2007 May « Rantings of an Angry Physicist

Archive for May, 2007

So you want to be a non-commutative geometer, Part I: Preliminaries

29 May 2007

OK, these are just my random notes on my studies of non-commutative geometry. I hope that someone who understands only vector calculus (and perhaps linear algebra and most certainly abstract index notation — tensors and all that jazz) alone could look through this series and become a non-commutative geometer.

So, before we begin, we should probably review: differential forms, (basic) abstract algebra, and some operator theory/Hilbert space geometry/fun stuff. This is more of a pragmatic introduction, so mathematicians will probably be shocked at this rather unorthodox introduction…but don’t worry, it’s only the preliminary material!

Differential Forms

OK, so, this is meant to be a brief review, and brief it shall be! Consider some continuous function y=f(x). A differential is defined as:

$dy = f'(x)dx$

where the prime denotes differentiation with respect to x here. $dy$ is a differential, of course! We can give it a special name like the “one form” (because there is one differential).

Now that we have a 1-form and a 0-form, and so forth, we can introduce an operator which is sort of like the cross-product from vector analysis. It is called the wedge product and it works for vectors as well as differential forms. The wedge product is defined as:

$\alpha\wedge\beta = \alpha^{[i}\beta^{j]} = \frac{1}{2}(\alpha^{i}\beta^{j}- \alpha^{j}\beta^{i})$

or for three vectors:

$\alpha\wedge\beta\wedge\gamma = \alpha^{[p}\beta^{q}\gamma^{r]} = \frac{1}{6} (\alpha^{p}\beta^{q}\gamma^{r} + \alpha^{q}\beta^{r}\gamma^{p} + \alpha^{r}\beta^{p}\gamma^{q} - \alpha^{q}\beta^{p}\gamma^{r} - \alpha^{p}\beta^{r}\gamma^{q} - \alpha^{r}\beta^{q}\gamma^{p})$

And so on and so forth (note that the pattern is 1/n! times the sum of the various permutations in the n indices similar to the definition of the determinant of a matrix). So it has the following properties:

$\alpha\wedge\beta = -\beta\wedge\alpha$ (Anti-symmetry)

$\alpha\wedge\alpha = 0$

One can think of it as a sort of generalization of the cross product. The geometric image that one should think of is (this is according to Penrose’s Road to Reality - what an utterly Platonic sounding title that is…) for say two vectors a and b, a $\wedge$ b is the oriented and scaled plane-element spanned by the independent vectors. For three vectors, it represents the oriented and scaled 3-volume element spanned by the independent vectors; and so forth. (See pages 212 to 216 of the Hard Cover first edition.)

There is this operator called the exterior derivative which essentially takes in a k-form, and returns a (k+1)-form. Suppose we had some k-form $\alpha = f_I dx_I$ where I ranges from 1 to k. We can find the exterior derivative of this:

$d\alpha = \sum_{i} \frac{\partial f_{I}}{\partial x_{i}}dx_{i}\wedge dx_I$

Note that there are some properties which are rather interesting:

$d(d\alpha ) = 0$

$d(\alpha\wedge\beta ) = d\alpha\wedge\beta + (-1)^{k}\alpha\wedge d\beta$ where $\alpha$ is a k-form (This is the leibniz property, remniscient of the usual derivative of a product in differential calculus)

Coincidentally wikipedia has a good page on exterior derivatives and wedge products…just something to go to for reference. It’s important to understand differential forms since non-commutative geometry begins with (and, as little of it as I have studied, it is) “quantum differential forms”. If you do not understand differential forms, you cannot understand non-commutative geometry!

I’d put up practice problems, but I’m lazy and pressed for time.

“But…but I don’t know differential forms! This is all greek to me!”

No worries! Here are some lecture notes, here’s something if you don’t know (or are rusty with) vector calculus.

Abstract Algebra

Now that you’ve mastered differential forms, it’s time to take on abstract algebra. Actually this isn’t so bad if you don’t mind learning definitions. If the term “abstract algebra” is absolutely new to you, go read Abstract Algebra by David S. Dummit and Ricard M. Foote. Once you have done that (be sure to do the exercises too!), go and read Algebra by Michael Artin. Then, finally, go and read Lie Algebras by Nathan Jacobson. That should equip you with the algebraic tools necessary to understand noncommutative geometry.

But suppose you live in the middle of nowhere with no access to the library and you’d still like to be a non-commutative geometer. Luckily, there are dozens of lecture notes available online for you…I’d try a whack at explaining it, but I’m exhausted and need to finish my term paper.

Operators and Hilbert Spaces

And so forth. Well, to start with, you need to understand linear algebra (linear operators, hermitian matrices, eigenvalues, etc.)…without understanding it, Quantum Mechanics becomes the fevered dream of a madman. So here are some lecture notes on linear algebra, notes on advanced linear algebra, more linear algebra, finite difference and spectrl methods in differential equations, differential equations, more differential equations, Hilbert Space fun, Hilbert Space Methods for Partial Differential Equations, (here is a semi non mathematical introduction to Hilbert Spaces for Quantum Mechanics, just what we’d like), bounded linear operators on a Hilbert Space, and if I may highly recommend for people absolutely new to this equipped only with knowledge of basic linear algebra: please read Griffith’s Quantum Mechanics! It’s a gem.

Posted in So you want to be a non-commutative geometer | 2 Comments »

Some Random Technical Papers to Read…

23 May 2007

This post is roughly divided in two: the physics half, and the operating systems half (both are fascinating subjects, it’s something one just has to accept).

Math/Physics

1) “Operating Order Ambiguities Versus Representation“

2) “Deformation Quantization of Relativistic Particles in Electromagnetic Fields“

3) “The inevitable nonlinearity of quantum gravity falsifies the many-worlds interpretation of quantum mechanics“

4) “Gauge Theory of the Star Product“

Operating Systems

1) “User-Level Socket-Based Checkpointing for Distributed and Parallel Computation“

2) “The Unix KISS: A Case Study“

Edit: Even More Papers to Look at!

1) “Virtualization: A double-edged sword“

A paper reviewing virtual machines and how god awful they are (I was hoping for more of an angry rant against Java, but what can one do?).

2) “Star Products for Relativistic Quantum Mechanics“

3) “Spin Description in the Star Product and the Path Integral Formalism“

4) “Is quantum field theory a genuine quantum theory? Foundational insights on particles and strings“

I can’t tell if this is a crack pot paper or a genius paper.

5) “Quantum Darwinism in quantum Brownian motion: the vacuum as a witness“

Posted in Uncategorized | 1 Comment »

General Covariance Generally Irrelevant In Quantum Gravity?

18 May 2007

Just a few quotes from Misner, Thorne, and Wheeler’s Gravitation (I love that book…it’s my Bible):

“By ‘prior geometry’ one means any aspect of the geometry of spacetime that is fixed immutably, i.e., that cannot be changed by changing the distribution of gravitating sources.“

(Italics are Misner, Thorne, and Wheeler’s; bold is my emphasis; this is on page 429)

“In this theory [Nordstrom's], the physical metric g (governor of rods and clocks and of test-particle motion) has but one changeable degree of freedom — the freedom in $\phi$ . The rest of g is fixed by the flat spacetime metric (’prior geometry’) $\mathbf{\mathit{\eta }}$ .“

(The underlining is my emphasis; this is on page 429)

“Mathematics was not sufficiently refined in 1917 to cleave apart the demands for ‘no prior geometry’ and for a ‘geometric, coordinate-independent formulation of physics.’ Einstein described both demands by a single phrase, ‘general covariance.’ The ‘no-prior-geometry’ demand actually fathered general relativity, but by doing so anonymously, disguised as ‘general covariance,’ it also fathered half a century of confusion.“

(emphasis is mine; this is on page 431 of Misner, Thorne, and Wheeler).

Hmm…how could an approach to quantum gravity ignore something so pivotal in classical general relativity?

For some strange reason, Misner et al. think that “the ‘no-prior-geometry’ demand actually fathered general relativity”; so that logically means that we ignore the demand when quantizing gravity.

What could possibly go wrong? (Famous last words.)

As “shocking” as it may sound, when you use the weak field approximation, you are using a prior geometry. Misner, et al., point out this is “acceptable” as an approximation if $|h_{\mu\nu}| << 1$ . (It’s even stated explicitly in equation (18.1) of their tome!)

Is it all right for calculations? Yeah, it’s a good approximation; would it be wise to use this for quantization? Not really, since “the ‘no-prior-geometry’ demand actually fathered general relativity” and the linear approximation uses a “prior geometry”.

It might be “acceptable” for the weak approximation, i.e. where $|h_{\mu\nu}|<<1$ . Would it be the “final version” of quantum gravity? No, it would be only an approximation to the “final theory” at certain, specific regions…even then, it may not necessarily look anything remotely like the final theory. It may very well only give the approximate answers in the “quantum weak field”.

Is there any reason as to expect the quantization of an approximation that ignores the cornerstone of classical general relativity to be the “final version” of quantum gravity?

“Well, all the other field theories were doing it…”

That excuse doesn’t work for drugs, and it won’t work for field theories.

If gravity worked like “any other field theory”, we wouldn’t be having this discussion, now would we? The plain fact of the matter is that gravity is different…and that’s a good thing!

“But Einstein used the linear approximations for finding the Newtonian potential, and he made the buses run on time with them. And he lived in rainbows with leprechauns and magic pixies.”

Yes, that’s great…the linear approximation is a good approximation at the right scale. It simplifies computations, but that does not make it equivalent to having the metric be fully dynamical.

It is great for classical computations where using the metric would be burdensome…it is rather terrible for the fine details. It is the fine details that we are interested in however.

No one said quantum gravity was easy, just glamorous.

“But you can bootstrap your way back up to the full dynamical metric tensor!”

Interesting point, but not always true…point-in-case: gravitons. They really aren’t exactly linearized gravity insomuch as they are a quantization of it. Logically, the classical limit should return the linearized gravity, which should return the full blown metric…but you have to first return to the classical limit if you want the full dynamical metric! So this doesn’t really tell us anything new if we quantize it…perhaps it makes the linear approximation more approximate, but it is still requiring a prior geometry at the quantum scale. That sort of defeats the whole purpose.

IF we had a final theory of quantum gravity, THEN we wouldn’t have this problem; we could easily see how to get from the quantization of the linear approximation to the final theory. However, since we don’t have the final theory, we can’t really get rid of the prior geometry from the graviton approach.

So, although this is certainly true in the classical case, for the quantization of the linear approximation it’s rather unclear. It still uses a background metric, which indicates the existence of the prior geometry. IF there were some way to get rid of the background metric, THEN I would have less of a problem; but you can’t, so I have more of a problem.

Giving hand wavy arguments that quantum gravity will look “nothing like” classical gravity is rather unsatisfactory too. This means the concept of a dynamical background goes completely out the window?! This seems unlikely, and to someone who learned general relativity before quantum theory, very unsatisfying.

Such is life I suppose.

Posted in String Rant | 4 Comments »

A Brief Special Thanks to…

17 May 2007

I just wanted to thank Dr. Woit for mentioning my blog on his site, it is a real privilege and honor to be mentioned on his blog; Dr. Carlip, for not killing me for writing this blog (if I were a professor in his spot, I wouldn’t know what I’d do); and last, but certainly not least, all the physicists and mathemagicians that have visited this site (I was really excited and honored that people like John Baez were here at least once).

I don’t know when I’ll post my next stuff, I’m intrigued right now with the problem of time and - actually thanks to some hints given by Dr. Carlip - I think I may have figured out a way around the problem. I also just realized that there are 24 days left to write my term paper, and I haven’t really started.

So, good luck to us all, and thanks for all the free fish!

[edit: this isn't my last post ever, I just am taking a break for a few days to finish my term paper!]

Posted in In Retrospect... | No Comments »

Anger is Freedom…or something…

16 May 2007

Every time someone makes a comment, I have to approve it. Coincidentally, a few of you might have your posts marked as “spam” by this automatic bayesian inference spamometer program, and I’m trying to work around it. But if your post hasn’t shown up lately, that’s probably why.

Anyways, I couldn’t help but notice one quote from another site:

I am really impressed by the openness and independence and mental energy of angryphysicist. I wish he/she would continue to write this blog and continue to evaluate physics as a free thinking insider.

How flattering, I couldn’t resist going to this fellow’s website and look at what s/he had to say. It was shocking to say the least.

Are calculations in General Relativity difficult? No. What is complex is the notation. Do you understand this? You need to eliminate the complex and cabalistic notation of General Relativity in order to make simple calculations. Calculations are not complicated, it is the notation which is complicated. And the notation obstructs easy calculations.

No, I don’t understand this. Since the calculations are so easy, perhaps you could enlighten us physicists with a large number of complex solutions to the Einstein field equation? Pretty please? The notation is irrelevant to the actual calculations, you could use various smiley faces if you wish.

Imagine someone who is so obsessed with the authority of IBM that he is still using vintage 1950 punch card driven IBM mainframe! This is your modern physicist. For physicists the authority of NewtonEinstein is sacred.

Hmm…I don’t think this fellow understood my criticism that String theory requires a prior geometry at all because the terrible irony is that I argued “Einstein is sacred, therefore there is no prior geometry by virtue of general covariance.”

My approach to quantum gravity is not by proposing outrageously nonsensical ideas (what do you take me for, a string theorist?). It’s the very opposite of that: be as cautious and conservative as possible! (This was actually impressed upon me back when I was studying with the retired rocket scientist in the CalTech library; he drilled this philosophy into me, and in retrospect that was a good thing.) Heck, I criticized the theory of the graviton as being too radical, and all it boils down to is simply applying the current QFT paradigm to General Relativity!

It’s not justified to criticize scientists who stick to theories that have not been falsified yet because they stick to theories that have not been falsified yet! Obviously Newton and Einstein were right at certain levels of approximation…otherwise we wouldn’t be taught their theories! When one can criticize scientists is when they stick to theories that have been falsified (another valid point of criticism is when a theory contradicts scientific materialism in my book).

<Just a little tangential rant about falsifiability and “faith” in current theories…

Nature works “as if” they were right at certain levels of approximation, so it’s “safe” to assume that they were correct until new unexplainable phenomena arises. Then it’s time for a paradigm shift. That’s what quantum gravity researchers are working on, because Einstein’s general theory breaks down at certain scales…most unsatisfactory!

Where things typically go awry is when there are two theories, A and B. A has more unnecessary premises than B and is reducible to B after certain manipulations…but offers no new predictions. That brings the important crucible question: “So what?” If there is no difference between the two, go with the one with fewer unnecessary assumptions. A contemporary example of a theory with more unnecessary premises that makes no new predictions is String Theory, making it impossible to falsify in a bad way.>

Physicist is stuck in the 18th century because even Einstein is Newtonian physics.

You mean Misner, Thorne, and Wheeler lied to me?! General Relativity is really Newtonian gravity?! Those mean geniuses! They hurt my feelings!

After studying General Relativity and moving indexes up and down according to Einstein convention for two decades poor physicists become science dead. After so much index tai chi no physicist can show any sign of science. I am just sorry for these doctors of physics.

But Kip Thorne has studied General Relativity for ages and he’s done very good work recently. Steve Carlip, too, is doing good work on Black Holes. And what about Wheeler?

There are always exceptions to the rules.

And I don’t really plan to stop doing General Relativity until I’m long gone…whether I do anything new and useful with it is still an open question. I doubt that I’ll make a difference (yeah, I’m cynical even with myself!)…given my rather odd entrance to the physics community online involving criticizing current theories and not offering a suitable alternative (I am, however, working naively on an alternative!) it’ll be hard to “make amends” with my faux pas. Such is life.

Posted in How...odd... | 92 Comments »

Why am I angry anyways?

16 May 2007

Some have pondered why this blog is “Rantings of an Angry Physicist” (as opposed to, say, “Rantings of an Ecstatic Physicist” :P), and there is a little story behind it.

If you go to my infamous “about” page, you’ll see in the second to last paragraph that at one point in time I wanted to be an economist (I saw it as the science of capitalism more or less). <An aside: It was during this time I was working my way through The Great Books of the Western World, and I went back to Hegel since I didn’t understand it the first time. This would be back in High School mind you. It was during this time that Ed Clark, the fellow mentioned in the previous post, said “Dialectics isn’t the science of change…it’s little more than complete bullocks! If you want to study that, study physics!” So I said to myself “Huh, I should study physics after this”. If it wasn’t for that comment, I probably wouldn’t have gotten into physics.> Well, after becoming disillusioned by the pseudo-scientific economists, I naturally became somewhat embittered towards their system where you either contribute to the pre-existing paradigm or you are insane.

I went on to something more scientific (there is little else that is more scientific than physics!). But there is a similar structure here it seems: you contribute to the pre-existing system(s) or you’re insane (or at least seen as somewhat of an odd hermit of an outcast — you know, like Roger Penrose). I’m back where I started! But I was promised that science doesn’t care about how you get your ideas or what your ideas are insomuch as how effective your ideas are at explaining and predicting phenomena (Hawking mentions this in A Brief History). I was lied to! Wouldn’t you be a little irate too?

The name “Angry Physicist” is a bit of a misnomer though, I’m rarely angry in person. But that’s the history behind my name as “Angry Physicist”, now you know!

Posted in How...odd..., In Retrospect... | 9 Comments »

Dr. Carlip’s Course on Quantum Gravity (pt 6)

15 May 2007

[Edit: This post is dedicated to Ed Clark, the fellow who first prodded me into physics; he survived the Hurricane Katrina, had a stroke, and I lost contact with him. He's in my thoughts to date.]

This basically continues the last lecture on quantization of constrained systems.

First Example: Electromagnetism

The electromagnetic tensor $F_{\mu\nu}=A_{\nu ,\mu} - A_{\mu ,\nu}$ has a gauge invariance; the following are mathematical niceties

$A_{\mu}\rightarrow A_{\mu} + \partial_{\mu}\Lambda$

$E^{i} = F^{i0}$

$B_{i} = \frac{1}{2}\epsilon_{ijk}F^{jk}$ .

$L = \frac{1}{4} F_{\mu\nu}F^{\mu\nu}$ .

Exercises: Demonstrate the following:

1) The canonical variables are $A_{i}$ , $\pi^{i}=E^{i}$ , $\{ A_{i}(x), E^{j}(x^{\prime}) \}_{t=constant} \sim \delta_{i}^{j} \delta^{3}(x-x^{\prime})$ /

2) $A^0$ is a lagrange multiplier and the corresponding constraint is the Gauss Law $\nabla\cdot E = 0$ .

3) The constraint generates gauge transformations:

$\{ \int\Lambda (x) \nabla\cdot E(x) d^{3}x, A_{i} (x^{\prime}) \} = A_{i}(x^{\prime}) + \partial_{i}(x^{\prime})$

$\{ \int\Lambda (x) \nabla\cdot E(x) d^{3}x, E^{j} (x^{\prime}) \} = 0$

Second Example: The Parametrized Nonrelativistic Particle

This second example is actually surprisingly relevant. [edit: I found this dissertation from 1995 by a one Mr. G. Ruffini, it's a good dissertation that's relevant here.] Recall the typical Lagrangian for a particle:

$L = p_{i}\dot{q}^{i} - H(p,q)$

The corresponding action is thus

$I = \int L dt = \int (p_{i}\dot{q}^{i} - H)dt = \int (p_{i}dq^{i} - Hdt)$

We introduce new variables $q^{mu} = (t, q^{i})$ , a new parameter $\tau$ which represents the time along the path of the particle (somewhat reminiscent of the proper time of relativity), $p_0 = -H$ . We can rewrite the action as:

$I = \int (p_{\mu}\frac{dq^{\mu}}{d\tau } - \lambda (p_0 + H))dt$ .

We have the gauge invariance: $\tau\rightarrow\tau^{\prime} = \tau + \delta\tau$

$\delta p_{\mu} = \frac{dp_{\mu}}{d\tau }\delta\tau$

$\delta q^{\mu} = \frac{d q^{\mu}}{d\tau }\delta\tau$

$\delta\lambda = \frac{d}{d\tau } (\lambda\delta\tau )$

Gauge invariance here is the same as parametrization invariance. The constraint is $C = p_{0} + H$ .

$\{ \epsilon (p_{0} + H), p_i \} = \epsilon\frac{\partial H}{\partial q^{i}} = -\epsilon\frac{dp_{i}}{dt} = (-\epsilon\frac{d\tau }{dt})\frac{dp_{i}}{d\tau }$

$\{ \epsilon (p_0 + H), q^i \} = -\epsilon\frac{\partial H}{\partial p_{i}} = -\epsilon\frac{dq^{i}}{dt} = (-\epsilon\frac{d\tau}{dt})\frac{dq^{i}}{d\tau}$

$\{ p_\mu , q^\nu \} = \delta_{\mu}^{\nu}$

$\{ \epsilon (p_0 + H), p_0 \} = 0 = \delta\tau\frac{dp_0 }{d\tau }$

$\{ \epsilon (p_0 + H), q^{0} \} = -\epsilon = \delta\tau\frac{dt}{d\tau } = \delta\tau\frac{dq^{0} }{d\tau }$

Recall the picture of the gauge orbits and so forth. Well, there are two approaches to quantizing this picture:

1) Reduced Phase Space Quantization the motto “Constrain then quantize”

2) Dirac Quantization the motto “Quantize then Constrain”

There are some theorems from symplectic geometry that says that reduced phase space quantization is almost always possible.

Reduced phase space quantization of the parametrized particle is seemingly elementary:

$I = \int (p_\mu \frac{dq^{\mu}}{d\tau} - \lambda (p_0 + H)) d\tau$ solve the constraint $p_0 = -H$ .

$I = \int (p_i \frac{dq^{i}}{d\tau} - H \frac{dt}{d\tau})d\tau$ gauge fix $\tau = t$ .

$I = \int (p_i\frac{dq^{i}}{dt} - H)dt$ which is the regular parametrized particle action that we began with.

In principle, Reduced phase space quantization is simple, in practice it is not always so simple. It can result in nonlocal conditions. Typically, gauge-fixing results in nonlocality. So the usual choice is Dirac quantization.

Example of Nonlocality in Electromagnetism

The gauge invariance is $A^{\mu}\rightarrow A^{\mu }+\partial_{\mu }\Lambda$ . Choose the Lorenz gauge (not to be confused with Lorentz, as in Lorentz transformation in special relativity!): $\partial_{\mu}A^{\mu} = 0$ .

We can rewrite the potential in two terms:

$A^{\mu} = \bar{A}^{\mu} + \partial^{\mu}\Lambda$ where $\partial_\mu A^\mu = 0$ .

The first term gives the line on the constrained surface where all the gauge orbits intersect the line once, and the second term involving lambda gives the gauge orbit. We can spot easily that

$\partial_\mu A^\mu = \Box\Lambda$

$\Lambda = \Box^{-1}(\partial_{\beta}A^{\beta})$

$\bar{A}^{\mu} = A^\mu - \partial^{\mu}\Box^{-1}(\partial_{\rho}A^{\rho})$

The second term ( $\partial^{\mu}\Box^{-1}(\partial_{\rho}A^{\rho})$ ) is a nonlocal expression.

Back to the Parametrized Particle (Dirac Quantization)

Recall the action:

$I = \int (p_\mu \frac{dq^{\mu}}{d\tau} - \lambda (p_0 + H))d\tau$

NOTE: The Hamiltonian is effectively zero, the parametrized Hamiltonian is $\lambda (p_0 + H)=0$ when parametrized; this is to be expected as gauge invariance is translation in $\tau$ .

Recall Unruh’s quantum clock that could possibly run backwards, it runs backwards if $H_{eff} > 0$ . Perhaps this parametrized particle has some deep meaning that avoids the problem. Note that for generally covariant systems $H_{eff}=0$ always!

Now we can introduce the wave function $\Psi [q^{\mu}]$ , and define the momentum operator as $\hat{p}_{\mu} = -i\hbar\frac{\partial}{\partial q^{\mu}}$ . There is an “auxillary Hilbert Space”, i.e. a Hilbert space not of physical degrees of freedom. We then impose the constraint
$(p_{0} + H) \Psi [q^{\mu}] = 0$ or equivalently

$(-i\hbar\frac{\partial}{\partial t} + H)\Psi = 0$ i.e. the Schrodinger equation is derived from this constraint.

We still have to define the inner product

$\langle \Psi | \Phi \rangle_{aux} = \int d^{4}q \Psi^{*}\Phi$ is not a correct inner product for the schrodinger equation, there are too many degrees of freedom. We need to gauge fix the inner product, e.g. choose $q^{0}=t$ , this gives

$\int d^{3}q\Psi^{*}\Phi |_{q_{0}=t}$ which gives the correct inner product.

There is a beautiful paper by Woodard on this, Enforcing the Wheeler-DeWitt Constraints the Easy Way. Classical And Quantum Gravity 10 (1993) 483.

Posted in Carlip's Class | No Comments »

Lubos Motl: What An Honor!

15 May 2007

I know I said I’d take the day off because I have to write an essay, but there are times when I have the urge to write prolific amounts.

The cause of this post is that Lubos Motl, the prophet of Strings, has written a prolific post on his blog dealing with the fallacious premises that I apparently begin with…unfortunately, I don’t really begin with them. Luckily String theory has the answer to everything! Kind of…

After reading through his post entirely, he either: 1) has terrible reading comprehension, 2) didn’t bother to read the posts (possibly skimmed them due to his importance as a string theorist), or 3) pulling red herrings to try and “prove” he’s right. It’s probably a combination of all three. Nearly a third to a half of the “myths” this guy “points out” is pulled out of his rectum. Well, I’m not afraid to say the emperor has no clothing (nothing’s stopped me so far).

It’s actually surprising to see a Harvard grad student not be able to read properly. No, not “surprise”, what would be the phrase I am looking for? “Horribly depressing beyond belief” yes that will suffice.

“Myth: The right theory of quantum gravity may be completely non-local“

I don’t know where Motl got this idea, perhaps he should try reading things a little closer. I previously stated:

“Worse, all the variables of general relativity are nonlocal (which isn’t necessarily a bad thing! Nonlocality is like cholesterol: there is the bad kind and the good kind, this is the good kind). Quantum Field theory deals with local variables.”

Dr. Motl rambles on to state that the correct theory of quantum gravity will be local, explains that nonlocality means that things move faster than light, and so forth. I think he needs to take this class more badly than me. There is a difference between the type of nonlocality in quantum theory which involves the wave function collapse in an EPR-like paradox transmitting information faster than light, and the nonlocality in the measurement of the position of Mercury via measuring the time it takes for radiation to hit Mercury and return to the observer with a nuclear clock. There are examples of nonlocal diffeomorphic quantities, e.g.

$\int R(x) \Delta^{-1}(x,x^{\prime}) R(x^{\prime})\sqrt{q(x)} d^{3}x \sqrt{q(x^{\prime})} d^{3}x^{\prime}$

Where $\Delta (x, x^{\prime})$ is a Laplacian quantity.

Overall there “appears to be” (uhoh, here’s another “big myth”) an inconsistency between locality and general covariance…this is the “Hole argument” in one of its many disguises. We now know better, perhaps this was what Motl was getting at but who knows.

“Myth: Quantization of gravity makes no sense at all“

Here Motl does the straightforward thing and attack the notion that quantum gravity makes no sense, what a surprise. OK, and…? He seems to be pulling this myth from his rectum.

“Myth: Gravity may be classical“

Hmm…again our dear prophet has some difficulties with his reading comprehension. The big push with semiclassical gravity was to assume gravity is classical and matter is quantized, then to see what you get.

“But but but this is impossible!” Well, semiclassical gravity is wrong (read: has been falsified), yeah. But it’s not really “impossible” (clearly not as it obviously has been formulated). As for it being physically impossible, we really couldn’t know or not…see Wittgenstein’s Tractatus for the impossibility of speaking logically of an illogical world.

“Myth: Only expectation values of operators follow their equations

The angry physicist writes something like the expectation values of Einstein’s equations, claiming that maybe, no other laws can be valid.”

Well thank goodness that Motl didn’t waste his time reading what I wrote, and instead just went directly to attack me! What a relief!

He is referring to the semiclassical gravity post, more specifically this section:

“We went on to discuss semiclassical gravity, where General Relativity is left as classical but the matter and energy is treated as quantum. He gave us another citation for this section (Page and Geilker, Physics Review Letters 47 (1981) p. 979. This changes Einstein’s field equation to
$G_{\mu\nu} = 8\pi \hat{T}_{\mu\nu}$
For the Einstein curvature tensor of spacetime $G_{\mu\nu}$ , and the energy-momentum operator $\hat{T}_{\mu\nu}$ . This does not make sense since we are equating a tensor to an operator (it’s like equating a matrix of numbers to a matrix of operators, doesn’t work!). So we could possible make it an Eigenvalue equation:
$G_{\mu\nu} \left| \psi\right> = 8\pi\hat{T}_{\mu\nu} \left| \psi\right>$
Another possibility is to have
$\langle \psi | G_{\mu\nu} | \psi\rangle = 8\pi \langle \psi | \hat{T}_{\mu\nu} | \psi\rangle$
which is the expectation value of the quantized stress-energy tensor is equal to the “expectation value” of the Einstein tensor. This approach is called semi-classical gravity and was seriously considered in the 1960s.”

Motl’s description of the passage seems to indicate he either didn’t read it well or he has poor reading comprehension. Thank goodness he attributed it to me rather than the authors who came up with the idea (they’re in the passage above too!). If you can’t find it or you’re Motl, it’s Page and Geilker, Physics Review Letters 47 (1981) p. 979.

“Myth: Gravity waves could have continuous energy“

Again, this myth is pulled from Motl’s rectum.

“Myth: Quantum stress-energy tensor isn’t conserved“

I don’t exactly know why but the angry physicist argues that the covariant divergence of “T_{mn}” is not zero in the quantum theory.”

Well, this actually was an interesting argument, since he’s arguing that semiclassical gravity (since Motl may be reading, perhaps I should explain what “semiclassical gravity” is: treat matter as quantized and spacetime as classical) apparently doesn’t have any real problem and that I made up this problem. Oh snap, I didn’t make it up: here’s a technical paper on the problem whose first section deals with what I covered. Could it be that Dr. Motl, prophet of String theory, mystic saint that instantly perceives the truth, is full of it? No, never!

“Myth: The only task is to add nice hats“

Yes, that is the only task, as I’ve stated repeatedly in every post. Drat, you found me out Motl! If only there were some way to have actually proposed this!

“Myth: In the context of singularities, the only goal of quantum gravity is to make things look finite“

There are other goals of quantum gravity with regards to singularities? What more could there be other than removing them?! Motl didn’t elaborate on the other goals of quantum gravity in the context of singularities, but he did maunder on about irrelevant topics.

“Myth: The Hilbert space of black hole microstates is universal“

I don’t think that I have even mentioned the Hilbert space of a black hole, much less its microstates. Motl appears to be unjustly attacking Dr. Carlip, but I just can’t let him going around attacking researchers who are doing actual work progressing quantum gravity (yeah, I’ve got Carlip’s back on this one, I owe it to the man for letting me be in his course).

Well, I “would” have Carlip’s back on this one if Carlip didn’t do such a crackerjack job in his technical papers! He wrote in a paper back in 2005: “The black holes we are interested in are not two dimensional, of course, and despite some interesting speculation [22], there is no proven higher-dimensional analog to the Cardy formula.” –emphasis added (The reference is to E. Verlinde, eprint hep-th/0008140) This seems pretty damning to Motl’s argument:

“This is really Steve Carlip’s myth but it naturally fits into this text. In string theory, one can calculate the entropy of huge classes of black holes and other black things with charges, angular momenta, and diverse topologies in various dimensions. The calculation typically reduces to the Cardy’s formula: the microscopic machine to get the right exponential degeneracy of states boils down to the same method of counting of states in a conformal field theory.”

Hmm…it appears that Steve Carlip has somehow magically foreseen this prophet’s criticism and added salt to injury by actually criticizing it quite well.

“Myth: The problem of time means that everyone must work with non-local observables all the time“

Yes, that’s what I scribbled down in my notes, and wrote in the blog. I actually am starting to worry about poor Motl, seeing that his reading comprehension is worse than my 10 year old sister’s.

“Myth: We don’t know how to renormalize wave functions, and thus cannot really know how to get probabilities

Probabilities are always obtained as squared absolute values of probability amplitudes and there is never a problem with the normalization of the thing that is properly called a wave function - the state vector. It can simply be defined to be normalized to one. What we call “wave function renormalization” in quantum field theory is actually a renormalization of the field operators, not the actual wave function. The word “wave function” is only used for these operators because they may be thought as arising from single-particle wave functions by the second quantization.”

Now this grabbed my attention! Largely because Motl appears to be ignoring the fact that traditionally time has played the important role of renormalizing the wave function. By making time a coordinate, you have a problem with the renormalization of the wave function, and then a problem with getting the probabilities.

“Myth: “We don’t know if quantum gravity is generally covariant” is a meaningful sentence“

This myth that Motl is criticizing is rather perplexing…well, Motl’s criticism is anyways.

Here he does away with the notion that a good theory incorporates the predictions of preceding theories, and it makes new definite predictions. Instead, a good theory of quantum gravity is dependent on preserving general covariance.

I happen to agree with him that general covariance is an important feature of quantum gravity, it may or may not look the same in the quantized theory as it does in the classical theory, but to say that all theories without general covariance are wrong is too hasty a move. Things may look phenomenally different at the quantum level than expected classically.

“Myth: The Hamiltonian is 0 in any Hamiltonian formulation of classical general relativity“

Here Motl is merely venting his rage against the ADM machine. Frankly it works, so there’s not much to contest about it; and as for the Hamiltonian being zero, that’s covered in Henneaux’s Quantization of Gauge Systems. Perhaps Motl will read it one of these days (is anyone else finding it depressing that a first year freshman has read it but a Harvard professor has seemingly forgotten all of it?).

“Myth: A major task for quantum gravity is to find a nice field redefinition“

Perhaps, perhaps not…it seems that a major task for quantum gravity is to quantize gravity, but I’m a “radical thinker” in this regard. This really isn’t a “major task” and it hasn’t been presented as such, I commented in the semiclassical gravity post:

“There are a variety of other complications, like field redefinitions. It’s ambiguous enough in quantum field theory, but now it’s extraordinarily ambiguous! This is not a “fatal” objection, but it introduces complications.”

But Dr. Motl can feel free to keep ignoring what I write for as long as he likes (and people wonder why I’m angry!).

“Myth: Ordering ambiguities are an independent problem of a local quantum theory“

I love this man, he is like a magician: pulling myths from his hat.

Myth: Perhaps we could abandon this notion of the graviton and *gasp* move forward?

The existence of gravitational waves has been proven by the pulsars so that even the Swedes are satisfied. And the existence of quanta of energy carried by these waves is essentially proven at the beginning of the text. When quantum gravity is defined at a technical level, the scattering matrix for gravitons is not only the most important set of observables we have but, in some sense, the only one.

Also, any simple attempt to show a contradiction about the existence of gravitons is a result of sloppy thinking. Gravitons neither violate laws of thermodynamics nor they create infinite recursion, and the first somewhat technical analysis one can make shows that they are philosophically analogous, with almost all details, to photons.”

Ah yes, proof of gravitational waves MUST ONLY logically have the conclusion that gravitons exist…because geons would be only too logical.

It’s rather comical actually that Motl rushes from the given proof of gravitational waves to the conclusion that there must be gravitons as if there were no other explanations out there.

“Myth: We should quantize the curvature instead“

Perhaps this isn’t a myth insomuch as it is an idea from a sleep deprived fellow trying to come up with something to quantize. This is new to Motl, but in science you can’t say “That’s bullocks!” and not offer a replacement. That’s the whole point of paradigm shifts! True, offering the curvature as something else to quantize wasn’t the best alternative, but that’s irrelevant to the fact that’s largely ignored: it is an alternative. Either I or - far more likely - someone else will offer something better…it’s something to research!

“Myth: Gravity must be treated as geometry, not a field“

I’m not saying that gravity “must” be treated as geometry. What I am saying is that given String theory’s failure as a theory, and the failure of quantization of gravity as a force, it now seems logical that we should take another different approach. But hey! Some guys like walking into a wall repeatedly.

Apparently this is a myth though, as our prophet has now declared it to be against the will of Nature.

“Myth: But a geometric approach is better, isn’t it?

In physics, the primary way of dividing theories is into correct theories and wrong theories. A general attempt to divide ideas and tools into geometric ones and non-geometric ones is typically ill-defined - it depends on the definition of “geometry” which is a matter of historical and social coincidences in mathematics rather than a matter of well-defined differences. Our understanding what geometry is has been evolving for centuries. More importantly, the approach that is labeled “more geometric”, whether or not the reasons behind this terminology are rational or not, doesn’t have to be “more correct”.”

Motl seems to have the bit between his teeth to “prove” that the geometric interpretation of general relativity is the wrong thing to quantize…apparently.

He gives no argument as to why other than “Well, you shouldn’t be picky over the mathematical formulation of a theory”…but it is formulated as a geometric theory! How can one be so blind or decoupled from reality to say “Well, yeah it’s formulated that way…but that doesn’t mean we have to quantize it that way!”?!

“Myth: Something’s wrong with the weak-field expansions because they’re against the philosophy of GR“

Oh how silly of me, the weak field approximation of General Relativity is obviously far superior to the full blown theory. That’s why the full blown theory works in the weak and strong fields whereas the weak field approximation works only in the weak field.

One serious problem that it neglects is the really important feature emphasized by Misner, Thorne, and Wheeler in chapter 17.6 of their infamous Gravitation phone book. The title of the section explains it all: “ ‘NO PRIOR GEOMETRY’: A Feature Distinguishing Einstein’s Theory from other theories of Gravity“! Perhaps our dear Motl forgets, but with the weak field condition, there is a background metric and a background (i.e. prior) geometry. Like it or not, that’s an important part of the theory of General Relativity…it has nothing to do with “philosophy” as Motl would lead one to believe.

Einstein himself relied on the weak-field expansions intensely. That’s how he derived the Newton’s potential - even though he may have been able to find the exact Schwarzschild solution, too. And Einstein has also derived the existence of gravitational waves from the weak-field expansions, even though he used to love Mach’s principle that disagreed with the existence of gravity waves.

Perturbative expansions are among the paramount tools of physics and, indeed, all of science. Whoever denies their critical importance shows that he’s not really interested in the true answers to well-posed physical questions. Don’t get me wrong: I think that the full non-linear equations of general relativity are prettier if printed on a T-shirt than some particular calculation in the weakly curved regime. Well, it’s because detailed calculations are almost always uglier than the fundamental laws. But the real importance of Einstein’s equations as well as other fundamental laws is to allow us to make calculations in concrete situations - and the weak-field situations dominate.

The fact that a weak-field calculation looks less elegant than the equations you started with doesn’t allow you to say that there’s something wrong with this calculation. Only simpletons could say that something is wrong - or not even wrong - because of these irrational reasons. And they, in fact, do. As Einstein has said, only two things are infinite - human stupidity and the Universe - and we’re not sure about the latter.

No one is denying that they are great tools for calculational purposes. What is being denied is that they are equivalent to the full theory; they are approximations that work at a certain scale and in a certain region in the gravitational field.

Quantizing an approximation to a theory isn’t really satisfying…why not quantize Newtonian gravity and say “Well, it’s really the same thing as General Relativity at the right velocities and distances”?

Is it just me or is Motl being hypocritical by appealing to Einstein here but criticizing me for appealing to Einstein’s “philosophy of General Relativity”?

Would quantizing the full blown Einstein field equations be harder than the perturbative method? Yes, significantly harder, but what did you expect from quantum gravity? A deus ex machina that magically falls out of the sky into your lap that gives you everything?

Further, perhaps more important, it shows a complete neglect for the importance of the full blown field equations, and the lack of a prior geometry. Perhaps our dear Motl should read up on his General Relativity before criticizing others on it.

“Myth: All the components should be first quantized in isolation

This is what not only the angry physicist but whole communities of people think.”

Hey look at me mom! I’ve one-upped Monsieur Molliere’s bourgeois gentilhomme: I’ve been quantizing components first in isolation all my life without knowing it!

I assume he’s apparently criticizing the use of the time components of four-vectors and tensors are constraints (I can only assume because he’s being as ambiguous as possible)…or else the canonical decomposition of spacetime into space and time. Both are more or less equivalent with regard to gravity, and yeah it doesn’t really bother me.

However, Motl’s seemingly Hegelian appeal for the “interconnectedness” of space and time (in other words his appeal to the philosophy of General relativity…where have we heard of this before?) seems misplaced. It’s not as though time is surgically removed, it’s given a new role; perhaps one could suggest the obvious that time is a man made abstraction (put that in your Hegelian pipe and have some sort of unity of opposites involving smoking it and not smoking it too).

Since the use of time components as Lagrange Multipliers “works”, it seems that Motl’s entire point is purely philosophical. Ironic he appeals to the philosophy of general relativity after criticizing those that appeal to it! Such is life.

Posted in String Rant | 6 Comments »

Dr. Carlip’s Course on Quantum Gravity (pt 5)

14 May 2007

OK, this will be the last post for a few days, I have to write an essay on the rising use of double speak in advertisement in late capitalism (I know, the subject is just an irresistible one: advertisement = propaganda). However, this should not take more than a day and Dr. Carlip is in Canada for a black hole conference this Tuesday the 15th, so I may be able to cram a few more lectures in later tomorrow.

References:

Rivers, Path Integrals In Quantum Field Theory, Chapter 6.5

Mayes and Dowker, J Math Physics, 14 (1973) 434

DeWitt-Merette (I am not sure about the spelling on that second name) et al., Phys. Reports 50 (1979) 255

<The GRW model (I think this is a somewhat legitimate pdf) is a “deterministic”, that is non-wave function and additional stochastic mechanisms, interpretation.

Gerard ‘t Hooft is working on classical models that stochastically reproduce quantum models (his work actually looks fascinating, especially with automata!).> Note that I write asides in those sort of brackets, and now a little something about the cited works.

Mayes and Dowker specifically deal with operator ordering and the ambiguities that lie therein.

DeWitte-Morette deal with stochastic equations, measurements with path integrals, etc.

The path in path integrals are typically not nice and aren’t really differentiable.

Wiener measure is the mathematical formalism about changing coordinates in path integrals.

Quantizing Constrained Systems

Begin with a Lagrangian that depends on $q$ and $\dot{q} = dq/dt$ . Recall, if you haven’t seen this before, that the action is:

$I = \int L(q, \dot{q}) dt$

Varying this action gives the equations of motion: $\ddot{q}=F(q,\dot{q})$ . There ,ay be terms with two time derivatives in the Lagrangian which can be “converted” into terms with one time derivative via integration by parts.

There is a general theorem by Ostrogradski that says in general Lagrangians with greater than one time derivative gives unstable systems, in general where the Hamiltonian is not bounded below so there is no minimum energy. Here there is another reference Carlip gave us (the man is like a wizard with his ability to seemingly conjure citations from thin air): Woodard, astro-ph/0601672.

The easiest way to turn this into a constrained system is with a new action:

$I^{\prime} = \int (L(q, \dot{q}) + \lambda C(q, \dot{q}) )dt$

where $\lambda$ = Lagrange Multiplier, then

$\ddot{q}=F(q,\dot{q})$ , $C(q,\dot{q})=0$ , $\dot{\lambda}=G(q,\dot{q})$

Just another note, the constraints in General Relativity is $G^{t}_{\mu} = 0$ .

In the canonical form, the constrained action takes the form:

$I^{\prime} = \int (p\dot{q} - H + \lambda^{\alpha}C_{\alpha}(p,q))dt$

First class constraints will be covered first (there are two classes of constraints: first and second; second class constraints are constraints which are not first class constraints). To have an intuitive picture of what constraints mean, it’s a surface in the phase space. One use of constraints is suppose we have N particles with a constant total energy and momentum, that would create a 4N dimensional surface in the phase space where the state of the system must be (supposing it were a relativistic system). Now, back to first class constraints, supposing we have several constraints:

$[ \hat{C}_{1}, \hat{C}_{2} ] = 0 \Rightarrow \hat{C}_{1} \left| \psi \right> = 0$ and $\hat{C}_{2} \left| \psi \right> = 0$ both cannot hold!

There are two conditions for a first class constraint (according to these notes of mine, note that I was rushing to jot all this down; Dr. Carlip appeared to be rather excited to teach this):

(1) $\{ C_{\alpha}, C_{\beta} \} = f_{\alpha\beta}^{\gamma}C_{\gamma}$

(2) $\{ C_{\alpha}, H \} = V_{\alpha}^{\beta}C_{\beta}$

These two conditions must be fulfilled for the constraint to be first class (first class constraints are easier than second class constraints in quantizing…I know from a nightmarish experience that sucked time away from my Winter Break; it was actually quite fun). If one or neither of these conditions are fulfilled, then you have second class constraints.

(First class) Constraints generate gauge-transformations, first class constrained systems are gauge invariant. Gauge choice is a coordinate choice on the surface in the phase space, I’ll cover this a little bit later on.

See the infamous Henneaux text for more information on quantization of gauge systems specifically, section 3.2.

Now we can find the variations of certain variables:

$\delta q = \{ \epsilon C, q \}$

$\delta p = \{ \epsilon C, p \}$

$\delta\lambda =$ to be determined.

There is a wee bit of interesting math that I have nervously scribbled down in my notebook which I’ll reproduce here (if it’s wrong…and it’s most likely wrong…feel free to inform me):

$\delta (p \dot{q} ) = \{ \epsilon C, p \} \dot{q} + p \frac{d}{dt} \{ \epsilon C, q \}$

$= \epsilon \dot{q} \frac{d C}{d q}+p \frac{d}{dt} ( -\epsilon \frac{d C}{d p} )$

$= -\frac{d}{dt} (\epsilon p\frac{\partial C}{\partial p} ) + \epsilon\dot{p}\frac{\partial C}{\partial p} + \epsilon\dot{q}\frac{\partial C}{\partial q}$

$= -\frac{d}{dt} (\epsilon p\frac{\partial C}{\partial p} ) + \epsilon\frac{d C}{d t}$

$= \frac{d}{dt} (\epsilon C - \epsilon p \frac{\partial C}{\partial p}) - \dot{\epsilon}C$

And $\delta H = \{ \epsilon C, H \} = \epsilon V C$ with one constraint:

$\delta C = \{ \epsilon C, C \} = 0$ . Now we can find the variation of the action:

$\delta I^{\prime} = ($ boundary terms $) + \int ( -\dot{\epsilon} C - \epsilon V C + (\delta\lambda )C)$ if $\delta\lambda = \dot{\epsilon} + \epsilon V$ , then $\delta I^{\prime} = 0$ .

In the phase space, there is a surface called the constrained surface (a fibre bundle - yeah I randomly spell words like a British fellow). We will continue talking about quantization of constrained systems next post too, and perhaps if I can doodle with GNUpaint well enough I might upload an image of the constrained surface in the phase space.

Posted in Carlip's Class | 2 Comments »

Categorical Logic, Message Passing, and Relational Quantum Mechanics

14 May 2007

Category theory to me is like a shiny new car to a fellow that can’t drive: it looks impressive and I want it! Category theory, though, has always been taught as little more than an elaborate game of connect the dots (there are the dots, “objects”, and then there are the arrows, the “morphisms”). Well that’s dandy, and I’m sure if I were a grad student in math I would immediately say “I understand everything now!” However, I’m not a grad student in math, so I won’t say it.

However, undaunted by this nauseatingly complicated game of connect the dots, I have attempted (more than once!) to worm my way through two books:

Categories and Sheaves by Masaki Kashiwara and Pierre Schapira, and

Topoi: The Categorical Analysis of Logic by Robert Goldblatt

Both are excellent, once you actually know category theory. They are, as far as a first year freshman is concerned, not all that great for an introduction.

Maybe it’s me, but I can’t really learn math that well that has no application. It seems to me to be little more than Platonic sophistry to list a series of definitions and at the end say “I’m a genius, woosh!” So I am looking into the introductions to categorical logic, hoping that it would actually having applications of categories in a concrete manner.

Luckily, I managed to find a good resource: some lecture notes from Steve Awody on Categorical Logic. I though they were very good notes, I learned quite a bit about categorical logic and type theory.

Still unsatisfied, I looked further for applications of category theory in programming…since I program, and I understand a little category theory, I thought it would be the perfect opportunity to learn more about the application of categorical logic. Coincidentally, I found this technical paper: The Logic of Message Passing. (If you work on a microkernel operating system, message passing is the bread and butter of the trickier parts of the kernel.) Perhaps one could think of a quantum system as a “program”, and the interactions as “passing messages”; this is just some random thought I had when I read the paper. If you haven’t realized by my about page, I think of random things from time to time.

Perhaps it would be worth-while to examine the relational interpretation of quantum mechanics like this; it has the possible advantage of context dependency. I actually looked into using domain relational calculus for relational quantum mechanics, and I think it might be worth while (an open query being “equivalent” to an eigenvalue problem, and closed queries being “equivalent” to some quantum logic like structure). Naturally, being pressed for time as an undergrad, I didn’t look into this approach very far…but it seems feasible.

Posted in Category Theory, Foundational Quantum Mechanics | 9 Comments »

Rantings of an Angry Physicist