Path integral quantum field theory, perhaps single handedly constructed by Richard Feynman, remains both elusive to undergraduates that wish to study the subject and immensely useful for performing the calculations found in quantum field theory due to the path integral formulation being manifestly symmetric between space and time. Here, we will enter the world of path integral quantum field theory by meticulously performing the calculations so that it is accessible to an advanced undergraduate.

# Why We Suck at Division & Subtraction Inherently

**Disclaimer**: I am not at all a neuroscientist. I’m just a guy that has an interest in it and saw a pretty cute connection.

Of the different forms of arithmetic, I believe that subtraction and division, which if we were being honest is just a glorified form of subtraction, are the most difficult manipulations to perform $-$ why, you might ask, well I believe the reason is due to object permanence.

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# A Weak Derivation of the Momentum Operator

Here is a cute trick to derive the momentum operator in quantum mechanics. Usually, we taken this for a given or a definition, however there are ways to prove the relationship. Here we explore the relationship between position and momentum using Fourier Transforms.

# Path Integral Free Propagator

In quantum field theory, the propagator gives a probability amplitude for a particle traveling from some point $latex (t_i,x_i)$ to $latex (t_f,x_f)$ with a certain energy and momentum. These propagators are the first steps into quantum field theory, I aim to bring these to the masses. The one that is most interesting and easy to grasp is the free propagator for a free field theory. The following derivation is inspired by Anthony Zee’s *Quantum Field Theory in a Nutshell* of UC Santa Barbara.

# Proof of Euler’s Formula

Around 1740, Leonard Euler discovered a formula that connected functions of complex arguments to trigonometric functions, effectively forming a link between analytic functions and geometric functions which eventually extended to topology, differential equations, and mathematical physics. All of this began with one simple formula, lauded by *Richard Feynman* as “the most remarkable formula in mathematics,” and it is

$$e^{ix}=\cos(x)+i\sin(x), \text{ with } x\in\mathbb{R}. $$