… relativity isn’t theory of relativity at all, it’s a theory of invariants (already Einstein made this observation, but it was too late, the current name had caught on). Some (and actually many) things in relativity are absolute.

– Marek

2011.06.19 Sunday ACHK

Choice in General Relativity: Metric, . Directly determines

Riemannian geometry; Geometrodynamics.

In all other interactions, by contrast, the basic variable is a Connection, i.e., a matrix valued vector potential .

Gauge theories: Connection-dynamics

Connections: Vehicles for parallel transport.

Classical singularities are gates to Physics Beyond Einstein.

— Introduction to Loop Quantum Gravity

— Abhay Ashtekar

2011.06.17 Friday ACHK

After all, G2 is the automorphism group of the octonions, and it has a 7-dimensional irreducible representation on the imaginary octonions; as explained in “week104” by Robert Helling, the octonions are secretly what let you write down the superstring Lagrangian in 10d spacetime.

— John Baez

2011.06.10 Friday ACHK

The difference between the classical and quantum dynamical equations is just hidden in the fact that rho,x,p are operators in quantum theory and they generally don’t commute with each other. But otherwise, the interpretation from classical physics may be pretty much directly extended to the quantum theory!

— Density matrix and its classical counterpart

— Lubos Motl

2011.06.09 Thursday ACHK

Instead, the energy is universally defined in such a way that it is conserved as a result of the time-translational symmetry; and the action is defined in such a way that the condition δS=0 (stationarity of the action) is equivalent to the equations of motion. These are the general conditions that define the concepts in general and that make them important; particular formulae for the energy or action are just particular applications of the general rules.

— Lubos Motl

2011.06.06 Monday ACHK

Fermat’s principle is the main principle of quantum electrodynamics where it states that any particle (e.g. a photon or an electron) propagates over all available (unobstructed) paths and the interference (sum, or superposition) of its wavefunction over all those paths (at the point of observer or detector) gives the correct probability of detection of this particle (at this point). Thus the extremal (shortest, longest or stationary) paths contribute into this interference most as they can not be completely canceled out.

— Wikipedia on Fermat’s principle

2011.06.04 Saturday ACHK

How does the particle find the right path?

Does it ‘smell’ the neighboring paths to find out whether or not they have more action?

In the case of light, when we put blocks in the way so that the photons could not test all the paths, we found that they couldn’t figure out which way to go, and we had the phenomenon of diffraction.

— Chapter 19. The principle of least action

— Volume 2. Mainly electromagnetism and matter

— The Feynman Lectures on Physics

2011.06.02 Thursday ACHK

Moreover, in the usual formulation of classical action principles, the initial and final states of the system are fixed, e.g.,

    Given that the particle begins at position x1 at time t1 and ends at position x2 at time t2, the physical trajectory that connects these two endpoints is an extremum of the action integral.

In particular, the fixing of the final state appears to give the action principle a teleological character which has been controversial historically. However, this apparent teleology occurs only because of the way in which the question was asked. By specifying some but not all aspects of both the initial and final conditions (the positions but not the velocities) we are making some inferences about the initial conditions from the final conditions, and it is this “backward” inference that can be mistaken for a teleological causal influence.

— Wikipedia on Principle of least action

2011.05.30 Monday ACHK

Here’s the link between particle interchange and rotation: Look at the twist you introduced into your strip of paper. There is another way to introduce that type of twist. Instead of rotating one end, switch the position of the ends of your strip, while keeping both ends of the strip facing up.

—

2011.05.27 Friday ACHK