, 1925)English translation:
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