QED and QCD theta angles, asymptotic boundary conditions in gauge theory

Warning: this post is for specialists.

I had reason to look back at the paper below recently, and thought I would write a longer post on the subject as I regularly see searches on "QED theta angle" and similar in my traffic logs. These readers may be unsatisfied with the standard textbook treatment of this topic.

The conventional thinking is that because FF-dual in QED is a total derivative, doesn't affect the classical equations of motion, and isn't related to any topological vacuum structure, it can't have physical consequences.

In the paper below I constructed gauge configurations in (3+1) dimensions that connect an initial configuration (e.g., vacuum state A=0) to two different final configurations A1 and A2 (e.g., in the far future). A1 and A2 differ by a gauge transformation (i.e., represent the same physical electric and magnetic fields), but the two (3+1) interpolating configurations are not gauge equivalent. By construction, the value of \int FF-dual is not the same when evaluated on the two (3+1) configurations. Thus the two trajectories have a relative phase weighting in the path integral which depends on the value of theta. This suggests that theta can have physical (though perhaps small and non-perturbative) effects, so it is indeed a fundamental physical parameter of QED and of the Standard Model of particle physics.

http://arxiv.org/abs/1107.0756

Theta terms and asymptotic behavior of gauge potentials in (3+1) dimensions

We describe paths in the configuration space of (3+1) dimensional QED whose relative quantum phase (or relative phase in the functional integral) depends on the value of the theta angle. The final configurations on the two paths are related by a gauge transformation but differ in magnetic helicity or Chern-Simons number. Such configurations must exhibit gauge potentials that fall off no faster than 1/r in some region of finite solid angle, although they need not have net magnetic charge (i.e., are not magnetic monopoles). The relative phase is proportional to theta times the difference in Chern-Simons number. We briefly discuss some possible implications for QCD and the strong CP problem.

The question of whether physical observables can depend on the value of theta QED is somewhat esoteric. However, the analysis raises the issue of asymptotic boundary conditions in gauge theory. One typically expects that local properties of a quantum field theory are independent of the choice of boundary conditions when the size of the system is taken to infinity. But topological or total derivative terms such as FF-dual seem to defy this expectation.

The gauge potentials required for the construction described above must have  A ~ 1/r  behavior for some region of solid angle. In d=4 Euclidean space, the assumption is often made to allow only potentials with faster than 1/r falloff. In this way one obtains a topological classification of gauge configurations (i.e., in terms of instanton number). However, in Minkowski space (d=3+1) there exist classical solutions of non-Abelian gauge theory (e.g., SU(2); discovered by Luscher and Schecter) that exhibit 1/r falloff and are the analog of the U(1) configurations I described above. These L-S solutions have fractional topological charge.

In the presence of fractional topological charges the gauge theory partition function no longer appears periodic in theta. This may have consequences for the Strong CP problem in QCD, which are briefly discussed in the paper above.

Note Added: Writing this blog post was beneficial -- thinking through the topic again allowed me to formulate the conclusions more clearly than in the original paper. I've replaced it on arXiv with a new version containing the additional material below. The observation at the end is related to Elitzur's Theorem -- gauge-variant operators must have zero average in the path integral.

Dept. of Physicists Can Do Stuff: Ashton Carter

Secretary of Defense nominee Ashton B. Carter on his education and early career in theoretical physics. His Wikipedia entry says he did postdocs at Rockefeller University and MIT.

... when I rather unexpectedly was accepted into a good college, Yale, I was determined to make the most of it. I disdained the “preppies” and other privileged students who seemed to regard college as an opportunity to enjoy freedom at long last. I was an intensely serious student, what would probably be called today a “grind.”

At Yale I ended up pursuing two entirely different majors – physics and medieval history. There was no relationship between them in my mind except that both fascinated me. I liked dusty archives, learning to decipher manuscripts in medieval script, and learning all the languages necessary to read the primary and secondary historical literature, especially Latin. I wrote a senior thesis on the use of Latin by contemporary monastic writers to describe the vibrant world of 12th century Flanders in which they lived. I also enjoyed English legal history and the foundations of the Common Law as established in the 11th through 13th centuries. I also did a lot of work on the hagiography of Saint Denis, patron saint of the French monarchy during its formative period in the 9th century.

Physics was entirely different: clean and modern, logical and mathematical. I was lucky enough to be asked by a professor to assist him on an experiment in elementary particle physics at the then-new Fermilab outside of Chicago, home of the world’s largest particle accelerator. I would fly back and forth from New Haven to Chicago, feeling very serious and very important. We were involved in the search for the quark, a sub-atomic particle then only theorized. I eventually wrote my senior thesis, which was later published, on the “charmed quark.”

As far as course choice was concerned, I had no interest in between the extremes of medieval history (history, language, philosophy) on the one hand, and science (physics, chemistry, mathematics) on the other. It may sound shocking to Kennedy School students, but I have taken exactly zero social science courses in my entire life. My arrogant view at the time was that life would eventually teach me political science, sociology, psychology, and even economics, but it would never teach me linear algebra or Latin. It seemed best to get my tuition’s worth from the other topics and get my social science for free!

The end of college brought the usual crisis of what to do next. Such a bimodal distribution of training and interests made the problem more acute. The default solution was to go to medical school, since my father was a physician and I had worked in hospitals back in Philadelphia.

Fortunately, I was rescued from this dilemma by the awarding of a Rhodes scholarship, entitling me to free study at Oxford University. Many Rhodes scholars pursue a second Bachelors degree or a Masters degree at Oxford, but I was still a man in a hurry. I decided I would use the free funding to get my doctorate in theoretical physics. Oxford did not have enough money to have world class experimental facilities in elementary particle physics, but it had a great theoretical physics department. All you need for theoretical physics is a pencil and paper and the ability to sit for hours of intense concentration with a page of equations in front of you. I worked on the theory of quantum chromodynamics, the quantum field theory then postulated to explain the behavior of nuclear reactions and the structure of the sub-atomic zoo of particles. Unfortunately, it was a mathematical theory so complex that its equations could not be solved. I found a way to solve its equations in certain special circumstances, thus allowing it to be tested against experiments. Oxford was a very lively intellectual community. The expatriate Americans would spend long hours debating the topics of the day. Much of my otherwise lacking social science training occurred by osmosis in the pubs with Rhodes friends.

I had no doubt, however, that I wanted a career of thinking and writing in academia, then meaning theoretical physics. I therefore went back to the United States to start to climb the academic ladder in physics, beginning in the usual way with a postdoctoral appointment. I wrote several papers. The one of which I am proudest and which is still frequently cited, was on “time reversal invariance,” the proposition that the world could run backwards according to the same laws by which it runs forwards. While this may seem like a bizarre question to ask, such a symmetry in nature, if it exists, is actually a very fundamental property of our universe.

A Turning Point

I was happily building an academic career in theoretical physics when a serendipitous opportunity arose which opened up an entirely new vista for me. The year was 1979 and the Cold War was ratcheting up to a new peak of tension and the nuclear arsenals to new levels of potential destructiveness. My field of physics dated itself to the wartime Manhattan Project, and many of the senior figures in my field had long participated in the furtherance, but also in the control, of military technology. It was their view that their successor generations had a responsibility to remain involved in these matters. Thus, several senior figures in the field urged me to take a one-year leave of absence from theoretical physics to join a study team of scientists being assembled at the Congressional Office of Technology Assessment. ...

See also Physicists Can Do Stuff and Wandering Physicists.

Theta terms and asymptotic behavior of gauge potentials

[ See QED and QCD theta angles, and asymptotic boundary conditions in gauge theory for more discussion. ]

This is the sequel to an earlier paper on physical effects associated with the QED theta angle. Unfortunately, field configurations which would yield theta-dependent effects are not realizable in the lab, based on general considerations discussed in the new paper. There could, however, be small non-perturbative effects sensitive to theta despite the lack of topology in QED (maps from S^3 to U(1) are all topologically trivial).

http://arxiv.org/abs/1107.0756

Theta terms and asymptotic behavior of gauge potentials in (3+1) dimensions

We describe paths in the configuration space of (3+1) dimensional QED whose relative quantum phase (or relative phase in the functional integral) depends on the value of the theta angle. The final configurations on the two paths are related by a gauge transformation but differ in magnetic helicity or Chern-Simons number. Such configurations must exhibit gauge potentials that fall off no faster than 1/r in some region of finite solid angle, although they need not have net magnetic charge (i.e., are not magnetic monopoles). The relative phase is proportional to theta times the difference in Chern-Simons number. We briefly discuss some possible implications for QCD and the strong CP problem.

Trento talk

My student David Reeb will be attending this meeting at ECT Trento next week:

Sign Problems and Complex Actions

workshop at ECT* Trento

Monday March 2 - Friday March 6 2009

organizers: Gert Aarts (Swansea University) & Shailesh Chandrasekharan (Duke University)

I would really have liked to go but my wife is giving a talk at Emory at the same time so I am stuck at home watching the twins. No Alps for me :-/

Here are the excellent slides for David's talk. The first section is a nice introduction to Monte Carlo methods in quantum field theory, which he made for a local seminar here in the ITS (for non-experts).

Sign problem in QCD

The revised version of our paper 0808.2987 is up on arXiv now. Special thanks to Kim Splittorff, Mark Alford, Bob Sugar, Phillippe de Forcrand and many others for comments. See earlier discussion.

On the sign problem in dense QCD

http://arXiv.org/abs/0808.2987

S. Hsu and D. Reeb

We investigate the Euclidean path integral formulation of QCD at finite baryon density. We show that the partition function Z can be written as the difference between two sums Z+ and Z-, each of which defines a partition function with positive weights. If the ratio Z-/Z+ is nonzero in the infinite volume limit the sign problem is said to be severe. This occurs only if, and generically always if, the associated free energy densities F+ and F- are equal in this limit. In an earlier version of this paper we conjectured that F- is bigger than F+ in some regions of the QCD phase diagram, leading to domination by Z+. However, we present evidence here that the sign problem may be severe at almost all points in the phase diagram, except in special cases like exactly zero chemical potential (ordinary QCD), which requires a particular order of limits, or at exactly zero temperature and small chemical potential. Finally, we describe a Monte Carlo technique to simulate finite-density QCD in regions where Z-/Z+ is small.

ECT in Trento

I'm off soon to the following meeting at ECT: European Center for Theoretical Studies in Nuclear Physics, which is located in the mountain town of Trento, in the Italian Alps. The picture above was taken just north of Trento. I'm excited to see the dolomiti!

I'm flying in and out of Venice -- any tips on what to do there or in Trento would be appreciated :-)

Meeting: The statistical model of hadron formation and the nature of the QCD hadronization process

program , poster , slides of my talk.

Dense nuclear matter: intuition fails!

I usually don't get into detailed physics exposition on this blog, but I thought I would make an exception with regard to the paper 0808.2987 which I recently wrote with my student David Reeb. (See earlier blog post here.)

In the paper we conjectured that there might be regions in the QCD phase diagram where the sign problem does not prevent monte carlo evaluation of the Euclidean functional integral. We rewrite the partition function as

Z = Z+ - Z-

where Z+ and Z- are sums with positive weights, and each define independent statistical ensembles. Defining Z+ = exp( - V F+ ), and similarly with Z-, so that F+ and F- are the (piecewise) analytic free energies of the two ensembles, we conjectured that

F+ < F-

is the generic situation. Note Z > 0 so F+ > F- is not possible, but they can be exactly equal: F+ = F- , which is where the sign problem is most severe (see below). Since the F's are analytic except at phase boundaries, we reasoned that if they are equal in a region they must be equal everywhere within that phase region. At mu = 0 we know Z- = 0, so we assumed that there would be a region of small mu where F+ < F- and that this region would extend into the mu-T plane.

It turns out this last assumption is probably wrong! We were unaware of results which strongly suggest that even at arbitrarily small (but positive) mu and small T, Z+ does not dominate Z. That is, in the thermodynamic limit F+ = F- exactly even at small nonzero mu. The order of limits matters: taking V to infinity for fixed nonzero mu (no matter how small) leads to large phase fluctuations. The only way to avoid it is to take mu to zero before taking V to infinity. (See 0709.2218 by Splittorff and Verbaaschot for more details. Note their results rely on chiral perturbation theory, so don't apply to the whole plane.)

It is quite strange to me that zero density QCD can only be reached in this way. The case we are most familiar with turns out to be the oddball.

To make a long story short, our conjecture is probably incorrect: what we thought would be "exceptional" regions in the phase diagram are the typical ones, and vice versa -- at least as far as anyone knows.

Note to experts: we used the term "sign problem" a bit differently than apparently it is used in the lattice community. We refer to dense QCD as having a sign problem even though we don't know for sure (i.e., for all mu and T) that Z is exponentially small in V due to cancellations (i.e., a "severe sign problem"). Our usage probably translates to "potential sign problem" -- the functional measure isn't positive, so potentially such cancellations can occur, although we do not know if in fact they do. We got a lot of emails from people who thought we were claiming to have a method for dealing with severe sign problems, but in fact we were claiming something else entirely: that there should be regions in the phase diagram where the sign problem is not severe.

Dense nuclear matter

New paper! Probably too technical to go into here, but it relates to our current inability to directly simulate dense nuclear matter (QCD at nonzero baryon density). When the number of quarks and antiquarks is equal, the functional integral representation of the partition function Z has good positivity properties and can be evaluated using importance sampling (lattice Monte Carlo methods). That is no longer true when the system has nonzero baryon number, as would be the case inside a neutron star or in nuclear matter.

We rewrite Z = Z+ - Z- , where Z+ and Z- have good positivity properties, and conjecture, based on arguments using the analytic properties of the free energy, that at most points of the phase diagram Z+ dominates Z-. At such points one can simulate the theory using Monte Carlo.

http://arxiv.org/abs/0808.2987 (paper available after 5 pm pacific 8.24.08)

Sign problem? No problem -- a conjecture

Stephen D.H. Hsu, David Reeb

We investigate the Euclidean path integral formulation of QCD at finite baryon density. We show that the partition function Z can be written as the difference between two sums, each of which defines a partition function with positive weights. We argue that at most points on the phase diagram one will give an exponentially larger contribution than the other. At such points Z can be replaced by a more tractable path integral with positive definite measure, allowing for lattice simulation as well as the application of QCD inequalities. We also propose a test to control the accuracy of approximation in actual Monte Carlo simulations. Our analysis may be applicable to other systems with a sign problem, such as chiral gauge theory.