Suppose a particle travelled one dimensionally. Classically, the particle’s behavior should be in line with Newton’s laws of motion. Quantum mechanically, however, one cannot determine the exact position and momentum of the particle and as such is reduced to using a probabilistic interpretation. Here we will explore the notion of a probability density and how one could derive such an artifact from a simple Taylor expansion argument. Continue reading Probability Densities in Quantum Theory

# Probability Densities in Quantum Theory

# Second Variation of an Action: Part I

So your advisor asks you, “what are the background fluctuations for your action?” If you don’t know what they are talking about, you’ve come to the right place. In the meantime, just smile and wave guys $–$ just smile and wave.

# Integration on Manifolds: Compact Support

Compact support for a metric of a manifold is often required to preform integration on the manifold. However, why is it required? Can one preform integration on a manifold without compact support? We will explore those questions here!

# Jhep For Students: A LaTeX Package

TL;DR: I created the *Jhep for Students* $\LaTeX$ package for students and you can find it here. Give it a whirl!

# Physics of Differentiable Manifolds: Part I

Often when trying to study general relativity, the most difficult aspect to understand rigorously is the underlying mathematics. Before anyone can do general relativity, the concepts of differentiable manifolds, smooth curves, vectors & co-vectors, tensors, tangent spaces and cotangent spaces, and many more have to be fully fleshed out. There are excellent texts out there that help achieve this goal, such as Sean Carroll’s Spacetime and Geometry. Here, however, we’ll provide a list of definitions with a few nuggets of information towards the end as a useful summary of the topics discussed in this post. Eventually, we will motivate how quantum field theory and general relativity can both be generated with concepts starting with a differentiable manifold with a metric.

Continue reading Physics of Differentiable Manifolds: Part I

# Play With Neural Networks: Tensor Flow

**Learn about how Neural Networks evolve and how to control them at TensorFlow. ****This blog has a through analysis of how neural networks work as well.**

# Dimensions of Various Lie Groups

How do you determine the dimensions of a Lie group? Why is the dimension of $O(n)$ the same as $SO(n)$? Similarly for $\text{GL}(n,F)$ and $\text{GL}^+(n,F)$, where $F$ is a field?

# Why Massive Particles are Slow and Lazy

Here we’re going to discuss the equations of motion for a charged particle in a curved spacetime with an electromagnetic tensor, $F_{ab}$, show why massive particles have a contracted $4-$velocity, $u^a$, that is constant along a charged-particle path, and why massive particles move slower than the speed of light.

# Cognition & Infants

**An incredible TED talk on the cognitive development of infants by, MIT associate professor Laura Schulz, can be found here. I highly recommend you check it out.**

# Philosophy: The Logic Borne From Consciousness

Here we present an argument of how the humanities, such as art and especially philosophy, are a by-product of logic, which is the essential utility of consciousness.

Continue reading Philosophy: The Logic Borne From Consciousness