476 Brazilian Journal of Physics, vol. 35, no. 2B, June, 2005 Exact Uncertainty Principle and Quantization: Implications for the Gravitational Field M. Reginatto Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany Received on 23 December, 2004 The quantization of the gravitational field is discussed within the exact uncertainty approach. The method may be described as a Hamilton-Jacobi quantization of gravity. It differs from previous approaches that take the classical Hamilton-Jacobi equation as their starting point in that it incorporates some new elements, in particular the use of a formalism of ensembles in configuration space and the postulate of an exact uncertainty relation. These provide the fundamental elements needed for the transition from the classical theory to the quantum theory. 1 Introduction imalist in nature: unlike canonical quantization, no a priori assumptions regarding the existence of a Hilbert space struc- ture, linear operators, wavefunctions, etc. is required. The Perhaps one of the most fundamental distinctions between sole “nonclassical” element needed is the addition of fluctu- classical and quantum systems is that a quantum system must ations to the momentum variable. The exact uncertainty ap- obey the uncertainty principle, while a classical system is not proach is thus rather economical, in that it appears to make subject to such a limitation. The degree to which a quantum use of the minimum that is required for the description of a system is subject to this limitation is given by Heisenberg’s quantum system. uncertainty relation, which provides a bound on the minimum uncertainty of two conjugate variables such as the position x The paper is organized as follows. In the next two sec- and the momentum p of a particle: it states that the product of tions, I briefly review the case of a non-relativistic particle: in their uncertainties must be equal to or exceed a fundamental section 2, I discuss classical ensembles and derive the equa- limit proportional to Planck’s constant, ¢x¢p ¸ ~=2. This tions of motion from an ensemble Hamiltonian; in section inequality, however, is not sufficiently restrictive to provide 3, I present the derivation of the Schrodinger¨ equation using a means of going from classical mechanics to quantum me- the exact uncertainty approach. In the remaining sections, I chanics. Indeed, in most approaches to quantization, no di- discuss the application of the exact uncertainty approach to rect reference is made to the need for an uncertainty principle. gravity. A recent paper [4] provides an overview of the ex- For example, in the standard approach to canonical quantiza- act uncertainty approach in quantum mechanics and quantum tion, one introduces a Hilbert space, operators to represent gravity that emphasizes conceptual foundations. Therefore, observables, and a representation of the Poisson brackets of in the last sections I focus instead on some technical issues the fundamental conjugate variables in terms of commuta- and on some issues of interpretation. In section 4, I introduce tors of their corresponding operators. Once this structure is the Hamilton-Jacobi formulation of general relativity and for- in place, one can show that the Heisenberg uncertainty rela- mulate a theory of classical ensembles of gravitational fields; tions are indeed satisfied – but the uncertainty relations are in section 5, I apply the exact uncertainty principle to the derived, and the uncertainty principle, instead of motivating gravitational field, which leads to a Wheeler-DeWitt equa- the method of quantization, seems to be little more than a tion. In section 6, I consider some implications of the exact consequence of the mathematical formalism. uncertainty approach for the description of space-time in the quantized theory. Finally, some further remarks concerning One can, however, introduce an exact form of the uncer- the application of the exact uncertainty approach to gravity tainty principle which is strong enough to provide a means of are given in section 7. going from classical to quantum mechanics – indeed, this ex- act uncertainty principle provides the key element needed for such a transition. In particular, the assumption that a classical 2 Classical ensembles of non- ensemble of particles is subject to nonclassical momentum fluctuations, of a strength inversely proportional to uncer- relativistic particles tainty in position, leads directly from the classical equations of motion to the Schrodinger¨ equation [1, 2]. This approach To motivate the introduction of classical ensembles in con- can also be generalized and used to derive bosonic field equa- figuration space, consider the possibility that, due to either tions [3]. The exact uncertainty approach is extremely min- practical or theoretical reasons, the position of a particle is M. Reginatto 477 not known exactly. To describe this uncertainty in position, where the fluctuation field f vanishes everywhere on average. one has to introduce a probability density P (x; t) on config- Denoting the average over fluctuations by an overline, and us- uration space. The description of the physical system is then ing the relations f = 0 and p = rS, the classical ensemble one that is given in terms of a statistical ensemble of parti- energy is replaced in this case by cles. The dynamics of such an ensemble may be described Z Ã ! jrS + fj2 using a Hamiltonian formalism, in the following way. For hEi = dxP + V an ensemble of classical, non-relativistic particles of mass m 2m moving in a potential V , the correct equations of motion can Z f ¢ f be derived from the ensemble Hamiltonian = H~ + dxP c 2m Z Ã ! jrSj2 H~ [P; S] = dxP + V : (1) Thus the momentum fluctuations contribute to the average ki- c 2m netic energy of this modified classical ensemble. What conditions on the fluctuation field f lead to the The equations of motion that follow from eq. (1), quantum equations of motion? It is sufficient to consider four principles [1, 2, 3] (i) Action principle: The equations @P ±H~ @S ±H~ of motion can be derived from an action principle based on = c ; = ¡ c ; @t ±S @t ±P an ensemble Hamiltonian. Then, the strength of the fluc- tuations is determined by some function of P and S and take the form their derivatives, i.e. f ¢ f = ® (x; P; S; rP; rS; :::). It µ ¶ will also be assumed that the equations of motion that re- @P rS + r ¢ P = 0; (2) sult are partial differential equations of at most second order. @t m (ii) Independence: if the system consists of two independent non-interacting sub-systems, the ensemble Hamiltonian, and @S jrSj2 + + V = 0: (3) therefore ®, decomposes into additive subsystem contribu- @t 2m tions. (iii) Invariance: the fluctuations transform correctly T Eq. (2) is a continuity equation for an ensemble of particles under linear canonical transformations (i.e., f ! L f for described by the velocity field v = m¡1rS, and eq. (3) is any invertible linear coordinate transformation x ! Lx.A the Hamilton-Jacobi equation for a classical particle with mo- sightly different principle was used in [1, 2]). (iv) Exact mentum p = rS. Note also that H~C is the average energy of uncertainty principle: the strength of the momentum fluc- the classical ensemble. The formalism of ensemble Hamil- tuations is determined solely by the uncertainty in position. tonians that I have used here, while very simple, turns out to Since P contains all the information on the position uncer- be quite general (see [5, 4] for more details on this formalism tainty, ® can only depend on P and its derivatives. and various applications). In particular, it can be shown that The first three principles are natural on physical grounds, both classical and quantum equations of motion can be de- and therefore relatively unconstraining. The fourth principle, rived using this formalism. The difference between classical however, by establishing a precise relationship between the and quantum ensembles can be characterized by a simple dif- position uncertainty (as described by P ) and the strength of ference in the functional form of the corresponding ensemble the momentum fluctuations, introduces a strongly nonclassi- Hamiltonians, as shown in the next section. cal element. It is therefore appropriate to refer to the fluc- tuations due to the field f as nonclassical momentum fluctu- ations. It can be shown that these four principles define the 3 Quantum equations for non- functional form of ® uniquely and lead to a quantum ensem- ble Hamiltonian of the form relativistic particles Z 1 jrP j2 H~q = H~c + C dx (4) To derive quantum equations of motion, one needs to add P 2m fluctuations to the momentum variable which satisfy an ex- where C is a positive universal constant. Furthermore, since act uncertainty principle, and modify the classical ensemble the term ® is proportional to the classical Fisher informa- Hamiltonian to take into consideration the effect of these fluc- tion, one can use the Cramer-Rao´ inequality of statistical es- tuations. There are, of course, further assumptions that need timation theory to derive an exact uncertainty relation that is to be made, and these are explained in this section – however, stronger than (and implies) the Heisenberg uncertainty rela- the main conceptual ingredients are the nonclassical momen- tion (see [1, 2, 3] for derivations of these results). tum fluctuations and the exact uncertainty principle. It is straightforward to show that the Hamiltonian equa- Consider then another type of ensemble, one for which tions of motion for P and S derived from eq. (4), the assumption of a deterministic relation between position ~ ~ and momentum (implicit in the classical relation p = rS) is @P ±Hq @S ±Hq = ; = ¡ ; no longer valid. Assume instead that the momentum is sub- @t ±S @t ±P ject to stochastic perturbations, and that the relation between are identical to the Schrodinger¨ equation for a wavefunction p and rS takes the form Ã, provided one defines p p p = rS + f ~ := 2 C;Ã := P eiS=~: 478 Brazilian Journal of Physics, vol.