Physical results should of course not depend on a particular choice of the gamma matrices. I think the solution to this puzzle is in the definition of the dual Dirac spinor. By definition, the dual spinor should transform as barpsi'=barpsi Z^-1, making the average quantities independent of the particular choice of Dirac matrices. One can write in general barpsi=psi^dag G, where G is some matrix. Clearly, G should transform as G'=Z^dag,-1 G Z^-1 and therefore does not belong to the Dirac algebra (it is kind of a metric in Dirac space). Now, one usually considers Z unitary, so that G can in practice be identified with a Dirac matrix.

Things become pretty clear if one writes explicitly the indices in Dirac space. Let us write a spinor in Dirac space as psi^a. A Dirac matrix maps a Dirac spinor onto a Dirac spinor, so one should write explicitly g^a_b. A dual spinor (belonging to the dual Dirac space) reads barpsi_a. It can be related to a Dirac spinor via a metric as follows barpsi_a=psi*^b G_ba. Clearly, G and g are different objects. One usually considers unitary transformations only and does not make any distinction between covariant and contravariant indices (i.e. matrix notation instead of tensor notation), leading to the identification of the metric G_ab with the Dirac matrix (gamma^0)^a_b.

let's try again the same exercise with and not (...|...), since the proper scalar product does not have the \gamma^0 in there. The (...|...) is not really a scalar-product, as it can be seen that (for instance) if \psi = L, then (L|L) = = 0. Physics-wise it would make sense to have a proper norm, as in = 1. So I would like to understand things from the perspective of for which there is some physical meaning.

I am not aware of any consistent physical interpretation you asked for in terms of the standard scalar product (i.e. Euclidean metric) and doubt there exists one. For example, the standard scalar product is never used in relativity (special or general). There is no generic physical meaning in the proper norm (ct)^2+x^2+y^2+z^2.

Restricting ourselves to Z unitary, one can always find an operator O such that =(L|O|L) and therefore find some physical interpretation for the standard scalar product. But if we consider Z non-unitary, the operator O (and thus the physical interpretation) will change with Z.

that is precisely the problem - anyone can choose another set of gamma's, related via a non-unitary transformation. Which scalar product is used matters physics-wise, but mathematics-wise you can have the one you prefer. There will always be some non-unitary Z that gives invariance problems to /.

For a real even dimensional vector space V with a non degenerate quadratic form g, consider the abstract Clifford algebra Cl(V,g). It has a unique [1] spinor representation \Delta and a carries a unique (up to real scalar) _but- _not_ _necessarily_ _positive_ _definite_ Hermitian form h such that the action of a vector in V is Hermitian.

This is the form that physicists write as

\bar\psi\phi = h(\psi, \phi).

If we already have a hermitian inner product ( , ) on \Delta then we can represent

h(\psi, \phi) = (\psi, H\phi)

for some ( , ) hermitian linear map H: \Delta \to \Delta which is determined by

(v\cdot)^* H = H (v\cdot)

where ^* is the hermitian conjugate with respect to (, ).

Now _choose_ an orthonormal basis e_i with g(e_i, e_i) = 1 for i = 1 .. p (the time like basis vectors) and g(e_i, e_i) = -1 for the remaining q = n - p (the space like basis vectors). Also _choose_ a matrix representation on C^{2^n/2} as we can. We now have gamma matrices

\gamma_i = \gamma(e_i)

and the standard inner product ( , ) = ( )^\dagger ( ), which however has nothing to do with the invariant hermitian form h except that as above h but now translated into matrices we can write

h(phi, psi) = phi^\dagger H \psi

for some Hermitian matrix H which is determined by

\gamma_i^\dagger H = H\gamma_i for i = 1, ... , n

If p = 1 and we have already chosen a matrix representation such that the \gamma_i are Hermitian for the time like and skew hermitian matrices for the spacelike basis vectors ((in the usual sense that \gamma_i^\dagger = \pm \gamma_i), then we see that H = \gamma_0 is a solution. Since we already know that the matrix H is unique up to scale it is the only solution unto scale.

Finally, note that the invariant hermitian form does actually determine a Hilbertspace of spinors _for_ _every_ (co-orriented) _space like_ _hyper surface_ X (not necessarily linear). This is because we now have a time like normal vector n and the hermitian form

(\phi, \psi) = h( phi, n\cdot \psi)

is postive (or negative) definite. In fact if you take the the hypersurface orthogonal to e_0 ( i.e. spanned by the spacelike basis vectors) and we have chosen our favorite matrix representation with the \gamma(e_i) = \gamma_i (skew) Hermitan matrices it is just the "standard" inner product. The functions X \to \Delta (or sections in a spinor bundle) with the inner product

= \int_X (\phi(x), \psi(x) ) \mu_{g|X}

gives you a countably infinite dimensional Hilbertspace to do quantum mechanics with.

[1] the isomorphism is only unique up scale however! The inability to choose one fixed isomorphisms is giving real problems if you want to have spinors in every point on a space time manifold and ultimately gives rise to spin and spin^c structures.

non-unitary choice = possible, but for that the scalar product needs be gauged for each system, and the non-sense with gamma^0 has to go