Dork Scratchings

In quantum mechanics a lot of emphasis is placed on the concept of  vector spaces.  One of the key tools is the ability to construct new vector spaces out of existing ones.  However, very often authors construct new vector spaces without explicitly saying what they have done, and the result can be confusing.  In this post I am going to attempt to summarize all the methods I have seen for constructing new vector spaces out of old, and point out where they are used in quantum mechanics. The building blocks What: Hilbert Spaces Why: To represent superpositions of classical states   The building blocks are always Hilbert spaces.  These are vector spaces over the complex numbers $\mathbb{C}$, with inner products and limits.  The pair $(V, \langle\cdot\lvert\cdot\rangle)$ is a Hilbert space if $V$ is a vector space over $\mathbb{C}$ $v\mapsto\langle u\lvert v\rangle$ is linear map $V\mapsto\mathbb{C}$ for any $u$ in $V$ $\langle u\lvert v\rangle ...

The Theoretical Minimum series by Leonard Susskind et al. My favourite books by a long way.  They do exactly what it says on the tin: Provide the minimum level of theory to start doing Physics properly. Prior to reading these books I had read a lot of pop-science and was always left with an uneasy feeling that I'd been duped. Having chosen to keep the audience broad the authors of most pop science struggle to communicate the concepts, and usually fall back on analogy - or wonder! - creating ambiguity and misunderstanding. (There is one notable exception - Feynman's QED: The Strange Theory of Light and Matter .) The Theoretical Minimum series side steps the problems faced by most science communicators by assuming the readers do have some tools under their belt.  Specifically basic calculus.   It's then able to take the quickest possible route to the edge of science .  (Okay, maybe not the edge, but it feels like it.)  Just don't skip any bits because t...

A non-euclidean surface with coordinate system Metrics are important in both the calculus of surfaces and in General Relativity.  These objects are essentially NxN matrices whose values are defined by a) the coordinate system used to parametize the space, and b) the particular point $\boldsymbol{x} = (x^i)$.  In general, theorems can be categorized as either intrinsic or extrinsic , where extrinsic means that information about how the space is embedded in a higher dimensional space is used, and intrinsic means that nothing other than the metric and it's partial derivatives were used.  The word intrinsic comes from the idea that a being which "lives" in the space can calculate the metric. So how does such a being find the metric?   2 & 3 dimensional spaces In three dimensions it's pretty straightforward: Impose a coordinate system $x^i = x^1, x^2, x^3$ on the space Define the covariant basis $\boldsymbol{e_i} = \frac{\partial{\boldsymbol{R}}...

I was looking up David Deutsch when I stumbled across this amazing lecture by the gloriously dorky Sidney Coleman.  It's a grainy recording of a talk he gave to the American Physical Society in 1994.  It's full of witty repartee as he talks about his philosophical struggles with QM prior to embracing Many Worlds.  He wonders, for example, who or what exactly can collapse a wavefunction.  The Copenhagen Interpretation doesn't - cannot - answer this.  Coleman describes going to a colleague and sharing his concern that maybe solipsism is the only consistent interpretation - that maybe he is the only one capable of collapsing wavefunctions.  The colleague puffs on his pipe and says: tell me, before you were born, was your father able to collapse wavefunctions?   Wonderful! Quantum Physics meets Psychoanalysis. After watching the video I made some notes on my favorite app. I'd wanted to understand Bell's Inequality/Bell's Theorem for some time, bu...

After a few abortive attempts to teach myself General Relativity I eventually realized that the best approach would be to master (as best as possible) Tensor Calculus on its own, and then tackle GR.  That sounds easy doesn't it!  Well, no, as it turns out.  There's certainly lots of material online, and there are books on the subject at the Cambridge University Press Bookshop, but in general they are all very bad at explaining the motivation behind the maths, or they get hung up on relating back to the building blocks of pure maths, thereby obscuring the vision. Eventually I found Pavel Grinfeld's excellent online course.  48 lectures (yes!) that take you by the hand and really do explain what Tensor Calculus is about, and how to do it.  I watched these during my lunch hours over the course of a couple of months.  I'm sure all the other cafe goers thought the man sitting in the corner with his headphones in was a complete dork, but I didn't care - I...