The Shoreline'14 April, 2014 | Page 40

quantum setting. Following WWII, rapid discoveries of a whole zoo of new elementary particles led theorists to postulate the existence of two more forces of nature - in addition to the already known electromagnetic and gravitational forces - known as the “weak” and “strong” forces. Theoretical work by giants of 20th century physics, such as Feynman, Gell-Mann and Gerard ’t Hooft among many others, led to a unified description of the weak force and electromagnetism in a framework known as the “electroweak” theory. This process culminated in the last quarter of the 20th century with the establishment of the Standard Model of particle physics which provides a unified - albeit, in some ways flawed - description of the weak, strong and electromagnetic forces as excitations of the quantum mechanical “vacuum”. Gravity, however, remains outside the grasp of such unified frameworks. Consequently we are left in the awkward situation where the most complete formulation of physical law must be written in the form: Standard Model + Gravity. Superpositions of Geometry The challenge in unifying gravity with quantum mechanics can be understood in the following way. The central feature of quantum mechanics is the principle of superposition; that the wavefunctions which describe two different particles or systems can overlap. Consequently two systems described by two different wavefunctions and can instead be treated as a single composite system with wavefunction Wavefunctions are just that - functions - and must therefore be defined on some set. In quantum mechanics the set is taken to be the co-ordinates (x,t) of the spacetime our system is embedded in. So we can always write down what the wavefunction of a particle passing through two slits at the same time must look like. Similarly we can write down the superposition of two particles in terms of their momenta, rather than their position, by working in the (p,t) basis instead of the (x,t) basis with the momenta p being related to the position x by the usual fourier transform: Regardless of whether we work in the position basis or the momentum basis, there is an implicit assumption at work in this prescription - the assumption of a flat background geometry for which we can assign a set of co-ordinates (x,y,z,t) to each point of the spacetime. Now, General Relativity teaches us that physics should be independent of the particular co-ordinates used to describe a system. Moreover, any theory which is consistent with GR must also be well-defined both on curved space as on flat space. It turns out that while we can perform quantum mechanical calculations to our heart’s content with wavefunctions defined on flat space, the case of wavefunctions living on curved space becomes tricky. The complications associated with spacetime curvature can be dealt with by resorting to sufficiently sophisticated mathematical methods. The resulting framework is known as Quantum Field Theory on Curved Spacetime (QFT-CS) – not a terribly memorable phrase. It was by using the methods of QFT-CS that Stephen Hawking obtained his historic result that a blac k hole must emit thermal radiation at a rate inversely proportional to its mass. However, even QFT-CS does not qualify as a theory of “quantum gravity”, as explained next. Quantum Mechanics is about assigning various attributes to a system and then constructing states of the system corresponding to each attribute. These states can be superimposed and the result- 38 The Shoreline ing system will manifest all non-intuitive phenomena associated with quantum behaviour such as interference and entanglement. As mentioned previously, gravity in the modern conception arises from the non-trivial geometry of a region of spacetime induced by some distribution of matter. Some of the geometric attributes that one can assign to a given region of spacetime are length, area, volume, angles etc. A theory of quantum gravity should be able to tell us how to write down the wavefunction, not defined on a given region of spacetime, but a wavefunction of a given region of spacetime, allowing us to construct states which correspond to superpositions of different geometries. However, as mentioned previously traditional quantum mechanics tells us only how to write down the wavefunction on a given geometry rather than of a given geometry. The traditional language of quantum mechanics is thus insufficient to describe quantum states of geometry. For the same reasons QFT-CS is also not a theory of “quantum gravity”. There the curved spacetime merely serves as an arena on which quantum states can be defined, but there is no notion of states of the geometry itself, rather than of the matter which moves about on that geometry. At present there are several approaches towards tackling the open question of writing quantum states of geometry. These include String Theory, Loop Quantum Gravity and Causal Dynamical Triangulations among others. Most laypersons with an interest in science have heard of String Theory, simply because it is the oldest of these approaches and thus also the most widely taught and practiced. LQG was born about a decade after String Theory and has only recently reached a level of maturity and acceptability as a valid physical theory. It would take us far afield to go into details - even at a non-technical level - of these approaches and their similarities and differences. I will try to briefly summarize the two approaches and the basic idea behind each one. New Paradigms The idea behind String Theory is that instead of a description of fundamental particles as point-like objects we should switch to a picture where the basic entities are extended one-dimensional objects called strings. These strings move and interact in some background spacetime. Requirements of physical and theoretical consistency restrict the number of dimensions of the spacetime in which strings can live to 26, 11 and 10 depending on the particular characteristics - fermionic, bosonic, open and closed - we choose to endow the strings with. The excitations of a string happen to include a part which can be identified with gravitons - which are excitations of the background geometry the string is propagating in. Though gravitons are often thought of as the quanta of the gravitational field, in the same way that photons are quanta of the electromagnetic field, this belief is only partially correct. As mentioned in previous sections, the gravitational field is characterized by geometric attributes such as lengths, areas and volumes. Therefore, quanta of the gravitational field should correspond to quantized lengths, areas and volumes, in the same way that a quantum of the electromagnetic field corresponds to a quantized amount of energy given by Planck’s relationship between the energy of a photon and its frequency, E=hv. However, the graviton picture does not predict any such relations - such as the area of a given region of spacetime and the frequency of a gravitational wave which passes through that region - between any fundamental geometric quantities and so cannot be said to provide a picture of quantum geometry. Moreover, gravitons are perturbations of the background spacetime which is, by default,