Large numbers and abstraction in mathematics – a musing

It’s been some time since I’ve written on this blog.  So it’s appropriate I talk about large numbers.  I currently teach sections of a mathematics class for liberal arts students.  This means that they are not typically Science, Technology, Education or Mathematics (the so called STEM majors).  It has its ups and downs, one of the ups being a flexible curriculum.  This semester I wanted to work on two themes, one was geometrical.  Starting with the geometry of experience 2d/3d I moved onto the geometry of 4d (in a very  limited way) by analogy.  There’s a great deal you can do if you work on thinking like a Flatlander, though it’s not easy to do so since we interpret visual data spatially.

The second theme of the semester and the one that is relevant to this post is finite to infinite.  We have experience with finite numbers, they describe quantities, lengths, percents etc.  Our knowledge of the finite equips us to generalize or abstract to the infinite.  Large numbers appear in our daily lives, often from the news.  Millions, billions and trillions are common place.  If we’re interested in science we often hear of yet larger numbers, numbers that describe facts or processes about our solar system, galaxy or the entire universe.  At some point we reach numbers which seem to contain, by their immensity, the entire universe.  It’s estimated that there are around 10^{80} protons in our universe and upwards of 10^{125} bits of information.  Are these numbers the largest that have any physical relevance?  We can certainly continue to add zeroes as long as we’d like. We can create notations that allow us to write down incredible numbers but  what do they represent?

After my students spent some time playing with numbers that described the astronomical I spent a week teaching them the basics of RSA encryption (blog post forthcoming).  This relies on the difficulty in factoring very large numbers (RSA modulus) into their prime factors.  It’s relatively easy for computers to multiply a couple of large primes and obtain a large number however.  Some of the more secure RSA encryption uses primes that are over 300 digits in length.  Division isn’t as hard as factoring and so a naive concern might be that some folks with ill intent would simply keep a list of large primes and then just check them against the RSA modulus.  No worries here, the Prime Number Theorem allows us to place an estimate on the number of 300 digit primes, which is around 10^{297}.  There are too many of these for this universe to contain!  Using millions of these primes every second wouldn’t even dent this list, this enormous list of enormous numbers.

I was a bit surprised by this number. Two avenues of thought resulted.  The first concerned what we mean when we do arithmetic on numbers of this size. The second concerns the existence of numbers.  Dealing with the first, when we learn to do arithmetic we do so in an experimental setting.  We have piles of counters to check that the addition or subtraction of numbers is consistent with the manipulation of quantities.  We can be confident that 2+5=7 because we can check it.  This suggests some shortcuts for things like 124+453 because we can manipulate the digits using these facts: 4+3=7, 20+50=70 and 100 +400=500 so we have 577. This sum could still be checked in principle, we don’t.  We’re well convinced by this time and carry on.

What does it mean to perform the following sum?

3x10^{184} + 5x10^{184}

 

This can  certainly still be thought of as a quantity of something, some number of permutations or positions in some game etc.  It no longer refers to a physical quantity however.  Not a number of atoms, not a number of stars or organisms etc.  Let me say that I don’t doubt that the operations used here are consistent with the operations used on integers that can represent heaps of things.  Rather I’m reminded how students are often taken back when they learn about i=sqrt{-1} which bears the unfortunate name of imaginary. In my teaching experience this seems to be too much for most and they become rather suspicious of what we mathematicians are doing.  This might be the first time they’ve come up against the concept of a number that’s not a quantity.  Of course there have been others, but this is the first they’ve noticed.  I make a point of pointing out that the irrational numbers are just as imaginary given the limits of measurement.  Irrational numbers are, in some sense, conveniences to smooth out the dense but pock marked number line of rational values.  Calculus would be something awful without the continuum.  Much in the same way that plenty of branches of mathematics would bemoan the loss of the complex plane.

Integers of any size seem benign.  The integers above are very large and it’s easier to come up with larger ones (the number of books in Jorge Borge’s The Library of Babel) that dwarf them.  I wonder though whether this familiarity with the digits of these large numbers make us slow to see their necessary abstraction.  We can talk about, prove results about, access and find numbers of 300 digits that are prime.  But we can never have them all.  There are too many, it would take too long and there’s too much information for the universe we live in.  So what’s a number?  I think I’ll find myself heading back for another attempt at Frege’s text.

I’m also reminded, though I can’t find the quote at the moment, of something Brouwer is supposed to have remarked.  I don’t understand a great deal about intuitionist mathematics but I remember reading something to the effect that he wrote about the application of mathematics to science.  The mathematics he was doing, which had slightly different foundations, would result in consequences incompatible with mainstream mathematics and science.  Brouwer said something along the lines of “it doesn’t matter if mathematics can or can’t be used in science, it’s not the job of mathematics to support science” or something to this effect.  I’ll be editing this paragraph for certain.

The second avenue of thought is about the existence of these numbers and I’ve already said a little.  In what sense do these large numbers exist?  I must caution the reader that I’m no philosopher and can only blunder about with these ideas and questions.  I repeat, in what sense do they exist.  How do you introduce them?  What are the examples of them?  I can show you examples of 7 and presumably, after a while, you see that the common thing among the piles or heaps I show you is their quantity which has the cardinality of 7.  How do I show you

44422180169906282030865231707744178205474059724209874047561419422925718096057797267971937899588020604289210726207367610233576023602937600936234

Where do I point?  What about numbers such as 10^{10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000} which cannot be written out in full in this universe.  What about Graham’s number?  We can work with these numbers, use them, prove things to be true and false about them but they cannot reside in our universe, they don’t describe physical things or they are just too big to fit in our tiny universe.

No conclusions, only thoughts.  Yours are appreciated.

 

Posted in Uncategorized | 2 Comments

Current musings and an apology

It’s nearing the end of the semester here and so things are getting busy.  It’s also spring vacation for the kiddos and so I’ll be spending more time with them.  To add to that, my better half’s family is up helping with the renovations and so it’s a very busy week.

However, I’ll give you this: “The Library of Babel” by Jorge Luis Borges (you ought to be able to find a pdf online easily enough) is a short story about a universe which is a library, an immense library.  The library houses every 410 page book based off a 25 character alphabet possible.  There are delightful implications and Borges’ writing is great.  This is a wonderful blend of mathematics and literature.  Once you finish that, and it’s quick, check out “The Unimaginable Mathematics of Borges’ Library of Babel” which is an easy and fun read.

I’ll be posting much more on this in the future as I hope to wrap up the liberal arts math course I teach with this short story and an exposure to large numbers.  Until next time.

Posted in Uncategorized | Leave a comment

Lie Groups and Lie algebras – a question

I can’t remember how this came up, I believe it involved an exchange with Shea Browne but it certainly has been motivated by readings.  You see, sometimes you’ll see a Lie Algebra defined all by itself, as a vector space with a certain product structure on it.  This has struck me as odd since on the other hand we know that the Lie algebra is a local linearization of a Lie group at it’s identity element.  Lie algebras can seem like a funny place to start because they also don’t uniquely determine a Lie group.  What gives?

I’ll advance a thought which is also a question.  Feel free to chime. If you don’t I’ll figure this out eventually and post it all the same.  Physics starts locally, which is expressed in its differential equations.  Things like vector fields on a manifolds have a bracket structure and so make it a Lie algebra, but those vector fields are differential operators and these allow us to express differential equations.   The process of solving that differential equation is to find a group of transformations that tell us what happens if you’re at a particular point on the manifold, where do you go from there?  The differential equation gives you a little information, where to go (infinitesimally) and how long to do so (infinitesimally) but the group (the solution) gives you a non-infinitesimal set of directions and times.  This information is still localized, I’ll need to review the existence theorems for solutions of differential equations and so maybe the fact that Lie algebras don’t uniquely determine Lie groups have something to do with this?

I know this isn’t the whole picture because you can also look at the Lie group of symmetries of a Lagrangian or differential equation so I’m missing a lot of the puzzle.  Thoughts?

Posted in Uncategorized | Leave a comment

What I am and what I am not.

The last few years have been humbling experiences.  I’ve been coming to terms with who I am and who I want to be, fortunately they’re not too different.

A few years ago I prepared for my comprehensive exams in physics.  I pored over old exams that dated back more than a decade.  While there were questions that repeated, the format seemed wide open.  Nearly anything (appropriate for the level of my department) could be asked.  How was I going to remember  all these different problems?  It’s easy while you’re taking courses to have tunnel vision.  Faculty generally encourage it by assigning problems from specific chapters of specific books with implicit assumptions about whether we should be thinking relativistic or quantum mechanical.

I threw away all the old exams.  What was the point of looking at problems?  I didn’t know what problems were going to be on the exam. I didn’t know if I would be able to solve the problems.  If the exams were any good I wouldn’t have seen the problems before anyway.  I spent a month (it’s all I had) organizing my understanding of physics.  I used concept diagrams, took up whole blackboards trying to tie together different results, when they held, what they meant and what they implied.  I organized my knowledge and the only real computations I did was to review some basic techniques.

I can’t avoid making this post sound dramatic, but that month was like an academic vision quest.  I realized how little I had learned as an undergrad and a graduate student.  I had earned good marks, but I had earned those marks working problems from specific chapters in specific classes.  That month was a wonderful synthesis of ideas and I felt ready for the exams that I would pass.  Three days, 7-8 hours a day.  I enjoyed it and I came out a slightly different person.

It’s affected the way I teach because I’ve come to realize how much I relied on remembering instead of understanding and discovering.  There’s a faculty member, who’s a friend of mine, in the physics department we’ll call him Dan.  I think he’s a poor teacher in some ways, great in others.  He would say something along the lines of “Don’t remember, think!”

With that one phrase he may have been a more valuable instructor than almost any other.  While I think he has trouble instructing he does do a good job of asking easy questions that require thought and resist remembering.  It’s uncomfortable to be faced with your limitations and shortcomings, but its invaluable.

Over the years I’ve begun asking my students simple conceptual questions that require almost no computation.  They might rely on units, simple pictures or other short conceptual connections.  I then witness almost complete failure in the presence of computational competence.  They can repeat, they can remember but they do not understand.

Who am I and what do I want to be?  I could make excuses or sell them as reasons why I am where I am and how I arrived here.  I’m much more of a mathematical physics aficionado than a mathematical physicist.  Analogous to enjoying good wine but not making it.  While I have no doubt that I’ll eventually solve some cute personal problems of interest and publish them, I have to say that I’m not disappointed that I haven’t.  While I once wanted to obtain a tenure track position I no longer do.

I’m fairly intelligent and have enough passion to compensate but realistically I have no talent.  Most researchers don’t.  The talented folk blaze paths, come up with ideas so profound you wonder sometimes how you missed them.  I’d rather work on understanding the great minds that are than carving out a small niche for myself.  I do aspire to greater understanding and depth in mathematics but also in recreational math and in teaching and in outreach mathematics to children and to bringing in more art (of all sorts) into both my teaching, my learning and my outreach.  A retiring mathematics faculty once chastised me for my wanderlust in mathematics and my inability to focus on any single area.  He said I wouldn’t amount to anything and in a very real sense he’s correct, I will not amount to much in the halls of mathematics but there’s more to success and accomplishment than academic accolades though I have great respect for those (and this particular faculty member’s).

Like a child there is much that I delight in, revel in fact.  I think that many of us seek faculty positions because we’re not sure where else to go.  We need down time, lots of it.  We generally can’t get enough of it.  We’re not using that time to run errands or do chores, we’re furiously attacking the beautiful immensity of mathematics, science, philosophy and art.  Faculty positions often let us share this enthusiasm with a captive audience which allows us to hear ourselves out loud without the stigma of talking to ourselves.  It allows us access to like minded individuals, colloquia, and lots of days off.  It generally pays reasonably well so we don’t have to live as paupers.

Adjuncts are paid much less than faculty but it’s not a bad job, neither is the position of instructor which is still much more junior to either assistant or associate professor.  There’s a bounty of spare time and some opportunities to talk to yourself in front of others.  You have access to intelligent and curious peers and a great library.

My goals are humble.  Live small, leave the mainstream of American consumerism.  Mind you, I don’t condemn it for it affords some of us to live in the eddies of its currents.  The goal is to live simple, live free, and trade money for time. Earn less have more.  What we thinkers all want are long luxurious expanses of time.  Time to write, time to walk, time to sit and stare letting the knots of tangled thoughts relax and  lower the impedance between our understanding and the thing we which to understand.  Develop that understanding and then share it.

So while I consider myself a mathematician, I’m not a good one nor talented.  I’m passionate, I love it, it is my an important part of who I am. This blog cannot be anything more, however, than the musings of an excited, poorly educated academic.  This is why I write, to understand, to share, to learn and be honest with both myself and those who wish to read.

Posted in Uncategorized | Tagged | 3 Comments

“On Space and Time” – part 3

I’ve finished “On Space and Time”  and my opinion of the book has only increased (see part 1 and part 2 for initial chapters).  There were three chapters remaining, one by Alain Connes, the founder of Noncommutative geometry, Michael Heller, probably the coolest Roman Catholic priest I’ve read and a short epilogue by John Polkinghorne, an Anglican theologian.

Connes really tries to bring his ideas down to the level of us mortals but he continually fails.  In fact as I’ll write later on, Heller’s chapter gives a much better description of Noncommutative geometry than the founder does.  Connes begins with the combined Lagrangians for general relativity and the Standard model.  He’s troubled by the fact that the Lagrangian for the Standard model is long, really long (there’s a nice picture of it in the chapter, but you can Google it and get the idea.  I’d post something but I wouldn’t be able to tell if I found the correct Lagrangian or not).  Though Connes is a pure mathematician he has a profound interest and respect for physics.  Via Noncommutative geometry he’s found a way to form a simpler functional that is defined on a product space M\times F where M is spacetime but F is a discrete space.  Lots of mathematics follows and I can only do it injustice at this time.  It does yield lots of predictions however and it would be interesting to see if any of these have been verified or refuted since the time of the writing in 2008.  I do want to include one quote however, Connes is discussing the simplicity of the result and the following sentence appears,

“Thus we imagine trying to explain this action principle to a Neanderthal man.”

It’s likely that you’ll take this out of context, but I thought it was funny because I imagine Connes would imagine me, the reader, along the lines of such a creature.

Michael Heller’s chapter is from the perspective of theology. For him, science informs theology and theology informs science.  This might be my second favorite chapter in the text.  Heller approaches a tender subject, God in science, masterfully.  As an atheist I was impressed by the quality of thought and found no objections, in fact, I would greatly enjoy to read more of his writing.

Heller begins by discussing the notion of “the God of the gaps.”  This is the tendency to see God in areas of human ignorance, such as the Big Bang.  He cautions against this approach as it only erodes faith and increases the likelihood of it becoming increasingly irrelevant as science advances and fills in such gaps.  He then spends some time reviewing some  theology, in particular the notions of time and how time relates to God.  St. Augustine and Aquinas are discussed and the notion of atemporality of God is introduced.  The idea that God doesn’t sit in the time stream, that such notions of past or future may not make sense in a discussion of God.  This discussion of time is one of the motivating factors for Heller’s interest in quantum gravity.  What is the nature of time?  General relativity had already altered our perspective of it, what will the future hold?  How will this conception of time affect or inform the theology concerning the temporality of God?  For Heller, science doesn’t compete with religion,

“But the correct theology is obliged to take into account what science has to say to us in this respect…Any theology that would choose to ignore this magnificent process is a blind way to nowhere.”

But he feels that science can benefit likewise and I can see his point,

“To see profit for science is perhaps less obvious, but we should take into account the fact that much of Western science, such as for instance Newton’s ideas, are imbued with things taken ultimately from theology, and it is better to be aware of this influence than not.”

Majid had similar sentiments in his chapter though of a more philosophical bent (without mention to theology) and this is certainly where I agree.  I’m aware only of my own ignorance to the mutual impacts of science and philosophy.  It affects our language, how we ask questions, which questions we ask and how we interpret the answers.  It can be no coincidence that the brightest minds of mathematics and science take issues of philosophy very seriously and the those who disparage philosophy are often the crudest or more mundane minds.

Heller is no mere priest.  He has a PhD in physics, specializing in cosmology and has published numerous scientific papers on general relativity, quantum mechanics and their unification.  As such it turns out that he gives a rather nice overview of noncommutative geometry and the consequences if spacetime is noncommutative at some scale.

Manifolds are usually specified with charts and atlases, apparently there’s another method which involves the algebra of continuous functions from the manifold to the real numbers.  I’m not aware of this method (yet).  It ends up being coordinate free which is enticing and it ends being more amenable to generalization.  If instead of functions to the reals you wre to look at the functions to a space of matrices then your function algebra would be noncommutative.  At the most basic level this is the meaning of noncommutative in noncommutative geometry.  This isn’t even the tip of the iceberg, this is akin to hearing a story about icebergs.  When you abstract like this you lose, apparently, any notion of ‘local.’  You can distinguish points, there aren’t any points.  This reminds me of Penrose tiles where you can’t tell which tiling you’re looking at from any finite section of it.  In fact any picture of a Penrose tiling you’ve seen occurs infinitely often in every Penrose tiling.  They are nonlocal and always described in introductions on noncommutative geometry.   I’ll let Heller say a bit more,

“One of the most striking peculiarities of noncommutative spaces is their totally global character.  In principle, no local concept can be given any meaning in the context of noncommutative geometry.  Typical examples of local concepts are those of ‘point’ and neighborhood of  a point.  In general, noncommutative spaces do not consist of points.  Let us look more closely at this peculiarity of noncommutative geometry.  The basic idea  of standard (commutative) geometry, in its map and atlas approach, is that every point is identifiable with the help of its coordinates.  When we prefer the functional method, we can identify a point with all functions that vanish at this point.  All such functions form what in algebra is called a maximal ideal of a given algebra of functions.  In other words, points are identified with maximal ideals of a given functional algebra.  And now a surprise!  It can be shown that for strongly noncommutative algebras there are typically no maximal ideals.  In such circumstances, there is nothing that could be identified with a point.  Similar reasoning could be used with respect to other geometrical local concepts (like neighborhoods).”

This hit home for me as I’ve been struggling on and off to penetrate the mysteries of noncommutative geometry.  Even the simple texts such as “Very Basic Noncommutative Geometry” by KhalKhali are dense and dispiriting.  Heller echoes Majid again by suggesting that it doesn’t matter if noncommutative geometry ends up being part of  quantum gravity, the benefit, conceptually, to both mathematics and physics, through the exchange of ideas and analysis of consequences is benefit enough.  This is just too cool not to look into.

What are the consequences of nonlocality?  Would the EPR paradox become trivial?

“The paradoxical behavior of two electrons from the EPR experiment is due to the fact that these two electrons, by their very quantum nature, explore the fundamental level in which there is no concept of distance in space”

Wild! He has a couple other examples I’ll leave to you to read about, but its provocative.  Heller even introduces the notion (lightly) of noncommutative probability that I was unaware of until this article.  The idea would follow from his above description of nonlocality.  Typically for probability theory you consider events as elements (points) in a sample space and assign probabilities to those.  I can’t imagine what noncommutative  version would be but I can see how the lack of locality could cause problems.

It’s getting late and I’m going to wrap this up.  If you’ve seen any of the Amazon reviews, many of them are dismissive of the last two chapters of the book.  Polkinghorne’s was short and not very enlightening I thought, though relevant.  Heller’s I thought was excellent.  I’ll be buying this book so I can transfer all my pencil marks out of the libraries copy.

Posted in Uncategorized | Leave a comment

“On Space and Time” – part 2

I’m still reading “On Space and Time” that I started reviewing last week.  I just finished the chapter by Roger Penrose entitled “Causality, Quantum Theory and Cosmology.”  Like Majid’s article, this one is a pleasure to read.

Penrose has always and still motivates me to learn GR much much much better. His article addresses his idea for the Big Bang and cosmology.  He’s a proponent of it but is honest about its short comings.  It’s called Conformal Cyclic Cosmology, or CCC for short.  This is the second “Conformal – awesome theory” that I’ve come across in the past few months (the other being conformal quantum field theories).  The idea seems to be to conformally map the Big Bang singularity onto a 3-surface that joins our spacetime (up until the big bang) to another 4-manifold.  This can be done either as a mathematical trick for some ease in calculations, or taken seriously as Penrose suggests.

Penrose is concerned about the arrow of time and the very low entropy (and phase space volume of the universe) at the Big Bang and he’s trying to find it a way to reconcile this with the remote future.  He has some unorthodox views, but again he’s honest about them, concerning information loss in black holes and quantum measurement.  He suggests that information is indeed lost in a black hole and so once it’s evaporated, in the remote future, that this information is truly gone.  This, as I understand, is contrary to the current view after the long standing debate between Hawking and Susskind (and others?).

This works for CCC which posits that our spacetime is sandwiched between 3-surfaces, one of them being the big bang and the other being the remote future.  Won’t the future go on forever?  Here’s where it gets really interesting.  He suggests that eventually there’ll be no massive particles left.  Once everything has been sucked into black holes and those black holes are colliding with one another and evaporating we’re left with a universe of pure electromagnetic and gravitational radiation.  The representative particles do not experience time and a clock in such a future could not exist. Penrose suggests that at such a time the universe is purely conformal (all the matters is casual structure not the rate of time passage). He notes (and I want to check this eventually) that out of the metric tensors 10 components, 9 are conformal and only 1 sets the scale for the passage of time.

The remote future is then thought of as a 3-surface along with some scalar field (not sure of this part, it generates dark matter or something?) that initiates a new aeon (a new big bang).  These surfaces are not identified, there’s no toroidal universe, it’s a steady progression of Bang, followed by expansion, followed by black holes and evaporation, followed by another aeon.

The link between conformal invariance and time is something I’d never considered. He notes that EM is entirely conformally invariant since photons don’t carry clocks (or at least those clocks don’t tick) which is very neat.  All that EM (and gravitational radiation) cares about is causality, the light cone structure of space which is conformal.  This is really neat!

Another neat bit is that he really gets his mileage out of the Weyl tensor that I have no familiarity with other than in name.  It motivates me to spend some time with it and I wonder if this article and Penrose’s ideas would be a nice way to illustrate the point and utility of the Weyl tensor?

Onto Connes’ essay, his stuff is always a doozy.

Posted in Uncategorized | Tagged , , | Leave a comment

Determinants and traces – 1

I wonder how many posts I’ll label “Subject matter – 1” and how many sequels I’ll end up actually writing.  Last time, I wrote that I had started thinking about the trace of a linear map.  I finally found a few moments, after the kiddo finally went to sleep, to play with an example.

Consider the Lie group of 2×2 invertible matrices with real entries, GL(2,\mathbb{R}).  The determinant is a differentiable homomorphism from GL to R.  As such, it’s differential is a map between tangent spaces of GL and R.  Let’s look at a rather particular tangent space, the Lie algebras \mathfrak{gl} and \mathbb{R}.  Let \vec{v}\in\mathfrak{gl}.  It’s 4-dimensional (just like GL_2).  Let’s use the following coordinates around the identity element of GL,

\begin{pmatrix}  a &b \\  c& d  \end{pmatrix}

To find a basis for our Lie algebra in terms of the coordinates around the identity we take such a matrix as above and differentiate in each direction and evaluate at I.  As an example consider the ‘a’ direction,

CodeCogsEqn (14)

Each direction will result in a basis vector for \mathfrak{gl},

CodeCogsEqn (13)

So, given arbitrary coefficients, an element of \mathfrak{gl} would look like,

\begin{bmatrix} a & b \\ c & d \end{bmatrix}

But this time with no constraint on the determinant.  \mathfrak{gl}\cong M(n,\mathbb{R}).  I don’t want to break the flow of this post but I’m going to do a Lie groups and Lie algebras -3 soon where I address some of my own questions from before.  Notice, for the moment, how easy it is to obtain the above elements of \mathfrak{gl} but that this vector space is isomorphic to \mathbb{R}^4 and that doesn’t give us diddly.  This is why we need some sort of product, otherwise there’s nothing special about any of the tangent spaces of GL.

So our typical tangent vector, using local coordinates, would look like,

\vec{v}=v_1\frac{\partial}{\partial a}+v_2\frac{\partial}{\partial b}+v_3\frac{\partial}{\partial c}+v_4\frac{\partial}{\partial d}

While our typical tangent vector in the Lie algebra of R would look like \vec{w}=w\frac{d}{dt} for coordinate t.  Let g:\mathbb{R}\to\mathbb{R}.  Denote the differential of the determinant as det*.  So det^*(\vec{v})=\vec{w}  The image, \vec{w} acts on functions of R as \vec{w}(g)=w\dot{g}(0) where the dot denotes differentiation with respect to t. Let’s figure out what w should be given an arbitrary \vec{v} in the Lie algebra of GL.  We’ll use the above mentioned function, g, as dummy input.

\vec{w}(g)=w\dot{g}=(det^*\vec{v})(g)=\vec{v}(g\circ det)

The important thing to remember in the next step is that when a tangent vector acts on a function that it’s not just differentiation, but differentiation and evaluation.  The tangent vectors in the Lie algebra differentiate functions on the Lie group and then evaluate at the identity element, which for us is when a=1, b=0, c=0, and d=1.

\vec{v}(g\circ det)=\vec{v}(g(ad-bc)=1\cdot\dot{g}v_1-0\cdot\dot{g}v_2-0\cdot\dot{g}v_3+1\cdot\dot{g}v_4

Which reduces to \dot{g}(v_1+v_4)  This is just the trace of

\begin{bmatrix} v_1 & v_2\\ v_3 & v_4 \end{bmatrix}

So one way of looking at the trace is as the differential of the determinant evaluated at the identity.  So it’s  a local linearization of the deteminant for values near the identity.  The determinant of the identity is 1, for any of these matrix groups.  But the trace for the Lie algebras is going to vary.  As a spoiler, the Lie algebra \mathfrak{sl} contains only matrices of trace zero.  So there’s more.  The determinant of a matrix also tells us how it scales the space its acting on and whether it acts as a reflection or not.  Does the trace give us any information concerning this?  We’ll see.

Posted in Uncategorized | Tagged , | Leave a comment