Summary of the key formula of QFT

[kdv]

When I learned QFT in school, I got lost among all the steps involved. There were the Feynman rules, wick's theorem, the interaction picture, the LSZ reduction formula, the S matrix, the T matrix, the transition amplitude matrix, and on and I lost track of the big picture pretty quickly. Later I did research in QFT but I never had to go back to the fundamentals and I did not have time to do so. Then I started teaching in a small college and I decided that I really wanted to get back to research but that I also wanted to understand tons of things that I had never completely assimilated. One thing that I wanted to do was to get a big picture of the computational steps of QFT. I was also bothered by the very starting point of QFT (the business of quantizing classical fields) but that's for another post.

Now I just want to present a technical overview of the pieces involved in going from a Lagrangian to observables such as decay rates and cross sections. I am not going to present the derivations, the goal is more to simply organize the big pieces, to present the big picture, which is what I was missing when I was a student.

This will be useless for someone beginning to learn QFT. It might be useful to someone who is in the process of learning the stuff and who is getting lost among all the steps like I used to be.


I don't have questions (for now) so this would probably belong more to the blogs but equations do not show up there so I decided to post here. I just hope that this could be useful to someone else some day.

I will simply present formula for the case of a scalar field and only for canonical quantization. I might dicuss the case of particles with spins and the equations in the path integral approach at som epoint if someone is interested but I am guessing very few people here will find much interest in this thread


I assume that we start with a scalar field Lagrangian which has been second quantized and we want to get from there to some observables. The actual process of quantization is nontrivial in my opinion and deserves a whole thread just to itself. But I won't discuss this here.

I have divided the process into 6 steps. With step 0 being the actual quantization which is, IMHO, the most subtle and confusing part. But after that, we start with

STEP 1: Introducing the S-matrix


We assume that at plus or minus infinite time the system is governed by the free theory so that we know the eigenstates which are free particles. We are interested in the transition from a certain set of free particles at t= minus infinity to another state of free particles at t= plus infinity. Given that, we introduce the S matrix element as

$$\begin{equation*} \addtolength{\fboxsep}{5pt} \boxed{ \begin{gathered} ~<\vec{p}_1 \ldots \vec{p_n}|S| \vec{k_1} \ldots \vec{k_m}> = ~<\vec{p}_1 \ldots \vec{p_n}|1+ i T| \vec{k_1} \ldots \vec{k_m}> \end{gathered} } \end{equation*}$$
and we will drop the ``1" part of the S matrix so we really only care about the iT part.

Step 2: The LSZ formula

The next step is to relate the amplitude given above to the expectation values of fields. The key trick is this: we assume that in the far future and far past the particles are free and can be described by momentum eigenstates. Then we replace one by one the creation or annihilation operators for states of definite momenta in terms of the field operator $\hat{\phi}$. It's nontrivial but here I give only the final result.

$$\begin{equation*} \addtolength{\fboxsep}{5pt} \boxed{ \begin{gathered} \Pi_{i=1}^m \int d^4x_i \, e^{-i (k \cdot x)_i} ~\Pi_{j=1}^n \int d^4y_j \, e^{+ i (p \cdot y)_j} \\ <0| T\{ \phi(x_1) \ldots \phi(x_m) \phi(y_1) \ldots \phi(y_n) \} |0> \\ = \biggl( \Pi_{i=1}^m \frac{i \sqrt{Z}}{k_i^2 - m^2} \biggr) \biggl(\Pi_{j=1}^n \frac{i \sqrt{Z}}{p_j^2 - m^2} \biggr) ~<\vec{p}_1 \ldots \vec{p_n}|i T| \vec{k_1} \ldots \vec{k_m}> \end{gathered} } \end{equation*}$$

What we are looking for is the last expression on the rhs. The equation gives a way to calculate it in terms of a vacuum expectation value of the field operators. The equations tells us that the integral on the left side will contain poles at the on-shell values, poles which must be removed in order to be left with the vacuum expectation value of the T matrix.

Step 3: expectation values in terms of free field operators

The next step is to find a way to compute the vacuum expectation value of a bunch of field operators as we have on the lhs of the previous equation. This is nontrivial because those field operators are the fields of the fully interacting theory.
This is where one introduces the time evolution operator and the interaction picture. The end result is


$$\begin{equation*} \addtolength{\fboxsep}{5pt} \boxed{ \begin{gathered} <0| T \{ \phi(x_1) \ldots \phi(x_n) \} |0> \\ = \frac{<0| T \{ \phi_I(x_1) \ldots \phi_I(x_n) ~\rm{exp} ( -i \int d^4x ~{\cal H}_I ) \} }{ <0| T \{ \rm{exp} (-i \int d^4x ~{\cal H}_I ) \} |0> } \end{gathered} } \end{equation*}$$
Note that the Hamiltonian appearing in the exponential is the interaction part of the Hamiltonian written in terms of interaction picture fields! So there should be two labels ``interaction" in principle1

Step 4: Wick's theorem


Now one expands the exponential and one has to deal with the expectation values of the time-ordered product of a bunch of products of fields. It's at this point that one introduces Wick's theorem and that one starts thinking in terms of Feynman diagrams. I used to know how to do contractions in LaTeX but I can't find the trick anymore so I will use a simpler notation. Basically, one simply writes the time ordered expectation value fo a bunch of fields as a sum over all the possible contractions and normal ordered products. For example,

$$\begin{equation*} \addtolength{\fboxsep}{5pt} \boxed{ \begin{gathered} T \{ \phi_1 \phi_2 \phi_3 \phi_4 \} = : \ \phi_1 \phi_2 \phi_3 \phi_4 + \, D_{12} : \phi_3 \phi_4: + D_{13} : \phi_2 \phi_4: \\ + \, D_{14} : \phi_2 \phi_3: + \, D_{23} : \phi_1 \phi_4: + \, D_{24} : \phi_1 \phi_3: \\ + \, D_{34} : \phi_1 \phi_2: + \, D_{12} D_{34} + \, D_{13} D_{24} + \, D_{14} D_{23} \end{gathered} } \end{equation*}$$

where the D's are of course the two point Green's function of the theory (the Feynman propagator) and the colon indicate normal ordering. Of course, sandwiching this between the vacuum kills off all the terms which are normal ordered so that in the previous example, we simply get:

$$\begin{equation*} <0|T \{ \phi_1 \phi_2 \phi_3 \phi_4 \}|0> = D_{12} D_{34} + \, D_{13} D_{24} + \, D_{14} D_{23} \end{equation*}$$

Therefore, we now have a way to compute all the vacuum expectation values of products of time ordered fields as they appear in the equations of step 3, in terms of different combinations of the two points Green's function.

What we now need is an explicit representation of this famous propagator!

 

[kdv]

Step 5: Evaluating the propagator

At this step one must evaluate explicitly the two-point Green's function, i.e. the propagator. It's here that the question of the $i \epsilon$ prescription enters and that one must choose how to deal with the poles. The end result (with the standard prescription for the epsilon) is

$$\begin{equation*} \addtolength{\fboxsep}{5pt} \boxed{ \begin{gathered} D(x-y) = \int \frac{ d^4 p}{(2 \pi)^4} ~ \frac{i}{p^2-m^2 + i \epsilon} ~e^{-i p(x-y)} \end{gathered} } \end{equation*}$$


Interlude

At this point, we have everything to compute the T matrix element. Using the explicit form of the propagator, we can compute any vacuum expectation value of time ordered products of fields using Wick's theorem. So we can calculate any term in the expansion of the exponential appearing in the equation of Step 3. From the LSZ reduction formula we therefore have a way to calculate the transition amplitude of any process. All we have now to do is to relate the transition amplitude to a physical process. That's the final step.

Step 6: Connection with observables

Finally, one relates the transition amplitude to things that are actually measured: cross sections and decay rates.

First, we introduce the so-called scattering amplitude ${\cal M}_{fi}$ by pulling out a Dirac delta and a normalization constant from the T matrix element, i.e. we define

$$S_{fi} = \delta_{fi} + i (2 \pi)^4 \delta^4 (p_i - p_f) ~ {\cal M}_{fi}$$

Now we introduce the differential n-body phase space

$$\begin{equation*} \addtolength{\fboxsep}{5pt} \boxed{ \begin{gathered} d \Phi \equiv (2 \pi)^4 ~ \delta^4(p_i-p_f) \Pi_{i=1}^n ~ \frac{d^3p_i}{(2 \pi)^3 2 E_i} \end{gathered} } \end{equation*}$$

In terms of this phase space factor, we have that the decay rate of a particle with energy $E_p$ is given by

$$\begin{equation*} \addtolength{\fboxsep}{5pt} \boxed{ \begin{gathered} d \Gamma = \frac{1}{2 E_p} | {\cal M}_{fi}|^2 d \Phi \end{gathered} } \end{equation*}$$

It's useful to write down the above formula in the special cases of a two body final state and in the rest frame of the decaying particle.
The result is

$$\begin{equation*} \addtolength{\fboxsep}{5pt} \boxed{ \begin{gathered} d \Gamma = \frac{1}{ 64 \pi^2 M^3 } \bigl(M^4 +(m_1^2-m_2^2)^2 - 2 M^2 (m_1^2+m_2^2) \bigr)^{1/2} | {\cal M}_{fi}|^2~ d \Omega \end{gathered} } \end{equation*}$$


Now consider cross sections. The general formula i s

$$\begin{equation*} \addtolength{\fboxsep}{5pt} \boxed{ \begin{gathered} d \sigma = \frac{1}{4 {\sqrt{(p_1 \cdot p_2)^2 - m_1^2 m_2^2}} } ~ | {\cal M}_{fi}|^2~ d \Phi \end{gathered} } \end{equation*}$$

Again, it's useful to write the explicit case of a $2 \rightarrow 2$ cross section in the center o fmass frame with the result (the labels 1 and 2 refer to the initial two particles):

$$\begin{equation*} \addtolength{\fboxsep}{5pt} \boxed{ \begin{gathered} d \sigma = \frac{1}{64 \pi^2 s} | {\cal M}_{fi}|^2 ~\frac{| \vec{p'} |}{| \vec{p} |} ~d \Omega \end{gathered} } \end{equation*}$$
A special sub-case of the above special case is elastic scattering in which case $m_1 = m_3$ and $m_2 = m_4$ and which implies that $| \vec{p'} | = | \vec{p} |$. In that case, we have

$$\begin{equation*} \addtolength{\fboxsep}{5pt} \boxed{ \begin{gathered} d \sigma_{elas} = \frac{1}{64 \pi^2 s} | {\cal M}_{fi}|^2 ~d \Omega \end{gathered} } \end{equation*}$$

 

[Avodyne]

Excellent summary.

The recent book by Srednicki (which can be downloaded from his web site) presents these steps in pretty much this order).

 

[pellman]

This was conceptually helpful to see all presented on one page. Thanks for posting, kdv.

 

[nrqed]

This was conceptually helpful to see all presented on one page. Thanks for posting, kdv.

You are welcome

kdv aka nrqed

 

[reilly]

kdv or nrqd ; why two names?Anyway, indeed a very nice summary of basic QFT. Thanks.

regards,
Reilly

 

[lbrits]

I would say that the "wick's theorem" part could be augmented by saying that the terms in the perturbation series can be obtained by taking functional derivatives of the generating function
$$Z[J] = \int\!D\phi\,e^{i S[\phi] + \int\, J \phi}$$
wrt $$J$$ and setting $$J = 0$$. This is a bit more modern and you don't have to bother talking about normal ordering and time ordering.

 

[nrqed]

I would say that the "wick's theorem" part could be augmented by saying that the terms in the perturbation series can be obtained by taking functional derivatives of the generating function
$$Z[J] = \int\!D\phi\,e^{i S[\phi] + \int\, J \phi}$$
wrt $$J$$ and setting $$J = 0$$. This is a bit more modern and you don't have to bother talking about normal ordering and time ordering.

True. But as I said at the beginning of my post, I was simply presenting the steps for the canonical quantization approach. I would have presented also all the steps in the path integral approach if there had been some interest.

 

[lbrits]

True. But as I said at the beginning of my post, I was simply presenting the steps for the canonical quantization approach. I would have presented also all the steps in the path integral approach if there had been some interest.

Granted. I just think the only place the interaction picture lives on is in old textbooks.

Edit: and i mean that in a constructive way =)

 

[nrqed]

Granted. I just think the only place the interaction picture lives on is in old textbooks.

Edit: and i mean that in a constructive way =)

Well, P&S covers it so it's not that outdated

Of course there is the question of whether only the PI approach should be taught. But operator formalisms are important in string theory and in quantum gravity so some ease with the operator formalism is important (but granted, not the whole interaction picture/wick theorem is required in those different applications)