Missing Matter

Apologies for the enormous delay. I was busy with research, but I've given some thought to your questions and believe my answers will be superior to what they would have been if I'd responded initially. Causal, to a physicist, is meant in the sense of special relativity. It means that the influence of one physical system on another cannot travel faster than the speed of light.
Quantum Mechanics is non-deterministic or statistical. Everybody knows what this means so I won't explain it.Quantum Field Theory is still probabilistic or non-deterministic (which ever word you want), however it is causal, in that nothing travels faster than light.The reason the word causal is associated with staying below the speed of light is because if the speed of light could be exceeded then the world would have no real past--->future flow of time that anybody could roughly agree on and would lead to time travel.QM might have random events but they are nicely ordered in a temporal sense, if you get my meaning, e.g. This atom randomly decayed before this other atom. So QM is causal. Or rather QFT is causal as standard QM doesn't have relativity built in.

I've come up with what I think is a good example of this stuff. It'll take a few posts. I'm going to exploit the fact that Quantum Mechanics, in mathematical sense, is more general version of a branch of maths everybody is familiar with. Probability.First I'll start with something simple, the Hilbert space of a classical coin toss.Let me take a fair coin, one that is equally likely to land heads or tails. If I have such a coin then I can measure several observables as physicists call them. For example if +1 is heads and -1 tails I could measure the average of all coin tosses. I could measure the average of the cube of each coin toss, e.t.c. Of course for a coin there aren't really that many interesting observables. Any way for each observable I'll get numbers for all my averages. From this list of averages I can figure out all statistical information about the coin that I could possibly want.If I use an unfair coin, one with a bias to heads for instance, then I'll get a different list of averages which would lead to different statistical information. This is to be expected since an unfair coin should be different statistically from a fair coin.Now for every single way the coin could be unfair (and the the one case when it is fair), that is for every different possible state of the coin, I'll get a different list of averages.So I can label each state of the coin by its list of averages. The collection of these lists of averages is then called the Hilbert Space of the coin. Since each list of averages belongs to one state of the coin, this Hilbert space is also known as a state space.If this makes sense I'll move on to QM (which is actually a very simple generalisation of all this) and then entanglement will be quite easy to explain. If fact spin and entanglement are actually just features of the quantum analogue of a coin toss.Edited by Son Goku, : Better title.

Yes, although I'll say it a different way for clarity.There are several different quantities or observables related to a coin that can have averages.To illustrate, consider the following list:
(Average of Coin Toss, Average of double coin toss, Average of thrice of coin toss, ......)
(+1 is heads, -1 is tails.)For a fair coin I get:
(0,0,0,......)For a biased coin I might get:
(0.66, 1.5, 2.25,.....)The first list is a list of averages for a fair coin, the second a list of averages for a biased coin. However it is the collection of all possible such lists that is the Hilbert space.So the Hilbert Space is a "space" of lists and any given list is a list of all the averages of different quantities. Well, the funny thing is, this isn't really an analogy. The above description is actually the Hilbert space of a coin toss.
As for if it would be different for a dime or a quarter, well two things. First, I'm not that familiar with what a dime and quarter are actually like. Looking at wikipedia they seem to be roughly similar circular coins. Second, the differences between them come down to things like composition and mass, e.t.c. The way I'm discussed things above, I didn't really look at these kinds of differences. However it doesn't matter, for if I was to take these things into account, dimes and quarters would behave like one of the biased lists, a different list for each. (Both lists would be very close to the fair coin list, since currency coinage is quite fair when it comes to a toss). However a list of averages is still a list and so it is in the Hilbert space. So you're correct with "the Hilbert space is powerful enough to describe all two sided coins". All you do is choose the right list.

My apologies for being so late to reply to this. Several theories predict such objects. In fact very simple extensions of the standard model of particle physics include such objects. Good observation. Dark Matter doesn't really change the Big Bang theory because adding it in doesn't really change which solution to Einstein's Field Equations describe the universe.
Okay technically it does change it, but it's still a Big Bang spacetime with all the same features, it only effects stuff such a galaxy rotation. There studies being started which will attempt to get some grip on how Dark Matter is distributed. Dark Matter is still clumpy like ordinary matter as it seems to surround normal matter galaxies. There would be some in our solar system, but not enough to really do anything. Yes, the extra dimensions in String Theory are small and curled up on themselves. The particles are simply more mathematically tractable. The objects described by another basis would be just as real. However I'll explain in my next post why we use particles.

My extremely slowly delivered series on Quantum Field Theory continues...So I've just described the Hilbert Space for a coin. The important points:
1. The list of all averages for a coin (see above) is called a state since it describes the coin.
2. The Hilbert space is the set of all such states, the set of all possible coins.Now obviously, related to what Stile said above, if two coins are identical statistically, or the exact same as far as a gambler is concerned, then they are identical in this Hilbert Space way of viewing things. So a Hilbert space doesn't really describe coins completely. However it does describe particles completely.Quantum Particles:
In Quantum Mechanics all have to do to my list of averages is make it noncommutative. What this means is actually quite simple to understand and is the cause of the uncertainty principle.Let's go back to the old classical example of a roulette wheel. Just like the coin I can make a Hilbert Space description of the Roulette Wheel. Let's say I'm working with a Roulette Wheel where you use two balls on every spin and one has to land on red and another has to land on black. Then one average, in my list of averages, is the average value of the black number multiplied by the red number. I'll call this quantity:
B.RNow obviously this is the exact same as the average value of the red number multiplied by the black number:
R.BSo, B.R = R.BSo in my list of averages I'll get two quantities which are the same, for example:
(..., 34, ......., 34,.......)
All quantities which are just things multiplied in a different order will have the same average.Now let's turn to a quantum particle. Two things we can measure are position, X, and momentum, P. Just like the Roulette wheel and the coin toss, I can make a list of averages. However position times momentum:
X.PWill not have the same average as momentum times position:
P.XQuantum Mechanics is probability where the order of multiplication matters, if the order of multiplication matters in mathematics we say it is noncommutative.Now in an experiment in order to measure X.P you would measure position and right down the result, then measure momentum and right down the result and then multiply them.To measure P.X you do the same thing, but measure momentum first. However because order matters, this tells you that as you measure X.P and your colleague measures P.X over and over again, the average result in the two cases will differ. Measuring momentum first is different from measuring position first.So the position of a particle is affected by measuring momentum before it. If we already knew the position of the particle before hand and then measured momentum and then measured position again, the second position measurement would be altered by the fact that a momentum measurement proceeded it. So you will get a different answer from the first time you measured the position, hence measuring momentum ruins your previous knowledge of position. It works the same with momentum. Measuring one ruins your knowledge of the other, hence the necessary uncertainty in position and momentum.All because nature cares about which way things are multiplied.Was this helpful?

I will now begin to add some bits and pieces that are necessary before I begin to introduce relativity into quantum mechanics.Thus far I have said that quantum mechanics is described by a Hilbert Space. The Hilbert space is a set of states and each state is a list of averages. What distinguishes quantum mechanics from regular probability is that these averages have different values for quantities which are different orderings of the same thing, like X.P and P.XTime Evolution:
The next important thing is time evolution. Obviously things evolve in time in the real world. This is also true in quantum mechanics. It occurs by the list of averages changing over time. So lets say I watch a system over a time period of three seconds. I could get the following (t is the time in seconds):
(0.4, 0.5, 0.01,.......) t = 1
(0.2, 0.8, 0.12,.......) t = 2
(0.3, 0.1, 0.94,.......) t = 3So the values of the averages are changing over time.
Since a "state" is a list of averages and the particle is changing averages, it is changing its state over time. Its moving from one state to another in the Hilbert space.How it changes its values is described by the Schrdinger equation, the central equation of quantum mechanics.Two or more particles:
If I have two particles the Hilbert space gets a little more involved, but not greatly so.
Basically the lists will be a lot longer because you have:
1. The averages of the every quantity to do with the first particle
2. The averages of the every quantity to do with the second particle
3. The averages of the every quantity that involves both particles, for example their total spin or total energy.For three particles one has:
1. Averages associated with one particle
2. Averages associated with any pair
3. Averages associated with all the particlesAnd so on if you have four, five, e.t.c.In the two particle case, lets say I have a list:
(...1...;...2...;...3...)
1,2,3 mean the different types of averages I mentioned earlier.Sometimes these two particle lists can be broken done into two separate one particle lists, so the particles can be viewed and understood separately.However some two particle lists can't be broken up like this, the particles must have a single combined list, they cannot be understood separately. This is entanglement.Now some notation for the next post. If I write a list like this:
{2}(.........)
I mean a two particle list.
{5}(.........)
is a five particle list and so on.The next post introduces special relativity and will explain why we need fields and not just particles.

Hey Stile, apologies for the ludicrously long delay teaching and research has me busy!Anyway on to your question, which will lead nicely into entanglement. In this case the multiplication is directly related to the order in which things are measured. However in some cases the multiplication can't really be interpreted that way, an example of which will be in my next post.

This post is a slight detour to explain entanglement and hopefully with the terminology I've introduced I can make it quite simple.In Quantum Probability 2, I described quantum mechanics as a probability theory which cares about multiplication. To use technical language a noncommutative probability theory.Obviously this allows different phenomena than regular probability, the best example being entanglement.The basic idea is the following experiment:
Two observers are at a great distance from each other, let's say ten light years. Half way between the observers there are two particles each of which must have a spin of 1/2 and the total spin must be 0, so they have spins in the opposite direction to the each other.The particles are sent off, one to one observer and one to the other observer. Observer 1 sees its particle spinning up and observer 2 sees its particle spinning down. Not too surprising, exactly what you would expect. Things can turn out differently of course and sometimes observer 2 will see their particle spinning up and observer 1 will see theirs spinning down.So far so good.Now what if observer 1 keeps measuring the spin in the up-down direction, but observer 2 measures it in the left-right direction.
There are four possible outcomes of this experiment:Observe 1: Up Observer 2: Right
Observe 1: Up Observer 2: Left
Observe 1: Down Observer 2: Right
Observe 1: Down Observer 2: LeftIf the observers do this experiment hundreds of times, they will see statistical correlations in their data. The spin in the left-right direction of one particle is correlated with the spin in the up-down direction of the other particle, despite the fact that the particles are widely separated by light years.At first one would imagine that one particle is "causing" an effect in the other particle, even though they are light years apart, somehow influencing its spin. This is what some have called "spooky action at a distance".However as anybody familiar with statistics has heard, correlation is not causation. In order to establish causation we need to perform a few other statistical checks beyond correlation. The particles fail these tests. So it is not causation which is occuring. The particles are not actually affecting each other's spins magically at a distance.Rather the fact is that QM is a new probability theory which allows new correlations. Correlations stronger than those in regular probability theory. Hence things can be correlated in a way that is impossible in good old "probability due to lack of knowledge". In some sense the particles are more closely connected initially than was possible in any previous theories, be they probabilistic or deterministic theories.
We say the particles are entangled, a new quantum phenomena.Edited by Son Goku, : Title

I will use my notation from Quantum Probability 3, where:
{2}(.....) is a two particle list
and
{5}(.....) is a five particle listNow I will introduce relativity into the mix. First of all a brief recap of the crucial insights of relativity.Rel1: In pre-relativistic physics there was a quantity called mass, which measured a body's ability to resist being accelerated. There was also energy, which measured the ability to do work. In relativity, mass becomes a form of energy. Hence a physical system can decrease its resistance to motion to gain an increase in its ability to do work and vice versa.Rel2: If two events are separated in such a manner that one would have to "travel-faster-than-light" to move between them, then they cannot communicate and can not have any possible influence on each other.Rel3: Space and time are unified into a single spacetime. This isn't too important for this post.Somehow one has to implement these points into quantum theory. To make quantum particles obey special relativity. This is the problem of relativistic quantum theory.
However there is also another problem, I have only described particles so far with quantum theory. I haven't dealt with fields, such as the electromagnetic field. This is the problem of the quantum theory of fields.
The solution to these problems turns out to be one and the same.First of all let's consider Rel1 and what it implies.
A very fast moving particle possesses high energy due to its motion. It is possible that this energy could be "turned into" mass to produce another particle. In general a system of many particles with lots of energy can lose some energy and gain some particles.
This means that I could start with a two particle list like
{2}(.......)
and eventually end up with a four particle list like:
{4}(.......)So I need a new way of evolving in time that allows me to move between lists.Now for Rel2. This means that my quantum theory can not allow influences to travel faster than light.So how can I move between different numbered lists and prevent things traveling faster than light?Now let's look at applying quantum mechanics to field, for instance the electromagnetic field. For a field I should have lists which give all the averages possible for all observables of a field instead of for a particle.
There are an absolutely huge number of lists in this case, since you can observe a lot more things with a field then with a particle. However if you look at these lists there is some structure inside.Basically some of the lists for the field behave as if nothing is going on, I'll call them a vacuum list:
{0}(......)
Another bunch of lists act like nothing is going on, except there is one "small lump" of mass/energy of the field moving around in a quantum way. It turns out that these lists act exactly like lists for one single particle, in fact there is no difference. These lists are the lists for a single particle. So I can use my old notation:
{1}(......)It turns out there are field lists which act like two particles:
{2}(......)
three particles:
{3}(......)
and so on.....However since all these lists are simply lists of the original field, they are all connected and over time they can move from one to the other, allowing particles to be created and destroyed.
Also since fields like the electromagnetic field naturally obey relativity, that is they never move faster than light, this fact is now automatically built into the lists.This is quantum field theory, where particles are seen to be a consequence of quantum mechanics and fields. Different fields give different particles, e.g.
Electromagnetic field => Photons
Dirac field => ElectronsNext I'll try to deal with some of the physics that comes out of quantum fields.I just wanted to ask, is any of this making any sense?

Just a brief explanation of this theorem since it comes up so often, for instance about three years ago RAZD started an interesting discussion on it.As I've already explained, Quantum theory is a noncommutative probability theory. Very early on people proposed that the probability in QM came from lack of knowledge or ignorance. Basically that there was a hidden theory underneath, a real honest old fashioned theory and QM was just some statistical approximation.However statistics due to ignorance is always (stress on always) commutative probability. Bell realised that this meant the only way you could say QM was a statistical approximation was if you could find some commutative probability theory which could simulate QM.In 1964 he proved that the only if the commutative probability theory was nonlocal could it simulate QM. That is only if it allowed transmission of information faster-than-light.This actually leads to a popular misconception about QM. QM is not nonlocal, it's totally local. Nothing is transmitted faster-than-light. Rather any theory which tries to replicate QM while claiming the probability is due to ignorance must be nonlocal.Anyway next post I promise I'll have actual physics.

I apologise for the eons it took to respond.... Yes, precisely. Yes. Sorry to be so short, but you are simply exactly correct. No, not entirely. Although we do know that if it is true, it must be pretty damn weird to the point where you would wonder if it was any better than QM.
First of all, if there were such a theory what we now call quantum mechanics would simply be a statistical approximation or probability due to ignorance, i.e. commutative probability. From Bell's theorem we know that commutative probability can only replicate noncommutative probability if it is nonlocal. So this theory would have information being transferred faster then light, but in such a way that this transfer of information can never be observed or used so that special relativity still holds. Which seems a little odd to me, to have some effect which for the consistency of the theory must never be observed.Also, due to the Kochen-Specker theorem (see my post "Interpretations") we know the theory must give up a few other things. Since you want it to be a good old fashioned theory that replicates QM it must give quantities definite values, i.e. something is really spin up or spin down. However the Kochen-Specker theorem says that you then have to give up value realism or noncontextuality.
Giving up noncontextuality would mean that what an experiment means depends on context, for example my experiment to measure an electron's spin in the z-direction in Dublin when Frank measures a electron's spin in the x-direction in Guatemala City is a actually a different experiment to me measuring an electron's spin in the z-direction when Frank measures it in the y-direction. i.e. I'm measuring a different quantity of property of the elextron in both experiments, because even though I'm doing the exact same thing the global context is different.
Giving up value realism would be equally bizarre, to quote the Stanford Encyclopedia of Philosophy: So we can't totally discount the possibility of these theories, but they have to be weird to the point where you actually wonder is it really any better then QM. They have to give up noncontextuality and locality or give up value realism and locality, which in my opinion makes them worse than QM.For instance one way you have a universe which talks to itself faster-than-light in a way that can never be detected and where what quantity one looks at depends on what everything else is doing. Not even the value of the quantity, but the quantity itself!
The other way you also have the faster-than-light communication and where just because some quantity is measurable doesn't mean that things related to it by trivial mathematics, like the square of the quantity, have a meaning or exist.
Are either of these really worse than QM? In QM everything is local, things don't depend on context, if I can measure something I can measure another thing related to it. What QM has is that probability is noncommutative, which is difficult to interpret and understand. Is this really worse than the above?A few people work on these alternate theories, but quite often it is a lot of hard work to replicate QM and in the end the theory is stranger than QM, harder to use and quite often (not always) can't be made to obey special relativity.Edited by Son Goku, : Small addition.

Although I am going to try and finish these posts sometime in the next thousand years, I just thought anybody here might like to check out:
Why the world needs quantum mechanics | Michael Nielsen
Michael Nielsen is quantum computation expert, one of the top in the field and author of one of the best text books on the subject. Here he tries to explain why our intuitive understanding of the world must be wrong and some new theory (e.g. quantum mechanics) is needed. Check it out!Edited by Son Goku, : Title

Alright, so as I promised about nine months ago (!), I'll start discussing some physics and how quantum field theory deals with it. In order to do this I will discuss one of the most famous calculations in Quantum Electrodynamics (QED), the calculation of the magnetic moment of the electron, which basically was what got Julian Schwinger his 1965 Nobel Prize. This will allow me to discuss a sticky issue in physics called renormalization.
Quantum Electrodynamics is the quantum field theory of electrons interacting with photons.Okay, so let's set up the problem. The magnetic moment of the electron is basically a measure of how much an electron is affected by a magnetic field. Just like charge decides how much it is affected by an electric field.Historically, before quantum field theory came on the scene, Dirac had calculated the magnetic moment of the electron to be 2 using an older theory of his creation.Schwinger attempted to get a more accurate answer using QED. To this he basically analysed how the list of probabilities for a single electron, {1}(......) in my notation above, change in the presence of a magnetic field. Inside those changes will be the electrons magnetic moment.Unfortunately the equations are far, far too difficult to solve. So people came with an idea, approximate QED. The idea is that you start off with the theory where electrons and photons don't interact and work things out there. Then you work out things in a theory where they interact once and once only for all time, then a theory where they interact twice, e.t.c. By adding these results together, you slowly approach the correct theory.Schwinger only consider the first two cases which affect the magnetic moment, namely one interaction and three interactions. These cases are symbolised by Feynman diagrams:
The one interaction diagram:

The three interaction diagram:

In the first term the solid lines are the electron and the wavy line is a photon from the magnetic field which causes it to move, the electron's reaction to this photon is its magnetic moment.In the second term, the electron emits its own photon, then gets kicked by the photon from the magnetic field and then reabsorbs the photon it emitted. This diagram has a loop in it and effects due to diagrams with one loop are called (incredibly) one-loop effects.Everybody could work out the first term, which gave the same result as Dirac, that the magnetic moment was 2.However the second term posed a problem, when calculated the answer was infinity. For over a decade nobody really knew what to do with this, perhaps QED was just nonsense. In fact most people thought this.However Julian Schwinger came up with a brilliant solution. In calculating these diagrams we have neglected something. In the calculations, the electron's charge "e" that appears is shifted by one loop effects itself and we can't compute this diagram without taking this into account. Pretending that the electron's charge stays the same is what gives the infinity. If we replace the electron's charge by its shifted value, then the diagram suddenly becomes finite.
Replacing the unphysical "unshifted" quantities with their physical "shifted" replacements is called renormalization and it is the procedure which saved quantum field theory.Now that the second diagram was finite, Schwinger simply added the two results together to get the following for the electron's magnetic moment:
2.0023228
Which is only slightly off 2. However very shortly afterward experiments showed this result was incredibly accurate.Today people have gone all the way up to diagrams with four loops to get:
2.00231930436170Which agrees with experimental measurements to ten parts in a billion, making this number the most accurately verified prediction in all of physics.

I've mentioned renormalization in my last post. It is the art of making apparently infinite expressions in quantum field theory into finite expressions by taking into account that certain quantities get shifted.In all of our theories today this works out. I'll explain the details of the theories of the Strong and Weak nuclear forces in another post, but first I'd like to continue a bit with Quantum Electrodynamics (QED) and discuss gravity a bit.In QED there are really only four quantities that get shifted, one is the electron mass and another is the electron charge. (The other two are a bit abstract so I'll leave them). When you properly take these into account the whole theory becomes finite for all diagrams*.The basic problem with gravity and the main technical reason behind the need for a new theory like String Theory e.t.c. is that this doesn't work for gravity. If you directly turn General Relativity into a quantum field theory (by using the quantising procedure I described in an earlier post) and then take into account the shifting of quantities, the theory is still infinite. Hence gravity does not appear to work as a quantum field theory, so we need some new ideas.*Proving this is quite a task however and was only really rigorously proven in 1988, fourty years after Schwinger had done it for one loop.

Now we turn to the Weak Nuclear force, or as it is sometimes called quantum flavorodynamics.The Weak Nuclear force is basically responsible for the vast majority of radioactive decay that we observe.
Unlike electromagnetism, which is communicated by the photon, a single massless particle, the Weak force is communicated by three massive particles, called W-, W+ and Z0. Exchanging, emitting or absorbing these causes a particle to change its "flavor" or species. So for example, if a down quark emits a W- it becomes an up quark.This is how a neutron can decay, for example a neutron is made of one up quark and two down quarks:uddThe down quarks spits out a W- and becomes a up:uud + W-And then the W- decays into an electron (e-) and a neutrino (v):uud + e- + vuud, is a proton, so in total we have gone from a neutron to a proton, electron and neutrino. The neutron has decayed.Another strange thing about the Weak nuclear force is that it is chiral. This means that not every particle can feel or "see" the Weak force. Specifically if a particle is spinning one way (left) it can see the Weak nuclear force and if it is spinning the opposite way (right) it can't. However this is quantum mechanics so a particle is usually both left and right with different probabilities, but all the probabilities to decay come from the left probabilities.Next the Strong Nuclear Force.