Syllabus of Math and Physics Doc

7/26/2019 Syllabus of Math and Physics Doc

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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 byour experimental corpus. There is also available a longer, far more detailed,

historically based followon !syllabus" on these topics. #s 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 $lato

conceived it. The intent of these documents is to provide a guided path to a more

mature theoretician%s grammar.

&hat is meant by a more mature theoretician%s grammar' (onsider a lifeguard,

sitting at his watch post at the beach, who detects a drowning person o) to the sidein 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. *e has in+nitely

many paths to chose from, but only one choice, or function that is, is optimal in

minimizing the intercept time. #n application of simple di)erential calculus renders

us this optimal function, giving us a particular solution, which, by the way, also

describes the refraction of light between two di)erent 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. ow consider another seemingly unrelated optimization problem. #

cable suspended between two poles of di)ering heights assumes a unique shape to

minimize its potential energy from an in+nite set of functions. -ind the curve

satisfying the stated condition. &hen you +nd this solution, you will have another

particular solution to this other optimization problem. $roceeding in this ad hoc

way, you will collect more and more special, individual, particular optimization

solutions to particular optimization problems. n this sense, when there is no rhyme

or reason between optimization problems and their solutions, your optimization

grammar is immature. /nly after you develop the calculus of variations, unifying

otherwise disparate optimization problems, will you have acquired a more mature,

more uni+ed 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.

#nother example of this kind of maturation is exempli+ed by the work of 0ernhard

1iemann shortly after the discovery of the +rst few non2uclidean geometries, e.g.,

0olyai3obachevskian hyperbolic geometry early in the 45thcentury, and elliptical

geometry 6Saccheri7. *e quickly generalized the small set of known geometries into

an in+nity of geometries through a small set of unifying concepts today falling under

the rubric of 1iemannian geometry. nterestingly, 0olyai mentioned in his work that

it is not possible to decide through mathematical reasoning alone if the geometry of


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the physical universe is 2uclidean or non2uclidean8 this, he stated, is a task for the

physical sciences, and indeed this is what the physicist #lbert 2instein succeeded at

advancing beyond the 2uclidean worldview when he developed general relativity to

describe astrophysical observations in the framework of curved spacetime. #s was

the case with 1iemann quickly generalizing a handful of recently developed

geometries into in+nitely many more geometries, it did not take long after theannouncement of general relativity for people to cook up endless many more

!geometric" theories beginning with the original 9aluza9lein theory, a clever +ve

dimensional curved spacetime construct devised to unify gravity and

electromagnetism. Today this type of theorizing continues unabated, seemingly

pellmell8 witness string theory and &eyl:irac theory to name but two.

-ortunately this growing bulk of theoretical constructs still remains bound together

by a relatively small thread of key physics and mathematical ideas and methods.

;nfortunately, 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 3ie algebras and 3ie groups.

They do not realize


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historical paths to quantum physics including those of -eynman andSchwinger. This book should probably be read concurrently during the+rst year of graduate school, if not at the completion of theundergraduate degree. &ithout these readings, or similar, the use ofthe principle of least action is little more than a physics gimmick.

2. $hysics> (lassical Mechanics

a. ewtonian :ynamics, 1. 0aierlien, McDraw*ill 0ook (ompany. This isa beautifully written, concise presentation of undergraduate classicalmechanics. 0aierlein covers everything essential for further graduatework, including perturbation theory, nonlinear oscillators, and3agrangian and *amiltonian mechanics. 1ead it shortly after theabove two texts on the calculus of variations.

b. (lassical Mechanics, *. Doldstein, #ddison&esley $ublishing (ompany.ncorporating all of the above readings, this book expands on 0aierleinand generalizes the grammar of physics underlying relativity andquantum +eld theories. /n pages F54 and F5H of the second edition,Doldstein describes how close *amilton came in 4IJF to the waveequation of SchrKdinger and de 0roglie of 45H@. n a more direct

manner than &. Courgrau and S. Mandelstam, Doldstein connects howvarious least action formulations of classical mechanics led to variousformulations of quantum mechanics.

c. Buantum Mechanics, Schaum%s /utlines. This is a pretty good, selfcontained introductory text on quantum mechanics. t contains manylittle typos, but +xing them is very educational and reassuring.

d. 0asic Buantum Mechanics, 9. Liock, 45@5. This book may be dicultto acquire, and the foundations you need in quantum mechanics arewell covered by the above Schaum%s /utlines text. *owever, Liockgives a very nice !lowbrow" approach to positronium and covers partialwave scattering in more depth than usual.

Mathematics=$hysics block 6(lassical and Buantum -ields7>

&hereas the +rst block recommends the mathematics readings prior to, or

concurrent with the physics readings, this block goes the other way around.

1. $hysics>a. ntroduction to 2lectrodynamics, :. Driths 6a classical +eld

theory7. # lot of the material in this text, as in any electrodynamicstext for physicists, deals with electrodynamics in media and otherareas which seem unrelated to B2:. The physical intuitiondeveloped, and the mathematical skills learned from reading theentire text are invaluable beyond Fpotential formulations of

radiation +elds in the usual gauges. This goes for, !(lassical2lectrodynamics," Hnded., G. :. Gackson, &iley. Gackson will give youa work out in mathematical physics, much of it related to specialfunctions.

b. Buantum Mechanics, Schaum%s /utlines


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d. 6#lso recommended7 Buantum Mechanics, (. (ohenTannoudAi, 0.:iu, and -. 3aloN. This text is far more detailed than the Schaum%stext, providing many more applications and mathematicalunderpinnings to quantum mechanics.

e. ntroduction To 2lementary $articles, :. Driths. This is a greatintroductory textbook to take you from experimental particle

physics to hands on practice with low order -eynman diagrams inelectrodynamics, weak, and strong nuclear interactions.

ote


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5. Droups, 1epresentations #nd $hysics, Hnded., *. -. Gones, nstitute of$hysics $ublishing. This was the +rst book that took me a long way intoboth understanding and being able to apply group theoretic methods toquantum mechanics and quantum +elds. #fter working through Gones,however, still felt there was a deeper plane of truth, or a better grammarif you will. There was still too much !genius", too much particularization.

0efore reading Gones, recommend as a minimal prerequisite anintroductory text on group theory at the Schaum%s outline level. personally like, !Modern #lgebra, #n ntroduction", Hnded., G. 1. :urbin,&iley. Cou need only cover the material up through group theory. Takewith you the notion of a normal subgroup when you proceed to readDilmore.

6. 3ie #lgebras n $article $hysics, -rom sospin to ;ni+ed Theories, H nd. ed.,*. Deorgi, -rontiers in $hysics. couldn%t have read this book without +rsthaving read and worked through Gones. Deorgi was dicult for me, butwhen cracked it, began to feel like was starting to understand thephysicist instead of the mathematician. deally, read the +rst fourchapters of 1. Dilmore%s text +rst. The Othchapter covers applications to

areas typically presented in graduate physics coursework. Then readGones, then Deorgi. There will be much less for you to have to accept by+at.

*istory> 6quoted from Dilmore%s text7 !This study of simultaneous di)erentialequations led 3ie to investigate continuous transformation groups, from which thetheory of 3ie groups emerged. 3ie groups have been studied so extensively in theirown right that their connection with partial di)erential equations is often overlookedand forgottenP


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647 The adAoint or regular representation, equivalent to the structureconstants.

6H7 The use of the secular equation and its roots, which lead to furtherinformation about the structure of a group. The information is summarized in the+rst criterion of solvability.

6J7 The use of a metric 6(artan9illing form7 on the vector space associated

with the 3ie algebra. The information is summarized in the second criterion ofsolvability.

6F7 The folding of the +rst two criteria in the (artan criterion.6O7 The exploitation of the root and metric concepts to give a canonical

structure to the commutation relations of the regular representation of semisimplealgebras.

%m hoping that !Droup theory and physics," S. Stemberg, (ambridge, which %veAust acquired, will also provide more help with the above toolset. B-T literature isrife with results derived from the above big guns.

#t this point you now have a path to the underpinnings of two maAor chunks of the

mature grammar of modern B-T theorizing> the principle of least action, and toolsfor studying the structure of 3ie groups and 3ie algebras 6and particle spectra7. Stillmissing is a deeper, more general understanding of the principle of minimalcoupling resulting from the requirement of local invariance of CangMills 3agrangiandensities. To maintain invariance under local transformations


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principle bundles. f anything, you have to read aber%s chapter for motivation,and %ve reluctantly come to appreciate all of the mathematics studied trying to getthrough aber, especially di)erential forms. #t this point began to see that thereis probably no end to physics theoreticians cooking up hypothetical universes thatdon%t necessarily have to have anything to do with what we perceive to be ouruniverse. 2ven theorizing over our own apparent universe is probably unlimited.

The creative degrees of freedom to cook up mathematical universes that behave atlow energy like what we observe seem in+nite. #s our experimental knowledgegrows, we exile certain theories of physics into the realm of mathematics, only toquickly create a whole new frontier of endless physicsbased possible universes.

This realization took the wind out of my pursuing my belief in 2instein%s dream of a+nal theory. 0y the way, found a pretty tidy review of di)erential forms online,namely, !ntroduction to di)erential forms," :. #rapua, H5. was never satis+edby any of the physics books purportedly written to teach forms.

Summary of mature grammar memes so far: 6physics > mathematics7

$rinciple of least action > (alculus of ?ariations.

$rinciple of minimal coupling > (onnections=principle bundles, forms.

$articles and -ields8 S;SC > 3ie%s theorems, their inverse theorems, and the

classi+cation problem8 graded algebras 61yder7

&orld lines &orld Sheets> Strings 6# -irst (ourse in String Theory, 0.

Lweibach, (ambridge ;niversity $ress.

do not touch on the concept of symmetry breaking to give mass to CangMills +eldtheories. 2ntry level B-T texts do a reasonably good Aob treating this. /ne area %mstill missing is that dealing with e)ective 3agrangians and renormalization theory.

The following two articles were strongly recommended to me as good primers. Themethods of the renormalization group and 2)ective 3agrangians. 1ead> 2)ective-ield Theories, #. ?. Manohar, arWiv>hepph5@@HHHv4 F Gune 455@. 2)ective -ield

Theory, #. $itch arWiv>hepph5I@JJv4 J Gune 455I.

#ll of the above memes wrapped up in a 3agrangian expressing least action,minimal coupling, geometry, algebra, topology, and algebraic topology, is howwe%ve come to think about our universe, and hypothetical universes. t%s an entrylevel, minimally mature grammar to muse about universes and existences in thesense of ewton> ! do not know what may appear to the world, but to myself seem to have been only like a boy playing on the seashore, and diverting myself innow and then +nding a smoother pebble or a prettier shell than ordinary, whilst the

great ocean of truth lay all undiscovered before me."

$S


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with the Lermelo-rankel axioms, like the 0anachTarski paradox, and thelimitations we%ve discovered to be inherent in mathematics as discovered by(ohen and DKdel in the followon work. aturally, also discuss some of theinherent limitations in physics that %ve come across. #s for an algebrabackground to Dilmore, all the background found necessary from my training inpure mathematics was little more than the concept of a normal subgroup. /nly

once Dilmore begins to develop the structure of 3ie groups does all of the drycrap on towers in a standard graduate text on algebra, such as 3ang, start tomake sense beyond symbol manipulation. n the end, studying mathematics forits own sake is great, but it can sure slow you down if you%re interested inphysics.

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 di)erential and integral calculus and=or a year of calculus based

physics. The followon 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%tstudy on your own. That this followon presents the underlying key historical

motivations and interlinking of the various subAect 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 6to the point of understanding what

a normal subgroup is7, and at least enough real analysis to understand very basic

point set topology up to what being *ausdor) means. # chemist, mathematician, or

engineer should familiarize himself or herself with classical mechanics at the Aunior

level up to the concept of 3agrangian and *amiltonian mechanics, e.g., 1alph

0aierlein, !ewtonian :ynamics", 45IJ. *e or she should also review one

semester%s worth of electricity and magnetism at the Aunior level, e.g., Driths,

!ntroduction to 2lectrodynamics", and at least one semester of quantum

mechanics at the Aunior or senior level. The Schaum%s /utline in Buantum

mechanics is good enough, especially if you bother to +x its many minor typos


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t was during my years working as a nuclear weapons physicist at 3os #lamos

ational 3aboratory that had both the time and reason to +nally put together a

uni+ed mapping of applied mathematics and physics foundations. ;nless you%re

happy to only run weapons codes written by others, and otherwise ape intelligence

with technical babble read o) $ower$oint slides, nuclear weapons physics forces

you to go back to every core area covered in a graduate physics program, and thensome. (ertainly 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, Ueeting, typically

unstable phenomenology. The digging and battling kindles an intuition between

theory and experimental !reality" which a good astrophysicist might develop, but

which a socalled +nancial physicist might not, excepting, of course, the deep

mathematical analysis underpinnings of Monte (arlo transport methods. The

followon 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 +rst part of my Aourney took around +ve years after +nished

my formal schooling. The stu) above on the core of physics and mathematics

underpinning our theoretical toolset regarding real and hypothetical particles, +elds

and universes with a mature grammar, took an additional +ve years of my time. n

all honesty, the process wasn%t really that linear.

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Syllabus of Math and Physics Doc