Unifying The Quantum Principle – Flowing Along In Four Dimensions

[/caption]

In 1988, John Cardy asked if there was a c-theorem in four dimensions. At the time, he reasonably expected his work on theories of quantum particles and fields to be professionally put to the test… But it never happened. Now – a quarter of a century later – it seems he was right.

“It is shown that, for d even, the one-point function of the trace of the stress tensor on the sphere, Sd, when suitably regularized, defines a c-function, which, at least to one loop order, is decreasing along RG trajectories and is stationary at RG fixed points, where it is proportional to the usual conformal anomaly.” said Cardy. “It is shown that the existence of such a c-function, if it satisfies these properties to all orders, is consistent with the expected behavior of QCD in four dimensions.”

His speculation is the a-theorem… a multitude of avenues in which quantum fields can be energetically excited (a) is always greater at high energies than at low energies. If this theory is correct, then it likely will explain physics beyond the current model and shed light on any possible unknown particles yet to be revealed by the Large Hadron Collider (LHC) at CERN, Europe’s particle physics lab near Geneva, Switzerland.

“I’m pleased if the proof turns out to be correct,” says Cardy, a theoretical physicist at the University of Oxford, UK. “I’m quite amazed the conjecture I made in 1988 stood up.”

According to theorists Zohar Komargodski and Adam Schwimmer of the Weizmann Institute of Science in Rehovot, Israel, the proof of Cardy’s theories was presented July 2011, and is slowly gaining notoriety among the scientific community as other theoretical physicists take note of his work.

“I think it’s quite likely to be right,” says Nathan Seiberg, a theoretical physicist at the Institute of Advanced Study in Princeton, New Jersey.

The field of quantum theory always stands on shaky ground… it seems that no one can be 100% accurate on their guesses of how particles should behave. According to the Nature news release, one example is quantum chromodynamics — the theory of the strong nuclear force that describes the interactions between quarks and gluons. That lack leaves physicists struggling to relate physics at the high-energy, short-distance scale of quarks to the physics at longer-distance, lower-energy scales, such as that of protons and neutrons.

“Although lots of work has gone into relating short- and long-distance scales for particular quantum field theories, there are relatively few general principles that do this for all theories that can exist,” says Robert Myers, a theoretical physicist at the Perimeter Institute in Waterloo, Canada.

However, Cardy’s a-theorem just might be the answer – in four dimensions – the three dimensions of space and the dimension of time. However, in 2008, two physicists found a counter-example of a quantum field theory that didn’t obey the rule. But don’t stop there. Two years later Seiberg and his colleagues re-evaluated the counter-example and discovered errors. These findings led to more studies of Cardy’s work and allowed Schwimmer and Komargodski to state their conjecture. Again, it’s not perfect and some areas need further clarification. But Myers thinks that the proof is correct. “If this is a complete proof then this becomes a very powerful principle,” he says. “If it isn’t, it’s still a general idea that holds most of the time.”

According to Nature, Ken Intriligator, a theoretical physicist at the University of California, San Diego, agrees, adding that whereas mathematicians require proofs to be watertight, physicists tend to be satisfied by proofs that seem mostly right, and intrigued by any avenues to be pursued in more depth. Writing on his blog on November 9, Matt Strassler, a theoretical physicist at Rutgers University in New Brunswick, New Jersey, described the proof as “striking” because the whole argument follows once one elegant technical idea has been established.

With Cardy’s theory more thoroughly tested, chances are it will be applied more universally in the areas of quantum field theories. This may unify physics, including the area of supersymmetry and aid the findings with the LHC. The a-theorem “will be a guiding tool for theorists trying to understand the physics”, predicts Myers.

Pehaps Cardy’s work will even expand into condensed matter physics, an area where quantum field theories are used to elucidate on new states of materials. The only problem is the a-theorem has only had proof in two and four dimensions – where a few areas of condensed matter physics embrace layers containing just three dimensions – two in space and one in time. However, Myers states that they’ll continue to work on a version of the theorem in odd numbers of dimensions. “I’m just hoping it won’t take another 20 years,” he says.

Original Story Source: Nature News Release. For Further Reading: On Renormalization Group Flows in Four Dimensions.

26 Replies to “Unifying The Quantum Principle – Flowing Along In Four Dimensions”

    1. It is b-eautiful, but shy and is currently hiding behind the a- and c-theorem, at least while d remains odd.

  1. Worst article I have read on UT, you should at least make an effort to make the material accessable to those of us with just an interest and not a PHD.

    1. I am somewhat sympathetic to those who do not like this particular article. However, you have to give Tammy a bit of slack here. The subject here is rather deep and it is not easy to write a layman’s article on this. I wrote this morning something which is meant to give a better flavor of this. There is an element of mathematics here that is not easy to relate. However, the subject of conformal fields is quite central to the foundations of physics. Further, conformal field theory is equivalent to anti de-Sitter spacetime, which our universe is the boundary of. A pursuit of this leads to a very strange realization of the universe.

      LC

    1. A rather heavy article, but I like it even if I don’t understand much. Just stare at fancy words. 😀 You don’t want to mess with it, it’s just a threory. Don’t try to understand it until it’s confirmed by LHC.

      The thing you should extract is that there are 4 fundamental forces in nature. Electromagnetic, weak force, strong force and gravity. Electroweak unifies the 1st two and IIRC it’s not a big problem. This article talks about unifying the 1st three and they have found a great principle. It works in 4D and you gotta check it out in LHC.

  2. A breakthrough can look like that, “the whole argument follows once one elegant technical idea has been established.”

    Keeping my vector’s crossed for this one. A definitive character of quantum theory is precisely that applications are diverse and so disconnected. I believe I read the other day that you can make a theoretical claim on that is what would be expected. [That site is currently down.]

    As for meaning, the particular result is technical since it is at the bleeding edge of research. That “the RG flux is irreversible” (from the paper) would mean that we see emergence (of new degrees of freedom) when going from low energy to high energy, as I understand it. This is lcrowell land I hope, I haven’t studied quantum field theory.

    However, it seems to me what it means in general terms is that we are assured that at higher energy new and more complex effective theories emerge. New degrees of freedom would mean more laws, particles, forces.

    An effective theory is a high level description when low level descriptions can’t be found or are complex. For example general relativity is known to be an effective theory which presumably has an underlying quantum description.

    * In sum, expect more nice (useful) theories! *

    In any case, they can predict what this result would be used for in simpler terms:

    “The a-theorem would help because given predictions from a theory at low energy, it would constrain what the predictions at high energy should look like, and vice versa.” [From the news release.]

    They point out potential near term uses at LHC and for condensed matter physics.

  3. Though a layman, I understand a LOT about physics and Cosmology and rarely have any trouble at all understanding 99% of what I encounter on sites like this (I have been subscribed here, BTW for a few years now). This article was COMPLETELY over my head, There were no brief parenthetical descriptions of the main topic or any of the other ‘headier’ vocabulary and seemed to be written for people that are at least close to grad student knowledge of the topic. Not being a “Troll”, just pointing out this piece missed a good amount of the more ‘lay’ minded target audience for this site, knowing Fraser’s mindset per publishing it.

    1. I am somewhat sympathetic to those who do not like this particular article. However, you have to give Tammy a bit of slack here. The subject here is rather deep and it is not easy to write a layman’s article on this. I wrote this morning something which is meant to give a better flavor of this. There is an element of mathematics here that is not easy to relate. However, the subject of conformal fields is quite central to the foundations of physics. Further, conformal field theory is equivalent to anti de-Sitter spacetime, which our universe is the boundary of. A pursuit of this leads to a very strange realization of the universe.

      LC

      1. Thanks for the response. I watched the video and ‘groked’ the concept more. It’s funny, coincidentally I just this last week watched the new Nova serial about Space itself and became more versed then on the topic. One point: I didn’t even realize this article was about the holographic universe concept. Adding a simple line to it that mentioned this, would help tie it in to people already with a morsel of that concept. In the future perhaps, stepping back from such a heady topic with mention of more ‘popular’ ideas that pertain could help people get a better foothold as they move conceptually forward into the newer territory. Thanks again!

      2. PS: perhaps even a small glossary of concepts to wiki on headier topics at the end would be helpful too, like your mention in the reply on this string of, ” conformal fields” led me to do just that.

      3. The topic of conformal field theory connects with the holographic principle. This article did not make mention of this. QCD of quarks and gluons are a form of string theory at low energy. The M-theoretic form of string theory is an extremely high energy version of QCD. This connects with what I wrote about the two energy scales.

        LC

  4. I clicked off this article and then came back to see if other commentors had found it as fruitless as I. Glad to see I’m in good company. Yes, it was poorly presented and probably shouldn’t even have been included at all as it really describes nothing that an amateur astronomy nut would even necessarily be interested in.

  5. This is rather up my ally. The theory here involves conformal symmetry, which has its origin in the theory of complex variables in two dimensions. It turns out this has implications in higher dimensions. In general conformal symmetry is a symmetry which rescales the size of things while preserving angles and orthogonality (perpendicular) conditions between lines. Zamolodchikov found that this structure works for quantum field theory in two-dimensions so that conformal symmetry leads to a rescaling of energy or momentum in such a way that it preserves the structure of the field. This is his famous c-theorem.

    Conformal symmetry comes from something called the Cauchy-Riemann theorem. I am going to hammer this out here in just this paragraph before going on with the physics here. This is undergraduate level mathematics, it is not hard, and anyone with an interest in physics should at least see this. It is basic mid-19th century mathematics. This pertains to complex variables, which are numbers like z = x + iy. This is usually depicted with y the vertical axis of imaginary numbers i = sqrt{-1}. Now suppose there is some function

    f(z) = u + iv

    which maps our complex number z to w = u + iv, another complex number. Ok fine, we then want to think about the derivative df(z)/dz = dw/dz. The question emerges about how one defines this derivative. In order to make sense it should be independent of how one approaches the limit, say along x or along y, or some combination. So I now break out this derivative

    df(z)/dz = (?u + i?v)/(?x + i?y)

    = ?u/?x + ?v/?y + i?v/?x + ?u/i?y,

    where 1/i = -i, which can be checked. The independence of the approach by x or z means the two real parts are equal ?u/?x = ?v/?y and the same for the imaginary part (multiplied by i) so ?v/?x = -?u/?y. These are the Cauchy-Riemann equations, and the assumption that the approach by x or y in this limit is independent means that angles are preserved by a map that obeys this condition.

    This is the start of the whole theory of conformal invariance, which applies to complex valued spaces, or even dimensional spaces with certain properties that emulate complex variables, such as by obeying the Cauchy-Riemann equations. In effect a spacetime which obeys conformal symmetry is one which looks the same under any sort of rescaling or blow up. This is a nice structure for physics, for it means any theory can be truncated at some momentum value such that what is removed may simply appear identical to what is being considered. The basic renormalization group equations came from the regularization of quantum electrodynamics spelled out by Feynman. In the two-dimensional case this conformal symmetry leads to equations which are identical in form to the flow of a fluid in two-dimensions. The Zhoukowski transformation is a conformal map w = (1/2)(z + 1/z) maps a circle to an airfoil cross-section, which describes an aircraft wing. The air flow over this surface is described by conformal lines. So this analogue with fluid flow is one reason the term renormalization group flow (RG flow) is used.

    For higher dimensions this description becomes more difficult, and the Cauchy-Riemann conditions are generalized to something called holomorphy. In QCD the rescaling of the theory at high energy, so called asymptotic freedom is an RG flow. Of course one thing that has to be realized is the RG flows have to cut off. This is because there are masses, and the mass of an elementary particle is a breaking of conformal symmetry. In naturalized units a mass is equivalent to the reciprocal of a length, and the rest mass of an electron or an up quark is fixed. RG flow is broken, or so it would seem.

    There is something lurking in the wings though. The highest energy scale is the Planck energy at around M_pl = 10^{19} GeV. The energy density of the universe is such that it gives rise to a cosmological constant ? = (8?G/3)?, and this energy density corresponds to an energy in a unit volume of M_? = 10^{-18}GeV. This reflects how the cosmological constant is so very small. So a mass scale defined by M = sqrt{M_?M_pl}is between 1 and 10 TeV in energy. This is about the energy scale of the LHC. So the universe may have two RG-flows. The first is the high energy or UV flow from the Planck scale to around 1-10 TeV, and the other is the low energy (IR) flow from M_? to 1-10 TeV. These two RG flows are then duals of each other. The two are holographic equivalencies which might exist as some consequence of quantum gravity.

    LC

      1. That is noble of you but, I don’t think this forum needs the extent too which you, and others now joining the bandwagon, point out minor spelling and punctuation errors. Again I am here to read the articles and “pertinent” commentary! JMHO.

      2. If you are going to criticize IVAN, you should at least use the proper form of to in your criticism. “extent too, should be “to” Sorry, but you asked for it…..

  6. Agreeing with the other commenters. After reading the first quote:

    “…for d even, the one-point function of the trace of the stress tensor on the sphere, Sd, when suitably regularized, defines a c-function, which, at least to one loop order, is decreasing along RG trajectories and is stationary at RG fixed points, where it is proportional to the usual conformal anomaly.”

    with absolutely no introduction to what we are talking about, I started to laugh thinking to myself, “Surely this is Star Trek technobabble.” I then scrolled down to see what the joke was. But no, it appeared the article was completely serious about providing extremely technical information with no context. C-function? Loop-order? Conformal anomaly? Temporal nexus? Space-time subspace inverters? Really? Really?? If this was a published piece in a technical journal that could be acceptable, but on Universe Today?

    Of course, this probably is a joke I’m just not getting and I’ll have egg on my face later :).

Comments are closed.