Dirac's Postulations: A Scalar Function or Four-Dimensional Vector?

[Raparicio]

Hello Friends,

About Dirac's postulation about non conmutative operators and the scalar function with 4 elements, some questions:

why this function is a scalar function? Couldn't it be a four-dimensional vector? Why?

Best Reggards.

 

[dextercioby]

Which scalar function with 4 elements are you talking about?Please make a specific reference.I have no idea what you're referring to.

Daniel.

 

[Raparicio]

HIlbert

Hello Dextercioby and forum,

How are u? I'm agreed to re-read u!

The question is this:

In Dirac's $$i \frac{\partial \Psi} {\partial t} =[\alfa (p-eA) + \beta m + e \O ] \Psi$$ or Schrödinger's $$i\frac{\partial\Psi}{\partial t} = \frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi}{\partial x^{2}} + V\Psi$$, the wave function is a Hilbert Space wave, I think. Are they vectors or scalars?
The Hilbert Space is a vectorial, or scalar space?¿?

My best reggards.

 

[dextercioby]

1.A Hilbert space is a complete VECTOR SPACE with scalar product over the field of complex numbers...

2.Schroedinger's wave-function
$$\Psi (\vec{r},t)$$
is a vector from the Hilbert space $$\mathbb{L}^{2}(\mathbb{R}^{3})\otimes \mathbb{R}$$.

3.Dirac's field
$$\Psi^{\alpha} (x^{\mu})$$
is essentially a 4-spinor (Dirac spinor,if u prefer) and is an element of the vector space of the representation $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ of the restricted Lorentz group.The algebric structure determined by these spinors is actually a Grassmann algebra with involution over the vector space mentioned earlier...

Once you quantize Dirac's field,the classical spinors become operators and that's another (quite complicated ) story...

Daniel.

 

[Raparicio]

thanks

Thanks you another time!