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DIARY 1997

- by Changhai Lu -

原 “航海日志” 栏目引言 <<

Jan 1, 1997, Wednesday

These days I have been reading some background materials about monopoles, but Kimyeong asked me to do a two-dimensional field theory in which he wants to find the SUSY partners of a Q-ball and its excitations. Contrary to his expectation, I have shown him that the uniform background charge present in the theory will destroy SUSY. My calculation also showed that the spectra of the bosonic and fermionic excitations do not match.

Jan 23, 1997, Thursday

One of the reasons Kimyeong believes there exists SUSY partners of Q-balls is that (in some sense) he treated Q-balls as states of quantized field. His argument is based on the Schrödinger representation of the quantum field theory.

This argument is not new to me since I have considered the same thing but in the canonical formalism. The question is: can we really treat a Q-ball state |φ_b> as a state of quantum field? I believe the answer is "no". (... ... mathematical details omitted ... ...)

Jan 30, 1997, Thurday

Things have changed a lot recently, Kimyeong finally gave up his belief on the existence of the fermionic zero mode in our original model. He asked me to generalize our model by adding another neutral scalar field (since our original model doesn't have large Q-balls). When calculations on the new model still produced non-symmetric bosonic and fermionic spectra, he seems no longer strongly insisted.

We also tried to add a term -1/6|Φ|⁶ to the superpotential of the model. In that case something unexpected was found and it finally led us to a systematic way to compute models with an arbitrary superpotential W(|Φ|²).

(... ... mathematical details omitted ... ...)

Jan 31, 1997, Friday

Although I'm quite experienced in playing with superfields now, I'm not quite optimistic on the physical significance of the whole problem. It seems to me that the expectation we had on this problem was wrong from the very beginning. But I have to leave it for Kimyeong to make a judgement.

Feb 3, 1997, Monday

Kimyeong finally asked me to go back to the monopole space, I think that is an alternative way to say good-bye to the Q-balls.

Feb 25, 1997, Tuesday

I have been reading several papers on Nahm's equations these days. Those papers are all very mathematical, but they really provided a nice way to compute multi-monopole solutions and moduli space metrics. Kimyeong asked me to find out the exact moduli space metric for two identical massive and one distinct massless monopoles in the case when a gauge group Sp(4) breaks into SU(2)×U(1). Our strategy is to embed the Sp(4) group into an SU(4) group for which the Nahm's equations are directly applicable.

Mar 9, 1997, Sunday

I have read a short paper of H. J. Schnitzer "The Quantum Field Theory of Physics and Mathematics". In this paper, the main difference between the two types of QFTs is stated as: QFTs in the mathematical sense is well-defined at any scale, while QFTs in physics is only well-defined up to a certain scale. So far as I know, rigorous QFTs in the mathematical sense only exist in 1+1 dimensions. On the other hand, I remember that exact conformal symmetries also exist only in 1+1 dimensions. It seems these two concepts are related to each other since in some sense the exact conformal symmtry ensures if a theory is well-defined at some scale, it will be well-defined at any other scale.

It has been conjectured for a long time that pure Yang-Mills theories might be the candidates of mathematical QFTs in 3+1 dimensions. Similar to the case of 1+1 dimensional QFTs, this might be related to the conformal invariance of such theories. Here the complexity is: the conformal invariance for (3+1)-dimensional Yang-Mills theories is - unlike its 1+1 dimensional analogue - broken at a quantum level. This probably is why it is so hard to prove such a conjecture.

Apr 3, 1997, Thurday

Kimyeong thought we have obtained basically everything we wanted to know about this Sp(4) monopole system. I will begin to write a draft paper on that.

Apr 9, 1997, Wednesday

I have finished a draft paper on the Sp(4) results and brought it to Kimyeong.

Apr 22, 1997, Tuesday

I'm doing some reading on differential geometry and twistor thoery.

From the pointview of fiber bundle theory, general relativity and Yang-Mills theories are fairely similar to each other. Geometrically speaking, the reason that makes their quantizations so different is the following: in the case of a usual Y-M theory, the fiber manifolds are not correlated to the base manifold (which is a four-dimensional pseudo-Riemannian manifold) while in the case of GR, the fiber manifolds are the tangent space of the base manifold, therefore is closely related to the base manifold. It is this subtle correlation that makes the quantization of GR so much more difficult.

Jun 2, 1997, Monday

Today is the first day of the TASI-97 summer school (hosted by University of Calorado at Boulder). The first four series of lectures are:

• J. Hewett (The Standard Model)
• P. West (Introduction to Supersymmetry)
• S. Dawson (MSSM)
• J. Gates (Superspace)

P. West and J. Gates are pretty good, but the other two (ladies) are not as good.

Jun 4, 1997, Wednesday

I have learned something about MSSM (Minimal Supersymmetric Standard Model) through the lectures. I have to say that I don't like this theory at all, even though it has a fairly good experimental fit comparable to that of the standard model. What makes it ugly is its introducing of a huge number of free parameters. What makes me even more unhappy is the fact that all the SUSY breaking terms in MSSM are the so-called soft SUSY breaking terms. Those terms are something that break SUSY explicitly ("soft" only refers to the fact that they still preserve the cancellations of the quadratic divergence). This kind of symmetry breaking is way too arbitrary and it is not quite different from directly throwing SUSY away (breaking a symmetry arbitrarily is not very different from not introducing it at all). Another aspect that makes MSSM questionable, in my opinion, is it predicts the mass of the lightest Higgs to be less than 120 Gev, which is almost ruled out by experiments (the standard model, on the other hand, wants it to be heavier than 135 Gev).

The reason that MSSM has a Higgs mass prediction radically different from the standard model (as an extension to the standard model, one might expect MSSM be able to re-produce standard model results under certain limit) is, in my understanding, due of the fact that the couplings (at least some of them) between SUSY particles and ordinary particles are related to the couplings of the standard model in a definite way, therefore can't go to the strict weak-coupling limit. On the other hand, in order to solve the hierarchy problem, it is required that the masses of the SUSY particles need to be in the region of 1 Tev, therefore no infinite mass limit can be taken as well.

Jun 5, 1997, Thursday

In today's MSSM lecture I realizeded that the number of the free parameters in MSSM can be highly reduced by assuming superparticle masses and coupling constants present in the soft SUSY breaking terms equal at the GUT scale. This reduced model (called CMSSM - constrained MSSM or supergravity inspired MSSM) contains only a few extra parameters beyond the standard model and yet preserves an excellent experimental fit. This slightly improved my feeling about MSSM.

Jun 9, 1997, Monday

A new week started, the speakers in this week are:

• D. Kaplan (Effective Field Theory)
• L. Randall (Super Cosmology)
• R. Kallosh (Gravity and Supergravity)

Kallosh's lecture is good.

Jun 12, 1997, Thursday

In a quantum theory, should central charges be quantized? In the simplest case of the N=2 SU(2) Super-Yang-Mills theory, we know that central charges represent electric and magnetic charges, both are quantized. I asked Prof. R. Kallosh, but she doesn't know the answer.

Jun 16, 1997, Monday

This week we have:

• D. Pierce (Radiative Corrections in SUSY)
• S. Lammel (Tevatron Searches)
• L. Hall (SUSY Flavor Problem)
• R. Mohapatra (Super GUTs)
• S. Thomas (Dynamic SUSY Breaking)

Thomas' lectures will start tomorrow. Among the four lectures we had today, Hall's and Mohapatra's are good (especially Mahapatra's).

Jun 17, 1997, Tuesday

Thomas' lecture is also not bad.

Jun 23, 1997, Monday

This is the last week of TASI, we have:

• P. Nilles (Superstring Phenomenology)
• D. Nemeschansky (Duality in Supersymmetric Yang-Mills Theory)

and two other series of lectures on experimental topics.

In Nemeschansky's lecture, I asked him about the relation between t'Hooft-Polyakov quantization condition and Dirac quantization condition. To my surprise, he was totally lost on this problem.

Sep 3, 1997, Wednesday

Kimyeong and I found that our previous idea of embedding G₂ into Sp(6) is wrong. The reason is fwo-fold:

(... ... mathematical details omitted ... ...)

It seems the simplest group meets our need is SO(7) which has the same Dynkin diagram as Sp(6) except that γ is the short root rather than the long root.

Sep 5, 1997, Friday

I've made some progress on the G₂ case. It is a lot more difficult to determine the parameters directly from the boundary conditions for G₂ than for Sp(4). (... ... details omitted ... ...) But it turned out that without getting through those complications we are already able to determine all the parameters except one.

Sep 9, 1997, Tuesday

So far all the efforts to fix the final parameter failed. We believe the metric of the four-dimensional hyperKähler quotient space M₄(ζ = 0) is (at least asymptotically) a Taub-NUT metric with a positive mass parameter.

Sep 19, 1997, Friday

Today in Brian Greene's quantum field theory lecture, a little question occured to my mind: usually when people demonstrate how the path integral formalism approaches a classical result, it is said that when the Planck constant ћ approaches zero, contributions from all paths except the classical one - the one that stablizes the action - are cancelled. Now what if there are more than one paths that stablize the action? It seems we will always have interference between those paths therefore no classical limit can be reached. Brian said usually we assume the uniqueness of such a path. In case when there are more than one, superselection rules will be applied to pick up a unique path.

A typical example is a particle moving on a sphere, a unique path is selected by specifying the energy, the direction of the initial motion, etc. These conditions will correspond to a set of superselection rules in the path integral formalism.

Oct 8, 1997, Wednesday

As we know, spontaneous symmetry breaking (SSB) happens only in quantum field theory with an infinite volumn. This has a direct consequence on cosmology: if the universe is closed, there won't be any strict electroweak and GUT transitions. In other words, so long as we believe in SSB, the universe can't be too small at the time SSB happened.

Oct 17, 1997, Friday

As we are still not able to make definite progress on both the MSB (Maximal Symmetry Breaking) Sp(4) problem and the G₂ problem, Kimyeong asked me to do something on calorons (calorons are periodic instantons - or equivalently instantons on S¹ × R³, the name "caloron" came from the word "calorie" since they appear in finite temperature gauge theories).

Oct 28, 1997, Tuesday

These days we have been discussing a problem in our paper pointed out by a referee. The problem is: when we constructed the hyperKähler quotient space M₄(ζ), we have used a U(1) action that turned out to be non-free (it has the spherically symmetric configuration as its fixed point). After discussion, we now realized a few subtleties about the boundary condition and the U(1) action. (... ... details omitted ... ...) The main results of our paper, however, is un-affected by these subtleties.

Dec 3, 1997, Wednesday

The building blocks of the Standard Model are combined together in a way that no piece can be missing. For instance, the cross-section of νν → W⁺W^- scattering will grow with s therefore violate unitarity in absence of the ZWW coupling term. This is usually stated as: unitarity requires ZWW coupling. There are many similar unitarity-based arguments in the standard model.

But it seems to me unitarity is a very general feature of quantum theories as long as the Hamiltonian of the theory is Hermitian (therefore the evolution operator - in Schrödinger picture - is unitary). In the example mentioned above, if there's no ZWW coupling, one can still have a theory with a Hermitian Hamiltonian therefore still have unitarity. If the tree level perturbation theory violates unitarity, it only means such a perturbative calculation is not applicable (at high energy). So in my opinion, those unitarity arguments in the Standard Model are not quite convincing.

>> 原 “航海日志” 栏目引言 | 二〇〇二年日记

posted on July 21, 2002
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