Dirac Equation and Spinor Field Transformations: Understanding the Basics

In summary, Dirac equation and friends can be used to derive useful details about a spinor field transformation.
  • #1
Marco_84
173
0
Dirac equation and friends :)

I was playing with Dirac equations and deriving some usefull details,
Note sure for a calculation, is all the math right?

Beginning:
we require for a pure Lorentz trasf that the spinor field trasform linearly as:

[tex]\psi'(x')=S(\Lambda)\psi(x)[/tex] (1)

where [tex]x'=\Lambda x[/tex] and [tex]S(\Lambda)[/tex] is a linear operator that we can write follow:

[tex]S(\Lambda)=exp(-\frac{i}{a}\sigma_{ab}\omega^{ab})[/tex] (2)

If we take the [tex]\dagger[/tex] and use Dirac gammas on (1) we obtain the transformation law for the dagger spinor:

[tex]\psi'(x')^{\dagger}=\psi(x)^{\dagger}S(\Lambda)^{\dagger}[/tex]

and using gamma zero we have:

[tex]\overline{\psi'(x')}=\overline{\psi(x)}\gamma^{0}S(\Lambda)^{\dagger}\gamma^{0}[/tex]

Now what i want to show is that:

[tex]\gamma^{0}S(\Lambda)^{\dagger}\gamma^{0}=S^{-1}(\Lambda)[/tex]

correct me in the following equalities if i make something wrong:

[tex]\gamma^{0}S(\Lambda)^{\dagger}\gamma^{0}=\gamma^{0}exp(\frac{i}{a}(\omega^{ab})^{\dagger}(\sigma_{ab})^{\dagger})\gamma^{0}=exp(\gamma^{0}\frac{i}{a}(\omega^{ab})^{\dagger}(\sigma_{ab})^{\dagger}\gamma^{0})[/tex]

Now using the properties of gamma zero on the matrices:

[tex]\gamma_{0}^2=Id[/tex]; [tex]\gamma_{0}\sigma_{ab}\gamma_{0}=(\sigma_{ab})^{\dagger}[/tex]

we get to:

[tex]\gamma^{0}exp(\frac{i}{a}(\omega^{ab})^{\dagger}(\sigma_{ab})^{\dagger})\gamma^{0}=exp(\gamma^{0}\frac{i}{a}(\omega^{ab})^{\dagger}\gamma^{0}\gamma^{0}(\sigma_{ab})^{\dagger}\gamma^{0})=exp(\frac{i}{a}\sigma_{ab}\omega^{ab})\equiv S^{-1}(\Lambda)[/tex]

The last follow from (2).

Am i correct??

thanks in advance.

marco
 
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  • #2
Sorry i wrote a instead of 4 in the S operator underneath the i...
regards
marco
 
  • #3
Marco_84 said:
Am i correct??

It looks right to me. And you can edit your posts here for 24 hours so you can fix things like the a versus 4 detail.
 
  • #4
thanx because my prof...Im sure will be very miticoulous on this "Not so important" calculus...
regards
marco
 

1. What is the Dirac equation and what does it describe?

The Dirac equation is a relativistic quantum wave equation that describes the behavior of fermions, specifically electrons, in a relativistic quantum field theory. It takes into account special relativity and spin, and is used to describe the behavior of particles moving at high speeds.

2. How was the Dirac equation developed?

The Dirac equation was developed by British physicist Paul Dirac in 1928, as a way to reconcile special relativity with quantum mechanics. He was able to combine the Schrödinger equation, which describes the behavior of non-relativistic particles, with special relativity to create this new equation.

3. What is the significance of the Dirac equation in modern physics?

The Dirac equation is a fundamental equation in modern physics, as it provides a framework for understanding the behavior of particles at high energies. It has been used to make predictions about the behavior of particles, such as the existence of antimatter, and has been crucial in the development of quantum field theory.

4. What are the "friends" of the Dirac equation?

The "friends" of the Dirac equation refer to other equations that are related to it and have similar properties. These include the Klein-Gordon equation, which describes the behavior of spinless particles, and the Proca equation, which describes the behavior of vector bosons. These equations are all part of the Dirac equation's broader family of relativistic quantum wave equations.

5. How is the Dirac equation used in practical applications?

The Dirac equation has many practical applications, particularly in the fields of particle physics and quantum mechanics. It has been used to make predictions about the behavior of particles in high-energy accelerators, and is also used in the development of technologies such as transistors and lasers. Additionally, the Dirac equation has been used in the development of quantum computing and quantum cryptography.

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