Representation of Lorentz group and spinors (in Peskin page 38)

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Discussion Overview

The discussion centers on the representations of the Lorentz group and spinors, particularly in the context of quantum field theory as presented in Peskin's text. Participants explore the implications of Lorentz transformations on spinors versus traditional 4-vectors, and seek resources for clearer understanding.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant expresses confusion about the treatment of Lorentz transformations and their relation to spinors, asking if these transformations involve rotations and boosts in spin space.
  • Another participant suggests that the Poincaré group, which includes the Lorentz group, is crucial for understanding relativistic quantum field theory and recommends Weinberg's work as a resource.
  • It is noted that while Lorentz transformations affect Minkowski space, they specifically involve boosting and rotating spinors rather than 4-vectors.
  • A participant seeks clarification on how Lorentz transformations affect spinors, indicating a desire for deeper understanding of the physical implications and distinctions between spinors and vectors.
  • Some participants mention various resources, including Ryder's notes and Lambert's treatment on SUSY, as helpful for understanding the topic.
  • A mathematical expression for the Lorentz transformation of Dirac spinors is provided, highlighting the construction of finite transformations from infinitesimal ones.
  • There is a discussion about the need for a comprehensive understanding of Lorentz transformations and their representation in spinor space, with references to multiple texts being explored.

Areas of Agreement / Disagreement

Participants generally agree on the importance of understanding how Lorentz transformations apply to spinors, but there is no consensus on the best resources or the clarity of Peskin's treatment. Multiple viewpoints on the nature of spinors versus vectors are expressed, indicating ongoing exploration and debate.

Contextual Notes

Participants mention various texts and resources, indicating that there may be limitations in their understanding based on the complexity of the material and the varying levels of detail provided in different sources.

Who May Find This Useful

This discussion may be useful for students and researchers interested in quantum field theory, particularly those seeking to understand the role of spinors and Lorentz transformations in theoretical physics.

silverwhale
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I am very confused by the treatment of Peskin on representations of Lorentz group and spinors.

I am confronted with this stuff for the first time by the way.

For now I just want to start by asking: If, as usual Lorentz transformations rotate and boost frames of reference in Minkowski space, are we considering now rotations and boosts in spin space?

And does anybody know about some clear treatments on spinors and representations of the Lorentz group?

Any help would be greatly appreciated!
 
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The best treatment about the Poincare group (it's the Poincare group not only the Lorentz group which is important in relativistic quantum-field theory!) can be found in

S. Weinberg, Quantum Theory of Fields, Vol. I

You also find a treatment in my QFT manuscript:

http://fias.uni-frankfurt.de/~hees/publ/tft.pdf
 
Ryder does a good job of discussing the spinor representation of the Lorentz group if I recall correctly. To answer your other question, you're still dealing with Minkowski space, but you're boosting and rotating spinors as opposed to the usual 4-vectors.
 
If, as usual Lorentz transformations rotate and boost frames of reference in Minkowski space, are we considering now rotations and boosts in spin space?
this does not make sense.are you asking about how Lorentz transformation affects spinors.In that case,your question can be answered.
 
bapowell said:
Ryder does a good job of discussing the spinor representation of the Lorentz group if I recall correctly. To answer your other question, you're still dealing with Minkowski space, but you're boosting and rotating spinors as opposed to the usual 4-vectors.

It's a dual view actually, you could well have one field configuration viewed by different inertial observers connected thorough a Lorentz/Poincare' transformations.
 
To andrien. Yes I am asaking exactly that!
How do Lorentz transformations affect spinors?

And thanks to everybody for posting very helpful links! I'll go to the library today to get the books and I'll take a look at vanhees's notes (how did you know that I can speak German? :-D )!
 
Well I think I do understand now what is happening (very generally) from dexterciobys post. Basically we still are in M spacetime, and we have intertial oberservers connected by LT but instead of looking at 4 vectors or tensors they are looking at spinors, am I correct?
 
I liked the treatment of this topic in Lambert's notes on SUSY very much. It also paves the way to SUSY for you ;)
 
silverwhale said:
Well I think I do understand now what is happening (very generally) from dexterciobys post. Basically we still are in M spacetime, and we have intertial oberservers connected by LT but instead of looking at 4 vectors or tensors they are looking at spinors, am I correct?

Yes.
 
  • #10
A finite transformation can be constructed from infinitesimal ones in case of lorentz transformation.lorentz transformation of dirac spinor is
ψ'=e(-iωμvSμv)ψ,it is the finite one constructed from infinitesimal ones.ωμv are antisymmetric and are 6 parameters representing boosts and rotations.S's are generators written in terms of pauli matrices.
 
  • #11
to andrien, that description I do know, it's just what Peskin writes in his book!

I am looking for a real understanding of lorentz transformations and their representation in spinor space. And especially what it physically means and what the distinction between spinors and vectors are, etc.

Ryder seems to be a very good introduction!

I'm still searching in vanhees's script for the relevant paragraphs.

at haushofer, do you mean the lecture notes by Neil Lambert?

I am still looking into Weinbergs book, I didn't go quickly through the chapter to see if I can find answers to my questions, but there is a lot of standard stuff in it which one can find in peskin and other resources too.

Well I'mm still working through the resources to resume, I'll post something when I find the good one (for me)!
 
  • #12
Yes, I ment those lecture notes :)
 
  • #14
Hello Everybody,

Sorry for replying now!

Van Hees notes and Ryder are good. I worked through Ryder and still working through Van Hees script right now.

Would you recommend going through Weinberg chapter 2.2 to 2.7? I don't have much time right now, so I thought better study it later.

Thanks for your support.
 

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