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  • Yahya Ashraf

A brief introduction to the unified field theory



Despite being very obvious on human scales, gravity remains a very enigmatic physical phenomenon. Since the eve of scientific thought, philosophers and scientists alike have pursued a greater understanding of gravity, but have never succeeded in reaching a fundamental theory that explains the mechanisms underlying the force. There have been a variety of approaches to quantising the gravitational field, including, but not limited to, loop quantum gravity (LQG) and string theory. These are the two most popular approaches, with significant amounts of research and this blog tries to explain the history of Loop quantum gravity.


Motivation to pursue a quantum theory of gravity


Quantisation refers to the process of taking something continuous (in the case of gravity this “something” is spacetime itself) and making it discrete, or “quantised”. Classical physics, such as mechanics, electromagnetism and relativity, are almost exclusively concerned with continuous variables. In contrast, quantum mechanics deals with bodies where variables, such as energy and angular velocity are discrete, and can only take on certain values. LQG and string theory rely on the quantisation of the gravitational field even though the methods through which they achieve it is different. There are several reasons put forward as motivation for the quantisation of gravity. Situations in black hole physics and in early universe models looking at the Big Bang, mandate a theory of quantum gravity, due to the existence of singularities. Singularities are points in spacetime where the curvature (or gravity as shown later) becomes infinite, and the laws of physics break down. These are a consequence of contradictions within general relativity and quantum mechanics, and because both need to be used in conjunction in these fields, it is absolutely required that these contradictions be solved. This is the primary motive for pursuing a quantum theory of gravity. Another motivation for the quantisation of gravity is scientific reductionism. This philosophy considers the idea that it should be possible to reduce complex interactions, as well as the theories that explain these phenomena, to a simpler, more fundamental theory. In a sense, scientific reductionism has culminated in the creation of the standard model, which is able to explain a large proportion of modern science, including most of quantum and classical physics - although as already noted it cannot explain gravitational effects.


General Relativity


General relativity (GR), the most contemporary classical theory of gravity, is an important step in our understanding of gravitational physics. In previous theories, such as Newton’s theory of universal gravitation, gravity was thought to be a force capable of acting instantaneously over long distances. It was defined as being the force between two masses, which is inversely proportional to the square of the distance between them - i.e the further the masses are from each other, the less force is applied to each of them by the other.

However, in general relativity, this is no longer the case. Instead, Einstein postulated that gravitation is not a force, and is instead the warping of spacetime around a mass. This also means that it does not act instantaneously, but rather at the speed of light.

More accurately, spacetime is defined as a manifold with four dimensions - three spatial dimensions and a one-time dimension. A manifold is a topological space that locally resembles “flat” Euclidean space. General relativity uses tools such as differential geometry and tensor calculus to define useful functions that can be applied to the manifold. One such measure is known as the metric (or the metric tensor), which in GR is broadly analogous to the gravitational potential of Newtonian gravitation.


Quantum Mechanics


Quantum mechanics, and by extension quantum theory, is a cornerstone of modern physics. The theory arose from what is known as the “ultraviolet catastrophe”. In order to solve the resulting contradiction between the classical model (where at UV wavelengths, the intensity of the radiation predicted by the classical models was infinite.) and observations which showed otherwise, Plank found that the phenomenon could be explained by a particle model, in which the energy of the photon (a particle of light) was proportional to the frequency: E = hf.

Over time, this theory was developed into what is now considered to be some of the most foundational ideas in physics, mostly confined to the study of very small scales. Particularly relevant to this discussion is the creation of quantum field theory (QFT). This esoteric theory is the basis for our understanding of the fundamental forces of nature. It led to the construction of the standard model of particle physics. However, one key development is a process known as renormalisation. A feature of many calculations within QFT is that parts of the calculation (terms) diverge to infinity - that is, they do not converge on a single solution, and instead give out infinite answers. This is a clear problem, as in reality, no physical quantity can be infinite.

Renormalisation, therefore, is the method by which these terms are made non-infinite. The main way this is done is by altering these terms to account for self-interaction. One of the significant issues with quantum gravity, however, is that it is non-renormalisable.


LQG


One theory of quantum gravity that seeks to remedy the issues surrounding non-renormalisation is loop quantum gravity (LQG). Its basis is rooted in general relativity, and it attempts to implement parts of quantum mechanics. Within the theory there are two fields of study - covariant and canonical LQG.

The development of LQG began from a reformulation of general relativity, known as the ADM Hamiltonian form. From this, it was realised that it was possible to further reformulate the theory, which culminated in a form known as the Wheeler-DeWitt equation. Modifications made by scientists such as Ashtekar and Sen led to the construction of Ashtekar variables, which allowed general relativity to be written in a simplified form. From this, it was realised that the Wheeler-DeWitt equation would give out solutions in terms of loops. This formed the foundation of both the canonical and covariant approaches to LQG.

As of today, Loop quantum gravity offers insight into aspects of cosmology, which indicates there is a direct application. For instance, it is possible to derive the equation for the BekensteinHawking entropy of a black hole, and progress has been made on other features of cosmology. Equally, another significant consequence of LQG is the resolution of singularities. The theory posits that singularities of any kind are not possible. In the case of a non-rotating black hole, the singularity is replaced by a transition surface, which then resolves itself into a “white hole”. A derivation for a rotating black hole model within LQG has also been found, and it may be possible to test the model using observational data. Other singularities, such as that of the Big Bang, have been shown to be the result of a “big bounce” where a previous universe collapsed inwards until it hit a certain length scale before it transitioned into the early universe.


String theory


String theory is another theory of quantum gravity, which has its basis in quantum mechanics. It builds upon existing knowledge from quantum field theory, including the standard model. String theory represents a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. For example, take a piece of wood. Inside it, you would find millions of atoms bonded together. Inside every atom, there is a nucleus. The atomic nucleus consists of nucleons—protons and neutrons. Protons and neutrons are made of quarks. The string theory states that inside quarks there are strings which vibrate at different frequencies causing the property of the substance. The different vibrational patterns of the string cause the specified property.

One application of string theory that is being heavily researched presently is that of black holes. These astronomical objects, which arise from general relativity, are not well explored. It has only recently been realised that there are different types of black holes - there are ones that are static, spinning and ones that have an electric charge. A realisation made by string theory is that it is possible to relate black holes to elementary particles. This at first seems unlikely, since there is a significant size difference between the two. However, these same fundamental properties (mass, charge and spin) are manifest in both situations and are the only things needed to completely describe black holes and elementary particles. In addition, unlike common descriptions, black holes are not perfectly black. A discovery made by Bekenstein and Hawking found that black holes emit radiation (this has now been derived from LQG).