I love how news outlets ‘sex’ up the news. In this article the author, when speaking of entanglement, claims that until now, “scientists have assumed such a marriage would endure forever.” We have? This seems to imply that all prior entanglement experiments have produced particles that are still entangled which is, of course, not true. The reality is that a pesky little thing physicists call decoherence has made long time- or distance-scale entanglement difficult. Decoherence can be caused by lots of things – noisy channels for example. So, perhaps in an ideal world with no noise, no second law of thermodynamics, etc. we might expect entanglement to last forever. Certainly there isn’t always a time dependence in the equations of entanglement (depending on what you’re doing). But the real world is very different. While I think the paper the article refers to is fascinating and ground-breaking, the MSNBC author has (not surprisingly) been a bit misleading about the whole thing.
Archive for January, 2009
For those of us working in quantum mechanics group theory is an indispensable tool. Group theory has an enormous range of application, however. In essence it is the language of symmetry. While to physicists it is thus obvious as to why it is important for the study of most modern physics topics, one may ask what its relation to visual symmetry might be. I am nearly finished reading Mario Livio’s fantastic book
The Equation That Couldn’t Be Solved: How Mathematical Genius Discovered the Language of Symmetry. In it he gives a terrific introduction to groups via its relation to visual symmetry!
So what is a group? Groups are actually related to sets, the fundamental building blocks of modern mathematics. Many people mistakenly assume that the elements (members) of sets are always numbers. In fact anything can be an element of a set. A set simply is a way to categorize items. A group is a set whose elements may be combined by some operation. The elements and the operation have certain properties that may essentially be summarized as
- Any combination of elements by the operation produces another element of the same set. This is called closure. For example, in the set of integers with the operation of addition, adding any two integers produces another integer.
- When combining three or more elements of the set using the operation, it does not matter how the operations are combined. For example, (3+5)+2=3+(5+2)=10. This is called associativity.
- The set contains one element (called the identity element) that, when combined with any other by the operation, leaves that other element unchanged. For example, working with the natural numbers (1,2,3…), if the operation is addition the identity element is 0. If the operation is multiplication the identity element is 1.
- Each element has an inverse that is also a member of the set. When you combine anything with its inverse you get the identity element. If your operation is multiplication, this implies that their product is 1. If it is addition, their sum is zero.
This last point is a crucial one since, from this, we can see that if the operation is multiplication the set of all natural numbers is not a group! Why? Take the number 6 for example. Its inverse is 1/6. But 1/6 is not a natural number! Now, there are ways around this. One way is to employ modular arithmetic, but there is not time or space to explain that here. (Note: if the operation is multiplication, then the set of all natural numbers, that is all positive integers, is still not a group since the inverse of 6, for example, would be -6).
Now if we’re trying to relate groups (which we have just defined) to symmetry, it would be helpful to have some idea of what symmetry is. In physics and mathematics, any time we perform an operation on something and it remains unchanged, our operation represents a symmetry. So, for example, the laws of physics should look the same regardless of how we look at them or where we look at them from (i.e. they should be the same to us as they are to people living in a different galaxy). This is the Principle of Relativity and forms the conceptual foundation of both special and general relativity theory. In nuclear physics we see symmetries in the types of particles that exist – each type seems to have a similar partner. So another way to look at symmetry is as a representation of balance.
How does this apply to visual symmetries though? Take the human body – in fact just about any animal or insect body – for example. It exhibits bilateral symmetry, sometimes referred to as mirror symmetry since one side is (roughly) the mirror image of the other. This objects that have this symmetry remain unchanged under two symmetry transformations – reflections (we’ll call this r) and the identity (meaning doing nothing and we’ll call this I). Does this set of operations constitute a group? Indeed it does! Let’s see why.
Our operation will simply be the combination of transformations (so it’s like addition). For example, we could write I º r where the º symbol just means we applied the I transformation first, then the r. Let’s make sure these transformations and the operation satisfy the four conditions of a group.
- First, let’s check to make sure that any combination gives another element of the group. So if we do nothing (I) and then do nothing (I) we’ve done nothing (I). In other words IºI=I. Now suppose we do nothing (I) then reflect one half of a creature’s body (r). That’s equivalent to just the reflection alone. In other words Iºr=r. So far so good. What happens if we reflect half the body and then reflect that same half again? It ends up back where it started which is just like doing nothing! Thus rºr=I. It seems as if any combination of I and r produces either I or r again which is exactly what we want.
- Can we combine them in any order? Sure! Does it matter whether we first do nothing (I) and then reflect (r) or reflect (r) first and then do nothing (I)? Nope! (Think about it.)
- Is there an identity element? Yep! It’s I (doing nothing).
- Does each element have an inverse? This might seem a bit trickier, but it isn’t. The definition says that combining an operation with its inverse should give the identity element. In our case, taking any element twice in a row does the trick, i.e. rºr=I and IºI=I.
Thus the symmetry transformations of animal and insect bodies form a group! In the process of understanding this we catch a glimpse of the beauty and elegance of mathematics. As Einstein said in one of my favorite quotes, “Pure mathematics is the poetry of logical ideas.”
Thanks go out to one of my students who just brought this to my attention: Chris Monroe at the University of Maryland (who happens to be on the Board of Directors of the APS group that I am newsletter editor for) and his team have successfully teleported a quantum state for the first time between atoms in unconnected enclosures. Trés cool.
All politics aside, there are some very cool things about our new President and the recent election.
- The new Prez is a black dude from Lincoln’s home state who will be sworn in on Lincoln’s bible.
- He’s a black dude from Chicago and if you don’t get that reference read your 1960s and 1970s history.
- He was still in high school when the original Star Wars came out which means he’s the closest any Prez has been to my generation (his kids are about the ages of mine, in fact).
- His brother-in-law is Craig Robinson, the men’s basketball coach at Oregon State University.
- He was born in Hawaii.
- The new VP is from Delaware (yay for small states!) and, though I can’t stand her, had she been elected Sarah Palin was from Alaska which would have been cool.
- The inauguration is the day after Martin Luther King, Jr. day.
- Obama is a true American mutt: Kenyan, English, Irish, and possibly Native American.
- Obama speaks Indonesian.
- The new Secretary of Energy is physicist Steve Chu whom I’ve written about before (as have most physicists).
- McCain and Obama have not held any grudges and the latter feted the former at a dinner last night. In today’s political climate, how rare is that? Hell, even Hoover and Roosevelt wouldn’t talk to each other and that was over 70 years ago.
- Obama plays basketball, which is the hippest of the four major sports.
- He’s also a technology addict.
- Update: He likes Spiderman comics. How cool is that??
What else did I miss?
Fed Chair Ben Bernanke, speaking before a group at the London School of Economics, defended the government’s handling of the bailout (maybe that should be plural at this point), saying “[t]he United States’ economic system is critically dependent on the free-flow of credit,” adding that “[i]t is like the economy’s oxygen.”
But wait. Isn’t credit how we got into this mess in the first place? I know, I know, it was bad credit and Bernanke’s talking about the good kind. But what is the good kind?
Credit has been a part of economics since time immemorial (see The Bible for plenty of examples). But the overabundance of credit in modern society has been criticized as creating phantom wealth. By doling out more credit than there is money to back it, lenders have been doing something (generally unintentionally) akin to a Ponzi scheme. Perhaps the underlying assumption was always that the Fed would be there to bail them out.
But where does the Fed get its money (especially since we’re not on the gold standard anymore)? From what I heard, the Fed indeed has the actual cash on hand to do this (i.e. they don’t need to print money). But if this bailout gets any bigger (reports are recently that the steel industry now wants some dough) the Fed may be forced to print money, especially if the Chinese and Russian banks are no longer in a position to lend us any. The problem with printing more money, of course, is that it leads to inflation because it’s essentially phantom money anyway.
I’m sure the Fed’s argument is that the infusion of cash will increase production capacity thereby increasing the amount of actual goods and services on the market and giving them some wiggle room to print money if needed (printing more money is generally alright if there are real-world items to back it with). But this is a top-down approach again. Through oversight the Fed presumably is aiming to make sure the money goes to tangibly increasing productivity. But, from my personal viewpoint at the bottom of the food-chain, I can’t see how it’s going to help me. Increased productivity is good, but only if there is a market for the product. People need money to buy products. Detroit may come out with some beautiful new well-made hybrid, but if I don’t have the money to buy it, it’s useless.
I don’t have an answer to the problem though I do think more money should be given to average Americans and less to corporations. I think we should let some of them fail and see what sort of innovations arise from their ashes. Frequently failure is a necessary step in progress. I think aspects of the software industry offer an interesting comparison. Many tech concepts have flamed out over the years but, frequently, bigger and better ideas have popped up out of what was left of them. While I understand why Detroit needs some sort of a bailout (since there would be an enormous ripple effect in the global economy), a more cautious approach might be wise. Otherwise, the government is simply going to be saving the economy by doing exactly that which led to the crisis in the first place – handing out cash with nothing to back it up.
As anyone who has read my blog for awhile or who runs into me at conferences knows, I have been working on a paper related to the Cerf-Adami inequalities for two and a half years now. The paper has gone through several drafts and been changed so dramatically from the first that its basic ideas appear in four separate papers on the arXiv here, here, here, and most recently here (which is usually not kosher, but read on…).
After bitching about the trouble I’d been having with this all the way back in April, and again in May, I was generously offered extensive help from Terry Rudolph, though he admitted he wasn’t an expert on entropies. Nonetheless, with his editing acumen and his taskmaster mentality having been applied to the paper for nearly six months (May through October) I was convinced I’d have more success. And, of course, I was wrong.
Now, to review, the following people have, at one time or another in the past few years, spent countless hours scouring this paper and offering very detailed suggestions: Terry, Ken Wharton, Barry Sanders, and Frank Schroeck. These guys aren’t lightweights and, as I did in the acknowledgments section of the paper, I thank them heartily.
But, no one, to this point, has been willing to spend the time to actually collaborate on this, i.e. serve as a co-author. Every effort I have made to obtain a co-author (hell, even to get someone to read it, with the exceptions listed above) has met with the all-too-typical, “I’d love to, and I think the idea is very interesting, but I’m just so busy.”
Even the final verdict of the PRA Editorial Board was an “I’m sorry, but…” sort of thing. The final sentence in the Editor’s report was “I am sure that from the idea as such, once this additional time is invested, one could make a splendid paper, an assessment that apparently our colleague Barry Sanders also shares.” Since the Editor signed his name I thought perhaps he might be willing to share authorship and help me finish this. See the last line of my previous paragraph for the response.
While one might assume that it is the topic that people are unsure about, I have had the same sort of trouble with my now year-and-a-half long excursion into the world of quantum communication on closed time-like curves (see my Christmas Day post). I have also received lukewarm to tepid responses regarding assistance on some work I’ve started on open quantum systems (an off-shoot of my CTC work), even from the person who suggested I start working on it in the first place!
What the f&^% do I need to do to get a goddamn collaborator? Oh, I’m sure I’ll now get a string of e-mail from crackpots offering assistance. But, it seems to me (now that PRA and probably JPA as well are out of the picture) there aren’t too many decent journals left out there. Why do I want to see it in a decent journal? Well, take these words from Terry on this very blog: “I’ve sent Barry the link to this page. Frankly I am surprised you didn’t get it published despite those things I said above. I’m also surprised Barry didn’t suggest the change of focus etc that I mentioned above. I’m pretty sure he’ll agree that it will make it much easier to read and to publish.”
(*&^@%*&^$_!(*^_(*^#^@^%*$#% What the FU&%!!!!
Update: Could it be my personality maybe? I mean, I’m a little weird and eccentric, yeah, but we’re physicists and mathematicians for crissakes! We’re all weird! Apparently I am just not the right kind of weird (especially in this day and age when being weird is now cool).
I was poking around in cyberspace looking for information on a quantum version of the Monty Hall problem and discovered that one version was co-authored by quantum game expert Derek Abbott. I additionally discovered that, back in 1999, Cosma Shalizi of Three-Toed Sloth fame bet Abbott that he couldn’t get the name “Monica Lewinsky” published in a serious article. Shalizi lost the bet (and thus owed Abbott a beer) when Abbott managed to insert the name into a paper published in the journal Chaos in 2001. Sadly, though many of Abbott’s great papers on quantum games are on the arXiv (including the one on the Monty Hall problem), the paper mentioning Monica Lewinsky is not. Oh Monica, Monica, wherefore art thou Monica?