Hello:
I was under the impression that gauge symmetry was a property of the Lagrange density. Here is the Lagrangian for EM written out in its components:
\begin{align*}
\mathcal{L}_{EM} &= J\cdot A +\frac{1}{2}\left(B^2-E^2\right) \quad eq.~1\\
&=\rho \phi - Jx Ax - Jy Ay - Jz Az \\...
Solid technical reason to skip quaternions: representing the Lorentz group
The Lorentz group is vital to modern physics. Minkowski figured out that the Lorentz boost was like a rotation in spacetime. Since quaternions do space rotations, it should be a simple step to do spacetime rotations...
Maxwell field equations
Hello:
I am glad my book is being put to use, that was the goal. This is an ongoing research project. I wish I could keep the book up-to-date, but there is only so much time in a day (I have a full time job, wife, and 2 year old).
Here is the right way to derive the...
The energy density of a gravitational system is NOT negative. Sure lots, hell, most people get this one wrong. This story was written up really well here, http://mathpages.com/home/kmath613/kmath613.htm" [Broken]. I recommend this page highly because it shows an honest struggle Maxwell faced...
Agreed. My question is how is that shown in math notation. If one takes an element of math symmetry U(1) and multiplies by an element of SU(2), the result belongs in the group SU(2). In the physics context, the electroweak symmetry U(1)xSU(2) is a different gauge symmetry than weak symmetry of...
Hello:
The gauge symmetry of the standard model is written in authoritative places like wikipedia :-) as U(1)xSU(2)xSU(3). This would have 12 elements in its Lie algebra corresponding to one photon, W+, W- and W0 or Z, and the 8 gluons. I recall reading discussions that such a...
Creatively seeing sine
Hello:
On occasion I pitch my quaternion animations to people who have written something on quaternion. The usual reply is no reply. Sometime we exchange a few emails, then go our separate ways.
This time I found out the fellow was visually impared. My animations...
Quaternions are not only Cl(0, 2)
Hello:
Clifford algebra = Geometric Algebra.
David Hestenes has promoted GA as a label over Clifford algebra because Clifford was one guy who worked on some aspects of this approach, while Geometric Algebra suggests what the math does. I only figured...
Mathematical fields
Hello Max:
We share a common suspicion about the focus of efforts in the most prestigious centers of study on the planet. Lee Smolin and Peter Woit are the biggest voices in the professional community. My focus is on constructing a viable alternative, so I like to stick...
Quaternions in Lagrangians
Hello:
Preparing the talk for Midwest Relativity Meeting was quite productive. It forced an interesting shift: I actually need to use quaternions for the action, tensors will not suffice. It is ironic that it has taken so long for me to insist on using quaternion...
Talk last weekend and this weekend
Hello:
Last weekend, I went to the Fall APS/AAPT meeting at the University of Massachusetts, Boston. They have a lovely campus, right next to the JFK library on the water. It was a sunny and warm day thanks to global warming.
The first speaker was also...
Gravity versus EM
Hello Max:
I do have a word description about these two forces that is based on looking at the equations and solutions for quite some time. Here goes...
Stuff in Nature is pretty well isolated from each other after a separation lasting 13 billion years. In an ideal...
Three definitions of a 4-force
Hello Lut:
I have used 3 definitions for force. One is the derivative of the 4-momentum with respect to the interval tau. I saw that in a discussion of the Lorentz force of EM in Jackson, the end of chapter 11. This is the dP/dt and dE/dx one. The second...
Bicycle wheels and disc galaxies
Hello:
In this post we will think about momentum for a disk bicycle wheel and a disk galaxy. The similarities between the two systems may give support to the relativistic rocket effect discussed here.
A disk has all of its mass in a plane. Both of the...
Dark energy and the relativistic rocket effect
Hello:
All forces must have a cause and an effect. For a disk galaxy, there is the Newtonian force, Fij Rr Cos(\theta), whose effect is the centripital motion. At the current time, this is all that is used to calculate the momentum profiles of...
Symmetry and torque forces
Hello:
Symmetry is useful to think about because it can pinpoint what can be ignored. Feynman emphasized looking for things that add up, knowing we can ignore things that cancel. This is what I drew on my board this morning:
The two Fv's point in opposite...
Disk galaxy momentum profile - the drawing
Hello:
To use Newton's law of gravity, masses and distances must be known. For a disk, one cannot treat it as an effective point mass. Instead, the galaxy needs to be chopped up into little bits, and add up each contribution. My plan is to do...
SUSY and dark matter
Hello Vanick:
Supersymmetric (SUSY) proposals have been with us for over 30 years. During that time, every time a new more powerful collider has come on line, part of the justification was the search for the SUSY particles. The same is claimed for the Large Hadron...
The plan
Hello:
Here is a picture of the plan I have for calculating the rotation profile of a galaxy:
I had included a higher resolution image in the previous post, but apparently it was too large to be accepted here (600x450 pixels, 57k works).
Doug
Disk galaxy momentum profile - the plan
Hello:
Personal stuff:
I spent the month of July drifting. I was bummed that neither the trip to Brazil nor the MIT GR course created lasting dialogs with the participants. Although I am familiar with not connecting, both events involved...
Hello Ghost_of_PL:
> most likely this is my last post on this subject
This discussion is becoming repetitive, so that is good news for general readers of this thread.
Your weakest statement is this assertion:
> Your matrix representation can fare no better since it also admits stand...
Square of a quaternion
Hello Ghost_of_PL:
There are only 4 slots in a quaternion, something that is obvious from the 4x4 real matrix representation used in the Mathematica notebook. Looking back, I not only used a variation on a common definition (which would normally have i, j, k for e_1...
Basis vectors and quaternions
Hello Ghost_of_PL:
Good old fashioned algebra rocks! The only way I really learn something is with paper and pencil (so I can erase a lot). I did not understand what your issue was, but now I have a better grasp of it. I did define what I meant by a...
Inverse of curved quaternions
Hello Ghost_of_PL:
Great question! I had not looked at all the details, and your post got me to think about them.
There is hope that the algebra represented in equations 11-15 will be a division algebra since as M approaches zero, the algebra approaches the...
Force
Hello:
I am taking the General Relativity course 8.06s at MIT's Professional Institute. It is great fun that I get to think and chat about physics for four days. It has been work the 4 vacation days and $2k for the class.
At lunch, one fellow said his biggest concern was with...
Cosmology
Hello Joe:
I have not been able to download the 2.8G torrent yet, so I will present some of my thoughts about cosmology independent of that video.
Cosmology has major problems, and they know it. I try and be careful in discussing these issues, to call it by the problem with the...
Picturing quantum field theory
Hello:
There are two groups that understand the odd ways of relativistic quantum field theory.
The first group are the elite people who hang around the physics departments of major research universities. They don't bother to teach quantum field theory to...
Screws and Quaternions
Hello:
How would you described the motion of a screw? It is a combination of linear and angular motion, with the angular motion perpendicular to the linear. At the meeting in Brazil, I learned the way most people handle this problem: they use dual quaternions. I had...
Hello Lut:
I was projecting to the minds of three dead men, Hamilton, Grassmann, and Clifford, whose lives where lived within the 1800's, well before the insights of 1905.
Doug
The geometric product
Hello:
Today I will discuss the geometric product, a tool used extensively for people applying Clifford algebras to problems in physics and robotics. Here is the definition:
ab = a.b ~+~ a \wedge b \quad eq~1
There are a number of great features to this...