I have been thinking about Ariel Caticha and Carlo Cafaro’s derivation of Newton’s Second Law from probability theory (http://arxiv.org/abs/0710.1071).  In this derivation, they assume three things:

1. There exist basic entities called particles
2. There is an intrinsic uncertainty about a particle’s position
3. There is change… not time necessarily, but change.

The first thing I like about this derivation is the fact that distance in space derives from distances between probability distributions describing particles at two positions.  Distance comes from the Kullback-Leibler Divergence, and leads directly to the Pythagorean Theorem.  More on that in another post… 

The derivation results in an expression describing an interaction between two particles and this results in changes in position which, when parameterized, leads to F = ma. 

The mass turns out to be related to the uncertainty in the position of the particle.  The more uncertain one is about a particle’s position, the more difficult it is to accelerate it (change its change in position).

uncertainty in position^2 = 1/mass

This is quite satisfying as it is reminiscent of Heisenberg’s Uncertainty Principle.

But what implications does this picture hold?
First, it gives us another viewpoint of mass.  Multiple equivalent viewpoints always leads to constraints.  These constraints can be used to figure out how things work… or at the very least to rule out possibilities.
(see my earlier post on E8 to get another picture of mass)

Well, photons have no mass.
So there is infinite uncertainty associated with the position of a photon.  In fact, I am not sure photons “travel”.  I prefer to imagine that they just “are”.  One way to imagine photons might be to see them as lines connecting two particles in spacetime.  The photon is spread out across space.

So now comes the important question.  Given this probabilistic picture, what can we learn about mass?  Do we need a Higg’s boson?  I would think not.  Except, today I spoke with Ariel, and the way that the Higgs imbues mass is by having the particles scatter off of it.  The more the particles scatter, the less certain we are of their position… and the more mass the particle has.  Maybe this is where the Zitterbewegung originates.  And maybe the Higgs boson is just a reflection of an intrinsic uncertainty.

Instead, I imagine that as these ideas are refined to describe more than Newtonian mechanics, and include spin, charge and flavor, a relationship between the particles and their masses would emerge naturally. 

Again, these are raw thoughts… not even half-baked… and should be treated as such.  My grandma always said that eating raw dough will give you worms. 
Just think what thinking raw thoughts might do!!!

Kevin Knuth
Albany NY

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Garrett Lisi recently posted a paper to the arxiv (arXiv:0711.0770v1 [hep-th]) titled “An Exceptionally Simple Theory of Everything”.  In it he proposes that the group E8 unifies all of the fields of the Standard Model with Gravity and predicts new particles.

In addition to the paper, here is an excellent blog that discusses the theory, its inspiration and implications:

BackReaction on Exceptionally Simple TOE

I cannot possibly do the topic justice, but allow me to attempt a watered-down, but relatively accurate description of what is going on here for those less versed in Quantum Gravity.  The basic idea is that the elementary particles have properties that obey certain symmetries.  These symmetries enable us to make diagrams that display the particles as vertices of symmetric graphs.  Below is one such picture showing the symmetries of the what is known as the Baryon Octet.  A Baryon is a particle made of three quarks.  One set of axes denotes the charge Q of the particle (in units of electron charge), and the other set of axes indicates the number of strange quarks that compose the particle.  The symmetries of the diagram represent the symmetries between the particle properties.   The two particles on the top are the familiar proton and neutron.

However, these symmetries only goes so far since the particles have different masses.  The fact that they have different masses breaks the symmetry.  This is known as the problem of symmetry-breaking.

When you see a picture like this, it seems like we have a complete theory.  However, some symmetries contain other symmetries.  For instance, if you consider an equilateral triangle, you can rotate it 120 degrees and it looks the same.  This symmetry happens to be contained within the hexagonal symmetry.  But the hexagonal symmetry has more structure.  It also is contained in the rotational symmetry of the circle.  If we are trying to find a symmetry that contains the known symmetry of the equilateral triangle, but also contains new properties that can explain phenomena that we currently poorly understand, there is no obvious solution.  There are many symmetries to consider (in this toy example the hexagon and circle are only two of them).

This is basically what is going on now in particle physics.  We know some of the symmetries, but not all of them.  The question is, which higher-order symmetries include and unite all of the symmetries we know about, while simultaneously introducing new symmetries that can explain effects that we currently have no theory for, such as uniting gravity with quantum mechanics.

Garrett Lisi is proposing that E8 does exactly that.  It apparently contains all of the familiar symmetries, and unites gravity and the Standard Model of particle physics in a very satisfying way.  As I understand it, this theory enables him to make testable predictions, which is a huge step forward.  We will see if it is right…

For now, here is a movie that shows the symmetry of E8 while performing rotations in multiple dimensions.  Here is a link to a much cleaner version (about 10 MB).

Particles related to the gravitational force are represented by green circles, particles related to the electroweak force are represented by yellow circles, particles related to the strong gauge fields are blue circles. The frame-Higgs particles are squares. Three generations of leptons (electrons, muons, tauons and their associated neutrinos) are yellow and gray triangles. Finally the quarks (rbg triangles) are related by triality (lines).

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I recall back to my first days in physics class where I read the quotation “Nature abhors a vacuum”.  The phrase was accompanied by an image of two teams of horses trying to pull two hollow hemispheres apart after the air was pumped out from inside.

These were called Magdeburg hemispheres and the experiment was performed by Otto von Guericke.  I didn’t realize this, but according to Wikipedia it was Aristotle’s theory Horror Vacui that von Guericke was trying to disprove.  This theory suggested that nature hates a vacuum and that the vacuum will suck material in to fill it.  Von Guericke demonstrated that it is not a sucking force, but a pushing force from the outside air.  However, as I recall from the physics text, the presentation was written as if this experiment was a demonstration of this “principle”.  I imagine that this misconception is why you still see the phrase commonly used today.  This is one of the science myths that keeps floating around.

A more accurate viewpoint is that Nature Abhors a Gradient.  It isn’t that there is a vacuum that is holding the hemispheres together.  Nor is it that there is air surrounding the hemispheres that is holding the hemispheres together.  Rather it is that there is an enormous gradient in the air density outside the hemispheres with respect to the inside.  The result of this gradient is a force.

Gradients in potential energy cause forces.

More generally, gradients in any scalar field result in generalized forces.

The gradient of the electric potential is the electric field.  The gradient of the gravitational potential energy is the gravitational force.  Over and over again these ideas reappear.  In physics, we learn them as separate concepts, and then later in the abstract topic of statistical mechanics we are expected to put it all together.  But each of these gradients resides in its own little box in the student’s cortex, and statistical mechanics is in its own little box.  Rarely do these ideas merge to form a unified concept.

Temperature gradients are responsible for our weather. 
Huge temperature gradients are responsible for hurricanes.  The temperature of the extremely warm air at the sea surface in the Gulf of Mexico drops quickly as one rises higher into the atmosphere.  This enormous gradient powers the heat engine known as a hurricane.  Nature abhors gradients, and will do something about them.  The gradients will result in forces that tend to eradicate the gradient.  Watching this NASA video of satellite imagery during the 2005 hurricane season, one can clearly see that hurricanes are designed to cool the sea surface.

Our weather here in Albany is due mainly to cyclonic storms that act to relieve the temperature gradient between the Earth’s equator and the poles.  Forces due to this gradient brings parcels of warm air north from the south and exchanges them with parcels of cold air from the north.  The spinning of the Earth results in the Coriolis force which deflects the northbound warm air eastward and the colder southbound air westward.  This creates a counter-clockwise rotating structure that brings warm air north on the eastern front and cold air south on the western front.  The result is that the extremely warm weather we had a few days ago had to be followed by the extremely cold weather as this cyclonic structure moved over us from the west.

However, the weather is not the only system driven by the Earth’s energy gradients.  Life itself is driven by these forces.  The “purpose” of life is to relieve gradients.  This is why forests are cooler, they are working to dissipate thermal gradients induced by incoming light heating the surface.  During the process of relieving these thermal gradients, the plants grow.  This excess organic matter results in residual chemical potential energy, which again creates a gradient.  The herbivores devour the plant matter to further relieve the chemical potential gradients.  This works up to a point, but again there is a residual chemical potential, of which the carnivores take advantage.  The cycles of life are driven by energy gradients.

But it doesn’t stop here.  There are other scalar fields that have gradients.  Wealth gradients result in generalized forces.  In cites where there are enormous gradients between the wealthy and the poor one sees violence.  Across the southern U.S. border there is another enormous gradient in wealth.  This gradient results in forces that drive Mexican immigrants into this country.  Building a wall won’t help because it doesn’t eliminate the gradient.  The gradient will merely increase until new forces become strong enough to eliminate it. 

How does one solve the problem? 
Eliminate the gradient.
Nature abhors a gradient.

Kevin Knuth
Albany NY

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One convergence that I am particularly interested in is this notion that the Laws of Physics are in reality a special application of the Laws of Inference.  The idea that is developing is that the laws of physics are not so much laws that describe the universe, but rather are laws that describe how an ideal observer, or a rational agent, could make predictions about the universe.  This sounds rather like some of the ideas about quantum mechanics with cruel observers and abused cats, but there are some critical differences.  Here I have said nothing about the necessity of an observer.  Instead, I posit that the laws of physics describe how an ideal observer would make predictions.  This distinction is crucial.

But this is a strange idea that we are taking right to the limit.  Even laws as seemingly physical as classical mechanics, such as F = ma, are not laws that describe the object being accelerated as much as they are laws about our making inferences about the object and its position or state of motion when it is interacting with another object.  (See Caticha’s MaxEnt 2007 paper)

This is not a particularly unusual idea when one considers thermodynamics, or more generally, statistical mechanics.  Here it is almost obvious that the theory is about our ability to make inferences about macrostates given that we can only measure macroscopic variables.  This is where entropy appears as a way to assign probability distributions subject to constraints, such as the total energy.  And it is here that it is most clear that entropy refers to the optimal state of knowledge of an ideal observer and not the observed system.  This becomes even more clear when one considers that a macrostate is not really a “state” at all, but rather is an equivalence class of microstates. 
(see my MaxEnt 2007 tutorial)

But this begs the question: What is a state?

My belief is that a state is a description.
State = Statement
That’s rather nice!

And in that sense, a macrostate is a statement.  Or more precisely, a macrostate is a disjunction of statements.  It is the statement that the system can be described by microstate a or microstate b or microstate c, etc; where stating that the system is in “microstate a” is a statement itself.

Statements, or descriptions, are clearly not properties of a system.  They are instead properties of observers caught in the act of describing a system.  This space of statements is the space in which physics takes place. 

In this sense, physics is all about information…
Information Physics. 

Kevin Knuth
Albany NY

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