Solve for Eigenvalues in QFT using Matrix Algebra | Ryder's QFT p.44

In summary, Homework Equations: Equation (2.94) is(\gamma^{\mu}p_{\mu} - m)\psi(p) = 0The Attempt at a Solution: Writing out all four components, and then taking the determinant and setting to zero, I get:m^4 - (E^2 - p^2)^2 = 0 or m^4 = (E^2 - p^2)^2Taking the square root once:\pm m^2 = E^2 - p^2 or E^2 = p^2 \pm m^2.And taking the square root again:E = \pm(p
  • #1
Jimmy Snyder
1,127
20

Homework Statement


On page 44 of Ryder's QFT, near the bottom of the page, it says:
Ryder said:
it is straightforward to show, by writing out all four components of (2.94), that the eigenvalues of E are:
[tex]E = +(m^2 + p^2)^{1/2}[/tex] twice,
[tex]E = -(m^2 + p^2)^{1/2}[/tex] twice,

Homework Equations


Equation (2.94) is
[tex](\gamma^{\mu}p_{\mu} - m)\psi(p) = 0[/tex]

The Attempt at a Solution


Writing out all four components, and then taking the determinant and setting to zero, I get:
[tex]m^4 - (E^2 - p^2)^2 = 0[/tex] or [tex]m^4 = (E^2 - p^2)^2[/tex]
Taking the square root once:
[tex]\pm m^2 = E^2 - p^2[/tex] or [tex]E^2 = p^2 \pm m^2[/tex].
And taking the square root again:
[tex]E = \pm(p^2 \pm m^2)^{1/2}[/tex]
and I end up with different eigenvalues than I am supposed to.
 
Last edited:
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  • #2
How did you calculate the determinant? I calculated it with some formulae, and I got only two eigenvalues. Maybe you should try using a software like mathematica to calculate it by brute force.

Let just hope that your Dirac matrices are same as mine. I used

http://www.stochasticsoccer.com/Clipboard01.jpg

then I used
[tex](\sigma_\mu p^\mu)(\sigma'_\mu p^\mu) = p_\mu p^\mu[/tex]
where [tex]\sigma^\mu = (1, \sigma^i)[/tex]
where [tex]\sigma'^\mu = (1, -\sigma^i)[/tex]
[tex]\sigma^i[/tex] are Pauli matrices.

I got [tex]m^2 = E^2 - p^2[/tex].
 
  • #3
kakarukeys said:
I got [tex]m^2 = E^2 - p^2[/tex].
Thanks for taking a look at this kakarukeys. I don't think your equation could be the determinant since there are supposed to be 4 eigenvalues, and your equation is only quadratic in E.
 
  • #4
At one stage of my calculations, I had ()^2 = 0 implies () = 0
so a 4th order eq became quadratic eq
 
  • #5
kakarukeys said:
At one stage of my calculations, I had ()^2 = 0 implies () = 0
so a 4th order eq became quadratic eq

So, each of your distinct eigenvalues is repeated.
 
  • #6
Thanks kakarukeys. I was making two errors. First of all, I had the wrong matrix for [tex]\gamma^0[/tex], and second of all, I was calculating the determinant incorrectly. With your help, I now get the following determinant:
[tex](E^2 - m^2 - p^2)^2[/tex] and setting this to zero gives the correct eigenvalues. Thanks George to you as well. Actually, I gathered the same meaning from message #4 as you did, but it's good to know that you have my back.
 

What is an eigenvalue in quantum field theory (QFT)?

In QFT, an eigenvalue is a scalar value that represents the possible energy states of a quantum system. It is obtained by solving the eigenvalue equation, which involves the system's Hamiltonian operator and the corresponding eigenvector.

How is matrix algebra used to solve for eigenvalues in QFT?

In QFT, the Hamiltonian operator is represented as a matrix and the eigenvalue equation is solved using matrix algebra techniques such as diagonalization or eigenvalue decomposition. This allows for the calculation of the system's energy states and corresponding eigenvalues.

What is the importance of eigenvalues in QFT?

Eigenvalues in QFT play a crucial role in understanding the energy states and dynamics of a quantum system. They provide information on the possible values of energy that the system can have and the corresponding probabilities of transitioning between these states.

Are there any limitations to using matrix algebra to solve for eigenvalues in QFT?

While matrix algebra is a powerful tool for solving for eigenvalues in QFT, it can become computationally challenging for large and complex systems. In such cases, other numerical methods such as perturbation theory or variational methods may be used.

How does the concept of eigenvalues relate to quantum entanglement in QFT?

In QFT, eigenvalues are used to describe the entangled states of a quantum system. Entanglement occurs when two or more particles are connected in a way that their properties are dependent on each other, and the eigenvalues of the system can reveal information about this entanglement.

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