Exploring the Dirac Fields in Srednicki (Ch88)

In summary: This is because \bar{N_R} = (0, \nu) and \gamma^{\mu}D_{\mu}N_R = \gamma^{\mu}(\partial_{\mu}\nu + ig_1 B_{\mu}\nu) = \gamma^{\mu}D_{\mu}\nu = D_{\mu}\varepsilon. Therefore, the full kinematic Lagrangian can be written as:L_{\text{kin}} = i\bar{N_L}\gamma^{\mu}D_{\mu}N_L + i\bar{\varepsilon_L}\gamma^{\mu}D_{\mu}\varepsilon_L + i\bar{\vare
  • #1
LAHLH
409
1
Hi,

In srednicki (ch88) he starts off considering the electron and associated neutrino, by introducing the left handed Weyl fields [itex]l, \bar{e} [/itex] in the representations (2,-1/2), (1,+1) of SU(2)XU(1).

The covariant derivaties are thus

[itex] (D_{\mu}l)_i=\partial_{\mu}l_i-ig_2A_{\mu}^a(T^a)_i^jl_j-ig_1(-1/2)B_{\mu}l_i[/itex] and [itex]D_{\mu}\bar{e}=\partial_{\mu}\bar{e}-ig_1(+1)B_{\mu}\bar{e} [/itex] where the T's are SU(2) gens and Y=-1/2I for l, and Y=+1 for the [itex]\bar{e} [/itex]. He then relabels the SU(2) components of [itex]l=\left( \begin{array}{c} \nu\\ e\end{array} \right) [/itex] and defining the Dirac field [itex]\varepsilon=\left( \begin{array}{c} e\\ \bar{e}^{\dagger}\end{array} \right) [/itex] and finally the Majorana feld for the neutrino:[itex]N=\left( \begin{array}{c}
\nu\\ \nu^{\dagger}\end{array} \right) [/itex].

Now the kinematic term in the Lagrangian starts life in terms of the Weyl fields looking like:
[itex] L_{\text{kin}}=i\nu^{\dagger}\bar{\sigma}^{\mu}(D_{\mu}\nu)+ie^{\dagger}\bar{\sigma}^{\mu}(D_{\mu}e)+i\bar{e}^{\dagger}\bar{\sigma}^{\mu}(D_{\mu}\bar{e}) [/itex]

My question is how exactly do I rewrite this in terms of the Dirac fields?

I believe if you define [itex] N_L:=P_L N=\left( \begin{array}{c} \nu\\ 0\end{array} \right) [/itex] then the first term can be expressed as [itex]i\bar{N_L}\gamma^{\mu}D_{\mu}N_L [/itex]. This is the case because [itex] \bar{N_L}=(0, \nu^{\dagger}) [/itex] so we get:

[tex] i\bar{N_L}\gamma^{\mu}D_{\mu}N_L =i(0, \nu^{\dagger})\left( \begin{array}{cc} 0 & \sigma^{\mu}\\ \bar{\sigma}^{\mu}&0\end{array} \right)\left( \begin{array}{c} D_{\mu}\nu\\ 0\end{array} \right) =i\nu^{\dagger}\bar{\sigma}^{\mu}(D_{\mu}\nu)[/tex]

The second term of the kinematic Lagrangian is similarly [itex] i\bar{\varepsilon_L}\gamma^{\mu}D_{\mu}\varepsilon_L [/itex] as far as I can tell, but then the third term does not seem to fall into such an expression?
 
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  • #2


Hi,

You are correct in your interpretation of the first two terms in the kinematic Lagrangian. The third term, however, can also be written in terms of the Dirac fields. This is because the Majorana field N can be expressed as a linear combination of the Dirac fields \varepsilon and \bar{\varepsilon}. In particular, we have:

N = N_L + N_R = \left(\begin{array}{c} \nu \\ \nu^{\dagger} \end{array}\right) + \left(\begin{array}{c} 0 \\ \nu \end{array}\right) = \left(\begin{array}{c} \nu \\ \nu \end{array}\right) = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)\left(\begin{array}{c} \nu \\ \nu \end{array}\right) = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)\varepsilon

Therefore, the third term in the kinematic Lagrangian can be written as i\bar{N_R}\gamma^{\mu}D_{\mu}N_R = i\bar{\varepsilon}\gamma^{\mu}D_{\mu}\varepsilon.
 

FAQ: Exploring the Dirac Fields in Srednicki (Ch88)

1. What is the Dirac equation and how does it relate to the Dirac fields in Srednicki (Ch88)?

The Dirac equation is a relativistic wave equation that describes the behavior of fermions, such as electrons, in quantum mechanics. The Dirac fields in Srednicki (Ch88) refer to the mathematical representation of these fermions in a quantum field theory framework.

2. Why are the Dirac fields important in particle physics?

The Dirac fields are important in particle physics because they provide a unified description of fermions and their interactions with other particles, such as bosons. They also allow for the prediction and observation of new particles, such as the Higgs boson.

3. How do the Dirac fields differ from other quantum fields?

The Dirac fields differ from other quantum fields in that they describe fermions, which have half-integer spin, while other fields, such as scalar fields, describe bosons with integer spin. They also have a unique mathematical structure that incorporates both negative and positive energy solutions.

4. What is the role of the Dirac spinor in the Dirac fields?

The Dirac spinor is a mathematical object that describes the spin and momentum of a fermion in the Dirac fields. It is a four-component complex vector that contains information about the particle's spin states, allowing for the prediction of its behavior in different spin configurations.

5. How does Srednicki (Ch88) expand upon the original Dirac equation?

Srednicki (Ch88) expands upon the original Dirac equation by incorporating the concept of quantum field theory and its mathematical framework. This allows for a more comprehensive understanding of the behavior of fermions and their interactions with other particles, as well as the prediction of new phenomena in particle physics.

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