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April 19, 2011 / zitterbewegung

Opening chapter of a short story

Chapter 1
Just another day rolling in as he so anxiously awaits the arrival of the Fall. At this he knew he must certainly act. The imperative was in place, his clock moving forward relentlessly. For all he had struggled with in his conquest to learn, the scuffle in the wee midmorning hours, reading and calculating by the mere faint glow of the dome light of his truck. Driven with an intrinsic burning desire to know, to understand; that which would only be privy to those of a most eclectic niche.
Soon the sun would dawn and with it the bustling of a new and busy day. Walking back through the kitchen of the restaurant liken with the hooded executioner of the medieval time, he careless sweeps aside the corpses of the nasty and retched creatures that once inhabited its walls.  He was the exterminator, the bug man, the one person who delivered a swift and agonizing demise to all vermin that might dare to enter his protected kingdom.
As he returns to this white behemoth of a vehicle stocked with all the tools of the killer’s trade, meditating once again at his notes. Asking, “Will Fall bring change?” The reality of his position materializes like the stench of regret as he proceeds to drive away. He had promised himself the year prior of such a return to purpose, a proclivity amongst his most amiable of peoples, academia. This lost promise which did arise from the tip of the twentieth Century to more than a decade, aged as the wheel of cheese lurking in a cellar.
Time and occupation with no meaning, a wage, a means to live another day; this was simple survival. Overlooking the insult of the snaring gaze of those obliged with financial favor, his was a world of pure intellectual curiosity. “Where is your ambition man?” Surly one of your cognitive pedigree should have the money by the tail. The common sickening and mindless parroting lore of Western society, such a common and overused theme, that Intellect predicates wealth. For him there is no greater annoyance than the fertile growth of ignorance.
For the autodidact the world is the educator. He was no different. “It is the goal of society to educate the common man”, this, a reveling statement with an almost Marxist theme. But what is the value of this endeavor? A most fortuitous irony he faced, he couldn’t learn from the very system with which he shared passion and sought employment. Where does one appeal when much is amassed, analyzed, churned, and processed? Can there be resolve in the beginnings that never were? Setting the sky as the limit he still had realize that everything of purpose starts on the ground.
There is only one difference between intention and accomplishment; that is action. In a world of doer’s he had been the passive observer. How much must one know of things before one tries? Certainly the lure of emotional stimulation is the key, so he thinks. How is it that natural design placed such a burden on those tasked to learn by induction, he asked. Symmetry must lie at the root of explanation for this most perplexing phenomenon. In a highly symmetrical system, a beehive for example, an attacking wasp would be confused by the regular pattern of hexagons in the comb. But if even one of these units is deformed and draws the attention of the wasp; the larvae inside is certainly doomed. So goes the struggle for the inductive thinker in a society populated with deductive types; he proposes.
Something so eargerly craved by most, yet it’s presence certainly evokes suspicion, fear, and anxiety. So any change in objective reality is feed as the dropper given to a infant, it is best served with reserve. Popularism and collective realism run hand in hand in the ranks of the deducers. The self-worth of an individual is therefore conferred by society as a whole, this is a blatant farce indeed! He quotes Jung,”Follow that will and that way with which experience confirms to be your own.” For his formative years, this might have been the less wiser of route. He and youth like him were confused and dissociated from the social norm of the day. A cage with a wonderful view rendering a sign, “Do Not Feed The Animal.”

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October 8, 2010 / zitterbewegung

Response from a curious onlooker

I was sporting the colors today while doing my weekly shopping at Harris Teeter.

The “colors” refer to our t-shirts on unique approach to a unified field theory that utilizes the most successful physical theory on the books, Maxwell! This was covered in a prior post. Go here for a look-see and get your shirt,

http://world.std.com/~sweetser/quaternions/tshirt/tshirt.html

Anyhow, I was asked by a rather attractive young lady what the meaning of the back of the shirt was, I told her it was a Lagrangian for a unified field theory. Completely expecting a typified casual response, I ventured onward, then suddenly was amazed by her reply…”you mean this would be like the equations of motions for putting gravity, electromagnetism, the weak and strong force into a single category?”

Following the complete stun effect I experienced, I replied with an assured, “yes indeed!” We conversed for a short time and I encouraged her to spread the word.

September 27, 2010 / zitterbewegung

Symmetric rewrite of Maxwell, a Unified Field Theory that’s quite a “GEM”

Introduction

The term “Symmetric Gauge theory” is suppose to be an oxymoron, and not to mention, a technical impossibility!

First Lesson: Gravity and Operator Calculus (the stuff quantum theories are built from) hate each other!

Actually as the story goes, Gauge theory is as “antisymmetric” as you can get, as it is based on an antisymmetric field strength tensor, known as the Yang-Mills tensor. Gravity on the other hand is expressed with the full symmetric Riemann tensor.

First, let’s explain what is meant by (anti)symmetric here! Well use simple example, say I wish to multily two numbers, x and y. I can say that the product of x and y is commutative,

xy = yx

this is a symmetric operation. But if we have,

xy = -yx

this an antisymmetic operation. These rules generalize to tensors.

All this leads to the fact that Gravity is very different from the other forces. One key aspect that creates this difference lies in the subject known as geometrodynamics, which in my opinion sounds like the description of the motion of a GMC economy car! But in reality, this literally means that space-time geometry is dynamical, so this thing we like to call Gravity is not a simple force as it were with the case of the the other three, but the actual change in the curvature of space-time itself!

Second Lesson: The first three (gauge) forces of nature use space-time as a “static” background, while gravity represents changes in the background at point to point.

Field Strength Tensors and Contractions

Let’s look at the field strength tensors mentioned above in more detail,

The Yang-Mills tensor is,

First note that with the formula for the hermiticity condition gives the ‘allowed’ rank of the tensors, (for the ambitious reader, this relates to the constrain coming from unitarity, i.e., the fact that the Yang-Mills tensor is based on the symmetry of the SU(n) group)

Of course excluding the special case of , for which we get the abelian group U(1) for Electromagnetism, Also this field is invariant under Lorentz transformation and possess space-time symmetry,

From purely mathematical considerations, the U(1) group is the 1-dimensional group over the complex numbers and forms a 2-d normed division algebra over the reals. Taking the appropriate numeric assignement and going from Greek to Latin indices gives the electric and magnetic fields,

At this point I should interject that Electrodynamics is a beautiful and elegant theory in the history of physics (not to mention successful as all hell in everyday life).

Considering the more generalized cases when    , gives the massive vector bosons and gluons, given by the gauge groups SU(2) and SU(3), respectively. These forces, unlike electromagnetism are strictly internal (quantum) symmetries.

All of the gauge theory assumes a metric of unity, .

That’s all good if you live in flatland. If you introduce space-time curvature, the metric is no long a constant, but becomes dynamical (i.e. it can deviate below unity due to curvature). The metric becomes a solution to a differential equation for 2nd order curvature. This is given by the two contractions of the Riemann tensor,

And forms the Einstein-Hilbert field equation,

Note that the general structure of the Riemann tensor is very much like the Yang-mills. But if we express the Christoffel symbol (connection coefficient) in terms of the metric,

It is obvious that the apparent similarity in the structure of the equations is a superficial one at most.

Everything in GR is about the metric! I can here Jan Brady now, “Metric, metric, metric!”

Enter Symmetric Gauge Theory – Go with the best, use Maxwell.

 Maxwell’s theory along with the weak and strong forces are antisymmetic in nature, as mentioned above. This has a direct correspondence with a simple but profound experimental fact,

For an antisymmetric field theory (tensor), like charges always repel. This is a first blow to the use of a  rank-1 field theory for gravity, as with gravity, like charges always “universally” attract.

Turns out that at the classical level (and perhaps the quatnum) this difficulty can be overcome by addressing a simple accounting issue with the way Nature may choose to multiply quantities (to quote The Stand-Up Physicist). But to use this practice, we are forced to abandon the use of tensors (like the ones above), as tensors can’t perform this feat.

Question is, “how does one do field theory without tensors?” the short answer lies in a mathematical field know as a quaternion. The quaternions are a 4-dimensional division algebra over the reals with the “built-in” definition of a norm, this comes in handy when desribing an inertial frame and a Lorentz invariant scalar (i.e. normally taken as the product to tw0 4-vectors, but our case the product to two quaternions). In 4-dimensions, quaternions share all the important properties of tensors.

The re-write is actually much easier than one might think, we start by defining the U(1) group as a quaternion over it’s own absolute value,

 

The group SU(2) is called the group of Unit Quaternions, it is obtained by taking the exponential of the same quaternion minus its conjugate,

We arrive at the fully quaternion by simply taking the product of 1st and 2nd result,

The Lie algebra of the SU(3) group is different from that of SU(2), we get the correct multiplication table by using a second exponential of a second quaterion minus it’s conjugate, but this time we “double conjugate” the first quaternion minus it’s conjugate to give SU(3),

 

One important comment is in order about quaternions vs. tensors. Tensors can be generalized to higher dimensions, where as quaternions (via their unique structure) makes them restricted to 4-dimensions. In a certain sense, we are ‘guided’ via accident or destiny to use this formalism to describe how fields in nature behave together.

So at this point we’ve got Special Relativity and Gauge theory nailed down pretty good. But as mentioned above, a symmetric theory is need to handle gravity.

 

Hypercomplex Numbers and the “Symmetrization of Maxwell”

 

Quaternion multiplication is as follows,

ijk = -1

ij = k, ji = -k …

On the other hand, multiplication with hypercomplex numbers is,

ij = k, but ji = k also!

jk = kj = -i

ki = ik = -j, end result

i^2 = j^2 = -k^2 = -1

Then choose the so-called “California representation,” so everyone multiplies out to just positive 1.

So is this a potential route to the quantization of gravity (exclusively in 4-D, BTW, the stuff that can be tested!)? I say, hells yea.

 

 

 

 

 

For further details and REALLY COOL ANIMATIONS, visit bit.ly/qmnserver

 

 

September 27, 2010 / zitterbewegung

Weak gun laws an excuse for gun crime?

I’m furious about what was just reported on our local ABC affiliate here in Charlotte, NC. The story was on the apparent increase in crimes involving weapons in the region. At the tail end of the story, the anchor said, “it’s believed that this increase in crime is a result of gun laws that are too weak.”

This is the DUMBEST THING I’ve ever heard! First, If I’m a person with a wrap sheet that commits crimes with handgun for instance, I’m not going to venture down to a dealer or shop, present my identification and submit to a background check. These people obtain weapons in back alleys and dark parking lots!

The damn media needs to stop the government sanctioned propaganda about gun laws and how many we don’t have! We have laws that work well enough if we as law abiding citizens must show a current form of picture identification and wait for our background to be run.

As a Civil Libertarian I felt compelled to post this. Well, back to townhall.

August 1, 2010 / zitterbewegung

The cycles of growing up

Like my Father bestowed upon me some moons ago, I too tell my Son that, “Common Sense is Uncommon.” Being a father now for nearly 18 years I’ve come to find that what my Dad was telling me about hard work, responsibility, and good judgement, was in fact truth. This brings me to the question, “Does anyone really ever grow up?”

I sent the majority of my Son Matt’s teen years as more the roll of a friend than Father. I personally grew up in a situation where you were expected to know how to conduct yourself in social event, etc. So in essence I had enormous freedom with the ranks of responsiblity at close bay. I think you have to give your Kids this kind of leverage if you expect an adult to emerge from the years of trials. Naturally my wife thinks I’m too much a buddy and not enough an enforcer. I think that using intimidation or trickery and deception are bad parenting.

My wife’s Father passed when she was 6, so she pretty much was raised by her mother (and sometimes older sister and her husband). She has an incredibly sheltered upbringing. I meet her at 23 and her mom had her name on a joint checking account with my wife. She also was still living with her mom when we dated. I was out on my own officially at 17, kinda the work and shack with whoever I could until I jonined the Air Force at 19.

It wasn’t until a few years ago that I began to realize that a lot of growing up is knowing when to be a kid. So growing up really doesn’t mean always taking a mature and responsible attitude and being ultra serious all the time, it means you know when and where you do and don’t have to do this. For the youth, this is a matter of induction and training. For the advanced in years, it’s a matter of timing.

August 1, 2010 / zitterbewegung

Physics Tattoos

Been considering a tattoo of a famous equation in physics. Getting a tattoo is no trivial matter indeed. Many people with them will tell you it’s no big deal at all, but It would seem to me that putting a “permanent” image on your body is not a light decision.

In making this decision I have thought out criteria, as follows,

1) The type of tattoo – in this case a physics tattoo, specifically the derivation of the wave equation from Faraday’s law.

2) Where? – I would prefer the upper left arm where it can be covered by a shirt sleeve if necessary.

3) The tattoo should be about something in which you are very passionate or care deeply about, and not just on a superficial level (i.e. passing fad).

Well, I safely passed all three of these criteria with flying colors. First, I’m very passionate about the subject of electromagnetism, pound for pound it’s my favorite area of physics. 

This is a rough idea of the design I want,

July 30, 2010 / zitterbewegung

Zero-Point Fiction, A wake-up call to the free energy community?

Well for all the folks in the “energy for nothing” club, it turns out that there is a growing body of literature that dismisses zero-point electromagnetic energy as a physically observable phenomenon. Every since the dawn of quantum field theory, this item has been heavily debated.

The vacuum of space-time boils like a huge cauldron with virtual electromagnetic fluctuations. John Wheeler referred to this as a quantum foam (in analogy to the bubbles in a beer). The ‘free energy gang’ has been spouting grandiose claims about this bounty of power since the time of Nicola Tesla (who by the way I happen to have a ton of admiration and respect for and I personally believe that many of Tesla’s ideas are taken out of context).

Well, it might be all for naught! This whole issue is likely to be a misinterpretation of experimental data about the way molecular electric fields behave, known as the Van der Waals forces. These forces are very significant at the surface of very small objects (e.g., a few hundred atoms). This point was raised in a paper in Physical Review Letters in 2005. Here the author points out that,

Casimir effect gives no more (or less) support for the reality of the vacuum energy of fluctuating quantum fields than any other one-loop effect in quantum electrodynamics, like the vacuum polarization contribution to the Lamb shift, for example.

Jaffe goes on to say,

The Casimir force can be calculated without reference to vacuum fluctuations…There is a long history and large literature surrounding the question whether the zero-point fluctuations of quantized fields are ‘‘real’’

The whole point here is that this view says that the ZPE is “strictly” a property of the vacuum that is NOT accessible by any means. Let’s take a quick look at what quantum mechanics says about the world in which we live.

When objects in the everyday experience move we say that they obey Newton’s laws of motion, the basis of classical mechanics. Take for instance the motion of a bowling ball, we can toss it down the lane and know with “certainty” that it will hit those pins (unless you’re a career gutter bowler). With quantum motion things are not so simple. When we let go of the ball it is like it vanishes into la-la land (this is actually called Hilbert space in the books). Let’s look at this from an example, 1) is the classical (Newtonian) example (that’s a risky shot for a novice, but an old pro like me can use those to rack up a 190 on my slow nights!) 2) is the quantum example. Notice that the moment you let go of the ball it just spreads out like a cloud. By the time it gets down to the pins it a complete matter of probability as to which pin it strikes!  (in other words, quantum mechanics and bowling don’t mix!)



The ‘quantum nature’ of the ball is described by a so-called wavefunction. As the name implies, this means that the ball exhibits wave-like behavior when travelling and particle-like behavior when touching (i.e. interacting) with something. For those of you concerned about the ball merely hovering outside the lane, we’ll suppose that the lane acts as a boundary. Also the lane has magical properties (i.e. it can’t collapse the wavefuction and let us see the ball rolling on the classical path).

 So what is all this zero-point stuff about anyhow?

The ZPE is the term used to represent the lowest possible state of energy in the physical vacuum of space-time. Consider the diagram below, notice the term with the 1/2 in front (and the fact that it resides BELOW the zero energy line!) it’s the ZPE,

This diagram is an obvious ladder of rising energy states. Since the terms are quanta of radiation or photons, then the ladder can be thought of as small additions of thermal energy as you climb the rungs. This was actual how Planck made has break though discover that radiation was quantized into discrete packets, and this lead to the resolution of the infamous ultraviolet catastrophe. If we examine the 1/2 term there seems to be no energy present at all, so no heat, hence the moniker zero-point energy or “motion at absolute zero.”

Next we need to draw a distinction between just quantum mechanics and quantum field theory. In quantum mechanics we do what is called the first quantization. This is because the usual quantities we measure, the canonical quantities like position, momentum and energy are transformed into differential operators.

On the other hand in quantum field theory, it is the field itself that gets quantized this is referred to as second quantization. By the way, the two types of quantization are completely unrelated and very different. It is unfortunate that they got labeled 1st and 2nd as this suggests that the 2nd somehow is a subsequent procedure to the 1st. below a classical field is quantized,

Notice the field lines are transformed into a cloud of virtual quanta. This picture of second quantization gives us the intuitive notion of how the virtual quanta interact with the electromagnetic fluctuations of the vacuum.

Another note on the difference between 1st and 2nd quantization is in order. Recall in quantum mechanics we already postulated the dual wave-particle nature of matter, the wave function. But the goal of a quantization program is to create discrete representations of forces or quantities related to forces, but not masses. Notice in the diagram above the particle or mass remains unchanged by quantization, this is because the mass is treated as the origin of the charge or source. This invariance of mass under quantization is related to the fact that mass is an invariant scalar quantity. This is a subtlety that I shall not cover here as it is unnecessary to the present discussion. In field-theoretic terms this is referred to as a bare quantity. When calculating interactions of quantum fields it is typical to remove the bare quantities by a process called renormalization, which is also beyond the scope of this post so I will not venture into it.

What I’m driving at with all this is that you can’t make the quantum level of reality ‘stop moving.’ Remember that an everyday object has a well-defined position in space, but a thing as small as an atom doesn’t. This example gets even more extreme if we look at the physical vacuum of space-time. It is a random bath of virtual quanta also referred to as ‘off-shell’ photons. A virtual quanta is like a photon that you can’t observe directly, yes, it is only endowed with a quantum mechanical nature, recall the cloud around the source above.

We actually need to draw another distinction at the is point in the game, that is that fact that the field composed of virtual quanta exists INDEPENDENTLY of any source. We can think of the existence of the quantum field with a source at the origin as a means of ‘organizing’ the zero point electromagnetic vacuum fluctuations. This matter is studied in the subject of quantum electrodynamics or QED which is concerned with how electrons (or charged particles in general) and photons interact. One prominent example describing the exchange of virtual quanta is when two atoms or electrons get close to each other this is described by a process called Moller Scattering.

This is a t-channel Feynman graph that represents this process with two electrons interacting at the tree level. By tree level, it means the process is calculated to the lowest order that can be experimentally measured. Notice the blue wavy line, that’s the virtual photon. You can think of this as the electrons pushing on each other with their respective electric fields.

The conventional way to understand zero point energy in quantum field theory is by the crossing symmetry of the creation and destruction operators,

Where the first one is the destruction operator and the second is the creation operator. The q and p are coordinates and momenta of the fields respectively. If the destruction operator acts on the vacuum, it destroys as quanta so the result is zero. Conversely if the creation operator acts on the vacuum it creates a quanta of momentum k,

That’s not the end of the story though. The creation and destruction operators are built out of positive and negative frequency components. So the beauty of this approach is the elegant analogy with Young’s double slit experiment in optics and the notion of interference. The zero-point energy is a result of the superposition of these positive and negative frequency component of the creation and destruction operators,

It is a common procedure to ‘remove’ the ZPE from calculations by the process called Normal Ordering. This is accomplished by separating the positive and negative products in the fields and placing all the positive to the left, and negative to the right. This is envisioned as letting the operators take a direct path as opposed to an exchange path (which would create the interference and ZPE).

[1]  http://arxiv.org/PS_cache/quant-ph/pdf/0509/0509071v2.pdf

[2] Quantum Field Theory: A Modern Introduction, M. Kaku

[3] Modern Quantum Mechanics, J. Sakurai