Make the Dirac Equation Consistent with Relativity

In summary, the free Dirac equation is given by ##(i\gamma ^\mu \partial _\mu -m)\psi = 0## and in order for it to be consistent with Relativity, the gamma matrices must satisfy ##[\gamma ^\mu ,\gamma ^\nu]=2g^{\mu \nu}##. To show this, one can equate the Dirac Hamiltonian ##\vec {\alpha} \cdot \vec p +\beta m## to the energy-momentum relation ##E^2=|\vec p|^2 + m^2## in natural units with the ##+---## metric. This leads to the conclusion that ##\beta^2=1## and ##
  • #1
sk1105
88
12

Homework Statement


The free Dirac equation is given by ##(i\gamma ^\mu \partial _\mu -m)\psi = 0## where ##m## is the particle's mass and ##\gamma ^\mu## are the Dirac gamma matrices. Show that for the equation to be consistent with Relativity, the gamma matrices must satisfy ##[\gamma ^\mu ,\gamma ^\nu]=2g^{\mu \nu}##.

Homework Equations


Dirac equation
Gamma matrices
##E^2=|\vec p|^2 + m^2## in natural units
We use the ##+---## metric.

The Attempt at a Solution


I know that the Dirac Hamiltonian is ##\vec {\alpha} \cdot \vec p +\beta m##, so I have equated it to the energy-momentum relation as follows:

##E^2 = (\vec {\alpha} \cdot \vec p + \beta m)^2 = (\vec {\alpha} \cdot \vec p)^2 + 2\beta m\vec {\alpha} \cdot \vec p + \beta^2m^2 = |\vec p|^2 + m^2##

It's clear that this leads to ##\beta^2=1##. I also know that we get ##\alpha^i\beta + \beta\alpha^i = 0##, although I'm less sure why. The bit that really puzzles me though is that we're supposed to get ##\alpha^i\alpha^j + \alpha^j\alpha^i = 2\delta^{ij}##, and I can't see how it follows from the equation above.

Then even with those three pieces I'm not sure how to arrive at the required commutator. Any help is much appreciated.
 
Physics news on Phys.org
  • #2
sk1105 said:
##[\gamma ^\mu ,\gamma ^\nu]=2g^{\mu \nu}##.
That should be an anticommutator.
sk1105 said:
The bit that really puzzles me though is that we're supposed to get αiαj+αjαi=2δij\alpha^i\alpha^j + \alpha^j\alpha^i = 2\delta^{ij}, and I can't see how it follows from the equation above.
Expand ##(\alpha \cdot \mathbf{p})^2## and make it equal to ##|\mathbf{p}|^2##

sk1105 said:
##2\beta m\vec {\alpha} \cdot \vec p##
By the way you shouldn't write it like that since you know that ##\beta## does not commute with ##\alpha_i##.
 
  • #3
blue_leaf77 said:
That should be an antimcommutator.

Yes you're right, my bad.

blue_leaf77 said:
Expand ##(α⋅p)2(\alpha \cdot \mathbf{p})^2## and make it equal to |p|2

That makes sense, I was just getting muddled with all the vector components. Thanks for your help.
 

Related to Make the Dirac Equation Consistent with Relativity

1. What is the Dirac equation and why is it important?

The Dirac equation is a relativistic wave equation that describes the behavior of spin-1/2 particles, such as electrons, in quantum mechanics. It is important because it successfully combines quantum mechanics and special relativity, providing a consistent description of these particles at high speeds.

2. What is the inconsistency between the Dirac equation and relativity?

The original Dirac equation did not take into account the effects of general relativity, such as the curvature of spacetime. This means that it is not fully consistent with the principles of relativity, which are essential for our understanding of the universe.

3. How can the Dirac equation be made consistent with relativity?

One way to make the Dirac equation consistent with relativity is to introduce a term known as the "spin connection" that takes into account the effects of general relativity. This term is added to the original Dirac equation, resulting in a modified version known as the "Dirac-Coulomb equation".

4. What are the implications of making the Dirac equation consistent with relativity?

Making the Dirac equation consistent with relativity has important implications for our understanding of fundamental particles and their interactions. It allows us to accurately describe the behavior of spin-1/2 particles in a relativistic framework, which is necessary for many applications in physics, such as particle accelerators and quantum field theory.

5. Has the Dirac equation been successfully made consistent with relativity?

Yes, the Dirac equation has been successfully modified to be consistent with relativity. The Dirac-Coulomb equation, which includes the spin connection term, has been extensively tested and is in agreement with experimental results. However, there are ongoing efforts to further refine and improve upon this equation to better understand the fundamental nature of particles and the universe.

Similar threads

  • Advanced Physics Homework Help
Replies
3
Views
962
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
708
  • Advanced Physics Homework Help
Replies
0
Views
594
  • Advanced Physics Homework Help
Replies
4
Views
911
  • Advanced Physics Homework Help
Replies
4
Views
810
  • Advanced Physics Homework Help
Replies
6
Views
368
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
8
Views
1K
  • Advanced Physics Homework Help
Replies
4
Views
861
Back
Top