Yogurts and the Quantum World

Hah! --a bizarre impossible connection you'd say. Not really, hear me out! When I was visiting Les Houches, attending a school devoted to Lattice Quantum Chromodynamics devoted to studying the quantum world in much more detail, there were lots of things that would make our life fun. One of those was the yogurts and diary products from DANONE, some diary brand.
Today, while coming back from the bank, I saw an advertisement of DANONE. Here in India!  India is fast becoming a market to expand into.
Well, what a weak link! But I had to mention this. Why exactly it caused me such an excitement when I saw the ad, I can't exactly express. Maybe my country and its people being courted by foreign companies to sell their product? Pride? Or folly that we have to depend on them, still? I wish I knew!

The Spending Spree in India

Recently, as most people have noticed, the Indian economy has experienced a boom, with a lot of good and bad effects. To sustain this growth, a spending spree seems inevitable. Let me tell you how I actually experienced it.
There is this travel company called Make My Trip, a pretty well known company in India. I had used this to book some air tickets previously. But, suddenly, a month ago they suddenly offered me a package--for which I had to a pay a service charge of about 7000, and in return they gave me several discount vouchers. They did explain all the terms and conditions--or as much of them as they thought would not put me off in accepting the offer. I was a bit sceptical and so were some of the people I consulted with. But it seemed an interesting thing to experiment with. So I have accepted the offer. The entire package gives you a lot of discounts, but it also necessiates that you spend a lot more than you planned to! I knew all this, but the real challenge to me is how to get the most by spending the least. Lets see how it goes--I have about an year.
But this is indeed what goes on all the time--everywhere---the entire credit system based on the Credit and Debit cards promises you more benefits the more you use them. The more you spend. The more debt you incur.
No longer a scary thought; but a reality to deal with!

Do you think if somebody smiled at you, you can make out if the smile was out of genuine pleasure of seeing you, or merely trying to be polite? Can you be sure? Ever?

It was a cloudy day, and we were there in some sort of a huge stadium. Not these modern stadiums you see in the movies, but in a pretty old fashioned one where the benches were made of wood. I don't exactly remember what game was being played, but I remember the moment when everything suddenly stopped and there was a great commotion in the skies. A huge object has started approaching us, the stadium. It was there in a sudden flurry, and before you could start wondering what it was--it revealed itself to be a huge spaceship. Pretty huge, I'd say, but I can't give you the exact dimensions. It was alien. How did I know it? I don't know, but somehow all of us were sure that it was an alien ship, though there was nothing in the make to suggest it to be so. And we didn't panic. We didn't try to run away. We just sat there, in wonder, gaping at the huge shape in front of us, that was slightly wobbling as it was trying to maintain it's height in the sky. And then, from nowhere, this other large ship appeared in the stadium, which was, by now I had realized, was far from being circular. It was like a curved runway, but made of earth. If the Earth was expecting to fight the aliens with this ship then it was going to be a poor job. The Earth-ship was equally unobtrusive as the alien one, but it was smaller in shape and more wobbly on the ground than the alien one was in the sky. But it made up to the sky--and then suddenly a flurry of smaller ships appeared on the runway and started taking off. We all waited and watched in wonder what would happen--what the Earth to offer, more? And then I suddenly asked myself, why do we have to fight? Why can't we ask these people, or however shaped they were, what they wanted?  I imagined that there should be a helicopter in white, waving white flags going to the alien ship. But what if the aliens didn't know the meaning of white? What if they blew the helicopter up thinking it was a sign of aggression? 


I was kept wondering. I don't know what happened next, since I think I woke up at that point. If you want to know what happened next, you could try going to sleep after watching the movie, "Race to the witch mountain", and maybe you'll get the same dream I had, or a much more interesting one. Anyway, that's one of the reasons I have trouble in getting up once I fall asleep. So many dreams, so many stories. I yet have to upload my photos and/or tell you about the trip to Lattice and Extreme QCD.

A tryst with VISA

There, there--hello again! Tryst with VISA-s can be very irritating; as I am sure people who apply to the US already know. But even then small countries, trying to cook up austerity drives to reduce their national deficits, can be so nettling at times! Anyway I am not trying to make general statements here about big things; but I am trying to point out a small case which exemplifies some general features on a much puny scale. Okay so here's my story.

I am going to travel to Italy and Germany for some conferences and both organizers have graciously provided me with all possible documents to help me get my VISA. But of course, the problem with the Italian consulate was that the person sitting with the phone (who was unfortunately an Indian) thought it was too much to listen to what I had to say--all he could happily say was---Go the the VFS and get rid of us. Okay, I decided, that is probably the best instead of wasting my time. But then it turned out that the great VFS, whose job is to provide information on what type of VISA to provide drew a complete bank look at my covering letter. "Sorry sir",---they very politely told me,"we have to send your application to the Consulate to decide on the type of VISA"--etc etc. You cannot imagine the feelings I had after I had travelled to that place wasting so much of my time and money and in the unbearable heat. So what do I do? Nothing, but meekly go back again the next day! And what do they do then---accept the form as it is--and now all because the Consulate has told them. What a big fallacy! And why, because some stupid Indian guy at the Consulate had thought it was beneath his status to talk to the ordinary public. Maybe, I should write a proper letter of complaint to the Consul General.
But anyway---sometimes I wonder whats the point--in today's world, it is the first of the many VISA related frustrations that I am  probably going to have! :( !

24+15

This was an interesting week for me..and extremely intense. There have been only several stints in my life when I have been so intensively involved. One thing was to make the SU3 program running; which I guess I already have described before. The other thing was finishing the rewriting of the draft of the paper on the screening masses. I managed to polish the paper quite a bit and also managed to write quite a few somethings on the introduction, and modify nearly all the figures. On the whole I think it looks quite nice! :) But let us see how much of the modifications I still have to do. The 24 + 15 in the title refers to the 24 hours of work it took on the paper followed by the 15 hours of sleep I needed to refresh myself.

Links and more

Yes, links and more...that was that was keeping my time occupied for the last few days. I did the upgrading for SU2 to SU3 and with some occasional help. But it was as rewarding as it was hectic! It really was ecstatic feeling that I had when I got correct results. Now I feel nice: I have experiencing in programming from scratch all scalar theories (Z2, O(N) models) and gauge theories (SU2,SU3) with a host of updating procedures! To me, quite an expeirence and expertise.
But this is only the beginning of the things that I have to finish before the month is over. Yes, let me go back...

Hunting for treasure

This is the first time I am trying to put some topic into my post. This essentially meant to focus something important that happened to me during the time in between the posts and doesn't necessarily mean that the post will be about the topic exclusively.

Well anyway, although the treasure hunt was important for me, since I was playing this time with a lot of friends, we lost by about a minute. It might be of interest that the winning team took about 40 minutes to decide that the sequence 3 14 15 ... has anything at all to do with the fabled $\pi$. But it was fun and it was tiring.

Back, I have a lot of catch up to do despite the fact that I worked pretty hard yesterday. The SU3 prog is coming up from the SU2 one; and am trying to think the correct tuning for the multilevel process. I was told that the optimum no of sublattice averaging in the latter case might go up to thousands but I didn't appreciate it properly until I saw the Koma et al paper that does the same. That does bring up the question where are we going to get so much time? Another thing that was pointed out, was that could we do any calculation in the strong coupling limit of the theory? Well, it is time I tried to understand the Laine et al paper much more seriously.

Have been away to Kolkata last week. On work. Therefore no time for play. Yes. Honestly, I mean it. But on the whole it was pretty enjoyable! Of course, the absence of Internet was pretty frustrating and the experience of getting absolutely drenched in the local storm "kalbaisakhi" was superb. However was anxious all the time because the tickets had not been bought, neither the registration not the hotel booking was done. By the way all this is for the LATTICE conference, to which I will be most probably going this time. South Calcutta seemed pretty much changed:beautification projects all around! The South City mall was big and nice, but not as great as some of the malls you can see in Bombay.
And yes, got to see a lot of wonderful sights from the plane. The sight of Bombay, as the plane was taking off----that of the Worli sea link was particularly wonderful. The roads were tubelights of light! So was Calcutta while getting down. It seemed so "cute"! While coming back, the most remarkable were that of the Western Ghats: you can make out where the ranges start, grow taller, some parts of the plateau sandwiched in between the mountains, and finally the mountains giving way to the narrow valley and finally the sea----mesmerising. You could see deep gorges when the river is in the mountain and ox-bow lakes as it is moving on a flat land: geography should be taught on the air like this!
Working hard on the Diffusion coefficient. Need to see how much the multilevel improves over the standard result.

A very tiring set of three days: you wont even imagine what we have to put up through at times! Want to attend the lattice conference and am breaking my head over getting the funds. Have written so many e-mails and have looked up so many different schedules; that if this thing doesnt come off in one piece, I'll probably blow up!
No physics for so many days. No physics posts till now. Very frustrating, indeed!

Last night was worth its weight in gold. Finally managed to finish a first draft of the paper on screening masses. It looked quite good to me. I was able to tie up quite a number of points. It remains to be seen, how much of it will be okay-ed by my guide.

Never realised till now that writing blogs could be so much fun,...and so much useful.
Am writing the first draft of the screeningmass paper. Has been a long day today; but not yet ended!

Electric Dipole Moment

Definition of Electric Dipole Moment

The Energy of a system of charges is defined as:

ε=∫ρΦ dV
where ρ is the charge density and Φ the potential both functions of coordinates.Choosing a particular origin and expanding the potential in a Taylor series about the origin:

ε=∫ρ{Φ(r₀)+x_i&part_i&Phi(r₀)+½ x_ix_j&part_i&part_j&Phi(r₀)+...}

ε=∫ρΦ(r₀)+d_iE_i+d_ijE_iE_j+...

where d_i is the Electric Dipole Moment and d_ij is the Induced Dipole Moment and E the electric field.

For discrete charges the dipole moment is

d=&Sigma_ir_ie_i

where i denotes the sum over the charges.

EDM for a system
Let P is the Parity operator.If P is conserved then the energy eigenstates are also
parity eigenstates.Under a parity transformation

&lang m|P⁺PdP⁺P|m &rang =-&lang m|d|m &rang
.
So the dipole moment must vanish.A non-vanishing dipole moment signals violation of parity.


Let us consider non-degenerate energy eigenstates characterised by only by its angular momentum operator J.

Under the action of time reversal,

T d T^-1 =d

T J T^-1 =-J

implying the EDM has to vanish.

So, the presense of an electric dipole moment would violate both time reversal and parity.

However, a system possessing non-degenerate eigenstates which are parity eigenstates as well can have a non vanishing dipole moment.This is because the energy shift is quadratic in &Delta² where &Delta is &lang+|zE_z|- &rang and therefore quadratic in E.So it is an induced dipole moment and doesnt imply the violation of time reversal.


When the system is in a degenerate state ( for example the ground state of a water molecule), the energy shift can be linear in E, but then it does not imply violation of time reversal invariance.This is usually called the permanent dipole moment.

Projection Operators

Permutation Operator: The Permutation Operator is defined by its action on a N-particle wavefunction as follows:
p_ijψ(...,i,...,j,...)= ψ(...,j,...,i,...) ( 1)
In general, p_ij²= e^2iφ
The order of the permutation group of order N is N!.Further, if the hilbert space of each single particle state is d,dimension of the total hilbert space formed is d^N.The particles are assumed to be indistinguishable.

Completely Symmetric & Completely Antisymmetric States and Projection Operators:
Notation: |i1,...,iα,...,iN> is a multiparticle state which means that particle no α is in a state denoted by |iα>.If the |i>'s form a complete orthonormal system in the space of N-particle system,the completely symmetric and completely antisymmetric basis states is defined by:
S_±|i1,...,iα,...,iN>=(1/√ N!) Σ_p(±)^pp|i1,...,iα,...,iN> (2)
By definition,it is clear that S_± are 1-dimensional subspaces.This we can also prove by writing down the projection operators onto these states.The trace of the projection perators is equal to the dimensionality of the hilbert space onto which they project as can be seen from the example of a three dimensional vector.From this elementary consideration, the following properties of the projection operators can be formulated:
1.Tr P < D,where D is the dimensionality of the hilbert space.
2.Tr P is always integral.
3.If Tr P =0 then it projects on to the null space.
4.If P_iP_j=&delta_ij and Tr P_i=1, then i can lie between 1 and D.
5.Iff 1 ≤ i ≤ D, then Σ _i P_i=I

Now, let us give some theorems and their proofs.
Theorem 1: The trace of a projection operator is equal to the dimension of the Hilbert space over which it projects.
Proof:Let the basis of an d dimensional Hilbert space be denoted as {|i 〉 }.The projection operator P_r which projects over a r dimensional subspace can then be written as Σ _k=1 ^r |k 〉 〈 k|.Then Tr P= Σ_l=1 ^D 〈 l| P |l 〉 = Σ_l=1 ^D 〈l Σ_k=1 ^r |k 〉 〈 k|l 〉 =Σ_k=1 ^r Σ_l=1 ^D 〈 l|k 〉〈 k|l 〉=Σ_k=1 ^r1 = r, since
〈 k|l 〉=δ_kl.

Theorem 2:The Projection Operator on to the completely symmetric and the completely antisymmetric states are given by
P_±=(1/√ N!)S_± (2)
Proof:From our definition of the S_± operators we know that there can only be a single symmetric or anti-symmetric state.So let us write
P₊=c₊ S₊.We need to find c₊ such that P₊²=P₊
Now, P₊²|i1,...,iα,...,iN&rang =(1/N!)c₊²&Sigma_qpqp|i1,...,iα,...,iN&rang
=(1/N!)c₊²&Sigma_phh|i1,...,iα,...,iN&rang
=c₊²&Sigma_hh|i1,...,iα,...,iN&rang
(since if pq=h, then p=hq^(-1) and has N! distinct elements.)
=c₊²(√ N!)S₊|i1,...,iα,...,iN&rang
=c₊(√ N!)P₊|i1,...,iα,...,iN&rang
which implies c₊=(1/√ N!)
Thus:
P_±=(1/N!) &Sigma _p(±)^pp|i1,...,iα,...,iN&rang

Theorem 3:Tr P_±=1
Proof: We shall prove this theorem only for the case of P₊.The other one involves more complicated algebra, but gives the same result.
Tr P₊=&Sigma{|i 〉 } &lang i1,...,iα,...,iN|(1/N!)&Sigma_pp|i1,...,iα,...,iN&rang
Only p=1 gives non zero contribution and there are N! such identities are generated due to the sum on p and hence we have the result Tr P₊=1

noethers theorem

1. we proved the noether's theorem( for every continuous symmetry with end points fixed characterized by m parameters, there are exactly m conserved quantities characterised by Q_k, where Q_k=(delta L /delta q_dot)*(delta q / delta alpha_k), alpha ks are parameters.) this we did by writing the transformed q_i's as functions of q_j and alpha_k, effecting the variation of the action,setting it to zero for a symmetry and using the lagrange equations.
2.next, we also considered the cases where the end points vary and noted that such a variation of end points causes the lagrangian to remain otherwise unchanged.this variation causes a variation of the coordiantes at the boundary, but along the trajectory.the limits of integration change.for the action to remain unchanged, the change of lagrangian due to this must equal the change of lagrangian due to the coordinate variation.a short calcualtion then yielded the conservation of energy.
3.we also saw that the alpha_k 's form a lie algebra
4.we derived the energy momentum tensor of the field.we noted that the tensor was not unique and so certain tensors(anti-symmetric in the last two indices) can be added to it to symmetrize it.
5.we derived its form for the classical electromagnetic field, checked that the tensor was symmetric and traceless.
6.it was also noted that this tensor cannot always be diagonalised; since under such a transformation the metric (which is not an identity metric, it is a minkowski metric) must also be preserved.

further points of discussion

1.we discussed the possibilty of whether all canonical transformations can be represented by rotations of phase space and reasoned that in general it cannot.the reasoning follows the lines of symplectic structure of the poisson brackets which implies that canonical transformations cannot be
in general represented by rotations of phase space.
2.the second pair of maxwell equations
3. we saw that quantum mechanics is a field theory in some sense.this is because we can always obtain the klein-gordon equation (which describes quantum mechanics) from an action which can be said to describe a classical field theory in 0+1 dimension.
4.some discussions on path integrals

interesting points of fields in lower dimensions

there are some points about electrodynamics you can rather easily see from the field tensor.

in 1+1 dim, the field tensor is a 2X2 matrix with only one independent component.this component is the 01 component:which is one space and one time.so this is an electric field.note that E is a polar vector and so is the 01 component; hence the electric field is a scalar in 1+1 dimension.

all components exhausted, there is no B in 1+1 dimension!!

this also follows from Biot-savart law; the force on a current due to another current acts in a direction prependicular ot the line joining them, which is not possible in 1 space.

now lets come to 2 +1 dim.the electric field has 2 components.

what about the magnetic field ?

recall biot-savart law.force is given as a cross pdt, it has the structure of Fj=εjkvkB so it turns out that it is a scalar.also since F and v are polar vectors,B must ont change sign under parity and hence it is a true vector.

Bianchi identity: suppose u have an antisymmetric object and u have an operator that acts linearly on the tensor , then the sum of all cyclic permutations of the tensor add up to zero.
Fij=DjAi – DiAj (1)
define tijk=DkFij (2)
then the bianchi identity is
Tijk=tijk+tjki+tkij=0 (3)
as can be shown by substituting 1 & 2 in 3.

Next,using the anti-symmetric properties of Tijk in 3 we see that only 4 non-trivial equations result.knowing that this involves no source terms, we suspect these to be the source free maxwell equations.there are the right number of them!so the bianchi identity are the source free maxwell equations

why do you need a four-vector potential to characterize the field in the action formulation?well,this is because you have E from a scalar potential and B from a vector potential.

note that the convention E= -div (phi) comes from the convention that a positive charge moves in the direction of lower field.

the sturcture of the bianchi identity, ie, of some linear operator acting on an antisymetric structure is the same as for maxwell equations.

ED#

proceeding further, we wrote down the action of the complete field+particle system and then used the variational techniques with respect to different cocordinates to get maxwell equations in four-dimensional form.symmetry arguments for gauge invariance yielded the continuity equation.the properties of the field tensor in 1+1 and 2+1 dim were also deduced.

Into Electrodynamics

With the notational ideas cleared up, i move into the covariant formulation of electrodynamics.But first, a short review of relativistic mechanics.
Using the principle of least action,the lagrangian is determined to be
L=-mc^2*sqrt(1-v^2/c^2)
Then we go ahead to find the expressions for momentum and energy in the case of a free particle.the euler-lagrange equations of motion(ELEOM) give us the condition that the four-velocity must be a constant.
LL makes the comment that the energy in relativistic mechanics is always positive and proportional to the mass, unlike in class mech where it can be either positive or negative.That would be because in class mech u do have the potential energy of interaction of a system of particles.but to do so in rel. mech. one has to explicitly include the momentum and energy of the field produced by the interacting particles.
A short review was also made of the angular momentum tensor, its antisymmetric nature.

Next, we come to the section where the field is viewed as an physical entity that mediates interactions between particles, since relativity allows only contact interaction.A particle imparts the information about its motion to the field surrounding it, which then propagate it far away.Also, since relativiy disallows the concept of rigid bodies, the so called elementary particles must be pointlike.
the next sections are more mathematical.in them we use the four-vector A to characterise the field, obtain the lagrangian, the generalised momentum,the hamiltonian.Then we proceed further to obtain the ELEOM and define the elecric and magnetic fields.interestingly, we see that the work done on the particle is only by the electric field E, since the magnetic field B is always perpendicular to the velocity v.
Equations of motion in classical mechanics are time reversal invariant; the same thing is true in electrodynamics under the transformation t-> -t; E -> E ; B -> -B which in turn implies phi -> phi; A -> -A; phi and A being the components of the four vector A.
then i studied gauge invariance and solutions for E,B and energy in constant electomagnetic fields.

four vectors

Thought it would be wise to recall some of the concepts before proceeding further.So,I started with revising the section on Four Vectors of sec 6 of Landau-Lifshitz(LL).
there is one thing i wanted to clarify.A quantity that is a product of a pseudoscalar and a scalar is a pseudoscalar; the product of a pseudoscalar with another pseudoscalar would be a scalar.I can extend the same argument to vectors and tensors; and then i would claim that (levi-civita_iklm)*(levi-civita_prst) is a true tensor.