Dependence of the affine coherent states quantization on the parametrization of the affine group
aa r X i v : . [ m a t h - ph ] M a r Dependence of the affine coherent states quantizationon the parametrization of the affine group
Andrzej Góźdź, ∗ Włodzimierz Piechocki, † and Tim Schmitz ‡ Institute of Physics, Maria Curie-Skłodowska University,pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland Department of Fundamental Research,National Centre for Nuclear Research,Pasteura 7, 02-093 Warszawa, Poland Institut für Theoretische Physik, Universität zu Köln,Zülpicher Straße 77, 50937 Köln, Germany (Dated: March 4, 2020)
Abstract
The affine coherent states quantization is a promising integral quantization of Hamilto-nian systems when the phase space includes at least one conjugate pair of variables whichtakes values from a half-plane. Such a situation is common for gravitational systems whichinclude singularities. The construction of the quantization map includes a one-to-one map-ping of the half-plane onto the affine group. Particular cases of this mapping define specificparametrizations of the group. Our aim is showing that different such parametrizationslead to unitarily inequivalent quantum theories. Depending on the Hamiltonian systemunder consideration, this dependence could potentially be used constructively. ∗ [email protected] † [email protected] ‡ [email protected] Typeset by REVTEX 1 . INTRODUCTION
The coherent states quantization can be applied to the quantization of Hamil-tonian systems when the physical phase space can be identified with a Lie groupacting on itself. This group is expected to have a unitary irreducible representationin a Hilbert space. The latter allows one to construct a resolution of the identity inthat Hilbert space, which can be used to map the observables of that system intoHermitian operators. This quantization method is especially useful in cases wherethe physical phase space includes at least one variable with a nontrivial topology,e.g. R + = { x ∈ R | x > } , as is common to gravitational systems.The affine coherent states (ACS) quantization is a special case of the affine integralquantization, where ‘affine’ refers to the symmetry group of the half-plane consistingof translations and dilations. Mathematical aspects of this quantization are pre-sented, for instance, in the review article [1]. ACS have been used in quantizationsof numerous physical systems. For a comprehensive review of known applications werecommend the references [2] and [3]. The proceedings of the conference on coherentstates [4] present recent developments.The ACS quantization of the so-called mixmaster solution of the Einstein equa-tions, examined by Misner [5], is widely discussed in [6]. It concerns the quantumdynamics of the diagonal Bianchi IX model. This quantization method, within thesemi-classical approximation, leads to the avoidance of the cosmological singularity.Recent results on quantization of the so-called Belinski-Khalatnikov-Lifshitz (BKL)scenario [7] are much more general [8, 9]. The classical BKL scenario was derived byconsidering the dynamics of the non-diagonal (general) Bianchi VIII and IX models,and describes the asymptotic approach to a generic spacelike singularity in gen-eral relativity. The quantum BKL scenario obtained within exact ACS quantizationshows that the gravitational singularity can be replaced by a quantum bounce as aconsequence of unitary evolution of the system, and suggests that quantum generalrelativity is free from singularities.Quantization methods usually come with quantization ambiguities. They are gen-erally undesirable as they lower the predictive power of the resulting quantum theory.The most well known example is probably the freedom to choose different factor or-derings in the usual canonical quantization procedure. In the ACS quantization thereis no factor ordering ambiguity, but one can choose with some freedom the so-calledfiducial vector (to be explained later). Quantization ambiguities need to be fixedby conceptual criteria and experimental data if available. On the other hand, the2mbiguities are what allow us to tailor the constructed quantum theory to satisfythese restrictions.Broad classes of quantization schemes suffer from ambiguities and mathematicalinconsistences (see, e.g. [10] and references therein). Since we have growing evidencethat the ACS quantization is capable of dealing with singular gravitational systems,it is reasonable to identify its limitations.In this article we wish to discuss a quantization ambiguity of the ACS quantizationwhich, as far as we know, has not been recognized in the literature before. One mayexpect that the result of quantization does not depend on how exactly one identifiesthe phase space with the Lie group, or in other words how one parameterizes thegroup, but this is not the case. In fact, different parametrizations lead to quantumsystems which are not unitarily equivalent, i.e. represent different quantum systems.The aim of this paper is showing explicitly this intriguing property of the ACSquantization.In what follows, we first present two parametrizations of the affine group used inthe literature and derive the two corresponding quantizations which turn out to beunitarily inequivalent. Next, we extend this result to the general case. Finally, weconclude. II. COMPARING TWO KNOWN PARAMETRIZATIONS
To make the present paper self-contained we first recall general ideas underlyingcoherent states (see, e.g. [11]). For a Lie group G , let U ( g ) , where g ∈ G , be aunitary irreducible representation of it in some Hilbert space H . One can take an(at this point) arbitrary | Φ i ∈ H , called fiducial vector, and act on it with U ( g ) asfollows | g i = U ( g ) | Φ i , (1)to construct a family of coherent states.Consider the operator O = Z G dµ ( g ) | g ih g | = Z G dµ ( g ) U ( g ) | Φ ih Φ | U † ( g ) , (2)where dµ ( g · g ) = dµ ( g ) is a left invariant measure on G . It is easy to see that O intertwines U ( g ) , U ( g ) · O = Z G dµ ( g ′ ) U ( g · g ′ ) | Φ ih Φ | U ( g ′− ) h = g · g ′ = Z G dµ ( h ) U ( h ) | Φ ih Φ | U ( h − · g ) = O · U ( g ) . (3)3s we know from Schur’s Lemma, any non-trivial intertwiner is a scalar multiple ofthe identity, i.e. O ∝ I H . The factor of proportionality has to be decided on a caseby case basis, and may come with a restriction on the fiducial vector | Φ i .The above means that if U is the unitary irreducible representation of G in theHilbert space H , the family of coherent states (1) can be used to define a resolution ofthe identity in H . The latter is of primary importance in the quantization proceduredescribed below.In what follows we discuss coherent states constructed as above from the affinegroup, affine coherent states (ACS), and show how they can be used in quantization.This procedure is called the affine coherent state quantization, and we give here ashort introduction to this method.In this section we compare the ACS quantization corresponding to two simpleparametrizations of the affine group. The general case is considered in the nextsection.Suppose the phase space of some physical system is a half plane, Π = { ( p, q ) ∈ R × R + } . It can be identified with the affine group G := Aff ( R ) by defining themultiplication law either by (see [8] for more details) ( p , q ) · ( p , q ) := ( q p + p , q q ) , (4)or by (see [1] for more details) (˜ p , ˜ q ) · (˜ p , ˜ q ) := (˜ p / ˜ q + ˜ p , ˜ q ˜ q ) . (5)Eqs. (4)–(5) define two different parametrizations of G . They correspond, respec-tively, to the two actions of this group on R + : x ′ = ( p, q ) · x := xq + p and x ′ = (˜ p, ˜ q ) · x := x/ ˜ q + ˜ p . (6)The affine group has two (nontrivial) inequivalent irreducible unitary represen-tations, [14] and [15, 16], defined in the Hilbert space H := L ( R + , dν ( x )) , where dν ( x ) := dx/x . For both parametrizations we choose the one defined, respectively,by U ( p, q )Ψ( x ) := e ipx Ψ( qx ) and U (˜ p, ˜ q )Ψ( x ) := e i ˜ px Ψ( x/ ˜ q ) , (7)where Ψ( x ) = h x | Ψ i and | Ψ i ∈ H . To make possible the comparison of the parametrizations used in [8] and [1], we rename thevariables ( p, q ) ∈ R + × R of [8] so that ( p, q ) ∈ R × R + , in the present paper, which fits thenotation of [1]. The representation defined in [1] is U ( p, q ) ψ ( x ) = e ipx √ q ψ ( x/q ) with the carrier space L ( R + , dx ) ,but takes the form U ( p, q ) ψ ( x ) = e ipx ψ ( x/q ) when acting in L ( R + , dx/x ) . Since the measure dx/x is invariant for dilations on R + , it is natural to use the latter space as the carrier space of theaffine group representation. Similarly, L ( R , dx ) is the natural carrier space for the representationof the additive group on R because dx is invariant on R . Z G dµ ( p, q ) := Z ∞−∞ dp Z ∞ dq/q and Z G dµ (˜ p, ˜ q ) := Z ∞−∞ d ˜ p Z ∞ d ˜ q , (8)where both measures in (8) are left invariant.Any coherent state can be obtained as h x | p, q i = U ( p, q )Φ( x ) or h x | ˜ p, ˜ q i = U (˜ p, ˜ q )Φ( x ) , (9)where the fiducial vector Φ( x ) = h x | Φ i , | Φ i ∈ H , is required to satisfy h Φ | Φ i = 1 .The resolutions of the identity in the Hilbert space H read Z G dµ ( p, q ) | p, q i h p, q | = 2 πA Φ I and Z G dµ (˜ p, ˜ q ) | ˜ p, ˜ q i h ˜ p, ˜ q | = 2 πA Φ I , (10)where A Φ = Z ∞ dxx | Φ( x ) | < ∞ . (11)Eq. (11) defines an additional condition we impose on the fiducial vector Φ( x ) .Making use of (10) one can (formally) map any observable f : Π → R into asymmetric operator ˆ f : H → H as follows: ˆ f := 12 πA Φ Z G dµ ( p, q ) | p, q i f ( p, q ) h p, q | , (12)or ˆ f := 12 πA Φ Z G dµ (˜ p, ˜ q ) | ˜ p, ˜ q i f (˜ p, ˜ q ) h ˜ p, ˜ q | . (13)Due to the above we have ˆ f Ψ( x ) = 12 πA Φ Z G dµ ( p, q ) e ipx Φ( qx ) f ( p, q ) h p, q | Ψ i , (14)where h p, q | Ψ i = Z ∞ dν ( x ′ ) e − ipx ′ Φ( qx ′ ) ∗ Ψ( x ′ ) , (15)and ˆ f Ψ( x ) = 12 πA Φ Z G dµ (˜ p, ˜ q ) e i ˜ px Φ( x/ ˜ q ) f (˜ p, ˜ q ) h ˜ p, ˜ q | Ψ i , (16)where h ˜ p, ˜ q | Ψ i = Z ∞ dν ( x ′ ) e − i ˜ px ′ Φ( x ′ / ˜ q ) ∗ Ψ( x ′ ) . (17)5o proceed in calculations we use in what follows the following identity definedin H and derived in App. A: I = Z ∞ dν ( x ) | x ih x | . (18)To get (15) and (17), we apply (18) to each of the left hand side of these equations,and Eq. (9) together with Eq. (7). Making use of the substitution q = 1 / ˜ q in Eq.(14), we easily obtain ˆ f Ψ( x ) = 12 πA Φ Z G dµ (˜ p, ˜ q ) e i ˜ px Φ( x/ ˜ q ) f (˜ p, / ˜ q ) h ˜ p, ˜ q | Ψ i . (19)Comparing (16) with (19) we can see that for a generic | Ψ i ∈ H we have ˆ f | Ψ i 6 =ˆ f | Ψ i , as in general f (˜ p, ˜ q ) = f (˜ p, / ˜ q ) . It means that these operators act quitedifferently in H .Are operators, however, unitarily equivalent? To answer this essential question,we compare the traces of both operators in some orthonormal basis {| e k i} of H tosee whether Tr( ˆ f ) = Tr( ˆ f ) . (20)Eq. (20) is satisfied if the operators ˆ f and ˆ f are unitarily equivalent, since theyhave the same trace: Tr( ˇ U ˆ f ˇ U − ) = Tr( ˆ f ˇ U ˇ U − ) = Tr( ˆ f ) , (21)where ˇ U is some unitary operator.The main part of the above verification is the rewriting: Tr( ˆ f ) = X n h e n | ˆ f | e n i = 12 πA Φ X n Z G dµ ( p, q ) h e n | p, q i f ( p, q ) h p, q | e n i = 12 πA Φ X n Z G dµ (˜ p, ˜ q ) h e n | ˜ p, ˜ q i f (˜ p, / ˜ q ) h ˜ p, ˜ q | e n i = 12 πA Φ Z G dµ (˜ p, ˜ q ) h ˜ p, ˜ q | X n | e n ih e n || ˜ p, ˜ q i f (˜ p, / ˜ q )= 12 πA Φ Z G dµ (˜ p, ˜ q ) h ˜ p, ˜ q | ˜ p, ˜ q i f (˜ p, / ˜ q )= 12 πA Φ Z G dµ (˜ p, ˜ q ) f (˜ p, / ˜ q ) . (22) The trace of an operator does not depend on the choice of basis. X n | e n ih e n | = I and h ˜ p, ˜ q | ˜ p ˜ q i = h Φ | U − U | Φ i = h Φ | Φ i = 1 . (23)Similarly, we get Tr( ˆ f ) = 12 πA Φ Z G dµ (˜ p, ˜ q ) f (˜ p, ˜ q ) . (24)Therefore, Eq. (20) cannot be satisfied, as f (˜ p, ˜ q ) = f (˜ p, / ˜ q ) , so that the consid-ered operators, ˆ f and ˆ f , are unitarily inequivalent. It further means that the twoconsidered affine group parametrizations, defined by (4)–(5), lead to quite differentquantum systems. III. CONSIDERING THE GENERAL CASE
In this section we consider an arbitrary parametrization of the affine group. Differ-ent parametrizations of the affine group can be implemented by a family of one-to-onetransformations χ : Π → Aff(R). Every function χ provides a parametrization of theaffine group by elements of the phase space Π as follows χ ( p, q ) = ( ξ ( p, q ) , η ( p, q )) ∈ Aff(R) . (25)To define a general parametrization, one can use as an intermediate step either (4)or (5). To be specific, we choose (4) so that we have the composition law in the form ( ξ ( p , q ) , η ( p , q )) · ( ξ ( p , q ) , η ( p , q ))= (( η ( p , q ) ξ ( p , q ) + ξ ( p , q ) , η ( p , q ) η ( p , q )) . (26)This determines uniquely the composition law for the new parametrization of theaffine group.The corresponding invariant measure can be obtained by the change of variablesin the first measure in (8): dξ dηη = (cid:20) η ( p, q ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( ξ, η ) ∂ ( p, q ) (cid:12)(cid:12)(cid:12)(cid:12) dp dq =: σ ( p, q ) dp dq . (27)7herefore, the ACS quantization of the phase space function f : Π → R yields ˆ f = 12 πA Φ Z G dξ dηη | ξ, η i f ( p ( ξ, η ) , q ( ξ, η )) h ξ, η | = 12 πA Φ Z G dp dq σ ( p, q ) | ξ ( p, q ) , η ( p, q ) i f ( p, q ) h ξ ( p, q ) , η ( p, q ) | . (28)Following the idea of the preceding section, we calculate the trace of ˆ f . If the traceof ˆ f is independent on the affine group parametrization, then the ACS method ofquantization is universal. But we have Tr( ˆ f )= 12 πA Φ Z G dp dq σ ( p, q ) f ( p, q )Tr( | ξ ( p, q ) , η ( p, q ) ih ξ ( p, q ) , η ( p, q ) | )= 12 πA Φ Z G dp dq σ ( p, q ) f ( p, q ) . (29)To obtain the second equality in the above equation we have used (23).Eq. (29) shows explicitly the dependence of the trace of the operator ˆ f on theparametrization (25) due to the term σ ( p, q ) in the integrant. Therefore, the ACSquantization scheme depends on the parametrization of the group manifold by thephase space variables. The implications of this dependence are discussed in theconclusions. IV. CONCLUSIONS
The result we have obtained is quite general. The dependence we have found canbe seen as an advantage of this method over other quantization schemes as it allowsone to construct quantum theories fulfilling specific requirements. However, this highflexibility may lead to low predictability of the resulting quantum theory. Therefore,an additional constraint on the choice of the parametrization, well motivated physi-cally, should be an essential element of the constructed integral quantization scheme.In the case of quantization of the BKL scenario [8] we have found [9] that the twosimplest parametrizations of the affine group, considered in Sec. II, lead to qualita-tively the same results. The difference concerns quantitative details not essential tothe main conclusion that a regular quantum bounce replaces the classical generic sin-gularity of the BKL scenario. In this case, the conceptual criterion of having regularquantum dynamics does not single out a parametrization, neither is this necessary.However, obtaining this result was made possible by taking a simplified form of theclassical Hamiltonian that describes the dynamics properly only in the close vicinityof the gravitational singularity. Considering the exact Hamiltonian would probably8ead to making use of the freedom in the choice of the group parametrization to meetthis conceptual criterion.In App. B we consider the issue of quantization of the affine group algebra. Itis shown that the parametrization of [8] fails in reproducing the classical algebra,whereas the one of [1] is successful. This is a further possible criterion for choosinga particular parametrization.The issue of mapping a classical observable f : Π → R into a self-adjoint operator ˆ f : H → H is of basic importance in any quantization scheme . It is highly prob-lematic if ˆ f is an unbounded operator (see, e.g. [17, 18]). In the case of the ACSquantization, the dependence on the group parametrization can be helpful. To bespecific, let us consider the norm of ˆ f defined by (28)): k ˆ f k ≤ πA Φ Z G dp dq k σ ( p, q ) | ξ ( p, q ) , η ( p, q ) i f ( p, q ) h ξ ( p, q ) , η ( p, q ) |k = 12 πA Φ Z G dp dq | σ ( p, q ) f ( p, q ) | k| ξ ( p, q ) , η ( p, q ) ih ξ ( p, qk ) , η ( p, q ) |k = 12 πA Φ Z G dp dq | σ ( p, q ) f ( p, q ) | , (30)where the projection operator | ξ ( p, q ) , η ( p, q ) ih ξ ( p, qk ) , η ( p, q ) | has the norm k | ξ ( p, q ) , η ( p, q ) ih ξ ( p, qk ) , η ( p, q ) | k = 1 .The inequality (30) shows that the symmetric operator ˆ f is bounded so thatself-adjoint if πA Φ Z G dp dq | σ ( p, q ) f ( p, q ) | < ∞ . (31)The open question is whether one can always find a parametrization of the groupmanifold which, due to the form of the Jacobi determinant σ ( p, q ) , fulfills the condi-tion (31). Each case needs a separate examination.In summary, one can say that the freedom in the choice of the group parametriza-tion in the ACS quantization is a free ‘parameter’ of this quantization scheme. How-ever, one should be conscious that different parametrizations lead to unitarily in-equivalent quantum systems.Our next papers will concern the ACS quantization of the Oppenheimer-Snyderand Lemaître-Tolman-Bondi models for gravitational collapse [19, 20], which havealready been quantized within the canonical approach. Possible agreement of theresults obtained by these two quite different methods may serve to demonstraterobustness of the obtained results. In this context, the possibility of choosing differentparametrizations of the affine group may turn out to be helpful. In fact, only a self-adjoint operator can represent an observable at the quantum level. CKNOWLEDGMENTS
This work was partially supported by the German-Polish bilateral project DAADand MNiSW, No 57391638.
Appendix A: Subsidiary identity
Let us consider two spaces of square integrable functions: K = L ( R , dy ) and K = L ( R + , dx/x ) , where G = ( R , +) is the additive group of real numbers andG = ( R + , · ) is the multiplicative group of positive real numbers.The measure dy is the invariant measure for G and dν ( x ) = dxx is the invariantmeasure for G .The logarithmic function gives an isomorphism between both groups: ln : G → G , y = ln( x ) . (A1)This means that one can transfer a part of the notions well defined on one group tothe second one. For example, the Dirac delta function δ ( y ) defined on R determinesthe Dirac type delta function δ R + ( x ) defined on R + : δ R + ( x ) = δ (ln( x )) = δ ( y ) , x > . (A2)This implies (renaming φ ( x ) = ψ (ln( x )) = ψ ( y ) ) that Z R + φ ( x ) δ R + ( x ) dν ( x ) = φ (1) . (A3)In both spaces K and K the position operator is defined as the multiplicationoperator (in K the additional constraint y > is required) ˆ xψ ( y ) = yψ ( y ) . (A4)The generalized eigenvectors of ˆ x corresponding to the eigenvalue x are the Diracdelta distribution δ ( y − x ) in K and, by making use of the logarithmic transformation,the distribution δ R + ( x − y ) in the space K . This and (A3) imply h x | φ i := Z R + dν ( y ) δ R + ( x − y ) φ ( y ) = φ ( x ) . (A5)Using the above property, one gets h φ | φ i = Z R + dν ( x ) h x | φ i ⋆ h x | φ i = h φ | (cid:26)Z R + dν ( x ) | x ih x | (cid:27) | φ i (A6)10or all φ and φ , which implies: Z R + dν ( x ) | x ih x | = I . (A7) Appendix B: Example of choosing a suitable parametrization
In this section we focus on the reproduction of the classical algebra of a subsetof observables of the theory. Usually, the prime example would be the canonicalcommutation relation [ˆ q, ˆ p ] = i ~ , but on the half line it is problematic: ˆ p is notstrictly speaking an observable in the quantum theory, since it cannot be made self-adjoint.Instead we want to focus on the classical relation { q, d } = q , where d := pq ,as was done for example in [3]. The quantum counterpart of this is called theaffine commutation relation. The operator ˆ d generates dilations, which in contrastto the translations generated by ˆ p can not push position eigenstates off of the halfline. This makes the affine commutation the natural replacement for the canonicalcommutation relation for quantum theories on the half line. In addition, this algebrais also of importance to some approaches to quantum gravity, see e.g. [12, 13].For simplicity we only consider real fiducial vectors, and restrict ourselves to thetwo specific parametrizations used in section II. A quick calculation shows that therespective position operators are given as ˆ q ψ ( x ) = 1 A Φ ψ ( x ) x , and ˆ q ψ ( x ) = B Φ A Φ x ψ ( x ) , (B1)where B Φ = Z ∞ dxx Φ( x ) , (B2)and the dilation operators as ˆ d ψ ( x ) = − iA Φ ddx (cid:18) ψ ( x ) x (cid:19) , and ˆ d ψ ( x ) = − i B Φ A Φ x ddx ψ ( x ) . (B3)This leads to [ˆ q , ˆ d ] = iA Φ ˆ q , and [ˆ q , ˆ d ] = i B Φ A Φ ˆ q . (B4)Apart from numerical prefactors depending on Φ( x ) , which could be absorbedthrough a rescaling of p , the parametrization of [1] fulfills the affine commutation11elation, and the one from [8] does not. Based on this criterion, one should hencechoose the former one. [1] C. R. Almeida, H. Bergeron, J.-P. Gazeau, and A. C. Scardua, “Three examples ofquantum dynamics on the half-line with smooth bouncing”, Annals of Physics ,206 (2018).[2] S. T. Ali, J.-P. Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Gener-alizations , 2nd Edition in Theoretical and Mathematical Physics, Springer, New York,2014.[3] J. R. Klauder,
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