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Dirac's postulations and QM

  1. Feb 15, 2005 #1
    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.
  2. jcsd
  3. Feb 15, 2005 #2


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

  4. Feb 15, 2005 #3

    Hello Dextercioby and forum,

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

    The question is this:

    In Dirac's [tex] i \frac{\partial \Psi} {\partial t} =[\alfa (p-eA) + \beta m + e \O ] \Psi [/tex] or Schrödinger's [tex] i\frac{\partial\Psi}{\partial t} = \frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi}{\partial x^{2}} + V\Psi[/tex], 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.
  5. Feb 15, 2005 #4


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

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

    3.Dirac's field
    [tex] \Psi^{\alpha} (x^{\mu}) [/tex]
    is essentially a 4-spinor (Dirac spinor,if u prefer) and is an element of the vector space of the representation [itex] (\frac{1}{2},0) \oplus (0,\frac{1}{2}) [/itex] 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...

    Last edited: Feb 16, 2005
  6. Feb 16, 2005 #5

    Thanks you another time!!! :wink:
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