This Week's Finds in Mathematical Physics (Week 38)
John Baez
John Baez
I've been busy, and papers have been piling up; there are lots of
interesting ones that I really should describe in detail, but I
had better be terse and list them now, rather than waiting for the
mythical day when I will have time to do them justice.
So:
1) Topological quantum field theories from generalized
6j-symbols, B. Durhuus, H. P. Jakobsen and R. Nest, Reviews in Math.
Physics 5 (1993), 1-67.
In "week16" I explained a paper by Fukuma, Hosono and Kawai in which
they obtained topological quantum field theories in 2 dimensions
starting with a triangulation of a 2d surface. The theories were
"topological" in the sense that the final answers one computed didn't
depend on the triangulation. One can get between any two triangulations
of a surface by using a sequence of the following two moves (and their
inverses), called the (2,2) move:
O O
/|\ / \
/ | \ / \
/ | \ / \
O | O <----> O-------O
\ | / \ /
\ | / \ /
\|/ \ /
O O
and the (3,1) move:
O O
/|\ / \
/ | \ / \
/ | \ / \
/ | \ / \
/ _O_ \ <----> / \
/ _/ \_ \ / \
/ _/ \_ \ / \
/_/ \_\ / \
O-----------------O O-----------------O
Note that in either case these moves amount to replacing one part of the
surface of a tetrahedron with the other part! In fact, similar moves
work in any dimension, and they are often called the Pachner moves.
The really *wonderful* thing is that these moves are also very significant
from the point of view of algebra... and especially what I call
"higher-dimensional algebra" (following Ronnie Brown), in which the
distinction between algebra and topology is largely erased, or, one
might say, revealed for the sham it always was.
For example, as explained more carefully in "week16", the (2,2) move is
really just the same as the *associative* law for multiplication. The
idea is that we are in a 2-dimensional spacetime, and a triangle
represents multiplication: two "incoming states" go in two sides and
their product, the "outgoing state", pops out the third side:
O
/ \
/ \
/ \
A B
/ \
/ \
/ \
/ \
O--------AB-------O
Then the (2,2) move represents associativity:
O O
/|\ / \
A | (AB)C A A(BC)
/ | \ / \
O AB O <----> O--BC---O
\ | / \ /
B | C B C
\|/ \ /
O O
Of course, the distinction between "incoming" and "outgoing" sides of
the triangle is conventional, and the more detailed explanation in
"week16" shows how that fits into the formalism. Roughly speaking, what
we have is not just any old algebra, but an algebra that, thought of as
a vector space, is equipped with an isomorphism between it and its
dual. This isomorphism allows us to forget whether we are coming or
going, so to speak.
Hmm, and here I was planning on being terse! Anyway, the still *more*
interesting point is that when we think about 3-dimensional topology and
"3-dimensional algebra," we should no longer think of
O O
/|\ / \
/ | \ / \
/ | \ / \
O | O and O-------O
\ | / \ /
\ | / \ /
\|/ \ /
O O
as representing *equal* operations (the 3-fold multiplication of A, B,
and C); instead, we should think of them as merely *isomorphic*, with
the tetrahedron of which they are the front and back being the
isomorphism. The basic philosophy is that in higher-dimensional
algebra, as one ascends the ladder of dimensions, certain things which
had been regarded as *equal* are revealed to be merely isomorphic. This
gets tricky, since certain *isomorphisms* that were regarded as equal at
one level are revealed to be merely isomorphic at the next level...
leading us into a subtle world of isomorphisms between isomorphisms
between isomorphisms... which the theory of n-categories attempts to
systematize. (I should note, however, that in the particular case of
associativity this business was worked out by Jim Stasheff quite a
while back: it's the homotopy theorists who were the ones with the guts
to deal with such issues first.)
Now, it turns out that in 3-dimensional algebra, the isomorphism
corresponding to the (2,2) move is not something marvelously obscure.
It is in fact precisely what physicists call the "6j symbol", a gadget
they've been using to study angular momentum in quantum mechanics for a
long time! In quantum mechanics, the study of angular momentum is just
the study of representations of the group SU(2), and if one has
representations A, B, and C of this group (or any other), the tensor
products (A tensor B) tensor C and A tensor (B tensor C) are not
*equal*, but merely *isomorphic*. It should come as no surprise that
this isomorphism is represented by physicists as a big gadget with 6
indices dangling on it, the "6j symbol".
Quite a while back, Regge and Ponzano tried to cook up a theory of
quantum gravity in 3 dimensions using the 6j symbols for SU(2). More
recently, Turaev and Viro built a 3-dimensional topological quantum
field theory using the 6j-symbols of the *quantum group* SU_q(2), and
this led to lots of work, which the above article explains in a
distilled sort of way.
The original Ponzano-Regge and Turaev-Viro papers, and various other
ones clarifying the relation of the Turaev/Viro theory to quantum
gravity in spacetimes of dimension 3, are listed in "week16". It's also
worth checking out the paper by Barrett and Foxon listed in "week24", as
well as the following paper, for which I'll just quote the abstract:
2) Spin networks, Turaev-Viro theory and the loop representation,
by Timothy J. Foxon, 19 pages in Plain TeX available as gr-qc/9408013.
We investigate the Ponzano-Regge and Turaev-Viro topological field theories
using spin networks and their $q$-deformed analogues. I propose a new
description of the state space for the Turaev-Viro theory in terms of skein
space, to which $q$-spin networks belong, and give a similar description of
the Ponzano-Regge state space using spin networks.
I give a definition of the inner product on the skein space and show
that this corresponds to the topological inner product, defined as the
manifold invariant for the union of two 3-manifolds.
Finally, we look at the relation with the loop representation of quantum
general relativity, due to Rovelli and Smolin, and suggest that the above
inner product may define an inner product on the loop state space.
(Concerning the last point I cannot resist mentioning my own paper on
knot theory and the inner product in quantum gravity, available as
"tang.tex" by ftp just like all the "week" articles, as described at the
end of this post.)
In addition to the papers by Turaev-Viro and Fukuma-Shapere listed in
"week16", there are some other papers on Hopf algebras and 3d
topological quantum field theories that I should list:
3) Involutory Hopf algebras and three-manifold invariants, by Greg
Kuperberg, Internat. Jour. Math 2 (1991), 41-66.
A definition of #(M,H) in the non-involutory case, by Greg Kuperberg,
unpublished.
Greg Kuperberg is one of the few experts on this subject who is often
found on the net; he is frequently known to counteract my rhetorical
excesses with a dose of precise information. The above papers, one of
which is sadly still unpublished, make it beautifully clear how "algebra
knows more about topology than we do", since various basic structures
on Hopf algebras have a pleasant tendency to interact just as needed
to give 3d topological quantum field theories.
4) Spherical Categories, by John W. Barrett and Bruce W. Westbury, 16
pages in AMStex available as hep-th/9310164.
The equality of 3-manifold invariants, John W. Barrett and Bruce W.
Westbury, 8 pages available in AMStex (you need BoxedEPSF for 3
postscript figures; use of BoxedEPSF is not essential) as
hep-th/9406019.
Let me quote the abstract for the first one; the second one shows that
the authors' 3-manifold invariants coming from spherical Hopf algebras
agree with Kuperberg's coming from involutory Hopf algebras when both
are defined.
This paper is a study of monoidal categories with duals where the
tensor product need not be commutative. The motivating examples are
categories of representations of Hopf algebras and the motivating
application is the definition of 6j-symbols as used in topological field
theories.
We introduce the new notion of a spherical category. In the first
section we prove a coherence theorem for a monoidal category with duals
following MacLane (1963). In the second section we give the definition
of a spherical category, and construct a natural quotient which is also
spherical.
In the third section we define spherical Hopf algebras so that the
category of representations is spherical. Examples of spherical Hopf
algebras are involutory Hopf algebras and ribbon Hopf algebras. Finally
we study the natural quotient in these cases and show it is semisimple.
5) Invariants of 3-Manifolds derived from finite dimensional Hopf algebras, by
Louis H. Kauffman and David E. Radford, 33 pages available in LaTeX as
hep-th/9406065.
This is paper also relates 3d topology and certain finite-dimensional
Hopf algebras, and it shows they give 3-manifold invariants distinct
from the more famous ones due to Witten (and a horde of mathematicians).
I have not had time to think about how they relate to the above ones,
but I have a hunch that they are the same, since all of them make heavy
use of special grouplike elements associated to the antipode.
6) Four dimensional topological quantum field theory, Hopf categories,
and the canonical bases, by Louis Crane and Igor Frenkel, available as
hep-th/9405183.
Work in 4 dimensions is, as one expect, still more subtle than in 3,
since again various things that were equalities becomes isomorphisms.
In particular, this means that various things one thought were vector
spaces --- which are *sets* that have *elements* that you can *add* and
*multiply by numbers*, and which satisfy *equations* like A + B = B + A
--- are now reinterpreted as "2-vector spaces" --- which are
*categories* that have *objects* that you can *direct sum* and *tensor
with vector spaces*, and which have certain *natural isomorphisms* like
the isomorphism A direct sum B and B direct sum A. In particular, using
Lusztig's canonical basis, Crane and Frenkel start with quantum groups
(which are Hopf algebras of a certain sort) and build marvelous "Hopf
categories" out of them. While they do not construct a 4d TQFT in this
paper, they indicate the game plan in terms clear enough that they will
probably now have to race other workers in the field to see who can get
the first interesting 4d TQFT... or perhaps something a bit subtler than
a 4d TQFT (e.g. Donaldson theory).
Finally, let me turn to a subject that is closely related (though
unfortunately this has not yet been made sufficiently clear), namely,
holonomy algebras and the loop representation of quantum gravity. Let
me simply list the references now; many of these papers were discussed
at my session on knots and quantum gravity at the Marcel Grossman
conference, so I promise to explain at some later time (and in some
papers I'm writing) a bit more about how the loop representation of a
gauge theory is interesting from the viewpoint of higher-dimensional
algebra!
7) A manifestly gauge-invariant approach to quantum theories of gauge
fields, by A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao, T. Thiemann,
contribution to the Cambridge meeting proceedings, 27 pages in LaTeX
available as hep-th/9408108.
Topological measure and graph-differential geometry on the quotient
space of connections, Jerzy Lewandowski, 3 pp., Proceedings of
``Journees Relativistes 1993'', 3 pages available as gr-qc/9406025.
Integration on the Space of Connections Modulo Gauge Transformations,
Abhay Ashtekar, Donald Marolf, Jose Mourao, 18 pages in LaTeX available
as gr-qc/9403042.
New loop representations for 2+1 gravity, by A. Ashtekar and R. Loll, 28
pages in TeX (1 figure in postscript, compressed and uuencoded)
available as gr-qc/9405031.
Independent loop invariants for 2+1 gravity, by R. Loll, 2 figures,
gr-qc/9408007.
Generalized coordinates on the phase space of Yang-Mills theory, by R. Loll,
J.M. Mour\~ao and J.N. Tavares, 11 pages in TeX available as gr-qc/9404060.
The extended loop representation of quantum gravity, C. Di Bartolo, R. Gambini
and J. Griego, 27 pages available as gr-qc/9406039.
The constraint algebra of quantum gravity in the loop representation, by
Rodolfo Gambini, Alcides Garat and Jorge Pullin, 18 pages in Revtex,
available as gr-qc/9404059.
--------------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics (as well as some of my research papers) can be
obtained by anonymous ftp from math.ucr.edu; they are in the
directory "baez." The README file lists the contents of all the papers.
On the World-Wide Web, you can attach to the address
http://info.desy.de/user/projects/Physics.html to access these files and
more on physics. Please do not ask me how to use hep-th or gr-qc;
instead, read the file preprint.info.
Archimedes Plutonium
Archimedes Plutonium
Sep 25, 2021, 2:28:56 PM
to sci.physics
SAVE// AP's 151st book looks like October at earliest to publish, for I have to include Kirchhoff's rules (laws).
TEACHING TRUE PHYSICS// 1st year College// Physics textbook series, book 4
by Archimedes Plutonium
Preface: This is AP's 151st book of science published. It is one of my most important books of science because 1st year college physics is so impressionable on students, if they should continue with physics, or look elsewhere for a career. And also, physics is a crossroad to all the other hard core sciences, where physics course is mandatory such as in chemistry or even biology. I have endeavored to make physics 1st year college to be as easy and simple to learn. Good luck.
Cover picture is the template book of Halliday & Resnick, 1988, 3rd edition Fundamentals of Physics and sitting on top are cut outs of "half bent circles, bent at 90 degrees" to imitate magnetic monopoles. Magnetic Monopoles revolutionizes physics education, and separates-out, what is Old Physics from what is New Physics.
The world needs a new standard in physics education since Feynman set the standard in 1960s with his "Lectures on Physics" that lasted until about 1990 and then AP's Atom Totality theory caused Feynman's Lectures to be completely outdated. And so much has changed in physics since 1960s that AP now sets the new world standard in physics education with this series of textbooks.
To be a Master of physics or Calculus or Mathematics, has to be seen in "signs and signals". Can you correct the mistakes and errors of Old Physics, of Old Calculus, of Old Math? If you cannot clean up the fakery of Old Physics, of Old Calculus, of Old Math, you have no business, no reason to write a physics, calculus or math textbook. There is an old legend in England about King Arthur, and the legend goes, that the King is the one who pulls Excalibur out of the iron anvil. Pulling the sword out of the anvil is a metaphor for Cleaning up all the mistakes and errors of Old Physics, of Old Calculus, of Old Math.
Should you write a textbook on Calculus, if you cannot see that the slant cut in a cone is a oval, never the ellipse? Of course not. Should you write a Calculus textbook if you cannot do a geometry proof of Fundamental Theorem of Calculus? Of course not. Should you write a physics textbook if you cannot ask the question, which is the atom's real true electron, is it the muon or the 0.5MeV particle that AP says is the Dirac magnetic monopole.
Feynman wrote the last textbook in 1960s to guide physics forward, and although Feynman did not clean up much of Old Physics, he did direct the way forward in that Electricity and Magnetism in his Quantum Electrodynamics was the way forward. It would have been nice for Feynman to have found that it is impossible for a 0.5MeV particle to be the atom's electron moving near the speed of light outside the proton of hydrogen and still remain an atom, thus all atoms collapse. It would have been nice for Feynman to say the muon is the real atom's electron and that the 0.5MeV particle was Dirac's magnetic monopole. But it just was not in the fated cards of Feynman's physics. Yet, his textbook served the leadership of physics from 1960 to 1990. Time we have the new replacement of physics textbook.
Now, in 2021, we need a new textbook that carries all of physics forward into the future for the next 100 years, and that is what this textbook is.
I will use Halliday and Resnick textbook as template to garner work exercise problems for 1st year and 2nd year college. For 3rd and senior year college physics I will directly use Feynman's Lectures and QED, quantum electrodynamics. Correcting Feynman and setting the stage that all of physics is-- All is Atom and Atoms are nothing but Electricity and Magnetism.
Much and most of 20th century physics was error filled and illogical physics, dead end , stupid paths such as General Relativity, Big Bang, Black holes, gravity waves, etc etc. Dead end stupidity is much of Old Physics of the 20th century. What distinguishes Feynman, is he kept his head above the water by concentrating almost exclusively on Electrodynamics. He remarked words to the effect== "QED is the most precise, most accurate theory in all of physics". And, that is true, given All is Atom, and Atoms are nothing but Electricity and Magnetism.
This textbook is going to set the world standard on college physics education. Because I have reduced the burden of mathematics, reduced it to be almost what I call -- difficult-free-math. I mean, easy-math. Meaning that all functions and equations of math and physics are just polynomials. All functions of math and physics are polynomials. Making calculus super super easy because all you ever do is plug in the Power rules for derivative and integral, so that physics math is able to be taught in High School. In other words, physics with almost no math at all-- so to speak, or what can be called as easy as learning add, subtract, multiply, divide.
What makes both math and physics extremely hard to learn and understand is when mathematics never cleans itself up, and never tries to make itself easy. If all of math can be made as easy as add, subtract, multiply, divide, no one would really complain about math or physics. But because math is overrun by kooks (definition of kook: is a person who cares more about fame and fortune than about truth in science), that math has become a incomprehensible trash pile and the worst of all the sciences, and because the math is so difficult, it carried over into physics, making physics difficult.
You see, one of the greatest omissions of science in the 20th and 21st century was the idea that math can be reduced to a Simplicity of education. That math need not be hard and difficult. Yet no-one in the 20th and 21st century ever had that idea of simplicity, (with the possible exception of Harold Jacobs) that math had run out-of-bounds as a science and was more of a science fiction subject for kook mathematicians. Had become absurdly difficult because of the reason that kooks gain fame and fortune on making math difficult. Mathematicians never thought their job was to make math simple and easy, instead, the kooks of math piled on more trash and garbage to make math a twilight zone of science.
When you make all of math be just polynomial equations and functions, you make math the easiest of the major sciences, which then follows up by making physics easy as possible.
--------------------------
Table of Contents
--------------------------
Part I, Introduction, and about physics.
a) About this textbook and series of Physics textbooks.
b) Brief history lesson of 20th century physics.
c) How we make the mathematics super easy.
d) Horrible concept of "charge" in Old Physics, and thrown out of New Physics.
e) How I have to use Biology DNA knowledge to unravel the physics light wave.
Part II, 6 Laws of EM theory.
f) The 6 laws of ElectroMagnetic theory and their Units, EM theory.
g) The four differential equations laws of EM theory.
h) Defining the units of Coulomb and Ampere as C = A*seconds; and the Elementary-Coulomb.
i) Faraday Constant Experiment in classroom.
j) Matching the physics Algebra of units with the physics Geometry of units.
k) The EM Spectrum, Electromagnetic Spectrum.
Part III, 1st Law of EM theory.
l) 1st Law of EM theory; law of Magnetic Monopole and units are B = kg/ C*s = kg/ A*s^2.
m) Series versus parallel connection of closed loop.
Part IV, 2nd Law of EM theory.
n) 2nd Law of EM theory; New Ohm's Law V = CBE, the Capacitor-battery law.
o) Review of Geometry volume in 3D and path in 2D.
Part V, 3rd Law of EM theory.
p) 3rd law of EM theory, Faraday's law, C' = (V/(BE))'.
q) Short history lesson of Old Physics, 1860s Maxwell Equations.
r) New Rutherford-Geiger-Marsden Experiment observing Faraday Law.
s) Math Algebra for making one physical concept be perpendicular to another physical concept.
t) EM laws derive the Fundamental Theorem of Calculus.
u) Principle of Maximum Electricity and Torus geometry so essential in Atomic Physics.
Part VI
v) 4th law of EM theory; Ampere-Maxwell law B' = (V/(CE))'.
Part VII
w) 5th law of EM theory; Coulomb-gravity law; E' = (V/(CB))'.
x) Centripetal versus Centrifugal force explained.
Part VIII
y) 6th Law of EM theory, electric generator law; differential equation of New Ohm's Law V' = (CBE)'.
z) Reinventing the Multivariable Calculus.
aa) Short Circuit.
bb) Atomic bomb physics.
--------
Text
--------
Part I, Introduction, and about physics.
AP, King of Science, especially Physics