Quantum Gravity

GR demonstrates the need for a quantum theory of gravity rather directly, as it relates $R_{\mu \nu}$ to $T_{\mu \nu}$ which is a quantum operator. In the usual approach to quantize gravity one needs to select a (vacuum) state for the metric which is a solution to the classical equations of motion and expand around it:

$$g_{\mu \nu} = \tilde g_{\mu \nu} + h_{\mu \nu} ,$$ (A.1)

where $\tilde g$ is the smooth background metric field. The $h_{\mu \nu} (x)$ represent the excitations of the gravitational field, i.e. the graviton. Being related to a Lorentz tensor field, the graviton is a spin-2 particle; see note []. It is also associated with a specific representation of the Poincaré group. In essence eq. (A.1) is a perturbative expansion of the metric. If the vacuum is taken as a flat Minkowski space then $\tilde g_{\mu \nu} = \eta_{\mu \nu}$ and equivalently $\Lambda = 0$. By inserting eq. (A.1) into the Einstein equations and ignoring higher order terms in $h$, one can derive a wave equation for the graviton analogous to Maxwell's equation for the vector particles of electromagnetism. A difference is that the gravitons can self-interact. Problems arise because the theory is non-renormalizable -- a death sentence for perturbative theories. Also, the canonical quantization schemes employing commutation relations of the field operators (seen in the next section)

$$[ h_{\mu \nu} (x) , h_{\mu \nu} (x^\prime) ],$$

break down because the concept of distance and intervals is only defined within the context of a (background) metric which now is also an operator. Not even the idea of semi-classical quantum gravity, i.e. retaining the gravitational field as a classical background and quantizing matter fields in the usual way, hence giving rise to the subject of QFT in a curved background space-time, gives real insight; [BiDa82]. The core issue is addressed by Ashtekar: `we must learn to do physics in absence of a background space-time geometry'; [AL96]. Unfortunately also string/M-theory has not yet been formulated in a background independent manner. In the words of Witten in [Wi92/93]: `finding the right framework for an intrinsic, background independent formulation of string theory is one of the main problems in string theory and so far has remained out of reach.' Further: `this problem is fundamental because it is here that one really has to address the question of what kind of geometrical object the string represents.' The idea that the relationship of an object should only depend on other objects and not on a background space is in fact somewhat familiar to Mach's principle, see item one of note [].

Nevertheless, the three most prominent approaches to non-perturbative quantum gravity, which are viewed as being very probable solutions, are:

1. String/M-theory (mentioned foremost in notes [1], [], [3], [], [], [], [], [], [13] and the second and third section of appendix A.7).
2. Ashtekar's new variables theory (item six of note []).
3. Quantum geometry.

Interestingly all three theories are believed to be linked to one another and so these three alternative lanes to unification may in fact be deeply related. This is the topic of two recent books; [Sm00] and [CH01]. However, the amount of work inspired by these subjects hints at the fantastic intricacies faced by this class of theory.

In 1995 Smolin and Rovelli took Ashtekar's non-perturbative approach a step further by introducing the ideas of so-called spin networks, developed by Penrose in the 1960s (see [Maj99]); [RoSm95] and [Sm97]. It is interesting to note that spin networks also appear in gauge theories, topological QFT's, conformal field theories and knot theory; compare with appendix B. But there is also a direct link to section 3 of appendix A.7 given by the fact that the so-called loop representation of quantum gravity (or loop gravity) yields a discrete picture of space-time; [BMS95]. The loop representation is a formulation of QFT suitable when the degrees of freedom of the theory are given by a gauge field or a connection, as is the case with the new variables theory. Spin networks can be employed to characterize quantum gravity in the loop representation.

There are many approaches to non-perturbative quantum geometry. Penrose's twistor theory developed in 1986 is an example; [PR86], [WW90] and [HT94] . Ashtekar bases his line of attack on canonical quantization and the absence of any background geometry. Interestingly, ideas originating in twistor theory and spin networks play a key role in this version of quantum geometry. String/M-theory also modifies the (general) relativistic notion that space-time is a smooth manifold. The geometrical structure used to describe the physics of string/M-theory is also referred to as quantum geometry, to note its distinctiveness from classical (differential) geometry. This term is somewhat arbitrary, as it depends upon which probe is used to study the underlying geometry. Hence a string or a D-brane will reveal a different geometry, in effect concluding the rich structure of the quantum geometry associated with string/M-theory. In addition, it is possible to study a kind of classical geometry arising within the context of non-quantum string theory, sometimes referred to as `classical stringy geometry'. So quantum geometry in this context is understood as the new geometrical discipline whose basic ingredients are the observables of string/M-theory, which in an appropriate limiting case reduces to standard geometry (algebraic geometry and topology). Hence the dilemma of pre-string theory, that although one had a gauge theory and a QFT for spin-1 force carriers there existed only a `gauge theory' (meaning GR) for spin-2 gravitons and no appropriate quantum field formulation, is maybe overcome. It appears that the corresponding `QFT' for gravity is quantum geometry. Consult [Gre97].

Reference [H-et.al.98] contains contributions from Ashtekar (quantum gravity and quantum geometry), Connes (non-commutative geometry), Smolin (spin networks) and in addition covers the topics of twistors among many other aspects of the geometry of space-time. Rovelli summarizes all the approaches to quantum gravity in [Ro98], among the topics are string theory, loop gravity, QFT in curved space-time, twistors, non-commutative geometry and topological QFT. Twistors are covered in [HSW99] within the context of integrable systems. Also, see [Ho95] for a note on supersymmetry and twistors. In [FGR97] an additional connection between quantum gravity and non-commutative geometry is given. It is also argued that a combination of GR and and quantum theory leads to the prediction that space-time cannot be a classical manifold and that the basic degrees of freedom should be associated with extended objects (e.g. strings). Quantum gravity and its philosophical implications are discussed in [BI99].