Iranian Journal of Physics Research, Vol. 5, No. 3, 200 5

On quantizing g ravity and geometrizing quantum mechanics1

Ali Mostafazadeh

Department of Mathematics, Ko ç University, 34450 Sariyer , Istanbul, Turkey E-mail: amostafazadeh @ku.edu.tr

(Received 18 March 2005 ; accepted 9 April 2005) Abstract We elaborate on some recent results on a solution of the H il bert-space problem in minisuperspace and discuss the conseq uences of making the (geometry of the) H il bert space of ordinary nonrelativistic quantum systems time -dependent. The latter reveals a remarkable similarity between and .

Keywords: quantum , Hilbert -space problem, geometric phase

1. Introduction The severity of the problems associated with Perhaps the most challenging problem facing theoretical canonical of gravity has led a number of physicists since the 1930 ’s has been the formulation of a researchers to take the op posite root, i.e., rather than consistent physical theory that encompasses Quantum trying to quantize GR they tried to geometrize QM, [ 8]. Mechanics (QM) and General Relativity (GR). The early This was mainly motivated by the idea that perhaps the steps toward addressing this problem were taken by origin of the above -mentioned difficulties is that QM Dirac who attempted to apply the rules of canonical itself is an approximate theory and that one must put QM quantization to GR . The fact that 4 out of 10 components in a geometric setting and seek for its nonlinear of the Einstein field equation are constraints necessitated generalization(s) [ 9]. The developments in this direction the development of a quantization sc heme that could have admittedly failed to lead to a concrete and handle the presence of constraints. This constituted one physically acceptable alternative to QM. Nevertheless, of the most significant contributions of Dirac to revealing the geometric structure of QM is an i nteresting . It is referred to as Dirac’s endeavor in its own right, especially in connection with constrained quantization scheme w hich he formulated in geometric phases and their widespread applications and the 1950 ’s, [1]. implications [ 10 ,11 ]. The appli catio n of Dirac’s scheme to GR required a Today the most promising candidate for a canonical Hamiltonian for mulation of GR which was made quantum theory of gravity is the so -called loop quantum available in early 1960 ’s [ 2]. The first concrete steps gr avity [ 12 ]. Similarly to its more popular rival, String toward establishi ng a canonical quantum theory of Theory, loop quantum gravity has not yet produced a gravity was subsequently taken by Bryce DeWitt [ 3] and concrete experimentally verifiable prediction. Therefore John Whe eler [ 4]. The result was an ingenious theory one cannot view it as a physical theory. As a that suffered from severe mathematical as well as mathematical theory with potential physical a pplications, interpretational problems [ 5,6]. These problems have however, it enjoys a higher level of rigor in its been the subject of numerous research articles for t he formulation than the String Theory. past 40 years or so . Yet a satisfactory solution of m any of them is still unavailable [ 7]. Among the most 2. Hilbert -space problem in minisuperspace important of these is the Hilbert -space problem that we quantum cosmology will elude to here. Consider a classical system that is time - ______Dedicated to the memory of Bryce Dewitt. 54 Ali Mostafazadeh IJPR Vol . 5, No. 3 reparameterization invariant. Then its classical Hamil - leave H invariant, i.e., there are ψ ∈H such that tonian K vanishes identically; the system has a first -class qˆi ψ ∉ H and pi′ ψ ∉H . This shows that the constraint K(q, p) = 0 , where q = (q1 qn ),..., and coo rdinate and momentum operators (qˆ , pˆ ) are not p = ( p1,..., pn ) are the unconstrained coordinate and i i momentum degrees of freedom. According to Dirac [ 1], physical observables. In particular, the eigenvalues qi′ of quantizati on of this system is performed in two steps. qˆi are not (measurable) physical quantities, and the First one quantizes ( q, p) as if there were no constraints. wave functions ψ q′)( do not represent physically This is done according to the well -known rules of meaningful entities. The attem pts to extract a canonical q uantization: q → qˆ , p → pˆ where i i i i ‘probability density’ out of the wave function ψ q′)( [qˆ , qˆ ] = [ pˆ , pˆ ] = 0 , [qˆ , pˆ ] = iδ . (1) i j i j i j ij therefore lack a logical basis. Then one impose s the constraint as the condition: The situation is quite different in the absence of such K( pˆ,qˆ) ψ = 0 . (2) a constraint. In that case H ′ is the physical H il bert This defines the physical state vectors ψ of the spa ce, and being Hermitian operators acting in H ′ , qˆi constrained system. and pˆi are physical observables. Therefore, we can Dirac ’s scheme involves an auxiliary (unconstrained) ide ntify q′ with the result of a qˆ -me asurem ent wh ose 2 2 Hil bert space H ′ , that is endowed with the L -inner probability density is proportional to ψ (q′) .1 product and furnishes the Heisen berg -Weyl algebra (1) When the constraint (2) is present, one must first with an irreducible representation, and a physical H il bert deter mi ne the physical position operator that unlike qˆ space H, which is defined according to eq. (2) as the kernel (null spa ce) of the quant um constraint acts in H (an d leaves it invariant) and then use its eigenvectors to define a new set of wave functions Kˆ := K( pˆ, qˆ) : H ′ → H ′ , whose modulus -square ca n be interpreted as the physical H := {ψ ∈ H ′ ˆ ψ = 0} = Ker(KK ˆ ). (3) probability density of localization of t he system. All this cannot be ac hi eved unless one solves the H il bert -space This equation identi ties H wit h a certain vector proble m. subspace of H ′ . It does not specify the inner product on In fu ll canonical qua ntu m gravity, in its traditional thi s su bspace. This lack of a presc ription to endow H metric formulation [ 3], the metric tensor gµν plays the with an inner product is known as the H il bert -space role of the coordinates q. In the Hamiltonian formulation problem. Once an appropriate inner product is given to the diffeomorphism invariance of the theory manifests H , one can define the physical observables of the itself as the prese nce of a Hamiltonian constraint which theory (as Hermitian operator acting in H ), outline a is related to ti me-repara meterization sy mmetry of the dynamics by selecting a Hamiltonian operator from theory and three momentum constraints that are linked among the observables, and employ the axioms of QM to with the diffeomorphism symmetry of the spatial deal with interpretational issues [ 13 ]. hypersurface in the ADM decomposition of the It is customary to use the coordinate basis {q′ } to spacetime . Quanti zation of the Hamiltonian constraint represent the state vectors and operators, leads to an infinite -dimensional analog o f eq . ( 5) which is known as the Wheeler -DeWitt equation; it is a q′ qˆ = q′ q′ , q′ pˆ = −i∂ q′ , ψ (q′) := q′ ψ , i i i qi′ functional partial differential equation plagued with a (4) number of problems [ 5,6,7,14 ]. n Imposing the extremely s trong and highly unrealistic where q′ = (q′,..., q′ ) ∈R and ψ =ψ q′)( is the 1 n simpl ifying assumption of spatial ho mogeneity, one is coordinate wave function associated with the state vector led to a finite -dimensional system in which the Wheeler - ψ . In this coordinate representation the quantum DeWitt equation may be writte n as a Klein -Gordon constraint (2) takes the form of a linear partial equation with a variable mass term: differential equation: 2 [∂ ′ + D]ψ (α′, β ′,...., β ′ ) = 0 (6) K(−i∂q′, q′)ψ (q′) = 0 ( 5) α 1 n where ∂ = (∂ ∂ ,...,∂ ∂ ) , and the physical H il bert where α′ and β ′,....,β ′ correspond to the eigenvalues q′ q1′ qn′ 1 5 space is identified with the solut ion space of this of the operators obtained by quantizing the 3-metric equation. In effect, the Hilbert -space problem is gij , βi′ with i > 5 represent matter degrees of freedom, equivalent to promoting the vector space of solutions of and D is a second orde r elliptic differential operator with eq . ( 5) into a Hilbert space . variable coefficients. We will view eq. (6) as a second It is important to note that in general the operators qˆi and pˆ do not commute with Kˆ . Therefore they do not ______i 2 1. It is equal to ψ (q′) f or a normalized state vector ψ . IJPR Vol. 5, No. 3 On quantizing g ravity and geometrizing quantum mechanics 55 order ordina ry differential equation defined on the As seen from the form of the classical Hamiltonian ~ Hilbert space H := L2 (Rn) by identifying D with a α′ - constraints (8) and (10 ), α is a time -like coordinate in ~ the minisuperspace. This has made the variable α′ that dependent Hermitian o perators D acting in H ′ . ~ appears in the Wheeler-DeWitt eq. ( 6) a popular choice Introducing ψ (α′) :=ψ (α′,.)∈ H , we then have for a time -variable in quantum cosmology. A surprising 2 recent result [ 18 ] is that no matter which positive - [∂ +D]ψ (α′) = 0. (7) α ′ definite inner product one endows the physical Hilbert Note that we label the coordinate operators acting in the space H with, the α′ -translations correspond to non - ~ 2 n ˆ Hilbert space H = L (R ) by βi and identify their unitary operators acting in H . This is a concrete common eigenvectors and the corresponding eigenvalues evidence that one cannot use α′ as a physical time by β ′ and βi′ . Then we have variable. Indeed the rather involved analysis of the Hil bert -space problem for the Wheeler-De Witt eq. ( 6) ψ (α′, β ′,..., β ′ ) = β′ ψ (α′) , 1 n shows that the problem has a rather trivial solution. One where we use . . to denote the L2 -inner pro duct on can identify the space of solutions H with the space ~ 2 n 2 n H . L (R )× L (R ) of initial conditions ψ (α0′ ),∂α ′ψ α0′ )( As an example, consider a n FR W model coupled to a of the W heeler-DeWitt eq. ( 7) and endow the vector real scalar field ϕ . Then (in natural units [ 15 ]) the space. L2 (Rn)× L2 (Rn) with a positive -definite inner classical Hami ltonian constraint has the form product. For example one may take the direct sum inner 2 2 4α 6α 2 n 2 n K = − pα + pϕ − κ e + e V (ϕ) = 0 , (8) product, i.e., i dentify H with L (R ) ⊕ L (R ) The choice of the initial value α′ of α′ and the choice of where α is the logarithm of t he scale factor: a = eα ,κ 0 2 n 2 n is the curvature index (= -1, 0, 1 for open, flat, and the inner product on L (R )× L (R ) are physically closed universes, respectively), and V is the self - irrelevant, for up to unitary equivalence H has a unique separable Hil bert space structure [ 19 ]. interaction potential for the field ϕ . In this case, all βi , The explicit form of an inner product on can be except β = ϕ are absent, and the corresponding H 6 easily written down [ 18 ] and physical observables may Wheeler -DeWitt equation is given by eq. ( 6) with be constructed. For example, there is an observable εˆ 2 4α ′ 6α ′ D = −∂ϕ′ − κe + e V (ϕ′). (9) that squares to 1, i .e., it has ±1 as eigenvalues . A much Another example is the mixmaster model [ 16 ,17 ] more difficult task is to formulate a correspondence whose Hamiltonian constrain t has the form principle that would associate classical observables to 2 2 2 4α the obtained quantum mechanical observabl es. This K = − pα + p + p + e [V (β1, β2 ) −1] = 0 , (10 ) β1 β2 requires certain amount of speculation. In Ref. [ 18 ], we where α is again the logarithm of the scale factor have outlined a proposal in which the classical analog of the quantum observable εˆ is the sign of the derivative of (a = eα ) that together with β and β parameterize the 1 2 the classical scale factor with respect to any class ical 3-metric, physical time variable. In this formulation the    β1 + 3β2 0 0  eigenvalues ±1 of εˆ correspond to the expanding and retracting universes, and in general the state vector for (g ) = e2α  0 β − 3β 0  , ij  1 2  the universe will have expanding and retracting  0 0 − 2β1  components.   In [ 18 ] w e also outline a proposal for the formulation βi with i> 2 are absent, and of the dynamics of the theory. It is based on the 1 4 V (β , β ) := e−8β1 − e−2β1 cosh(2 3β ) + Schr ödinger time -evolution determined by a 1 2 3 3 2 Hamiltonian operator H acting in H that is obtained by 2 quantizing an associated reduced classic al Hamiltonian. 1+ e4β1 [cosh(4 3β ) −1]. (11 ) 3 2 In practice, this involves selecting a classical time - Again the associated Wheeler-DeWitt equation is given variable, finding a corresponding reduced classical by eq. ( 6) with Hamiltonian that would yield the classical dynamical 2 2 4α ′ equation for this time -variable, and finally quantizing the D = −∂ − ∂ + e [V (β1′, β2′ ) −1]. (12 ) β1′ β2′ latter to obtain H. Note that in these examples, and more generally for We wish to emphasize that the Hilbert space problem other spati ally homogeneous cosmological models, the is not a major obstacle in quantizing minisuperspace operator D appearing in the Wheeler-DeWitt equation is models. The difficulty lies in the formulation of a identical with the standard Hamiltonian operator for a correspondence principle that associates to each quantum non -relativistic particle moving in Rn and interacting observable a classical counterpart. This is also of basic with a certain α′ -dependent potential. importance for our proposal [ 18 ] for defining the dynamics. The theory is still far from satisfactory for it is 56 Ali Mostafazadeh IJPR Vol . 5, No. 3 plagued with the notorious multiple choice problem Furthermore, the requirement that .,. be positive - [5,6]: Different choices for the classical time variable η seem to lead to dif ferent quantum systems, and it is not definite, i.e., for all nonzero ψ ∈ Η , ψ ,ψ η > 0 , clear how one should avoid or interpret this apparent implies that η must be a positive -definite operator (its violation of the time -reparameterization invariance. A spe ctrum must be strictly positive.) possible way out is to identify an equivalence relation between these quantum systems and associate physical The operator η may be conveniently used to reality to the quantities that are common to all members quantify the choice of the geometry of the Hilbert space; of a given equivalence class. The feasibility of this it is referred to as a metric operator. In the following we approach can be decided only after a detailed will take the inner product . . of Η as a time - investigation of some concrete models. Unfortunately, independent reference inner product and characterize the this has not been possible even for the simpl e models dynamical geometries of the H il bert space in terms of such as eqs. (8) and (10 ) for technical reasons . time -dependent metric operators. Now, suppose that the dynamics of a quantum system 3. Hil bert space with changing geometry is dete rmined through the Schr ödinger equation: What made Albert Einstein the most influential scientist d of the 20 th century was that not only he had the courage i ψ (t) = ψ (tΗ ) , (15 ) dt to unify the concepts of time and space, but he was willing to consider the revolutionary idea of making the where H : H → H is a given possibly time -dependent geometry of spacetime dynamical. and not necessarily Hermitian operator. Then the Recently, within a totally remote subject, it was condition that H generates a unitary time -evolution for a discovered that one could devise a unitary quantum ti me -dependent inner product defined by the metric system using a seemingly non -Hermitian Hamiltonian, operator η t)( i.e. , requiring that for any pair 2 3 d suc h as Η p += ix , provided that its spectrum was ξ (t) , ζ (t) of solutions of eq. ( 15 ), ξ,ζ = 0 , ( ) η(t) real [ 20 ,21 ]. This was only possible at the expense of dt adopting a non -standard inner product on the we find d Hilbert space of the system. The ensuing theory is itη()= Ηη† () tt − ηΗ (). (16 ) termed as pseudo -Hermitian quantum mechanic s dt [20 , 22 , 23 , 24 , 25 , 26 ]. It provides a general method of This is the n on -Hermitian extension of the Liouville -von constructing the most general inner product that renders Neumann equation. It reduces to the following Liouville - the Hamiltonian self -adjoint and consequently restores von Neumann equation for a Hermitian Hamiltonian H: the unitarity of the dynamics. It has direct applications in d i η(t) = [Η ,η]. (17 ) relativistic quantum me chanics of scalar fields and dt quantum cosmology [ 18 , 27 , 28 ] and many other areas This equation has two interesting ramifications. [29 ]. In particular, in its application in quantum 1. The inver tible mixed states (density matrices) used cosmology, one is forced to deal with explicitly time - in quantum statistical mechanics define acceptable dependent Hamiltonians which would generate unitary metric operators, for they are positive -definite solutions time -evolutions pr ovided that the inner product of the of the Liouville -von Neumann equation. Hilbert space be made time -dependent as well. This 2. Each acceptable metric operator (that satisfies eq. observation subsequently led the present author to (17 )) is a dynamical invariant in the sense of Lewis and investigate the consequences of having a quantum Riesenfeld, [ 32 ]. In particular, the eigenvalues of η (t) system defined on a Hilbert space with a time -dependent inner produc t and hence a dynamical geometry [ 30 ]. The are time -independent and one can obtain a complete set resemblance to the passage from special to general of solutions of the Schrödinger eq . ( 15 ) out of relativity needs no clarification. eigenvectors of η t)( . The geometry of a Hilbert space is determined by its The connection between metric operators and inner product. It is well -known [ 31 ] that any inner dynamical invariants is particularly interesting because product (.| .) on a given vector space ν may be obtained there is a well -known formulation of geometric phases in terms of dynamical invariants [ 33 ]. In view of results of from a given inner product . . according to [34 ], it is the metric operator η t)( that determines the (.| .) . η. == : .,. , (13 ) η geometric phases. This observation reveals, in a rather where η :ν →ν is an invertible operator satisfying direct manner, the geometric nature of these phases without appealing to the conventional geometric . η. = η. . . (14 ) formulation based on the classifying U (1) princ ipal That is if we label the inner product (Hilbert) space bundle over the projective Hilbert space [11 , 35 ]. obtained by endowing ν with . . by H , then as an Another interesting conclusion that may be drawn out of the above -described connection is that one can operator acting in , η is self -adjoint or Hermitian. H postulate a different or more general dynamical equation IJPR Vol. 5, No. 3 On quantizing g ravity and geometrizing quantum mechanics 57 for the metric operator, determine the geometric phases In connection with of GR, we for evolving states using the eigenvectors of the solution argued that at least for the simplified quantum of this equation, and postulate a method to identify the cosmological models the H il bert -space problem may be dynamical phases for an evolving state. This would lead avoided by trying to d irectly endow the solution space of to a generalization of quantum mechanics which shares the Wheeler-DeWitt equation with an arbitrary inner the same geo metric structure. In Ref. [ 30 ] we offer a product. The difficulty lies with interpreting the quantum proposal for such a generalization based on Lindbald’s observables or alternatively relating them via a master equation [ 36 ]. correspondence principle to the classical observables. Finally, given a Hamiltonian there are an infinity of This approach is the opposite of the one taken in the so - acceptable metric operators (parameterized by the initial called group -averaging scheme [ 37 ]. It has the advantage value η0 of η(t) ). There is a permutation group G that of being explicit and easy to compute [ 18 ]. relates this metric operators. The freedom to choose η On the subject of the geometrization of QM, we 0 elucidated the role of the geometry of the Hilbert spa ce is a rather trivial quantum mechanical analogue of the and showed that it was possible to consider quantum diffeomorphism invariance of GR. Similarly, the systems with dynamical H il bert spaces. After all, the perm utation group G plays in QM a similar role as the geometry of the Hilbert space is as unobservable as the diffeomorphism group does in GR. state vectors themselves. What is observable is the The natural extension of the ideas presented above is transition amplitudes and expectation v alues which to attempt at introducing a particular type of nonlinearity involve both the state vector and the metric operator. in QM by allowing the metric operator to be state - Promoting the role of the geometry of the Hilbert space depende nt. Whether and how this can be achieved is the from an idle observer to a real player in the quantum subject of a future investigation. game reveals some remarkable resemblances between QM and GR. It also opens up t he way to formulate a 4. Conclusion nonlinear extension of QM which might be more Quantization of GR and geometrization of QM are both convenient to be unified with GR in a yet-to -be - as attractive and intriguing subjects for research as they discovered consistent quantum theory of gravity. were initially considered in the 1960 s and 197 0s. In this paper we elaborated on some recent developments in Acknowledgment these subjects. I wish to thank Nematollah Riazi for inviting me to contribute to this special issue of IJPR .

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