Background Independence for the local Xray technologist

[marcus]

A PF poster recently asked about the Background Independence feature
that distiguishes classical 1915 General Relativity from Stringy theories.

She suggested that the explanation be at the "basic Xray tech" level.

That probably means starting with a short exerpt from Smolin article
in January 2004 SciAm, "Atoms of Space and Time". I am a little unsure about what Xray technologists know. Maybe a bunch of physics already. Still let's start with Smolin.


-------quote page 68----

...In the mid-1980s a few of us...Ashtekar...Jacobson...Rovelli...decided to reexamine the question of whether quantum mechanics could be combined consistently with general relativity using the standard techniques. We knew that the negative results from the 1970s had an important loophole. Those calculations assumed that the geometry of space is continuous and smooth, no matter how minutely we examine it, just as people had expected matter to be before the discovery of atoms.

Some of our teachers and mentors had pointed out that if this assumption was wrong, the old calculation would not be reliable.

So we began searching for a way to do calculations without assuming that space is smooth and continuous. We insisted on not making any assumptions beyond the experimentally well tested principles of general relativity and quantum theory. In particular we kept two key principles of general relativity at the heart of our calculations.

The first is known as background independence. This principle says that the geometry of spacetime is not fixed. Instead the geometry is an evolving, dynamical quantity. To find the geometry, one has to solve certain equations that include all the effects of matter and energy. Incidentally, string theory, as currently formulated, is not background independent; the equations describing the strings are set up in a predetermined classical (that is, nonquantum) spacetime.

The second principle, known by the imposing name of diffeomorphism invariance, is closely related to background independence. This principle implies that, unlike theories prior to general relativity, one is free to choose any set of coordinates to map spacetime and express the equations. A point in spacetime is defined only by what physically happens at it, not by its location according to some special set of coordinates...

...By carefully combining these two principles with the standard techniques of quantum mechanics, we developed...[the means]...to do a calculation...
That calculation revealed, to our delight, that space is quantized. We had laid the foundations of...loop quantum gravity...

------end of exerpt-----

The principle is Background Independence because that is the feature that takes work to arrange for, when building a theory.

The automatic thing to do when you construct a theory or any kind of mathematical description of nature is to start with some kind of set geometry---some type of graph paper in effect.
Then when you have constructed the theory it will be apt to depend on the geometry you started with. Background dependency is not a feature so much as something that happens to a theory if you aren't careful.

A favorite background geometry in "Quantum Field Theory" is called "Minkowski space". It is basically a kind of 4D graph paper with a certain distance&angle function (a "Minkowski metric") that AE made prominent in 1905.

The hard thing, which AE achieved in 1915, is to construct a theory, where you don't start with any preliminary geometry, no "metric" at all, in other words----you just have an amorphous continuum to begin with and you say how geometry (distances and angles) emerges and evolves in relation to the matter and energy flowing around in it.

In Relativity (meaning General, from this perspective the 1905 "special" business doesn't really count) the continuum ends up merely being a mathematical convenience used in defining the geometry. Things that happen, material occurrences like A meeting B, are what define the places and times. Individual points in a continuum don't have physical meaning since any point would do as well as any other for some event to happen at.

So what's a good link?
Well, you could try some of Rovelli's "Quantum Gravity" (a draft is online) skipping any mathematical formulas and obscure terminology. In other words, read for the downtoearth examples and whatever is in plain language.

Page 7 of Rovelli's book (1.1.3 "the physical meaning of general relativity", 1.1.4 "background independent quantum field theory")

Page 47 of Rovelli's book (2.2.5 "general covariance")

Page 53 (2.3.2 and surrounding pages)

Diffeomorphism invariance is discussed in the "general covariance" section---they mean about the same thing.

Other people here may have ideas of good entry-level discussion of BI and DI.

Maybe the source of any mystery is this: if you have some experience with mathematical models of nature then it seems like a miracle that AE's 1915 could be BI and DI----you can scootch around the space and the stuff in it any old way and the equation keeps on working! You can build the theory on wet kleenex (look ma no graph paper!) and
it creates beautiful structures growing from the amorphous blob by the magic of differential equations. Etc Etc Etc. But if you have
NOT played around previously with differential equation models of nature then BI and DI probably don't seem miraculous or remarkable or very noticeable, even. This is why its hard to explain them. They are pretty simple but as soon as I tell you what they are you probably wonder "is that all?"

Oh yes, consequences. Nondescript as they may seem BI and DI have
profound consequences (besides making GR, the prevailing theory of gravity, fundamentally incompatible with conventional Quantum Mechanics and the like) I mean desirable consequences--getting rid of unwanted infinities (eliminating divergences & singularities), getting discrete Planck-scale area and volume spectrums. But discussing such consequences makes it seem as if there is a choice about whether to have those features in ones fundamental picture of nature.

well maybe someone else will supply some better links for the basic Xray technologist. anyway that's all for now.

 

[chroot]

marcus: I edited the thread's title. I assume you actually meant "Background," not "Backyard," yes?

- Warren

 

[marcus]

Originally posted by chroot
marcus: I edited the thread's title. I assume you actually meant "Background," not "Backyard," yes?

- Warren

Well, how about Barnyard Independence

good old rugged frontier self-reliance and all that
a patriotic note?

 

[Tsu]

Originally posted by marcus
Well, how about Barnyard Independence

good old rugged frontier self-reliance and all that
a patriotic note?

Well, I already have a really good grasp on Barnyard Independence. Our friends sheep always used to escape, as well as our neighbors cows and our very own goats were total escape artists...

HEY!...HOLY SMOKE! I was just talking with Ivan about all of this and he gave me some real good translations and descriptions for a bunch of things...all my loose threads have tightened up...I THINK I GET IT! (at least sort of...) WOW! SO COOL! GOTTA GO READ AND THINK MORE! SO MUCH INFO...SO LITTLE TIME! THANKS, EVERYONE! KEEP POSTING! I'M A READING FOOL, NOW! I'M SO EXCITED!

 

[marcus]

Tsunami your interest in quantum gravity is a delight and warms the heart but it takes actual questions to stir the people at these forums into motion. If you don't ask a question that catches the attention of somebody or other then nothing happens. Earlier you asked what was the BI principle and whether it had anything to do with the popular singer-entertainer Michael Jackson. This seemed reasonable enough and answers were forthcoming.

By the way do you think we should call it Background, Backyard, or Barnyard Independence. Your recent post suggests you prefer the term Barnyard Independence.

 

[Tsu]

Originally posted by marcus
Tsunami your interest in quantum gravity is a delight and warms the heart but it takes actual questions to stir the people at these forums into motion. If you don't ask a question that catches the attention of somebody or other then nothing happens. Earlier you asked what was the BI principle and whether it had anything to do with the popular singer-entertainer Michael Jackson. This seemed reasonable enough and answers were forthcoming.

By the way do you think we should call it Background, Backyard, or Barnyard Independence. Your recent post suggests you prefer the term Barnyard Independence.

Actually, it was Ivan who suggested that Michael Jackson was a possible candidate as a prime example of diffeomorphism invarience! (He cracks me up BAD! ) He was explaining the relationship of DI to background independence - and doing a fine job of it, I might add. It actually helped to break the words down into some semblance of meaning for me!

Yes, I do believe I prefer 'Barnyard Independence'. It reminds me of our goofy escape-artist goats that we used to have!

More questions:
I've been trying to understand supersymmetry. Do you have some magic words that will turn on THAT lightbulb for me?

How does it fit in with LQG and ST?

BTW, I just LOVED your analogy of the universe to the kitchen sink (in another thread SOMEWHERE - I'm so all over the place here I don't remember where it was)! It was 'elegant'!

 

[Ivan Seeking]

Hey, I just help to interpret the posts made by Marcus.

He da man!

Jeff too but we won't get into that.

 

[marcus]

Ivan, we have to rotate
explaining things to the local Xray technologist is
fun and the job should circulate
Let's ping selfAdjoint for the next round
(unless you have some other suggestions)
it should be someone whose dream of glory is
to rival John Baez as an explainer and of course
would never admit to this. Dont you think sA qualifies

 

[Tsu]

Originally posted by marcus
Ivan, we have to rotate
explaining things to the local Xray technologist is
fun and the job should circulate
Let's ping selfAdjoint for the next round
(unless you have some other suggestions)
it should be someone whose dream of glory is
to rival John Baez as an explainer and of course
would never admit to this. Dont you think sA qualifies

Sounds good to me! I enjoy sA's posts very much! 'Annoyance Distribution' is what we call student training rotations in my line of work. I've googled John Baez - LOVE his cousin, Joan (she really IS his cousin according to Wikipedia!) but there is a WHOLE LOT on him! Do you know of any particularly good 'explaining' links?

So, is Jeff the one I should hit up for M-Theory? {edit: HA! I can hear him screaming "Nooooooooo!" from here!} You know, I really don't want to waste anyone's valuable time with such piddley little stuff as my interest in basics...SO...

I would like to suggest sticky's at the top of each forum with lists of links for beginners like me to assist them in learning these basics. With something like this, we might even be able to come in with some halfway intelligent questions! (ie. "GR for Dummies") A lot of us don't have the time to take formal classes for this type of thing, and a short (OK. i realize nothing about this stuff is going to be 'short'...), basic, self-study module (list of links) would be so awesome! Then I could quit bothering you 'heavy-weights'. (This does NOT let YOU off the hook, Ivan! I plan on bothering YOU a LOT! QUITE a lot...)

 

[Tsu]

OK! I see that there are 67 views as opposed to 8 replies in this thread. Those of you who are popping in here to see just how dumb I can sound or if I fall on my face - start posting some links for me! Make yourself useful! I can laugh at myself all BY myself! I don't need your help! I NEED LINKS! (please and thank you)

cc: MY Thread

 

[Ivan Seeking]

Originally posted by marcus
Dont you think sA qualifies

He da udder man!

I have tremendous respect and admiration for a number of our members...but I'll never admit it! Oh yes, I just did.

You have taught more physics to Tsunami in a few weeks than I have managed in twenty years! Don't know...maybe I alienate her with the pop quizzes...and she was really unhappy with that last report card that I issued.

 

[selfAdjoint]

I just do so love to be volunteered, especially with flattery!

Let me tackle one of the X-ray technologist's likely problems with diffeomeorphism invariance. What is a diffeomeorphism?

So to get a picture in our minds let's take a curved surface. It could be a sphere, or a torus, or a hilly landscape. Now we set up a coordinate system on it; since the surface is curved or hilly, the coordinate llines will be curved or wiggly too, since they have to track the surface.

Now the coordinates we have set up are just arbitrary. Somebody else could come along and set up a different coordinate system. Maybe instead of a North and South pole on the sphere they prefer an East and a West pole. Or maybe instead of x-y coordinates they prefer polar coordinates, or maybe - well you can see there's an infinite number of ways someone might set up a coordinate system on that one surface. And really folks, you can calculate the curvature of the surface at a point from the coordinates (Gauss proved this) but that curvature doesn't truly depend on the coordinates; all the valid coordinates should give the same value.

This raises the question, what is a valid coordinate system? And that is where diffeomorphisms come in. Suppose you have one good coordinate system [sidebar: on a sphere there is no perfect coordinate system. You will always have at least one singularity; latitude & longitude has two, at the poles (longitude becomes undefined). This is a deep topological fact. end sidebar] But suppose you have one that satisfies you, and you want to know which other ones will be that good.

Well say you good coordinates are x and y, and the new ones will be x' and y' (this doesn't limit my choices, any coordinate system on a surface will have two coordinates and if I name them x and y that doesn't mean they're cartesian). Now the change of coordinates gives you a couple of functions x' = f(x,y) and y' = g(x,y), just saying that each of the new coordinates at any point depends on both the old coordinates of the point.

Now to keep the coordinates nice we want to avoid sudden jumps. That means to functions f and g have to be continuous, and we also want to have no kinks or corners in the coordinate line, so we also want the derivatives of f and g to be continuous. And in fact let's go all out and demand that f and g have continuous derivatives of all orders. Sometimes that is called smooth. It means the change of coordinates will itself not introduce any kind of singularity (jump or kink or whatever) at any level. There may be singularities in the new coordinates, but that will only be because there were singularities in the old coordinates. The property of having continuous derivatives of all orders defines a set called $$C^{\infty}$$, so we would say $$f,g \in C^{\infty}$$

So a change of coordinates with that smoothness property is called a diffeomorphism. And the idea that properties of the manifold - like curvature, don't change when you do any diffieomorphic change of coordinates is diffeomorphism invariance. If the physics on the manifold is related to its curvature, then that should be diffeomorphism invariant too.


Two cases where physics is diffeomorphism invariant: General relativity and the worldsheet of a (super)string.

 

[Tsu]

Originally posted by selfAdjoint
I just do so love to be volunteered, especially with flattery!

Let me tackle one of the X-ray technologist's likely problems with diffeomeorphism invariance. What is a diffeomeorphism?...

Well, OK... so if "we would say (your equation which I don't know how to write on this board)", does that mean that the woman peyaba and the man peyaba and the tantan cole baca lemongrass? And the leedy root, goody root, belly root - UH! And the fearless granny scratch scratch?

OK! GOT IT!

Um, did I mention that I hate math? The farthest I got was trig - a look at a Math Analysis book sent me into seizures secondary to severe twisty face.

So! - we have two apples...

Actually, Ivan helped me out a lot in translating the Math-ese. (Ya know... I LOVE x-ray. It can't go around corners.)

It's all as clear as mud now!

(God, what am I doing here? )

Can any of this be even SLIGHTLY comprehended without higher math, or am I just trying to blow away a dead bear, here?

 

[Ivan Seeking]

Not to worry sA,
We have talked and she gets it. She just needs some time to digest the information and to straighten out her face again. It's all twisted up.

 

[selfAdjoint]

If anyone has a question, I'd be delighted to (try to) answer it. This is like the basic stuff you want to get clear in your head before you go on.

 

[Tsu]

Well, OK! Background Independence! I think I've GOT IT!
(the rain in spain stays mainly on the plain...)