Spinnor said:
From
http://arxiv.org/pdf/0912.2560.pdf we are given,
Λ = e(iθµνMµν) , which in the 1+1D spacetime simplifies to
Λ = e(iθM)?
The same Λ then acts on both 2 dimensional vectors in 2 dimensional Minkowski spacetime, but also the space of 2 component complex spinors?
In this case is Λ the "concrete" matrix? If so what are the abstract elements in this case?
Historically, which representation came first?
Thanks for your help!
OK, this will help be a bit more concrete. The ##\Lambda## are elements of the Lorentz group, while ##M## is the generator of the Lorentz algebra. A specific choice of ##M## as a matrix (along with the vector space it acts on) is what is called a representation of the Lorentz algebra (it is also a representation of the group). The 2-dimensional representation of ##M## as a 2x2 real matrix acting on a real vector space would be referred to as the fundamental representation for this Lorentz group.
Now ##\Lambda## is how the Lorentz group acts on spacetime vectors:
$$ x^\mu \rightarrow {\Lambda^\mu}_\nu x^\nu.~~~(*)$$
The action on spinors will not in general be the same. To describe how the Lorentz group acts on other types of objects, let's be a little abstract and discuss a general case where we have a function ##\Phi_a(x^\mu)## defined on spacetime. The index ##a## labels the definite representation that the function transforms in. So if ##\Phi_a## was just a scalar-valued function (a real or complex number), then it would belong to the trivial representation. But ##\Phi_a## can also be a spinor, vector or other tensor-valued function. The Lorentz transformation rule for ##\Phi_a## when the coordinates are transformed by (*) is
$$\Phi_a(x) \rightarrow L_{ab}(\Lambda) \Phi_b (\Lambda^{-1}x).$$
Here ##L_{ab}(\Lambda)## is an appropriate matrix in the representation for ##\Phi## with parameters determined by ##\Lambda##. In the argument of the function, we have ##\Lambda^{-1}## appearing because this is an active transformation. The new value of ##\Phi## at the point ##x##, which we could call ##\Phi'(x)##, must be compared to the value of ##\Phi## at the point ##\Lambda^{-1}x## that was mapped to the point ##x##.
To complete the expression, I will note that we can reexpress
$$ \Phi_b (\Lambda^{-1}x) = e^{i\theta_{\mu\nu} J^{\mu\nu}} \Phi_b(x),$$
where ##J_{\mu\nu}## is a differential operator
$$J_{\mu\nu} = i (x_\mu \partial_\nu - x_\nu \partial_\mu).$$
You can compare these to the differential operators for angular momentum in quantum mechanics and their familiar action on the spherical harmonics. In fact our description of ##\Phi_a(x)## is completely analogous to spherical harmonics, which are functions of the points on the sphere and have indices labeling what representation of ##SO(3)## they belong to.
Now if we put this all together, we find that
$$\Phi_a(x) \rightarrow L_{ab}(\Lambda) e^{i\theta_{\mu\nu} J^{\mu\nu}} \Phi_b(x).$$
If we are considering a spinor as a type of function on spacetime, then we need to determine what ##L_{ab}## is. However, as I mentioned in a previous post, the fundamental spinor representation in 1+1 space time is actually just one real dimensional. The reason for this is analogous to the way in 3+1 a Dirac spinor (4 dimensional) can be projected onto 2 Weyl spinors (2 dimensional). The notes that you linked discuss that. In 1+1, what would have been a 2 dimensional Dirac spinor projects to a pair of real Weyl spinors (which are technically called Majorana-Weyl).
The upshot is that, for the basic spinor in 1+1, the ##L_{ab}## is just the identity. Under a LT, the only transformation on the spinor comes from the ##e^{i\theta_{\mu\nu} J^{\mu\nu}}## part having to do with the coordinates themselves.
