Re: Question on road to reality

Benjamin wrote:

Hello group, at page 629 of Penroses Book road to reality, Penrose
claims that one can break down the dirac equation to two "ingredients"
alpha_a and alpha_b : In the references Penrose says that he is not
gone to the detail here ,how the decomposition of the dirac equation
that he presents was calculated exacly, but he refers to some papers
in which I also haven't found this decomposition.

Does one know what he did at page 629.

The sad thing with this book is that it is a firework of nice ideas, but
it's neither a popular physics book nor a good text book. I have no
idea to which kind of reader the author addressed his writing. Anyway,
I like to read in it. If one knows about the stuff Penrose is
explaining one can get nice unconventionel insight into the subject.

Penrose's attempt to explain the Dirac equation on page 628 ff is
anything but new. It's the good old Weyl-spinor stuff. Usually one
writes small greek letters as indices and the two sorts of spinors are
distinguished by putting dots on the indices of one type of the Weyl
spinors.

The mathematics is the following: One looks for finite-dimensional
linear representations of the Lorentz group or rather its Lie algebra.

The Lorentz group contains the rotation group as a sub group. Then we
know from the quantum theory of angular momentum, which is nothing than
representation theory of the rotation group SO(3) that there exists a
covering group SU(2) of the rotation group, and this group is what one
has to use to realise representations of rotations and angular momentum
(including the spin of particles), because one rather looks for ray
representations than for linear representations of the symmetry groups
in quantum theory.

The most simple representation of SU(2) is its
two-dimensional "fundamental representation", and the objects
transformed are the Pauli spinors of spin-1/2-particles in
non-relativistic quantum mechanics.

This representation can be extended to a representation of the covering
group of the Lorentz group. To be more precise it's the compound of the
Lorentz group which is connected continuously with the identy (the
neutral element of the group), which is the special orthochronous
Lorentz group consisting of all Lorentz matrices with determinant +1
which does not change the direction of time. The covering group turns
out to be SL(2,C), the special linear group of the two-dimensional
complex vector space, i.e., all comples 2 x 2 matrices with determinant
1.

The SL(2,C) has the property to leave the skew-symmetric product

(chi,psi)=eps_{ab} \chi^a \psi^b

invariant. Here \chi and psi are two elements of C^2 and

eps_{12}=-eps{21}=1, eps_{11}=eps_{22}=0.

These spinors build obviously a representation of SL(2,C), namely its
fundamental representation. Let {T^a}_b \in SL(2,C). Then the
transformation is

chi'^a={T^a}_b \chi^b (and the analogue for psi).

Now there is another two-dimensional representation of SL(2,C), namely
that of the complex conjugated spinors. These are the ones you give
indices with a dot, or as Penrose does with a prime. They transform by
the rule

chi'^{a'}=({T^a'}_{b'})^* chi^{b'}

where the star indicates complex conjugation.

For the rotation group (or more precisely its covering group SU(2)) the
complex conjugate transformations build the same representation as the
fundamental one (up to isomorphy). Thus in the theory of angular
momentum you have only one representation in two dimensions.

In the context of the SL(2) the conjugate complex representation is not
isomorphic to the fundamental representation, and thus you have 2 sorts
of spinors which usually are called Weyl spinors to distinguish them
from the Pauli spinors.

As I said this point of view takes into account only the proper
orthochronous Lorentz group. Now, we like also to describe the discrete
symmetries of space time, i.e., time reversal and spatial reflections.
E.g., the electromagnetic interaction preserves parity, i.e., it is
invariant under spatial reflections, and to describe these, as it turns
out one needs both types of Weyl spinors in the theory, because a
spatial reflection maps a spinor with undotted indices in one with
dotted indices and vice versa.

Thus, to get a proper representation of the full orthochronous Lorentz
group (i.e., the Lorentz group with matrices which do not change the
direction of time but may have determinant -1, as the space reflection)
the two Weyl spinors are combined into a four-component object which is
called Dirac spinor, and the corresponding fields describe Dirac
particles like electrons (in the standard model the massive leptons and
the quarks are for sure Dirac particles, while this is not so clear for
the neutrinos, which are usually treated as Dirac particles, but they
may be identical with their anti particles, and then you get socalled
Majorana spinors, but that's another story).

In connection with space reflections it turns out that the Weyl spinors
are the parts of the Dirac spinor with a certain chirality
("handedness"). The one type describes "left handed", the other
type "right handed" Dirac spinors, and a space reflection interchanges
these two chiralities as it should be.

For more details, see my quantum field theory script on my home page p
270 ff:

http://theory.gsi.de/~vanhees/publ/lect.pdf

--
Hendrik van Hees Texas A&M University
Phone: +1 979/845-1411 Cyclotron Institute, MS-3366
Fax: +1 979/845-1899 College Station, TX 77843-3366
http://theory.gsi.de/~vanhees/faq mailto:hees@xxxxxxxxxxxxx

.

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