Ray optics made refraction rustic enough, as path of the light is represented by a straight line. This paves in surplus excitement to the human mind in search of rationale for every thing that it can conceive.
1) The speed of light in rarer(1) and denser(2) medium be v and c respectively. 2) For the sake of brevity light ray travel for same time in both the medium.
Logical offsets : i > r i,r <= 90 deg Therefore, Sin(i) > Sin(r)
The introduction of sine function as the consequence of Snell's law, where
Sin(i)/Sin(r) = n
n = (x1/x2)*(d2/d1) = (x1/x2)*((d2/t)*(t/d1)) = (x1/x2)*(v/c)
"Light ray travelling for same time in both the medium" was very much essential for the mathematical frame work for the three equations above. Lets analyse what happens if times were not equal. Now I will change adjustment 2) -> 2') (I will put it as adjustment rather than assumption).
2') Light travels for time t1 in medium (1) and time t2 in medium (2).
Rationale A :
Even if t1 not equalling t2 the ratio v/c do not get altered. This is 'physical' argument which steers the mathematical frame work. Therefore,
n = (x1/x2)*((d2/t2)*(t1/d1)) = (x1/x2)*(d2/d1)*(t1/t2)
Rationale B :
From 2') Light travels for time t1 in medium (1) and time t2 in medium (2).
n = (x1/x2)*((d2/t2)*(t1/d1)) = (x1/x2)*(d2/d1)*(t1/t2)
The above steps are mathematically inconsistent, because I am introducing a new ratio (t1/t2), where t1 not equalling t2, yet I am claiming it to equal 'n'.
Perspective :
Can rationality of human thought conceive every thing the nature provides ?
This makes me wonder, even if the laws of nature may be written in the language of mathematics but the limitations of the language nests the loss of many things which go untold.
I recently saw an old friend for the first time in many years. We had been Ph.D. students at the same time, both studying science, although in different areas. She later dropped out of graduate school, went to Harvard Law School and is now a senior lawyer for a major environmental organization. At some point, the conversation turned to why she had left graduate school. To my utter astonishment, she said it was because it made her feel stupid. After a couple of years of feeling stupid every day, she was ready to do something else.
I had thought of her as one of the brightest people I knew and her subsequent career supports that view. What she said bothered me. I kept thinking about it; sometime the next day, it hit me. Science makes me feel stupid too. It's just that I've gotten used to it. So used to it, in fact, that I actively seek out new opportunities to feel stupid. I wouldn't know what to do without that feeling. I even think it's supposed to be this way. Let me explain.
For almost all of us, one of the reasons that we liked science in high school and college is that we were good at it. That can't be the only reason – fascination with understanding the physical world and an emotional need to discover new things has to enter into it too. But high-school and college science means taking courses, and doing well in courses means getting the right answers on tests. If you know those answers, you do well and get to feel smart.
A Ph.D., in which you have to do a research project, is a whole different thing. For me, it was a daunting task. How could I possibly frame the questions that would lead to significant discoveries; design and interpret an experiment so that the conclusions were absolutely convincing; foresee difficulties and see ways around them, or, failing that, solve them when they occurred? My Ph.D. project was somewhat interdisciplinary and, for a while, whenever I ran into a problem, I pestered the faculty in my department who were experts in the various disciplines that I needed. I remember the day when Henry Taube (who won the Nobel Prize two years later) told me he didn't know how to solve the problem I was having in his area. I was a third-year graduate student and I figured that Taube knew about 1000 times more than I did (conservative estimate). If he didn't have the answer, nobody did.
That's when it hit me: nobody did. That's why it was a research problem. And being my research problem, it was up to me to solve. Once I faced that fact, I solved the problem in a couple of days. (It wasn't really very hard; I just had to try a few things.) The crucial lesson was that the scope of things I didn't know wasn't merely vast; it was, for all practical purposes, infinite. That realization, instead of being discouraging, was liberating. If our ignorance is infinite, the only possible course of action is to muddle through as best we can.
I'd like to suggest that our Ph.D. programs often do students a disservice in two ways. First, I don't think students are made to understand how hard it is to do research. And how very, very hard it is to do important research. It's a lot harder than taking even very demanding courses. What makes it difficult is that research is immersion in the unknown. We just don't know what we're doing. We can't be sure whether we're asking the right question or doing the right experiment until we get the answer or the result. Admittedly, science is made harder by competition for grants and space in top journals. But apart from all of that, doing significant research is intrinsically hard and changing departmental, institutional or national policies will not succeed in lessening its intrinsic difficulty.
Second, we don't do a good enough job of teaching our students how to be productively stupid – that is, if we don't feel stupid it means we're not really trying. I'm not talking about `relative stupidity', in which the other students in the class actually read the material, think about it and ace the exam, whereas you don't. I'm also not talking about bright people who might be working in areas that don't match their talents. Science involves confronting our `absolute stupidity'. That kind of stupidity is an existential fact, inherent in our efforts to push our way into the unknown. Preliminary and thesis exams have the right idea when the faculty committee pushes until the student starts getting the answers wrong or gives up and says, `I don't know'. The point of the exam isn't to see if the student gets all the answers right. If they do, it's the faculty who failed the exam. The point is to identify the student's weaknesses, partly to see where they need to invest some effort and partly to see whether the student's knowledge fails at a sufficiently high level that they are ready to take on a research project.
Productive stupidity means being ignorant by choice. Focusing on important questions puts us in the awkward position of being ignorant. One of the beautiful things about science is that it allows us to bumble along, getting it wrong time after time, and feel perfectly fine as long as we learn something each time. No doubt, this can be difficult for students who are accustomed to getting the answers right. No doubt, reasonable levels of confidence and emotional resilience help, but I think scientific education might do more to ease what is a very big transition: from learning what other people once discovered to making your own discoveries. The more comfortable we become with being stupid, the deeper we will wade into the unknown and the more likely we are to make big discoveries.
While the registration for the school is closed, we welcome you to
participate in the technical programme and interact with the
participants of the school (there are about seventy outstation
participants).
List of colloquia (Time: 4:00PM to 5:30PM on dates indicated)
Date : 2nd July, 2011
Speaker : Jainendra Jain
Title : "Exotic Emergent Particles in Fractional Quantum Hall Effect"
Date : 5th July, 2011
Speaker : Klaus von Klitzing
Title : "30 years of Quantum Hall Effects"
Date : 7th July, 2011
Speaker : Shun-Qing Shen
Title : "Surface-edge states and half-quantized Hall conductance in
topological insulator"
Date : 11th July, 2011
Speaker : Steven H.Simon
Title : "Topological matter and why you should be interested"
Date : 12th July, 2011
Speaker : Rahul Roy
Title : "Topological insulators and adiabatic cycles"
From the prologue of "Chaos and Life- complexity and order in evolution and thought":
About a hundred thousand years ago, perhaps as many as a hundred and fifty thousand or as little as fifty thousand, the first man was born. In the eyes of his parents he must have been an ugly baby, an outcast from the brood and from the tribe from the moment of his birth. He was lucky to survive, for monstrous births were not generally suffered to live in that time and many more like him would have perished, and would in other times and places perish, by accident or design, and without further consequence. He was different from his tribe: his features to them seemed curiously unnatural: angular, sharp, and distorted. His gracile frame seemed spindly and ill adapted to survive in the rigorous climate, which could vary every few years, alternating between humid–hot and freezing–dry conditions, interspersed with a temperate mediocrity hardly more favorable, since it ill suited the growth of the plants on which the tribes depended. To the society of the race he was a useless individual—an idle jack—spending much time in seemingly depressed, introverted contemplation; staring into space; fiddling with pieces of stone and slate; or making marks in the earth with a stick. Perhaps his social experience was the origin of the story of the Ugly Duckling. The first Outsider, he must have been familiar with loneliness in a time when loneliness was a difficult condition to achieve and an undesirable one. Let us call him Adam. Adam sat for long periods between gathering plants or catching animals, when he was not shivering too much to think or too exhausted by the sun’s heat. And in those periods when he was able to reflect, he contemplated the great mystery that was his existence in the world. He could speak in a way—the gift of tongues—that the tribe could not, but he had no one to speak to. He made peculiar guttural and explosive sounds, and he seemed to gesture, in apparent madness, at the beasts and plants and rocks, and even at the sky, as he made these strange noises. This was the birth of language, and he was naming the things around him.
-Richard J. Bird
"The more things change, the more they stay the same." is a line I've read quite often. I think the truth of this statement is reflected in the excerpt above. Many of us here, I'm sure, will be able to empathize with "Adam", because thinking, leave alone independent thinking, is not something that is encouraged, or sometimes even allowed today. In fact, the conventional "System", particularly the education system, seeks to stamp it out of our minds as quickly and as thoroughly as possible. So much for evolution.
Here is an extract from the book "The Dancing Wu Li Masters":
""What does physics have in common with enlightenment? Physics and enlightenment apparently belong to two realms which are forever separate. One of them (physics) belongs to the external world of physical phenomena and the other of them (enlightenment) belongs to the internal world of perceptions. A closer examination, however, reveals that physics and enlightenment are not so incongruous as we might think. First,
there is the fact that only through our perceptions can we observe physical phenomena. In addition to this obvious bridge, however, there are more intrinsic similarities.
Enlightenment entails casting off the bonds of concept ("veils of ignorance") in order to perceive directly the inexpressible nature of undifferentiated reality. "Undifferentiated reality" is the same reality that we are a part of now, and always have been a part of, and always will be a part of. The difference is that we do not look at it in the same way as an enlightened being. As everyone knows(?), words only represent (re-present) something else. They are not real things. They are only symbols. According to the philosophy of enlightenment, everything (everything) is a symbol. The reality of symbols is an illusory reality. Nonetheless, it is the one in which we live. Although undifferentiated reality is inexpressible, we can talk around it (using more symbols). The physical world, as it appears to the unenlightened, consists of many separate parts. These separate parts, however, are not really separate. According to mystics from around the world, each moment of enlightenment (grace/insight/samadhi/satori) reveals that everything—all the separate parts of the universe—are manifestations of the same whole There is only one reality, and it is whole and unified. It is one.
We already have learned that understanding quantum physics requires a modification of ordinary conceptions (like the idea that something cannot be a wave and a particle). Now we shall see that physics may require a more complete alteration of our thought processes than we ever conceived or, in fact,
than we ever could conceive. Likewise we previously have seen that quantum phenomena seem to make decisions, to "know" what is happening elsewhere (page 62). Now we shall see how quantum phenomena may be connected so intimately that things once dismissed as 'occult' could become topics of
serious consideration among physicists. In short, both in the need to cast off ordinary thought
processes (and ultimately to go beyond thought altogether), and in the perception of reality as one unity, the phenomenon of enlightenment and the science of physics have much in common. Enlightenment is a state of being. Like all states of being it is indescribable. It is a common misconception to mistake the description of a state of being for the state itself For example, try to describe happiness. It is impossible. We can talk around it, we can describe the perspectives and actions that usually accompany a state of happiness, but we cannot describe happiness itself. Happiness and the description of happiness are two different things. Happiness is a state of being. That means that it exists in the realm of direct experience. It is the intimate perception of
emotions and sensations which, indescribable in themselves, constitute the state of happiness. The word "happiness" is the label, or symbol, which we pin on this indescribable state. ' Happiness" belongs to the realm of abstractions, or concepts. A state of being is an experience. A description of a state of being is a symbol. Symbols and experience do not follow, the same rules. This discovery, that symbols and experience do not follow the same rules, has come to the science of physics under the formidable title of quantum logic. The possibility that separate parts of reality (like you and I and tugboats) may be connected in ways which both our common experience and the laws of physics belie, has found its way into physics under the name
of Bell's theorem. Bell's theorem and quantum logic take us to the farthest edges of theoretical physics. Many physicists have not even heard of them. Bell's theorem and quantum logic (currently) are unrelated. Proponents of one seldom are interested in the other. Nonetheless, they have much in common. They are what is really new in physics. Of course, laser fusion (fusing atoms with high-energy light beams) and the search for quarks generally are considered to be the frontiers of theoretical physics. In a certain sense, they are. However, there is a big difference between these projects and Bell's theorem and quantum logic.
Laser fusion research and the great quark hunt are endeavors within the existing paradigms of physics. A paradigm is an established thought process, a framework. Both quantum logic and Bell's theorem are potentially explosive in terms of existing frameworks. The first (quantum logic) calls us back from the
realm of symbols to the realm of experience. The second (Bell's theorem) tells us that there is no such thing as "separate parts ". All of the " parts " of the universe are connected in an intimate and immediate way previously claimed only by mystics and other scientifically objectionable people. The central mathematical element in quantum theory, the hero of the story, is the "wave function".
The wave function is that mathematical entity which allows us to determine the possible results of an interaction between an observed system and an observing system. The celebrated position held by the wave function is due not only to Erwin Schrodinger, who discovered it, but also to the Hungarian mathematician, John von Neumann. In 1932, von Neumann published a famous mathematical analysis of quantum theory called 'The Mathematical Foundations of Quantum Mechanics '. In this book von Neumann, in effect, asked the question, "If a wave function , this purely abstract mathematical creation, actually should describe something in the real world, what would that something be like?" The answer that he deduced is exactly the description of a wave function that we already have discussed (page 73). This strange animal constantly would change with the passage of time. Each moment it would be different than the moment before. It would be a composite of all the possibilities of the observed system which it describes. It would not be a simple mixture of possibilities, it would be a sort of organic whole whose parts are changing constantly but which, nonetheless, is somehow a thing-in-itself. This thing-in-itself would continue to develop indefinitely until an observation (measurement) is made on the observed system which it represents. If the observed system is a photon propagating in isolation the wave function representing this photon would contain all of the possible results of the photon's interaction with a measuring device, like a photographic plate (For example, the possibilities contained in the wave function might be that the photon will be detected in area A of the photographic plate, that the photon will be detected in area B of the photographic plate and that the photon will be detected in area C of the photographic plate).
Once the photon is set in motion the wave function associated with it would continue to develop (change) according to a causal law (the Schrodinger wave equation) until the photon interacts with the observing system. At that instant, one of the possibilities contained in the wave function would actualize and the other possibilities contained in the wave function would cease to exist. They simply would disappear. The wave
function, that strange animal that von Neumann was attempting to describe, would collapse. The collapse of this particular wave function would mean that the probability of one of the possible results of the photon-measuring-device interaction became one (it happened) and the probability of the other possibilities became zero (they were no longer possible). After all a photon can be detected only in one place at a time.
The wave function, according to this view, is not quite a thing yet it is more than an idea. It occupies that strange middle ground between idea and reality, where all things are possible but none are actual. Heisenberg likened it to Aristotle's potentia (page 66). This approach has unconsciously shaped the language and therefore the thinking of most physicists, even those who consider the wave function to be a mathematical fiction, an abstract creation whose manipulation somehow yields the probabilities of real events which happen in real (versus mathematical) space and time. ""
- [Ref 4]
This is to announce the following Physics Colloquium.
(NOTE THE SPECIAL DATE)
Title: "Elementary Particles, Strings and Black Holes"
Date and Time: Thursday 20th, 4:00pm
Speaker: Bernard De Wit
Affiliation: Institute for Theoretical Physics,
University of Utrecht,Netherlands
Abstract:
Progress in the theory of elementary particles gave rise to the discovery
of the standard model and of string theory. Some of these developments
will be described, both from a historical and from a more conceptual point
of view. They have had a major impact on our thinking about quantum
gravity, and especially about black holes. The more recent advances in the
theory of black holes in the context of supergravity and string theory
will be explained.
'Einstein, Picasso; The Art of Science and the Science of Art'
a talk by
Prof. Arthur I. Miller
Emeritus Professor of History
& Philosophy of Science,
University College of London
Date & Time: Friday, 14th Jan 2011, 4:00 pm
Venue : Centre for Contemporary Studies Seminar Hall Indian Institute of Science, Bangalore 12
Note our new premises : Former JNCASR, near Health Centre)
All are cordially invited
Tea/Coffee will be served at 3:30 p.m.
Abstract: Almost simultaneously, in the first decade of the 20th Century
Albert Einstein discovered relativity and Pablo Picasso cubism. How - and
why? This fascinating story involves their turbulent personal lives; the
high drama of their struggles to achieve new ideas in the face of
opposition from contemporaries; and the unlikely sources for their
creative leaps, ignored by everyone else. To fully understand what
happened involves coming to grips with wide-ranging questions such as: Are
there similarities in creativity between artists and scientists? What do
artists and scientists mean by 'aesthetics' and 'beauty' ? Can we unravel
creativity at its highest level?
About the speaker: Arthur I. Miller is a well known historian and
philosopher of science. In his own words, he has always been attracted by
"what is the nature of...." questions, and this affinity made him shift
gears from physics to the history of science. He is the author of several
popular level as well as academic books, "Deciphering the Cosmic Number:
The Strange Friendship of Wolfgang Pauli and Carl Jung" being the latest.
Prof. Humphrey J. Maris from Brown University, USA will be visiting IISc from Jan 10th till Jan 21st. He will deliver five lectures; please find the time and venue of the seminars below. More details will follow.
1) Tuesday, Jan 11, 4PM, Physics Lecture Hall-1 The physics of supersolid helium
2) Friday, Jan 13,2.30 PM, Physics Lecture Hall-1 The physics of nucleation processes in liquids and solids
3) Monday: Jan 17 4PM, Physics Lecture Hall-1 Experiments with single electrons in liquid helium and tests of measurement theory in quantum mechanics
4) Wednesday: Jan 19 4PM, Faculty Hall Ultra high frequency ultrasonics and studies of phonon dispersion and solitons
5) Friday: Jan 21 4PM, ECE Golden Jubilee Technology transfer, universities and small business
Humphrey Maris was a student at Imperial College, receiving his PhD degree in 1963. He was a postdoc at Case Institute, before joining the faculty at Brown University. He has been a visiting fellow at the University of East Anglia, Chalmers Institute, and CNRS Grenoble, and visiting professor at the Centro Atomico in Bariloche, the University of Tokyo, the University of Hokkaido, and the Ecole Normale in Paris. He has received the Senior Humboldt Award, the 2007 Klemens prize for phonon physics, and Brown University awards for Technological Innovation and for science teaching. His research interests include phonon physics, ultrasonics, and liquid helium.
DEPARTMENT OF PHYSICS
INDIAN INSTITUTE OF SCIENCE
BANGALORE 560 012
Astrophysics Seminar
Speaker: Pratyush Pranav
Kapteyn Astronomical Institute, Groningen
The Netherlands
Title: PERSISTENT BETTI-TOPOLOGY OF THE COSMIC WEB ALPHASHAPES
Date & Time 13.1.2011 at 4.00 p.m.
Venue: Lecture Hall - I of Physics Department
Abstract:
We analyze the topology of Cosmic Web defined by the pervasive
anisotropic network of structures like clusters, filaments, walls and
voids, and characterized by an inherent multi-scale mass distribution.
For a discrete representation of the underlying mass distribution by
galaxies and dark matter halos, Alphashapes provide us the description
of the intuitive notion of shapes. Following their calculation we may
directly infer the Betti numbers, the prime parameters for quantifying
the tpology of the corresponding surface. In 3-D these represent the
number of independent components, loops and voids enclosed by the
corresponding manifold. In order to differentiate between significant
structure and topological noise, we introduce the concept of
persistence. We demonstrate the sensitivity and discrimination power of
this description with respect to different cosmologies. In the ongoing
work we extend this analysis to continuous surfaces such as that of the
CMB. It might provide a critical step towards constraining primordial
non-Gaussianity.
Using latex is very easy, you just need to know the commands.
Usha ma'am, satyajit, and Raghu are already experts at it :).
Here are a few examples:
$ s = \int L ( q, \dot{q} , t ) dt $
To type this equation, one has to type & s = \int L ( q, \dot{q} , t ) dt &
You just have to replace & with $$.
Any latex commands can be used by enclosing them between $$ .
" There are two kinds of sufferers in this world: those who suffer from a lack of life and those who suffer from an over-abundance of life. I've always found myself in the second category.
When you come to think of it,almost all human behaviour and activity...is not essentially any different from animal behaviour.The most advanced technologies and craftsmanship... bring us, at best, up to the super-chimpanzee level. Actually, the gap between,say, Plato or Nietzsche and the average human...is greater than the gap between that chimpanzee and the average human.The realm of the real spirit, the true artist, the saint, the philosopher, is rarely achieved.
Why so few? Why is world history and evolution not stories of progress...but rather this endless and futile addition of zeroes? No greater values have developed. Hell, the Greeks years ago were just as advanced as we are. So what are these barriers that keep people... from reaching anywhere near their real potential? The answer to that can be found in another question, and that's this:
Which is the most universal human characteristic--
fear or laziness?"
-From the movie "Waking Life"
I was thinking of making the blog more technical and this will require us to be able to type equations. So does anyone know if Latex can be incorporated on the blog? Is there any html script or something using which this can be done?
Please let me know thanks.
For a while now, the correspondence between a mathematical structure and the way nature behaves has puzzled and amazed me. In other words, Why is it that I can write down equations on a sheets of paper and predict something about a natural object?
Newtonian Mechanics had a certain mathematical structure and Maxwell's Electrodynamics had a certain mathematical structure, and these two structures couldn't co-exist. Just by trying make the equations co-variant, we were able to get great insights into the way nature worked. We could get testable predictions which we were ofcourse verified by experiment.
Quantum Mechanics brought in another mathematical structure which takes into account the relationship between the observer and the observed, and gives us the nature of a measurement.
There may be domains where experimental physics cannot yet reach. There may be some underlying physical phenomena which cannot yet be detected by experiments. To probe into those domains, as of now, we may only have mathematical tools at our disposal.
Here is a paragraph from a book on Quantum Mechanics by Bohm ( Ref 3) that I think is worth mentioning:
" Physicists believe that there is something in nature, or in each restricted domain of it, that may be "understood"; that there is a structure in nature. To "understand" means to bring this structure into congruence with some structure in our mind, with a structure of thought objects, with a structure that has been created by our minds. For physics, this structure of thought objects is a mathematical structure. So to understand part of physical nature means to map its structure on a mathematical structure. To obtain a physical theory, then, means to obtain a mathematical image of a physical system. For the domains of quantum physics the mathematical structures are algebras of linear operators in linear spaces. The discovery of this, the fundamental properties of the algebra, and the other basic assumptions of quantum mechanics was a very difficult process. "
I just hope we have the capability to recognise the true underlying structure of nature( however naive this may sound).
P.s. The references are given in the column on the right.
Professor Roger Blandford
Kavli Institute for Particle Astrophysics and Cosmology
Stanford
Date : Wednesday, 8th December 2010
Time: 6.00 p.m.
Venue: Faculty Hall, Indian Institute of Science, Bangalore
Abstract: Professor Subrahmanyan Chandrasekhar, or "Chandra" as he was widely known, was a singular scientist and intellectual. Blessed with formidable mathematical ability and legendary powers of concentration, he was a scientific leader over an unequalled suite of the most challenging astrophysical disciplines. Although he may be most famous for his youthful discovery of a mass limit for white dwarfs and its famous corollary that black holes must exist, for which he was awarded the 1983 Nobel Prize, his lifetime contributions to mathematical physics, astrophysics and even the humanities, are even greater. The range and durability of his scholarship was memorialised in the naming of the finest imaging X-ray telescope ever launched. Vignettes from his life will be interspersed with a description of some of the amazing discoveries made by Chandra X-ray Observatory.
About the speaker: Roger Blandford is director of the Kavli Institute for Particle Astrophysics and Cosmology. He oversees research that seeks to answer some of our great cosmic questions: What powered the Big Bang? What are dark matter and dark energy? What is happening around black holes?. He is a Fellow of The Royal Society and the American Academy of Arts and Sciences, and a Member of the National Academy of Sciences.
Centre for Quantum Information and Quantum Computing
Indian Institute of Science
Bangalore
Announces its lecture
By
Dr. Prabha Mandayam
Institute for Quantum Information
California Institute of Technology, Pasadena, USA
Title: " Uncertainty and Complementarity in Quantum Cryptography: Security in the Noisy Storage model"
On Wednesday the 8th December, 2010 at 4.00 pm
Venue: Lecture Hall - I, Department of Physics, Indian Institute of Science.
Abstract
Historically, uncertainty relations have played an important role in our understanding of quantum mechanics. Recently, entropic uncertainty relations have gained prominence in the field of quantum information. In particular, they play a crucial role in analyzing the security of quantum cryptographic protocols such as quantum key-distribution and information locking. While optimal entropic uncertainty relations have been obtained for two measurements, not much is known in a general setting with more than two outcomes. We outline a novel construction of symmetric mutually unbiased bases (MUBs), using the 2n generators of the Clifford algebra in dimension d = 2^n. These bases satisfy the symmetry property that they are cyclically permuted under a unitary transformation. We prove a lower bound for entropic uncertainty relations for any set of MUBs and show that symmetry plays a central role in obtaining tight bounds. Finally, we describe a two-party protocol based on recent cryptographic models -- the "bounded-storage model" and the "noisy-storage model" -- whose security is directly related to a lower bound on the average entropy of complementary bases. While the security of most cryptographic systems in use today is based on the premise that certain computational problems are hard to solve for the adversary, these models are based on the physical assumption that no large-scale reliable quantum storage is available to the cheating party. Our protocol achieves security in the bounded-storage model even if the cheating party can store all but a constant fraction of the transmitted information, thus resolving an open question on the achievable physical limits of this model. In the general setting of the noisy-storage model, our protocol extends the range of storage devices for which security can be achieved.
I saw an interesting video on how groups were used by Garret in his model for a Theory of Everything. Here is the video. The group he used is an exotic Lie group commonly known as E8.
Most of us have group theory in our degree course. Some time ago, I was pretty critical about pure math and its abstract nature ( having nothing to do with reality). I treated them as just mental fantasies, some of which tends to have a structure that agrees with the way nature nature worked.
I went around telling people about some paradoxes in set theory and how Godel's theorem implies that any system based on precise axioms and set of rules is fundamentally flawed. One famous paradox is Russel's paradox.
I now realize that how I felt back then was wrong. I felt that way partly because I am not good at Math and partly because I my thinking was very naive. Which is not really a bad thing, I always learn from my naiveties.
I am now doing a project in Lie Groups under a professor in my college. And I visit a person named Dr. Aravinda from TIFR. He's helping me out with Topology and Differential Geometry. This is amazing because two years ago I didn't think I would find anyone working on these topics ( and also willing to guide me).
Anyway, I thought I'll put up something here that I found worth reading. This is the preface to one of the books that he suggested:
" As a graduate student in experimental physics, I found the study of group theory considered to be a useless "high-brow" luxury. Furthermore all attempts to follow a lecture course resulted in a losing battle with cosets, classes, invariant subgroups, normal divisors and assorted lemmas. It was impossible to learn all the definitions of new terms defined in one lecture and remember them until the next lecture. The result was complete chaos.
It was a great surprise to find later on that (1) Techniques based on group theory can be useful; (2) They can be learned and used without memorizing the large number of definitions and lemmas which frighten the uninitiated.
Angular momentum is presented in elementary quantum mechanics courses without a detailed analysis of the lie group of continuous rotations in three dimensions. The student learns about angular momentum multiplets and coupling angular momenta without realizing that these are irreducible representations of the rotation group.He also does not realize that the algebraic properties of other lie groups can be applied to physical problems in the same way as he used angular momentum algebra, with no need for classes, cosets etc. . . . . . . "
He then goes on to talk about further applications in creation and annihilation operators, and also talks about quasi spin etc..
The book is Lie groups for Pedestrians by Harry Lipkin.
Group theory is actually fun and beautiful ( a word I rarely use) , but again college takes the life out of it. I am actually lucky that this course is handled in my college by an expert in this topic, but even he is constrained because the college requires you to rush through all the theorems within one semester and one is not given enough time to understand the subtleties. But it is always best continue to reflect on it even after the course is done.
P.S: The weekend sessions have been going on. Everybody has just been too busy to put up the summary. We have had three sessions within a span of four days. Karthik made quite a lot of connections in classical mechanics during his class on friday. I conducted a test on saturday on whatever has been done, to judge the understanding so far. Today was Shruthi's session on QM, which was again taken up mostly by Karthik to make known previously unspoken things.
