This is the Hilbert space for the Dirac spinor and state vector.

[kof9595995]

I believe Dirac spinors are not in any Hilbert space since it has no positive definite norm. However one QM axiom I learned told me any quantum state is represented by a state vector in Hilbert space, so what is happening to Dirac spinor?Or is it just that the axiom is not for relativistic QM?

 

[Bill_K]

Considering ψ(x,t) to be a function, we expand it in plane waves,

ψ(x,t) = ∫√(m/E) (u^pexp^i(p·x-ωt) + v^pexp^i(p·x-ωt)) d³p

The normalization is ∫ψ(x,t)ψ(x,t) d³x = ∫ (u_pu_p - v_pv_p) d³p

and as you point out this is not positive definite, due to the "negative energy states" v.

In the second quantized version u and v become operators which obey anticommutation relations. But even so, if you interpret u and v as annihilation operators the theory has negative energy states and an indefinite norm.

The solution is to reinterpret v as a creation operator: v_k = w_k. Then v_pv_p ≡ w_pw_p = 1 - w_pw_p and the norm is

∫ψ(x,t)ψ(x,t) d³x = ∫ (u_pu_p + w_pw_p) d³p - ∫1 d³p

The infinite negative part is discarded, leaving us with a positive definite norm.

 

[Funky_Functor]

Dirac spinors are plane wave solutions to the Dirac equation that relies on a background metric (which is actually a Pseudo-Riemannian metric, i.e. not positive definite).
We also know that the plane wave solutions form a basis of a (infinite-dimensional) Hilbert space, which has it's own inner product that is positive definite.
So basically the answer to your question is that there are two 'metrics' at work: the positive definite metric of the Hilbert space and the pseudo-riemannian metric introduced by the Dirac equation.

 

[George Jones]

The norm is positive-definite, even without moving to operators. I don't know if this is readable, but try

http://www.math.niu.edu/~rusin/known-math/01_incoming/QFT .

 

[kof9595995]

I see, I was reading some history of relativistic QM and I should've switched my mind to field theory, thank you!

EDIT:This is a reply to Bill_k and I'm now reading George's material, and see if it'll make a difference:)

 

[Bill_K]

George, Unfortunately I disagree with your article. Perhaps I 'm misunderstanding it, but it says,

The inner product on spinor wave function space G = { f | f : R^4 -> D }
is defined to be

/
<f|g> = |f(p)^+ gamma^0 g(p) d mu ,
/

where, as above, the integration measure is given by

d mu delta( p^mu p_mu - m^2) theta(p^0) dp^0 dp^1 dp^2 dp^3 .

What I do not want to see is the theta function. For sure this is Lorentz invariant, and for sure it is positive definite, but you've made it so by excluding the negative frequency components entirely, which is exactly what must not be done.

A theory without antiparticles is acausal. Things look Ok as long as you talk only about the free field, but any interaction will bring in the Green's function, and it will be nonzero outside the light cone. Antiparticles are the cure to this problem and must be included.

If I've misunderstood, I apologize and am ready to be corrected on this.

 

[George Jones]

Yes my state space is only for a single particle, but I think it can be extended to (free)antiparticle and multiparticle state spaces by appropriate direct sums and tensor products. I could be wrong, though, as I never really learned quantum field theory. I have a reference that treats this nicely, but it's terse, tough going, so it might take me some time (if at all) to figure this out.

 

[George Jones]

What I do not want to see is the theta function. For sure this is Lorentz invariant, and for sure it is positive definite, but you've made it so by excluding the negative frequency components entirely, which is exactly what must not be done.

Yes, the Hilbert space constructed in post#4 is the Hilbert space for a single particle. Denote this space by $H_+$. Construct the multiparticle Fermionic Fock space $F_1$ from this single particle state space. Annihilation and creation operators for particles operate on $F_1$.

Now use an appropriate step function to restrict to the negative energy mass shell, and construct a Hilbert space $H_-$ as the Hilbert space of single antiparticle states. The charge conjugation operator $C$ maps $H_\pm$ to $H_\mp$.

Antiparticles have positive mass, so, for the space multi-antiparticle states, construct another Fermionic Fock space $F_2$ (separate from $F_1$) from the Hilbert space $H_+$. Use the the standard annihilation and creation operators together with $C$ to annihilate and create positive mass antiparticle states in $F_2$ that correspond to negative energy states in $H_-$.

Finally, the total Fock space, a genuine Hilbert space, is

$$F = F_1 \otimes F{_2} .$$