Missing Matter

Dark Matter interacts either not at all or very weakly with other matter through electromagnetic, strong or weak forces. Hence you can't see it and it passes through normal matter.
It does however interact gravitationally and hence its combined mass influences the motion of the cosmos.

Neither do we! Establishing that a specific new particle found in the collider is directly related to Dark Matter is exceptionally difficult. It could easily be argued that any weakly interacting particle discovered in the LHC are not the as the Dark Matter particle, simply because there would be no direct connection.One way of telling would be if the particle that was created was very stable, since Dark Matter doesn't seem to decay.

I would say it's unlikely for a few reasons. The first being and I'm not sure how to say this, the proton is not really "made" of three quarks. Rather the proton is a state produced by the interaction of about eleven different quantum fields. Three of those fields, if they didn't interact with the others, that is if they were free, would have excitations which we call quarks. So those fields are called the quark fields. However since the fields do interact, this picture isn't accurate and the fields never possess those excitations we call the quark particle. Usually when calculating things however we start of by having the fields being approximately free and so we use this fictitous notion of quark.
I hope that makes sense. In a way asking if the quark is made of anything wouldn't be sensible. One could ask if the quark fields are the results of fields interacting on a lower level. This is unlikely though, if the quark fields are made of something it's probably not a field.

Whoops! I somehow forgot about one of the most important experimental results in quantum chromodynamics.

It's both. A particle resulting from the interactions of fields. Particles are basic excitations of the fields and provided that the fields are not too "active" they may be used to provide a description of the fields detailed enough to compare with experiment. Concentrating on the particles also makes calculations simpler and means terms in the calculation can be represented by diagrams. This was Feynman's method. No, the extra dimensions of String theory arise from different considerations.

Hopefully I can outline things in decent way, although I've never actually been in String research.First of all you should know of a process called quantization. This is a standard mathematical procdeure where one turns a classical theory into a quantum theory. It twins a classical theory with a quantum theory, essentially giving a quantum version of that classical theory.Now when a physicist speaks of something like quantum field theory and String theory, we are not to be taken at our word. The objects these theories describe are not fields or strings. Rather those names refer to the kind of classical object described by the classical twin of these theories. For instance quantum field theory should be labelled the less catchy "Theory which is the quantum twin of a classical theory that describes feilds".Too cut a (very) long story short, several people for various reasons, came to the conclusion that a theory of Strings was the best way to incorporate gravity into a quantum mechanical framework.In order to find such a String Theory physicists used the technique of quantization to start with a classical theory describing strings and end up with its quantum twin which should describe gravity. However it was found that if one sticks to four spacetime dimensions this doesn't really work. The quantization process turns gives an inconsistent theory as the twin to the classical theory of strings. If one wants a mathematically healthy theory to be the result of the quantization process, you must crank up the number of dimensions.

Well I've an idea on how to explain things.Let's start with quantum mechanics and the Schrodinger equation.First of all, what happens when one tries to describe a single particle moving through space in classical mechanics? Usually it is done by finding what the position of the particle q is at time t. This information is encapsulated in the function q(t). You can find q(t) for a specific situation by using the Lagrangian or the Hamiltonian. (However I won't discuss them much further.)In quantum mechanics however the particle doesn't have a specific position q at a time t, so we can't use something like q(t). Instead we have the probability amplitude that the particle is found at q. The probability amplitude that a particle is to be found at q is labelled (q). The probability amplitude can change over time so we denote it (q,t) instead to indicate this. Obviously we need a way of finding what the value of (q,t) is for some choice of q,t. This is provided by the Schrodinger equation.Now for fields things move up a notch. Classically a field, like the electromagnetic field, is an object spread over space and has a different value, indicating its strength, at every point in space. The strength at a point in space x is labelled (x). If one knows all these values for every point you know everything about the field. Assuming you have this total knowledge you just call the field . Total knowledge of field is called a configuration. Here are four different configurations of a field:

If the field was the ocean, then the configuration is the specific form of the ocean at the current time.
Now just like the case of quantum mechanics, if you add in quantum considerations and move to quantum field theory then a field can't have precise configuration, labelled . Instead all you can have is the probability amplitude that the field is in the configuration . This is labelled (). Just as before the probability amplitude can change and so we have to include time (,t). The values of (,t) are found using a kind of "super" Schrodinger equation that is almost impossible to solve. We solve it approximately using diagrams known as Feynman diagrams to keep track of terms in the approximation.Does this make any sense?

I understand what you mean, the field wavefunctional is rarely discussed outside the ultra-rigour style approaches of Glimm and Jaffe, Frohlich, etc. However without it I can't go from QM to QFT the way I'm trying to. I'm just going to see how it goes, maybe it won't work. Although I'm doing this from an observation that field theory seems to make more sense to people after they can explicitly connect it back to QM, even if that's not how field theory is done in practice. Usually it clicks after one hears of the wavefunctional and then one can say "Oh Yeah!, just like QM". For instance I've heard many times "an operator on what?" The answer of course is the s, "just like QM". I think it is helpful to talk about the s pedagogically, but of course different fields (in the academic sense) will require different pedagogical approaches. I'm not going to go near String theory, as I said I don't know the subject. Hopefully though I can give a good summary of QFT.

That's all perfectly correct. Good to see I've written a solid first message to refer back to.

Loads, this is one of my specialities. Unfortunately if I go in to it we'll both start sounding like jargon cannons. So I'll leave my technical comments hidden below. I'll probably put up my next big post sometime in the next few days. See what you think!

Just wanted to say I haven't forgotten this. I've a few posts lined up for the weekend. They basically cover how quantum field theory deals with particles and the issue of renormalization.

Weekends have been known to move forward several days.....

Hilbert Space

Anyway, in my last post I discussed how we can understand quantum field theory. We use (,t), the probability amplitude for a field to be in a given configuration at time t. First of all probability amplitudes are square roots of probabilities. So when I discuss them below the numbers should be squared to get the probability.However quantum field theory is meant to describe particles. How are particles to be described with (,t)? The answer lies in a construction called a Hilbert Space. Bypassing several decades of functional analysis, I will define a Hilbert space very simply.Obviously you can have several different (,t). To explain this let me go back to the picture from the last post:

Here I'm looking at four different field configurations. A sample (,t) is where I have an equal probability amplitude.
(,t) = (0.5, 0.5, 0.5, 0.5)
This says that there is an equal chance of the field being in any one of the configurations. However it could be that there is a much greater probability amplitude to find the field in the fourth configuration. For this case I could use:
(,t) = (0.25, 0.25, 0.25, 0.9013) (The last figure is rounded, so the squares will not add to 1 exactly).Hopefully the point is clear. (,t) is a big long list of probability amplitudes to be in different field configurations. In reality there are an infinity of field configurations so (,t) is infinitely long.A Hilbert space is then the set of all these lists or the set of all (,t). (Basically Hilbert Space is a space where each "point" in the space is a specific (,t))
It is this space that the mathematics of quantum theory centers around.

Particles

Now the genius of using such a space is that it allows one to work on things in a concrete way. Using mathematical devices known as operators, I can find not only the probability amplitudes to be in a field configuration from the list (,t), but also I can extract the probability amplitudes to possess certain quantities such as energy, momentum, angular momentum and spin.Then one notices that certain (,t) have specific classical properties. Some (,t) have a definite (enough) momenta, are localised (enough) in space and have a specific spin so that certain lists have particle like properties.Using the Hilbert space structure I can use this to rewrite everything. Instead of having a list (,t) to be in a specific field configuration at time t, I can rewrite things in terms of new lists (n,p,s,t). The probability amplitude to have n particles with momentum p and spin s. Basically this is just the old lists (,t) rewritten to be probability amplitudes for possessing particle quantities rather than field configurations.Since we're working with quantum theory and probability amplitudes the particles are automatically quantum mechanical. Also since the original field configuration probability amplitudes obeyed relativity and causality, the new particle ones will as well. Voila!, a theory of relativistic quantum particles.Depending on the field we used at the beginning we get a different kind of particle when we rewrite the lists. If we use the electromagnetic field, we get photons and if we used something called the Dirac field we'd get fermions or matter particles.(If anybody is familiar with linear algebra, a Hilbert space is just a type of vector space and rewriting things in terms of particle quantities is just choosing another basis for the vector space.)Next time Feynman diagrams and renormalization.

To make this interesting I present a counterexample to both our positions. (Although the counter example to my position is a bigger problem than the counter example to yours.)First the counterexample to my position. Which is the "double well" quartic interaction scalar field theory or the Higgs field with one component. Here I can build a Hilbert space and define the Hamiltonian of the theory on it. I later discover that scattering processes with initial states in this Hilbert space have no final states in the space. However, if I take the fields and use their algebra, I find that the fields have three other representations, besides the one I'm using. If I direct sum the Hilbert spaces of the three other representation and my original one, then indeed all initial states have final states in this larger Hilbert space. Hence there is certainly no way a Hilbert space first approach is going to achieve this. I must look at the representation theory of the fields.Second the counterexample to your position. This is the fact that even though the fields have different representations (infinitely many) in most interacting field theories only a few or even only one are actually correct. In all others the Hamiltonian will not be defined. This means that many interacting theories are associated with just one representation and hence a specific Hilbert space. The theory is tied to a certain Hilbert space.(Warn me if you feel this has strayed too far away from standard physics and into axiomatic field theory. Unless you enjoy axiomatic field theory.)Edited by Son Goku, : Less generic title

For anybody interested....The discussion between myself and cavediver can be understood in the language of my message #56. I mentioned mathematical devices called operators which you can use to extract information from lists in the Hilbert space.It turns out (to cut things short) that all the information in a Hilbert space can be gotten at by the operators and that, in a certain sense, all the information is already in the operators to begin with. In fact one could start with the operators and use their information to build the space. IIn the language of message #56, instead of using operators to extract information from the lists, one can extract information from the operators and rebuild the lists. This causes a question to arise, which comes first the operators or the Hilbert space?It's a big discussion in mathematical physics and investigation of it has lead to the discovery of new physical phenomena.My view is the older conservative one, which may be considered unusual as I am younger than cavediver.

Yes Yes There is always an infinite number of points, even in my simpler example of only four possible states. Basically this is because any amplitude can be written as:
(a,b,c,d) Where a,b,c and d are the probability amplitudes for each field configuration. a,b,c and d can be any real number and since there are an infinity of such numbers there is an infinity of lists. Basically the number of dimensions of a Hilbert space is equal to the number of classical states. A classical bit of information can be either 0 or 1, but a quantum bit of information (a qubit) is a list of probability amplitudes for being in either 0 or 1. Such as (a,b). Where a is the probability amplitude for 0 and b is the probability amplitude for 1. So as you can see the amount of numbers in the list is basically dictated by the number of classical states.
For a field or even a normal particle, there is an infinite amount of states. Hence an infinite number of dimensions.The only slight qualifier is for properties like energy. To take the example of a hydrogen atom, classically there can be any number of energy states. An electron can have any energy it wants in the classical theory.In the quantum theory however one finds that the lists of probability amplitudes only list the probabilities for states of certain energies, some classically allowed energy states have no probability amplitude and hence the system has no chance of possessing that energy, an energy that was allowed in the classical theory.
For a hydrogen atom the fact that only certain energies can be obtained is what actually prevents an atom from collapsing.However roughly:
dimension of Hilbert space = number of classical states. They certainly aren't.