Dirac Gamma Matrices: Is Invariance Under Lorentz Transformation?

In summary, Dirac Gamma matrices are a set of four 4x4 matrices used in quantum field theory to describe the behavior of particles with spin 1/2. They are important because they are invariant under Lorentz transformation and are used to describe the spin of particles in different reference frames. They have a wide range of applications in theoretical and experimental physics, including the Standard Model of particle physics, condensed matter physics, and cosmology. They are not applicable for particles with spin other than 1/2, as different mathematical techniques and matrices are used for those cases.
  • #1
eoghan
207
7
Hi!
I can define
[itex]\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3[/itex]
I know that the four gamma matrices [itex]\gamma^i\:\:,\;i=0...3[/itex] are invariant under a Lorentz transformation. So I can say that also [itex]\gamma ^5[/itex] is invariant, because it is a product of invariant matrices.
But this equality holds:
[tex]\gamma ^5=\frac{i}{4!}\epsilon_{\mu\nu\rho\sigma}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}[/tex]
and this expression is not invariant!
So, is [itex]\gamma^5[/itex] invariant or isn't it?
 
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  • #2
[itex]\gamma^\mu[/itex] transforms like a 4-vector.
 
  • #3
But [itex]\gamma^5[/itex] transforms like a pseudo-scalar because of [itex]\epsilon_{\mu\nu\rho\sigma}[/itex]
 

1. What are Dirac Gamma matrices?

Dirac Gamma matrices, also known as Dirac matrices, are a set of four 4x4 matrices that were introduced by physicist Paul Dirac in his work on the relativistic wave equation for particles with spin. They play a crucial role in the mathematical formulation of quantum field theory and are used to describe the behavior of particles with spin 1/2, such as electrons.

2. What is the significance of invariance under Lorentz transformation in relation to Dirac Gamma matrices?

Invariance under Lorentz transformation means that a physical system remains unchanged when observed from different reference frames that are moving at a constant velocity relative to each other. The Dirac Gamma matrices are important because they are invariant under Lorentz transformation, meaning that they can be used to describe the behavior of particles in different reference frames and maintain the same physical properties.

3. How do Dirac Gamma matrices relate to the concept of spin in quantum mechanics?

Spin is a quantum mechanical property that describes the angular momentum of a particle. The Dirac Gamma matrices are used to describe the spin of particles with spin 1/2, such as electrons. They are also used in the Dirac equation, which describes the behavior of spin 1/2 particles in relativistic quantum mechanics.

4. Can Dirac Gamma matrices be used to describe particles with spin other than 1/2?

No, Dirac Gamma matrices are specifically designed to describe particles with spin 1/2. For particles with different spins, different mathematical techniques and matrices are used. For example, particles with spin 1 are described using the Proca equation, while particles with spin 0 are described using the Klein-Gordon equation.

5. What are some practical applications of Dirac Gamma matrices in physics?

Dirac Gamma matrices have a wide range of applications in theoretical and experimental physics. They are used in quantum field theory to describe the behavior of particles with spin 1/2, such as in the Standard Model of particle physics. They are also used in condensed matter physics to describe the behavior of electrons in materials. In addition, they have applications in cosmology, astrophysics, and other areas of physics where the effects of special relativity are important.

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