Problem Related to Photons with Mass

[diffidus]

Homework Statement

I am trying to understand example 13.6 given in the book 'Quantum Field Theory for the Gifted Amateur' by Lancaster and Blundell. The problem is: Consider a massive photon in the rest frame so that P^μ=([m,0,0,0). If we boost the particle in the z direction to P^μ=([E,0,0,p) can we calculate the product of the polarisation vectors:

Relevant Equations

I have included the relevant equations below. I tried multiplying the matrices together but could not get the correct answer. Is there anybody out there that can help?

Before boost we have

Then using the Lorentz boost:

I want to calculate:

I tried multiplying the matrices together but I never get the stated answer which should be:

 

[Hill]

I get the stated answer except the minus signs. Just using the result of the previous example:

 

[diffidus]

I get the stated answer except the minus signs. Just using the result of the previous example:
View attachment 338712

Hill

So I think you got your result by using the previous example values for ε as you have said. Then, I assume you carried out the product sum with with λ =1,3. Thanks for this answer, it gets me closer but then there is still the annoying minus signs, which makes me think there must be another subtlety and that worries me.

 

[Hill]

the annoying minus signs

I think they are a typo.

 

[diffidus]

I think they are a typo.

That would be great, but I am not certain since in the book that I refer to, it also gives the product involving εε in terms of a projection operator:

and that

When I use this method it gives the matrix including the minus signs. So I am still not confident. Anyhow, thanks for your effort, it is really appreciated.

 

[Hill]

When I use this method it gives the matrix including the minus signs.

Could you show how it gives the minus signs? I don't see it.

 

[diffidus]

Could you show how it gives the minus signs? I don't see it.

Apologies - I looked back at my calculation and it appears that I managed to get a minus sign in there. I have redone the calculation and I now get positive values and so I am now completely on the same page as you. In the matrix quoted in the book - it must be a typo. Thanks very much for your help in solving a frustrating problem.

 

[diffidus]

Could you show how it gives the minus signs? I don't see it.

I feel a bit like a bad penny now but I have looked over my calculation and I can now see where the minus signs come from.
The problem is given in terms of a boost in the z-direction:

From this we get:

Which can be calculated independently using:

Using this second method the minus signs in the A matrix arise since in the projection formula the momentum vectors are covariant and so should be:

So for example:

Similarly for A30.
The bad news is this still leaves me with the original problem! The good news, however, is that I think I have spotted the issue.

In the original problem equation 13.29 gives the boost matrix as (multiplied by 1/m):

However, I think there is a typo here as it should be:

When this boost matrix is used and followed through the rest of the derivation everything is resolved - phew!

Once again thanks for your help with this it has helped me a lot as it got me to think about my calculations and, naively, I never considered typos.

 

[Hill]

I feel a bit like a bad penny now but I have looked over my calculation and I can now see where the minus signs come from.
The problem is given in terms of a boost in the z-direction:
View attachment 338787
From this we get:

View attachment 338788
Which can be calculated independently using:

View attachment 338789
Using this second method the minus signs in the A matrix arise since in the projection formula the momentum vectors are covariant and so should be:
View attachment 338790
So for example:

View attachment 338791
Similarly for A30.
The bad news is this still leaves me with the original problem! The good news, however, is that I think I have spotted the issue.

In the original problem equation 13.29 gives the boost matrix as (multiplied by 1/m):

View attachment 338794
However, I think there is a typo here as it should be:

View attachment 338795

When this boost matrix is used and followed through the rest of the derivation everything is resolved - phew!

Once again thanks for your help with this it has helped me a lot as it got me to think about my calculations and, naively, I never considered typos.

Thank you for clearing this point. I think now that there is no typo at all, and the resolution for the minus sign in the original problem is the same: the polarization vectors in 13.30 are contravariant, but they are covariant in 13.40. This gives us the minus signs in the answer, doesn't it?

 

[billtodd]

A photon doesnt have a mass. it's massless.

 

[Hill]

A photon doesnt have a mass. it's massless.

Yes. However,

(Lancaster, Blundell, Quantum Field Theory for the Gifted Amateur)

 

[billtodd]

I remember this Lagrangian called proca lagrangian density. otherwise I cannot recall.