I can't figure out if these are matrices or numbers

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Discussion Overview

The discussion revolves around the interpretation of various quantities in the context of quantum field theory, specifically regarding whether certain expressions involving Dirac spinors represent matrices or numbers. The scope includes conceptual clarification and technical explanations related to particle physics and the Yukawa interaction.

Discussion Character

  • Conceptual clarification, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether the fermion propagator in momentum space is a matrix while the boson propagator is a number, expressing confusion over the notation used in Peskin and Schroeder.
  • Another participant suggests that the interpretation depends on the type of particles being described, noting that Dirac spinors require the SU(2) representation and that the sum over u\bar{u} results in a 4x4 matrix.
  • A later reply indicates that \bar{u} u is likely a number, depending on the representation used, and discusses the implications of row and column vector multiplications.
  • One participant mentions the potential confusion arising from the use of the Kronecker delta, which is typically the identity matrix but can sometimes appear as a number in different contexts.
  • Another participant clarifies that \bar{u} involves the conjugate of U and relates to the gamma 0 matrix, reinforcing the idea that it should yield a number.

Areas of Agreement / Disagreement

Participants express varying interpretations of the quantities involved, with some agreeing that \bar{u} u is a number while others highlight the context-dependent nature of these representations. The discussion remains unresolved regarding the definitive classification of certain expressions.

Contextual Notes

Participants note the dependence on specific representations and the potential for confusion due to notation differences. The discussion does not resolve the mathematical steps or assumptions underlying the interpretations.

Matterwave
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Hi, since Peskin and Schroeder pretty much suppresses the indices in every equation, I am now unable to tell if a lot of these quantities are matrices or numbers. I try to look back, but I still can't seem to figure all of these out. In the Yukawa interaction, for example, the fermion propagator (in momentum space) is a matrix right? While the boson propagator is simply a number?

Also, I see a lot of expressions like [itex]\overline{u}u[/itex]. Are these numbers or outer product matrices? I am really confused on these. If that's a number, then would [itex]u\overline{u}[/itex] be a matrix? u is the dirac spinor.

Is the big M a matrix or number? (In spinor space)
 
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Just a quick answer;

This depends what kind of particles you are trying to describe. For the dirac spinor indeed you are right. Fermions ( spin half guys ) need the SU(2) representation ie the pauli matrices.

The sum over u\bar{u} will give you a 4x4 matrix. Such as \slash p + m with some normalization.

What you should do is figure out what u, v are from looking at the dirac equation. Then you can see their Tensor ranks and figure it out from there.

Hope this helps a bit.
 
So it seems then that \bar{u} u is a number right? It depends on how the book wants to represent these right. If ubar is a row vector and u is a column vector, then row*column is a number whereas column*row is a matrix right? I do know what a Dirac spinor is, but sometimes I have trouble with the notation. For example, the kronecker delta is usually the identity matrix, but I see sometimes that like ostensibly what I would think would be a number is written as a kronecker delta...so I get confused...
 
Yeah.

To convert \bar{u} = conjugate of U ( ie from column to row or whatever ) times the gamma 0 matrix. So one is 'covariant' and one is 'contravariant'.

It it should be a number like you say. Yeah I guess it can be a bit confusing.
 

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