Talks

Domains were introduced in computer science in the late 1960\'s by Dana Scott to provide a semantics for the lambda calculus (the lambda calculus is the basic prototype for a functional programming language i.e. ML). The study of domains with measurements was initiated in the speaker\'s thesis: a domain provides a qualitative view of information expressed in part by an \'information order\' and a measurement on a domain expresses a quantitative view of information with respect to the underlying qualitative aspect. The theory of domains and measurements was initially introduced to provide a first order model of computation, one in which a computation is viewed as a process that evolves in a space of informatic objects, where processes have informatic rates of change determined by the manner in which they manipulate information. There is a domain of binary channels with capacity as a measurement. There is a domain of finite probability distributions with entropy as a measurement. There is a domain of quantum mixed states with entropy as a measurement. There is a domain of spacetime intervals with global time as a measurement. In this setting, similarities between QM and GR emerge, but also some important differences. In a domain, if we write x <= y, then it means that x carries information about y, while x << y is a stronger relation that means x carries *essential* information about y. In GR, the domain theoretic relation << can be proven to be timelike causality. It possesses stronger mathematical properties than << does in QM. However, by an application of the maximum entropy principle, we can restrict the mixed states in consideration and this difference is removed: the domains of events and mixed states are both globally hyperbolic -- where globally hyperbolic is a purely order theoretic idea that just happens to coincide with the usual notion in the case of GR. Along the way, we will see domain theoretic ways of distinguishing between the Newtonian and relativistic notions of time, how to reconstruct the topology and geometry of spacetime in a purely order theoretic manner beginning from only a countable set, see that the Holevo capacity of a unital qubit channel is determined by the largest value of its informatic derivative and have reason to wonder if distance can be defined as the amount of information (capacity) that can be transmitted between two points.

Symmetry principles in physics are a very powerful guiding principle. Sometimes they are so powerful that they can determine a theory completely. This talk will be a tour from the Standard Model of particle physics to string theory compactifications using mostly symmetry arguments.

Taking our intuitive understanding of the quantum world gained by studying a particle in a one-dimensional box, we generalize to understand a quantum harmonic oscillator.
Learning Outcomes:
• Introduction to the classical physics of a ball rolling back and forth in a bowl, a simple example of a very important type of bounded motion called a “harmonic oscillator.”
• The quantization of allowed energies of a harmonic oscillator: even spacing between energy levels, and zero point energy.
• Being able to sketch the allowed wavefunctions and particle probability patterns of a quantum harmonic oscillator, including a new phenomenon called “tunnelling.”

By applying our understanding of the quantum harmonic oscillator to the electromagnetic field we learn what a photon is, and are introduced to “quantum field theory” and the amazing “Casimir effect.”
Learning Outcomes:
• Understanding that classical electromagnetic waves bouncing around inside a mirrored box will exist as standing waves with only certain allowed frequencies.
• How each of these standing waves oscillates harmonically, and thus why – at the quantum level – their energies must be discrete, which is interpreted as the presence of a discrete number of photons.
• What the zero point energy of the electromagnetic field represents, and its relationship to a remarkable property of the quantum vacuum called the “Casimir effect.”

Space obeys the rules of Euclidean geometry. Spacetime obeys the rules of a new kind of geometry called Minkowskian geometry.
Learning Outcomes:
• Triangles in spacetime obey a Pythagoras-like theorem, but with an unusual minus sign.
• The true nature of time as geometrical distance in spacetime.
• How to analyse and resolve the Twins’ Paradox using spacetime diagrams in combination with Minkowskian geometry.

Learning to use Minkowskian geometry to understand, very simply, a variety of aspects of Einstein’s spacetime.
Learning Outcomes:
• How a straight line is the longest path between two points in spacetime.
• How a light particle experiences space and time: its journey from one location in the universe to another involves zero spacetime distance, and is thus instantaneous!
• How Einstein’s special relativity has no difficulty handling accelerated observers.

A discussion of how to synchronize clocks that are separated in space, and how this leads to the relativity of simultaneity.
Learning Outcomes:
• Understanding that clock synchronization is a physical process, and exploring various methods of synchronization using spacetime diagrams.
• How to measure distance with a clock: the concept of radar ranging distance.
• A profound realization about the nature of spacetime: Events that are simultaneous for one observer might not be simultaneous for another.

A more in depth discussion of what the Heisenberg Uncertainty Principle is trying to tell us about the nature of reality.
Learning Outcomes:
• Understanding the strong interpretation of the HUP: “Particles cannot simultaneously possess a definite position and a definite momentum.”
• Why the classical question: “Given a particle’s initial position and momentum, what is its position and momentum as some later time t?” makes no sense in the quantum world.
• Richard Feynman’s remarkable sum over paths interpretation of quantum mechanics.

Highlighting the essential difference between the classical and quantum worlds.
Learning Outcomes:
• A recap of what we’ve learned so far.
• Understanding that in the classical world we have either “particle moving to the right” OR “particle moving to the left.”
• Understanding that, in the quantum world, OR can be replaced with AND: “particle moving to the right” AND “particle moving to the left.”