Diary of a Quantum Automata

December 21, 2009

The (not so) simple pendulum: Part III

Filed under: the world around — vivisheksudhir @ 1:51 am

I started this series of posts about the simple pendulum from humble beginnings, talking about the exact solutions of a bob oscillating from a light string. In the second part, the “field theory” of the exact simple pendulum led to solitons and such interesting stuff. But of course, by that point, the simple in the title started becoming obscured. Unfortunately, that trend will continue on this post.

The motivation for this post is twofold – firstly, I started reading up on some Seiberg-Witten theory and found it enchanting and so beautiful, which took me through duality and other related esoterica; secondly, I fell ill to blogging-sickness, I had to post again!!

It has always been a source of fascination for me to hear about the ever increasing complexity of the mathematical machinery underlying much of today’s theoretical physics (at least, high-energy theory, which has slowly percolated to condensed matter theory as well). But then there are flashes like Seiberg-Witten, that provides a continuous and expansive canvas which interpolates between various disparate vertices of physics. And this unity, when it does become apparent, makes all the mathematical toil worthwhile in the end.

Steering back to the actual point of the post, the sine-Gordon Lagrangian, with a slight re-parametrization,

L_{sG}= \frac{1}{2}(\partial_\mu \theta)(\partial^\mu \theta)+\frac{\alpha}{\beta^2} \left( \cos (\beta \theta) -1 \right),

has the equation of motion,

\partial_\mu \partial^\mu \theta + \frac{\alpha}{\beta} \sin \beta \theta = 0.

The equation has soliton, anti-soliton and multi-soliton solutions. In fact, the general (1+1)-d N-soliton solution is given by,

\cos \beta \theta (x,t) = 1+2 \partial_\mu \partial^\mu \ln \det [M_{ij}] ,

where the N-dimensional matrix M has elements given by (here v_i are arbitrary constants),

M_{ij}= \mp \frac{2}{\sqrt{\frac{1-v_i}{1+v_i}} +\sqrt{\frac{1-v_j}{1+v_j}}} \cosh \frac{1}{2} \left( \frac{x\sqrt{\alpha}-v_i t}{\sqrt{1-v_i^2}}+ \frac{x\sqrt{\alpha}-v_j t}{\sqrt{1-v_j^2}}\right).

One of the fundamental characteristics of the sine-Gordon equation (and of very many other PDE’s) is that its small excitations (in the \theta \rightarrow 0 limit), the ‘mesons’, and its non-perturbative solutions, the solitons, form disconnected solution sets. This means that there exists no perturbative scheme,which to arbitrary orders of perturbation, gives solitons as the solutions. This means that the regime of the small perturbative parameter and that of the large perturbative parameter cannot be accessed, starting from the other. In fact, this is one of the reasons why non-perturbative QFT (particularly QCD) is given importance.

But it so happens that at times, the strong coupled regime of one theory is equivalent to the weak coupling regime of another theory. Perhaps the earliest reported use of this duality principle was in the determination of the critical temperature of the 2D Ising model via the so-called Kramers-Wannier duality transform [cf. P. Ruelle, Phys. Rev. Lett. 95, 225701 (2005)]. This strong-weak duality is termed S-duality and is a central theme to the development of the Seiberg-Witten theory.

Now, it so happens that the sG theory, which is basically a theory of bosons, is S-dual to the Thirring model, which is a theory of fermions! The Thirring model is an extension of the Dirac theory of the massive fermion to include self-interactions; the specific Lagrangian being,

L_T= \bar{\psi}( i\gamma^\mu \partial_\mu - m ) \psi-\frac{g}{2}(\bar{\psi}\gamma^\mu \psi)^2 ,

where \psi = (\psi_{+},\psi_{-})^T is a two-component spinor and the \gamma^\mu satisfy the (1+1)-d Clifford algebra (here \eta is the Minkowski metric),

\lbrace \gamma^\mu , \gamma^\nu \rbrace = 2 \eta^{\mu \nu}.

The proof of the duality starts from considering the bosonisation ansatz,

\psi_\pm (x) = \exp \left( \frac{2\pi}{i\beta}\int_{-\infty}^{x} \theta (x') dx' \mp \frac{i\beta}{2}\theta (x) \right) .

Now, the bosonic sG theory has the (equal time) bosonic canonical commutation relations,

[\theta(x),\theta(y)]=0=[\dot{\theta}(x),\dot{\theta}(y)]

[\theta (x), \dot{\theta}(y)]=i\delta(x-y).

Then, a few steps of algebra later, employing the bosonisation ansatz, one can calculate the products of the spinor components. This gives the familiar fermionic anti-commutators,

\lbrace \psi (x), \psi (y) \rbrace = 0

\lbrace \psi (x), \psi^\dagger (y) \rbrace \sim \delta (x-y).

These results prove the essential duality of the theories via bosonisation. It also shows the intimate symmetry between certain bosonic and fermionic theories. Of course, these days, this symmetry is better understood in the form of super-symmetry.

(To be contd.)

December 18, 2009

Shankar-isms

Filed under: physics — Tags: , , , — vivisheksudhir @ 11:28 am

I am finally getting down to the serious phase of university applications and it is already close to the US deadlines, so I had to rush a few things today. Finally most of it is done and before hitting bed, I wanted to mail Prof. Shankar at Yale about some application stuff. Anyway, that never happened… for now. But something else did, and thats why I am posting after a really long time.

Prof. Shankar is a pretty impressive prof, maybe as charismatic as Feynman or Coleman. My first acquaintance with his work comes through his textbook ‘Principle of Quantum Mechanics’, one of the most lucid expositions of the subject yet! I never thought that a text on quantum mechanics could be written that is so rigorous and clear at the same time. Most interestingly, it is talks about the path integral formalism in chapter 4 or 5, thats a real adrenaline-rush for any physics junkie. From then on, I have been a huge fan of the text and it is definitely on my list of must read texts for a budding physicist.

But the point of this post is not physics. It is about personality… the classroom persona of Prof. Shankar. And here I reproduce some of the funniest Shankar-isms (the rest of ‘em can be found in his personal webpage).

Ok, here goes….

  • “This is a very important day. You can forget your birthday, forget anniversaries, but you need to remember this day, because this is the day that you will learn Newton’s Laws”
  • “There’s not that much material that I can teach you, actually. I can write all the physics equations in one corner of the blackboard, and then all you need is an IQ of 5000 and you’re set!”
  • “You can add vectors, multiply a vector by a number, flip vectors – the fun is just endless”
  • “So Newton said ‘I will go invent integral calculus.’ After all, he just invented differential calculus the other day, so why shouldn’t he?”
  • “This problem in your book says that a physicist is hiking up the Alps. You know that’s a joke, right?”
  • “Let’s say the physicist gets stuck while climbing, and you want to send him something. It may be food, or since it’s a physicist, he might say ‘Send me my Wolfson and Pasachoff (our textbook)! I haven’t read it in two days!’ “
  • “I’m gonna go home and pick a day for the midterm, and you’ve gotta let me know if this is a problem. So if you’re getting married that day, bring in your spouse-to-be, and if you’re gonna have a heart transplant, I wanna see the new heart”
  • “So today we will do the problem that makes most people never want to do physics again”
  • “That’s the beauty of teaching- for 1 hour of the day you don’t feel like a complete idiot because you realize that there are many people worse off than you”
  • “In this first problem, there is a car driving along a cliff, and the car just jumps off. This person has decided to end it all. Now, we want to know at what time the car hits the ground. This is the beauty of physics, because if this were a psychology class we’d want to know why the person was jumping, but we are simply concerned with how long it takes.”
  • “Say you’re firing a rocket launcher. What angle should you fire it at for maximum range? Say you fire it straight up. The good news is that it’s going to be up in the air for a very long time. The bad news is that it’s going to land on your head”
  • “When you’re doing problems on the blackboard your intelligence is proportional to your distance from the board, so now I’m at an all time low”
  • “See, one reason why the Americans fought the British is because they couldn’t stand their units. You know they have something called a slug? I mean, what is a slug? I don’t know, and I’m proud of it!”
  • “You could write a law and think it’s correct, and then you’d publish a bunch of papers, and eventually you’d realize that your parents are the only ones reading them and then you’d know that you were wrong. Now, on the other hand, if your friends are reading your papers, your enemies are reading your papers, and then your enemies are stealing what you’ve written in your papers, then you’ll know that your law is correct”
  • “I’ve gotta be nice to my students, because one day one of you could be my physician. I could be lying flat on my back, and you could be coming up to me with a mask on and a knife in your hand. I’d say ‘What about my anesthesia?’ and you’d say ‘What about that formula sheet you promised me?’, so that’s why I try to treat you guys nicely”
  • “What I see is, the mathematicians tell us what the rules are, and then it’s our job to break them”
  • “The thing is, nature doesn’t care whether you like something or not – you just have to suck it up”
  • “Has everyone in here seen an integral? Good. Because I didn’t have a backup plan”
  • “The question you have to ask yourself is, if your professor drops dead in the middle of his lecture, will you be able to finish deriving the equation he started? If so, then you know you’re doing okay”
  • “Today we are going to talk about rigid bodies. Like Al Gore.”
  • “If we throw a cat up in the air it will be moving its arms and legs all around, and that’s not rigid. We want a rigid body, like a dead cat”
  • “If you miss class you should talk to someone, because I don’t go straight from the book. If you read the whole book you run the risk of learning something you don’t need to know. And who wants to do that?”
  • “Come on now, make big diagrams! If you want to save trees, do it on your own time – not in my course!”
  • “But if you’re from Harvard, you think the center of the universe is here, in Cambridge”
  • “If you put your hand on a hot plate, you should say ‘Wow, these molecules are fast!’. That’s what I want you to say from now on, not ‘Ouch!!’ ”
  • (Shankar “quoting” Carnot) “No engine can beat my engine”
  • “The Earth’s whole mass – you, me, China – everything is pulling it down”
  • “You may be questioned by the Mafia someday. And it’s standard practice for them to lower you down in this tank of water. So when you’re pushing out on the walls, trying to save yourself, think to yourself – how many Newtons am I applying to this wall? – because you’re causing pressure!”
  • Shankar: “Any suggestions on how to make up for the missed class?”

Student: “Put the lecture online?”

Shankar: “How can I convey the full force of my personality online?”

  • *messes up demonstration*

“You know, some of us go into theoretical physics for a very good reason”

  • “All we need to solve this problem is Gauss’s Law and several large hand-waving arguments”
  • (just before leaving on a 10 day trip) “See you suckers! I’m going where the sun is shining and it’s 75 degrees all day!!! Bye!”
  • “No, not strings, strings are chapter 9600 of this course”
  • “If you are a complete moron, you will take your wire all the way, but as the limit of moron is infinite, you will have wasted all your wire on a loop of zero area”
  • “There are some things you can always look up, like your social security number or your birthday, but the trig identities you gotta know”
  • “I know most days after 50 minutes you guys go into little convulsions and send me not so subtle hints that time is up – but not this day”
  • “When you have an i on the bottom you replace it with a negative i on the top. This has been known since Biblical times- an i for an i
  • “The act of observing an electron is very traumatic for that electron. Right now I’m getting hit by millions of photons. I’m taking it like a man. But for the electron, this is not the same”

Hmmm… A Feynman in the making? No, a Shankar in the making, thats what is more like it!

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