2 Gravity/String Quantization Schemes

There is an intriguing similarity between the problem of quantizing gravity, as described by General Relativity, and that of quantizing a relativistic string, or any higher dimensional relativistic extended object. In either case, one can follow one of two main routes:

1. a quantum field theory inspired-covariant quantization, or
2. a canonical quantization of the Schrödinger type.

The basic idea underlying the covariant approach is to consider the metric tensor $g _{\mu \nu} (x)$ as an ordinary matter field and follow the standard quantization procedure, namely, Fourier analyze small fluctuation around a classical background configuration and give the Fourier coefficients the meaning of creation/annihilation operators of the gravitational field quanta, the gravitons. In the same fashion, quantization of the string world-sheet fluctuations leads to a whole spectrum of particles with different values of mass and spin. Put briefly,

$$g _{\mu \nu} (x) = \hbox{background} + \hbox{\lq\lq graviton''}$$

$$X ^{\mu} (\tau , s) = \hbox{zero-mode} + \hbox{particle spectrum} \quad .$$

Against this background, one may elect to forgo the full covariance of the quantum theory of gravity in favor of the more restricted symmetry under transformations preserving the ``canonical spacetime slicing'' into a one-parameter family of space-like three-surfaces. This splitting of space and time amounts to selecting the spatial components of the metric, modulo three-space reparametrizations, as the gravitational degrees of freedom to be quantized. This approach focuses on the quantum mechanical description of the space itself, rather than the corpuscular content of the gravitational field. Then, the quantum state of the spatial $3$-geometry is controlled by the Wheeler-DeWitt equation

$$\left[ \hbox{Wheeler-DeWitt operator} \right] \Psi[G _{3}] = 0$$ (1)

and the wave functional $\Psi [G _{3}]$, the wave function of the universe, assigns a probability amplitude to each allowed three geometry.

The relation between the two quantization schemes is akin to the relationship in particle dynamics between first quantization, formulated in terms of single particle wave functions along with the corresponding Schrödinger equation, and the second quantization expressed in terms of creation/annihilation operators along with the corresponding field equations. Thus, covariant quantum gravity is, conceptually, a second quantization framework for calculating amplitudes, cross sections, mean life,etc., for any physical process involving graviton exchange. Canonical quantum gravity, on the other hand, is a Schrödinger-type first quantization framework, which assigns a probability amplitude for any allowed geometric configuration of three dimensional physical space. It must be emphasized that there is no immediate relationship between the graviton field and the wave function of the universe. Indeed, even if one elevates $\Psi [G _{3}]$ to the role of field operator, it would create or destroy entire three surfaces instead of single gravitons. In a more pictorial language, the wave functional $\Psi [G _{3}]$ becomes a quantum operator creating/destroying full universes! Thus, in order to avoid confusion with quantum gravity as the ``theory of gravitons'', the quantum field theory of universes is referred to as the third quantization scheme, and has been investigated some years ago mostly in connection with the cosmological constant problem.
Of course, as far as gravity is concerned, any quantization scheme is affected by severe problems: perturbative covariant quantization of General Relativity is not renormalizable, while the intrinsically non-perturbative Wheeler-DeWitt equations can be solved only under extreme simplification such as the mini-superspace approximation. These shortcomings provided the impetus toward the formulation of superstring theory as the only consistent quantization scheme which accommodates the graviton in its (second quantized) particle spectrum. Thus, according to the prevalent way of thinking, there is no compelling reason, nor clear cut procedure to formulate a first quantized quantum mechanics of relativistic extended objects. In the case of strings, this attitude is deeply rooted in the conventional interpretation of the world-sheet coordinates $X ^{\mu} (\tau , s)$ as a ``multiplet of scalar fields'' defined over a $2$-dimensional manifold covered by the $\{ \tau , s \}$ coordinate mesh. According to this point of view, quantizing a relativistic string is formally equivalent to quantizing a two-dimensional field theory, bypassing a preliminary quantum mechanical formulation. However, there are at least two objections against this kind of reasoning. The first follows from the analogy between the canonical formulation of General Relativity and $3$-brane dynamics, and the second objection follows from the ``Schrödinger representation'' of quantum field theory. More specifically:

1. the Wheeler-DeWitt equation can be interpreted, in a modern perspective, as the wave equation for the orbit of a relativistic $3$-brane; in this perspective, then, why not conceive of a similar equation for a $1$-brane?
2. the functional Schrödinger representation of quantum field theory assigns a probability amplitude to each field configuration over a space-like slice $t = const.$, and the corresponding wave function obeys a functional Schrödinger-type equation.

Pushing the above arguments to their natural conclusion, we are led to entertaining the interesting possibility of formulating a functional quantum mechanics for strings and other $p$-branes. This approach has received scant attention in the mainstream work about quantum string theory, presumably because it requires an explicit breaking of the celebrated reparametrization invariance, which is the distinctive symmetry of relativistic extended objects.

All of the above reasoning leads us to the central question that we wish to analyze, namely: is there any way to formulate a reparametrization invariant string quantum mechanics?
As a matter of fact, a possible answer was suggested by T. Eguchi as early as $1940$ [7], and our own elaboration of that quantization scheme [5] is the topic of Section 3.