Dork Scratchings

In 1964 John Stewart Bell proposed an experiment to determine whether the results of quantum measurements were truly random, or governed by hidden variables, i.e. state that exists prior to the measurement, but which we don’t have access to. The experiment involved creating a large number of EPR pairs, and firing them at two observers, Alice and Bob, who measure their photon’s polarisation, choosing the $\updownarrow$ direction or the $\nearrow\llap\swarrow$ direction at random. Determining the result of the experiment involves doing a complex statistical calculation to see if something called Bell’s inequality is satisfied or violated. The Bell experiment was first performed by in 1982 by Alain Aspect, and the result, as most commonly interpreted, is that hidden variables can only exist if Quantum Mechanics is non-local, i.e. if it supports faster-than-light causality! Some time after Bell proposed his experiment, Greenberger, Horne, and Zeilinger suggested an alternativ...

What's going on right now, 70,000 light years away on the nearest galaxy to our own, Canis Major?  Or put it another way, if we're at $x = y = z = t = 0$, what's happening at $y = z = t = 0$ and $x$ = 70,000 light years? According to Einstein's special theory of relativity it depends who you ask.  The most natural coordinates to use depend on your inertial frame.  Sitting in frame $S$, I may use the coordinates $x,y,z,t$ but someone in frame $S'$ would need to use $x',y',z',t'$ in order to observe the universe operating according to the same physical laws.  To make it more concrete, imagine $S$ and $S'$ share their $y$ and $z$ axes, and $S'$ is moving at velocity $v$ along the $x$ axis of $S$.  In this case the transform for converting between the two reference frames is $$ \begin{align} x' &= \frac{x - vt}{\sqrt{1-\frac{v^2}{c^2}}} \\ y' &= y \\ z' &= z \\ t' &= \frac{t - \frac{vx}{c^2}}{\sqrt{1-\frac{v^2}{c^...

MWQT$^3$ I've just discovered Quantum Tic Tac Toe .  This is a brilliant game designed to give players an intuition for Quantum Mechanics without requiring them to learn any complicated mathematics. Superposition Instead of playing in just one square each player gets to play in two squares at once.   Here player X has played in the 1st & 2nd squares on their first move, the 3rd & 6th squares on their 2nd move, and the 8th & 9th squares on their 3rd move.  Similarly player O has played on two squares each move, and the marks are subscripted to indicate when in the game play they were laid down. Entanglement Ultimately each pair of marks will be replaced with a single mark and the board will look like ordinary Tic Tac Toe - which is how we are able to determine a winner.  But the final location of, say, $O_2$ may need to depend on the final location of, say, $X_1$ if we are to avoid ending up with squares with multiple marks in them.  In this case...

Top left box evolves into a superposition of the other two When introduced to quantum physics the first example we encounter is usually that of a single particle.  We are shown that experiment demonstrates a particle left alone evolves into a superposition of states.  These states may be position states or they may be momentum states, or if the particle has spin it may be a superposition of spin states.  It doesn't matter, the point is that fundamental particles can be in a superposition of states.  That's because they are small, we are told, so you wouldn't expect them to behave like big things do, we are told. The next thing we are shown is how the Schroedinger equation governs the evolution of this superposition.  The particle is not usually in every state equally, it is more in one position state (or momentum state, or spin state) than it is in another.  The distribution over these so-called basis states evolves with time and the ...

Clockwise from top left: Occam, Deutsch, Everett, and Dirac Occam's razor The anthropic principle Forward reasoning Lack of any consistent alternative 1. Occam's razor Hugh Everett's 1956 thesis The Theory of the Universal Wavefunction opens with a mathematical summary of the then widely accepted Copenhagen Interpretation. "... there are two fundamentally different ways in which the state function can change:    Process 1:  The discontinuous change brought about by the observation of a quantity with eigenstates $\phi_1, \phi_2,...,$ in which the state $\psi$ will be changed to the state $\phi_j$ with probability $\lvert(\psi,\phi_j)\rvert^2$. Process 2: The continuous, deterministic change of state of the (isolated) system with time according to a wave equation $\frac{\partial \psi}{\partial t} = U\psi$, where $U$ is a linear operator." The 1st process is commonly known as the "collapse of the wavefunction" and ...

(Probably mad, but worth saving for posterity) Today's post follows on from this previous post  in which I argued that the entire universe is in a superposition of states. I showed how this can lead to the impression that only the system under study is in a superposition, whilst the rest of the world is in a single - classical - state.  To summarize that argument: Reality is a superposition of states for the entire universe, each of which can be thought of as a complete classical block universe. We don't directly experience the whole universe, so let's arbitrarily place a closed surface around us to demarcate what we "directly" experience.  Remember that this is a surface in 4 dimensional spacetime and so is itself 3 dimensional.  Also remember that it is completely arbitrary: if you like you can let it contain your entire body for your entire life; or it could just be your brain for some duration.  You could even let the surface enclose e...

Most Quantum Mechanics courses try to avoid imposing any kind of interpretation. This makes sense since the interpretation of QM is controversial but the mathematics is not. Unfortunately a little bit of interpretation always sneaks in through the back door. Whether you're being taught the Schroedinger Equation, Feynman path integrals, or QFT, the assumption is always that you can divide reality into That Which Is Under Study and the Rest Of The World... and that the nature of reality in the two parts is entirely different. If it's the Schroedinger Equation being taught That Which Is Under Study is represented by a state vector in a Hilbert Space that evolves with time; with Feynman Path integrals That Which Is Under Study is the set of all legal Feynman diagrams which complete the picture by joining neatly with the  diagram for The Rest Of The World; if it's QFT then the nature of reality inside That Which Is Under Study is a single state vector which can be converted in...

A review of Our Mathematical Universe (Max Tegmark) Our Mathematical Universe was my summer holiday reading this year.  But it turned out to be much more than just something to keep me occupied while lounging on the beach.  This book has changed my conception of reality.  The themes in the book are similar in nature to those in The Fabric of Reality by David Deutch - another one of my favourite books.  However, instead of restricting the argument to the parallel universes predicted by Everett's Universal Wavefunction, Tegmark takes us on a tour of four levels of multiverse.  More than this, he provides an overarching theoretical framework for understanding them based on what he calls "the A-word" (because using its full name is guaranteed to get your paper rejected). The argument goes like this.  Whenever we find Nature appears finely tuned to make self-aware life possible, then there are 3 possible explanations It's a fluke! It's design! (by a...

The idea of wave particle duality has been around for a long time, and is often used to explain phenomenon like the double slit experiment .  The idea goes like this: the particle (electron, photon, whatever) behaves like a wave some of the time - like when it is passing through the slits - and like a particle at other times - such as when it hits the screen and produces a flash in a single location.  This explains how it can behave as if it went through both slits at the same time despite being in just one location whenever we check - e.g. by making it collide with a surface.  The change from wave like behaviour to particle like behaviour is called the collapse of the wavefunction . This idea raises many questions, chiefly: what does and what does not collapse the wavefunction?  For example, if you replace the screen with a mirror then a photon continues to exhibit wave behaviour after bouncing off of it.  The proponents of the wave particle duality theo...