Updates
--> March 19: New update below on “internal symmetries”.
--> March 2: New lectures notes available.
--> February 17: New literature suggestions below.
--> February 3: Links to the scanned notes are now available below.
--> February 2: New update below!
Purpose of the project
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-Learn the very basics of group theory and Lie algebra. Understand the difference
between finite and infinite groups, discrete and continuous groups, as well as the
meaning of expressions like simple group, subgroup, quotient group, Lie group...
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-You should also understand how group theory is used to describe different areas of
physics. We will focus on the importance of group theory for particle physics.
Literature
- Main literature: “Group Theory: A Physicist’s Survey” by Pierre Ramond
- Another very useful reference that we will look at is “Semi-simple Lie algebras
and their applications” by Robert N. Cahn. An electronic version of the book is freely
available from the author’s website:
http://phyweb.lbl.gov/~rncahn/www/liealgebras/texall.pdf
Along the way I will also give additional references which are useful
for certain aspects of the topic.
Tasks
You will be guided in your work by 3 tasks, in increasing order of complexity:
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(1)Describe the group theoretical background to the concept of “spin” in quantum mechanics.
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(2)Describe the relation between the representation theory of SU(3) and the particles
called quarks and mesons.
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(3)Describe the representation theory for SU(3)xSU(2)xU(1) and its relation to the standard
model of elementary particles.
We’ll go through these in steps, starting with (1). We might not make it all the way to (3),
but that’s ok. The main point is that we learn something and have fun along the way!
Lecture notes
Here are links to my scanned notes for the lectures:
- Notes 1 (Introductory stuff, basics of groups, Lie groups etc.)
- Notes 2 (Basics of Lie algebras, representation theory of sl(2,C))
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-Notes 3 (Representation theory of the Poincaré group)
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-Notes 4 (Semisimple Lie algebras, Chevalley-Serre presentation, roots and weights)
- Notes 5 (The Poincaré algebra, irreducible representations, 1-particle states)
Beware that these notes have not been proof read and are bound to contain typos; use at your own risk.
Summaries, additional literature suggestions and guidance
March 19.
Below follows some comments and literature suggestions aimed to guide you into the
second general topic of your project, namely internal symmetries.
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-So far, we have mainly talked about symmetries of physics which are in one way or another related to
spacetime, the ultimate example being the Poincaré group which acts on the coordinates on spacetime,
thereby inducing transformations of the physical fields. Such symmetries are often said to be external. Internal symmetries, on the other hand, are characterized by the fact that they do not act on spacetime itself, but only
act directly on the fields.
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-Particles in nature are classified according to their transformation properties under both external and external
symmetries. The Coleman-Mandula theorem states roughly that the most general symmetry group of a quantum field
theory is a direct product of the Poincaré group ISO(3,1) and some internal symmetry group G. This implies that particles correspond to irreducible representations of
ISO(3,1) x G
Recalling that irreps of direct sums of groups are given by tensor products between the irreps of the factors,
we deduce that all particle representations can be described in terms of tensor products between irreps of
the Poincaré group and irreps of the internal symmetry group G.
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-As a first illuminating example, one can look at electrodynamics, or Maxwell theory. This describes the dynamics
of a photon, which is represented by a vector field A_\mu. The Lagrangian is given by the square F_{\mu\nu}F^{\mu\nu} of the field strength F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu. Since all Lorentz indices are contracted, the Lagrangian is manifestly invariant under the Lorentz group. But this theory has an additional internal symmetry, corresponding to shifts A_\mu --> A_\mu + \partial_\mu \Lambda, where \Lambda is a scalar. This is the internal
U(1) gauge symmetry of Maxwell theory.
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-Another important example is that of weak isospin. This describes the proton and the neutron as two facets of a
single particle called the nucleon, which transforms in the fundamental representation of SU(2). Note that this is
a different SU(2) from the one that is responsible for the spin of the electron. It turns out that the isospin SU(2)
is not an exact symmetry of Nature, but is broken when electromagnetic interactions are turned on.
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-Another important approximate symmetry is the so called flavor SU(3) symmetry. This is the SU(3)-symmetry
which organizes bound states of two quarks, known as mesons. The eight lightest mesons are classified into 4 irreducible representations of isospin SU(2): pions ([1] = 3-dimensional spin j=1 irrep of SU(2)), kaons ([1/2]), antikaons [1/2] and eta ([0]). Gell-Mann observed that these 8 particles can be organized into the 8-dimensional adjoint representation of SU(3). This is called the flavor SU(3). Note that this is still different from the standard model SU(3), which is called color SU(3)!!
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-The color SU(3) is the exact symmetry of QCD, the theory of strong interactions. Quarks fall in the 3-dimensional fundamental representation of color SU(3).
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-The full internal symmetry group of the standard model of particle physics is
G = SU(3) x SU(2) x U(1)
where SU(3) is the color group of QCD, while SU(2) x U(1) is the symmetry group of the electroweak interactions. Note that these are not the isospin SU(2) or the U(1) symmetry of Maxwell theory. For instance, the U(1) symmetry of Maxwell
theory arises as a certain non-trivial U(1) subgroup of SU(2) x U(1).
- The above discussion makes it very clear that the same underlying group and its representations can appear in very different physical contexts!
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-Now, to some literature where more details can be found. For an overview of the concept of external and internal symmetries I recommend again the survey by Gieres:
http://arxiv.org/abs/hep-th/9712154v1
which you will probably find it easier to cope with this time around. See in particular sections 3.1-3.4.
A very nice overview of the group theory behind the standard model can be found in section 2 of the following review paper
The algebra of Grand Unified Theories, John Baez & John Huerta
John Baez also has a series of very nice expository notes on related stuff, for example:
(note that these are really elaborate homework problems for a course)
For some more details on the relation between color SU(3) and flavor SU(3), see also chapter XVI of the book by Georgi.
February 17.
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-As I already mentioned, the stuff about the Poincaré group that I talked about today
was taken from Chapter 2 of the book
“The Quantum Theory of Fields - Vol I Foundations” by Steven Weinberg.
You can find it in Chalmers main library; it is currently available.
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-In regards to a more precise and mathematical treatment of Lie groups and Lie algebras
I recommend some very good lecture notes by Kirillov, available here:
http://www.math.sunysb.edu/~kirillov/mat552/liegroups.pdf
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-Another source for Lie algebras and representation theory from a more
mathematical perspective is the book:
“Introduction to Lie algebras and representation theory” by J. E. Humphreys
This book is a real classic; it is very good, but also a bit heavy... You can probably
find it in the mathematics library, or you can borrow a copy from me.
February 2.
Below follows some follow-up remarks from the meeting today, and some suggestions
for further reading.
- First, a remark on algebra. So, the mathematical field of “algebra” is huge, and is not really
one field but is a common name for a number of different things. The simplest is perhaps that
which you learn in high school, namely by representing numbers by letters and performing various
“algebraic operations” on them, such as division, multiplication etc.
The object that I was referring to loosely today as “an algebra” should rather be called an algebra over a field.
Here the word “field” refers to a number field, e.g. R, C etc. An algebra A over some field F is a vector
space V over F equipped with the additional structure of a bilinear product, i.e. a map: V * V --> V.
This bilinear product is further required to satisfy various requirements, or axioms, such as distributivity, i.e.
(x+y) * z = x * z + y * z
x * (y+z) = x * y + x * z
where x, y, z are vectors in V, and scalar compatibility, i.e.
(ax) * (by) = ab (x * y)
where a, b are scalars, i.e. valued in the field F. Very often one also imposes the condition that the bilinear
product is associative; the algebra is then referred to as an associative algebra. If for any two elements
x, y in the algebra A, we have x * y = y * x, then A is called a commutative algebra.
So, to conclude, a Lie algebra is an associative algebra where the bilinear product is the Lie bracket [ , ].
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-Now let me mention some places where you can read more about representations of the Lorentz group.
First, as I said during our meeting today, chapter 11.1 in Ramond is a pretty good starting point.
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-Another excellent resource is the book “Quantum Field Theory” by Mark Srednicki, for a which a pre-publication
draft is freely available from the author’s homepage:
http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf
Don’t go ahead and read the entire book; what you are after is found in chapter 2 (basics of the Lorentz group)
and chapter 33 (representation theory of the Lorentz group). These chapters can be read independently of the
rest of the book. Eventually chapter 34 will also be relevant but I suggest you ignore that for now.
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-Finally, for a very nice and clear discussion of the isomorphism SO(1,3) = SL(2,C)/Z_2 I recommend the
following lectures by Matteo Bertolini:
http://people.sissa.it/~bertmat/susycourse.pdf
Now, again, these notes are mainly about stuff that doesn’t concern you so don’t be scared off by the title.
The stuff you are interested in is contained in chapter 2.1, so restrict your attention to those few pages.
Let me emphasize that a lot of the things that you will read in the above references is stuff that I did not
have time to cover in my lecture today, but which I had planned to explain before you delve into the
literature. So, if it feels a little heavy, don’t get frustrated; I will try to help you out next time, and maybe
give another lecture on this topic if you feel that you need it.
January 26.
Today we started to discuss Lie algebras, with emphasis on the example of su(2). We also discussed
the complex Lie algebra sl(2,C), which corresponds to a complexification of su(2). From this point
of view, su(2) is said to be a compact real form of sl(2,C). I briefly mentioned another real form of
sl(2,C), namely sl(2,R) which is obtained by simply restricting to linear combinations over the real
numbers. This second real form is non-compact.
I also started to talk about representation theory of Lie algebras. Roughly speaking, representation theory
is all about representing the algebra in terms of matrices acting on a vector space. I showed the explicit
example of the fundamental representation of sl(2,C), which corresponds to the action of 2x2 complex
traceless matrices acting on the complex 2-dimensional vector space C^2.
Some further remarks, and suggested reading:
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-I showed how to describe the fundamental representation of sl(2,C) in terms of the action on the symmetric
algebra S^{1}(C^2), generated by the product v_j=\epsilon_1^{1-j}\epsilon_2^{j}, with j=0,1. This showed
explicitly that the 2-dimensional representation is described in terms of two weights: v_0 and v_1. Aside from
the trivial representation this is the smallest representation of su(2). I suggest that you analyze in a similar
way the representations obtained by acting on the k:th symmetric algebra S^{k}(C^2), which is generated by
v_j = \frac{1}{(k-j)! j!} \epsilon_1^{k-j} \epsilon_2^{j}, j=0,1,...,k
In this way you will be able to find all higher dimensional representations of sl(2,C)! The representation
corresponding to k=2 is also especially important; it is called the adjoint representation. What is its
dimension?
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-For further reading about su(2) and its representation theory, an excellent resource is chapter I in the
book by Cahn, available here:
http://phyweb.lbl.gov/~rncahn/www/liealgebras/texall.pdf
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-Chapters 5.1 - 5.3 in Ramond are also very good when it comes to su(2) and its relation to angular
momentum and spin.
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-Finally, the old book by by Georgi, “Lie algebras in Particle Physics”, is really good. See chapter III for
a detailed discussion of su(2). You can probably find it in the library; if not, let me know and I will copy
parts of it for you.
January 22.
Here follows some pointers to things that are worth looking closer into.
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-I mentioned that Lie groups can also be viewed topologically as manifolds. For example, the
group U(1) is equivalent to a circle S^1. This was done by thinking of an element of U(1) as an
operator that acts on complex numbers by multiplication by an exponential e^{i \theta}. The
real variable \theta corresponds to a rotation angle and may therefore be viewed as a coordinate
on S^1. Now, here is something to think about: What is the difference topologically between the
following two exponentials:
e^{i\theta}
e^{\theta}
If the first corresponds to S^1, then what is the second? Is it compact?
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-Remember the example of the group SU(2) that I showed during the last meeting. First verify that
the matrix U that I wrote down satisfies all the axioms of a group. Then look up your old QM mechanics
books and find the form of the 3 Pauli matrices \sigma_i, i=1,2,3. Verify that the matrix U’ defined
as follows:
U’ = \exp\{i a_1 \sigma_1 + i a_2 \sigma_2 + i a_3 \sigma_3\},
where a_1, a_2, a_3 are arbitrary real numbers, also satisfies the same group axioms. In the above formula
you have to use the matrix exponential; if you don’t know how that works, see for instance MathWorld:
http://mathworld.wolfram.com/MatrixExponential.html
This is a different way of looking at the group SU(2), namely through exponentiation of its Lie algebra,
denoted su(2). So, the matrices in the exponential correspond to elements of the Lie algebra, while the full exponentiated matrix U’ is an element of the group SU(2). This relation will be extremely important for your
project, and will be discussed in more detail during our next meeting. As an appetizer, see the next point
below.
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-For any two matrices X and Y, define the following operation:
[X, Y] = X Y - Y X
The bracket [ , ] is called the commutator. Work out the commutators between the Pauli matrices, i.e
compute:
[\sigma_1, \sigma_2]
[\sigma_1, \sigma_3]
[\sigma_2, \sigma_3]
What can kind of conclusions can you draw from the result? Keep the property of “closure” in the
group axioms in mind when analyzing the result.
January 19.
Thanks everyone for a nice first meeting! In the coming weeks you should work up some
feeling for the concept of a group, and play with some of the examples I gave, e.g. unitary
groups SU(n), orthogonal groups SO(n) etc.
I suggest to focus on the small examples, i.e. study U(1), SU(2), SO(2), SO(3).
As we discussed the first project goal is to understand the group SU(2) from the point of
view of quantum mechanics.
Before we get there we must develop some more tools for how to look at groups, such as Lie algebras,
which I will introduce during the next meeting. It would also be good if you review some basic
stuff about spin in quantum mechanics.
When it comes to literature, I would recommend reading the following parts of the book by Ramond:
2.0; 2.1; 2.2 (only pages 6-8); 2.4.1
Another very useful source is the review article “An elementary Introduction to groups and representation”¨
by Brian Hall, which is freely available at this link:
http://arxiv.org/pdf/math-ph/0005032v1
Here I suggest you read the following chapters
1; 2.1; 2.2 (until page 14).
Note that chapter 3 in the article by Hall introduces the concept of “matrix exponential” which is
what I alluded to towards the end of the meeting; this is the basic relation between Lie groups and
Lie algebras: a Lie group is like the exponential of a Lie algebra. I will go through this in some
detail next time, but you can glance at the beginning of that chapter beforehand if you
have time.
Also don’t forget, Wikipedia is in fact not too bad of a source! Especially in mathematics, many
of the articles in Wikipedia are pretty good. So don’t hesitate to try for instance the entries on
“Groups” or “Spin” to get some more inspiration and guidance.
January 5.
Before the first meeting it would be good if you could get a basic idea of what a
symmetry is, and why it is interesting for physics. For this, you can take a look at
the following references:
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-Chapter 1 (page 1-3) in the book by Ramond (see below).
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-Chapter 2 in “About Symmetries in Physics” by F. Gieres, available at this link:
http://arxiv.org/pdf/hep-th/9712154v1
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-Chapter 52-1, 52-2, 52-3 in “The Feynman Lectures on Physics” vol 3.