Ursula Franklin

At the International Conference on Women in Physics (ICWIP) 2014, held in WLU, Waterloo Canada, Ursula Franklin had a book signing event. Frail and hardly audible a 92 year old woman had the audience listen to her ideas about feminism, science and peace. I was one of the organizers, and had been busy behind the scenes getting the workshops scheduled in time, and was not in a state of mind to appreciate her in details. However, just her presence and the reverence in the audience showed that she was very important. Two years later, she died at the age of 94, and coincidentally Michael Steinitz asked me and Adriana Predoi-Cross to edit a commemorative Festschrift in her honour. I had my sabbatical that year, and readily agreed. As I researched and corresponded with her colleagues, family and staff at University of Toronto, it became obvious that she was greatly respected and appreciated. We sought scientific articles from researchers who identified with her ideals and though the time frame was tight, we managed to get 12 submissions. We followed those with articles written for and about her, and the Festschrift now has 18 contributions, all dedicated to this great woman.

Ursula Franklin Commemorative Festschrift

Recently, I gave a talk on her at the undergraduate conference in physics (CUPC). Her ideas on pacifism, feminism, and analysis of technology resonated with mine, and this is very rare. A person to be inspired with, a role model to cherish, unbelievably true. One of her observations on war stuck me to the core :war earlier was fought in battle fields and men came home from the war, with the modern technology all that changed, and war came home; (modern war bombs civilian cities) Ursula argued that in the post world war scenarios, no conflict had been resolved by war, a destructive and extremely costly activity which served no purpose. She actively called for a stop to use of tax payer’s money for military purposes and an end to the generation of `unknown enemies’ to perpetuate the military.
But, anyways, here are the slides of the talk; ursulafranklinppt

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The culture of Science and a 200 year old Legacy

Its hard to say where it all began, the first wheel, the first spark of electric static, the measurement of Earth’s radius, the first papyrus, the first first printing press, the first fraction, or that theorem. However, we currently agree that science has existence as a human endeavour. There is a culture of science, an indescribable quality of that activity, in which we belong to no country, no time, no age or no religion. If you assimilate science according to the country from which the it come from, or from the identity’ of the authors name be it gender or race, well then that’s not science.  However, overall there are features, there are distinctive elements which define some groups. And it is undoubted that much of modern science was developed in Europe. Isaac Newton was attributed with one of the first attempts to describe a rational causation for dynamics in nature. Newton’s laws of motion bring a logical certainty to natural phenomena, out from the hands of gods and their saintly followers. Or when we observe how C. A. Coulomb quantified interactions of lets say a positively charged glass rod and a negatively charged object, he was contributing in a way someone observing the same static Electric phenomena in India was not at that time of human history.

This brings me to the roots of my scientific education, Presidency College, Kolkata. Even though the milieu in our times was rather terrible, our courses did not provide us much, the name carried with it a great scientific legacy. The college grew up when the British were still ruling India, and Kolkata was their Indian capital. European science and its influence was easily carried from thousands of kms. Jagadish Chandra Bose, observed that plants are alive, and discovered the first radio waves. M. N. Saha wrote the first equation for thermal ionisation in astrophysics, and Satyen Bose sent his paper on the Bose-Einstein statistics to Einstein. The pre-independence era is a roll call of who's and who of important contributors to international science. This legacy continued for sometime, the year we entered, Prof. A. K. Raychowdhury, famous for his Raychowdhury Equation‘ which formed the basis of Hawking-Penrose singularity theorems retired.

In 1994, when we joined this esteemed line of great scientists, much of this legacy was in the fade out. The elitism of the college was in question, and the academic cluster had been disbanded. The mantra in our times was `escaping the system’ rather than staying and struggling, and very soon we were planning our exits. Some of us who wished to stay in the country were advised to leave Kolkata, try in the south, Bombay, Pune. Our era was that of skepticism and cynicism of the educational system Kolkata had to offer, and we were arrogant, and disrespectful. Except for one lone Shyamal Sengupta, who taught us statistical mechanics, and told us stories which inspired us towards research, we forgot every aspect the college had to offer in 1991-1994.

About 22 years later, thus it was a pleasant surprise that the college had rebounded from that bleak phase.  I went there to attend the Bicentennial celebrations, a series of lectures and debates in January 2017. As a ordinary visitor, I walked into the solid pillars of the physics department, and recounted the many hours of the classes we had sat through. There was no women’s washroom, at that time, one of the indicators, that there were very few women in the two hundred years of the existence of the now converted University. But, here now, there was a door with `women’ written on it? I snapped with my cell phone, and stared ruefully at the class which was ongoing inside one of the class rooms, really the Physics department had modernized…

Asif woken from a reverie, I turned around to see a young person smiling at me, `Are you a Physics alumni? I am too from 2001 batch’, soon we got several young people around us, and of course we started telling each other what we love most, physics.  I invited myself to a colloquium, and tell them the story of coherent states and more.

I was shown around the Physics department, the faculty had offices! In our times, the faculty had used Almirah’s to cordon off some space as office'. I saw research labs with functional instruments and was told frontier research happens in Graphene. In our times, Ghoshals nuclear physics lab still existed, but was considered inhabited by ghosts and dusts, no one had used it for decades. Now there was a upgraded computer lab, where students learnt coding...in our times we spent answering questions from a 10 year answer code’ to pass our 4 hour centrally conducted exams.

A department had been raised from almost extinction, there was money and a future. As a reflection, the faculty told me that students were respectful, wanted to work in ernest science.

As I walked out after my colloquium, and had taken up one and a half hours of the students time, talking to them about quantum gravity and nature of reality, I thought the legacy of discoveries should continue.

I though had a sinking feeling that in those two hundred years, the only two women who were projected by the Presidency University were two Mrs. Sens.





But as a true scientist I said lets move on and find the answer to the next big puzzle for humans, which in my colloquium I projected as `finding the origins of consciousness.’




The march of human science continues, and indeed there are different interpretations of reality, and what we label as scientific reality. The quantification of Coulomb’s law has lead to many advances of science and technology. However, it does not matter how we measure nature, if we can use it to build new reality.  The local reality of every human brain can open up new avenues of scientific discovery anywhere in the universe. It is important that we observe, persist, and discuss natural phenomena. Who knows what consciousness is and how the discovery of the origins of that might influence our future history?

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Celebrating 100 years of General Relativity

In 1915 Albert Einstein presented a new theory of gravity, to the Prussian Academy of Sciences, which changed our perception of physical reality. 100 years later, the theory has retained its success despite many tests, and is yet mysterious in many ways. Bizarre objects such as black holes, wormholes seed galaxies and promise reality of time travel, yet we are searching for direct evidence of `gravity waves’ ripples of space-time carrying gravitational energy. These waves should be detected as they are same as light waves, the carriers of electromagnetic energy.

On December 10th, 2015, the department of physics and astronomy at University of Lethbridge celebrated the 100th anniversary of the discovery of this theory as an outreach activity. The event was well attended by members of the public, and of the university community. The entire event was very enjoyable and fielded curiosity about futuristic science as seen in popular movies like Interstellar. There were many question about time travel, e.g. could we travel to the past? Where did the wormhole get its name and so forth. The day ended with a joke that your neighbour could be a visitor from the future, as time travel exists theoretically and is inevitable.

The discovery of General Theory of relativity has always puzzled historians of science, as there was no experimental compulsion or evidences from nature for a new theory of gravity. In 1905, Einstein, then a Swiss patent clerk, had published four seminal papers, one of which was about the constancy of the speed of light. This theory, known as special theory of relativity (STR), required that time dilated and length contracted in fast moving systems, e.g. rockets. STR has now been verified, and indeed astronaut’s clocks click slower than those on earth by a few milli-seconds. However, STR leads to many paradoxes, one of which was that of the rotating disk.

In a rotating disk, the velocity of rotation increases with distance from the center, and thus for a fast rotating disk, the circumference would contract in length according to STR. The radius, being perpendicular to the direction of velocity would remain unchanged. The Euclidean theorem of the circumference to radius ratio of 2π, would no longer be true. How does one explain this conflict with Euclidean geometry? Einstein realized that a rotating disc is an accelerating system, the centripetal acceleration directed towards the center of the rotation makes this a non-inertial system, outside the scope of STR. Einstein began a solo journey to find the General Theory of Relativity, the theory of non-inertial accelerating systems, which would imply non-Euclidean geometry.

In addition to this, Einstein was inspired by Machian ideas of relativity. In physics measurements are defined with respect to observers and frames are as important as the physical event. Using Gedanken or thought experiments, Einstein formulated his `weak’ and `strong’ equivalence principles. According to this, gravitational fields can be replaced by accelerating frames. This is easy to see, the zero-gravity planes, now popular with tourists, simply take everyone to a sufficiently high level, and then `free-fall’ in earth’s gravity, at `g’ the acceleration due to gravity. The result is that everyone in the plane gets few minutes of `floatation’ or zero gravity as in a space-ship.

However, till 1911 Einstein had not made much progress. At that time, he started discussing with Marcel Grossmann, a mathematician and a classmate from ETH Zurich, who initiated him into principles of differential geometry, and the work of Bernhard Riemann, a French mathematician. General Relativity, slowly took shape, and it was formulated by equating energy momentum of matter with curvature and geometry of space-time. This was `gravity’ and the new physics was covariant in accelerated frames. The curved path taken by planets around the sun, the motion of galaxies is the easiest path or shortest path taken by these in the curved geometry created by massive objects.

General Relativity’s advent as a new theory was soon made credible as it predicted the yet unaccounted for 43 arc seconds (about 0.0119 of a degree) per century perihelion precession of Mercury. The bending of light due to Sun was also calculated correctly by the theory and confirmed by observations in 1919. The latter made Einstein particularly popular as newspapers headlined `Revolution in Science-New Theory of the Universe-Newtonian Ideas Overthrown.’

The presentation of the results in 1915 was soon followed by publication of an exact solution to the Einstein equation, found by K. Schwarzschild, in 1916. This solution, characterized by an event horizon, and a central singularity was named the `black hole’ by J. Wheeler in 1967. Anything which falls inside the black hole event horizon is lost to the outside world, and is destroyed at the central singularity. These bizarre gravitating objects were soon astrophysical realities. In 1930 S. Chandrasekhar had defined his `Chandrasekhar limit’ for gravitational collapse: all stars heavier than 1.39 times the mass of our sun collapse under their own gravity when they use up their fuel and form black holes. Today there are thousands of black hole candidates in the sky, including one at the center of our Milky way, the Sagittarius A* which might have seeded the formation of the galaxy.

The concept of the `big-bang’ and the Universe starting from nothing was a concept developed from the initial work of Lemaitre, Einstein and Friedman from 1917-1930. The current expanding universe solution is due to Robertson-Walker (1935). However, we are still investigating the exact curvature of the spatial Universe. Data from WMAP experiments, Planck satellites suggests that we have a slight positive curvature. In the 1990’s new observations showed that the Universe was not expanding at a uniform rate as predicted, but rather accelerating. This mystery baffled scientists and the current explanation is that there is dark matter and dark energy apart from visible matter which drive the cosmology and cause the acceleration.

Gravitational lensing is also one of the most beautiful astrophysical observations we owe to General Relativity. Light/Electromagnetic wave get bent due to massive compact objects and form multiple images of stars which are behind these compact objects. Einstein’s ring is like a `necklace formation’, one of the first of these was due to a radio source MG1131+0456 discovered in 1988 by Hewitt.

However, the biggest mystery of the theory is the elusiveness of the Gravity waves. Contrary to perceptions, gravitation is one of the weakest forces in nature, 10-35 times weaker than electromagnetic interactions. Theoretically Einstein had discussed ripples in space-time, the waves as in a sea of space-time which will carry away energy from gravitational processes such as galactic mergers, black hole collisions. Indirect evidence exists for these waves, but despite building of ground based interferometers for decades, we have not detected even one direct signal to date. The hope for observing these is now on the shoulders of European Space Agency which is involved in building e-LISA. This space-based antenna will be launched in 2034, with interferometer arms millions of Kms long. Ground based LIGO interferometer has also started functioning with increased sensitivity in September 2014. The hope remains that gravity waves will be detected soon.

As Einstein said in 1919 when General Relativity’s prediction of the bending of light due to Sun was confirmed by observations, for if it had not been “Then I would feel sorry for the dear lord. The theory is correct anyway.’’

At the University of Lethbridge the Theoretical Physics group is also involved in quantizing gravity, finding the `quanta’ of gravity such as the graviton. We ask questions like is space-time coarse grained or discretized at quantum length scales? As General Theory of Relativity is so much different from other theories of nature, maybe we need an Einstein brain to solve for quantum gravity.

Einstein-Online http://www.einstein-online.info/
Wikipedia-Online Encyclopedia: https://en.wikipedia.org/wiki/Albert_Einstein
ESA: http://www.esa.int/ESA
LIGO: https://ligo.caltech.edu/
Arundhati Dasgupta, Talk at 100 years of General Relativity, University of Lethbridge, 2015

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Art, Music and Physics


Goedel-Bach-Escher, the triangle of Logic-Music and Art, a well debated topic.
But here I am writing this not to debate the trio, but to relate to it from a feminist perspective: …all about eve again!

On October 20th, the women scholar speaker series at University of Lethbridge hosted a talk, a `conversation’ on Women in Science. The speaker was Dr. Lynne Maquat, a well established biochemist from University of Rochester Medical Center. She has some nice stories to share, including about the skepticism of women’s importance in science. She started in 1970’s as a graduate student, and the discouragement was severe. She fought her way up, she is now accomplished, and recognized. She leads the University of Rochester Graduate Women in Science organization which serves various networking purposes. One interesting story was that a male student had filed a lawsuit as men were being excluded from such activities. When she clarified that there was no such intention, the lawsuit was dropped.

At the end of the talk, I asked her what did she suggest for attracting girls to physics? Her suggestion was `Try art’.

That set me on some trail and quest for finding arty physics. It was coincidental that I was teaching the method of images in my Electromagnetic theory course that day, and Griffiths thinks there is some art in that!

The method of images can be `artistic’ : a system of charges and grounded conductors can be replaced with a bit of imagination and physics with image charges which produce the same boundary conditions. These `image’ charges do not exist, but simulate the reality. The saving grace is the uniqueness theorem: The same potential function represents the sphere of influence of the image and the real configuration as they satisfy the same boundary conditions. There is a unique potential function which fits a given boundary value.



(Method of Images, source: MIT web)

At the end of the class I asked the women in the class does this appear artistic and comfortable to you than the sometimes `scary videos’ we watch of Faraday’s cage?. From their smiles I guessed the answer was `Arn’t we already converted?’ Even the music in the tesla coils is quite a jar..
Singing Tesla Coils.

Having said that there Art, Music and Physics are interconnected. Harmony and sound-waves is music, optical imagery is art, and symmetry is Physics. Not everyone realizes that the tesla coils are singing to preserve a U(1) symmetry, the symmetry of a circle.

But coming to the initial question about women and art and physics, young kids 7-8 year old will all love the singing tesla coils, so gender conditioning hasn’t happened yet. The divide becomes obvious at high school level (at least according to Canadian statistics), where girls are almost half the number of boys in Physics 30. I tried to ask some high school girls at a open University event, Why not Physics? The answers came with a giggle and a shrug: We’re not just into physics…Physics is not us…

About attracting Girls to physics, maybe I shall try George Clooney….or a Marylin Munroe…(sorry I don’t know, the current Munroe, definitely not Sandra Bullock, of Gravity fame, she’s a different sort…)

Currently, most research institutes have artists in residence, check out the new Perimeter Institute chalkboard item .

However, most scientific visualization is very attractive, as they exist. Check out numerical relativity,
http://ccrg.rit.edu/research/numerical-relativity, or just Chandra images of the Universe. Chandra is a X-ray telescope and the images are color coded images.

Sometimes I ask Maple to draw Riemann surfaces and that is art for me.

Image source: http://www.apmaths.uwo.ca/~rcorless/frames/PICTURES/riemannw.jpeg

Try the Wolfram Riemann surface generator:
Riemann Surface Generator

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The Importance of being Non-Linear

It is well known that every-equation humans have solved on this earth, are linearized to make life technically easier. However, nature is inherently non-linear and simple problems e.g. like the pendulum swinging in Earth’s gravitational fields behave according to linearized `sinusoidal’ wave predictions due to `small angle approximations’.

Continuing with our favourite, the pendulum bob: As a small mass `m’ hangs from a string fixed to a stand, it presents to us a `simple pendulum’. This instrument was used in our grandfather clocks to keep time. The motion is also well know, the pendulum swings to and fro or oscillates with a fixed amplitude and time period. The Newton’s law from which we derive this sinusoidal behaviour is however inherently non-linear. The right hand side of the Newton’s law: m X acceleration= force, has a sin x as one has to take the `sine component of the gravitational field’. This makes the differential equation non-linear, and approximating sin x= x in the small x approximation, we obtain our linearized equation. The solution to this linearized equation is sinusoidal. But the moment we relax the idealized small angle approximation, we are in the non-linear regime. Thus we are at the edge of the sea of non-linearity…perpetually..


(Image credit: Seiko clocks)

Not only that, many of our processes in complex nature, cannot be written as Newton’s laws. The fractal florets of a broccoli are examples of patterns which form and deform in nature which do not evolve using Newton’s piecewise linear functions.



(Image Credit: Sekkap Orchard)

The large scale structure of our universe, the grand stars and galaxies are also not formed from linearized approximations to cosmology. Even though Einstein’s equation is non-linear, the complexity in the universe is yet to be explored, to be solved. If we succeed then we can predict the weather perhaps.

However, systems exist, where predictability is never plausible. Non-linear differential equations which we have numerically solved show bifurcation points, incommensurate periodicity, curves and paths which never close on each other or follow regular patterns. An example of such `chaotic’ system is the double pendulum.

Pages from doublependulum
(Image Credit: Arundhati Dasgupta using Maple for Physics 4200)

It is time to forget the linear past and take a plunge into the non-linear future…

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Women in Physics Educationl

this week to complete the task of  writing on women who have contributed to my personal journey on education, I decided to talk about the women who taught me Physics.

though I was a very avid physics student and got good grades most of the good teachers I admired and remember are of male gender. At that time gender was irrelevant, however today I am disappointed to find no influence on my early education from female teachers.  There was a reason of course, when the sciences were split into the different streams of : physics, chemistry and biology, in the sixth class, ( I joined Ashok Hall Girls School in Kolkata halfway in fifth grade) we briefly had a very good teacher, Mrs. Haldar. But she left us immediately to relocate to some other city.  After that we never had one permanent physics teacher till the time we graduated.

In my higher studies, I did encounter one very good teacher, very clear and very confident,  probably Arunima Mukherjee, the head of the department of physics of Lady Brabourne College Kolkata.

My experience might be unique, but the two I remember for their brief teaching Mrs. Haldar and Arunima Mukherjee  they were rare.

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Women in Education II

Women and feminism is a contentious issue, gender stereotypes are `prisons we choose to live inside’ (A Doris Lessing book).

I began my schooling in Jaipur, a beautiful city in western India. Architectural splendour and remnants of feudal past, tales of valour and sacrifice, a surreal introduction to modern education. I began my primary schooling in a non-descriptive school Saint Anne’s Academy. The only teacher I remember from there is Jenny, and a friend named Mukul. It was a co-ed school, and I guess we learnt nursery rhymes, and the loo was always dirty.

One of Jaipur's main attractions, the Hawa Mahal is also known as the 'Palace of Winds'.

One of Jaipur’s main attractions, the Hawa Mahal is also known as the ‘Palace of Winds’.

Curiously, my rather anti-elitist parents decided to admit me into a rather expensive girl’s school, Maharani Gayetri Devi Girl’s High School.  The school was founded by and named after the queen of Jaipur, a very elegant Gayetri Devi, a politician and parliamentarian in post independent India.


Coincidentally this years CBSE topper has the same name as this Jaipur Queen. 55ff38bc-ca49-49dc-a24e-b2473d4b1519wallpaper1

The school was built to train royal princesses to transform into beautiful well groomed queens for their kings. This school thus had courses for horse riding, swimming, dancing, sewing, etc amongst of course the regular curriculum. When the school opened to public, it became famous for its balanced approach to education. I remember every morning at the school steps the house prefects would examine our uniform and shoes. For reasons unknown we would call senior girls `Jijas’  and wear naughty boy shoes. I remember these morning episodes as quite often I would be stopped by some jija to have my shoe laces tied. 71jWkc0CdwL._UL1500_ I also remember dancing lessons to `Madhu gandhe bhara’ a rabindrasangeet item and `nanha munha rahi hun’. I remember stiching Tea Cosys, Table Clothes, and embroidering an Ostrich (why?) on a green bag.

The school flourished and produced beautiful students.  Not to anger some of my friends, one of whom is Chief of Robotic Surgery somewhere in the US: its most infamous student is a British spy, a modern day Mata Hari.

I have some beautiful memories, of touching Gayatri Devi’s sari pallu when she came on a regular inspection, or getting a free lunch when the school dog Pannah ate my lunch box. Pannah was a german shepherd; Mrs. Lutter’s dog, the first Principal of the school who had died the previous year.

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Women in Education

In early 1900, it was very difficult for women to attain education, at least in India. When things appear very difficult in 2015, I refer to my grandmothers. Particularly my mother’s mother’s sister, Jyoti Prabha Dasgupta. She lived in the ground floor of my maternal ancestral home, a life dedicated to women’s education. A dedication unthinkable in today’s rather cynical world. In fact she was not alone, occasionally the house was filled with her friends and sisters, never married to devote their lives to the profession of teaching. They were not exactly nuns, they did not see god as their saviour, rather they saved the society and raised the `better half’ to be self sufficient and resilient, to dream of discovering the world and build new futures. After a long career of teaching in girls’ schools in various cities, Jyoti Prabha founded the Multipurpose Girls High School in Kolkata, India. She was the principal of this school till retirement.


She and her many friends, retained so much faith and belief in their values, in those dark times of pre-independence chaos, that it is unnerving. She used to travel in trains with a hand fan in those days, to beat the heat, but quite often had to use it to scare thieves who would pull at her trunk, amongst other things. There are many instances of her bravery and courageous journey of a austere life.


My childhood memories involve her telling me many stories, and one curious source of goodies. All the women in my family worked sometime or the other in women’s welfare home in Kolkata, where they sheltered destitute women. This home which Jyoti Prabha helped to manage, had a canteen, and I would get treated to delicious cookies from there. When Jyoti Prabha died, it was a Sunday. The school still sent the school bus to collect girls from their homes and sing hymns for her. I regret that at that age, the only thing which I appreciated was that there was a big obituary for her in the News papers. Who would ask her today to narrate her life, write a biography? Make a movie….

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Measurement and Quantum Mechanics

Quantum Mechanics revolutionized our perception of measurement.  The principle of uncertainty broke the deterministic nature of how systems evolved, and also changed our ways to `quantify’ the universe by measuring physical quantities like length, mass, time, force.

Initially a new paradigm was introduced to explain the existence of the black body spectrum.


This skewed `bell jar’  shape of the distribution of intensity of radiation as a function of wavelength could not be explained using  laws of physics which existed at that time.  One had to assume that the energy was being emitted as `discrete’ packets of energy: each packet with energy as integer multiples of a very small number times the frequency of the emitted radiation. This `quantum’ version gave the correct shape. This discovery is due to Max Planck, and the small number is known as Planck’s constant. (~10^(-34) J-s). Thus the evidence of the `quantum’ can only be found at small length scales, as Planck’s constant defines the physics.

Why was energy quantized? This question led to a 30 year journey into the unknown, with new axioms of physics and fascinating new philosophical implications.  A very funny set of anecdotal stories can be found in the book `30 years that shook physics’ by George Gamow describing the journey.

What emerged though is not yet understood completely, however in the process we gave away our confidence in measurement of natural phenomena.

In a quantum measurement one can never be sure of the result, but one can only predict probabilities. Is that the limitation of our instrument, or nature is limited at quantum length scales?

Why probabilities? There are various interpretations, about why we cannot be sure of the measurements, the one which is again mind boggling is the many worlds interpretation due to Everett.

Our Universe, and our conscious interpretation of that is not unique, there are infinite versions of the reality, and our measurements pick up one of these ……..

Learn more about this and more in the Quantum Mechanics Course Spring 2015.


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Grassmann numbers


It is strange that there are two types of particles in the world, one set known as Bosons and the other set the Fermions. You can put as many of the Bosons in one `state’ and they will be together. The Fermions, have `exclusive’ behavior, in the sense that no two fermions can be put in the same state. This exlusion principle was discovered by Wolfgang Pauli, and is known as `Pauli’s exclusion Principle’.

The electron, the proton essentially behave as fermions, and they are described in field theory using Grassmann numbers. These are strange objects, they are not typical numbers, but mathematical structures, which play the same role as numbers. If you square a Grassmann number it is zero. These variables `anticommute’ . The electron wavefunction can be derived using a `path integral’ over a Grassmann, and thus they are useful to describe physics. They are also used to obtain determinants useful technique in physics and mathematics. The Grassmanns are also used to describe a symmetry of particles known as supersymmetry which transforms Bosons to Fermions, and vice-versa. This symmetry however has not been observed in nature, though if realised will solve some of the Gauge hierarchy problem. I worked with Grassmann numbers and it was great fun. Let z be a Grassmanian, a function in z f(z)= A +Bz and thats it! No higher powers of z they are all zero. If there are two Grassmann numbers z and y then zy+yz=0 (anticommute!) as opposed to numbers who commute (i.e. zy-yz=0). Now let me pose a puzzle, what is the inverse of a Grassmann number?

(The image is stolen from http://www.cika.co-googled!)

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