What are the possible dimensions of representation of SL(2,C)?

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Discussion Overview

The discussion centers on the possible dimensions of representations of the SL(2,C) group, particularly in relation to their applications in physics, such as Lorentz transformations and particle representations. Participants explore the nature of these representations, their dimensionality, and their connections to other groups like SU(2).

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that there are only two inequivalent two-dimensional representations of SL(2,C), which relate to the Lorentz transformation of left and right Weyl spinors.
  • Questions arise regarding the differences between the representations (1/2,1/2) and (1/2,0)+(0,1/2), with one participant suggesting that the former describes an electron (or positron) while the latter describes a photon, highlighting the role of spin.
  • One participant notes that the representation (m,n) is (2m+1)(2n+1) dimensional, leading to a 4x4 matrix for (1/2,1/2), and speculates on its relation to traditional Lorentz transformation matrices.
  • Another participant mentions that representations of SL(2,C) are derived from representations of SU(2), suggesting that the pair (m,n) indicates a tensor product of (2m+1)-dimensional and (2n+1)-dimensional representations of SU(2).
  • References to literature, such as works by Moshe Carmeli and Willard Miller, are provided for further exploration of the topic.

Areas of Agreement / Disagreement

Participants express differing views on the nature and implications of the representations of SL(2,C), with no consensus reached on specific details or interpretations.

Contextual Notes

Some assumptions about the relationships between representations and their dimensionality remain unverified, and the discussion does not resolve the mathematical details of the representations.

paweld
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Is it true that there are only two inequivalent two-dimensional representation of
SL(2,C) group and they are responsible for Lorentz transformation of left and right
Weyl spinor.
 
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What's the difference between representation (1/2,1/2) and (1/2,0)+(0,1/2) of
SL(2,C)?
 
It's simple, one describes an electron (positron), the other a photon. It's the spin which makes a difference. Electron 1/2, photon 1.

One can also say that the photon transforms irreducibly wrt the SL(2,C) group, while the electron not.
 
Thanks. Do you happen to know how the matricies of general representation (m,n)
look like. As far I know the representation (m,n) is (2m+1)(2n+1) dimensional so
these matricies should be also (2m+1)(2n+1) dimensional. In case of (1/2,1/2)
it gives 4x4 matricies which are probably somehow related to traditional matricies of
lorentz transformation of spacetime points. But on the other hand I heard that
the representation of SL(2,C) are derived from representation of SU(2) and the pair
(m,n) says probably that this representation of SL(2,C) is a tensor product (?) of
(2m+1)-dimensional and (2n+1)-dimnsional representation of SU(2). Does anyone know
the details?
 
paweld said:
Thanks. Do you happen to know how the matricies of general representation (m,n)
look like.

I don't know, but I can point you to the vast literature on this issue. Try one of Moshe Carmeli's books on group theory and General Relativity. The introduction treats SL(2,C) extensively.

As far I know the representation (m,n) is (2m+1)(2n+1) dimensional so
these matricies should be also (2m+1)(2n+1) dimensional. In case of (1/2,1/2)
it gives 4x4 matricies which are probably somehow related to traditional matricies of
lorentz transformation of spacetime points.

Absolutely correct.

But on the other hand I heard that
the representation of SL(2,C) are derived from representation of SU(2) and the pair
(m,n) says probably that this representation of SL(2,C) is a tensor product (?) of
(2m+1)-dimensional and (2n+1)-dimnsional representation of SU(2). Does anyone know
the details?

Willard Miller's book on group theory deals with the connection between SO(3), restricted Lorentz, SU(2) and SL(2,C) and the way the finite dim. of these Lie groups are related.
 

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