The Good Will Hunting Experiment at Harvard

16 Mar 2009

7 replies [Last post]

Mon, 03/16/2009 - 00:12

doug

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Hello:

I conducted the "Good Will Hunting" experiment Monday, Feb. 2 at Jefferson Labs at Harvard University. My boss let me go early to see "Confronting the Dark Energy Crisis in Fundamental Physics" by Prof. Christopher Stubbs. He is an experimentalist and chairman of the physics department. The talk was packed, the mix of his stature and style - he doesn't mind using sarcastic digs at our current state of confusion. He put on a good, informative show.

Before the talk, they have tea and cookies in the library on the 4th floor. This is a social disaster for me since I know no one. I sit on the edge, say nothing. Instead, I decided to write out my proposal for a new unified field theory on a public blackboard, to see if anyone would notice. Here is what I wrote

Start with the best - Maxwell

$S_{EM}=\int \sqrt{-g}d^4 x(-J^{\mu}A_{\mu} - \frac{1}{4}(\nabla^{\mu} A^{\nu}-\nabla^{\nu} A^{\mu})(\nabla_{\mu} A_{\nu}-\nabla_{\nu} A_{\mu}))$

Upgrade tensors, allow * & / - ie quaternions

$S_{qEM}=\int \sqrt{-g} d^4 x(-J A - \frac{1}{4}(\nabla A - (\nabla A)^*)(A \nabla - (A \nabla)^*))$
$...(0, -E + B)(0, -E - B)=(B^2-E^2,2 E \times B)$

Upgrade to SU(2) symmetry

$S_{qWeak} = S_{qEM}:A\rightarrow exp(A - A^*)$

Upgrade to U(1)xSU(2) symmetry

$S_{qElectroweak} = S_{qEM}:A\rightarrow \frac{A}{\left |A \right |} exp(A - A^*)$

Upgrade to U(1)xSU(2), SU(3) symmetrty

$S_{qStrongElectroweak} = S_{qEM}:A\rightarrow (\frac{A}{\left |A \right |} exp(A - A^*))^* \frac{B}{\left |B \right |} exp(B - B^*)$

Upgrade quaternions to hypercomplex numbers

$q:i^2=j^2=k^2=-1\quad ij=-ji$
$q=\begin{bmatrix} t & -x & -y & -z\\ x & t & -z & y\\ y & z & t & -x\\ z & -y & x & t \end{bmatrix}$
$hc:i^2=j^2=k^2=+1\quad ij=ji$
$hc=\begin{bmatrix} t & x & y & z\\ x & t & z & y\\ y & z & t & x\\ z & y & x & t \end{bmatrix} \quad \%\quad Eigenvalues$
$S_{hcG}=\int \sqrt{-g} d^4 x(-J \boxtimes A^* - \frac{1}{4}(\nabla \boxtimes A^* - (\nabla \boxtimes A^*)^*) \boxtimes (A \boxtimes \nabla^* - (A \boxtimes \nabla^*)^*))$

Apply Euler-Lagrange to 6

$\rho=-\nabla \cdot \frac{\partial A}{\partial t} + \nabla^2 \phi$

Static solution if phi is constant

$\begin{align*} \rho &= -\nabla \cdot \frac{\partial A}{\partial t} + \nabla^2 \phi \\ &= \nabla(\partial \phi - \Gamma) = -\nabla \Gamma \end{align*}$
solution is
$g_{\mu \nu}=\begin{bmatrix} e^{-\frac{2 GM}{c^2 R}} & 0 & 0 & 0\\ 0 & -e^{\frac{2 GM}{c^2 R}} & 0 & 0\\ 0 & 0 & -R^2 & 0 \\ 0 & 0 & 0 & -R^2 sin^2 \theta \end{bmatrix}$

Metric passes all 1st order PPN tests
Quadrapole is the lowest mode of emission for a gravity wave
Metric differs from GR for light bending around the Sun (0.8μarcsec more bending)

Needless to say I have not heard back from Harvard.

Mon, 05/04/2009 - 16:59

Lowell

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Re: The Good Will Hunting Experiment at Harvard

Doug,

$60 a minute to talk to chickens, man you are serious!

GEM allows one to swap in different symmetries because the potential is more than a 4-vector, it is a division algebra

H is a 4-D Normed Division Algebra over the Reals! Now all i've got to do is define this in technical terms and...

One thing I have failed to understand is why objects like H's or even O's for that matter have been completely resigned to math departments. I get the basic notion of how much more versitle an H is than a 4-vector. From what I've read' H's got the boot when guys like Gibbs and Heaviside hailed vectors as supreme creatures and they made the Maxwell equations shrink in volume considerably. Although the two formulations are supposed to be equal, is there not greater 'insight' into things (e.g., the way the components interact) when you put Maxwell in H's? When one takes the div and curl of an H is this even similar to a vector? I could hunt this info down, but your very good at explaining this kind of stuff.

Question about the D formula,

$D_{\mu} = \partial_{\mu} - i g_1 \frac{Y}{2} B_{\mu} - i g_2 \frac{\tau^i}{2} W^i_{\mu} - i g_3 \frac{\lambda^a}{2} G^a_{\mu} \quad eq 1$

when you put this up did you have to type in everything by hand from [image source] or did you start at [title] and type the actual formula? I thought that mathtex should have the correct HTML command stuff at the front. I really do need to mess with this tool a little.

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Lowell

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Lowell

Mon, 05/04/2009 - 23:56

doug

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Pretty input

Hello Lowell:

As webmin, I just made your post look pretty with two obscure steps. Below the text box, there should be a blue underlined link that says "input format". You must click on that, then choose the "Full HTML" format. Voila! The default does not appear to handle anything right, and I don't want to waste time debugging Drupal filters. Choose input format->Full, and your post will look better.

Usually I just type tex stuff. I will sometimes go to codecogs.com. At some point, I will try to get this editor to call that codecogs tool which is done very well.

Now to the substance...

Being a researcher, trying to do new things with quaternions, I have not been obsessive about getting all the details of the twisted history of quaternions. Few know that like most other things, Gauss discovered quaternions first, wrote about them in one of his notebooks, and did not publish on the topic. Hamilton - the physics hot shot of the day - tried to find them for ten years (Gauss probably got to them in five minutes being Gauss). He wrote up the story of the discovery which has its own history in the history of math. Rodrigues did it independently for rotations. A school grew around Hamilton, the folks taking it very seriously. Yet they could not explain something simple: what does it mean to multiply two quaternions together? Heavyside was well-known for being heavy handed. He and some others ripped into the quaternion school. It was every bit as nasty as the discovery of complex numbers. The complex numbers side won, while the quaternion folks lost. Gibbs gutted quaternions so well, claiming he knew what div, grad, curl and all that look like, that only math history wonks learn about them, along with game programers and rocket scientists to do 3D rotations a la Rodrigues.

Maxwell had big time respect for Hamilton. He tried to write his proposal using quaternions, but it did not work. The first edition had a section where he wrote three vectors as pure quaternions, quaternions where the scalar is zero. It was a chapter devoid of insight.

I think that section was dropped by the third edition. In the introduction, he speculated that someday someone would figure out how to write all this stuff using quaternions. It is one of my most outragious accomplishments that I filled a request of James Clerk Maxwell. I should say that Peter Jack did find my first road to those equations a year before me using a combination of commuting and anticommuting operators. It is my current belief that finding the action using real-valued quaternions was a deeper and more useful connection to EM because it generalized to the other forces of the standard model, and by switching quaternions to the hypercomplex, may reach out to gravity.

I have not made enough animations with div, grad, and curls to say I understand them. The more harsh view: anyone who talks about div, grad, and curl doesn't know how to tell a complete story of Nature. One can generate those with the quaternion operator $(0, \nabla)$ acting on a pure quaternion function, (0, F) (note: I am using the quaternion ASCII convention, where a capitalized symbol means a 3-vector). Shift the reference frame, and one has $(\frac{\partial}{\partial t}, \nabla)(g, F)$ . Since one can always change the inertial reference frame, I look at equations and see if they contain the fab 5: a time derivative of the scalar, a negative div, a grad, a curl, and a time derivative of the 3-vector. This is one reason I am such a consistent jerk insisting on writing all the parts of a quaternion out. It is now easy for me to spot incomplete stories, a fun trick to apply repeatedly.

Doug

Tue, 04/28/2009 - 21:00

dougsweetser

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The quadrapole

Hello:

The quadrapole moment for an isolated source is due to conservation of energy and momentum, and the fact that gravity has only one sign. There are isolated electric dipoles because there are two signs for electric charge. For gravity, all an isolated source can do is wobble like a water balloon.

People who add another field into general relativity run into trouble because that additional field can store both energy and momentum. Therefore one can have a dipole moment for the gravity waves which are shown not to exist by observations of binary pulsars.

Sun, 05/03/2009 - 16:35

Lowell

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Re: The quadrapole

Do you mean isolated electric monopoles? Sources and Sinks for charge.

Gravity obviously has only a sink (as it sucks!) What I'm have difficulty understanding is how a gravity wave is actually modeled. Don't the equations tell you in the limit of a static solution that it's just the local curvature of space-time? Then to do wave mechanics you have to assume a change in density of space-time between two points of potential.

How do the people studying GR and cosmology come to the conclusion that gravity can oscillate in a T-mode? Is this to give it a velocity at or near c?

<em>People who add another field into general relativity run into trouble because that additional field can store both energy and momentum. Therefore one can have a dipole moment for the gravity waves which are shown not to exist by observations of binary pulsars.
</em>
Is this an attempt to "extract" a linear conversation of energy-momentum from a gravitating system?

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Lowell

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Lowell

Mon, 04/27/2009 - 15:29

technogeeky

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Re: The Good Will Hunting Experiment at Harvard

It is important to note that the "even" representation (or, more specifically, the use of hypercomplex numbers) and the resulting quadrapole is just the *minimum* configuration; depending on the particular restrictions used, there may be a finite or infinite number of pole configurations.

In terms of Clifford algebras (which the even representation is not!), if you use the following restriction:

closure (under multiplication)

Then the resulting space is a multivector spanned by 2^k bases (where the bases are):

In the Clifford algebra, this results in: ij = -ji, and i(jk)=+(ij)k.

Note however, in the above "even" representation that this would *not* be true; instead, ij = ji and the i(jk) = ij(k) = ijk. I am not totally sure of this, however.

As Doug says, this needs to be analyzed in more detail. It is possible that this representation implies an infinite number of configurations (greater than a quadrapole).

-tg

Sat, 04/25/2009 - 13:29

Lowell

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Re: The Good Will Hunting Experiment at Harvard

I suppose Mini Driver didn't make a cameo appearance?

In step 3 it looks like you equate Maxwell and Yang-Mills, i.e., U(1) and SU(2). Do quaternions give you the ability to combine gauge theory in this manner?

Concerning statement 10.quadrapole is the lowest mode of emission for a gravity wave

Is this due to the fact that gravity in it's simplest representation is a second order phenomenon? So a quadrapole would be one inward traveling wave and one outgoing wave, right?

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Lowell

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Lowell

Tue, 04/28/2009 - 21:40

doug

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U(1), SU(2), and SU(3)

Hello Lowell:

The way these groups are handled in the standard approach to the standard model is to stapling another chicken to the others. You might say that farmer-like phrase is garbage, but that reflects the "covariant derivative". Let me write it from page 77 of "Modern Element Particle Physics: The Fundamental Particles and Forces" by Gordon Kane:

$D_{\mu} = \partial_{\mu} - i g_1 \frac{Y}{2} B_{\mu} - i g_2 \frac{\tau^i}{2} W^i_{\mu} - i g_3 \frac{\lambda^a}{2} G^a_{\mu} \quad eq 1$

The players include three coupling factors (the g's), the symmetry generators Y, tau, and lambda, and the potentials, which have internal symmetries, the i and a parts. I will be spending about $600 on Monday to complain about this for ten minutes (a national APS meeting in Denver), and expect a bad ROI.

Instead of adding in whatever we see, GEM allows one to swap in different symmetries because the potential is more than a 4-vector, it is a division algebra. At this point, I do not know how to solve any problems with the weak force, so I cannot say if my formalism has a chance of working. Pros would immediately point to how gamma matrices have to be used. Yet gamma matrices can be written with quaternions (a cool bit of research done by others, it is all 16 products of the form i q j, where the i and j take on the 4 values of the basis vectors, honest!). The part I am least clear on is how to handle the coupling constants in a meaningful way. Sure, I could just add them on, but that is like stapling a chicken.

Doug