Download Syllabus of math and physics doc

A pedagogic picture is worth a thousand theorems
This brief pedagogical outline of key ideas, text books, and literature concerns pinning down the minimal
set of physics ideas and mathematical tools required to understand the Standard Model and its various
extensions through a generalized grammar of fundamental physics and mathematical constructs
compelled on us by our experimental corpus. There is also available a longer, far more detailed,
historically based follow-on “syllabus” on these topics. As in this outline, the follow-on document also
lists the good text books and key references from the literature, the key ideas and methods, several
logical approaches to pursue these readings, but also the innate dead ends and fundamental impasses
inherent in physics and mathematics that proscribe our wayward pursuit of absolute, ideal Truth as
Plato conceived it. The intent of these documents is to provide a guided path to a more mature
theoretician’s grammar.
What is meant by a more mature theoretician’s grammar? Consider a lifeguard, sitting at his watch post
at the beach, who detects a drowning person off to the side in the water, he must make an optimal
decision of how much to run on the sand, a fast process, and how much to swim in the water, a slow
process. He has infinitely many paths to chose from, but only one choice, or function that is, is optimal
in minimizing the intercept time. An application of simple differential calculus renders us this optimal
function, giving us a particular solution, which, by the way, also describes the refraction of light between
two different media, e.g., between air and water, where the speed of light, like the life guard, is faster in
the former and slower in the latter. Now consider another seemingly unrelated optimization problem.
A cable suspended between two poles of differing heights assumes a unique shape to minimize its
potential energy from an infinite set of functions. Find the curve satisfying the stated condition. When
you find this solution, you will have another particular solution to this other optimization problem.
Proceeding in this ad hoc way, you will collect more and more special, individual, particular optimization
solutions to particular optimization problems. In this sense, when there is no rhyme or reason between
optimization problems and their solutions, your optimization grammar is immature. Only after you
develop the calculus of variations, unifying otherwise disparate optimization problems, will you have
acquired a more mature, more unified grammar capable of treating a large, general class of optimization
problems including many of those posed by physics. This calculus of variations is part of a more general
theoretician’s grammar.
Another example of this kind of maturation is exemplified by the work of Bernhard Riemann shortly
after the discovery of the first few non-Euclidean geometries, e.g., Bolyai-Lobachevskian hyperbolic
geometry early in the 19th century, and elliptical geometry (Saccheri). He quickly generalized the small
set of known geometries into an infinity of geometries through a small set of unifying concepts today
falling under the rubric of Riemannian geometry. Interestingly, Bolyai mentioned in his work that it is
not possible to decide through mathematical reasoning alone if the geometry of the physical universe is
Euclidean or non-Euclidean; this, he stated, is a task for the physical sciences, and indeed this is what the
physicist Albert Einstein succeeded at advancing beyond the Euclidean worldview when he developed
general relativity to describe astrophysical observations in the framework of curved spacetime. As was
the case with Riemann quickly generalizing a handful of recently developed geometries into infinitely
many more geometries, it did not take long after the announcement of general relativity for people to
cook up endless many more “geometric” theories beginning with the original Kaluza-Klein theory, a
clever five-dimensional curved spacetime construct devised to unify gravity and electromagnetism.
Today this type of theorizing continues unabated, seemingly pell-mell; witness string theory and WeylDirac theory to name but two.
Fortunately this growing bulk of theoretical constructs still remains bound together by a relatively small
thread of key physics and mathematical ideas and methods. Unfortunately, with physicists and
mathematicians having gone hog wild, we are losing this thread. The swell of theoretical
particularizations issuing from this small set of ideas and methods overwhelm us. Most doctoral level
physicists, for example, have learned quite a lot about special functions, but they have probably only
picked up a little, disparate knowledge about Lie algebras and Lie groups. They do not realize—certainly
most of them don’t have to—that these two areas are actually very tightly linked together by the
powerful theorems of Sophus Lie, and their inverse theorems. Special functions, Lie groups, Lie
algebras, commutators, and much more that goes into the particles and fields of the Standard Model
and beyond actually go hand in hand in a relatively simple and unified grammar. So where are we
heading? To the sources of this small unified grammar, at least those that worked for me.
Mathematics/Physics block I (Least Action):
1. Mathematics: Calculus of variations
a. Calculus of Variations, L. D. Elsgolc, Dover Publications. Originally written in Russian,
this book was first published in English in 1961. Using clear notation, Elsgolc develops
the calculus of variations side-by-side with ordinary differential calculus. Starting with a
challenge to Isaac Newton, this calculus originated from extremization problems in
physics, e.g., least time, maximum entropy, least action. The Standard Model, general
relativity, string theories, to name but a few, are expressible in terms of least action.
Ideally this book should be read before graduate work in physics, around the time junior
level mechanics has been covered.
b. Variational Principles in Dynamics and Quantum Theory, W. Yourgrau and S.
Mandelstam, Dover Publications. Tracing the evolution of the concept of the innate
economy of nature (least action) from the Greeks through to Fermat’s principle of least
time and Maupertuis’ le principe de la moindre quantité d’action (least action) in 1744,
this book traces the development of the equations of Lagrange, Hamilton, HamiltonJacobi, etc., in classical mechanics and electrodynamics to the various historical paths to
quantum physics including those of Feynman and Schwinger. This book should probably
be read concurrently during the first year of graduate school, if not at the completion of
the undergraduate degree. Without these readings, or similar, the use of the principle
of least action is little more than a physics gimmick.
2. Physics: Classical Mechanics
a. Newtonian Dynamics, R. Baierlien, McGraw-Hill Book Company. This is a beautifully
written, concise presentation of undergraduate classical mechanics. Baierlein covers
everything essential for further graduate work, including perturbation theory, nonlinear
oscillators, and Lagrangian and Hamiltonian mechanics. Read it shortly after the above
two texts on the calculus of variations.
b. Classical Mechanics, H. Goldstein, Addison-Wesley Publishing Company. Incorporating
all of the above readings, this book expands on Baierlein and generalizes the grammar
of physics underlying relativity and quantum field theories. On pages 491 and 492 of
the second edition, Goldstein describes how close Hamilton came in 1834 to the wave
equation of Schrödinger and de Broglie of 1926. In a more direct manner than W.
Yourgrau and S. Mandelstam, Goldstein connects how various least action formulations
of classical mechanics led to various formulations of quantum mechanics.
c. Quantum Mechanics, Schaum’s Outlines. This is a pretty good, self-contained
introductory text on quantum mechanics. It contains many little typos, but fixing them
is very educational and reassuring.
d. Basic Quantum Mechanics, K. Ziock, 1969. This book may be difficult to acquire, and the
foundations you need in quantum mechanics are well covered by the above Schaum’s
Outlines text. However, Ziock gives a very nice “lowbrow” approach to positronium and
covers partial wave scattering in more depth than usual.
Mathematics/Physics block II (Classical and Quantum Fields):
Whereas the first block recommends the mathematics readings prior to, or concurrent with the
physics readings, this block goes the other way around.
1. Physics:
a. Introduction to Electrodynamics, D. Griffiths (a classical field theory). A lot of the
material in this text, as in any electrodynamics text for physicists, deals with
electrodynamics in media and other areas which seem unrelated to QED. The
physical intuition developed, and the mathematical skills learned from reading the
entire text are invaluable beyond 4-potential formulations of radiation fields in the
usual gauges. This goes for, “Classical Electrodynamics,” 2nd ed., J. D. Jackson, Wiley.
Jackson will give you a work out in mathematical physics, much of it related to
special functions.
b. Quantum Mechanics, Schaum’s Outlines—see block I above.
c. (Strongly recommended) Modern Quantum Mechanics, J. J. Sakurai. Group theory
begins to enter the picture in the study of permutation symmetry and Young
Tableaux.
d. (Also recommended) Quantum Mechanics, C. Cohen-Tannoudji, B. Diu, and F. Laloë.
This text is far more detailed than the Schaum’s text, providing many more
applications and mathematical underpinnings to quantum mechanics.
e. Introduction To Elementary Particles, D. Griffiths. This is a great introductory
textbook to take you from experimental particle physics to hands on practice with
low order Feynman diagrams in electrodynamics, weak, and strong nuclear
interactions.
Note—From all of the particulars of a. through e., one begins to get the feeling that there should be a
more general, overarching grammar to physics, otherwise physics begins to seem more like a set of
unexplained voodoo prescriptions.
2. Quantum Electrodynamics, W Greiner and J Reinhardt, 3rd ed., Springer. Another
pedagogical text, this book presents, detail by detail, QED the old fashioned way, the way
people, including Feynman, first developed QED. I believe this is an essential read if you
want to understand QFT, including some of the early issues with divergences.
3. Quantum Field Theory, 2nd ed., L. H. Ryder—This is a very well written introductory text on
QFT up through introductory supersymmetry (SUSY). Ryder surveys relativistic wave
equations and Lagrangian methods, the quantum theory of scalar and spinor fields, and
then the guage fields. In chapter 3, Ryder carefully explains the principle of minimal
coupling by requiring local invariance of, firstly, the Lagrangian for the complex scalar field
before moving on treat Yang-Mills fields. Also in chapter 3, Ryder points out the parallels
originating from parallel transport in general relativity in the framework of a manifold, and
parallel transport in the algebra of quantum fields in the framework of continuous Lie
groups. Is this parallel an accident? Is there a deeper level of grammar? The answers to
these questions are not easily found in popular QFT texts, nor in books on the mathematics
of differential geometry, group theory, or algebraic topology. In fact, most books
purportedly treating these areas of mathematics for the physicist also fail to deliver the
deeper grammar.
Answers from history:
4. Lie Groups, Lie Algebras, and Some of Their Applications, R. Gilmore, Dover, was originally
published in 1974. In the same sense that the two books on the calculus of variations,
Elsgolc 1961, and W. Yourgrau and S. Mandelstam 1968 provide the fundamental least
action underpinnings of classical and quantum physics, Gilmore provides the foundations to
the prescriptions in standard QFT books. The physicist’s book, “Lie Algebras In Particle
Physics, From Isospin to Unified Theories,” 2nd ed., H. Georgi makes a ton more sense after
Gilmore. If I had stumbled across Gilmore sooner, I probably wouldn’t have spent years
pouring over a ton of pure mathematics texts, never quite understanding how to bridge
pure algebraic topology back to quantum fields. The following books should probably be
read in reverse order from the way I found and read them. They are:
5. Groups, Representations And Physics, 2nd ed., H. F. Jones, Institute of Physics Publishing.
This was the first book that took me a long way into both understanding and being able to
apply group theoretic methods to quantum mechanics and quantum fields. After working
through Jones, however, I still felt there was a deeper plane of truth, or a better grammar if
you will. There was still too much “genius”, too much particularization. Before reading
Jones, I recommend as a minimal prerequisite an introductory text on group theory at the
Schaum’s outline level. I personally like, “Modern Algebra, An Introduction”, 2nd ed., J. R.
Durbin, Wiley. You need only cover the material up through group theory. Take with you
the notion of a normal subgroup when you proceed to read Gilmore.
6. Lie Algebras In Particle Physics, From Isospin to Unified Theories, 2nd. ed., H. Georgi,
Frontiers in Physics. I couldn’t have read this book without first having read and worked
through Jones. Georgi was difficult for me, but when I cracked it, I began to feel like I was
starting to understand the physicist instead of the mathematician. Ideally, read the first
four chapters of R. Gilmore’s text first. The 5th chapter covers applications to areas typically
presented in graduate physics coursework. Then read Jones, then Georgi. There will be
much less for you to have to accept by fiat.
History: (quoted from Gilmore’s text) “This study of simultaneous differential equations led Lie to
investigate continuous transformation groups, from which the theory of Lie groups emerged. Lie groups
have been studied so extensively in their own right that their connection with partial differential
equations is often overlooked and forgotten[—a bad thing!]. So it is sometimes quite a shock to learn
that many of the differential equations of mathematical physics are expressions of the Casimir invariant
of some Lie group in a particular representation and moreover, that all the standard special functions of
mathematical physics are simply related to matrix elements in the representation of a few of the
simplest Lie groups. It is safe to say that Lie group theory provides a unifying viewpoint for the study of
all the special functions and all their properties.” I suggest you put aside your modern books on
mathematical physics, go back in time and read:
7. A Course of Modern Analysis, E. T. Whittaker, 4th ed., poorly reprinted by Cambridge University Press.
The first edition dates back to 1902. Once you understand the three theorems of Lie, and their inverses
in R. Gilmore’s text, you will come to appreciate deep mathematical physics and analysis as a unified
whole with the help of 4th ed. Whittaker. There are a handful of very expensive books directly tying Lie’s
work directly to mathematical physics. I recently found a good book online, “The Lie theory approach to
special functions,” W. Miller, University of Minnesota, 2010.
I must admit that the material in Gilmore after chapter 5, on the general structure theory of Lie groups,
is difficult going. Ironically, once you understand the first 4 chapters of Gilmore, you probably will get a
lot of help from Jones and from Georgi to work through the remainder of Gilmore, especially from
chapter 7 onwards, where Gilmore starts trotting out of the big guns to demolish the structure of an
algebra into its irreducible components, namely:
(1) The adjoint or regular representation, equivalent to the structure constants.
(2) The use of the secular equation and its roots, which lead to further information about the
structure of a group. The information is summarized in the first criterion of solvability.
(3) The use of a metric (Cartan-Killing form) on the vector space associated with the Lie algebra.
The information is summarized in the second criterion of solvability.
(4) The folding of the first two criteria in the Cartan criterion.
(5) The exploitation of the root and metric concepts to give a canonical structure to the
commutation relations of the regular representation of semisimple algebras.
I’m hoping that “Group theory and physics,” S. Stemberg, Cambridge, which I’ve just acquired, will also
provide more help with the above toolset. QFT literature is rife with results derived from the above big
guns.
At this point you now have a path to the underpinnings of two major chunks of the mature grammar of
modern QFT theorizing: the principle of least action, and tools for studying the structure of Lie groups
and Lie algebras (and particle spectra). Still missing is a deeper, more general understanding of the
principle of minimal coupling resulting from the requirement of local invariance of Yang-Mills Lagrangian
densities. To maintain invariance under local transformations—the steps being presented very clearly in
Ryder—requires the addition of extra terms. In the case of the complex scalar field, the extra terms
correspond to electrodynamics expressed in the potential formulation—hence why I suggest you
understand electrodynamics at least at the level of Griffiths. Your radar should be on, looking for a
more mature grammar to the principle of minimal coupling, especially after reviewing basic Kaluza-Klein
theory. Kaluza-Klein theory was an early attempt to unify electromagnetism with gravity, and it goes
like this: An extra fifth spatial dimension can be understood to be the circle group U(1) as
electromagnetism. Electromagnetism can be formulated as a guage theory on a fiber bundle, namely
the circle bundle with guage group U(1). Once this geometrical interpretation is understood, it is
relatively straightforward to replace U(1) by a general Lie group. Such generalizations are often called
Yang–Mills theories in flat spacetime, as opposed to curved spacetime in Kaluza-Klein theory. Note that
Kaluza-Klein theory (in any (pseudo-)Riemannian manifold, even a supersymmetric manifold) can be
generalized beyond 4 spatial dimensions. So is there a good book out there to view this approach in a
more unified way? Yes.
8. Geometry, Topology and Physics, M. Nakahara, Graduate Student Series in Physics, chapter 9. I read
the first four chapters before skipping to chapter 9. To me, chapter 9 seems fairly self-contained.
However, by the time I happened upon Nakahara, my background in mathematics was far beyond my 36
hour masters degree in pure mathematics, and I also knew what the goal was beforehand thanks to
another book, namely, “Topology, Geometry, and Gauge Fields, Foundations,” G. L. Naber. Naber sucks.
Naber is part of the reason I overdid mathematics, but Naber put the goal, the mature grammar in easy
to understand words. “…These Lie algebra-valued 1-forms…are called connections on the bundle (or, in
the physics literature, guage potentials).” The guage fields in QFTs are connections over principle
bundles. If anything, you have to read Naber’s chapter 0 for motivation, and I’ve reluctantly come to
appreciate all of the mathematics I studied trying to get through Naber, especially differential forms. At
this point I began to see that there is probably no end to physics theoreticians cooking up hypothetical
universes that don’t necessarily have to have anything to do with what we perceive to be our universe.
Even theorizing over our own apparent universe is probably unlimited. The creative degrees of freedom
to cook up mathematical universes that behave at low energy like what we observe seem infinite. As
our experimental knowledge grows, we exile certain theories of physics into the realm of mathematics,
only to quickly create a whole new frontier of endless physics-based possible universes. This realization
took the wind out of my pursuing my belief in Einstein’s dream of a final theory. By the way, I found a
pretty tidy review of differential forms online, namely, “Introduction to differential forms,” D. Arapua,
2009. I was never satisfied by any of the physics books purportedly written to teach forms.
Summary of mature grammar memes so far: (physics : mathematics)

Principle of least action : Calculus of Variations.

Principle of minimal coupling : Connections/principle bundles, forms.

Particles and Fields; SUSY : Lie’s theorems, their inverse theorems, and the classification
problem; graded algebras (Ryder)

World lines  World Sheets: Strings (A First Course in String Theory, B. Zweibach, Cambridge
University Press.
I do not touch on the concept of symmetry breaking to give mass to Yang-Mills field theories. Entry level
QFT texts do a reasonably good job treating this. One area I’m still missing is that dealing with effective
Lagrangians and renormalization theory. The following two articles were strongly recommended to me
as good primers. The methods of the renormalization group and Effective Lagrangians. Read: Effective
Field Theories, A. V. Manohar, arXiv:hep-ph9606222v1 4 June 1996. Effective Field Theory, A. Pitch
arXiv:hep-ph9806303v1 3 June 1998.
All of the above memes wrapped up in a Lagrangian expressing least action, minimal coupling,
geometry, algebra, topology, and algebraic topology, is how we’ve come to think about our universe,
and hypothetical universes. It’s an entry level, minimally mature grammar to muse about universes and
existences in the sense of Newton: “I do not know what I may appear to the world, but to myself I seem
to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a
smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered
before me.”
PS—The mathematicians over do it and the physicists under do it. How much pure math do you
have to study to start R. Gilmore? Point set topology begins with abstracting the topological
properties of the real line, e.g., Hausdorff separability. This field then proceeds on far past what a
physicist needs to get started. The connection between the concept of closed and bounded to many
theorems, like the Bolzano-Weierstrass theorem is also important, but it leads to foundations
problems in mathematics. I cover the history of these sticky issues with the Zermelo-Frankel
axioms, like the Banach-Tarski paradox, and the limitations we’ve discovered to be inherent in
mathematics as discovered by Cohen and Gödel in the follow-on work. Naturally, I also discuss
some of the inherent limitations in physics that I’ve come across. As for an algebra background to
Gilmore, all the background I found necessary from my training in pure mathematics was little more
than the concept of a normal subgroup. Only once Gilmore begins to develop the structure of Lie
groups does all of the dry crap on towers in a standard graduate text on algebra, such as Lang, start
to make sense beyond symbol manipulation. In the end, studying mathematics for its own sake is
great, but it can sure slow you down if you’re interested in physics.
The audience: The intended audience spans across people with a varied, but minimal, entry level
background. The bare bones entry level is for those with no less than a year of differential and integral
calculus and/or a year of calculus based physics. The follow-on work both outlines and motivates what
mathematics and physics areas you shall require in order to proceed, the standard types of books, and
the corresponding courses found at colleges and universities, not that you can’t study on your own.
That this follow-on presents the underlying key historical motivations and interlinking of the various
subject matters makes it worth a look at. The ideal minimal entry level is a good undergraduate degree
in physics with at least an introductory course in modern algebra (to the point of understanding what a
normal subgroup is), and at least enough real analysis to understand very basic point set topology up to
what being Hausdorff means. A chemist, mathematician, or engineer should familiarize himself or
herself with classical mechanics at the junior level up to the concept of Lagrangian and Hamiltonian
mechanics, e.g., Ralph Baierlein, “Newtonian Dynamics”, 1983. He or she should also review one
semester’s worth of electricity and magnetism at the junior level, e.g., Griffiths, “Introduction to
Electrodynamics”, and at least one semester of quantum mechanics at the junior or senior level. The
Schaum’s Outline in Quantum mechanics is good enough, especially if you bother to fix its many minor
typos—a great exercise. Some members of the intended audience may already have possession of all of
the requirements, but haven’t been shown the connections because these are being lost to history.
The apology: This outline and the follow-on work will only guide you to a connected path to more
mature grammar for studying QFTs up through introductory level graduate and introductory postgraduate texts—more than enough to ponder over universes. I wrote this stuff because I learned black
magic and voodoo in school when what I sought was understanding. Instead of understanding, I learned
for example of a prescription, or a spell if you will, for turning classical physics quantities like total
energy and total angular momentum into quantum mechanical operators, leading to quantum
mechanical differential equations with quantized eigenstates. I had to press on and prepare for
qualifying exams.
It was during my years working as a nuclear weapons physicist at Los Alamos National Laboratory that I
had both the time and reason to finally put together a unified mapping of applied mathematics and
physics foundations. Unless you’re happy to only run weapons codes written by others, and otherwise
ape intelligence with technical babble read off PowerPoint slides, nuclear weapons physics forces you to
go back to every core area covered in a graduate physics program, and then some. Certainly you have
to pin down thermodynamics, statistical physics, electricity and magnetism, mathematical methods of
physics, nuclear physics, numerical simulation of complex, coupled systems such as stars, and to a
degree some portions of quantum mechanics for deriving a few, limited equations of state, a little
special relativity for computing corrections to certain classical results, and a skill at unifying a wide
variety of as yet poorly understood, fleeting, typically unstable phenomenology. The digging and
battling kindles an intuition between theory and experimental “reality” which a good astrophysicist
might develop, but which a so-called financial physicist might not, excepting, of course, the deep
mathematical analysis underpinnings of Monte Carlo transport methods. The follow-on work attempts
to provide a list of what core knowledge and underlying references are important for developing good
physics intuitions no matter what you do for a living. This first part of my journey took around five years
after I finished my formal schooling. The stuff above on the core of physics and mathematics
underpinning our theoretical toolset regarding real and hypothetical particles, fields and universes with
a mature grammar, took an additional five years of my time. In all honesty, the process wasn’t really
that linear.