Tag Archives: fields

Third step in the Triplet Model is up.

Advanced particle physics today:

Today I’m continuing the reader-requested explanation of the “triplet model,”  (a classic and simple variation on the Standard Model of particle physics, in which the W boson mass can be raised slightly relative to Standard Model predictions without affecting other current experiments.) The math required is pre-university level, just algebra this time.

The third webpage, showing how to combine knowledge from the first page and second page of the series into a more complete cartoon of the triplet model, is ready. It illustrates, in rough form, how a small modification of the Higgs mechanism of the Standard Model can shift a “W” particle’s mass upward.

Future pages will seek to explain why the triplet model resembles this cartoon closely, and also to explore the implications for the Higgs boson. 

Please send your comments and suggestions!

Triplet Model: Second Webpage Complete

Advanced particle physics today:

I’m continuing the reader-requested explanation of the “triplet model,” a classic and simple variation on the Standard Model of particle physics, in which the W boson mass can be raised slightly relative to Standard Model predictions without affecting other current experiments.

The math required is pre-university level, mostly algebra and graphing.

The second webpage, describing what particles are in field theory, and how the particles of one field can obtain mass from a second field, is ready now. In other words, the so-called “Higgs mechanism” for mass generation is sketched on the new page.

Meanwhile the first page (describing what the vacuum of a field theory is and how to find it in simple examples) is here. 

Please send your comments and suggestions, as I will continue to revise the pages in order to improve their clarity.

Triplet Model: First Webpage Complete

Advanced particle physics today:

Based on readers’ requests, I have started the process of explaining the “triplet model,” a classic variation on the Standard Model of particle physics, in which the W boson mass can be raised slightly relative to Standard Model predictions without affecting other current experiments.

The math required is pre-university level, so it should be broadly accessible to those who are interested.

My guess is that I’ll structure the explanation as four or five webpages, and will put up about one a week. The first one, describing what the vacuum of a field theory is and how to find it in simple examples, is here. Please send your comments and suggestions, as I will continue to revise the pages in order to improve their clarity.

Modern Physics: Increasingly Vacuous

One of the concepts that’s playing a big role in contemporary discussions of the laws of nature is the notion of “vacua”, the plural of the word “vacuum”. I’ve just completed an article about what vacua are, and what it means for a universe to have multiple vacua, or for a theory that purports to describe a universe to predict that it has multiple vacua. In case you don’t want to plunge right in to that article, here’s a brief summary of why this is interesting and important.

Outside of physics, most people think of a vacuum as being the absence of air. For physicists thinking about the laws of nature, “vacuum” means space that has been emptied of everything — at least, emptied of everything that can actually be removed. That certainly means removing all particles from it. But even though vacuum implies emptiness, it turns out that empty space isn’t really that empty. There are always fields in that space, fields like the electric and magnetic fields, the electron field, the quark field, the Higgs field. And those fields are always up to something.

First, all of the fields are subject to “quantum fluctuations” — a sort of unstoppable jitter that nothing in our quantum world can avoid.  [Sometimes these fluctuations are referred to as “virtual particles”; but despite the name, those aren’t particles.  Real particles are well-behaved, long-lived ripples in those fields; fluctuations are much more random.] These fluctuations are always present, in any form of empty space.

Second, and more important for our current discussion, some of the fields may have average values that aren’t zero. [In our own familiar form of empty space, the Higgs field has a non-zero average value, one that causes many of the known elementary particles to acquire a mass (i.e. a rest mass).] And it’s because of this that the notion of vacuum can have a plural: forms of empty space can differ, even for a single universe, if the fields of that universe can take different possible average values in empty space. If a given universe can have more than one form of empty space, we say that “it has more than one vacuum”.

There are reasons to think our own universe might have more than one form of vacuum — more than just the one we’re familiar with. It is possible that the Standard Model (the equations used to describe all of the known elementary particles, and all the known forces except gravity) is a good description of our world, even up to much higher energies than our current particle physics experiments can probe. Physicists can predict, using those equations, how many forms of empty space our world would have. And their calculations show that our world would have (at least) two vacua: the one we know, along with a second, exotic one, with a much larger average value for the Higgs field. (Remember, this prediction is based on the assumption that the Standard Model’s equations apply in the first place.)  An electron in empty space would have a much larger mass than the electrons we know and love (and need!)

The future of the universe, and our understanding of how the universe came to be, might crucially depend on this second, exotic vacuum. Today’s article sets the stage for future articles, which will provide an explanation of why the vacua of the universe play such a central role in our understanding of nature at its most elemental.

Which Parts of the Big Bang Theory are Reliable, and Why?

Familiar throughout our international culture, the “Big Bang” is well-known as the theory that scientists use to describe and explain the history of the universe. But the theory is not a single conceptual unit, and there are parts that are more reliable than others.

It’s important to understand that the theory — a set of equations describing how the universe (more precisely, the observable patch of our universe, which may be a tiny fraction of the universe) changes over time, and leading to sometimes precise predictions for what should, if the theory is right, be observed by humans in the sky — actually consists of different periods, some of which are far more speculative than others.  In the more speculative early periods, we must use equations in which we have limited confidence at best; moreover, data relevant to these periods, from observations of the cosmos and from particle physics experiments, is slim to none. In more recent periods, our confidence is very, very strong.

In my “History of the Universe” article [see also my related articles on cosmic inflation, on the Hot Big Bang, and on the pre-inflation period; also a comment that the Big Bang is an expansion, not an explosion!], the following figure appears, though without the colored zones, which I’ve added for this post. The colored zones emphasize what we know, what we suspect, and what we don’t know at all.

History of the Universe, taken from my article with the same title, with added color-coded measures of how confident we can be in its accuracy.  In each colored zone, the degree of confidence and the observational/experimental source of that confidence is indicated. Three different possible starting points for the "Big Bang" are noted at the bottom; different scientists may mean different things by the term.

History of the Universe, taken from my article with the same title, with added color-coded measures of how confident we can be in our understanding. In each colored zone, the degree of confidence and the observational/experimental source of that confidence is indicated. Three different possible starting points for the “Big Bang” are noted at the bottom; note that individual scientists may mean different things by the term.  (Caution: there is a subtlety in the use of the words “Extremely Cold”; there are subtle quantum effects that I haven’t yet written about that complicate this notion.)

Notice that in the figure, I don’t measure time from the start of the universe.  That’s because I don’t know how or when the universe started (and in particular, the notion that it started from a singularity, or worse, an exploding “cosmic egg”, is simply an over-extrapolation to the past and a misunderstanding of what the theory actually says.) Instead I measure time from the start of the Hot Big Bang in the observable patch of the universe.  I also don’t even know precisely when the Hot Big Bang started, but the uncertainty on that initial time (relative to other events) is less than one second — so all the times I’ll mention, which are much longer than that, aren’t affected by this uncertainty.

I’ll now take you through the different confidence zones of the Big Bang, from the latest to the earliest, as indicated in the figure above.

Continue reading

Brane Waves

The first day of the conference celebrating theoretical physicist Joe Polchinski (see also yesterday’s post) emphasized the broad impact of his research career.  Thursday’s talks, some on quantum gravity and others on quantum field theory, were given by

  • Juan Maldacena, on his latest thinking on the relation between gravity, geometry and the entropy of quantum entanglement;
  • Igor Klebanov, on some fascinating work in which new relations have been found between some simple quantum field theories and a very poorly understood and exotic theory, known as Vassiliev theory (a theory that has more fields than a field theory but fewer than a string theory);
  • Raphael Bousso, on his recent attempts to prove the so-called “covariant entropy bound”, another relation between entropy and geometry, that Bousso conjectured over a decade ago;
  • Henrietta Elvang, on the resolution of a puzzle involving the relation between a supersymmetric field theory and a gravitational description of that same theory;
  • Nima Arkani-Hamed, about his work on the amplituhedron, a set of geometric objects that allow for the computation of particle scattering in various quantum field theories (and who related how one of Polchinski’s papers on quantum field theory was crucial in convincing him to stay in the field of high-energy physics);
  • Yours truly, in which I quickly reviewed my papers with Polchinski relating string theory and quantum field theory, emphasizing what an amazing experience it is to work with him; then I spoke briefly about my most recent Large Hadron Collider [LHC] research (#1,#2), and concluded with some provocative remarks about what it would mean if the LHC, having found the last missing particle of the Standard Model (i.e. the Higgs particle), finds nothing more.

The lectures have been recorded, so you will soon be able to find them at the KITP site and listen to any that interest you.

There were also two panel discussions. One was about the tremendous impact of Polchinski’s 1995 work on D-branes on quantum field theory (including particle physics, nuclear physics and condensed matter physics), on quantum gravity (especially through black hole physics), on several branches of mathematics, and on string theory. It’s worth noting that every talk listed above was directly or indirectly affected by D-branes, a trend which will continue in most of Friday’s talks.  There was also a rather hilarious panel involving his former graduate students, who spoke about what it was like to have Polchinski as an advisor. (Sorry, but the very funny stories told at the evening banquet were not recorded. [And don’t ask me about them, because I’m not telling.])

Let me relate one thing that Eric Gimon, one of Polchinski’s former students, had to say during the student panel. Gimon, a former collaborator of mine, left academia some time ago and now works in the private sector. When it was his turn to speak, he asked, rhetorically, “So, how does calculating partition functions in K3 orientifolds” (which is part of what Gimon did as a graduate student) “prepare you for the real world?” How indeed, you may wonder. His answer: “A sense of pertinence.” In other words, an ability to recognize which aspects of a puzzle or problem are nothing but distracting details, and which ones really matter and deserve your attention. It struck me as an elegant expression of what it means to be a physicist.

Quantum Field Theory, String Theory and Predictions (Part 9)

Today I continue with my series of posts on fields, strings and predictions.

During the 1980s, as I discussed in the previous post in this series, string theorists learned that of all the possible string theories that one could imagine, there were only five that were mathematically consistent.

What they learned in the first half of the 1990s, culminating in early 1995, is that all five string theories are actually little corners of a single, more encompassing, and still somewhat mysterious theory. In other words, after 30 years of studying various types of theories with strings in them, they ended up with just one!

On the one hand, that sort of sounds like a flop — all that work, by all those people, over two decades, and all we got for our efforts was one new theory?

On the other hand, it’s very tempting to think that the reason that everyone ended up converging on the same theory is that maybe it’s the only consistent theory of quantum gravity! At this point there’s no way to know for sure, but so far there’s no evidence against that possibility.  Certainly its a popular idea among string theorists.

This unique theory is called “M theory” today; we don’t know a better name, because we don’t really know what it is. We don’t know what it describes in general. We don’t know a principle by which to define it. Sometimes it is called “string/M theory” to remind us that it is string theory in certain corners.

Fig. 1: M theory is a set of equations that, depending on how they are used, can describe all known consistent  string theories and 11-dimensional supergravity, as well as many more complex and harder to understand things.  Only at the corners does it give the relatively simple string theories described in my previous post.

Fig. 1: A famous but very schematic image of M theory, which is a set of equations that, depending on how they are used, can describe universes whose particles and forces are given by any one of the known consistent string theories or by 11-dimensional supergravity.   Only at the corners does it give the relatively simple string theories described in my previous post.  More generally, away from the corners, it describes much more complicated and poorly understood types of worlds.

Note that M theory is very different in one key respect from quantum field theory.  As I described in the second post in this series, “quantum field theory” is the term that describes the general case; “a quantum field theory” is a specific example within the infinite number of “quantum field theories”. But there’s no analogue of this distinction for M theory. M theory is (as far as anyone can discern) a unique theory; it is both the general and the specific case.  There is no category of “M theories”. However, this uniqueness, while remarkable, is not quite as profound as it might sound… for a reason I’ll return to in a future post.

Incidentally, the relationship between the five apparently very different string theories that appear in M theory is similar to the surprising relationships among various field theories that I described in this post. It’s not at all obvious that each string theory is related to the other four… which is why it took some time, and a very roundabout route involving the study of black holes and their generalizations to black strings and black branes, for this relationship to become clear.

But as it did become clear, it was realized that “M theory” (or “string/M theory”, as it is sometimes called) is not merely, or even mainly, a theory of strings; it’s much richer than that. In one corner it is actually a theory with 10 spatial (11 space-time) dimensions; this is a theory with membranes rather than strings, one which we understand poorly. And in all of its corners, the theory has more than just strings; it has generalizations of membranes, called “branes” in general. [Yes, the joke’s been made already; the experts in this subject had indeed been brane-less for years.] Particles are zero-dimensional points; strings are one-dimensional wiggly lines; membranes are two-dimensional surfaces. In the ordinary three spatial dimensions we can observe, that’s all we’ve got. But in superstring theory, with nine spatial dimensions, one doesn’t stop there. There are three-dimensional branes, called three-branes for short; there are four-branes, five-branes, and on up to eight-branes. [There are even nine-branes too, which are really just a way of changing all of space. The story is rich and fascinating both physically and mathematically.] The pattern of the various types of branes — specifically, which ones are found in which corners of M theory, and the phenomena that occur when they intersect one another — is a fantastically elegant story that was worked out in the early-to-mid 1990s.

A brane on which a fundamental string can end is called a “D-brane”. Joe Polchinski is famous for having not only co-discovered these objects in the 1980s but for having recognized, in mid-1995, the wide-ranging role they play in the way the five different string theories are related to each other. I still remember vividly the profound effect that his 1995 paper had on the field. A postdoctoral researcher at the time, I was attending bi-weekly lectures by Ed Witten on the new developments of that year. I recall that at the lecture following Polchinski’s paper, Witten said something to the effect that everything he’d said in his presentations so far needed to be rethought. And over the next few months, it was.

DBranes

Fig. 2: In addition to fundamental strings (upper left), string theories can have D-branes, such as the D string (or D1-brane) shown at lower left, the D particle (or D0 brane) shown at lower right, or the D2-branes shown at right. There are also D3, D4, D5, D6, D7, D8 and D9 branes, along with NS5-branes, but since they have more than two spatial dimensions I can’t hope to draw them. There are no strings or D-branes, but there are M2-branes and M5- branes, in the 11-dimensional corner of M theory. A D-brane is an object where a fundamental string can end; therefore, in the presence of D-branes, a closed string can break into an open string with both ends on a D-brane (center and right).

The fact that string/M theory is more than just a theory of strings is strikingly similar to something known about quantum field theory for decades. Although quantum field theory was invented to understand particles in the context of Einstein’s special relativity, it turns out that it often describes more than particles. Field theory in three spatial dimensions can have string-like objects (often called “flux tubes”) and membrane-like objects (often called “domain walls”) and particle-like blobs (“magnetic monopoles”, “baryons”, and other structures). The simplest quantum field theories — those for which successive approximation works — are mainly theories of particles.  But flux tubes and domain walls and magnetic monopoles, which can’t be described in terms of particles, can show up even in those theories. So the complexities of M theory are perhaps not surprising. Yet it took physicists almost two decades to recognize that “branes” of various sorts are ubiquitous and essential in string/M theory. (We humans are pretty slow.)

Notably, there are contexts in which M theory exhibits no string-like objects at all. It’s the same with particles and fields; simple field theories have particles, but most field theories aren’t simple, and many complicated field theories don’t have particles. It can happen that the particles that would be observed in experiments may have nothing to do with the fields that appear in the equations of the theory; this was something I alluded to in this article. I also earlier described scale-invariant quantum field theories, which don’t have particles. Quantum field theories on curved space-time don’t have simple, straightforward notions of particles either. Quantum field theory is complex and rich and subtle, and we don’t fully understand it; I wrote seven posts about it in this series, and did little more than scratch the surface. String/M theory is even more complicated, so it will surely be quite a while before we understand it. But specifically, what this means is that what I told you in my last article about “simple superstring theories” is simply not always true. And that means that the first “vague prediction of string theory” that I described might not be reliable… no more than overall predictions of simple field theory, all of which are true in the context of simple field theories, but some of which are often false in more complex ones.

By the way, those of you who’ve read about string theory may wonder: where is supersymmetry in my discussion? Historically, in all these developments, the mathematics and physics of supersymmetry played an important role in making it easier to study and confirm the existence of these branes within string/M theory. However, the branes are present in the theory even when supersymmetry isn’t exact. One must not confuse the technically useful role of supersymmetry in clarifying how string/M theory works for a requirement that supersymmetry has to be an exact (or nearly-exact) symmetry for string/M theory to make sense at all. It’s just a lot harder to study string/M theory in the absence supersymmetry… something which is also true, though to a somewhat lesser extent, of quantum field theory.

To be continued… next, how are quantum field theory and M theory similar and different?

Quantum Field Theory, String Theory, and Predictions (Part 7)

Appropriate for Advanced Non-Experts

[This is the seventh post in a series that begins here.]

In the last post in this series, I pointed out that there’s a lot about quantum field theory [the general case] that we don’t understand.  In particular there are many specific quantum field theories whose behavior we cannot calculate, and others whose existence we’re only partly sure of, since we can’t even write down equations for them. And I concluded with the remark that part of the reason we know about this last case is due to “supersymmetry”.

What’s the role of supersymmetry here? Most of the time you read about supersymmetry in the press, and on this website, it’s about the possible role of supersymmetry in addressing the naturalness problem of the Standard Model [which overlaps with and is almost identical to the hierarchy problem.] But actually (and I speak from personal experience here) one of the most powerful uses of supersymmetry has nothing to do with the naturalness problem at all.

The point is that quantum field theories that have supersymmetry are mathematically simpler than those that don’t. For certain physical questions — not all questions, by any means, but for some of the most interesting ones — it is sometimes possible to solve their equations exactly. And this makes it possible to learn far more about these quantum field theories than about their non-supersymmetric cousins.

Who cares? you might ask. Since supersymmetry isn’t part of the real world in our experiments, it seems of no use to study supersymmetric quantum field theories.

But that view would be deeply naive. It’s naive for three reasons. Continue reading