QFT Journal Club 2: Propagator Theory

Time: 10/3/2009

Speaker: James Murray

Outline:

• Non-relativistic propagators in quantum mechanics.
• Propagators of scalar field.
• Propagators of fermion field.
• The perturbative solution of propagators.

Reference:

Bjorken & Drell, Relativistic Quantum Mechanics, Chapter 6.

Comments:

Only part of the comments were made in the meeting.

• Propagators as Green’s Functions

The mathematical meaning of propagators are just Green’s functions of the equations of motion (EOM) of the particles or fields.

Where is a linear differential operator.

We learned in mathematical methods of physics that the relationship between Green’s functions and the general solutions of the EOM with source depends on the order of derivatives in the EOM and also on boundary conditions. Generally, if an EOM is second-order in the time derivative, we need to know the initial conditions of both the solution and the first order time derivative of the solution. In non-relativistic quantum mechanics (NRQM), we only have the first order time derivative, so it’s legitimate to write down a general solution of the wave function as

This is just an application of Huygens’ principle for evolution of waves. Then in relativistic quantum mechanics (RQM), the Klein-Gorden equation is in the second order of time derivative, but we used the same evolution function for the wave functions. Why? Because we set the initial condition as

You may notice that there is no boundary terms in space either, because they are also set to be 0.

The Green’s function itself also depends on boundary conditions. It’s remarkable that ‘time ordered’ is just a kind of boundary conditions. In NRQM, we input the step function as a boundary condition that . This gives us the retarded Green’s functions. In RQM, the boundary condition becomes for and for , where is the on-shell Green’s function without this time-ordered condition. is just the Feynman propagator.

In RQM or QFT, we only deal with the simplest case with the simplest boundary conditions. As a contrary, in condense matter physics, there may be many weird boundary conditions and the problem becomes much more complicated.

• Propagators as Correlation Functions

There is another name for propagators which is more commonly used in statistical mechanics: two-point correlation functions. Propagators are just a kind of correlation functions. Since it correlates two states in space-time, it describes the evolution, ie, propagating of the particle or field. In condense matter physics, the more commonly used correlation functions are only defined in pure space, where time is not a variable. Although they have different properties, their essence is the same: for a given system, if we know the probability amplitude of the ‘particle’ at one point in the ’space’, a correlation function gives us the amplitude of the ‘particle’ at another point under such conditions. Here the ‘particle’ can also mean field or abstract states, and ’space’ can also mean space-time or any abstract phase space.

By understanding its physical meaning, it’s not difficult to write down the general and abstract formula for two-point correlation functions:

Where x and x’ are space-time indices and are indices for all internal degrees of freedom. are a complete set of states.

• How physical is a propagator?

In QM, we learned that some quantities are physical and some are not. For example, wave functions are not physical, because they can have different phases for the same physics; but the modulus squared of a wave function or of an inner product of wave functions are physical, because they mean probabilities which can be measured directly. Generally speaking, a physical quantity respect the symmetry of the system, ie, it must be some symmetric representations of the symmetry group, while an unphysical quantity does not necessarily have this constraint.

We’ve seen that propagators are amplitudes, so the question ‘how physical a propagator is’ can be translated into the question ‘ how physical an amplitude is’. In QM, an amplitude is almost physical in the sense that there is only an unphysical phase in the amplitude. So if we impose Lorentz symmetry into the system, it’s natural to require that amplitudes, then also propagators to be Lorentz invariant. However, this is not always the case in RQM. The propagator of a scalar field is Lorentz invariant because the two state correlated are also Lorentz invariant. But we’ll see later that a propagator can be a tensor which is Lorentz covariant, eg, fermion propagator, or it can even have no Lorentz covariance property, eg, photon propagator in Coulomb gauge. This is because each component of the propagator describes the correlation of two components of the field, and even the field components can be unphysical as we’ll see in gauge field theory.

• Classical and Quantum Propagators

As long as there is a linear differential equation, there is a set of Green’s functions. Physically it means propagators exist in every wave system, no matter it’s quantum or classical. Quantum mechanics (including relativistic quantum mechanics) treats single particles as waves, so it’s also called Wave Mechanics. In this sense, QM is essentially the same as a classical field theory (Of course, QM is a special field theory in the sense that it has only the first order of the time derivative). For example, the Klein-Gorden equation is a quantum equation for relativistic scalar particles, but it’s also a classical equation for relativistic scalar fields. Then what’s the difference between quantum and classical theories? They have different propagators. We use retarded propagators in classical field theories while Feynman (time-ordered) propagators in relativistic quantum mechanics. They have different boundary conditions in time according to different physical conditions. For classical field theory, we do not want any particle or anti-particle to be produced, so we have propagators which start at an initial time point and always propagate forward in time with a positive pole energy. For quantum mechanics, we tried to keep the same property, yet realized that in order to combine quantum mechanics with relativity, we have to end up with a weird propagator in which some negative energy propagates backward in time. Then we have to accept the concept of anti-particles and realized that RQM is in itself inconsistent, because we start with a single particle system yet end up with a system in which particle number is not conserved. The only solution is to further treat particle wave functions as field operators, then we come to quantum field theory. This is called the second quantization. In fact, there is another way leading to QFT, that is to quantize classical field directly, in which all the symmetries and dynamical variables are already prepared.

It is remarkable that the uncertainty principle and the ‘off-shell’ of a propagator are not exclusively the properties of quantum mechanics. In fact they are properties of waves and also exist in classical field theories. As there are sources, the EOM of free waves no longer holds at all time so the dispersion relation changes, and that is the cause of ‘off-shell’.

• The Role of Propagators in Perturbative Solutions of the EOM

In principle, if we can exactly solve the EOM of a system with a certain boundary condition, all dynamics and hence the state of the system at any future time is determined, then our work is done. However, most of the EOM’s are not easy to be solved analytically, especially when there are interactions. We have seen it in QM that there are only a few exactly solvable systems. So we introduced a powerful tool: perturbation methods. In QFT, solving EOM becomes even more difficult because the EOM with interaction is usually nonlinear and the interaction always depends on time. To use the perturbation method, we have to make two assumptions: first, in order to use the superposition principle of Green’s functions, we assume that the interaction only happens in a finite region of space-time and we are only interested in initial and final states which are far from the interaction region and can be considered as free particles for which the EOM is linear; Second,  we assume that the interaction is small and can be expanded perturbatively, then as in time-dependent perturbation theory in QM, we can calculate the correlation functions iteratively, with different interaction points connected with propagators of free particles (Note: free does not mean on-shell, it just means it’s the propagator solved from the EOM without interactions ).

• The structure of propagators in QFT

Now we’ve seen two kinds of propagators, the one for scalars:

and the one for spinors:

Where is a matrix carrying spinor indices.
We see that the spinor propagator equals a scalar propagator, which provides the pole and time-order structure, multiplied by some matrix in spinor space. This structure is very general. No matter the field is a scalar, a spinor or a vector field, each component (polarization) of the field propagates like a scalar, then we only need to sum over all the components to get a total propagator, that is the why the numerator of the spinor appears. Let’s use this manner to look at a vector propagator:

A vector field can be written as
Where is the polarization basis and is the ‘coordinate’ on this direction which is a pure number. To calculate the propagator, we take as a delta function at point x. According to the definition of correlation functions mentioned earlier,

The propagating between polarizations are orthogonal, so in momentum space

then we have
This is the the general structure of vector propagators and is gauge dependent for massless vector field so we see that the propagator is also gauge dependent.

QFT Journal Club 1: Groups and Group Representations in Physics

Time: 9/19/2009

Speaker: Chris Brust

Outline:

• Definition of groups: 4 axioms.
• Finite groups: defined by a multiplication tables.
  Example: Permutation group
• Group representations: mapping a group to a set of matrices.
  trivial rep, faithful rep, reducible and irreducible rep, unitary rep
  Example: 3 (and the only 3) irreducible rep of , 1 faithful but reducible rep of
• Lie groups: continuous groups which can also be described as a manifold.
  Commonly used Lie groups in physics:
  
• Group properties: Isomorphism(btwn two groups), Abelian and non-Abelian, compactness, connectedness, simply connected or not.
  Examples: Poincare group
• Lie algebras: defined by 3 axioms.
• Generating a group from an algebra and vice versa.
  Example: Heisenberg algebra and group

Comments:

• Very nice talk, informative and well organized. Thank you, Chris.

• Why do we need group theory in physics?

It’s all about symmetry. Symmetry plays a significant role in modern physics. From crystal lattice in condense matter to elementary particles in high energy physics,  it is its symmetric structure that causes the system’s rich phenomena, and almost all we care about in theory is related to its symmetry realization and breaking. Group theory is an indispensable math tool for describing symmetries. A system has some symmetry means the Hamiltonian or Lagrangian is invariant under the transformation of the corresponding group. So we get some rigid mathematical form for this symmetry and studying this system becomes studying the Hamiltonian or Lagrangian under such constraints. Buy using this tool, we can even lift different specific physical systems to some abstract structure and find their common properties, as happened again and again in the history of physics. One of the most remarkable example is the 2008 Nobel price for physics: Nambu was awarded for his work on spontaneous symmetry breaking in superconductor which latter played a vital role in particle physics.

• Why are group representations so important in physics?

Groups are only some abstract math f0rms. To connect math to physics, we need one more step: to find some specific representations of the group. Different systems may have the same symmetry, but their constituents can have very different behaviors under the symmetric transformation. Some may not change, some may exchange identities with each other, some may shift by some values, but the Hamiltonian or Lagrangian is invariant under all these changes. In math language it means, they are in different representations of the same group. For example, the ones that are kept unchanged are in trivial representations, ie., . In particle physics, the role of representations is even more obvious: the nature has only one fundamental physics law, which means the groups that describe all the matters in the universe are the same, but why are there so many different species of fundamental particles with different spins and interactions? They are distinguished by different representations. Different spins and momenta are distinguished by rep’s of Poincare group, different interactions are distinguished by rep’s of gauge groups.

• Why do we also need Lie algebra?

There is a most important class of groups called Lie groups, which played a central role in studying QFT. Lie groups describe continuous symmetric transformations, eg., Lorentz transformation, translation and gauge transformations. However, usually we only care about *local* properties of a system, ie., how it behaves under some infinitesimal transformations. That’s where Lie algebra comes out. In geometric language, Lie groups can be taken as manifolds, each group element is a kind of ‘translation’ on the manifold and the generators of a Lie algebra are a set of basis of the manifold. (To imagine it, you can use ordinary vector space as a analogy.) By studying the properties of the basis, we can know the properties of the whole manifold, but wait, not all properties of the manifold are included in the basis. The same Lie algebra may generate different Lie groups. For example, and are equivalent both as Lie algebras and groups (ie., they are isomorphic); While and have the same algebra, they are different groups ( is simply connected but is not. ). This is because some discontinuous symmetry distinguishes their global properties[1]. Fortunately, in QFT, usually the local property says everything about physical observables we care about, eg., cross-section of collision, life time, etc. So we do not need to be too serious on distinguishing Lie groups and Lie algebras.

• The first step of constructing a quantum field theory
  - One example of group representation theory used in QFT

One of the most important Lie groups in QFT is of course the Poincare group, which carries the physical meaning of special relativity. To make a relativistic quantum mechanics, we only need to let each of the group elements act on a vector(state) of a Hilbert space which satisfies all the axioms of quantum mechanics and get another vector in the same space,

That means this Hilbert space is a symmetric space under the transformations of the group. So we have relativity and quantum mechanics both satisfied. Then our task is to find all the possible representations of the group and do experiments to see what representations are chosen by the nature, ie., what species of particles do exist in nature. Mathematically we can prove that translation and Lorentz transformation can be disentangled, and further, the only irreducible representations of Lorentz group are spin-half-integer particles. Now we find all possible kinds of elementary particles in the nature! (Assuming relativity and quantum mechanics are correct, of course.) In reality, we see only spin-1/2 , spin-1 and spin-2 elementary particles, but who knows spin-0 and spin-3/2 elementary particles exist or not, they may be waiting for us on the LHC[2].

Till now we only discussed Poincare symmetry for free particles. Most of the interactions are related to gauge symmetry and they can be studied in a similar manner.

Now we’ve learned the first step of constructing a general quantum field theory: determine all the symmetries of the system, find and select certain representations of the symmetry groups, and write down a Lagrangian which is invariance under the symmetric transformations by using these representations as degrees of freedom.

—————————————————

[1] For your information, is in fact isomorphic to , where means an action of orbifold. )

[2] Elementary particles with spins higher than 2 are theoretically forbidden for some deeper reasons.

[3] Text books on group theory suggested by the speaker:

Georgi, Lie Algebras in Particle Physics

M.S. Dresselhaus (Author)

› Visit Amazon’s M.S. Dresselhaus Page

Find all the books, read about the author, and more.

See search results for this author

Are you an author? Learn about Author Central

M.S. Dresselhaus, G. Dresselhaus, A. Jorio, Group Theory: Application to the Physics of Condensed Matter

Micheal Tinkham, Group Theory and Quantum Mechanics

Discussions on Entropy

Information entropy is in fact a more fundamental definition than our familiar Baltzmann entropy in stat mech.

When we use Boltzmann energy , where W is the total number of microscopic states of the system with a given macroscopic state, we assume that all microscopic states has the same probability. This is true for most thermal systems. However, if the probability of each micro state is not the same, we have to modify the definition of entropy, then we have Gibbs entropy, which is equivalent to Shannon (information) entropy: , where is the probability of a state , and the sum takes over all ’s. You can check that if all are equal, this definition goes back to Boltzmann entropy.

So we can have a ‘modern’ interpretation of thermal entropy: the amount of entropy means the amount of information we need to input into this system to determine its micro state for a given macro state. In other words, order means predictability, and for a specific state, the predictability means its probability. If a thermal system has a larger number of micro states, then each state would have smaller probability, which means we have a smaller chance to predict the right micro state, that’s why this system has a higher entropy.

A very useful lesson from thermal theory for us to understand order and disorder in information theory is: For two systems with the same macro state, the one with independent sub systems has higher entropy than the one with correlated sub systems. For example, ideal gas system has higher entropy than interacting gas system with the same micro state. We can use this intuition into information theory. For example, compare two pages of paper with the same number of letters on it. The letters on one page is totally random, while the other page is a well written article. So we say the letters on the first page are independent while the letters on the second page have some correlations with each other. To determine the micro structure of the first page, we have to put in the information of each letter, with probability for each one. To determine the micro structure of the second page, we only need to put in the information of each word, with the probability larger than , where n is the number of letters in the word, since we know that there are some combination of letters which are definitely not a word. So now you can calculate the probability of each micro state, the second should be larger than the first. So we say the second page is more ‘predictable’, and the information comes from correlation. It’s interesting that we have just interpreted information entropy by using thermal entropy, for smaller probability of a micro state means larger number of micro states.

But there are 2 cases in which we can only use information entropy. One is, as I mentioned, when the probability of each micro state is not the same; Another one is non-equilibrium process. I am not familiar with either of them. Does any one know any examples of these cases?

The following is Lightsaber’s explanation of entropy in non-equilibrium process:

Let me start with non-equilibrium situations. In my lecture, I mentioned that “Information is …… boundary condition.” Many thought that “boundary condition” was a phrase I randomly picked, it’s not. Actually, I was referring to the non-equilibrium cases. Non-equilibrium is featured by non-uniform distribution and time-variance. By possessing the information of its spatial distribution, the entropy of the system is reduced. When equilibrium is established, all the information become no longer valid, and the entropy increases. That is why the establishment of equilibrium is always entropy-increasing. This piece of information could be valid only at a particular time, or be valid during the entire time interval we study. In the former case, it’s an “initial value condition” in PDE language, but it’s but a boundary condition in time domain anyway.

Starting from the simplest example in non-equilibrium thermodynamics: diffusion. If we connect a full bottle of nitrogen dioxide (bottle A) and a full bottle of air (bottle B) with a glass tube, we can see the red color gradually propagates, until the gas in both bottle has the same color. This is a entropy-increasing process. At the very beginning, we do know (know = possess a piece of valid information) that there’s neither air in A nor air in B. With the knowledge, the entropy is relatively low. In the language of probability theory as Shannon used, the probability of (an infinitesimal domain) in bottle A being filled with NO2 is 1, while it is 0 in bottle B. After the establishment of equilibrium, this piece of information become completely invalid, and the entropy is larger.

However, how can we describe the dynamical process (the formal term is “transport process” I think) between the start and the equilibrium? Can we use information theory to process it? I believe so.

In my PERSONAL opinion which hasn’t appeared on any reference material I’ve read so far, the introduction of fuzzy mathematics will be a possible way to solve it. Darthmaverick is an outstanding expert in this area, but I can discuss to the best of my knowledge. We can define a “membership function of validity” for any piece of information, which is dependent on time. This membership function can be determined as “The probability that the piece of information is true”. For example, for the statement “Bottle A is full of NO2 without air”, we can define the membership function as “P(an infinitesimal domain is filled with NO2)-P(an infinitesimal domain is filled with air)”. At the starting of transpotation, this membership function is 1, and it becomes 0 eventually. It can be proven (though I haven’t done it myself) that this membership function could be constructed to be proportional to the “entropy decreasing capability” of the corresponding information, as in the example above.

The utilization of this measure is still to be explored. After all, I come up with this combination of non-equilibrium thermodynamics, information theory and fuzzy mathematics independently, and it’s expected that some original work can be done following this direction.

That’s all for now, thank you.

As to the cases in which “the probability of each micro state is not the same”, intuition told me that it’s EQUIVALENT to the non-equilibrium cases, or uniquely corresponds to one. I wish I could prove it mathematically, but I cannot do it in a rigorous way. Possibly it’s wrong anyway.

One of the possible way of proving it is to consider the symmetry of the system. The unsymmetry of different microstates vs the broken symmetry the macroscopic system, what’s the connection?

Plus, according to LOT2, a closed system tends to maximize its entropy. Can a system reach the GLOBAL maximum of entropy without eliminating the unsymmetry among microstates?

I wish I could find an example in which symmetry among microstates is essentially and permanently broken. If it exists, and it can be stablized given time, my hypothesis in the 1st paragraph is wrong.

Then CoolPro asked an interesting question (in Chinese):

我想问问，ABCD四个球在正方形四顶角上
状态1，A朝东，B朝西，C朝南，D朝北。
状态2，B朝东，C朝西，D朝南，A朝北。

如果这时候算W，那上边两个状态是否可以算作相同的地位，从而总共算两个状态？即W=1+1+····

可是实际上，东西南北这个方位信息是我们给的，我们如果不给这个信息的话，这两个状态就是一个状态。

也就是说，给的信息越多，状态就越多，最终决定状态是否相同的是信息，但是信息的量，是怎么掌控的？我可以无限制地加信息，那熵就会无限制地减小。那么一个确定的系统的熵是可以无限多个的。
这是否跟我们的“事实”相违？一个确定的系统，却有不同的指标·····感觉很矛盾，估计只能在人类意识涉及到的地方才存在了

My answer is:

你说得不错，熵的确定确实依赖于我们给定的信息。我的印象是，在解决热学体系问题时，如何确定熵确实是一个很不简单的问题。确定一个系统的熵之前，我们需要定义这个宏观系统，即给定这个系统的一套完整独立的宏观参数，比如温度、能量、体积等，也包括LS提到的那些边界条件。在确定了这些信息之后，我们只需要问，还需要多少微观状态的信息才能把这个系统唯一地确定下来，这就是熵。但这里就有一个问题，也是你问到的问题：什么叫“唯一”？我怎么知道两个微观态是不是同一个？原则上，我们永远也不知道，因为一个体系的自由度多少取决于我们看这个系统所用的尺度，尺度越小，分辨越精细，自由度也就越多。同样两个状态，在大尺度上看可能是一样的，但在小尺度上看就有区别。所以可以理解，对同一个客观系统，当我们用的尺度越大时，算出来的熵越小，反之，熵越大。当然，“尺度”这个概念可以再抽象一下，变成我们关心的自由度和不关心的自由度。在你的例子里，第一种情况我们关心方位，尺度小，所以熵大，第二种情况不关心方位，尺度大，所以熵小。还有一个典型情况就是粒子全同性问题，全同粒子体系是比可区分粒子体系的熵小的。

所以说，系统是由信息确定的，即便我们面对的是同一个客观系统，只要我们预设的信息不一样，系统就不一样。这样做的有效性在于：我们通常只关心一个系统在演化中的熵变，而不关心它的绝对熵值，在不涉及系统自由度的动力学变化的时候，不同定义的熵只相差一个常数，而演化之间的熵变是唯一确定的。这可以类比量子场论中的重整化：当我们在不同程度上忽略更精细的结构时（比如加上一个动量截断），算出来的物理量也相差一个常数（尽管这个常数依赖于截断），但不会改变物理体系的变化方式（即参数在不同能标间的跑动以及由此算出来的观测量值）。

当然，并不是所有情况都是安全的。当体系的自由度发生动力学变化时，以前所假设的“模糊”自由度下所定义的熵就失效了。用场论的语言说，就是有效场论失效。比如说一团气体，我们先用分子做自由度，分子内部的不同结构在我们眼里对应同一个态。但后来由于能量升高，分子内部的结构变化开始影响到分子之间的相互作用，我们就不得不把分子自身变化的熵变计算到总熵里面，也就是说，这个时候可能就需要取原子或者原子团做自由度，把不同分子结构当成不同状态了。当然一个原则上一样方向却相反的例子是粒子全同性，我们从前以为粒子是可区分的，结果在计算气体混合熵的时候出现Gibbs样谬，才发现粒子应该是全同的，我们区分得过于精细，而这并非真实的状况。至于你提到的四个球的问题，是个很好的例子。如果我们把它想象成一种分子结构，当这个分子在三维空间里自由悬浮的时候，两个状态可以通过旋转变成一样，所以它们是同一种状态；而当这个分子被固定在二维平面上的时候，他们就是不同状态。所以说，怎么确定自由度还要看我们的物理体系会受到那些因素的影响，并不是那么地随意。

一个附带问题：压缩信息时熵怎么变化？
答：无损压缩熵不变，有损压缩么。。。如果按照“不同自由度，不同系统”的说法，有损压缩后自由度减少，由此算得的熵应该是更小，但显然不合理，因为比较两个不同方式定义的系统是没有意义的。所以，我们应该拿同样的自由度来比较，都用压缩前的自由度，有损压缩相当于是增加了被压缩的那部分状态的可能状态数，所以熵应该更大。

Another piece of LS’ comments:

A microstate is a specific, detailed configuration that includes the state of all the particles inside. For an N-particle system, it’s one single point in the 6N-dimensional phase space. For example, for a system described by canonical ensemble, its equilibrium macrostate is consist of numerous microstates, EACH OF WHICH obeys Gibbs distribution (cuz each microstate contains a complete set of information about all the particles and thus has its OWN distribution), and has the same probability as each other. If we change the parameters (like T), the equilibrium state will be consist of another set of microstates, but each of them will still obey Gibbs distribution and will has the same probability as each other.

My response:

To my knowledge, Gibbs distribution (or more commonly called ‘Gibbs measure’, or ‘Boltzmann distribution’) means the probability of a microstate of a system, not the distribution of the particles in this system. If this is not GL means, that’s fine, we don’t need to debate on terminologies. But I do have some more comments. The phase space you mentioned is for microcanonical ensembles, in which the density of states is a constant, which means all microstates, no matter what macrostates it correspond to, have the same probability. This is true for microcanonical ensembles (or isolated systems), but not for canonical ensembles, which is exactly described by Gibbs measure. In the phase space of a canonical ensemble, the density of a certain microstate is not a constant, it can repeat for many times, and the density or probability of the microstate is proportional to the number of microstates of the reservoir which ‘coexist’ with it. When you consider this and do the calculation, you get the Gibbs measure.

We should distinguish ensemble language and distribution language. The former one is only interested in microstates of the whole system. It is so abstract that it never cares about what kind of system it is or the fate of a single particle in it, while this fate or distribution probability of a single particle is just the essential focus of the distribution language, and it’s result varies with the types of systems(classical, quantum, interacting or not, etc. ). And further, a given microstate in ensemble language has no ‘probability distribution’, cuz it’s totally determined, the so-called distribution is just a description of this state, and it can be way off the Boltzmann distribution, eg, some higher energy level may have more particles than some lower energy level. Some of us may confuse ‘Boltzmann distribution’ in ensenble language with that in distribution language. They have exactly the same mathematical form, but have very different meanings: one is for microstates of a system ensemble, one is for particles in a single system. Why they have the same form is just a coincidence, because ensembles are defined as classical and independent with each other, which is the same property of the particles in a classical non-interacting thermal system. But see, we have other kinds of thermal systems, eg, quantum boson or fermion system, and they do not obey Boltzmann distribution.

LHC Physics Mini Workshop

LHC is coming soon, every one in particle physics is excited. And you can smell in the air how crazily LHC-driven they are. The mini workshop of LHC physics was held at UMD, for model builders to discuss the phenomenology of some model which are expected to be tested on LHC.

Quirks:
Markus Luty and Roni Harnik
Both of them talked about “quirk”(sounds like the bird who calls “quark, quark, quark” catches a cold). Quirk is the simplest extension of SM gauge symmetry. It’s a QCD inspired SU(N) field, with it’s strong coupling scale . So if you stretch a quirk-antiquirk pair, the energy reserved in a unit length is , unlike QCD, it is less than the energy required for producing another quirk-antiquirk pair, so the string will never broken. It can be stretched to macro scale, depending on the cutoff. Because of the string, this pair can oscillate like a spring. This gives the quirk very rich phenomenology. The two talks focused on the detection signal of quirks, they argued about different life time of the string before it annihilates and the stuff shaken off during the oscillation.
Markus’ quirks also carry QCD colors. So they will be surrounded by quarks and gluons to form a so called “brown muck”. When the string is oscillating, soft pions are the dominating shaken off particles. They lifetime of the string is estimated by using WKB approximation of the wave function of a quantum oscillator, and then considering the energy and angular momentum change by shaking off pions
Roni actually studied squirks, which are superpartners of quirks. They are uncolored under QCD. And in his “folded SUSY” model, squarks and quirks are orbifolded out by symmetry, leaving quark and uncolored squirks to be superpartners. This scenario can be realized in extra dimension. Since the squirks are uncolored, the dominating shaken off stuff are soft photons and glueballs(I do not understand why there are still gluons even though they are not colored.). And the life time is longer since the energy taken away by photons are much lower.

Loopy fermion mass:
Patrick Fox
This is most precise “posdiction” of fermion mass I have ever seen, which is too good to be true. But it’s interesting and maybe useful to my project. In this model, it assumes that only top quark (or one quark which we call “top”) gets mass at tree level, and all the other uptype fermions get their mass at loop level.
The conventional yukawa term is forbidden by a new U(1) symmetry of Higgs, so another U(1) charged scalar field is introduced to form a 5 dimensional operator, whose UV completion is the propagator of a massive U(1) charged fermion . The UV completion allows only one type of quark coupled to and , so after this U(1) symmetry is broken by vev of , only one type of quark obtain mass. But my question is, is there any symmetry to forbid other types of fermions in this coupling?
The next work is more generic. By introducing a QCD and EW charged scalar field r and a set of coupling constants, the following fermions get mass at 1,2,3,4,5 loops level respectively: . The generated mass are very close to the real values, by varying the couplings only between 0.3 and 3. This is the most beautiful part of the model.
The down type masses are much more messier. They have to introduce several weird fields and coupling terms, but the couplings are still order one. And the CKM matrix is in the right form. Neutrino mass is not explained in this model. But it can be done via seesaw mechanism.
The most intriguing LHC signals will be those colored and charged scalar fields. The constrain of available data is , which gives no hope to see them at LHC. But the constrain for a scalar field for downtype mass is .

Dark Energy – Ten Years

There is a symposium held by STScI to review the ten years’ progress in cosmology after the discovery of accelerating expansion of the universe. I missed some very important talks on Monday, not only because one of the speakers was Witten but also it was the only series of physical topics (rather than observational ones) of dark energy in these 4-day talks. I only attended several short talks(mostly 15 minutes each). Some topics for record:

1 First Results from the WiggleZ Galaxy Redshift Survey
Chris Blake (Swinburne University of Technology)
The WiggleZ project at the Anglo-Australian Telescope is a large-scale redshift survey of UV-selected emission-line galaxies. The survey is mapping a co-moving volume of approximately 1 Gpc^3 at a significantly higher redshift (0.5 < z < 1.0) than has been previously achieved by projects such as the 2dFGRS and SDSS. The main science goal is to use baryon acoustic oscillations in the galaxy clustering pattern as a standard ruler to measure the cosmic distance scale and expansion rate to z=1 and hence perform a robust test of the cosmological constant model. The survey is approximately 50% complete and is scheduled to finish in 2009. I will introduce the project and present initial results on the clustering, environments and luminosity function of high-redshift star-forming galaxies. I will also discuss forecasts for testing dark energy models with WiggleZ in the context of current and future cosmological datasets.

2 The Dark Energy Indicator: A Measure of Deviations of w from -1″
Ruth Daly (Penn State University)
The dark energy indicator provides a tool to measure deviations of the equation of state of dark energy from -1 over the redshift range from zero to one. The indicator is model-independent, and will be shown for the most recently available supernova and radio galaxy data sets. The preliminary results are consistent with a constant equation of state w of -1 from a redshift of zero to about one.
Note: This is interesting and may also be useful. The usual way we constrain the cosmological parameters, eg. , , is model dependent, ie., take a particular model with undetermined parameter and calculate the expected observational curve and fit with the data. This talk provided a way to draw the parameters directly from data. The only assumption is RW metric and GR. From the general Friedman equation, we can express the coordinate distance and it’s first and second derivatives (wrt z) with those parameters. By analyzing the distance-redshift curve directly, we can fit the parameters. In order to determine w, which can vary with z, we need to construct another parameter called “dark energy indicator” s, which is 0 for w=-1. The fitted s with supernovae data is 0 for z<1, but when z goes close to 1, there is a bump. It’s still unclear whether this bump is caused by systematics or it has any physical meaning.

3 Uncorrelated Estimates of Dark Energy Equation of State
Asantha Cooray (UC Irvine)
I will give a talk on some of the recent work we have done on how to extract and establish equation of state with supernovae and other cosmological data.
Note: There are three subtopics. I can only remember two. One is about the measurement of spectrum by putting many many filters for each wavelength. Another one is that type Ia supernovae actually have two subtypes with different light curved, ie., the distribution of time difference of maximum and some certain fraction of luminosity has two peaks. And these two subtypes’ population change with redshift differently, one increases with z, but the other one decreases. So the precision of measuring distance by using the time difference (cosmological time dilution) will be reduced by a factor of 2-3. Adam Riess complained soon. He said by investigating low z supernovae, we can get enough information to distinguish these two types. Well, he is the quasi-Nobel on supernovae, no one doubt he can do that.

4 Inflation and Dark Energy: Is There a Connection?
Scott Watson (University of Michigan)
We now have convincing evidence that both today and in the very early universe, the cosmic expansion went through a period of acceleration. A natural question arises: Are these periods of acceleration connected? I will briefly review past attempts to address this question, as well as more recent attempts motivated by the string landscape. In particular, I will discuss a crucial theoretical difficulty that arises in constructing such models, due to the vast range of energy and length scales involved. I will also discuss the possible experimental signatures that may arise if such models can be realized.
Note: If the acceleration today is due to the same mechanism as inflation, w should not be a constant, because the effective w of inflation is changing in order to end inflation at some time. Theoretically, the scalar field may not be elementary dynamical field. One model is ‘cascade universe’ with a stair of vacuums, and more generally they are inspired by string landscape. They difficulty of these models: I can’t remember. Think about these possible difficulties: no slow roll condition, no fluctuation seeds, unnatural(well, this is the weakest one), etc.

Nima’s Talk

Nima Arkani-Hamed may be the most famous and interesting name in the new generation of physicists. I attended his talk today in UMD, and his words are so impressive that even though I can’t understand the details, I feel deeply interested and attracted.

His title is ‘quantum gravity: possible and impossible’. In all, his purpose is to convince us that because of gravity, IR and UV are not totally decoupled as we thought before, gravity can contribute to high dimensional operators in EFT, and that not all IR effective field theory have a consistent quantum gravity theory as its UV completion, no matter it is string theory or not. The most important criteria he uses is the Weak Gravity Conjecture, which says that gravity must be the weakest long range (U(1)) force, and an effective theory should break down even below plank scale, at , where g is the gauge coupling, otherwise this system will decay into a black hole. A sharper form of WGC is that any matter must have . So if an IR theory violate WGC, it can never find a consistent quantum gravity theory as its mother theory.

So what EFT may be on the ‘dead’ list? One most important example is the inflation models which produce gravity wave, such as chaotic inflation. If gravity wave is generated, we require that the scalar field vev must be greater than plank scale. Naively, this can be realized in a 5 dimensional model with a very small 5D gauge coupling and it seems very natural. However, no one can find it in string theory, no matter what form they tried, there is always something, such as diliton, coming out to destroy it. Is there any general reason why they failed? The answer may be yes: it violates week gravity conjecture. There is another model named ‘N-flation’, with N species of scalar field triggering inflation. It is also problematic because it violates species bound which comes from constrain of entropy of a black hole. Moreover, if considering loop correlation of , we have which also leads to violation of WGC for large N. So Nima boldly guess that we can not see gravity wave in CMB, otherwise, it will be a big crisis.

Another interesting model he talked about was Euclidean warm hole. It was proved by Coleman that warm hole theory can be a local theory with global symmetry breaking, which can be realized with axion coupling with gravity. However, as chaotic inflation, we can not find it in string theory. And the obstacle is also WGC. To realize a warm hole, we just need to find a geodesic longer than , which means we have a energy scale higher than plank scale and the corresponding U(1) gauge coupling is weaker than gravity. Further, he said that if we realize it in a AdS/CFT world, it will violate unitarity of CFT, which I do not understand.

Nima also talked about some other models, checking them with WGC. Especially, he mentioned RS. He said we can not ignore the compacted in when we go down from string theory. Otherwise if we take a brane near the UV side with a string on it, the corresponding particle(gauge boson) will have mass which can be larger than . We need the extra-extra dimensions to ‘dilute’ the mass of the particle. I do not understand how either, too ’string’.

Nima is really a good speaker. He makes each point very clear, and goes through the details very smoothly. Speaks fast, writes faster.

Also Sprach David Gross

David Gross was invited to our department for 3 days. During one public lecture, one colloquium and some informal chat, there are something he said which may be interesting to hear. One thing to remind you, Gross is a very conservative reductionist, so you may not like some of his words. Welcome to leave your comments. I just quote them (not word by word) in a random order:

Things I learned these days

• Neother Theorem is the Lagrangian form of Ehrenfest Theorem which is in Hamiltonian form. There are two correspondences (not very exact) :

1. The Lagrangian is invariant under some symmetry corresponds to the Hamiltonian commutes with some operator (symmetry group’s generator)
2. The Neother charge is a time constant corresponds to the expectation value of the symmetry operator is a time constant, actually, the Neother charge is just the expectation value of the generator.

• From Kaplan’s Colloquium (11.29):

1. EW theory must break down because of 4W interaction, just like 4 fermion interaction must break down, which is caused by violating unitarity. And the solution is similar, 4 fermion need a gauge boson to change 4 vertex to 3 vertex, and 4W need a Higgs (or other new particle?) to change 4 vertex to 3 vertex.
2. MSSM has more than 100 parameters! ( I heard this for the first time)
3. Higgs mechnism can be traced back to Schwinger, who proved that massive gauge bosons do not necessarily violate gauge symmetry, by introducing a scalar field, but did not mention symmetry breaking. And there were other guys’ work following this, showing symmetry breaking, until Higgs pointed out the existence of a scalar particle following the referee(highly suspected to be Schwinger)’s suggestion. So came the name ‘Higgs particle’.

• From Tom’s seminar (12.3)

SUSY can be broken explicitly at UV (elementary sector) but emerges accidentally at IR (composite sector). The symmetries at lower energy are more than that at higher energy, which seems blizzard, but not. From 5D view, IR have more gauge symmetry and so more degrees of freedom just because UV and IR are two vacuums separated by the bulk (or domain wall); from 4D view, the different symmetries become global symmetry so it will not take more degrees of freedom, and the low energy global symmetry is always broken at high energy. (Because of gravity)

旧文重贴——A fantastic week

string2006已经过去一年多了，当时写了这篇文章，但是由于某些原因，贴上网时删了一些重要的内容。现在重新贴出完整版，算是给这个久不更新的博客充数。

A fantastic week
2006-6-25

2006弦论国际会议圆满结束，我也终于可以酣畅淋漓地看球睡觉了。这个星期的收获真是不小，尽管报告没听懂多少，但由于一个幸运的机会，我为一家科普杂志当了一回临时翻译，从追星族升华成记者，得以和Gross, Witten, Strominger 这些大师们“亲密接触”，还前所未有地训练了口语能力。尽管累一点，牺牲了看球时间，但过得很开心，因为能跟一群朋友愉快地合作。再次感谢Whitefalcon同学，每天都主动热情地帮我解决吃饭问题，让我这个没有经费注册的小本成功地从头蹭到尾，今天听说一顿70块，感觉真是赚大了（尽管远远没有吃回来），不过也有弊端，就是回学校食堂一看，胃口全无 ~_~||| 同时感谢张旭同学，给我这个对新闻工作一窍不通的书呆子很大的帮助和鼓励，并且十分佩服他广泛的兴趣和旺盛的精力。还要感谢该杂志的编辑pp姐姐张迪，给我带来这么好的机会，也给了我很大的鼓励。此外还有Susy、Tririver、yushe、菜菜等同学，合作愉快，呵呵~~~~~

本来打算开完会后把这几天的采访花絮和八卦趣闻都写下来，太多了，而且那股兴奋经已经消退了很多，就写写我对几位大师的印象吧。

Strominger
Strominger留给我的印象，政治比物理多。菜菜形容他长得像憨豆同学，圆圆的脸，圆圆的鼻子，笑起来表情丰富的眼睛，还有像小孩子一样的天真神态。他在人民大会堂作报告用的幻灯片上，画着许多搞笑的小人图片，而且他居然让布什同学掉进黑洞，一看就是一左派。后来采访的时候发现，他原来曾经是一个地地道道的美国左派愤青，不但在美国搞过人民公社，而且还跑到中国来学大寨，直到现在还同情社会主义。他那时刚上哈福，为了来中国，还学了中文，现在还可以秀上几句，以至于采访时出现我用英语问他用中文答的奇观。当他用中文慢慢地谈起70年代搞公社失败的那段往事时，他的眼眶发红了，听李老师说，他这段往事藏得很深，或许这是他第一次向媒体说出来。我可以感受到他年轻时的理想在他的一生中占有多重的分量，到现在还无法释怀。当我问他为什么从政治转向物理时，他说在他想做的所有事情中，只有物理最清晰，你知道该做什么，该怎么做出贡献，而且，他选择物理似乎还有一点政治失败后的避世情怀。我发现自己在这一点上跟他很像，尽管我什么都没经历过，但我同样是一个关心政治并曾经把政治当作自己理想的人，而我放弃政治转向物理的理由之一，便是Strominger说出的话，我同样认为，物理是最清晰，最能够把握的东西。

Gross
Gross曾被李老师比喻成甘道夫，现在看来不但地位相似，形象也相似，满头银发，气质非凡，不过我觉得他有时更像圣诞老人，和蔼可亲，幽默风趣，而且很喜欢长篇大论地谈话。（人民大会堂的报告就严重超时☺）对Gross的采访是我发挥得最好的，他说话很清晰，聊了很多物理，以至于后来几乎从采访变成答疑了。Gross是一个彻底的还原论者。当我们问他对Landscape和人则原理的看法时，他态度坚决，相信我们的科学既然已经可以把生命这样复杂的对象归结到只含一个参数的原子物理，把原子核归结到只含三个参数的标准模型，还原论就一定不会在宇宙学常数这个问题上栽跟头，因此一定要努力去kill掉Landscape这种silly idea。插一个花絮：后来我“用心良苦”（or 居心叵测？）地去问Gross，Polchinski和Susskind为什么没来，他的表情很诡异：“no reason”。这两位同学是大力提倡Landscape的，而我发现这次来的重量级人物似乎都是不太相信Landscape的人，如果它俩来了，这次弦论会议应该会更热闹。另外他对学习物理的见解很让我受启发。我一直很羡慕爱因斯坦那个年代的物理学生，以为他们学的比我们少很多，那时又没有相对论又没有量子力学，似乎本科学完后就马上可以到达科学前沿了。而我们现在，本科学完四大力学还差得远呢，从场论到弦论，要想爬上巨人的肩膀先要翻越物理和数学的重重高山，长此以往，人类如果还不延长寿命的话，以后就学一辈子好了，哪儿还能做研究啊？Gross对这个问题给出了很好的回答。他说其实以前的物理学生学得并不比我们少，因为当一个理论还没有发展成熟的时候，是相当复杂的，而当我们有了更好的理解后，能用更简单的描述把问题说清楚，并且有更明白的老师来教，就没有那么高深了。比如爱因斯坦时代，麦克斯韦方程在大学里是不讲的，因为太高深了，但现在却成了本科必修。SU(3)群刚出来的时候也是连大学教授学着都头痛的理论。我想除此以外，他们学习的经典物理肯定比我们多得多，比如经典力学，经典统计，好多计算的技巧我们现在都不讲了。看来我们完全没有借口抱怨现在学物理太难了，因为物理一直都这么难。我们又谈到了量子力学基本概念的理解，Gross认为我们其实已经理解了量子力学，只不过我们从小受到的都是经典物理教育，经典观念太根深蒂固了，以至于硬往量子上套而出现了问题，他希望以后能在中学甚至小学教量子力学。其实我也想过有没有可能不学那么多经典物理，直接学量子和相对论，不过我认为不可能跳过经典物理（尽管可以少讲一些，中学生在反复计算杠杆滑轮上浪费了太多时间），因为人的大脑发育有一个过程，我们认识世界都是从感性到理性，小孩子更是以感性为主，而经典力学都是我们日常生活中能直接接触到的，应该作为正常认知教育的第一步，就像胎儿的发育一样，学习应该也有类似的“重演律”。没想到Gross的思想这么革命，竟然想直接教量子力学，说到这里，两眼放光，脸上带着恶作剧的笑容，说那些孩子长大后一定会变得Crazy。他确实是一个革命的教育家，曾让自己的女儿8岁时看费曼物理讲义，他想让她成为理论物理学家，但是失败了，因为她物理毕业后转行做了历史学家。巧的是，Gross的学生，也就是大名鼎鼎的Witten，却是从历史转行做的物理，这真是一个奇妙的duality。

Witten
采访Witten比较失败，因为我太紧张了。紧张的原因，不仅仅是因为仰慕，还因为Witten说话声音比较沙，而且语速很快，不容易听懂，后来整理录音时发现在做高难度的听力测试，如果说Strominger是四级，Gross是六级，那Witten的难度甚至超过Toefl。还有一个最重要的原因，那就是Witten的性格似乎不太容易亲近。他回答问题很简短，问两句答三句，跟他老师Gross完全是两个极端，而且当他回答问题时满脸笑容，一说完话马上就严肃下来，搞得我总是以为自己说错了话惹他不耐烦了。整理录音时我才发现，他其实是在很认真地回答问题，而且还有些风趣，结果我们没有听懂，都没有回应，可能他也比较失望。由于还等着他去参加新闻发布会，我也三言两语压缩问题，结果整个对话时间只有十几分钟。一直想知道Witten对人则原理的态度，于是便问他是否相信，他的回答是：“I prefer not, but I don’t know.”没有Gross坚决，但看来还是跟Gross一条战线上的。采访完后来我才听说，Witten的父亲是JHU毕业的博士，曾在JHU任教，而且Witten是Baltimore人，本科曾在JHU读了一年物理，申请转哈福被拒，然后才到一个小学校去学历史的。这样看来，我还是Witten的半个校友呢，呵呵，可惜当时竟然没敢跟他说我要去JHU。Witten确实够牛，本科期间一个人自学物理，还能成功地申请到Princeton的博士，望尘莫及啊。

Hawking
Hawking是我中学时代的偶像，虽然现在已经不是了，但我仍然很敬佩他。由于Hawking的传奇色彩，受到了媒体最广泛的关注，我估计这次来的牛人中，公众唯一认识的就是Hawking同学。人民大会堂报告时，记者们蜂拥到台前拍照的情形真是恐怖，仿佛见到外星人一样。不过“神童Yau”（又是菜菜的天才发明）对记者的训斥更让我印象深刻，一句别扭的粤语普通话“你们比香港记者的素质差多了！”立刻让我脑子里出现那句经典的“Too young, too simple, sometimes naïve！”大会堂的报告很水，没有实质性内容，基本上就是抄写自己书上的话，用语音合成器挨个放，以至于我们怀疑Hawking同学是不是在台子上睡着了。后来那次记者招待会也很有意思，问Hawking的第一个问题是他喜欢中国的什么，他回答：“我喜欢中国文化，中国食品，我最喜欢的是中国女人，她们非常漂亮。”真够坦率☺ 我们当然没有机会采访Hawking，甚至没有机会照相，但我曾在他旁边站着看了很久，当时他就被放在友谊宾馆贵宾楼的大厅里，只有吴忠超和他的护士陪着。他正在准备自己的报告，我走到他身后去，看他通过眼皮眨动控制屏幕上的选词来拼凑自己的文章，大概一分钟只能选三四个词。看着他这样费力地工作，我真的很难想象是一种什么样的毅力，让他不但能够忍受全身瘫痪的折磨，而且还能在那不能动弹的躯体中养着这样一颗思想驰骋的大脑。物理界有些人对Hawking不以为然，认为他只不过是仗着自己的残疾挣点名气。对此我只想说，不懂广义相对论的人没有资格评价Hawking，他在70年代做出的贡献是有目共睹的；至于说现在的名气，无论他现在的物理工作在大多数人看来是否有意义，只要看看他是在什么境况下继续追求着物理的理想，哪怕是哲学的理想，就应该认同这样的名气对于他绝不为过。当然，名气有一个弊端，就是对他的猜想的简单理解和过分迷信造就了一帮民科，但是，在他的影响下，不是同样也鼓舞了一批有志青年投身于物理事业吗？至少我是其中之一。

Pseudo-Greene（这算花絮）
本来还想采访Greene，就是著名的弦论科普书《The Elegant Universe》的作者。我只看过他在李老师书上的照片，还有BBC拍的科普片中他的形象，可惜除了很帅，都不记得长啥样了。在超弦会议上，我们看到一个人身材修长，头发直立，穿着夹克，一幅很酷的样子，以为他就是Greene。他们派我去联系采访，一次我看他跟Strominger一起走，也不知道哪儿来的胆子，隔着老远就冲他大叫：“Professor Greene!……”他回头看了一眼又继续走，后来犹豫了一下又停下来，一脸茫然地望着我，而Strominger却在一旁很开心地笑着。我这时才发现有些不对，补问了一句他是不是Greene，回答显然是否定的，而且他很不服气地说：“Greene比我老得多。”失败，原来是个pseudo-Greene。为了确认真Greene，我们决定听完周三早上Greene的报告再找他，这次总不会错了吧。当那位Greene站在台上时，我们惊讶的发现他竟是个秃头，跟照片上的形象大相径庭。我心里还感慨：这才没过几年啊，怎么老这么快，真是人生沧桑啊~~~~报告完后，我便去找他约采访，他很高兴地答应了，但是最后却来一句：“我想知道你们要找的是不是Greene? 我不是Greene，他有事来不了，我来替他做的报告……”疯了，又是一个Pseudo-Greene！原来Greene同学根本就没有来中国。其实挺对不起这位教授的，当时他热情地说“我是很高兴接受采访的”，结果却让他空欢喜一场，真是太不好意思了。没办法，我不是编辑，决定不了采访谁。

写到这里，可以告一段落了。兴奋了这么久，也该沉静一下了。其实，真要说这次弦论会议给我最大的收获，不是蹭饭，不是照片，不是八卦，而是一句话——“临渊羡鱼，不如退而结网”。弦论会议结束后，这句话就开始在我脑子里反复回响，仿佛到了这时，我的身份才从“记者”回到了物理学生。对三位大师的采访，那些听不懂的报告，N多牛人的光辉事迹，理论物理学家特有的幽默，似乎都凝结成这句话飘荡在我记忆的上空。我迫不及待地想要钻进那些奇妙的理论里去欣赏她的美，也期待着能尽快成为这个可爱的理论物理学家群体中的一员。无论是理论本身的美，是探索理论的生活方式，还是这群探索者们神奇的思想，都深深地吸引着我。但我明白，前途漫漫，我还需要付出足够的努力脚踏实地往前走，需要学习的东西太多了，我应该加快步伐。

另外还有一个意义重大的副产品：我更加明确以后的方向了。2008年，我或许能在CERN再次聆听这群世界上最聪明的人的报告，希望那时我能全部听懂。☺