- #1
CharlieCW
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This is one of the problems I'm currently working on but understanding how to deduce the Feynman rules for this case would give me a better idea on how to do it for more general cases besides ##\phi^4## theory (which is the example commonly covered in books like Peskin and Greiner).
1. Homework Statement
Consider the interaction term ##\frac{\lambda}{3!}## for the Klein-Gordon lagrangian. Find the Feynman rules in position space to calculate the correlators of ##N## points ##G_N(x_1,...,x_N)##. Translate the same rules to momentum space, and draw all the diagrams of ##G_2(x_1,x_2)## up to quadratic order.
$$\langle \Omega |T[\phi_1...\phi_n]|\Omega\rangle=Lim_{T\rightarrow \infty}\frac{\langle 0 |T[\phi_1...\phi_nexp(-i\int^{T}_{-T} dt H_{int})]|0\rangle}{\langle 0 |exp(-i\int^{T}_{-T} dt H_int)|0\rangle}$$
$$\langle 0 | T{\phi_1...\phi_n} | 0 \rangle=(sum \ of\ all\ terms \ with\ fully\ contracted \ \phi 's)$$
$$H_{int}=\frac{\lambda}{3!}\phi^3$$
Following the recipe for the calculation of Feynman rules for ##\phi^4## theory in Peskin, I started with the correlator ##G_2(x_1,x_2)## and considered only the numerator:
$$\langle 0 |T[\phi_1 \phi_2 exp(-i\frac{\lambda}{3!}\int^{T}_{-T} dt \phi_y^3)]|0\rangle$$
Expanding the exponential, for ##\lambda=0## obviously we recover the free propagator:
$$-i\frac{\lambda}{3!}\int dt\langle 0 |T[\phi_1 \phi_2)]|0\rangle=K_F(x_1-x_2)$$
For the first order ##\lambda=1##, we have an uneven number of field operators:
$$-i\frac{\lambda}{3!}\int dt\langle 0 |T[\phi_1 \phi_2 \phi_y^3)]|0\rangle$$
Which according to Wick's theorem, thus become zero (since the expectation value of non-contracted terms is zero when they act on the free vaccum).
For the second order ##\lambda=2##, we have the first non-zero term since the number of operators is even:
$$-(i\frac{\lambda}{3!})^2\int dt \langle 0 |T[\phi_1 \phi_2 \phi_{y1}^3 \phi_{y2}^3)]|0\rangle$$
Where, for example, the first valid term is given by applying the contractions which are then traduced to the free propagators:
$$\phi_1\phi_2 \phi_{y1} \phi_{y1} \phi_{y1} \phi_{y2} \phi_{y2} \phi_{y2} \rightarrow K_F(x_1-x_2)K_F(y_1-y_2)K_F(y_1-y_2)K_F(y_2-y_2)$$
From this term, the corresponding Feynman diagram should be: one line that connects points ##x_1## and ##x_2##, plus one loop diagram that connects the ##y's## (vaccuum bubble). We can interchange the terms and produce ##6!## which effectively cancel the factor ##6!## we introduced before.
In the end, when dividing by the denominator, all the vacuum terms are canceled so effectively we only care about the connected non-vaccum terms, i.e., in this case the surviving term is the propagator of ##x_1## to ##x_2##.
From this, I think I can deduce the following Feynman rules:
*Each line accounts for a propagator term and contributes a factor of ##K_F(x-x')##
*Each vertex contributes with a factor of ##-i\lambda\int dy##
*For each external point, we have a factor of ##1##
However, I don't know if this is valid as I'm ignoring all the uneven powers of ##\lambda##.
I read online from Srednicki that the rules should also indicate that through each vertex should only pass 3 lines, which I don't know how to deduce from above. Moreover, I don't know how to the find a formula for the symmetry factors like in the ##\phi^4## case.
I've consulted Srednicki (http://chaosbook.org/FieldTheory/extras/SrednickiQFT03.pdf) as well as Dermisek (http://www.physics.indiana.edu/~dermisek/QFT_08/qft-II-1-2p.pdf), but they treat the problem using path integrals (which I still haven't studied for QFT), and I couldn't find a source that works with Wick's theorem like Perskin, which is the approach I need to take.
1. Homework Statement
Consider the interaction term ##\frac{\lambda}{3!}## for the Klein-Gordon lagrangian. Find the Feynman rules in position space to calculate the correlators of ##N## points ##G_N(x_1,...,x_N)##. Translate the same rules to momentum space, and draw all the diagrams of ##G_2(x_1,x_2)## up to quadratic order.
Homework Equations
$$\langle \Omega |T[\phi_1...\phi_n]|\Omega\rangle=Lim_{T\rightarrow \infty}\frac{\langle 0 |T[\phi_1...\phi_nexp(-i\int^{T}_{-T} dt H_{int})]|0\rangle}{\langle 0 |exp(-i\int^{T}_{-T} dt H_int)|0\rangle}$$
$$\langle 0 | T{\phi_1...\phi_n} | 0 \rangle=(sum \ of\ all\ terms \ with\ fully\ contracted \ \phi 's)$$
$$H_{int}=\frac{\lambda}{3!}\phi^3$$
The Attempt at a Solution
Following the recipe for the calculation of Feynman rules for ##\phi^4## theory in Peskin, I started with the correlator ##G_2(x_1,x_2)## and considered only the numerator:
$$\langle 0 |T[\phi_1 \phi_2 exp(-i\frac{\lambda}{3!}\int^{T}_{-T} dt \phi_y^3)]|0\rangle$$
Expanding the exponential, for ##\lambda=0## obviously we recover the free propagator:
$$-i\frac{\lambda}{3!}\int dt\langle 0 |T[\phi_1 \phi_2)]|0\rangle=K_F(x_1-x_2)$$
For the first order ##\lambda=1##, we have an uneven number of field operators:
$$-i\frac{\lambda}{3!}\int dt\langle 0 |T[\phi_1 \phi_2 \phi_y^3)]|0\rangle$$
Which according to Wick's theorem, thus become zero (since the expectation value of non-contracted terms is zero when they act on the free vaccum).
For the second order ##\lambda=2##, we have the first non-zero term since the number of operators is even:
$$-(i\frac{\lambda}{3!})^2\int dt \langle 0 |T[\phi_1 \phi_2 \phi_{y1}^3 \phi_{y2}^3)]|0\rangle$$
Where, for example, the first valid term is given by applying the contractions which are then traduced to the free propagators:
$$\phi_1\phi_2 \phi_{y1} \phi_{y1} \phi_{y1} \phi_{y2} \phi_{y2} \phi_{y2} \rightarrow K_F(x_1-x_2)K_F(y_1-y_2)K_F(y_1-y_2)K_F(y_2-y_2)$$
From this term, the corresponding Feynman diagram should be: one line that connects points ##x_1## and ##x_2##, plus one loop diagram that connects the ##y's## (vaccuum bubble). We can interchange the terms and produce ##6!## which effectively cancel the factor ##6!## we introduced before.
In the end, when dividing by the denominator, all the vacuum terms are canceled so effectively we only care about the connected non-vaccum terms, i.e., in this case the surviving term is the propagator of ##x_1## to ##x_2##.
From this, I think I can deduce the following Feynman rules:
*Each line accounts for a propagator term and contributes a factor of ##K_F(x-x')##
*Each vertex contributes with a factor of ##-i\lambda\int dy##
*For each external point, we have a factor of ##1##
However, I don't know if this is valid as I'm ignoring all the uneven powers of ##\lambda##.
I read online from Srednicki that the rules should also indicate that through each vertex should only pass 3 lines, which I don't know how to deduce from above. Moreover, I don't know how to the find a formula for the symmetry factors like in the ##\phi^4## case.
I've consulted Srednicki (http://chaosbook.org/FieldTheory/extras/SrednickiQFT03.pdf) as well as Dermisek (http://www.physics.indiana.edu/~dermisek/QFT_08/qft-II-1-2p.pdf), but they treat the problem using path integrals (which I still haven't studied for QFT), and I couldn't find a source that works with Wick's theorem like Perskin, which is the approach I need to take.