Gauging my Knowledge!!!

Many people have asked me and I’ve asked many people why a photon has only two polarizations/ degrees of freedom (DoF) and not four! The ‘qualitative’ answer which most people (including me) gave and with which I was satisfied for quite some time consists simply of two parts:

• one DoF reduced due to gauge invariance (4-1=3);
• another DoF removed due to masslessness of photon (3-1=2)!

But after sometime, hearing this mundane answer did not satisfy me or the others anymore (basically others but lets not digress)! So I took it upon myself to settle this once and for all. You may ask why go to such lengths to answer an age-old question of which everyone knows the answer which is ‘DoF for a massless vector gauge field in D dimensions is (D-2)’. Well, there are two reasons:

• As PvN stresses in his classes it is always good to have a concrete example worked out than always deal with generalities/ abstractions!
• In GS’s QFT class, I got to know the ‘answer’ which one of the many people (lets call him PJ) does not agree to happening… go figure!

There is one more (2+1=3) very important reason: after reading this page by WS, I got the urge to test the Unicode stuff myself. So I post here the answer starting ‘from scratch’. This will probably be the only place in which I present Math and Physics simultaneously and roughly in equal proportion. [As should be obvious from my second reason, the following is ‘shamelessly’ lifted from notes of GS’s QFT class. Though any mistakes below would be entirely mine due to the well-known phenomenon of Lost in Translation.]

Lets start from the Lagrangian and Equations of Motion (EoM) for the massless vector gauge field A_μ:

ℒ=-¼F^μνF_μν

EoM: ∂_μF^μν=0 ⇒ □A^ν-∂^ν(∂_μA^μ)=0

We know that ℒ is invariant under following Gauge Transformation (GT) where λ is a gauge parameter:

GT: A'^μ(x)=A^μ(x)–∂^μλ(x)

We can use GT to change EoM: Given A^μ(x) that solves EoM, construct λ(x) such that □λ=∂_μA^μ which implies ∂_μA'^μ=0. A thing to note is that A'^μ(x) still has same F^μν and EoM, i.e. A'^μ is physically equivalent to A^μ.

Now we gauge fix the Lagrangian, i.e. give up manifest gauge invariance but keep all the physically distinct solutions:

ℒ=-¼F^μνF_μν–½α(∂_μA^μ)²

EoM: □A^ν-(1-α)∂^ν(∂_μA^μ)=0

where α is a (real & non-zero) gauge fixing parameter.

Acting with ∂_ν on EoM gives α□(∂_μA^μ)=0 ⇒ ∂_μA^μ=0.

Now we are left with D EoM & 1 Lorentz condition:

□A^μ=0 & ∂_μA^μ=0.

Solving the first equation gives:

A^μ(x)=a^μe^-ik.x where k²=0 with k⁰≡ω_k≡|k|

(Notation: k is a D-vector and k is a (D-1)-vector!)

Lorentz condition enforces k.a=0. This suggests the general solution: a^μ=Ck^μ+ε^μ where ε^μ should satisfy ε⁰=0 & ε.k=0 (as is easily checked!) This already suggests 2 DoF have vanished; lets make it concrete by gauging A^μ using

λ(x)=iCe^-ik.x ⇒ A'^μ(x) = A^μ(x)–∂^μλ(x) = ε^μe^-ik.x.

Now, it is obvious that a vector gauge field has only (D-2) DoF since ε⁰=0 & ε.k=0. In other words, timelike and longitudinal components are not physical, only transverse components (usually called polarizations in 4-D and hence the symbol ‘remaining’ is ε!) are physical and count as DoF!

Thus, we see that loosing two DoF is not so straightforward as the argument at the beginning of this post suggested; there is a subtlety as to how gauge invariance and masslessness conspire to get rid of these DoF. If you are wondering where the masslessness was used(!), keep reading…

Finally, lets look at the massive vector field and observe how longitudinal component remains physical!

ℒ=-¼F^μνF_μν+½m²A_μA^μ (Not gauge invariant!)

EoM: □A^ν-∂^ν(∂_μA^μ)+m²A^ν=0

∂_ν EoM ⇒ m²(∂_νA^ν)=0

Here also we are left with D EoM and 1 Lorentz condition:

(□+m²)A^μ=0 & ∂_μA^μ=0.

But that is as far as we can go since no GT can reduce DoF any further (basically due to m² in EoM) and so we are left with one more DoF compared to massless case, i.e. D-1 DoF.

I hope this satisfies some (if not all) of the many people I referred to earlier. And if you (the reader) have a succinct/ concise/ elegant proof (but explicit as done here), you are welcome to post it in the comments or tell me via any other communication means you may find reasonable!

So, that is it for this month and if you have not read this ‘ground-breaking’ paper in fundamental mathematics; you are missing out something very important in your life!

Time variation of a fundamental dimensionless constant

My Obsession with Sketchbook

Before I start talking about the story behind my title and complete my post quota for the month of March, let me say I am ‘glad to tell you’ that I have my first paper @ SUNYSB (numerically 3^rd) put up on arXiv.org. [I may write about this in some detail in next month’s post.] Now that we are done with the formalities, lets get back to the informal stuff.

As you might remember from my previous post, I was (am!) quite hooked to the anime/manga called ‘Sketchbook’. I mean how can you not be hooked to this brilliant piece of work when the ‘darkest’ character (Kokage Kuga) is asked to laugh after being ‘complimented’ for having a poker face!

In fact, one of my friends says he’ll also get ‘hooked’ to this one after realizing there is a nature-lover (Nagisa Kurihara) in this manga! That said I’m so hooked that while working on the problem of  ‘Gauge Invariance…’ (check previous post to understand what I’m talking about!), I inadvertently started sketching the main character (Sora Kajiwara) on the same piece of paper! If you don’t believe me, here’s the sketch (the better one!):

What I was trying to sketch was this: [:)]

or this!

Now that I’ve wasted some of your precious time with irrelevant ranting, please get back to whatever you were doing before arriving to my relatively short post this month. Obviously, I’ll leave you with a last link as usual!

Nawaz Deobandi recited poems for 40 minutes on-stage!

Experiencing Death!!!

27 Jan: While taking an afternoon nap on this day, dreamt of chest's right side hurting a bit... slowly the pain seemed to worsen, so put one finger on a rib and pressed slightly... it didn't help much... the pain continued worsening... more and more and more... also it dawned that breathing had become difficult... increasingly difficult...

Suddenly, revelation hit... ITS NO DREAM!!! ITS REAL... indeed it was... excruciating pain in right side of the chest... difficulty breathing... can't breathe properly... can't inhale completely... pain shoots up while inhaling... as if the ribs are poking in the right lung!!! stand up slowly... STILL it pains while breathing... revelation hit again... DAMMIT ITS REAL!!! ITS NO DREAM...

Removed T-shirt... thinking maybe it soothes pain while breathing in short 'pulses' to reduce stinging chest pain... checked right-chest: huge depression on right-chest... now pretty sure of ribs digging in lungs... flush of despair, anxiety & pain... body heats up... start sweating... vision blurs... zooming through white space... like the stars screensaver but no white dots in black sky, just white streaks in white background!!! closed eyes...

Thoughts (in order of appearance!!!) start surfacing:

Is this 'Dying'??? (breathing excruciatingly painful...)

Should go to Infirmary... but that is too far!!! (can't even move my legs properly now...)

Should recite Navkar Mantra like mother tells to do under such circumstances!!! start doing so... stop after 4-5 times and thinking of lying down... (STINGING PAIN) lie down in opposite direction wrt earlier... feel cool breeze from a very slightly open window... (is it soothing???)

Now thinking of chugging Bryonia - homeopathic medicine for all ailments: cough, cold, fever... (probably would also work in case of dying of pain!!!)

HELL, HAVE NOT YET PROVED GAUGE INVARIANCE OF N=2 PROJECTIVE SUPER YANG-MILLS LAGRANGIAN... CAN'T DIE BEFORE SOLVING THIS PROBLEM*!!!

Slowly the pain disappears and the depression on chest seems to have been filled... lungs no more constrained... can inhale fully... open eyes slowly... vision restored!!! stand up... can breathe normally... no pain... flush of relief, Relief & RELIEF!!!

AFTERTHOUGHTS: The rumored depression on my chest still exists: it seems to be an indispensable part of my emaciated body!!! It seemed huge maybe because of pain, fear, delusion or whatever (leaning towards right makes the depression deeper!!!) and while lying down it does disappear which is normal!!! The pain may have been due to CO2 poisoning or lower O2 levels (but symptoms don't quite match!) owing to several factors: (i) the quilt (pretty low temperature outside!) was tightly wrapped around my face forming 'huge walls'; (ii) the 'wrong' inclination of the floor (or bed) meant my head lay lower than my feet; (iii) almost sealed room with only a slightly open window near my feet. The recovery may have been due to the  access to 'fresh air' after reorienting myself! After thinking of all this crap, I lie down again on the bed and laugh out loud for quite some time... enjoying the Death Experience!!!

Lastly and most importantly (and obviously) like in Rurouni Kenshin, the moral of the story is: There is Nothing Stronger than the Will to Live!!! :)

After reading through this 'grim' post, do lighten up by reading

Hidamari Sketch

*Great resolve now that I think back, though as of this day I'm yet to solve it!

Tasting Indian Food!!!

After being in US for nearly 17 months, homesickness has started to set in. And most of it is because of not getting 'typical' Indian food (the vegetarian portion is being considered here if you do not realize!) as seen in this image.

Look at all the Sweets: (1) Agra Petha, (2) Gazak, (3) Gulab Jamun, (4) Kaju Barfi/Roll, (5) Cake... & Namkeens: (6) Samosa, (7) Dahi Wada, (8) Cheese Toast, (9) Chhole, (10) Noodles... Maaannnn, if I were not as abstemious as I am, I'd be running out of towels to wipe off all my drool here in the apartment!!! So why am I  bringing this 'delicate' subject up in this post??? Good question indeed!!!

That is because I'm a self-constituted/confessed Sweet-Critic/Connoisseur. As a matter of fact, I taste all the sweets brought to home before anybody else. Now you may be thinking that's because I'm the youngest of all!!! (I hate these logical reasons...) Hell No... there do exist younger ones in the house (once in a while) but I am always able to snatch the first bite, Hell Yeah!!! Who do you think I am??? No one kids with me  when sweets are involved and that includes adults, kids and gods! There is a running gag in my family about offering 'प्रसाद prasad (offerings)' first to me and then to Ganesh!!!

OK, that's too much ranting about My Deserved Holiness (MDH: I wonder how many will get the pun in here)... Lets get back to the topic of this post: food and pick up from the first sentence of the last paragraph: ...seur. I tend to include any food item made of Paneer also in the definition of sweets much like the following usage of word 'sweet':

Someone: Hey, I got accepted/appointed/promoted/married to...

Someone else: Sweet!!!

So I could not stop myself from picking this "Amy's Indian Mattar Paneer" packet when I caught sight of it at Pathmark in the aisle (or 'path' as they like to call it there!) having frozen Pizzas.

Being a Critic/Connoisseur, I just could not let go of this opportunity of tasting something whose origin is so delightfully explained on the back of the packet!

Carefully note that only Mattar Paneer is written on the packet's front but the image shows Rice (pulao) and Garbanzo Beans (chana masala) also. I thought they might be providing serving suggestions but then I read the ingredients:

It shows all the ingredients shown in the front image but still I did not realize/believe that I would get all the three items in one packet - I was hoping it would just have Mattar Paneer. Noting so many organic things in the ingredients, you feel doubtful of all the things you have eaten your entire life till now!!! Also, noteworthy is the fact that Amy states SPICES* in the ingredients and then in the footnote says (no hidden ingredients)!!! This can probably be produced as an example of  a figure of speech called Ironymoron.

I b(r)ought it home, opened the packaging and I was left with my mouth agape. It had all the three food items and arranged almost in the same format as shown in the image on the front of the box. Even the number of Paneer pieces was same (6)! I microwaved it and tasted all the three items in one gulp. That aroma, flavor, texture was as close as it could be (here in US!) to what I would get at home or a 'proper' Indian restaurant like those run by Haldiram's!

Let me explain first the 4 aspects which I use to classify Sweets (food in general) before I proceed further:

• Aroma: The scent which you get even before you put the sweets in mouth. The smell of rose water or vanilla essence is the best!
• Flavour: This has to be experienced by yourself and the best way to describe it is the feeling of uuummmm... as the taste gets unleashed in your mouth.
• Texture: The quality of the sweet which is experienced as you chew it and roll it on your tongue. The best is when either the whey is rubbery and you realize you are chewing it longer than usual or the khoa dissolves in your mouth even before you realize you didn't need to use your teeth!
• Bliss: This is the rarest aspect one ever considers but since I'm a connoisseur, I got to go ahead of the mere mortals and say something  unusual. This is not the quality of the sweet itself as such but the feeling you get after devouring your favourite sweet one piece at a time. Obviously, it has to be prepared in the best possible way such that you can close your eyes and murmur the immortal line: "Today's a good day to Die!".

Now lets check how many aspects of the Amy's 're-creation of Indian food' appeal to me:

• Aroma: Excellent apart from the fact that I don't like Cumin's scent in my food!
• Flavour: Excellent again. The mildly spiced curry had the correct composition of Spices - the way I'm used to!
• Texture: Pulao and Chana Masala were perfect but it is the hardest aspect to infuse in Paneer, at least one that satisfies me! The raw Paneer used felt a little 'stiff' (my vocabulary of adjectives is limited so I cannot depict what I felt accurately but those thinking that microwaving it for some more time would have made it tender are wasting their thoughts!) which could have been remedied (I guess!) if they were fried which masks the 'stiffness' a little. But then maybe I should not expect much from frozen foods!
• Bliss: It made me close my eyes but instead of murmuring the immortal line, it made me murmur the mortal one: "I'm not dying till I get to eat home-cooked food from my mother's hands one more time"!

So, what is the result of my assessment? I'd give 4.5/5 to this 'would have been magnificent' product... '.5' deducted because price is too damn high (~$6) and the whole thing is not even enough for one meal. Generally I can buy 3 individual packets of Vegetable Pulao, Chana Masala & Mattar Paneer from Desi Store for that amount and use each packet for nearly two meals... obviously I use some 'extra' Chapatis or Parathas!!!

I realize now that this was a pretty long post and Kudos to you my dear reader for be(ar)ing with me till now! So, here I end my review of an American-Indian Food!!!

As is the norm, Check these Indian Sweets out so you know what you are missing on this Earth which can make you feel like you are in Heaven:

List of Indian Sweets

HAPPY NEW YEAR

As mentioned in previous post some months ago... here's the most anticipated post of December which is basically to wish you -one of the very few readers of my blog: A Happy New Year!!!

Enjoy the new day, week, month and the year that is 2009: Do you realize: 2009=2+0+0+3^2=23!!! (after taking trace, removing common factors and changing the definition of '+' & '^' wherever necessary...)

The Number 23

On Hiatus

New posts will appear most probably in December'08...

Enjoy your life till then!!!

[:)][:D][=))]

Another Manga Ends!!!

Elfen Lied with 107 chapters.

EL is not to be read in English but German which when translated in English means Elves' Song!

Now you might be thinking why would I be interested in a 'musical' manga and you cannot be more wrong than that in your entire life. [Though I watched such an anime 'Nodame Cantabile' earlier but that was due to entirely different reasons, similar to Sketchbook.]

This manga is not about Music; its about Humans, their lives and how they treat other 'lives' - portrayed with pessimistic point of view (in my opinion) and negative connotations (which is not that far from reality!).

Wikipedia has this to say about the anime which is also more than true of the manga:

Elfen Lied involves themes of social alienation, identity, animal cruelty, revenge, child abuse, jealousy and the value of humanity. The series employs graphic violence and nudity, ...

The original manga did not end now (if my post suggests that!), but the scanlated one did. The original ended in August 2005 nearly 3 years ago... I wonder why the delay in scanlating chapters!

Now I am thinking of what to write about the manga EL. The first manga post was about Inuyasha and I sort of sketched out the last chapter in some detail which was not well-received by my reader. Yes, I have One! The second manga post was about Sketchbook and I wrote about my views of the anime, basically why I enjoyed watching it.

I cannot do any of the above for this one. Why? Because:

• Reviewing the last chapter of EL does not mean much here. In case of Inuyasha, the 'happily ever after' scenario in the last chapter involved (almost) all characters and was meaningful as an end for the story. However, it is not so here. The ending is sort of different in this case and elaborating on it is equivalent to spoiling the ending of a worth-reading story.
• Why I liked EL is again a hard question to answer. One might be tempted to say 'nudity, obviously'! Well, that may be true in one's case or for some other manga like 'Golden Boy' but not so here in my case. And I don't like graphic violence, either. So, the only logical conclusion would be something from the themes.

Indeed, that is so - the interplay between humanity and an 'alien entity'; their views of each other; the value they ascribe to each other - is something that is beautifully depicted in EL. The schizophrenic personality of the main character - 'human side' trying to fit in the human world and 'alien side' (for lack of a better term!) trying to annihilate the entire human civilization - was something that was gripping and probably the main reason why I could not stop reading this manga till it concluded.

Now I think I've rambled enough in this post since I did not have any intention of doing that when I started typing this post. Wow, too many I's in the previous line! OK, so I conclude this post here.

As usual, provided below is the link to more stuff/info so that you can form your own opinions about this series and probably waste some time like I'm doing right now.

ELFEN LIED

Sketchbook

Obviously, it does not mean I've started a new hobby of sketching and thus bought a sketchbook. Nope... definitely not!

What it refers to is the Manga/Anime I came across: Sketchbook.

The manga is in a comic strip style so I thought: nothing much to get excited about. But, I read it anyway because the art was good and humorous. The humour as such is not entirely in the story or the situations portrayed but the 'Japanese way' of depicting emotions of characters. It is used extensively and achieves a new height in this manga which is refreshing and riveting!

But then reading got over soon since the chapters have barely 7-8 pages on average and 22 scanlated chapters do not take too long to read! So I ventured ahead, opened Wikipedia to get to know how many chapters are there altogether. It says 5 volumes have been released and I'd just started the 2nd volume so I was thrilled to know there will be lots more pages to read in the future...

Apart from this, I also saw the info that Sketchbook has been animated into 13 episodes titled "Sketchbook ~Full Color'S~". I thought how can you animate a comic strip which actually does not have an expressed theme into something worth viewing? I remembered Garfield's two movies which were good but compared to the strip itself, those definitely lacked something. In other words, its not the same: the (animated) movies can not really match the one-day (usually) adventures of free-willed Garfield.

But realizing that watching animes is a part of my hobby and since I watch only 'One Piece' nowadays, I decided to give it a try. I watched the first episode, then second and then third. I could not stop myself: the anime was too damn good... much better than the strip-styled manga. Apart from having the 'obvious advantages' of a different media (colour & video), the anime truly captured the theme of having no theme in particular. The simplicity of the usual events in the life of characters, their interactions among themselves and their art club activities were presented so intricately and (possibly) accurately that I was left gaping at my laptop's screen and awe-inspired to develop a quote by myself: "Simplicity is Beauty".

It seems like a take on the title of Ian Stewart's book "Beauty is Truth" but who cares. When I asked Shantanu about his views on above-mentioned quote he quite hesitatingly agreed by saying 'in acads.. yes because my mind can only understand simple things..' It seemed as if he anticipated my next question to be about 'naked beauty' or 'bare bottom' if he had not answered my query in that particular manner!

Anyway,  we are digressing here so lets get back to 'Sketchbook'. So, I recommended the first episode to him as he was out looking for a good movie to watch and waste some time. I hoped he would enjoy it. Rather, he said 'was boring; there was nothing in it...' I was too optimistic and naive in thinking that someone watching anime for probably the first time would enjoy it just based on the art and would ignore the null theme of the story!

So, I started thinking why did I like it and he not... First lets analyze his situation:

• First time (or so) watching an anime;
• Video is subbed, got to keep track of subtitles, thus loosing some of the nuances of the art;
• Not familiar with 'chibi' art style (generally used to depict character's moods in animes) which is used extensively and mostly generates the humour in 'Sketchbook';
• Obviously, no theme as such: the episode follows a shy & quiet girl and her art club 'friends' trying to find something interesting to draw - a typical day of their life: no action, romance, mystery, comedy or even drama!

This probably explains a lot - at least I'm satisfied with this explanation. Now then, lets analyze my situation:

• Definitely not my first time watching animes;
• I've watched more than enough of subbed videos to bother about this fact;
• Chibi art style is one of the many reasons I like animes;
• This fourth point got me thinking for a while - why do I like this 'so simple' anime?

Then I remembered Rutherford's words: "I am always a believer in simplicity, being a simple person myself. " Though Shantanu's earlier words also profess similar convictions, I guess quoting Rutherford makes people believe my 'knowledge' is not confined to an extremely narrow potential well. Thinking back about my schooldays, I see why I like this anime... its because my daily routine back then (even now!) was also as simple as (actually more so!) these character's. So I enjoyed seeing the 'routine' stuff with no 'ulterior motives' from third person's perspective. It is really fun (or pleasing) to know that someone exists out there (Totan Kobako) who also 'thinks' like me! :)

If you want to know what I'm talking about and decide on your own about the so-called 'simplicity' and 'beauty' of this anime then go ahead and watch

Sketchbook ~Full Color'S~ Ep-01

Another take on shayari by introducing...

Khamakhaan Hyderabadi or Khamakhawah Hyderabadi i.e.

खामखाँ हैदराबादी या खामख्वाह हैदराबादी!!!

Some beautiful & thought-provoking shers/ghazals by the above-mentioned poet are in store today. This is the first post of August and hoping the second one will appear before the end of this month. Though I don't have any readable material right now to write about in a week or so!

Let me not bother you with my 'predicament' in this post and get to the subject matter as fast as possible which is to post some of the videos I discovered on YouTube...

These videos do not have good audio but bear with it and after sometime you will get good at filtering out the static and picking-up the 'real signal'. Just to whet your appetite, one sher (?) is written below every video in which it appears:

Khamakha... Hyderabadi - I

मोहब्बत में मरते तो हैं खामखाँ सब,
जनाजा किसीका न देखा निकलते हुए
[Mohabbat mein marte to hain khamakhaan sab,
Janaaja kisika na dekha nikalte hue.]
Also,
माशूक़ हो हसीन ये जरूरी नहीं, मगर
आशिक़ को चाहिए की वो अंधा जरूर हो
[Maashuq ho haseeen ye jarooree nahin, magar
Aashiq ko chahiye ki wo andha jaroor ho.]

Khamakha... Hyderabadi - II

कर्ज़ भी मांगो तो मांगो इतनी खुद्दारी के साथ,
कर्ज़ देने वाला देकर तुमसे शर्मिंदा रहे
मैं भी एक दिन कर्ज़ लौटा दूंगा सबका खामखाँ,
लेने वाला मेरे देने तक अगर जिंदा रहे
[Karz bhi maango to maango itni khuddari ke saath,
Karz dene wala dekar tumse sharminda rahe.
Main bhi ek din karz lauta dunga sabka khamakhaan,
Lene wala mere dene tak agar jinda rahe.]

Khamakha... Hyderabadi - III

बुजदिल है वो जो जीते जी मरने से डर गया,
एक मईच था जो काम ही कुछ और कर गया
जब मौत आकू मेरेकू करने लगी सलाम,
मैं वालैकुम सलाम बोला और मर गया
[Buzdil hai wo jo jeete jee marne se dar gaya,
Ek maeech that jo kaam hi kuch aur kar gaya.
Jab maut aaku mereku karne lagee salaam,
Main waalaikum salaam bola aur mar gaya.]

Khamakha... Hyderabadi - IV

बेहतर है मौत खामखाँ, देती तो है कफ़न,
'गर ज़िन्दगी का बस चले, कपड़े उतार ले
[Behatar hai maut khamakhaan, deti to hai kafan,
'Gar zindagi ka bas chale, kapade utaar le.]
Also,
बीवी को सौत दे या मुझे मौत दे, ख़ुदा!
उठते नहीं हैं हाथ, मेरे इस दुआ के बाद!!
[Biwi ko saut de ya mujhe maut de, Khuda!
Uthte nahin hain haath, mere iss dua ke baad!!]

The following videos have better audio and many shers recited above can be heard clearly here! There are some new ones also. So enjoy...

Khamakha... Hyderabadi - A

Khamakha... Hyderabadi - B

Khamakha... Hyderabadi - C

Khamakha... Hyderabadi - D

That is all for today, I guess. Don't have any more wisdom to share. If you would like to hear some more, here is the new Beginning:

Girgit Ahmedabadi