NEQNET: Non-equilibrium Phenomena

107. New “Ender in Exile” by Orson Scott Card

Posted by Dmitry

at November 21, 2008

Sorry for the off-topic but I am actually a HUGE fun of the Ender series by Orson Scott Card. I think I’ve read the original “The Ender’s game” around 2000 and
Card immediately acquired another infinitely faithful worshipper - every single book in the series was bought afterwords, with some disappointments, but without second thought - once I saw his new book in a book store, I grabbed it. Who knows why? Maybe because teaching philosophy in the Space Academia was remarkably similar to the way boys are taught in Eton nowadays ( and the way we were taught at Landau Institute) How Card presented the Ender’s and Peter’s life (I never actually had any warm feelings to Valentine), how he explained what is essential and what is important in this life, what problems one may expect once globally accepted as Military Genius - all this entered so strongly into the resonance with my cognitive model, that I was first astounded, second ashamed of myself, and third - got filled with energy

But anyway… enough sentiments. I am glad to announce that Orson Scott Card has just released a new book in the Ender’s series! Have you ever wanted to learn what happened with teen Ender Wiggin after he left the Earth and how was he dealing with the Xenocide responsibility and the guilt in the death of Bonzo and Stilson while he still was so young? Do you want to know how did young Peter Wiggin evolve into the Hegemon of the Earth?… How did Valentine manage to reconcile her sacrifice of her own life to help the Ender to carry his burden?… What happened to other team members of the Ender’s “jeesh”? I know - if you love Card’s intelligent prose, you cannot help taking a look to the new “Ender in Exile“. And let me assure you, while “Xenocide” and Shadow series both were somewhat weaker than the two original novels, Orson Scott Card is back in all his shining power

Fun, SF

Related posts:

No related posts

106. Criteria for confinement. Wilson loop - getting more technical

Posted by Dmitry

at November 21, 2008

Last time we have discussed a bit the behavior of the Wilson loop expected in the confinement and deconfinement phases and have concluded from simple physical considerations that the first one corresponds to the area law, while the second - to the perimeter law. Let us now show directly that the Wilson loop VEV satisfies the area law for a large rectangular contour. This derivation will allow us to get familiar with several interesting features of the Wilson loop variable.

For simplicity we will choose the gauge $A_{0}=0$ and the contour with the length along the $x_{1}$ spacelike direction and the length along the timelike direction. The Wilson loop VEV is given by

$\langle{}0|M(C)|0\rangle=\langle{}0|{\rm Tr}\left(P\exp\left(i\int_{L}^{0}dx^{1}A_{1}(x_{1},T)\right)\times$
$\times{}P\exp\left(i\int_{0}^{L}dx^{1}A_{1}(x_{1},0)\right)\right)|0\rangle=$

$=\langle0|{\rm Tr}\left(P\exp\left(-i\int_{0}^{L}dx^{1}A_{1}(x_{1},T)\right)P\exp\left(i\int_{0}^{L}dx^{1}A_{1}(x_{1},0)\right)\right)|0\rangle.$

The vector potential operator $A^{\mu}(x,t)$ is the matrix (in the fundamental representation of the group SU(N) ), so is the Wilson loop operator

$P\exp\left(-i\int_{{\cal C}}dx^{\mu}A_{\mu}\right)$ ,

understood as the usual exponent of the matrix. By the way, it is clear why we have to introduce -ordering: the operator exponents $\exp(iA_{\mu}dx^{\mu})$ taken at different points of the spacetime do not commute with each other, since SU(N) is the non-abelian group.

Exercise. What is the value of the commutator of two such exponents taken at different points?

We can rewrite the trace above in the form

$\langle0|M(C)|0\rangle=\sum_{m,m'}\langle0|(M^{*}(L,T))_{mm'}(M(L,0))_{m'm}|0\rangle$ , (1)

where the operator

$M(L,t)=P\exp\left(i\int_{0}^{L}dx^{1}A_{1}(x_{1},t)\right)$ .

Next, we can relate the operator $M^{*}(L,T)$ to M^*(L,0) with the help of evolution operator. We have

$\langle0|M(C)|0\rangle=\sum_{m,m'}\langle0|(M^{*}(L,0)e^{i\hat{H}T})_{mm'}(M(L,0))_{m'm}|0\rangle$ .

At the next step we insert the complete set of excited states between the operators
$M^{*}(L,T)$ and M(L,0) to find

$\langle0|M(C)|0\rangle=\sum_{m,m',n}\langle0|(M^{*}(L,0)e^{i\hat{H}T})_{mm'}|n\rangle\langle n|(M(L,0))_{m'm}|0\rangle$ .

Since $e^{i\hat{H}T}|n\rangle=e^{iE_{n}T}|n\rangle$ ,

we conclude that

$\langle0|M(C)|0\rangle=\sum_{m,m',n}|M(L,0)|^{2}e^{iE_{n}(L)T}$ , (2)

where M(L,0) is the matrix element

$M(L,0)=\langle n|(M(L,0))_{m'm}|0\rangle$ .

(Basically, we wrote the expansion for the propagator of the complex field over the complete set of eigenstates of the Hamiltonian $\hat{H}$ .)

What do we learn from the exression (2)? There are actually several important lessons. First of all, let us consider a Euclidean version of the theory (that can be obtained by exchanging $T\to iT$ in formulae above). The expression for the VEV of the Wilson loop is given by

$\langle0|M(C)|0\rangle=\sum_{m,m',n}|M(L,0)|^{2}e^{-E_{n}(L)T}$ .

For very long contours with $T\gg E_{0}^{^{-1}}$ the only important contribution is given by the ground state:

$\langle0|M(C)|0\rangle\sim{\rm Const.}e^{-E_{0}(L)T}$ ,

where

${\rm Const.}=\sum_{n,n'}\langle0|(M^{*}(L,0))_{mm'}|0\rangle\langle0|(M(L,0))_{m'm}|0\rangle=$
$={\rm Tr}|\langle0|M(L,0)|0\rangle|^{2}$ ,

i.e., trace of the VEV squared of the Wilson line, connecting heavy quark and antiquark. The vacuum energy $E_{0}(L)$ is clearly given by the interquark potential
V(L) - recall that the quarks are infinitely heavy and therefore static, and the only contribution into the overall energy of the system comes from their potential energy. It is worth noting in this respect that we can only talk about the interquark potential, if the quarks are non-relativistic (even better - static).

If the potential V(L) is growing linearly, we find that the VEV of the Wilson loop satisfies the area law, famous criterion of confinement first proposed by Kenneth Wilson.

The second lesson is that area law does not hold for short contours, since excited states start to give larger contribution into (2). It is physically clear what happens at very small . The SU(N) gluodynamics is an asymptotically free theory, and the effective coupling $g^{2}$ gets small for $L<\Lambda_{YM}^{^{-1}}$ - the theory becomes effectively free or, as we often say, enters the Coulomb phase.

Exercise. Try to estimate the order of magnitude of the gap in the spectrum of the Hamiltonian $\hat{H}$ .

The third, very important, lesson is that we don’t actually expect area law for the Wilson law if we take two infinitely heavy bosons charged under SU(N)!

Exercise. Think a bit why it is so. If you will be unable to figure it out, then I shall explain this in the one of the next posts. But you see… from my point of view deriving things yourself (or at least being able to derive them) is equivalent to having the best fun ever Theoretical physics (and QCD in particular!) may become a huge source of fun in this respect

And if you want a hint, consider representations for various spins and figure out what the trace really means in the Wilson loop.

To be continued.

75. Chaos in YM and confinement

101. Confinement. Extremely naive introduction

26. Eye on ArXiv: 24 Apr 2008 - NSR superstring measures

Best posts

105. And to scare you even more…

Posted by Dmitry

at November 20, 2008

Overall traffic (all US roads and streets) has dropped down 4.4% (equivalent to 10.7 billion vehicle miles) for September 2008 (compared to September 2007). This sounds somewhat funny considering that the price for gas in US is down by about 50% from its maximum. What would that mean?

Via Andrei Simonov.

Entered Apprentice, Personal

Related posts:

No related posts

104. World crisis - looking for a job?

Posted by Dmitry

at November 20, 2008

US unemployment claims surged to a 16-year high, government data showed Thursday, adding to growing alarm as companies worldwide shed workers in the global economic downturn.

Initial jobless claims soared to a seasonally adjusted 542,000 in the week ended November 15, the highest level since July 1992.
The weekly US jobless claims report by the Labor Department offered more evidence the world’s largest economy appeared to be sliding into a deep recession.

France Presse

Just a week ago the number of jobless hit 515000. According to the FED forecast, unemployment rate will reach 7.4% in November, highest in recent years. From the beginning of this year, Americans lost about 1.2 million jobs in total.

Do you, guys, want to discuss what is going to happen with jobs in science in forthcoming years? On Uncertail principles Chad Orzel is trying to develop the physics professor strategy for working under conditions of cut in science funding. So far, the strategy consists of two steps: 1) Back your students up and 2) Invest in your community.

Chad, I am afraid the forthcoming turmoil will wipe out not only students and many of us, postdocs, but some professors, too.

Entered Apprentice, Personal

61. Rant: LHC, global crisis and switchers

62. Two decisions and the worth of US economy

26. Eye on ArXiv: 24 Apr 2008 - NSR superstring measures

Best posts

103. Criteria of confinement. Wilson loop - physical discussion

Posted by Dmitry

at November 19, 2008

It is often said that the most physically relevant criterion of confinement is the behaviour of the potential between two fermions: confinement implies the linear growth of potential between two charges with distance. Is it really so? In a gauge theory with fermions in the fundamental representation and the gauge group SU(N) ( N=3 corresponds to quantum chromodynamics), the string of chromoelectric field connecting a heavy quark $\Psi$ and antiquark $\bar{\Psi}$ may break down if the distance between the fermions becomes sufficiently large. This is due to the production of light quark-antiquark pair in the strong chromoelectric field (I estimate the maximal length of the chromoelectric string in this post.)

As a result, the interquark potential gets screened by light quark-antiquark pairs at large distances in analogues of quantum chromodynamics and does not grow at $L\to\infty$ .

By the way, people, who know QCD much better than me, I have a question to you - do you know how to derive $L_{\rm max}$ smarter?

But let me turn back to the original criterion of confinement. When exactly the unbounded growth of the interquark potential is equivalent to the quark confinement? Let us turn the light fermions off. The resulting theory is pure gluodynamics - SU(N) gauge theory, and its vacuum structure can be tested by infinitely heavy charges (potential between heavy test charges gives the gluon propagator) in the fundamental representation. If the distance between test charges grows large, the string of chromoelectric field cannot break down, since gluons in adjoint representation cannot form an object which is in the fundamental representation of the gauge group. Then, infinite growth of potential between test charges would imply confinement.

Following Wilson, let us consider a closed contour ${\cal C}$ and introduce the quantity

$M({\cal C})={\rm Tr}\left(P{\rm exp}\left(\int_{{\cal C}}A_{i}dx^{i}\right)\right)$ , (1)

where denotes the -ordered (along the contour ${\cal C}$ ) exponential

$P{\rm exp}\left(\int_{{\cal C}}A_{i}dx^{i}\right)={\rm \lim}_{\delta x_{i}\to0}\prod_{i}(1+A_{\mu}(x_{i})\delta x_{i}^{\mu}).$ (2)

The quantity $M({\cal C})$ is called the Wilson loop, and its correlation functions can say quite a lot about the vacuum structure of the theory. For example, let us take a rectangular contour ${\cal C}$ of the length in the timelike direction and the length in the spacelike direction (let us take it along -axis for simplicity). Then, the VEV of the Wilson loop

$\langle0|M(C)|0\rangle\sim\exp(-iE(L)T)$ (3)

carries the information about potential between two test charges.

Exercise. Check that the -ordered exponent (2) is not gauge invariant, while its trace is.

Indeed, we can immediately see from simple physical considerations that the Eq. (3) holds. Let us take a test charge $\Psi$ and carry it along the closed contour ${\cal C}$ . The corresponding quantum mechanical amplitude is given by $\langle{}0|M(C)|0\rangle$ . On the other hand, we can interpret this process as the creation of quark-antiquark pair, its propagation during some finite time and subsequent annihilation. It is clear from the form of the evolution operator that the corresponding correlation function can be written as $\exp(-iE(L)T)$ , where E(L) is the interaction energy between quark and antiquark. Indeed, in the limit $T\to\infty$ only the ground state of the Hamiltonian contributes into the expression for the amplitude. The energy of the ground state only depends on the distance between the quarks - the latter are heavy and only propagate in time.

If the theory is in the confinement phase, then $E(L)\sim\Lambda_{YM}^{2}L$ (interquark potential grows linearly!), and the VEV of the Wilson loop is

$\langle0|M(C)|0\rangle\sim\exp(-i\Lambda_{YM}^{2}LT)=\exp(-i\Lambda_{YM}^{2}S)$ , (4)

where is the minimal area spanned by the contour ${\cal C}$ . Further, since the spectrum of elementary excitations (gluons) in the theory has a mass gap, we conclude that

$\langle0|M(C_{1}\cup{\cal C}_{2})|0\rangle=\langle0|M(C_{1})M({\cal C}_{2})|0\rangle$ , (5)

i.e., the area law (4) should hold for arbitrary contours, not just rectangular ones in the confinement phase.

Exercise. What would happen in a theory without mass gap? Is it possible to have confinement in such a theory?

Now, suppose that our theory is not in the confinement phase but in the Coulomb phase instead - that is, the potential between two test charges behaves as $V(L)\sim-\frac{g^{2}}{L}$ , where is the coupling constant. Using Stokes theorem, we can write for the VEV of the Wilson loop

$\langle{}0|M(C)|0\rangle=\exp\left(-ig^{2}\int_{{\cal C}}dx^{\mu}\int_{{\cal C}}dy^{\nu}g_{\mu\nu}\frac{1}{2\pi^{2}(x-y)^{2}}\right).$ (6)

For $T\gg{}L$ this expression can be reduced to

$\langle0|M(C)|0\rangle\sim{\rm exp}(-i{\rm Const.}T)$ , (7)

i.e., the Wilson loop VEV behaves according to the perimeter law.

To be continued…

Want to know more before I wrote the next post?

Nice brief introduction into Wilson loops can be found in Peskin, Schroeder, “Introduction to quantum field theory”, which is an extremely good text book on QFT anyway. If you want something more advanced, then definitely Polyakov’s book is the best choice (you need the Chapter 5 entirely devoted to confinement).

Fellow Craft, Quantum field theory, Quark confinement

75. Chaos in YM and confinement

101. Confinement. Extremely naive introduction

41. Peter Woit: What will you do if string theory is wrong?

85. Hard thermal loops: what is it?

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Next Page >

Did not find what you were looking for? Try

107. New “Ender in Exile” by Orson Scott Card

106. Criteria for confinement. Wilson loop - getting more technical

105. And to scare you even more…

104. World crisis - looking for a job?

103. Criteria of confinement. Wilson loop - physical discussion

Search

Readability

Topic

Lectures

Recent Posts

Recent comments

Meta

Blogs: Physics

Top commentators

What I'm Doing...

107. New “Ender in Exile” by Orson Scott Card

106. Criteria for confinement. Wilson loop - getting more technical

105. And to scare you even more…

104. World crisis - looking for a job?

103. Criteria of confinement. Wilson loop - physical discussion

Search

Readability

Topic

Lectures

Recent Posts

Recent comments

Meta

Tags

Blogs: Physics

Top commentators

What I'm Doing...