Diary of a Quantum Automata

September 28, 2009

‘The Character of Physical Law’

Filed under: close to my heart, physics, science — Tags: character of physical law, feynman, Gell-Mann, symmetry — vivisheksudhir @ 10:56 am

For the past couple of days, I have been light-reading through the transcripts of Feynman’s ‘Messenger Lectures’ he gave at Cornell. The transcript is published as a book titled, ‘The Character of Physical Law’ and consists of the seven lectures,

The law of gravitation, an example of physical law
The relation of mathematics to physics
The great conservation principles
Symmetry in physical law
The distinction of past and future
Probability and uncertainty – the quantum mechanical view of nature
Seeking new laws

First of all, Feynman is one of my all-time idols in physics and I have devoured his lectures and popular books with great enthusiasm. For me, he is a startling example of a “pure physicist”, a term generally used only for mathematicians. To illustrate the point, let me recall a popular anecdote.

Perhaps most of you know Murray Gell-Mann, the Nobel winning physicist who was a contemporary of Feynman (or perhaps slightly younger). Once Gell-Mann went to a class full of graduate students to teach them the fine art of quantum field theory, something that is full of abstract and complex mathematics for the freshman grad student. As Gell-Mann started his rather long and tedious exposition of QFT, one of the grad students asked if there’s an easier way to do this and in response, Gell-Mann answered in the affirmative. On further questioning, he confided that there’s “Feynman’s way”, which consists of writing down the Lagrangian, stepping back from the balckboard, staring at it for a few minutes, and finally, writing down the answer.

For someone like Gell-Mann to say that about Feynman, one can only imagine the the amount of talent and insight posessed by Feynman. And a significant portion of his raw physical insight pours out through the pages of this book. Filled with illuminating analogies and profound implications, it is a wonderful exposition of the symmetry principles and conservation laws that have played such a central role in much of 20th century physics.

This is something that every physicist should read and any layman can understand, but perhaps the physicist will be in a better position to appreciate the true value of some of the ideas and see the hidden pictures behind some of the words.

***

“Nature uses only the longest threads to weave her patterns, so each small piece of her fabric reveals the organization of the entire tapestry”

“What is it about nature that lets this happen, that it is possible to guess from one part what the rest is going to do? This is an unscientific question: I do not know how to answer it, and therefore I am going to give an unscientific answer. I think it is because nature has a simplicity and therefore a great beauty”

-Richard Phillip Feynman

in ‘The Character of Physical Law’

September 27, 2009

The (not so) simple pendulum: Part II

Filed under: the world around — Tags: anti-kink, kink, Klein-Gordon, sine-Gordon, soliton — vivisheksudhir @ 8:45 pm

In the previous post I was talking about the pendulum and its integrability and so on. In effect, what I was doing was to set a stage for this post. After having solved the pendulum attached to a single particle, the next level of logical extension would be to consider some associated “field theoretic pendulum” where each point of the physical space is some kind of pendulum bob.

To do this consistently, one looks at the Lagrangian (which has been stripped of certain constants pertaining to the zero point of the potential) of the single particle pendulum,

$L' = \frac{1}{2}\dot{\theta}^2 - \omega^2 \cos \theta$ ,

which on going through the Euler-Lagrange equation gives the correct equation of motion,

$\ddot{\theta}+\omega^2 \sin \theta = 0$ .

Now, let us see how the above Lagrangian can be suitably (and in the easiest possible manner) extended to the field case.

If we allow for the (1+1)-d pendulum field to oscillate at speeds close to the speed of light i.e., allow for relativistic physics, then one is constrained to make the action Lorentz invariant. This implies that the Lagrangian itself is Lorentz invariant, except for a finite boundary term. As it stands, $L'$ is not so; in particular, the kinetic term $\dot{\theta}^2$ isnt, while the potential term, $\cos \theta$ is just fine.

The simplest possible adjustment that makes the Lagrangian satisfy Lorentz invariance in (1+1)-d is the addition of a quadratic space derivative term. This gives the new field Lagrangian (actually Lagrangian density),

$L = \frac{1}{2}\left( \dot{\theta}^2 - (\theta_{x})^2 \right) + \omega^2 \cos \theta$ .

For reasons that will become obvious in a while, the potential term is re-parametrized, an inconsequential constant is added and the whole Lagrangian is written in covariant form viz.,

$L= \frac{1}{2}(\partial_\mu \theta)(\partial^\mu \theta)-\frac{m^4}{\lambda} \left[ \cos \left( \frac{\sqrt \lambda}{m}\theta \right) -1 \right]$ .

Before finding out the equations of motion of this Lagrangian, let us briefly pause to observe something. In the limit $\frac{\sqrt{\lambda}\theta}{m} \rightarrow 0$ , the potential term can be approximated by its leading order Taylor expansions to give the Lagrangian,

$L_1 = \frac{1}{2}(\partial_\mu \theta)(\partial^\mu \theta) -\frac{1}{2}m^2 \theta^2$ .

Retaining the terms to next to leading order, we get,

$L_2 = \frac{1}{2}(\partial_\mu \theta)(\partial^\mu \theta) -\frac{1}{2}m^2 \theta^2 +\frac{\lambda^3}{4!}\theta^4$ .

Having obtained three field Lagrangians, let us see how their respective field equations look like. Again using the appropriate Euler-Lagrange equation,

$L: \,\,\, \partial_\mu \partial^\mu \theta +\frac{m^3}{\sqrt \lambda} \sin \frac{\sqrt \lambda}{m}\theta =0$ ,

$L_2: \,\,\, \partial_\mu \partial^\mu \theta +m^2 \theta =0$ ,

$L_4: \,\,\, \partial_\mu \partial^\mu \theta +m^2 \theta -\frac{\lambda^3}{3!}\theta^3 =0$ .

The equation in the center is immediately recognized as the (1+1)-d Klein-Gordon equation, the one above it is named the sine-Gordon equation in analogy and the last one is the Klein-Gordon equation of the $\phi^4$ field theory; and they all come from the same root, the sine-Gordon Lagrangian $L$ .

Historically, the Klein-Gordon equation originated as the first attempt at a relativistic single-particle wave equation. It was abandoned due to difficulties in interpreting its non-positive Noether currents as probability currents. Later, Pauli and Weisskopf resurected the scalar KG equation by interpreting these non-positive probability currents as the “charges” of some spin 1 particle.

A startling insight into the superficial nature of scalar fields is clearly obvious from such an interpretation. The actual particle that Weisskopf saw as a possible candidiate was the Yukawa meson. But Yukawa had based his meson theory on a massive vector boson (adding a mass coupling term to the covariant electrodynamic lagrangian). Under a choice of the Coulomb gauge for this theory, one recovers the KG equation as the field equation for the scalar potential. Thus, in this particular example (and in very many others), it is seen that scalar field theories come out as some specialized part of a broader, more general vector field theory.

It can be shown, via second quantization, that the KG equation actually breaks down to a system of quantum harmonic oscillators for each mode of the field variable.

The KG equation with the nonlinear $\theta^3$ term is a textbook example of a self-interacting field theory. As far as I know, the second quantized self-interacting theory is solved by perturbative expansions only. This forms, usually, the first introduction of a physicist to the world of Feynman diagrams and vertex Green’s functions. In some sense, this self-interacting KG equation can be considered an anharmonic oscillator and at the most fundamental level, anharmonic oscillators are models of self-interacting scalar particles!

Finally, the sine-Gordon equation. The equation itself is named after a pun on the KG equation and it was, in this (1+1)-d form, studied first in the context of solitary waves. But the general lineage of the equation strectches back even further. In differential geometry, the sG equation was studied as early as the 19th century. It comes out as a form of the Codazzi-Mainardi compatibility equation for pseudospherical surfaces.

Integrating the sine-Gordon

Let us consider the (1+1)-d sG equation in the form,

$\theta_{tt}-\theta_{xx}+\frac{m^2}{\beta}\sin \beta \theta =0$ .

We are looking for travelling wave solutions, so using the ansatz, $\theta(x,t) =\theta (x-vt)$ , one converts the above PDE into,

$(v^2 -1) \theta_{uu}+ \frac{m^2}{\beta}\sin \beta \theta =0$ ,

where, $u=x-vt$ is one of the characteristic curves of the travelling wave. Integrating this once, we obtain,

$\frac{v^2 -1}{2}\theta^2_u + \frac{2m^2}{\beta^2} \sin^2 \beta \theta - \frac{m^2}{\beta^2} = C$ .

From the previous post, this is nothing but the first integral of the pendulum!! (Except for some some constants which can be adjusted) This connection between the travelling wave solution of a field equation and its underlying single-particle theory is something quite general and obvious.

Similar to the method followed for the pendulum, making the substitution, $\zeta = \sin \frac{\beta \theta}{2}$ , we obtain,

$(\zeta_u)^2 = \frac{m^2}{1-v^2}(1-\zeta^2) \left( \frac{C+m^2/\beta^2}{2m^2/ \beta^2} -\zeta^2 \right)$ .

This has three solutions, two of which are super-luminal (the ones which are elliptic cnoidal wavefunctions) and the other, a sub-luminal wave. Due to physical considerations, we would like the travelling wave to go with $v < 1$ i.e., at speeds less than the speed of light (sub-luminal). This fixes $C = - \frac{m^2}{ \beta^2}$ , to give

$\theta_u = \frac{2m}{\beta \sqrt{1-v^2}} \sin \frac{\beta \theta}{2}$ ,

giving the solution,

$\theta(x,t)= \pm \frac{4}{\beta} \tan^{-1} \left[ \exp \left( \frac{m (x-vt-x_0)}{\sqrt{1-v^2}} \right) \right]$ .

The above solution with a positive sign is called a kink, while the one with the negative sign is called an anti-kink. This is deeply rooted in the fact that the solutions have a topological “charge” associated with it.

Interpreting these soliton solutions as particle states, one can have a kink anti-kink bound state, its called a breather. Kinks and anti-kinks can scatter off each other, or from a breather.

Various other generalizations of the sine-Gordon have been studied, an early review of which can be found in Il Nuovo Cimento A, 38, 4.

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Diary of a Quantum Automata

September 28, 2009

‘The Character of Physical Law’

September 27, 2009

The (not so) simple pendulum: Part II

Integrating the sine-Gordon

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