You will find truth more quickly through delight than gravity. Let out a little more string on your kite. ~ Alan Cohen!
General Relativity and quantum field theory are by far the most successful scientific theories. They are however, if not impossible, extremely hard to reconcile and the main obstacle has to do with renormalizability. In a nutshell, unlike all other particles, whether in a first or second-quantizational context, the graviton does not respond to charge directly: it is massless, because the gravitational force theoretically has unlimited range and must be a spin-2 boson, and so, responds directly to mass and energy, and since they carry energy, they respond to themselves: hence, they self-gravitate. The problem with that is quantum physics ‘tells’ us that gravitons are particles as well as waves: but, and that will be a central issue, particles are theoretically construed as 0-dimensional pointlike entities; so a pointlike graviton gravitates more strongly as you get closer and closer to it. But its gravitational field can only be interpreted as the emission of other gravitons (call those ‘children’-gravitons and the original one the ‘parent’). The gravitational field very close to the ‘parent’ graviton is extremely strong: that implies that the ‘children’ gravitons have enormous energy and momenta, by the uncertainty relation. The trouble now is that ‘children’ gravitons will also themselves emit gravitons (‘grand-children’), and the whole process, by non-renormalizability, runs away and one cannot keep track of the effects of such hyper-exponential gravitonic emissions, and one has to deal with actual infinities that cannot be eliminated. There is good news: enter String Theory.
The key philosophical point is that ontologically, particles are not interpreted as 0-dimensional manifoidal ‘points’ in an abstract mathematical structure, but as vibrational modes of strings that have non-zero dimension in actual physical ‘space’ and ‘time’. As strings dynamically ‘move’ – propagate – through space and time, they define a world-sheet which is isomorphic to a 2-dimensional Riemannian manifold, denoted by , where the and represent the string space and time coordinates that define the world-sheet, which is a two-dimensional manifold describing the embedding of a string in spacetime. Now, and are subject to quantum fluctuation and the only way these fluctuations can be consistent with a quantization of is if spacetime satisfies the equations of General Relativity – hence, the existence of , where is the string world sheet, plus quantum physics implies a finite theory of quantum gravity: QP + = Quantum Theory of Gravity. This seems magical and it is. Let us dig deeper though.
Recall that energy and time are related by the Heisenberg uncertainty principle: let denote that relation. On the tangent bundle of , in 2nd-quantization form, vectors and where represent the spacetime coordinates of have, by , an ambiguation problem: ‘t’ and ‘s’ as they occur in do not satisfy the quantum fluctuation of since only makes sense in a classical GR ontological framework of 0-dimensional point-particality, whereas ‘t’ and ‘s’ as they occur in and are non-classical and imply non-zero-dimensionality of time and space. What gives? A deeper look always repays. In the QP + setting, the metric is , deriving the area gives us
where the Lagrangian is given by
Now note that by , after disambiguation of the spatial coordinates as they occur in , by substituting in and , as one must, in order for the supersymmetric covariant derivative(s)
to determine the dynamics of string …