##### Intro to Quantum Field Theory

When we begin learning about matter, we are introduced to the notion that the fundamental building blocks of it are atoms, and that smaller particles such as neutrons and protons make up these atoms. While this is not untrue, it does not paint the entire picture. What are these particles actually like in the real world? Are they waves, particles, or perhaps a combination of both?

With the advent of quantum mechanics in the early 20th century, physicists started to describe these building blocks as having properties of both particles and waves. A physicist named Erwin Schrodinger arguably proved the most important with his development of the Schrodinger Wave Function, better known as just the wavefunction. This wave function describes the quantum state of a quantum system, and allows the probability of various measurements of a quantum system being true to be derived from it. He also developed the Schrodinger Equations that describe the aforementioned wavefunction, and this helps predict and find the probability of the events that can occur in a quantum system. This equation was flawed as it did not work in relativistic contexts, which was improved on by Paul Dirac later.

The fundamental particles, specifically the electron exhibit properties of both a particle and wave as shown in the double-slit experiment. This wave-particle duality can best be described currently, through the framework of Quantum Field theory which described these particle-waves as just localized vibrations of a quantum field, and in turn, affects other quantum fields. For example, a vibration in an electron field also affects the photon field, which is well known through physics as the photoelectric effect. Adding on to photons, they are arguably the most well-known product of a quantum field, they are the quanta of the electromagnetic field. These quantized photons make up light and other electromagnetic forces. From our macroscopic field of view, light is seen as smooth and continuous however looking closer at it, we can see that it is actually made up of those discrete photons.

Back to the fields, they extend infinitely through space and can be thought of as the backbone of matter, as we know it. An interesting component of these fields is that even when “empty”, there are particles and antiparticles within that annihilate each other upon contact, releasing a great amount of energy. For perspective, and as a side note, 1 kilogram of antimatter reacting with 1 kilogram of matter would produce over 1000 times as much energy as the nuclear bombs used in Hiroshima and Nagasaki combined, via Einstein’s famous equation of mass-energy equivalence, E=mc², which additionally is also directly related to Quantum Field Theory through its use of Special Relativity. This tangent aside, and back to the photons and electrons, Quantum Electrodynamics describes electromagnetism as interactions of electrons and photons, in matter and energy, through the lens of the relativistic model of Quantum Field Theory.

Back to the Schrodinger Equation, physicist Paul Dirac derived a new equation from the former which is creatively called the Dirac Equation. This equation describes the electron and other ½ spin particles within a relativistic, quantum mechanical context and is therefore prominently used is therefore aforementioned Quantum Electrodynamics This equation also accurately predicted the existence of antimatter, through the use of negative energy states. Through the framework of Quantum Field Theory, we can simply state that matter is an excitation of an invisible field. Back to Quantum Electrodynamics, light is an excitation of the electromagnetic field and acts as a harmonic oscillator. This means that like a guitar string, the force applied on it increases its frequency as an Integer multiple of an original quantized frequency which is the photon in this case. Dirac innovated by describing en fields through the framework of the quantized photons. Each particle of matter in this field was produced from photon oscillations. His use of calculus summed oscillations in the different quantum states, which was essentially counting photons per state. The notion of these harmonic oscillators is critical to Quantum Mechanics in general and can be applied to all quantum fields.

Back to the Schrodinger Equation, physicist Paul Dirac derived a new equation from the former which is creatively called the Dirac Equation. This equation describes the electron and other ½ spin particles within a relativistic, quantum mechanical context and is therefore prominently used is therefore aforementioned Quantum Electrodynamics This equation also accurately predicted the existence of antimatter, through the use of negative energy states. Through the framework of Quantum Field Theory, we can simply state that matter is an excitation of an invisible field. Back to Quantum Electrodynamics, light is an excitation of the electromagnetic field and acts as a harmonic oscillator. This means that like a guitar string, the force applied on it increases its frequency as an Integer multiple of an original quantized frequency which is the photon in this case. Dirac innovated by describing en fields through the framework of the quantized photons. Each particle of matter in this field was produced from photon oscillations. His use of calculus summed oscillations in the different quantum states, which was essentially counting photons per state. The notion of these harmonic oscillators is critical to Quantum Mechanics in general and can be applied to all quantum fields.

To get into the fields themselves, fields are just mathematical structures that have values for each point in space, rather than having a set location, as a particle would. Fields aren’t made up of anything, instead, they make up everything in the universe. Classical field theory describes fields as what make up the universe, but Quantum Field Theory states the Universe is made up of wave function that is derived (not as in derivative/differential) from the configuration of the classical fields. Fields can be represented with φ(x) with φ being the wave function or phi. This equation can be said to be given classical field configuration. The quantum wave function can be said to be a composite function of the classical configuration. The wave function is Ψ(φ(x)). It can be said to be in the form of G(F(x))in which G(x) is (x) and F(x) is φ(x). The probability of viewing a set configuration is the absolute value of the wave function squared. It can be simply expressed as the equation P(φ(x)) =|Ψ(φ(x))|². Every possible wave function complex value produces a complex value in the form of a+bi in which b is the square root of -1. A and B are complex numbers. This absolute value of a complex number. can be expressed as the expression a² + b². Then the square of this would just be a²+b², An alternate way of finding the absolute value squared of a+bi is in this equation, |a+bi|2=(a+bi)(a-bi)in which a-bi is the conjugate of a+bi which allows this complex value to become real. This is the equivalent of squaring any real number which has a positive product.

The classical field configuration can be described as a sum of modes in which modes refer to certain vibrational patterns that move in the manner of a sine wave. A sine wave is a graph of the function F(x)= sin(x). A diagram is pictured here. It can also be expressed as the expression; Kφk(h). The energy of φk(h) is proportional to h². This a harmonic oscillator in which energy disproportionate to displacement squared, like a guitar string. Quantizing in this context means looking for all solutions to this wave function. To solve a simple harmonic oscillator one must first find the lowest energy state of it, also called the ground state in which the particles are not excited which is represented by 0(x). Energy is potential and gradient (changing) energy. The normalization condition is represented by the integral |0(x)|2dx=1, in which 1 is to the sum of all the possible probabilities where the particle could be located. The graph for 0(x) has a shape resembling a hill in which the integral (area under the curve) rises a bit above 0 but tapers down at both -x and +x and has a horizontal asymptote of 0. The graph for 1(x) is half positive and half negative and as a vector can be thought of as perpendicular to 0(x). The increasing modes give more waves and therefore shorter wavelength. For energy level N(x), each energy state when graphed will intersect the x-axis N times and wave function N(x) has properties of a wave function with N particles. This has been an intro to Quantum Field Theory and the wave functions and math behind it. Further articles will go deeper into wave functions and particle interactions, with some more complex math used.

Sources:

Webb, Richard. “What Is Quantum Field Theory?” New Scientist, …..www.newscientist.com/definition/quantum-field-theory/.

“What Is Quantum Field Theory?” David Tong -- What Is Quantum Field Theory?, …..www.damtp.cam.ac.uk/user/tong/whatisqft.html.