Jason Allen Blood

Feb

General Relativity in Matrix Formalism

Hello, and welcome to a new stream of articles I plan to post regarding matrices and how that can apply to Physics as a possible replacement for pure vectors / tensors / spinors. The goal here is to show that by simple assumptions regarding the matrices in question, we can derive properties of the natural world that was postulated before.

Our world is not as simple as we see around us. In a way, the world of three space dimension and one time dimension is really a projection from a much more (pun intended?) complex world. I’m going to assume a basic knowledge of special and general relativity here, so if you aren’t familiar with those theories you’d be better served first by looking into them.

In relativity, time and space are part of one mathematical construct called space-time. The idea is that time is really another space dimension, with the only “weirdness” coming from the fact that time is of an opposite sign in the metric. In other words, the relationship between time and space is hyperbolic, while between two space dimensions is follows standard Euclidean geometry. In this way, while we have one mathematical construct (R^4) we really have to assume that time is the dimension that stands out. Where does this assumption come from? Well, that’s all described by Einstein so I won’t go into it. Suffice to say, the original assumption of the constant nature of the speed of light leads to this geometry.

Quantum physics requires us to consider mathematical entities that contain complex valued elements. For the purposes of this first article, we will look only at vector fields, which fall directly under the realm of the tensor mathematics used on general relativity.

Consider the all important line element from general relativity:

$l^2 = g_{\mu \nu} A^{\mu} A^{\nu}$

If we were to allow A to contain complex values, then the above definition would create complex values for the square of the length. We don’t (at this point) have any sense of meaning for this, and so the natural requirement is to adjust the above definition so that the length squared is still a real number. We can accomplish this via a Kahler requirement:

$l^2 = g_{\mu \nu^*} A^{\mu} A^{\nu^*}$

With the additional requirement that the metric is Hermitian:

$g_{\mu \nu^*} = g_{\nu^* \mu}$

Thankfully, we can rewrite these equations in a much more friendly way with matrices:

$l^2 = A^+ g A$

with

$g = g^+$

Moving forward we will take the convention of using lower case letters for covariant elements and upper case letters for contra-variant elements. Our affine connection, defined as a more general way to describe a “constant vector”, can also be written in matrix format:

$dA = dx^{\mu} \Gamma_{\mu} A$

Giving us a relationship between the metric matrix and the affine matrices:

$d(l^2) = dx^{\mu} A^+ \left ( \frac{\partial g}{\partial x^{\mu}} + g\Gamma_{\mu} + \Gamma^+_{\mu}g \right ) A = 0$

This has a simple matrix solution for the affine connection:

$\Gamma_{\mu} = -\frac{1}{2}g^{-1}\frac{\partial g}{\partial x^{\mu}} + ie\phi_{\mu}$

The extra unknown field is a direct result of allowing vectors to be complex valued, and corresponds to phase shifts in vectors as they are transported from one place to another. In this way, the geometry must be extended to include not only a metric matrix but also a phase field. These two together define our geometry. We can also write the matrix equivalent of the curvature tensor:

$R_{\mu \nu} = \frac{\partial \Gamma_{\mu}}{\partial x^{\nu}} - \frac{\partial \Gamma_{\nu}}{\partial x^{\mu}} + \Gamma_{\mu} \Gamma_{\nu} - \Gamma_{\nu} \Gamma_{\mu} + ieF_{\mu \nu}$

Where F is defined as:

$F_{\mu \nu} = \frac{\partial \phi_{\mu} }{\partial x^{\nu}} - \frac{\partial \phi_{\nu} }{\partial x^{\mu}}$

Using some simple calculus, we can rewrite the curvature matrices as functions of the metric and the phase-torsion field F:

$R_{\mu \nu} = -\frac{1}{4}\left ( \frac{\partial g^{-1}}{\partial x^{\mu}} \frac{\partial g}{\partial x^{\nu}} - \frac{\partial g^{-1}}{\partial x^{\nu}} \frac{\partial g}{\partial x^{\mu}} \right ) + ieF_{\mu \nu}$

We also have:

$R^{+\mu \nu} = -\frac{1}{4}\left ( \frac{\partial g^{-1}}{\partial x_{\mu}} \frac{\partial g}{\partial x_{\nu}} - \frac{\partial g^{-1}}{\partial x_{\nu}} \frac{\partial g}{\partial x_{\mu}} \right ) - ieF^{\mu \nu}$

We can put these two together to create a hermitian singular matrix which we associate with the Lagrangian of our theory:

Feb

The Purpose of Life

Dare I say I know what the meaning of life is? Dare anyone? If you are a religious person, that is if you believe in some variation of the notion of a God, then inevitably your purpose is tied to whatever God’s plan is for you. This is less a statement of revelation as it is a definition. ”God’s Plan” for us is the purpose of our lives. Thus, for the religious among us, the know your purpose is to know God’s plan. While there are many versions of “God” in modern society, ranging from traditional Judeo-Christian ideas to post-modern ideas to my very own (to be posted later), we will stick with the one postulated by Christianity. This is not me conceding the truth of this idea, but given my upbringing it is the definition to which I am most familiar.

So, here is what I know as well as I know it, and the chain of logic I have applied to reach my end conclusion regarding God’s plan for mankind (and thusly yourself). I am going to paraphrase a lot here, and certainly you are welcome to expand in your own mind on anything I quote as needed, but most people in this country are familiar with the stories of the Bible enough to get the idea. There are two primary types of beings associated with the stories of the Bible: Angels and Man. Angels were created first by God, and were built to love Him. That is, they have no free will as it pertains to this. God created them for the express purpose of loving Him, and while this worked for a time, He ultimately concluded it was not enough. Forced love, after all, has less meaning. Thusly, God created Man.

Man was given the most important and powerful gift of all God’s creations: free will. But why? Well, as indicated with the Angels, free will was granted because God wanted beings to love Him via their own choice. This form of love is more fulfilling and meaningful. Naturally, this also meant some among us would NOT love God, but God had the wisdom to know that the benefits of true love outweighed to loss of souls against Him. Ultimately, thus, the primary purpose of Man’s existence (and hence ours) is to use our God given free will to find a true love for God.

So, that said, what does that ultimately mean to people? How should we live our lives in order to achieve something that is easy to write yet hard to accomplish? Understand that this notion not only explains WHY we are here, but also explains the way God behaves. Why is He so hard to detect in daily life? Why not just appear before us and make Himself known? To do so would defeat our very purpose, and so God remains “behind the scenes”. He does not interfere directly in people’s lives in any obvious way, opting instead to stay hidden within the fabric of creation.

The first and frankly most important thing we can take from this revelation is the importance of our free will. To abandon it, in any form, is to turn our backs to God. Many spiritual organizations, all created by Man, attempt to bring us closer to God but have the unfortunate side effect of encouraging people to abandon their free will. ”Faith” as a word and idea is actually counter God’s plan for us. He does not want us to assume He exists. He wants us to FIND Him. He does not want us to believe in Him, he wants us to KNOW. This is something that can only be accomplished via exhaustive searching and questioning. Many people will make the mistake of confusing faith with knowledge, of following others rather than leading themselves to the answers. It is a natural tendency of people to be blind followers, but this is NOT what God wants from you. No man, woman, or child needs a church or a minister to find and know God. They need only the world, their own heart, and that most precious of all gifts: free will.

Secondly, we must question all the supposed truths contained within religious documents. God would never write a book. God would never appear directly before the masses. God is all around us, part of us, between us, yet He exists nowhere. He is not a white bearded being floating in the sky. He is the spirit of the wild, He is the soul of the universe. As is often done in Man’s existence, ideas and notions become distorted by the greedy, selfish, and power hungry. These people will always leverage our faults to their own gain, and our biggest fault is the willingness to abandon our free will; to relegate responsibility for our own actions to another. There is only ONE truth: God. The Bible is NOT the word of God, it is the words of men trying to interpret Him. Don’t look for God using someone else’s glasses, use your own. Does it feel right to hate people you don’t know simply because someone else tells you to? Does God really care about the outcome of Football games, or your gambling efforts? Would God create a gay man only to hate him?

This leads to an important idea here. Since our purpose is to use our free will to find love for God, it means that all other aspects of life exist for the purposes of serving that goal. Challenges in life, triumphs, defeats, all work to serve that purpose. All people, regardless of upbringing or specific beliefs, have the same ultimate opportunity. Things and behaviors that don’t directly interfere with others achieving that goal cannot be considered wrong. This then gives a REASON why things like murder are wrong. To kill another person is to stop their chance to achieve the true love that God seeks. Free will is a truly great power, and like power can be used for good and evil. To use free will for evil purposes is to use it in a way not intended by God. That considered, choosing fatty foods over healthy, or choosing a gay lifestyle over others, is not a choice that ultimately matters in the grand scheme. God does not want us to interfere in the free will of others.

Libertarianism, the idea that one should decide their own fate and be responsible for themselves without government interference, is a direct extension of this idea. The opposite, Fascism (government or religious), is the antithesis of what God wants from Man and is thus an evil notion. True Americanism, a variation of Libertarian principles, is in fact directly in line with what God wants and gives people the maximum opportunity to fulfill their life’s purpose.

Oct

Windows 8 and Microsoft’s Legacy

So, I’ve been working with Windows 8 now for a couple days and I’m sure you are all dying to know how I feel about it. Well, let me start out by saying that I am currently clicking “Update” in WordPress every couple of sentences as I write this, because Windows 8 freezes every few minutes. That’s right, freezes. As in, nothing works. No mouse, no keyboard, nothing. I literally have to hard reboot in order to get back up and running.

I’m hopeful that a slew of Windows Updates that I’m installing will fix this issue since it makes the computer virtually useless, not that the update really made the computer any more useful to begin with. So, let’s jump in with what I like and dislike, and then get into a fun discussion about Microsoft as a company.

Windows 8 likes:

Not really a Windows 8 specific thing, but MS did get it right when they paired the menu of a program with the actual window for the program (this only applies to Windows 8 in windowed mode). The shared menu bar in OS X is stupid and annoying. It makes hot swapping between apps slow and clunky.
I like the active tiles. The desktop area on a Mac is basically wasted space. While older Windows allowed users to clutter their desktops with junk (allowing the dipshits among us to showcase themselves by the piles and piles of crap there), the tile interface is cleaner and forces the user to start thinking about management of their stuff in a better way overall. While those who are used to the cluttered desktop will need to change how you “organize” your stuff, and that may be painful, who cares about you anyway. You suck and need to get a clue, so now’s your chance!
IE 10 is a step in the right direction. While it still has rendering issues, it’s far better than any other “browser” they have every put out there. Too bad it took over 15 years for them to jump on this whole Internet standards thing.

Windows 8 dislikes:

Now, I like minimalism is user interface design. I think that getting too fancy often just clutters things. Windows 8, however, went WAY overboard here. The mail app is so basic looking it’s actually visually confusing. You can’t tell where one message in the list starts and the other ends, and the display just feels totally soulless. Adding subtle gradients and lines between rows, across the board, would help everything.
The whole mouse in the upper right corner to bring up a menu thing is fucking dumb and annoying. If you are like me, you have multiple monitors. That means trying to line up your mouse just right on the left monitor to bring this menu us is hard to do.
Swapping between apps in metro view is not clear and clunky. Why wouldn’t you have a metro-esque bar across the bottom showing the running apps? Even a way to quickly close them? Fucking duh!
Here is a bit of insight, having been a Microsoft developer now for nearly 20 years. Windows 8 is just Windows 7 with a Silverlight wrapper. The core of the operating system is exactly the same, and as a result, is riddled with all the same issues that have plagued Microsoft since they first developed Windows (remember Windows 95? Yeah, that’s right, this is really the same damn operating system). Now, Microsoft will never admit this, going as far as trying to claim they re-invented tons of shit, but I know differently. I know its the same god-awful architecture that is riddled with bugs, easily hacked, and basically a fucking joke through and through. Expect issues, LOTS of issues.
OS still can get viruses? Yup. Fucking stupid in this day and age. OS X and Linux have had this issue licked for a long time. Not like Microsoft has to actually invent anything here, they just need to copy what works. But, naturally, they snub their noses at the consumers and continue on with what they perceive as “cheaper” than replacing their shitty software with something that works. The ineptitude of this company is truly staggering.
Metro uses a common application menu. Remember how I noted that a common menu, like the one OS X uses, is fucking annoying and stupid? Yeah, so Microsoft took the only thing they got right in Windows and tossed that out in metro view. Now, in a full screen / tablet user interface one might be inclined to think this was a good idea. After all, in this style of interface you don’t really ever have more than one app up at a time. Problem is, #1 Windows isn’t JUST an operating system for tablets, and #2 Apple was able to do away with that on their devices and it worked just fine. Menus are designed for mice anyway, so the reality is there shouldn’t be a menu PERIOD, let alone a single shared one. You see, one of the biggest design no-nos is re-purposing interactive elements (buttons, links, etc) for different effects. It confuses users. Having a single “Settings” button that changes what it does depending on the app is confusing. Having different Settings buttons, one per app, or a global settings area where the user can drill into a specific app is more clear and user friendly. Context for a user should never be implied, but should always be chosen. Windows 8 fails at this massively.
App store. There is nothing here. App store concept is good, glad to see them get on that bandwagon, but there own apps aren’t even in there. It would be like Nintendo releasing a new console but not a new Mario game.
Flat, flat, flat. Everything is boxy and solid colors. BORING. This isn’t cool, it’s dumb.
Back button? Anyone? App navigation is fucking awful.

I could actually talk forever about how bad this OS is. IF it were a new kernel, and not just Windows 7 wrapped in Silverlight, I would be able to give Microsoft credit for trying something new. But, they pooched it. Unlike Google, who straight ripped off Apple, Microsoft looked like they were trying something new. That’s actually a good thing! BUT, the issue with Windows really isn’t it’s lack of sexy panels or overall interface. It’s always been, and continues to be, the crappy software architecture it’s built on.

Some of you will complain that Windows has to support more hardware yadda yadda for reasons as to why they have issues. Frankly, that’s just bullshit. Microsoft has been running the show for so long now they don’t have excuses. They could have forced manufacturers to adhere to any standards they wanted, making everything easier for them and work better overall, but they didn’t. It is clear form their choices (and lack thereof) that they have just been content to make money retreading the same crap over and over again. The “don’t fix it if it ain’t broken” business philosophy. Problem is, it IS broken. I’ts been broken for decades. They’ve really coasted by because their hasn’t been a decent alternative.

So, while Microsoft has been happy raping their customers, things have changed right from underneath them. The cost of hardware has dropped, allowing companies like Apple to creep into the PC market with lowing costs. The way people use their computers has also changed, in part due to lower hardware costs with diminished returns, and in part due to the internet. As hardware got fast and cheap enough, people stopped seeing any significant gains in upgrading their memory or processor. As their use of the computer changed, specifically to more internet based activities that generally use less power on the machine, the need for high end computers diminished. All these forces play perfectly into Apple’s business model of producing a quality overall computing experience along with treating a computer like a consumer appliance.

People no longer care about high end customization of their computers. A few basic choices do well. They plan on “upgrading” through full device replacement every few years again due to the diminished return issue. Is it really worth spending hundreds of dollars to get a 20% processor boost when all you do is read email and browse the web? While some of us, myself included, DO care about such gains due to how we use our computers, we are the minority.

Microsoft has fallen asleep at the wheel, frankly barely being awake from the beginning, and have not found a way to adjust their overall business model for the times. The hardware they are now producing (ex: Slate) is clunky, heavy, and not in tune with their only strong market. So what should Microsoft do?

Microsoft should become the Apple of the business market. Apple, for whatever reason, has little to no interest in this market place. Microsoft still has a footing here, and could really seize control should they make a few changes. First, develop a real operating system. Windows is garbage. Toss the kernel and start over, using SMALL team of developers, and model your architecture after Linux. Secondly, get into the hardware game. Make business-ready devices like Apple is doing on the consumer side. Yes, that might even mean getting into chip manufacturing. When you control the hardware, you can make the software work SO much better.

Oct

Square Root of Relativity

It should be noted that the line element in general relativity is always written in a square format. This has an unfortunate consequence of loss of geometric information. The square root of this format may in fact be either positive or negative, but in fact can be written in an even more general way:

$ds = \sigma_{\mu}dx^{\mu}$

Here we have replaced the metric tensor with a set of 4 2×2 matrices. This still leaves a total of 16 unknowns. We can write the relationship between the metric and these matrices as follows:

$2g_{\mu \nu} = \hat{\sigma}_{\mu} \sigma_{\nu} + \hat{\sigma}_{\nu} \sigma_{\mu}$

The hat here represents the adjoint matrix. If we use the Pauli matrices as a basis, we can rewrite these equations as follows:

$ds = \gamma_{\mu}^{a} \sigma_{a} dx^{\mu}$
$g_{\mu \nu} = \gamma_{\mu}^{a} \gamma_{\nu}^{b} \eta_{ab}$

The gamma values are often referred to as “tetrads”. $\eta$ is just the Minkowski metric, though it is not in fact a tensor. The tetrads transform as vectors under a coordinate change, but the elements marked with regular letters don’t change under coordinate changes as they are in fact just constants. The 4 vectors represented by the tetrad gamma holds all the information for our geometry.

Let’s go ahead and look at what happens when we take the differential of the length of a vector, assuming the vector has zero covariant derivative (is a constant vector):

$dl = d(\gamma_{\mu}^{a} \sigma_{a} \psi^{\mu}) = \sigma_{a} dx^{\tau}(\frac{\partial \gamma_{\mu}^{a}}{\partial x^{\tau}} \psi^{\mu} + \gamma_{\mu}^{a} \frac{\partial \psi^{\mu}}{\partial x^{\tau}}) = \sigma_{a} dx^{\tau}(\frac{\partial \gamma_{\kappa}^{a}}{\partial x^{\tau}} + \gamma_{\mu}^{a} \Gamma_{\tau \kappa}^{\mu}) \psi^{\kappa}$

Since this vector is constant, we will assume it’s length is also constant, meaning the above value must be zero. This then gives us an equation relating the Christoffel symbols and the tetrads (we have dropped the tilda over the Pauli matrices, but they are just the static matrices).

$\frac{\partial \gamma_{\kappa}^{a}}{\partial x^{\tau}} + \gamma_{\mu}^{a} \Gamma_{\tau \kappa}^{\mu} = 0$

This is just the equation for 4 constant contravariant vectors. Plugging this back into our equation for the metric tensor gives us back our original equation relating it to the Christoffel symbols.

Jul

On The Differential Geometry of C^4

Let us suppose we are working in a 4 dimensional complex space. We will define the metric tensor as e, and construct the space so that this metric a Hermitian matrix.

$\epsilon^{+} = \epsilon$ (1)

From this space, we will also define a projection to the real space using the following mapping (written as a matrix equation):

$\left \{ x_0, ix_j \right \}=Z^{+}\gamma_{\mu}Z$ (2)

We have chosen this particular projection so as to allow the four space-time variables to be both positive and negative. Dirac originally multiplied the gamma matrices with gamma 0 to force the time element to be positive, as this projection was used for the probability density. Here, we have yet to deduce anything quantum mechanical. Rather, we are only creating a single C^4 space and defining 4 real variables by the above equation. With this projection in mind, it is possible to write the derivatives with respect to Z and Z+ in terms of the 4 real variables x:

$\frac{\partial }{\partial Z} = \frac{\partial x_{\mu}}{\partial Z}\frac{\partial }{\partial x_{\mu}}=Z^{+}\gamma_{\mu}\frac{\partial }{\partial x_{\mu}}$ (3)

The presence of the i in front of the space coordinates cancels out in our derivative. Also, we have assumed the gamma matrices do not dependent on Z. This is not true in the more general case and will (as we will see later on) require us to introduce the C^4 affine connections. For now, let us assume our space is flat.

We can take a further derivative with respect to Z+, and assuming our tensors and metric in C^4 depend only on Z we get:

$\frac{\partial^2 }{\partial Z^{+} \partial Z} \equiv \bigtriangledown^2 = \gamma_{\mu}\frac{\partial }{\partial x_{\mu}}$ (4)

Jumping ahead a bit, lets go ahead and write a C^4 equivalent of a wave equation:

$\bigtriangledown^2 \psi = \gamma_{\mu}\frac{\partial \psi }{\partial x_{\mu}} = k\psi$ (5)

Jul

Karma Police

Had the seed of an idea for a new book…

The Karma Police are a secret group whos purpose is to enforce balance (karma) by righting wrongs. In essence, when the system fails to achieve justice, these guys come in and punish those who deserve punishment. They focus on higher profile cases, and have different departments for different levels of offenses.

The story centers on our protagonist, an agent who is looking to retire early. He starts off my interviewing a guy who he ultimately plans to be his replacement. The guy’s partner was recently killed and there was also a case he was involved with where the punished was wrongly being punished, making him question if what they were doing was indeed the right thing to do. He decided, after long torture sessions to let the man go, bringing him under high scrutiny by the group.

Initial series of events (not written, but chronilogical):

Man is “working” a perp who supposedly was killing hookers, but had to be let go due to the police fucking something up (technicality). During the course of working him, he begins to question if the man in fact did the crime, and with his partner’s help, they help the man escape.
Man is questioned heavily by the group, and it is decided that he should retire early. He was one of the first agents of the group and is highly respected, so retirement is the best way for everyone to save face. His partner is reassigned, and he is told to hire and train his own replacement.
After some searching, Man reads about cop who’s brother was gunned down by ATF in a different city. The cop was very vocal that his brother was innocent, and the ATF claimed he had a gun. His brother was shot over 20 times, and despite a trial, the ATF agents were acquitted. The cop resigned in protest.
Man contacts the ex-cop and has him brought in for a job interview. He is told it is a high-paying mercenary job doing security for a big-wig. During interview, ex-cop is asked what he would do given the chance to be alone in a room with the leader of the ATF group that killed his brother. No consequences. Ex-cop is weary, but answers honestly, that he would choke the life from the man. The interview then moves to the “next room”. Ex-cop discovers that the scenario described is in fact real, and as the last part of the interview is given a chance to kill the man. He does, and is hired.
News comes in that the Man’s ex-partner was brutally murdered. Despite being asked to focus on training his replacement (though the ex-cop is NOT told he is replacing someone), he investigates.
The two are told to work another case, as part of the new guy’s training, where they stake out the home of a guy suspected by police to be a pedophile. The man is a high profile lawyer working under the DA and so prior investigations have been lax. During the stakeout, Man spends time reviewing information about his partner’s killing. There is a piece of evidence that reminds him of the guy they brought in and subsequently let go. Man realizes that he is responsible for his own partner’s death by helping the man they thought was innocent.
Pedophile is actually guilty, and they decide to bring him in for punishment.
When they return, it is discovered that another agent was found dead. Man discusses case with Director, revealing how and why he thinks the killer was the man they released. Other agents overhear conversation and there is a fight with the second dead agent’s partner. Agents demand Man be expelled immediately. He is, and is sent home.
While home, Man gets a threat call from the killer. He reveals that he was in fact innocent, but that their group was evil and needed to be taken out. He will kill each and every agent until they are shut down for good. Man tells him that he will pay.
ex-cop comes by next day (maybe few days) to check up on Man. He tells him he still wants to be trained by him, and that he doesn’t care if he gets in trouble. He figures finding the killer would be the best training he could ever get. Man tells him the reason why they are looking for the guy and what happened. He laments the moral ambiguity of it all.

May

Giving Vectors Phase

1. Adding Phase

Let us dive quickly in and explore a basic idea: ascribing a “phase” to vector fields. Instead of writing a vector $\psi_{\mu}$ a simply 4 real components, we will allow the vector to take on an additional property which we call it’s phase:

$\psi_{\mu} \equiv B_{\mu}e^{i\theta}$ (1.1)

The real-valued set B acts just like a normal vector, whereas the phase element $\theta$ is a scalar function. While we have added a new degree of freedom for vectors, it is important to note that we have NOT added another dimension to the space. This new degree of freedom will require us to adjust some definitions from differential geometry. For example, to ensure that distances are phase free (real valued), we will require the metric is a Hermitian matrix. The added degree of freedom also requires us to add an additional set of “coordinate changes” and expand on the affine connection to handle possible phase changes while a vector moves through a space.

$g_{\mu \nu}\rightarrow g_{\mu \nu^{*}} = g_{\nu \mu^{*}}$ (1.2)

$l^{2} = g_{\mu \nu^{*}} \psi^{\mu} \psi^{\nu^{*}} = g_{\mu \nu}B^{\mu}B^{\nu}$ (1.3)

$d \psi^{\mu} = \widehat{\Gamma}^{\mu}_{\nu \tau} \psi^{\nu} dx^{\tau}$ (1.4)

Which, if we expand, gives:

$d B^{\mu} = \Gamma^{\mu}_{\nu \tau} B^{\nu} dx^{\tau}$ (1.5)

$d \theta = A_{\tau} dx^{\tau} \theta$ (1.6)

And hence:

$\widehat{\Gamma}^{\mu}_{\nu \tau} = \Gamma^{\mu}_{\nu \tau} + i \theta A_{\tau} \delta^{\mu}_{\nu}$ (1.7)

Feb

Radial Complex Space

This is just the beginnings of a notion, taken from my small amount of work extending a vector field to a complex valued-vector field. What if, instead of writing complex numbers in the form x + iy, we write them in a radial form:

$z_{\mu}=x_{\mu}e^{i \theta_{\mu}}$

Here x would have the same meaning as the original x, but $\theta$ would be the dimensional “phase”. Since we are mostly interested in vectors, we could extend vectors to these new complex entities. Let’s look at some formula related to coordinate changes:

${z}'_{\mu}={x}'_{\mu}e^{i {\theta}'_{\mu}}$

${x}'_{\mu} = \frac{\partial x_{\mu}}{\partial {x}'_{\nu}}x_{\nu}$

We can also look at some properties of the inner product:

$l^2=g_{\mu \nu^*} \zeta^{\mu} \zeta^{\nu^*}$

We can guarantee this value is real (so as to maintain measurable lengths) by assuming the metric is hermitian. This then yields, after replacement:

$l^2=Re(g_{\mu \nu}) A^{\mu} A^{\nu}cos(\theta^{\mu} - \theta^{\nu})$

where

$\zeta^{\mu}= A^{\mu} e^{\theta^{\mu}}$

and

$g_{\mu \nu^*} = g_{\nu \mu^*}$

Jan

The Parton Model

Let me start out by saying the term “parton” has been used before. I didn’t know this when I first starting using the term, but it was officially a pre-cursor to the quark. Feynman used parton to describe a model of the inner structure of hadrons. Here, for our purposes, parton will be in reference to something else entirely.

To begin with, I started taking a look at the Dirac equation to see if there was a way to write it in a form that didn’t require 4×4 matrices. This was entirely due to my desire to figure out a form of the equation that works in general relativity and my immediate discovery that 4×4 matrices contain far too many degrees of freedom. It didn’t take long to discover Weyl’s variation of the equation which separates the 4 spinor into two coupled 2-spinors.

The coupling between them is via the particle’s rest mass. This means also that massless fermions can be described entirely by 2-spinors. ”Rest Mass” is really just a way of grouping self energy into a single term. An ideal theory would allow us to predict the rest masses of all the fundamental particles via some more descriptive interaction between the two 2-spinors. That method though would imply that the break up of the 4-spinor into two fields (though intimately tied) is more than just a mathematical trick, but rather an expression of something far more fundamental.

Normally though when we combine two spinors together we get a vector (combining two fermions gives a boson). This new combination cannot be via this normal mechanism. Instead, we will opt to write particle fields as a combination of two 2-spinors. These 2-spinors we will call “patrons” and we will asset that all fundamental particles are a combinations of two patrons. Bosons (vectors) are the particles that cause fermions to change state facilitating an interaction, and thus we can easily see that they would be written as a combination of a parton and anti-parton.

We therefor have a new way to refer to particles that is a consistent mathematical structure for both fermions, bosons, and their anti-particles.

$\psi =\left \{ p_L, p_R \right \}$

$A =\left \{ p_L, \bar{p}_R \right \}$

Here we have made a visual distinction between the parton on the left verses the parton on the right. This is important as the exchange of left and right do not yield the same particle.

$\bar{A} =\left \{ p_R, \bar{p}_L \right \}$

Using this, let’s write out the Dirac equations for partons

$\sigma_{\mu}\partial ^{\mu}p_L + \frac{iE_0}{\hbar c}p_R = 0$

$\hat{\sigma}_{\mu}\partial ^{\mu}p_R + \frac{iE_0}{\hbar c}p_L = 0$

These can be combined to recreate the Dirac equation

$\begin{pmatrix} 0 & \hat{\sigma}_{\mu} \\ \sigma_{\mu} & 0 \end{pmatrix} \partial^{\mu}\binom{p_L}{p_R} + \frac{iE_0}{\hbar c}\binom{p_L}{p_R} = 0$

Spin States

Suppose a parton has a spin-state we define / create. For our example, let us ascribe partons with a spin called “iso-spin” that is either up or down.

Dec

Intro

There are as many ideas about what God means as there are people. To me, God is that thing that gives our lives focus and meaning, even if it is something we dream up ourselves. Many people think having a multitude of talents is a good thing, something they wish they had, but it’s really not. How can we define ourselves and choose a guiding star when there are a million of them and they all look the same?

Me, I had a star once, but as it turns out everything was aligned against it. People with nothing at stake will make vacant comments, empty suggestions based on half ideas and truths, but they will never really know. The road I was on, the one that wound through the dark forest, was just a dead end. My life, for all that it could have been worth, is essentially over.

Instead, all I can really offer is this, my journal. I hope that by reading this, you can learn from my errors. Most are the same things I’ve seen others do, the same stupid mistakes we all make. To me, in reflection, the most important lesson I wish to teach is only this: don’t waste it.

All that you have, whatever that might be, be it talent or love or drive, or even little things, NEVER let it go to waste. Use it, share it, even give it away. It’s all you can ever do.

Jason Allen Blood

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