Fock spaces semantics and number of particles

[TheBlackNinja]

I was not formally introduced to this math, so I appreciate corrections but I'll give my impressions.

The fock space for a particle with space H is
(c, H, HxH, HxHxH, ... )

1: What is a hilbert space for a single particle? I believe know what a hilbert space is, but for 'a particle'.. what is the base of this space? what to its vectors mean?

2: Operaions on Direct sums of spaces are defined as parallel operations over vector of those spaces. If each position on a tuple does not have a single vector associated(or it has?), like a basis vector, what would mean in the fock space, for example, (0, 1 ,1 , 0 ..0..), (0, 2 ,1 , 0 ..0..), (0, 1 ,1 , 0 ..0..) or (0, 0.5 ,0 , 0 ..0..)

3: How would fock states be denoted as tuples of the fock space? like, would |2> be (0,0,1,0...) ?

4: What does it mean for a state to have a 'well defined number of particles'? that all except one coefficient on a fock tuple is non zero?

 

[mpv_plate]

1: What is a hilbert space for a single particle? I believe know what a hilbert space is, but for 'a particle'.. what is the base of this space? what to its vectors mean?

You still have to define the basis: choose if your Hilbert space is describing spin or momentum or position. The phrase "Hilbert space for single particle" does not tell anything about the representation; it only says that we are describing just one particle (and not a system of several particles).

 

[TheBlackNinja]

Ok, so let {|a>, |b>,|c>} be a base of vectors for describing position in space and time and |psi>=|b>+|c> be the wavefunction of a particle. What is the "particles Hilbert space" in this case?

 

[Bill_K]

The fock space for a particle with space H is (c, H, HxH, HxHxH, ... )

Not quite. Fock Space is a quotient space of what you have written. The symmetry/antisymmetry that goes with identical particles is built into the definition of Fock Space. Whereas in H ⊗ H, |1>|2> and |2>|1> are two distinct states, Fock Space contains one state which is the symmetric/antisymmetric linear combination of these two.

 

[TheBlackNinja]

Ok, its (c, H, S(HxH), S(HxHxH), ... ) where S is a symmetrizing or anti-symmetrizing according to the particle type. Its impossible to edut.

This answer helps me(Lubos's)

http://physics.stackexchange.com/questions/30751/what-is-the-single-particle-hilbert-space

So the "Hilber space for a single particle" is the set of all states on my original space which describe a single particle. I still don't understand |1> for example - this vector alone describes all single particle states?