tom thinks

date 2000-10-23:12:19

Physics The argument from invariance is one of the basic forms of argument in modern physics, yet it isn't one that gets talked about much. It tends to come up only in the most fundamental areas, whereas the bulk of any subject is in its applications. If you have an undergraduate degree in physics then the odds are the only time you've seen the argument from invariance is in relativity and possibly an advanced quantum course, but it really should be more widely known than that. Engineers use it sometimes, particularly in dimensional analysis, but they seem to treat it with a bit of embarrassment, as if it wasn't a real argument. But it is.

The basic idea is simple: if reality doesn't change under some transformation, your description of reality had better not change under that transformation either.

Like many simple ideas, this one has consequences that aren't obvious.

The earliest form of this argument is found in Newton's concern with the relativity of motion. Newton was a theist, and believed that space and time, being manifestations of God, were absolute. He had a physical argument about space, which involved a spinning bucket of water and the curvature of the water's surface. Because the curvature didn't depend (he thought) on anything but the rotation of the bucket in absolute space, he argued this proved absolute space existed. We like this argument, and so it get's mentioned in elementary physics texts. Newton's argument about time, which says that time is a manifestation of God's will, and is therefore absolute, is much more rarely mentioned.

It's a curious feature of Newton's physics, though, that his equations of motion are invariant under changes in position and velocity. If I move my billard table two feet to the left, the balls will still behave the same way relative to the table's edges, and if I write the equations that describe them in terms of their distances from some point on the surface of the table, my equations will stay the same even though the table has moved. Everything is invariant under the transformation of the table's position.

Likewise, if the table were put on a smoothly moving train, nothing would change either.

The first cracks in this ediface came when it was realized that Maxwell's equations, which describe electro-magnetism, were not invariant under changes in velocity. They seemed to say that observers moving at different speeds would see different things, but this didn't seem to be born out by experiment.

There were two ways out of this trap, the first of which is actually more typical of the way the argument from invariance is used. The first way is to assume the existence of a new entity that makes the system invariant. The second way is what was actually used in the case of relativity, which is to come to a new understanding of the nature of the transformation, and show that the actual transformation is different from the one you thought you were making.

In relativity, Einstein showed that whereas people thought that if you were moving close to the speed of light and increased your velocity by a little you would then be moving faster than the speed of light, what actually happened was you would be moving a little bit closer to the speed of light, not faster than it. The reason why the equations of motion weren't invariant under simple linear addition of velocities was not because the equations of motion were wrong, but because velocities don't add linearly!

Einstein's solution has a powerful unifying effect that the alternate explanation didn't have, but the alternate explanation -- that there exists an entity called the luminferous ether -- seemed a good deal simpler to many people at the time. The ether was endowed with properties, so went the story, such that it's effect on moving objects was to make them shorter in the direction of motion, and to make time appear to run slower, so that if all observers would just write their equations of motion relative to the co-ordinate system where the ether was at rest, everthing would be invariant again.

The problem was that the ether's properties, although they flow very naturally from the equations of electro-magnetism, also made this speculative entity absolutely undetectable, and physicists don't like undetectable entities. So Einstein's solution was much preferred. An excellent discussion of the classical ether theory is given in David Bohm's book Special Relativity, which is available as a Dover Paperback.

A more typical example of the argument from invariance is the Dirac equation. Paul Dirac developed a wave equation that was invariant under the transformations of special relativity, that describes electrons. The equation had two strange properties. The first was that it allowed negative energy solutions. In classical physics these would be summarily thrown away, but Dirac didn't see any physical grounds for doing so, and although his initial belief that they described protons was wrong, it turns out that they describe positrons, the anti-particle to the electron. This is an example of how you can get more out of an equation than you put in -- Dirac wanted an equation that was only constrained by the requirements of special relativity, and once he had that equation he felt bound to take it's solutions seriously, however strange they seemed, because there didn't seem to be any other equation that would do the job.

The second strange property of the equation was that if you transformed it by rotating the co-ordinate system it was described in, the form of the equation changed. This is bad, because the universe at large is famously invariant under rotations.

The mathematician Emy Noether showed in the early 1900's that when a system was invariant under a given transformation, there was a physical property that was conserved (whose value didn't change.) Systems that are invariant under translations conserve momentum, systems that are invariant under changes in time conserve energy and systems that are invariant under rotations conserve angular momentum. If Dirac's equation changed under rotations, it meant it described a world where angular momentum is not conserved, and not the world we live in, where it is. This would be bad.

Dirac didn't give up, however. He realized that he could add a new term to the solutions of his equation that wouldn't stop it from being a solution, and would also exactly cancel the effect that a rotated co-ordinate system would have. The existence of this term had an odd effect -- it meant that even when the object described by the equation was standing still, not rotating or anything, it would have some angular momentum. This is the intrinsic angular momentum of the electron -- which is observed experimentally -- and it is the "entity" introduced to make solutions to the Dirac equation invariant under rotations.

In modern particle physics the argument from invariance is used to predict the existence of particles in various theories. It turns out that we want our equations to be invariant under gauge transformations, which describe a change in the phases of the waves involved. In the everyday world, the change from daylight-savings to standard time is a gauge transformation -- everyone changes the phase of their clocks by an hour, so all the relationships between clocks stay the same.

Forcing the equations of particle physics to be invariant under gauge transformations has lead to the prediction (and subsequent experimental discovery) of the W and Z bosons that mediate the weak nuclear force. It's lead to the prediction of a great many particles that don't exist, too, but that's what we have experimentalists for.

There's a whole 'nother aspect to all this, which is the "creative" use of the argument from invariance, as exemplified by Jayne's work on the nature of probability, that I'll get to later.

SelfConciousness What's the point of writing all this?

I'm doing this for love, and for myself, as a way of keeping track of my own notes and ideas and thoughts and doings. I write about stuff that interests me. A lot of this is stuff I was writing to Carolyn about anyway, and this is a way of sharing a bit of that with the world. If people read and enjoy this, good. If they don't enjoy it, who cares? No one forced them to read it.

The writing here is rough and unpolished -- there's virtually no editing, and obviously no spell checking. This is a place to experiment with ideas and to accumulate notes on stuff I'm working on, letting people see the act of creation in progress.

I've written pretty much every day since I was fourteen years old, starting on October 13th, 1976, and this is just an extension of that. One of the few bits of advice everyone from Nabokov to Rand gives to young writers is that you should write every day. Given my years of it, you'd think I'd be better at it, eh? But there's too much you're not supposed to say, and I let that hold me back. Well, I'm done with that.

This is also an exersize in courage. I haven't published very much because I haven't had the guts to expose that much of myself, and that makes for bad writing. But I've got some stuff now that I really want to publish, and to do that I've got to be able to put it out into the world and say: this is me, reject it if you like. That isn't easy, and routinely hitting the "publish" key on this wonderful journal-entry program that Carolyn has created helps get me over that.

Poem There is no fate
No tide that moves our lives
Nor any harbour safe
Secure from all the storms
All surprises
Where effortless at anchor we may rest

We sail the widest seas
Run before the gale
Fret in stale sargasso's grip
Or whirl in maelstrom's hurricane of water
Sucked beneath the surface
Where monsters lurk

We may keep station with a close companion
With whom we share this world
Other times we sail alone
Or in keeping with a fleet
Yet always voyaging on
Without harbour, without cease

Reading The Count of Monte Cristo, the new Penguin edition, unabridged (of course) translated by Robert Buss. I read The Three Musketeers last summer and enjoyed it, and am enjoying this even more. The plot is twisty, the narrative swift, and the thought that Dumas wrote this in daily installments awe-inspiring. Crime, punishment, revenge, sex, drugs, money.... Something for the whole family.

Metaphysics by Richard Taylor. Picked this one up because Carolyn recommended it. The prose is light and he's more interested in presenting the issues than making a case, and sometimes the issues are pretty silly, or have premises (like eliminative reductionism, in the case of his presentation of materialism) that cry out to be explicated and eliminated.

Hitchhiker's Guide to the Galaxy by Douglas Adams. Rereading this to review for Enlightenment -- it's at least as funny as I remember, although I'll still argue that Terry Prachett is funnier over the long haul. But this was Adams' first book, and it's a good deal funnier than the earlier Discworld stories.

Find Enlightenment

date	2000-10-23:12:19
Physics	The argument from invariance is one of the basic forms of argument in modern physics, yet it isn't one that gets talked about much. It tends to come up only in the most fundamental areas, whereas the bulk of any subject is in its applications. If you have an undergraduate degree in physics then the odds are the only time you've seen the argument from invariance is in relativity and possibly an advanced quantum course, but it really should be more widely known than that. Engineers use it sometimes, particularly in dimensional analysis, but they seem to treat it with a bit of embarrassment, as if it wasn't a real argument. But it is. The basic idea is simple: if reality doesn't change under some transformation, your description of reality had better not change under that transformation either. Like many simple ideas, this one has consequences that aren't obvious. The earliest form of this argument is found in Newton's concern with the relativity of motion. Newton was a theist, and believed that space and time, being manifestations of God, were absolute. He had a physical argument about space, which involved a spinning bucket of water and the curvature of the water's surface. Because the curvature didn't depend (he thought) on anything but the rotation of the bucket in absolute space, he argued this proved absolute space existed. We like this argument, and so it get's mentioned in elementary physics texts. Newton's argument about time, which says that time is a manifestation of God's will, and is therefore absolute, is much more rarely mentioned. It's a curious feature of Newton's physics, though, that his equations of motion are invariant under changes in position and velocity. If I move my billard table two feet to the left, the balls will still behave the same way relative to the table's edges, and if I write the equations that describe them in terms of their distances from some point on the surface of the table, my equations will stay the same even though the table has moved. Everything is invariant under the transformation of the table's position. Likewise, if the table were put on a smoothly moving train, nothing would change either. The first cracks in this ediface came when it was realized that Maxwell's equations, which describe electro-magnetism, were not invariant under changes in velocity. They seemed to say that observers moving at different speeds would see different things, but this didn't seem to be born out by experiment. There were two ways out of this trap, the first of which is actually more typical of the way the argument from invariance is used. The first way is to assume the existence of a new entity that makes the system invariant. The second way is what was actually used in the case of relativity, which is to come to a new understanding of the nature of the transformation, and show that the actual transformation is different from the one you thought you were making. In relativity, Einstein showed that whereas people thought that if you were moving close to the speed of light and increased your velocity by a little you would then be moving faster than the speed of light, what actually happened was you would be moving a little bit closer to the speed of light, not faster than it. The reason why the equations of motion weren't invariant under simple linear addition of velocities was not because the equations of motion were wrong, but because velocities don't add linearly! Einstein's solution has a powerful unifying effect that the alternate explanation didn't have, but the alternate explanation -- that there exists an entity called the luminferous ether -- seemed a good deal simpler to many people at the time. The ether was endowed with properties, so went the story, such that it's effect on moving objects was to make them shorter in the direction of motion, and to make time appear to run slower, so that if all observers would just write their equations of motion relative to the co-ordinate system where the ether was at rest, everthing would be invariant again. The problem was that the ether's properties, although they flow very naturally from the equations of electro-magnetism, also made this speculative entity absolutely undetectable, and physicists don't like undetectable entities. So Einstein's solution was much preferred. An excellent discussion of the classical ether theory is given in David Bohm's book Special Relativity, which is available as a Dover Paperback. A more typical example of the argument from invariance is the Dirac equation. Paul Dirac developed a wave equation that was invariant under the transformations of special relativity, that describes electrons. The equation had two strange properties. The first was that it allowed negative energy solutions. In classical physics these would be summarily thrown away, but Dirac didn't see any physical grounds for doing so, and although his initial belief that they described protons was wrong, it turns out that they describe positrons, the anti-particle to the electron. This is an example of how you can get more out of an equation than you put in -- Dirac wanted an equation that was only constrained by the requirements of special relativity, and once he had that equation he felt bound to take it's solutions seriously, however strange they seemed, because there didn't seem to be any other equation that would do the job. The second strange property of the equation was that if you transformed it by rotating the co-ordinate system it was described in, the form of the equation changed. This is bad, because the universe at large is famously invariant under rotations. The mathematician Emy Noether showed in the early 1900's that when a system was invariant under a given transformation, there was a physical property that was conserved (whose value didn't change.) Systems that are invariant under translations conserve momentum, systems that are invariant under changes in time conserve energy and systems that are invariant under rotations conserve angular momentum. If Dirac's equation changed under rotations, it meant it described a world where angular momentum is not conserved, and not the world we live in, where it is. This would be bad. Dirac didn't give up, however. He realized that he could add a new term to the solutions of his equation that wouldn't stop it from being a solution, and would also exactly cancel the effect that a rotated co-ordinate system would have. The existence of this term had an odd effect -- it meant that even when the object described by the equation was standing still, not rotating or anything, it would have some angular momentum. This is the intrinsic angular momentum of the electron -- which is observed experimentally -- and it is the "entity" introduced to make solutions to the Dirac equation invariant under rotations. In modern particle physics the argument from invariance is used to predict the existence of particles in various theories. It turns out that we want our equations to be invariant under gauge transformations, which describe a change in the phases of the waves involved. In the everyday world, the change from daylight-savings to standard time is a gauge transformation -- everyone changes the phase of their clocks by an hour, so all the relationships between clocks stay the same. Forcing the equations of particle physics to be invariant under gauge transformations has lead to the prediction (and subsequent experimental discovery) of the W and Z bosons that mediate the weak nuclear force. It's lead to the prediction of a great many particles that don't exist, too, but that's what we have experimentalists for. There's a whole 'nother aspect to all this, which is the "creative" use of the argument from invariance, as exemplified by Jayne's work on the nature of probability, that I'll get to later.
SelfConciousness	What's the point of writing all this? I'm doing this for love, and for myself, as a way of keeping track of my own notes and ideas and thoughts and doings. I write about stuff that interests me. A lot of this is stuff I was writing to Carolyn about anyway, and this is a way of sharing a bit of that with the world. If people read and enjoy this, good. If they don't enjoy it, who cares? No one forced them to read it. The writing here is rough and unpolished -- there's virtually no editing, and obviously no spell checking. This is a place to experiment with ideas and to accumulate notes on stuff I'm working on, letting people see the act of creation in progress. I've written pretty much every day since I was fourteen years old, starting on October 13th, 1976, and this is just an extension of that. One of the few bits of advice everyone from Nabokov to Rand gives to young writers is that you should write every day. Given my years of it, you'd think I'd be better at it, eh? But there's too much you're not supposed to say, and I let that hold me back. Well, I'm done with that. This is also an exersize in courage. I haven't published very much because I haven't had the guts to expose that much of myself, and that makes for bad writing. But I've got some stuff now that I really want to publish, and to do that I've got to be able to put it out into the world and say: this is me, reject it if you like. That isn't easy, and routinely hitting the "publish" key on this wonderful journal-entry program that Carolyn has created helps get me over that.
Poem	There is no fate No tide that moves our lives Nor any harbour safe Secure from all the storms All surprises Where effortless at anchor we may rest We sail the widest seas Run before the gale Fret in stale sargasso's grip Or whirl in maelstrom's hurricane of water Sucked beneath the surface Where monsters lurk We may keep station with a close companion With whom we share this world Other times we sail alone Or in keeping with a fleet Yet always voyaging on Without harbour, without cease
Reading	The Count of Monte Cristo, the new Penguin edition, unabridged (of course) translated by Robert Buss. I read The Three Musketeers last summer and enjoyed it, and am enjoying this even more. The plot is twisty, the narrative swift, and the thought that Dumas wrote this in daily installments awe-inspiring. Crime, punishment, revenge, sex, drugs, money.... Something for the whole family. Metaphysics by Richard Taylor. Picked this one up because Carolyn recommended it. The prose is light and he's more interested in presenting the issues than making a case, and sometimes the issues are pretty silly, or have premises (like eliminative reductionism, in the case of his presentation of materialism) that cry out to be explicated and eliminated. Hitchhiker's Guide to the Galaxy by Douglas Adams. Rereading this to review for Enlightenment -- it's at least as funny as I remember, although I'll still argue that Terry Prachett is funnier over the long haul. But this was Adams' first book, and it's a good deal funnier than the earlier Discworld stories.