# S-Matrix Elements in Path Integral Formalism

+ 4 like - 0 dislike
446 views

I have a question related to the connection between the S-Matrix elements and the path integral formalism. In order to formulate the question, I will just work with a scalar field theory for simplicity.

Let us assume that we are given an action $S[\phi]$. In the path integral formalism, we can now define the generating functional $$Z[J] \propto \int \mathcal{D}\phi ~ e^{i S[\phi] + \int d^4x~ \phi(x) J(x)}$$ and calculate arbitrary vacuum expectation values $$\left<0| \phi(x_1) \ldots \phi(x_n) | 0 \right>$$ using functional derivatives with respect to the source $J$. I also know how to calculate vacuum expectation values in the "canonical quantization formalism" (Wick's theorem etc.). So far so good.

Usually, we are not interested in vevs but rather in S-matrix elements such as $\left<p_1, \ldots, p_n|q_1, \ldots, q_m \right>$ where $p_i$ and $q_j$ are outgoing and ingoing particle momenta. Furthermoe, the transition between $S$-matrix elements and vevs is also clear to me: this is just given by the LSZ reduction formula. So in principle, we are now good to go: we can calculate everything in the path integral formalism and eventually relate this to actual matrix elements using the LSZ formula.

Now come my actual questions:

1. It seems that there is a more direct relation between the S-matrix elements and the path integral formalism. In fact, on the Wikischolar article about the Slavnov-Taylor identities (written by Dr. Slavnov himself) it is stated that the $S$ matrix can be written as $S = Z[0]$. Where does this come from and how is it to interpret? I am confused because I thought that $S$ was rather a matrix (whose entries, i.e. matrix elements are numbers) and $Z[0]$ is just a number (an evaluated integral). So to me, thsi reads like "matrix = number"... Furthermore, if this equation holds true, how can we obtain the $S$ matrix elements from there?

2. Even more confusingly, there seems to be another relation to the $S$-matrix element. I have found this in Weinberg Vol. II, chapter 15.7 around equation (15.7.27). There, we have an action that is of the form $I + \delta I$ (the context is here that $I$ is the gauge fixed action of a non-Abelian gauge theory and $\delta I$ is the change due to a small variation in the gauge-fixing condition, but this does not really matter here). It says then: It is a fundamental physical requirement that matrix elements between physical states should be independent of our choice of the gauge-fixing condition, or in other words, of $\delta I$. The change in any matrix element $<\alpha|\beta>$ due to a change $\delta I$ in $I$ is $$\delta <\alpha|\beta> ~\propto ~<\alpha|\delta I|\beta>.$$ So now, there seems to be even a relation between the action and the $S$-matrix elements. How does this fit into the entire picture?

My QFT exam is coming up, so thanks a lot for your answers!

This post imported from StackExchange Physics at 2014-08-07 15:19 (UCT), posted by SE-user user56643
retagged Aug 7, 2014

+ 2 like - 0 dislike

IMHO, in the prof Slavnov article, the action path integral formula $(3)$ for $S$ should be understood with constraints about initial and final states. So, in fact, it is a matrix $S_{ij}$.

However, this is not the case for the formula $(4)$ for $Z[J]$. It is a path integral without constraints.

For your second question, just note that $e^{i (S+\delta S)} \approx e^{i S}(1+i\delta S)$, so, with the different definitions of matrix elements and Green functions, you get your result.

This post imported from StackExchange Physics at 2014-08-07 15:19 (UCT), posted by SE-user Trimok
answered Aug 5, 2014 by (950 points)
Trimok: Could you be a tad more precise in what you mean by "with the different definitions of matrix elements and Green functions"?

This post imported from StackExchange Physics at 2014-08-07 15:19 (UCT), posted by SE-user PPR
@PPR : As indicated in the question, you get $S$-matrix elements by applying LSZ reduction formulae to Green functions (vacuum expectation values). And vacuum expectation values are just successive derivates of $Z$ relatively to the currents $J(x)$ (up to a global constant)

This post imported from StackExchange Physics at 2014-08-07 15:19 (UCT), posted by SE-user Trimok
+ 0 like - 0 dislike

For your question 1.: note that the integral in $Z[J]$ is performed over only paths connecting the initial state $q_i$ to the final state $q_f$, i.e., $Z[J]$ actually depends on $q_i$ and $q_f$. So, you can view it as a matrix element of $\hat{S}$, namely, $S_{ij}$.

This post imported from StackExchange Physics at 2014-08-07 15:19 (UCT), posted by SE-user Hai-Yao Deng
answered Aug 5, 2014 by (0 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.