Derivation in Srednicki's which got me puzzled.

  • Thread starter MathematicalPhysicist
  • Start date
  • Tags
    Derivation
In summary, the conversation discusses the derivation of a term in equation 13.16, which involves the use of equation 13.12 and 13.15. The conversation also mentions the use of Mandelstam parameters and the evaluation of an integral using residues. It is noted that the two "propagators" appearing in the equation will be different due to the different "mass" values of the variables involved.
  • #1
MathematicalPhysicist
Gold Member
4,699
371
Hi. on page 95 , I am not sure how did he derive the second term on the RHS of equation (13.16).

http://books.google.co.il/books?id=...V4QSnhYFA&ved=0CBsQ6AEwAA#v=onepage&q&f=false

I mean if I plug back I should get:

[tex] i\theta(x^0-y^0)\int_{4m^2}^\infty \rho(s) \int \tilde{dk} e^{ik(x-y)} + i\theta(y^0-x^0)\int_{4m^2}^\infty \int \tilde{dk} e^{-ik(y-x)} [/tex]

And since [tex] \int \tilde{dk} e^{\pm ik(x-y)} = \int \frac{d^3 k}{(2\pi)^3 2k^0} e^{\pm ik(x-y)} = \frac{\delta(x-y)}{2k^0}[/tex]

But I don't see how do we arrive from all of the above to the epression: [tex] \frac{1}{k^2+s-i\epsilon}[/tex] in the integral, anyone cares to elaborate?

Thanks.
P.S
I forgot to mention that: [tex] k^0 =\sqrt{\vec{k}^2 +s} [/tex]
where s is one of Mandelstam parameters, I believe it's standard in the literature.
 
Last edited:
Physics news on Phys.org
  • #2
Wait a minute I have a mistake.

It should be: [tex] \int \frac{d^3 k }{(2\pi)^3 2k^0} e^{\pm ik(x-y)} = \frac{\delta^3(\vec{x}-\vec{y})}{2k^0} e^{\pm (-ik^0)(x^0-y^0)}[/tex]

I still don't see how did he get to this term in equation 13.16. :-(
 
  • #3
Let's have a look at the first term of [itex]<0|T \phi(x) \phi(y)|0>[/itex]

It is:

[itex] \theta(x^0 - y^0) <0|\phi(x) \phi(y)|0>[/itex]

Making use of 13.12

[itex] \theta(x^0 - y^0) (\int \bar{dk} e^{ik(x-y)} + \int_{4m^{2}}^{∞} ds \rho(s) \int \bar{dk} e^{ik(x-y)}) [/itex]

Similiarly for the 2nd term:

[itex] \theta(y^0 - x^0) (\int \bar{dk} e^{-ik(x-y)} + \int_{4m^{2}}^{∞} ds \rho(s) \int \bar{dk} e^{-ik(x-y)}) [/itex]

Now add them... the 1st terms will give you what you have from equation 13.15
Also you can put the thetas in the integrals and use 13.15 also for the second terms, getting the integral of rho(s) out...

So you will have something like (it's a sketching not the result):
[itex] \int F(k^{-2}) + \int F(k'^{-2}) \int \rho [/itex]
from where you can take out a common factor... of course the 2 k's you have, also as the author points out, are not the same- the one has "mass" m, while the other has "mass" s... So the two "propagators" appearing will be different.
 
Last edited:
  • #4
Also about the integral you ask...
It's easier to try to evaluate:
[itex] \int \frac{d^{d}k}{(2 \pi)^{d}} \frac{1}{k^{2}+m^{2}-i \epsilon} e^{ik(x-y)} [/itex]

using residues to end up in the RHS of 13.15...
In fact the residues are going to give you some positive energies moving forward and some negative energies moving backward.
 
  • #5
Ok, I think this expression is derived in the previous sections, I just didn't notice that the second term was included in the square brackets. My bad, sorry for that.

Thanks.
 

What is derivation in Srednicki's?

Derivation in Srednicki's refers to the process of mathematically deriving equations and formulas in the field of quantum field theory. It involves using principles from classical mechanics and quantum mechanics to derive equations that describe the behavior of subatomic particles and their interactions.

Why is derivation important in Srednicki's?

Derivation is important in Srednicki's because it allows scientists to understand and predict the behavior of subatomic particles and their interactions. This is essential for advancements in quantum field theory and for applications in fields such as particle physics and cosmology.

What are some common challenges in understanding derivation in Srednicki's?

Some common challenges in understanding derivation in Srednicki's include the use of complex mathematical equations, the need for a strong understanding of classical and quantum mechanics, and the abstract nature of quantum field theory itself. It may also require familiarity with advanced mathematical concepts such as group theory and differential equations.

How does derivation in Srednicki's differ from other fields of physics?

Derivation in Srednicki's differs from other fields of physics in that it specifically focuses on the behavior of subatomic particles and their interactions. This requires the use of principles from both classical and quantum mechanics, as well as advanced mathematical techniques. It is also a highly theoretical field, with many concepts and equations being difficult to directly test or observe in experiments.

What are some practical applications of derivation in Srednicki's?

Some practical applications of derivation in Srednicki's include advancements in particle physics and cosmology, as well as potential applications in technology such as quantum computing. It also helps scientists better understand the fundamental nature of the universe and the behavior of matter at the smallest scales.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
913
  • High Energy, Nuclear, Particle Physics
Replies
8
Views
747
  • High Energy, Nuclear, Particle Physics
Replies
5
Views
944
  • High Energy, Nuclear, Particle Physics
Replies
3
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
15
Views
2K
  • Advanced Physics Homework Help
Replies
4
Views
873
  • Advanced Physics Homework Help
Replies
4
Views
349
Replies
3
Views
561
  • Advanced Physics Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
558
Back
Top