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Sunday, June 27, 2010

Virasoro algebra

Black hole thermodynamics

From Wikipedia, the free encyclopedia

In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. Much as the study of the statistical mechanics of black body radiation led to the advent of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.

 It is important that ones is able to see the progression from abstraction to a interpretation of foundational approach.

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Andy Strominger:
This was a field theory that lived on a circle, which means it has one spatial dimension and one time dimension. We derived the fact that the quantum states of the black hole could be represented as the quantum states of this one-plus-one dimensional quantum field theory, and then we counted the states of this theory and found they exactly agreed with the Bekenstein-Hawking entropy.See:Quantum Microstates: Gas Molecules in the Presence of a Gravitational Field

See:Microscopic Origin of the Bekenstein-Hawking Entropy

Of course I am interested the mathematical framework as it might be compared to some phenomenological approach that gives substance to any theoretical thought.

For example, Tommaso Dorigo is a representative of the type of people who may affect the general distribution of "subjects" that may grow at CERN or the Fermilab in the next decade or two. And he just published a quote by Sherlock Holmes - no kidding - whose main point is that it is a "capital mistake" to work on any theory before the data are observed.See:Quantum gravity: minority report

I think you were a little harsh on Tommaso Dorigo  Lubos because he is really helping us to understand the scientific process at Cern. But you are right about theory in my mind, before the phenomenological approach can be seen. The mind need to play creatively in the abstract notions before it can be seen in it's correlations in reality.

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Virasoro algebra

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Group theory
Rubik's cube.svg
Group theory
In mathematics, the Virasoro algebra (named after the physicist Miguel Angel Virasoro) is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in string theory.

Contents


Definition

The Virasoro algebra is spanned by elements
Li for i\in\mathbf{Z}
and c with
Ln + L n
and c being real elements. Here the central element c is the central charge. The algebra satisfies
[c,Ln] = 0
and
[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}.
The factor of 1/12 is merely a matter of convention.
The Virasoro algebra is a central extension of the (complex) Witt algebra of complex polynomial vector fields on the circle. The Lie algebra of real polynomial vector fields on the circle is a dense subalgebra of the Lie algebra of diffeomorphisms of the circle.
The Virasoro algebra is obeyed by the stress tensor in string theory, since it comprises the generators of the conformal group of the worldsheet, obeys the commutation relations of (two copies of) the Virasoro algebra. This is because the conformal group decomposes into separate diffeomorphisms of the forward and back lightcones. Diffeomorphism invariance of the worldsheet implies additionally that the stress tensor vanishes. This is known as the Virasoro constraint, and in the quantum theory, cannot be applied to all the states in the theory, but rather only on the physical states (confer Gupta-Bleuler quantization).

Representation theory

A lowest weight representation of the Virasoro algebra is a representation generated by a vector v that is killed by Li for i ≥1 , and is an eigenvector of L0 and c. The letters h and c are usually used for the eigenvalues of L0 and c on v. (The same letter c is used for both the element c of the Virasoro algebra and its eigenvalue.) For every pair of complex numbers h and c there is a unique irreducible lowest weight representation with these eigenvalues.
A lowest weight representation is called unitary if it has a positive definite inner product such that the adjoint of Ln is Ln. The irreducible lowest weight representation with eigenvalues h and c is unitary if and only if either c≥1 and h≥0, or c is one of the values
 c = 1-{6\over m(m+1)} = 0,\quad 1/2,\quad 7/10,\quad 4/5,\quad 6/7,\quad 25/28, \ldots
for m = 2, 3, 4, .... and h is one of the values
 h = h_{r,s}(c) = {((m+1)r-ms)^2-1 \over 4m(m+1)}
for r = 1, 2, 3, ..., m−1 and s= 1, 2, 3, ..., r. Daniel Friedan, Zongan Qiu, and Stephen Shenker (1984) showed that these conditions are necessary, and Peter Goddard, Adrian Kent and David Olive (1986) used the coset construction or GKO construction (identifying unitary representations of the Virasoro algebra within tensor products of unitary representations of affine Kac-Moody algebras) to show that they are sufficient. The unitary irreducible lowest weight representations with c < 1 are called the discrete series representations of the Virasoro algebra. These are special cases of the representations with m = q/(pq), 0<r<q, 0< s<p for p and q coprime integers and r and s integers, called the minimal models and first studied in Belavin et al. (1984).
The first few discrete series representations are given by:
  • m = 2: c = 0, h = 0. The trivial representation.
  • m = 3: c = 1/2, h = 0, 1/16, 1/2. These 3 representations are related to the Ising model
  • m = 4: c = 7/10. h = 0, 3/80, 1/10, 7/16, 3/5, 3/2. These 6 representations are related to the tri critical Ising model.
  • m = 5: c = 4/5. There are 10 representations, which are related to the 3-state Potts model.
  • m = 6: c = 6/7. There are 15 representations, which are related to the tri critical 3-state Potts model.
The lowest weight representations that are not irreducible can be read off from the Kac determinant formula, which states that the determinant of the invariant inner product on the degree h+N piece of the lowest weight module with eigenvalues c and h is given by
  A_N\prod_{1\le r,s\le N}(h-h_{r,s}(c))^{p(N-rs)}
which was stated by V. Kac (1978), (see also Kac and Raina 1987) and whose first published proof was given by Feigin and Fuks (1984). (The function p(N) is the partition function, and AN is some constant.) The reducible highest weight representations are the representations with h and c given in terms of m, c, and h by the formulas above, except that m is not restricted to be an integer ≥ 2 and may be any number other than 0 and 1, and r and s may be any positive integers. This result was used by Feigin and Fuks to find the characters of all irreducible lowest weight representations.

Generalizations

There are two supersymmetric N=1 extensions of the Virasoro algebra, called the Neveu-Schwarz algebra and the Ramond algebra. Their theory is similar to that of the Virasoro algebra.
The Virasoro algebra is a central extension of the Lie algebra of meromorphic vector fields on a genus 0 Riemann surface that are holomorphic except at two fixed points. I.V. Krichever and S.P. Novikov (1987) found a central extension of the Lie algebra of meromorphic vector fields on a higher genus compact Riemann surface that are holomorphic except at two fixed points, and M. Schlichenmaier (1993) extended this to the case of more than two points.

History

The Witt algebra (the Virasoro algebra without the central extension) was discovered by E. Cartan (1909). Its analogues over finite fields were studied by E. Witt in about the 1930s. The central extension of the Witt algebra that gives the Virasoro algebra was first found (in characteristic p>0) by R. E. Block (1966, page 381) and independently rediscovered (in characteristic 0) by I. M. Gelfand and D. B. Fuks (1968). Virasoro (1970) wrote down some operators generating the Virasoso algebra while studying dual resonance models, though he did not find the central extension. The central extension giving the Virasoro algebra was rediscovered in physics shortly after by J. H. Weis, according to Brower and Thorn (1971, footnote on page 167).

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Wednesday, June 23, 2010

Stephen Hawking At PI Institute



Waterloo, Ontario, Canada, June 20, 2010 - In a public address before a packed audience at Perimeter Institute for Theoretical Physics (PI), Prof. Stephen Hawking, PI Distinguished Research Chair, recounted his research, life and times, saying that it has been a glorious period to contribute to our picture of the universe. Prof. Hawking is conducting private research activities at PI this summer, in what is expected to be the first of many visits.  

See: Stephen Hawking on Perimeter Institute and Special Places & Times for Scientific Progress

About TVO

TVO is Ontario's public educational media organization and a trusted source of interactive educational content that informs, inspires and stimulates curiosity and thought. TVO's vision is to empower people to be engaged citizens through educational media. The TVO signal can be found across Canada on Bell TV channel 265 or Shaw Direct channel 353. You will also find TVO on channel 2 via cable or over-the-air in most areas of Ontario.


Prof. Hawking's lecture will air on TVO on:
     - Sunday, June 20 at 8:00 pm and
       12:30 am EDT
     - Saturday, June 26 at 6:00 pm EDT
     - Sunday, June 27 at 5:00 pm EDT
     - Tuesday, July 6 at 10:00 pm EDT

TVO is viewable across Canada on Bell TV channel 265, Shaw Direct channel 353, and channel 2 in most areas of Ontario. 

Tuesday, June 22, 2010

Einstein Tower

Just wondering when the Einstein Tower was built?

See:Science Park "Albert Einstein" Potsdam

The connection to the design of the tower and the comment on pueblo design sparked familiarity with a image of a tower on the edge of the grand canyon and my posting on the Old One. 13.7 blog just recently had a blog posting on the religiosity of Einstein.

Desert View Watchtower was built in 1932 and is one of Mary Colter's best-known works. Situated at the far eastern end of the South Rim, 27 miles (43 km) from Grand Canyon Village, the tower sits on a 7,400 foot (2,256 m) promontory. It offers one of the few views of the bottom of the Canyon and the Colorado River. It is designed to mimic an Anasazi watchtower though it is larger than existing ones.[18]

I was wondering if there was some correlation that inspired Einstein with the Einstein Tower with that architectural design of the native culture?

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It is designed to mimic an Anasazi watchtower though it is larger than existing ones
Picture of Einstein was in 1931 while tower was 1932?

Anyway, I thought this picture important from a mandalic understanding of giving a historical example of what can be embedded in the very soul of an individual, as if this is an example of the foundations of mathematics depicted even historically cast in design and what is common among human beings today in their foundational search for meaning.



Fred Kabotie (c.1900 - 1986) was a famous Hopi artist. Born Nakayoma (Day After Day) into the Bluebird Clan at Songo`opavi, Second Mesa, Arizona, Kabotie attended the Santa Fe Indian School, and learned to paint. In 1920, he entered Santa Fe High School, and commenced a long association with Edgar Lee Hewett, a local archaeologist, working at such excavations as Jemez Springs, New Mexico and Gran Quivira. He also sold paintings for spending money.

In 1926, Kabotie moved to Grand Canyon, Arizona, working for the Fred Harvey Company as a guide. After various other jobs and travel, he was hired in 1932 by Mary Colter to paint his first murals at her new Desert View Watchtower.

Kabotie went on to a distinguished career as a painter, muralist, illustrator, silversmith, teacher and writer of Hopi Indian life. He continued to live at Second Mesa. Kabotie was instrumental in establishing the Hopi Cultural Center and served as its first president.

Fred's son Michael Kabotie (born 1942) is also a well-known artist.

Source: Jessica Welton, The Watchtower Murals, Plateau (Museum of Northern Arizona), Fall/Winter 2005. ISBN 0897341325

Saturday, June 05, 2010

Quasicrystal and Information

Consequently, a universe where time is real must be loveless. I don't like that idea.Impressions from the PI workshop on the Laws of Nature

Quasicrystals are structural forms that are both ordered and nonperiodic. They form patterns that fill all the space but lack translational symmetry. Classical theory of crystals allows only 2, 3, 4, and 6-fold rotational symmetries, but quasicrystals display symmetry of other orders (folds). They can be said to be in a state intermediate between crystal and glass. Just like crystals, quasicrystals produce modified Bragg diffraction, but where crystals have a simple repeating structure, quasicrystals are more complex.

Aperiodic tilings were discovered by mathematicians in the early 1960s, but some twenty years later they were found to apply to the study of quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography and solid state physics. Quasicrystals had been investigated and observed earlier[1] but until the 80s they were disregarded in favor of the prevailing views about the atomic structure of matter.

Roughly, an ordering is non-periodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled; i.e. the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two dimensions. The ability to diffract comes from the existence of an indefinitely large number of elements with a regular spacing, a property loosely described as long-range order. Experimentally the aperiodicity is revealed in the unusual symmetry of the diffraction pattern, that is, symmetry of orders other than 2, 3, 4, or 6. The first officially reported case of what came to be known as quasicrystals was made by Dan Shechtman and coworkers in 1984.[2] The distinction between quasicrystals and their corresponding mathematical models (e.g. the three-dimensional version of the Penrose tiling) need not be emphasized.
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What Is Information? by Stuart Kauffman
Put briefly — and Schrodinger did not say so guessing his intuition is up to us — I think his intuition was that an aperiodic crystal breaks a lot of symmetries, therefore contains a lot of (micro) constraints that can enable an enormous diversity of real and organized processes to happen physically. This idea of organized processes seems to be hinted at in his statement that the aperiodic crystal would contain a microcode for (generating) the organism. I have inserted “generating”, and this is the set of specific processes aspect of information that I think we need to incorporate into our idea of what information IS.  I think Schrodinger is telling us both a deeper meaning of what information “is”, and part of how the universe got complex — by repeatedly breaking symmetries that enabled organized processes to happen that both provided new sources of free energy and enabled the breaking of further symmetries.