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The Schrödinger equation on a finite graph
One of the most important discoveries in the history of science is the structure of the periodic table. This structure is a consequence of how electrons cluster around atomic nuclei and is essentially...
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It looks like the finite graph model is not just a toy model! It’s called a continuous-time quantum random walk and is used in quantum computing in a way similar to how random walks on graphs are used...
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The representation theory of SU(2)
Today we will give four proofs of the classification of the (finite-dimensional complex continuous) irreducible representations of (which you’ll recall we assumed way back in this previous post). As a...
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The Heisenberg picture of quantum mechanics
In an earlier post we introduced the Schrödinger picture of quantum mechanics, which can be summarized as follows: the state of a quantum system is described by a unit vector in some Hilbert space (up...
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Poisson algebras and the classical limit
In the previous post we described the Heisenberg picture of quantum mechanics, which can be phrased quite generally as follows: given a noncommutative algebra (the algebra of observables of some...
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Hilbert spaces (and dagger categories)
Hilbert spaces are a particularly nice class of Banach spaces. They axiomatize ideas from Euclidean geometry such as orthogonality, projection, and the Pythagorean theorem, but the ideas apply to many...
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Banach algebras, the Gelfand representation, and the commutative...
Banach algebras abstract the properties of closed algebras of operators on Banach spaces. Many basic properties of such operators have elegant proofs in the framework of Banach algebras, and Banach...
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Noncommutative probability
The traditional mathematical axiomatization of probability, due to Kolmogorov, begins with a probability space and constructs random variables as certain functions . But start doing any probability and...
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Finite noncommutative probability, the Born rule, and wave function collapse
The previous post on noncommutative probability was too long to leave much room for examples of random algebras. In this post we will describe all finite-dimensional random algebras with faithful...
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What’s a fire, and why does it – what’s the word – burn?
I was staring at a bonfire on a beach the other day and realized that I didn’t understand anything about fire and how it works. (For example: what determines its color?) So I looked up some stuff, and...
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