In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A.

— Wikipedia on Indicator function

2010.02.05 Friday ACHK

The analogy between the physics of superfluid helium and general relativity is well known. The mathematics that describe these systems are essentially identical so measuring the properties of one automatically tells you how the other behaves.

— Recreating the Big Bang Inside Metamaterials, The physics arXiv blog

2010.02.01 Monday ACHK

… so the principle of event symmetry states that the equations governing the laws of physics must be unchanged when transformed by any permutation of spacetime events.

More detailed studies have shown that different string theories in different background spacetimes can be related by dualities. There is also good evidence that string theory supports changes in topology of spacetime.

— Wikipedia on Event symmetry

2010.01.29 Friday ACHK

In quantum field theory, continuous fields are replaced with a more complex structure that has a dual particle-wave nature as if they can be both continuous and discrete depending on how you measure them.

— Wikipedia on Event symmetry

2010.01.28 Thursday ACHK

So the quantum-theoretic study of spin is pretty much the same as the study of an abstract free particle moving around on S^3, satisfying Schroedinger’s equation. The energy levels of this abstract particle correspond to the spin of the rotating body.

Geometrically, SU(2) is the same as what we mathematicians call S^3, or the 3-sphere – which is the unit sphere in R^4.

— Spin and the Harmonic Oscillator, John Baez

2010.01.17 Wednesday ACHK

However, the path integral formulation is also extremely important in direct application to quantum field theory, in which the “paths” or histories being considered are not the motions of a single particle, but the possible time evolutions of a field over all space.

Much of the formal study of QFT is devoted to the properties of the resulting functional integral, and much effort (not yet entirely successful) has been made toward making these functional integrals mathematically precise.

— Wikipedia on Path integral formulation

2010.01.25 Monday ACHK

In relativistic theories, there is both a particle and field representation for every theory. The field representation is a sum over all field configurations, and the particle representation is a sum over different particle paths.

— Wikipedia on Path integral formulation

2010.01.24 Sunday ACHK

This formulation has proved crucial to the subsequent development of theoretical physics, because it is manifestly symmetric between time and space.

Quantum field theory

The path integral formulation was very important for the development of quantum field theory. Both the Schrodinger and Heisenberg approaches to quantum mechanics single out time, and are not in the spirit of relativity.

— Wikipedia on Path integral formulation

2010.01.23 Saturday ACHK

Category theory is a unifying theory of mathematics that was initially developed in the second half of the 20th century. In this respect it is an alternative and complement to set theory. A key theme from the “categorical” point of view is that mathematics requires not only certain kinds of objects (Lie groups, Banach spaces, etc.) but also mappings between them that preserve their structure.

In particular, this clarifies exactly what it means for mathematical objects to be considered to be the same.

— Wikipedia on Unifying theories in mathematics

2010.01.11 Thursday ACHK

… more prescriptive. It frames a possible template for any mathematical theory: the theory should have nouns and verbs, i.e., objects, and morphisms, and there should be an explicit notion of composition related to the morphisms; the theory should, in brief, be packaged by a category. There is hardly any species of mathematical …

— When is one thing equal to some other thing?, Barry Mazur

2010.01.20 Wednesday ACHK

I should explain that “Yang-Mills” theories are nonlinear generalizations of electrodynamics which can describe forces such as the weak and strong interactions.

— How Far Are We from the Quantum Theory of Gravity?, R. P. Woodard

Yang–Mills theory is a gauge theory based on the SU(N) group. It was formulated by Yang and Mills in 1954 in an effort to extend the original concept of gauge theory for an Abelian group, as was quantum electrodynamics, to the case of a nonabelian group with the intention to get an explanation for strong interactions.

— Wikipedia on Yang–Mills theory

2010.01.19 Tuesday ACHK

The dynamics of photons in the double-slit experiment describes the relationship between classical electromagnetic waves and photons, the quantum counterpart of classical electromagnetic waves, in the context of the double-slit experiment. Superficially, it may look like the dynamics of a photon can be completely described by the classical Maxwell’s equations with only a reinterpretation of the classical field as a probability amplitude for the photon, however this notion is fraught with danger and ultimately leads to contradictions. One should not simply assume that the electromagnetic fields are a wave-function for the photon. For one thing, they are real and thus contain both positive and negative frequency components, which cannot be reconciled with the requirement for Schroedinger wavefunctions which are complex, positive frequency only. In addition, the electromagnetic fields are observable (e.g. with an oscilloscope) while Schroedinger wavefunctions are not observable, even in principle. Clearly, then, the fields are not wavefunctions, are physical, observable fields, rather than merely what you take the modulus-square of to obtain the probability of finding a photon somewhere. The existence of some “wavefunction of the photon” is not a fully settled issue.

— Wikipedia on Photon dynamics in the double-slit experiment

2010.01.17 Sunday ACHK

In 2003, Edward Witten proposed to marry twistor and string theory by embedding the topological B model of string theory in twistor space. His objective was to model certain Yang-Mills amplitudes. The resulting model has come to be known as twistor string theory.

Witten (2004) built on this insight to propose a way to do string theory in twistor space, whose dimensionality is necessarily the same as that of 3+1 Minkowski spacetime. Hence twistor string theory is a possible way to eliminate the need for more than 3 spatial dimensions when doing (super)string theory.

— Wikipedia on Twistor theory

2009.12.30 Wednesday ACHK