What is Quantum Field Theory?

A brief post today explaining the motivations for and success of Quantum Field Theory. Here at RTU we’ve given you a fair bit of exposure to the quirks and eccentricities of the Quantum Mechanical world (check out Laws of Quantum) but today we’ll take the theory of the tiny to its next great leap in understanding – Quantum Field Theory. Caveat; I will not explain the formulation of the QFT, solely the motivation for the theory and how it fuses important theories of the universe we have already come across. The main reason for this being I am still learning it myself and to explain qualitatively would be a very difficult task.

There are different ways to think about the importance of QFT, firstly we we can think of it as the extension of Quantum Mechanics from a system of few particles to a system of many particles. Quantum Mechanics can explain accurately the behaviour of one particle and therefore it can only operate with a limited number of degrees of freedom. (A degree of freedom of a physical system is a variable that is necessary to characterise the state of a physical system. For example a system that is confined to move in one direction with a fixed velocity has 2 degrees of freedom). As such QFT extends QM so that we are able to handle systems of many particles and infinite degrees of freedom.

Quantum Field Theory can also be thought of as the reconciliation of Quantum Mechanics and Special Relativity. The Schrodinger Equation – (the fundamental law for the evolution of Quantum Mechanical states in time) cannot obey the requirements of relativistic theories. Special Relativity, a relativistic theory as the name would suggest, requires that physical laws of nature are invariant under certain transformations (namely Lorentz transformations). For example a law of nature in one reference frame must look exactly in the same in a different reference frame that was shifted say shifted in position or boosted by a certain velocity. However the Schrodinger Equation is not invariant under such transformations and a quantum mechanical state will not evolve in exactly the same manner as one in a different frame. Additionally, a second clash between Quantum Mechanics and Special Relativity occurs, when particles have velocities close to the speed of light, as here Quantum Mechanics breaks down. But QFT allows us to work in relativistic frames which is extends our understanding of the world of the tiny enormously, as often tiny particles are able to move at very high speeds.


This diagram illustrates the two points above, N stands for the number of degrees of freedom, SRT for Special Theory of Relativity. 

Quantum Field Theory treats particles as excited states of an underlying field (see ‘What is a Field‘ for an introduction to the concept of a field). In QFT,  quantum mechanics interactions among particles are then described by interactions among the underlying quantum fields. The notation of the theory combines classical field theory, special relativity and quantum mechanics a new overarching manner. QFT was the pivotal rung in the ladder to elevate our understanding of the tiny into the realm of the fast moving whilst also extending our ability to be able to analyse systems with many particles and infinite degrees of freedom.

QFT is a wonderfully successful theory and one of modern physic’s great accomplishments. It is an effectively field theory and is widely believed to be a good low-energy approximation to a more fundamental theory which could take the physics towards the final frontier of incorporating General Relativity with the quantum world.




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