The appearance of degrees of freedom can potentially be an illusion. A measurement of a quantum system is ultimately an entanglement process. You entangle a system with another, which removes the superposition of the original system and puts it into an entanglement. So if you measure a two state system with another two states system, which I break out below, a naïve assumption is there are four degrees of freedom. However, in reality there are only two. So this issue is related to the occurrence of classical physics and the measurement problem.
A spin system has in the basis of the Pauli matrix σ_z the states |+> and |-> for spin up and down. The Pauli matrix acts on these states as
σ_z| +/-> = +/-| +/->.
Now these states are complex numbers, which means there are 2 variables for each state and thus 4 altogether. However, there are constraints, such as the probability Born rule 1 = P_+ + P_-, P_ +/- = |a_ +/-|^2 for a state |ψ> = a_+|+> + a_-|->, and irrelevance of a phase in real valued measurements. So this reduces the number of variables from 4 to 4 - 2 = 2. That is just what we would expect.
Now let us consider two spin systems, say two electrons. The use of electron spin state is not concrete, for these arguments hold just as well for polarization direction of photons. So we have two sets of states and operators {σ_z, | +/->}^1 {σ_z, | +/->}^2 denoted with an additional index i = 1, 2 and we still have
σ ^i_z| +/->^i = +/-| +/->^i.
We can form two independent states |ψ>^i = a^i_+|+>^i + a^i_-|->^i for the two spin systems. For each there are 4 variables and 2 constraints. This gives 4 degrees of freedom in total. Yet we can compose these spin states in various ways. One way of doing this is
|ψ> = (1/sqrt{2})(|+>|-> + e^{iφ}|-> |+>),
where I have dropped the index i, and we just implicitly see the first and second | +/-> as i = 1 and 2. This makes reading things clearer. The e^{iσ} is a phase which for it equal + and - the state |ψ> is not an eigenstate of σ^i and is an eigenstate of σ ^i respectively. So these are singlet and triplet state configurations. This is an entangled state. If you have access to | +/->^1 then you also have access to | +/->^2, and this holds no matter how far apart these states end up as. You can entangle two electrons by overlapping their wave functions. One that is done you can separate them arbitrarily far and they are still entangled.
Now let us count the degrees of freedom for this state. We have again 4 variables for each | +/->}^i but now we have one constraint from Born rule and another from the entanglement state. So you have 6 independent variables with 4 constraints giving 2 in total. This is the basic bipartite entanglement. There are also n-partite entanglements, such as the W and GHZ states.
So assume there are states given by SO(32) or E_8xE_8, which number 496. One of those states is an electron, another is an up quark and so forth. In the entire superspace each of these particle zig-zags through all space and has a configuration in a vast number of forms. These multi-threaded paths, in how Feynman originally thought of the path integral, are then eigenstates in a pure state. However, the observable universe is an entanglement with states in the rest of the superspace, and each particle is a local entanglement with particles in the whole "zig-zag." So we have this illusion of there being a vast number of degrees of freedom, with a huge number of duplicate particles. However, they may all the same particle. An electron running through you computer is the same as an electron being pushed as an exiton in photosynthesis in a leaf in Gabon and is identical to an electron in an accretion disk transport and the same electron everywhere.
Some of the machinery I am building up should lead to this understanding. At least this is one of the objectives.
Cheers LC
