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.

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[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)

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

Micheal Tinkham, Group Theory and Quantum Mechanics

概率问题

一个运动员在药检时结果呈阳性（即怀疑服用违禁药品）。药品检验器材的厂方提供的信息是：在阳性结果里有1%的失误率，在阴性结果里有20%的失误率。在 已经受检的20000名运动员中，有300人结果呈阳性。请问这位运动员确实服用了违禁药品的几率是多大？（提示：不是99%，我也搞不懂为什么）。

Update:

这是一个条件概率问题。关键在如何理解厂方信息。所谓“阳性结果里有1%的失误率”是什么意思？是不是说一个人如果被检验成阳性，他吃药的几率就是99%？从字面理解好像是这样，但仔细想想，这个数据不是厂方可以提供的，因为要想得到这个数据，我们需要一堆事先并不知道是否吃药的样本人群，让他们去测验，拿到结果后再跟他们是否吃药的真实情况比较，但是否吃药显然跟整个人群的吃药率有关，因此这个量不是能够独立测量的。但另一个量是只跟仪器有关，跟人群吃药率无关的，那就是一个事先确定吃药或没吃药的人的检查结果，厂方正是用这个方法来定义和测量仪器灵敏度的。用数学语言说，如果用P(A|B)表示条件B成立的情况下A的概率，那么厂方提供的是P(阳性|未吃药）=1%，P(阴性|吃药)=20%，而我们想要求的是P(吃药|阳性）。最后答案是32%。

Diagonalize the mass matrix

When we diagonalize the mass matrix, we use unitary bilinear transformation. Why unitary? Because we want no cross terms in the kinetic part, which can only be guaranteed by unitary transformation. Why bilinear? Because the left hand particles and the right hand particles are in principle independent, and we should not expect them to follow the same transformation. We know that the usual linear diagonalization is unique (if exists) from the secular equation. Now we can prove that the bilinear unitary diagonalization is unique in the terms of molds, but each eigenvalue can take an arbitrary phase. In the case of mass matrix, we require all the eigenvalues to be positive real numbers, so the diagonalization of the mass matrix is unique.