A Small Insight on Quantum Field Theory

I’ve been thinking about quantum field theory for a while now, trying to
understand it.. Here is a small insight I recently made.

One way of thinking about the configuration space is through the field
perspective. There are observables at every point in space, they obey the
Klein-Gordon equation with nonlinear perturbations. Taking the Fourier
transform, you get at every momentum a set of harmonic oscillators, and the
interactions are couplings between them. Quantizing this, each oscillator gets
an integer spectrum which corresponds to how many particles of a given type
there is in a given momentum. So now we get the particle perspective in momentum
space.

Another perspective, which is usually used in less rigorous descriptions of QFT,
is the particle presentation in position space. For example, this is implicit
in the “cloud of virtual particles” intuition. It seems to work, but nobody uses
it in calculation. Presumably it can described precisely by taking the Fourier
transform of the momentum space particles to get position space particles. Like
the field presentation, it seems to be manifestly local.

My small insight is this: there is a direct way to describe this perspective in
terms of the field perspective. It is this: a field at a given point over time
behaves somewhat like a harmonic oscillator. Take the basis derived from the
fixed energy states of these harmonic oscillators to get position-space particle
presentation.

Some consequences:

• A massless field doesn’t behave like a harmonic oscillator at a fixed position because there is no spring constant. This explains a claim I heard that position
  space doesn’t work for massless particles. However, extrapolating the creation
  operator as the spring consant goes to zero, it looks like something similar can
  be made to work when thinking of particle count as an index for a
  Taylor-expansion-like decomposition of the wavefunction into polynomials.
• This position space perspective is not Lorenz-invariant. This partially
  explains the claim I heard that position space also doesn’t work for massive
  particles in the relativistic theory. It also explains why nobody uses this
  perspective seriously.

Why Scientists Don’t Write Poetry

A pillow’s softness, from how it behaves:
To forces it reacts yet it conforms.
The patient repitition of a wave,
Transforming an atom from form to form.
How everything existing had begun,
Matter from matter moving fast away.
How everything is hidden in the sum
Of amplitudes of all the different ways.
Yet all the things I do ’till now express
Are mere fragments of a deeper core.
How would I love to faithfully address
The full wonders with all of their galore.
Yet when I try those feelings to compose
My poetry is hightened into prose.

Propositions as types in Foundations to Constructive Analysis

I was reading Foundations of Constructive Analysis by Errett Bishop, and I came across this intersting passage:

It is not strictly correct to say that a real number is an element of . An element of consists of a real number and a positive integer , such that , because an element of is not presented until both and are given. One and the same real number can be associated with two distinct (but equal) elements of . Nevertheless we shall continue to refer loosely to a positive real number . On those occasions when we need to refer to an for which , we shall take the position that it was there implicitly all along.

In other words, the proposition really stands for the type
of all with . Bishop does not appear to be aware of the
propositions-as-types interpretation in this book, but he uses it implicitly.

A fun arithmetic fact

1/27 = 0.037037037…

1/37 = 0.027027027…

Aerodynamic computing

Here’s an idea I have: A computer made entirely out of wind. I thought of this
by contemplating how to build machinery inside a plasma. Every existing
machinery depends on it being possible to make a stable structure using solids,
but that doesn’t work in these extreme conditions. So the alternative
possibility I see is using wind currents.

I actually think something like this could work. The design I’m thinking of
would look something like this: There is one big vortex in the middle. Orbitting
it are many smaller vortex loops. The ones closer to the center orbit faster
than there the ones farther out, and they interact with each other in many
complicated ways. All of these vortices eventually die out, but hopefully they
have the chance to orbit many times beforehand. Using some cleverness (i.e. I
have no idea how this part is going to work) one can make a pattern like this
self-correcting. Once a design is complicated and stable, it’s not far out from
being Turing-complete. What you need to do is add components which have two
different stable states, and then make a system where a component changes it’s
state based on other components. While rough details like this look reasonable,
again, I don’t really have any idea how this is going to work.

One piece of evidence indicates that this is difficult or impossible: Namely,
that it could disprove the conjecture of Navier-Stokes existence of solutions.
The conjecture says that if you start with a complicated combination of wind
currents, you can always extrapolate what they would do in the future without
something ridiculous like an infinite amount of air in the same spot.
The reason it would fail is that in fluid dynamics, a smaller version of any state can always
be made which is both quicker and uses less energy. This property is called
supercriticality. This means that a wind-based computer can make a smaller
version of itself. The smaller version would make an even smaller version in
less time, and eventually an infinite number of computers would be made in a
finite amount of time. Now this conjecture is one of the Clay Millenium
Problems, so you’d expect someone to have tried this approach. The fact that
they failed shows that filling in those “I have no idea how this works” is tough
or impossible.

Does life need to be made out of a liquid.

This post is inspired by this article. In it Isaac Asimov discusses various ways life can exist different from our hydrocarbons in water (in cells) form. All of his alternatives consist of macromolecules immersed in a liquid. I’m not too concerned about questioning the macromolecules aspect, because while I can imagine there being an alternative to macromolecules on what makes structure and holds genetic information, I can’t think of any good ones. What I concerned about is them being immersed in a liquid. This is a common assumption in astrobiology; people often talk about about ammonia based life because it seems like a valid alternative liquid. However, I don’t see why it’s necessary. Why can’t things be immersed in a gas? One guess is that it has something to do with the fact that polarity of the liquid. Water is polar, and this influences biochemistry a lot. For example, more things dissolve in water than in a vacuum or in air. I think that the polarity of water has to do with it. Still, this isn’t convincing. So, is there a genuine reason here, or are astrobiologists merely being stupid?

The Electron Fluid

So I had this blog initialised for a long time, but it didn’t have anything. Now let me put this article I had written. I actually wrote this a few months ago, but I’m only putting this on the internet now.

 

THE ELECTRON FLUID
by Itai Bar-Natan

Many of you know of a particle known as the electron. They are the one of the
primary components of an atom, along with the nucleus. Some of you may know that
electrons aren’t /quite/ particles. They obey the laws of quantum mechanics,
which means that sometimes they behave like a wave. Indeed, if you learn chemistry
you’ll quickly find out that electrons act in very unparticle-like ways.

Actually, considering electrons to be particles was a mistake in the first place.
They are really more like a liquid.

Let me clarify. I’m not some kind of crackpot who thinks they revolutionized the
entire field of chemistry. I know the electron is not actually a liquid. It’s not
particles either. It’s a quantum field. That said, quantum fields are fairly
complicated objects, and physicists and chemists like to pretend that the electron
is a particle because it is simpler to explain. That is what I seek to overthrow.

In the rest of this article, I will describe many aspects of chemistry and physics from the
perspective that electron is a liquid. For some of you what I say will be entirely new.
For others, you may already know what I have to teach, but will be delighted to
see just how much more sense it makes from this new perspective.

First let me remind you the basics of electricity. Everything has a charge: positive,
negative, or neutral. Positive objects attract negative objects and repel
other positive objects.  Negative objects attract positive objects and repel
other negative objects. Neutral objects are not affected by either postive or
negative objects.

Now, ordinary matter is made out of two materials: electron, which is a negatively
charged liquid, and nuclei, which are positively charged particles. Some nuclei
have larger charges, and some have smaller. All nuclei have a charge which
is a multiple of the proton charge.  Electron accumulates on a nucleus
until its charge cancels out with the nucleus’s charge, making a neutral atom.
While nuclei with larger charge have more electron around them, they also
squeeze it more strongly, leading every atom to have approximately the same size
You’d expect that electron fills up around the nucleus in a spherical shape, getting
as close to the charge as possible. However, for complicated reasons, it tends
to fill up in more complicated patterns, often making the atom lopsided. Rather,
it has a tendency to fill up in what are called orbitals. These are regions
surronding the atom that electron tends to completely fill before filling
up other regions. The ones closest to the nucleus, called 1s and 2s, are
perfectly spherical. The next orbital, however, called 2p, has more of an
oval shape, so when electron goes on there the atom becomes more unsymmetrical.
Beyond that is a long and complicated list which I don’t want to get into
too much detail.

Sometimes electron fills up around more than just a single atom. You see,
in a lone atom, the electron is closer to any surronding objects than the
nucleus. This means that although the atom is neutral, it can attract other
atoms. The other atoms would move close to the atom until their electron
gets mixed up. This is called a chemical bond. Sometimes when this happens
the electron ends up distributed differently among the two atoms than
it was when they were seperate, so that one atom has too little electron
to be neutral and the other has too much. Then the bond is called an
ionic bond. When the atoms remain neutral individually, it is called
a covalent bond.

Now is a good time to mention how electron acts like particles. You
see, electron can’t exist in any possible amount. Rather, the
amount of electron must always be a multiple of a certain quantity,
called a quanta of electron. This quanta has exactly enough charge
to cancel out the charge of a proton, so an atom can always become
neutral while having electron with the right number of quanta. When
a quanta of electron is isolated, it behaves a lot like a particle.  Physicists
doing particle collion experiments with them are justified in thinking of them
as particles in the context of particle collision. They also behave like
particles in a plasma, where the electron quanta form a gas. That said,
I think that in the context of ordinary chemistry, the electron quantum field
behaves a lot more like a liquid than a gas.