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...