1 Introduction

At its most fundamental level, research in theoretical high energy physics means research about the nature of mass and energy, and ultimately about the structure of space and time. It may even be argued that the whole history of physics, to a large extent, represents the history of the ever changing notion of space and time in response to our ability to probe infinitesimally small distance scales as well as larger and larger cosmological distances.

Figure: History of physics shows that conflicting theories eventually merge into a broader and deeper synthesis. Will $M$-Theory lead to a unique supersynthesis of quantum theory, gravity theory and supersymmetry?

The ``flow chart'' in Figure 1 summarizes the dialectic process which has led, through nearly twenty five hundred years of philosophical speculation and scientific inquiry, to the current theoretical efforts in search of a supersynthesis of the two conflicting paradigms of $20$-th century physics, namely, the Theory of General Relativity and Quantum Theory. In that Hegelian perspective of the history of physics, such a supersynthesis is regarded by many as the ``holy grail'' of contemporary high energy physics. However, the story of the many efforts towards the formulation of that synthesis, from supergravity to superbranes, constitutes, in itself, a fascinating page in the history of theoretical physics at the threshold of the new millennium. The early `$80$s excitement about string theory (``The First String Revolution'') followed from the prediction that only the gauge groups $SO (32)$ and $E(8) \otimes E(8)$ provide a quantum mechanically consistent, i.e., anomaly free, unified theory which includes gravity [1], and yet is capable, at least in principle, of reproducing the standard electro-weak theory below the GUT scale. However, several fundamental questions were left unanswered. Perhaps, the most prominent one regards the choice of the compactification scheme required to bridge the gap between the multi-dimensional, near-Planckian string-world, and the low energy, four dimensional universe we live in [2]. Some related problems, such as the vanishing of the cosmological constant (is it really vanishing, after all?) and the breaking of supersymmetry were also left without a satisfactory answer. The common feature of all these unsolved problems is their intrinsically non-perturbative character. More or less ten years after the First String Revolution, the second one, which is still in progress, has offered a second important clue into the nature of the superworld. The diagram in Figure 1 encapsulates the essential pieces of a vast mosaic out of which the final theory of the superworld will eventually emerge. Among those pieces, the six surviving viable supermodels known at present, initially thought to be candidates for the role of a fundamental Theory of Everything, are now regarded as different asymptotic realizations, linked by a web of dualities, of a unique and fundamentally new paradigm of physics which goes under the name of $M$-Theory [3]. The essential components of this underlying matrix theory appear to be string-like objects as well as other types of extendons, e.g., $p$-branes, $D$-branes, ..., (any letter)-branes. Moreover, a new computational approach is taking shape which is based on the idea of trading off the strongly coupled regime of a supermodel with the weakly coupled regime of a different model through a systematic use of dualities.
Having said that, the fact remains that $M$-theory, at present, is little more than a name for a mysterious supertheory yet to be fully formulated. In particular, we have no clue as to what radical modification it will bring to the notion of spacetime in the short distance regime. In the meantime, it seems reasonable to attempt to isolate the essential elements of such non-perturbative approach to the dynamics of extended objects. One such approach that we have developed over the last few years [4], [5], [6], is a refinement of an early formulation of quantum string theory by T. Eguchi [7], elaborated by following a formal analogy with a Jacobi-type formulation of the canonical quantization of gravity.
Thus, our immediate objective, in the following Section, is to illustrate the precise meaning of that analogy. In Sections 3.1 and 3.2 we discuss our quantum mechanical elaboration of Eguchi's approach in terms of "areal'' string variables, string propagators and string wave functionals. This discussion, which can be easily extended to $p$-branes of higher dimensionality, enables us to exemplify a possible relationship between $M$-theory and the quantum mechanics of string loops in section 4. Section 5 is divided into two subsections where we discuss the functional Schroedinger equation of "loop quantum mechanics" and its solutions in order to derive the Uncertainty Principle for strings as well as its principal consequence, namely, the fractalization of quantum spacetime (Subsection 5.1). We then conclude our discussion of the structure of spacetime in terms of an effective lagrangian based on a covariant, functional extension of the Ginzburg-Landau model of superconductivity (Subsection 5.2).