Posts tagged with parallel transport

Are you a physicist and want to learn intermediate microeconomics as quickly as possible? Here you go.

Minute 18

• Goods = vector space
• Price = covector
• Expenditure = their inner product
• Foliate the vector space by hypersurfaces convex to the origin with codimension 1. Indifference surfaces / isoutility surfaces.
• (no local minima/maxima, ever-increasing)
• Look at the inverse images, given a particular choice of price = budget constraint. Affine hyperplanes of codimension 1, translated from the origin, which are all based on the kernel of the pricing vector.
• The central dogma: agents spend up to their budget constraint reaching the highest level surface intersecting with the convex hull.
• People buy the unique basket whose tangent space at the basket to the indiffference space is equivalent to the kernel of the pricing vector in force.
• The space of all such baskets, given any income level but the same pricing system, is called the Engel curve.
• Minute 34: income vs substitution effects


Minute 31. For the economists in the audience. This is a really good point. We measure the inflation from period to period by some formula like

$\large \dpi{200} \bg_white \frac {\text{Prices}_{t+1} \ast \text{Quantities (consumption choices)}_t} {\text{Prices}_t \ \ \; \ast \text{Quantities (consumption choices)}_t}$

What’s up with multiplying prices from timepoint 2 against quantities from timepoint 1? That doesn’t really make sense does it. If prices changed in the next period then that induced a response in purchasing behaviour.

Not to mention that e.g., hats have fallen out of fashion for men since a century ago—so the price of hats no longer merits a high weight in the basket of what price increases are killing the budgets.

What we really want to do is use a connection. That gives us parallel transport across timepoints.

(Source: pirsa.org)

In the loop quantum gravity approach, space-time is quantized by a procedure that encodes it in a discretized structure, consisting of spin networks and spin foams.

A spin network consists of an oriented embedded graph in a 3-dimensional manifold with edges labelled by SU(2) representations and edges labelled by intertwiners between the representations attached to incoming and outgoing vertices. These representations relate to gravity in terms of holonomies of connections, and the formulation of Einstein’s equations in terms of vierbein, or tetrads, and dual co-tetrads.

Thus, to a spin networks, or the 1-skeleton of a triangulation by tetrahedra, one assigns operators of quantized area and volume, coming from counting intersection points of a surface, or 3-dimensional regions, with the edges or vertices of the spin network with a multiplicity given in terms of the spin representation attached to the edges and the intertwiners attached to the vertices.

SOURCE: Listening to Golem

Loop quantum gravity is one of a few general frameworks that may eventually form the basis of how physicists think of the smallest scales of time and distance.

(These frameworks are sometimes called Theories of Everything but they’re really just thoughts-on-the-way-to-theories of brief-and-tiny matter.)

###### GLOSSARY
• SU(2) is the group of 2×2 unitary matrices with determinant 1. They have the form
$\dpi{200} \bg_white U = \begin{pmatrix} \alpha & -\bar{\beta} \\ \beta & \bar{\alpha} \end{pmatrix}$
SU(2) is just like the unit quaternions, which represent rotations in 3-D. Here are the pieces that make up SU(2) .
$i\sigma_x = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$$i\sigma_y = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$$i\sigma_z = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}$
• 3-dimensional manifold — any shape that can be made with dough. Including that if you stretch the dough, the outside and the inside stretch.
• oriented graph — things like this:
• embedded oriented graph — oriented graphs sculpted in 3-D
• edge — arrows in the above pictures
• spin network — a graph like above and each circle has value  or −½
• representation — every group can be represented as a matrix
• holonomy — how to move things in parallel within the dough (curved 3-manifold)
• connection — parallel transport
• tetrad or vierbein — a 4-D spacetime frame of reference

I love randomly throwing out dense mathematical statements from theoretical physicists like this. Sometimes maths people sound like wizards canting magical runes.