The Future of Quantum Field Theory Andrei Oprea March 28, 2021 1 Introduction Quantum field theory (QFT) is one of the most accurate theories in physics and is essential to understanding how the universe works. The first ever quan- tum field theory was Quantum electrodynamics discovered by Paul Dirac in the 1940s.1 In quantum electrodynamics, Dirac unified special relativity and quantum mechanics in one equation that described the electron and the elec- tromagnetic field. This led to a greater understanding of particle physics and the future development of the strong and weak forces being described as fields. Quantum field theory has expanded much since then, using multi billion-dollar experiments such as CERN to confirm the Higgs mechanism an idea which has huge implications in cosmology, particle physics and Quantum Gravity. 2 Background in QFT Quantum field theory is a framework which describes particles as vibrations of a field and each particle has its own unique field. A field is a region that has a value at every point in space that can change over time. There are two categories of fields, which are: vector and scalar fields. A vector field has a magnitude and direction at every point in space whereas a scalar field just has a magnitude at every point in space. The particle vibrating in a field is called a quantum fluctuation. Quantum fluctuations are the temporary change in the amount of energy in a region of space prescribed by the Uncertainty Principle. The Uncertainty Principle states that you can never fully define position and momentum or energy and time simultaneously. This principle only applies at the subatomic level and below.23 Another important idea in QFT are virtual particles. Virtual particles are mathematical tools to explain how information is transmitted between quantum fluctuations. The reason why you cannot calculate these interactions perfectly is because when two quantum fluctuations interact, they will make a cycle of forwards and backwards reactions which are impossible to calculate. During 1SpaceTime, The First Quantum Field Theory, 2017 2SpaceTime, The Nature of Nothing, 2017 3Hilgevoord, 2001 1 these interactions, quantum fluctuations will transmit energy, momentum and quantum properties that can be described in a packet called a virtual particle (this means that quantum fluctuations exchange virtual particles). The hope is to approximate the state of a field at that interaction by summing up all the possible packets and combinations.4 In Quantum field theory, energy is represented as a vertical arrow pointing upwards with different levels of energy. These levels show how many particles are in one quantum state in a system. Most fields stay in a vacuum state, which is the lowest possible amount of energy that a system can have. This energy is called the zero-point energy. Because of the uncertainty you cannot fully define energy or time, so you get a blur of energy states, which means that the minimum energy is an average of the probabilities of the kinetic energy states of a system. 3 Recent developments One of the most recent and important advancements in QFT was the discovery of the Higgs boson. On July 4th, 2012, the Higgs boson was discovered at the Large Hadron Collider, in Geneva Switzerland, and confirmed the existence of the Higgs field. The Higgs field is a scalar field that extends throughout all of space and gives mass to particles (also providing mass to its own particle, the Higgs Boson) but it is special because it has a high value at every point in space whereas most fields have a value of 0.567 (It is important to note that all particles are mass less in the beginning). The Higgs field gives particles mass when a mass less particle interacts with the Higgs field which makes it bounce through the field and so the particle then goes slower than the speed of light or may even be static, showing that the particle has mass. Photons are the force carriers of the electromagnetic force and Gluons are the force carriers of the Strong force. These particles for example, do not interact with the Higgs field so they remain massless.89 4 Quantum Gravity and QFT Quantum Gravity is a huge field of physics trying to unify Quantum mechanics and General Relativity into one theory and is where Quantum field theory is applied. The two main contenders for a theory of Quantum Gravity are String theory and Loop Quantum Gravity. General Relativity is a theory that describes space-time as a 4-dimensional fabric (3 dimensions of space and 1 dimension of time). This fabric can warp and curve when masses move along space. This 4Jaeger, 2019 5Weinberg, 2012 6Garisto, 2020 7Greene, How the Higgs Boson Was Found, 2013 8O'Keefe, 2019 9Gibney, 2018 2 curvature is measured by a metric (called the metric tensor) and depends on the mass and energy of the object that is curving space. The results of General Relativity show that the curving of space gives gravitational effects. So, in summary, gravity is transmitted through the warping of space time. What is incompatible about Quantum Mechanics and General Relativity? General Relativity is a theory where gravitational fields are a continuous struc- ture however Quantum Mechanics is a theory in which fields are discontinuous and are defined by quanta, which are discrete quantities of energy, so there is no version of a gravitational field in Quantum Mechanics. In quantum field the- ory, forces act through nearby regions through the exchange of distinct quanta which is not possible when uniting with gravity. There are a couple of theories competing to resolve this incompatibility, out of which String theory and Loop Quantum Gravity are the most developed.10 String Theory is a structure that unifies Quantum Mechanics and Gen- eral Relativity, where particles are replaced by tiny one-dimensional loops and strings. String theory describes how these strings move and interact; however, it requires that there be extra dimensions that are represented by complex struc- tures. The theory postulates that these dimensions are so small that we cannot see them. String Theory solves a key problem with Gravity and Quantum Fields. The problem is that it is difficult to unite Gravity with Quantum field theory because the regular length scale of the Gravitational force is tiny, which is near the Plank scale (1.6 · 10−35), so that the QFT assumption for point like particles leads to infinities that are unable to be solved. However, String Theory resolves this problem as the lengthened interactions of strings evade such infinities.111213 Loop Quantum Gravity (LQG) on the other hand, is a theory that de- scribes space as a network of microscopic loops \like grains of space" that are all connected. These loops are in quantized chunks which allow you to control what possible values different states have and LQG is a procedure for building a Quantum Field Theory from a Classical field theory. LQG is a Canonical theory which means that it preserves the basic structure of QFT (through symmetries) and extends the theory to be able to quantize gravity onto smaller scales. LQG manages to do this by using a network (called a Spin Network) that incorporates General Relativity and represents the states and interactions of particles and fields. 141516 10Odenwald, 2003 11Khulmann, 2006 12Greene, The Elegant Universe, 2000 13Greene, The Hidden Reaility, 2012 14(Rovelli, What is time? What is space?, 2014 15Rovelli, The Order of Time, 2018 16(Rovelli, Reality Is Not What It Seems, 2017 3 5 Conclusion Quantum Field Theory has played key roles in parts of physics that were previ- ously unknown such as describing quantum phenomena using a modified concept of fields so that we can understand the nature of space and creating possible ways to make a quantum theory of gravity. QFT has enabled us to understand the universe to the greatest degree of accuracy ever and it has brought some of the most abstract and important tools used throughout physics today. However, this field of study does not end there. Many researchers are now exploring the maths behind Quantum field theory and its huge range of applications. References SpaceTime, P. (2017, June 28). The First Quantum Field Theory. Retrieved November 20, 2020, from YouTube: https://www.youtube.com/watch?v=ATcrrzJFtBY SpaceTime, P. (2017, October 19). The Nature of Nothing. Retrieved November 15, 2020, from YouTube: https://www.youtube.com/watch/X5rAGfjPSWE Hilgevoord, J. a. (2001, October 8). The Uncertainty Principle. Retrieved February 7, 2021, from Stanford Encyclopedia of Philosophy: https://plato.stanford.edu/entries/qt-uncertainty/#HeisArgu Jaeger, G. (2019, February 7). Are Virtual Particles Less Real? Entropy, 17. Retrieved February 2021, from https://inspirehep.net/files/950b86f8228f12776c88fec26aaaaaa0 Weinberg, S. (2012, July 13). Why the Higgs Boson Matters. Retrieved November 29, 2020, from The New York Times: https://www.nytimes.com/2012/07/14/opinion/weinberg-why-the-higgs-boson- matters.html Garisto, D. (2020, August 6). Higgs Boson Gives Next-Generation Particle Its Heft. Retrieved November 30, 2020, from Scientific American: https://www.scientificamerican.com/article/higgs-boson-gives-next-generation-particle- its-heft/ Greene, B. (2013, July). How the Higgs Boson Was Found. 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