“In almost all textbooks, even the best, this principle is presented so that it is impossible to understand.” (K. Jacobi, Lectures on Dynamics, 1842-1843). I have not chosen to break with tradition.

— V. I. Arnold, Mathematical Methods of Classical Mechanics, footnote, p. 246

2011.04.05 Tuesday ACHK

Again, the time reversal of macroscopic processes can only exist if we “saturate” the second law of thermodynamics: if the entropy stays constant. The time reversal of such processes keeps the entropy constant, too. These processes are not real processes because nothing much is changing. Instead, they describe a physical system at equilibrium.

— Lubos Motl

2011.04.04 Monday ACHK

The odd thing is that there are a lot of mathematical connections between string theory and the loop representation. Gradually, as time went on, I became more and more convinced that maybe the layfolk were right – maybe the loop representation of quantum gravity really WAS string theory in disguise, or vice versa. This made a little embarrassed by how much I had been making fun of string theory.

I decided to write a paper about this, and as I did some research I was intrigued to find more and more connections between the two approaches, to the point where it is clear that while they are presently very distinct, they come from the same root, historically speaking.

So what I’m hinting at, in brief, is that a bunch of category theory may provide the links between modern string theory with its conformal fields and the loop representation of quantum gravity. This is not as outre as it may appear. The categories being discussed here are really just ways of talking about symmetries (see my stuff on categories and symmetries for more on this). As usual in physics, the clearest way to grasp the connection between two seemingly disparate problems is often by recognizing that they have the same symmetries.

September 11, 1993
This Week’s Finds in Mathematical Physics (Week 18)
John Baez

2011.03.31 Thursday ACHK

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of, and angle between, tangent vectors.

— Wikipedia on Metric tensor

In general relativity, the metric tensor (or simply, the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as distance, volume, curvature, angle, future and past.

— Wikipedia on Metric tensor (general relativity)

2011.03.25 Friday ACHK

Often, when struggling with a book or paper, it’s not you that’s the problem, it’s the author. Finding another source can quickly clear stuff up.

— Michael Nielsen 

2011.03.20 Sunday ACHK

The core concept of general-relativistic model-building is that of a solution of Einstein’s equations. Given both Einstein’s equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold.

Matter and geometry must satisfy Einstein’s equations, so in particular, the matter’s energy-momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.

— Wikipedia on General relativity

2011.03.18 Friday ACHK

In mathematics and physics, n-dimensional anti de Sitter space, sometimes written AdSn, is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. It is the Lorentzian analogue of n-dimensional hyperbolic space, just as Minkowski space and de Sitter space are the analogues of Euclidean and elliptical spaces respectively.

It is best known for its role in the AdS/CFT correspondence.

— Wikipedia on Anti de Sitter space

2011.03.16 Wednesday ACHK

It is the most successful realization of the holographic principle, a speculative idea about quantum gravity originally proposed by Gerard ‘t Hooft and improved and promoted by Leonard Susskind.

— Wikipedia on AdS/CFT correspondence

2011.03.12 Saturday ACHK

T-duality is a symmetry of string theory relating small and large distances. T-duality is not present in ordinary particle theory, indicating that strings experience spacetime in a way that is fundamentally distinct than the way particles do. It relates different string theories that were thought to be unrelated before T-duality was understood. T-duality preceded the Second Superstring Revolution.

— Wikipedia on T-duality

2011.03.07 Monday ACHK

String theory 9

U-duality is a symmetry of string theory or M-theory combining S-duality and T-duality transformations. The term is most often met in the context of the “U-duality (symmetry) group” of M-theory as defined on a particular background space (topological manifold). This is the union of all the S- and T-dualities available in that topology.

The narrow meaning of the word “U-duality” is one of those dualities that can be classified neither as an S-duality, nor as a T-duality – a transformation that exchanges a large geometry of one theory with the strong coupling of another theory, for example.

— Wikipedia on U-duality

2011.03.04 Friday ACHK

Dualities

Before the 1990s, string theorists believed there were five distinct superstring theories: open type I, closed type I, closed type IIA, closed type IIB, and the two flavors of heterotic string theory (SO(32) and E8×E8). The thinking was that out of these five candidate theories, only one was the actual correct theory of everything, and that theory was the one whose low energy limit, with ten spacetime dimensions compactified down to four, matched the physics observed in our world today.

It is now believed that this picture was incorrect and that the five superstring theories are connected to one another as if they are each a special case of some more fundamental theory (thought to be M-theory). These theories are related by transformations that are called dualities. If two theories are related by a duality transformation, it means that the first theory can be transformed in some way so that it ends up looking just like the second theory. The two theories are then said to be dual to one another under that kind of transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.

— Wikipedia on String theory

2011.02.28 Monday ACHK

Infinitesimal is a process, not a state; 

infinity is a process, not a state.

— Me@2011.02.19

2011.02.19 Saturday (c) All rights reserved by ACHK

Spin is not related to the 3D real space wavefunction transformation.

What is the fundamental origin of spins?

Or, what is the extra dimension?

Don’t we live actually in a 3+1 D space instead of 3D?

“Rotation symmetry” in 3+1 D: Lorentz symmetry.

Wavefunctions with spins are actually the basis for the representation of rotation transformations in 3+1D.

Fundamentally, spins arises from the special relativity.

Quantum mechanics + relativity –> Relativistic quantum mechanics, or more precisely, quantum field theory (in relativity, a force has to be carried by a field to avoid instantaneous interaction.)

— Page 37, general angular momentum and spin

— 2007-08 PHY5410 Quantum Mechanics II

— Professor Renbao Liu

2011.02.16 Wednesday ACHK 

In 1954, Chen Ning Yang and Robert Mills proposed to generalize these ideas to noncommutative groups. A noncommutative gauge group can describe a field that, unlike the electromagnetic field, interacts with itself. For example, general relativity states that gravitational fields have energy, and special relativity concludes that energy is equivalent to mass. Hence a gravitational field induces a further gravitational field. The nuclear forces also have this self-interacting property.

— Wikipedia on Introduction to gauge theory

2011.02.15 Tuesday ACHK

Spin is a consequence of Relativity.

— Richard P. Feynman

2011.02.14 Monday ACHK 

Identical particles 2

Surprisingly, gauge symmetry can give a deeper explanation for the existence of interactions, such as the electrical and nuclear interactions. This arises from a type of gauge symmetry relating to the fact that all particles of a given type are experimentally indistinguishable from one other.

— Wikipedia on Introduction to gauge theory

2011.02.13 Sunday ACHK