Olive Diary

At the end of the last post we saw that the bare coupling in lattice gauge theory is a bad coupling to do a perturbative expansion in. If we use a good coupling, like $\alpha_{V}$ we will find a much more convergent perturbative series. This is crucially important if we want to determine things like improvement coefficients to good accuracy. An open question at the end of the last post was *why* the bare lattice coupling is bad, that’s the subject of this post.

To understand this, we need to understand how gauge fields are represented in the lattice. In lattice QCD, the matter fields are assigned to the sites of the lattice, and the gauge fields “live” on the links. This is very much in keeping with the geometric picture of gauge theory, where you have a parallel transporter ($U$) which allows you to compare matter fields at two different points. That is,
$$
\psi(x+\hat{\mu}) = U(x+\mu,x) \psi(x)
$$
In continuum gauge theory we have
$$
U(x+\mu,x) = P exp(ig \int_{x}^{x+\mu} dz^{\nu} A_{\nu}(z))
$$
Here P stands for path ordering.

This is actually much simpler on the lattice. On the lattice, there is a “shortest path” namely one link. So the parallel transporter simplifies to
$$
U(x+\mu,x) = U_{\mu}(x) = exp(i g a A_{\mu}(x))
$$
These parallel transporters (or link fields) are what one builds lattice actions out of. For example, a simple forward covariant derivative on a fermion field might be
$$
D_{\mu} \psi(x) = U_{\mu}(x)\psi(x+\mu) - \psi(x).
$$

Notice that the connection between the link field $U$ and the gauge field $A$ is non-linear. This is where the problems come in. Consider the expectation value of a single link field $(U)$. Expanding the expression for $U$ we find
$$
(U) = 1 - 2 \pi \alpha(\pi/a) a^{2} (A_{\mu}(x) A_{\mu}(x)) + …
$$
I’ve used (A) = 0, and evaluated the strong coupling at the cutoff $\pi/a$. This looks fine, in the formal continuum limit (a->0) the lattice corrections vanish. But all is not well! There’s a dangerous beast lurking in the loop integral (AA). Let’s look at what it is.

The expectation value (A A) can be written as the integral over the momentum space gluon propagator
$$
(A A) = \int_{0}^{2\pi/a} d^{4}q / qhat^{2}
$$
where
$$
qhat^{2} = 4/a^2 \sum_{1}^{4} sin^2 (q_{\mu} a/2).
$$
In the limit where $a$ is small, we can replace $qhat^{2}$ by $q^{2}$, and we find that the integral is
$$
(A A) ~ 1/a^{2} + O(1)
$$
This is the disaster, because when we substitute this back into our expression for the expectation value of $U$ we find
$$
(U) = 1 - 2\pi \alpha(\pi/a) a^{2} (1/a^{2} + O(1)) + …
$$
The $1/a^{2}$ from the loop integral cancels the $a^{2}$ from the expansion of the exponential leaving a lattice artifact suppressed only by the coupling constant, rather than the lattice spacing. The coupling constant is small (0.2) but not really small (like $\alpha a^{2} ~ 0.01).

It is factors like these (called gluon tadpoles) which spoil lattice perturbation theory in the bare coupling. The $\alpha_{V}$ coupling doesn’t suffer from these corrections because tadpole effects come in an 2nd order in $\alpha$. It would be nice to have a way to systematically account for these effects. This method, called tadpole improvement (or mean-field improvement) was proposed by Lepage and Mackenzie, and is now in wide use.

The idea is to use link fields that have the tadpole effects largely canceled out of them. To do this you take all the link fields you have $U$ and multiple and divide by some average link field $u$. That is
$$
U \to u U/u = u \tilde{U}
$$
The new fields $\tidle{U}$ are much closer to their continuum values, since the average link field division cancels out most of the tadpole effects. The factors in the numerator $u$ can be absorbed into couplings and masses. When combined with a good coupling (such as $\alpha_{V}$) the agreement between perturbation theory and non-perturbative simulations (of short distance quantities) is much better.

There is one last thing one can do to improve the perturbative series, namely set the scale of the coupling in a more sensible way. That will be the subject of my next post.

The United States has this fascinating relationship with inflation — there is this school of thought that if we disrespect it, belittle it, then perhaps it will go away.

That is not, unfortunately, the case in the real world. I guess the thought process is that if we collectively pretend there is no inflation, the Fed won't have to raise rates.

Let's look at last week's PPI as an example. We saw a hefty increase in component prices in March. There was a big 4.1% rise in energy, with gasoline up +8.7%. Food "only" saw a 1.7% rise in prices. If we ignore these two life necessities (Inflation ex-inflation), the core was unchanged. A 1.2% decline in light motor truck prices mitigated the increases everywhere else.

Here's the good news: if you forego consuming food and energy, and stick to buying light trucks, you have no inflation worries.

Elsewhere in the world, where real data matters and rhetorial games are not a substitute for real economic analysis, they do have inflation worries. The UK's version of inflation (CPI is out Tuesday in the US), showed sharp price rises:"The Office for National Statistics said output prices rose 0.6% on the
month in March to stand 2.7% higher than a year earlier, up sharply
from 2.2% the month before, a figure analysts had expected to be
repeated this time.

The figure spooked financial markets and the pound rose
to its highest in three months against the dollar of above $1.991.
Markets broadly expect Bank of England's monetary policy committee to
raise interest rates again next month, taking them to 5.5%.

Analysts said the figures suggested firms were managing to put up their
prices in the wake of a year of steep rises in energy costs. Core output price inflation, which excludes volatile
food and energy prices, also picked up more than expected to an annual
2.9%, the fastest rate of growth since June 2006." Note that Core Inflation prices are almost an afterthought, not the main focus of the inflation report.

We noted earlier in the year that The Office for National Statistics allows consumers to calcualte their own personal inflation rate, based on a variety of factors. I'd love to see the BLS try that trick . . . 

>

Source:
Factory prices fuel inflation fears
Ashley Seager
Guardian Unlimited, Monday April 16, 2007
http://business.guardian.co.uk/story/0,,2058237,00.html

Conspiracy at this year’s Brit Awards? Lily Allen, who went home empty-handed from the event, has said there was one. On arrival she told MTV: “I already know I won’t win any awards. I bet I’ll get nothing with me being on EMI. 70 per cent of the artists here tonight are on Universal. I’ll leave you with that.” It really doesn’t matter who wins this stuff. Lily recently said her favorite way to unwind on tour is to watch pornography. She told Heat mag, “When I travel around the world there often isn’t anything to watch on TV, especially when they don’t speak English. But watching porn is fine as it doesn’t need words. It’s sort of universal.” Heh. Keith Allen has insisted that his daughter will not be drawn in by addictions, despite the current trend among her contemporaries. “There is no way she’ll go to rehab,” he simply said. “She’s got too much of an old head on young shoulders - and she’s seen from her parents the consequences when you drink too much.” This lady will be fine. Bring on album number two.

Get ready another oldie reunion! Boyzone are set to reform at a charity gig this year. They will kick off with a concert in September in aid of the Red campaign, which raises money for the fight against Aids in Africa. Manager Louis Walsh let slip about it during an appearance on Magic Music TV on Thursday. They have also persuaded a once reluctant Ronan Keating to take part. The band split in 1999 and upon witnessing the success of Take That, the boys have been inspired to give their career a second chance. Picture of you in my mind, aye..

Michelle Ryan has landed the title role on NBC’s drama pilot “The Bionic Woman,” an updated version of the 1970s series that starred Lindsay Wagner. Ryan will play a bright woman who is transformed into a technological wonder with $50 million worth of bionic components implanted in her body. “The Bionic Woman” was originally a spin-off from “The Six Million Dollar Man,” which starred Lee Majors as Col. Steve Austin. Read more.. [MichRyan-Bionic]

Scarlett Johansson at Harvard as the Hasty Pudding Woman of the Year (15/02/2007)

Sophie Ellis Bextor - Catch You (live on Popworld)

Looks like Norah Jones has to lose some weight for a movie role. She told the British Glamour mag. “I was asked to lose a few pounds. First I got really mad, then I got nervous, thinking, ‘Oh no! I’m fat!’ It was a horrible thing to hear but I understand the game. I was just amazed that it really was an issue, and it made me have a lot of sympathy for really skinny actresses… But then, I still think they need to eat a cheeseburger.” Jones breaks onto the big screen in the upcoming My Blueberry Nights. [Norah’s-Weight]

A wealthy cocaine dealer, who terrified Nadine Coyle when he burst into her home few weeks back, has been found guilty of helping run a multi-million pound drugs empire. Jason Anderson and his gang had been hunting for a safe house to double as a narcotics cash’n'carry for trade clients. Agent insider Marc Fisher told them Nadine had moved out of her luxury apartment on a “gated and guarded” estate. But unfortunate for them, she was there for a further fortnight. All were remanded in custody for reports and will be sentenced later this month. From their ‘whole lotta history’ of crime, this incident will stick out like a very sore thumb.

Monica Bellucci & Elsa Pataky - “Manuale d’Amore 2″ Photocall in Madrid (15/02/2007)

Evangeline Lilly and her sis leaving a grocery store in Hawaii (15/02/2007)

Keira Knightley strips for her new film - only to have her breasts belittled. Her character’s husband from ‘Silk’ says her breasts are “small” and “like a child’s”. A source said: “She is nude in a couple of scenes, once during a love scene and another when she leans over her husband with her gown gaping open and he remarks that her breasts are small and like a little girl’s.” Keira plays Helene, the wife of 19th century silkworm smuggler Herve who is in love with a Japanese baron’s concubine. What a horrible husband to have.

LOST Season 3 Ep 8 - Flashes Before Your Eyes
Wow! It’s another brilliant eppy, brotha! We finally learned what happened when the fail-safe device got activated and discover more about Desmond’s magical prediction powers. Charlie and Hurley were called upon by Desmond to meet with Sayid/Locke who revealed the bad news about Mr Eko. While Locke was explaining details to them, Desmond sensed something and ran like mad to the deep ocean to save a drowning Claire, which he did. Hurley then told Charlie about Hume’s power to predict the future. Their plans to make him drunk with a bottle of ‘MacCutcheon’ as to get the truth out was a failure. Only after Charlie dissed Desmond did he get a violent reaction that leads to a flashback… It was simply the longest flash-’back’ in the history of the show. Scenes from last season’s finale where Desmond used the fail-safe key on the device were revealed. Next scene after the key got turned was of some random flashes and then of Desmond lying down on the floor with blood.. nah, paint all over him. Out pops a shocked Penelope who told him to remain still while she wipes the red paint off him. This setting was in his old apartment and strange things began to happen that caused Desmond to have random flashes of events related to him and the island. Many egs, including numbers: 1:08 showing up on his clock; delivery to Widmore offices for “8:15″ (4 8 15..) and the microwave beeping noise is the same noise as the warning noise for entering the numbers. He then went for an interview with Penelope’s dad, Charles Widmore but in actual fact what Des really wanted was an approval for a hand in marriage to his daughter. Evil dad’s response was pouring one glass with MacCutcheon Whiskey while leaving the other empty along with some clever rejecting lines to the dismay of Des. The painting in Widmore’s office has the word ‘Namaste’ backwards, along with a polar bear and a picture of an upside down buddha statue. Hmm.. those stuff do look kinda familiar. A dejected Des walked out of the building and while he was on the streets a familiar voice from a busker singing ‘Wonderwall’ came out to be none other than Charlie! Of which he didn’t recognise Des when asked and caused another flash-back/forward of Charlie in the hatch talking to him. A crazed up Hume approached his teacher friend Donovan and told him about things happening on the island; time-travel queries which was dismissed. “Make Your Own Kind Of Music” was played on the jukebox and Desmond’s prediction on what is going to happen in the pub was wrong. Most freaky moment was in the jewellery store where owner Ms. Hawking (from ‘The Others’ movie) revealed some future events like him breaking up with Penelope, sailboat race and button pushing to a shocked Desmond. Is she “The Oracle” in lost universe or the actual voice of the island? After being told that he can’t escape his future, Des broke up with Penny and then carried on with the events that led to him ending up at the island. How come Penny has another copy of the pic they took by the bridge? Back to the island, Des confessed that it wasn’t Claire’s life who he was saving but Charlie! After he turned the key, Des appears to be sent back to the past as he’s moving in with Penny with knowledge of the future. The most important thing is that it is fate that he ends up on the island and turns that key, and nothing he tries to do in the past will change that. The universe will “course correct” and find another way to force things to happen on their prescribed path. There was speculation this wasn’t a flashback but an actual event being relived. Listen to the end of the conversation Donovan has with a female student before seeing “The Scot” and then the conversation he has with Des afterwards, where he even admits he’s time traveling. Desmond’s oracle was named Ms. Hawking, and last week Aldo was reading Brief History of Time by Stephen Hawking? Coincidence? How did he end up being in US and receiving ‘The Elizabeth’ boat after being dishonourably discharged from the Royal Scots? Did Des really travel back in time? Is his past changed by his time travel? Did the event where the two guys on snowy Alps hired by Penny has anything to do with knowledge from this new changed reality?

Kasabian have revealed that they are busy at work on their new album and have got ‘angry’. Chief song-writer Serge Pizzorno made the revelation on Xfm’s breakfast show with Lauren Laverne. He said: “We’re angry, so it’s gonna be angry. It’s completely different to the first two. More like Bosnian folklore music.” They also snobbed the Brit Awards as they’re jetting off to Europe for a gig. He added: “We’re gonna be in Belgium, so I don’t think we’re gonna win it. We haven’t been asked to do the satellite link up or anything. George Michael’s gonna win it.” Wahahaha.. GM’s on the nomination list?

As predicted, Liam had some interesting comments upon hearing about his old pal who was recently admitted to a rehab centre. He told The Sun: “What’s his f***in’ problem, man? We all know what it is - he’s a f***in’ drama f***in’ queen. If you’ve got a f***in’ problem, why do you want the whole world knowing about it? He has to be on the front f***in’ pages, doesn’t he? Just sort your f***in’ self out. You make a f***in’ crap album then want everyone to feel f***in’ sorry for you. F***in’ tosser!” Back to you, Robb..

Katharine McPhee - Gillette Fusion Power Phantom Razor Unveiling (16/02/2007)

Debbie Harry is auctioning off one of her red stilettos - all to raise money for charity. She threw the shoe into the crowd at a Virgin Media launch party in London. You can place your bid for the unique piece of memorabilia here. [Debbie-Auction] The auction is part of a series hosted by Virgin Unite - a charitable side of the Virgin group raising money for global problems.

Women gamers enjoy more sex? Carried out by GameTart - an online games rental company - the survey revealed that women who played video games have sex 4.3 times a week on average compared to just 3.2 times a week for those who do not. The rather interesting results came about after the firm asked 200 women their thoughts on the subject. According to GameTart, “We wanted to see how the recent influx of the likes of Pink PSP’s and DS Lites would affect gamers’ sex lives across the country.” They added, “More promising for newer players was the fact that many women we interviewed who have only recently started playing games said that they now have sex more often than before.” Maybe it’s time to forget about drinks and dinners..

Welcome to Hitmyn.net. This is your first post. Edit or delete it, then start blogging!