Probability Densities in Quantum Theory

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

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.

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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.

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