Calculating Terms in $\bra{P'}\phi^4\ket{P}$

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In summary, the conversation discusses the calculation of terms involving annihilation and creation operators in the context of an integral. By commuting these operators, it is possible to show that the integrand scales as ##k^{-2}##, indicating that the integral will be divergent even with only one integration measure.
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Diracobama2181
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Homework Statement
I am trying to show that ## \bra{P'}\phi^4 \ket{P}##is divergent, where ##\ket{P}## is a free spin 0 boson.
Relevant Equations
##\ket{P}=a^{\dagger}(k)\ket{0}##
##\phi=\int \frac{d^3k(\hat{a}e^{-ikx}+\hat{a}^{\dagger}e^{ikx})}{2\omega_{k}(2\pi)^3}##
##\omega=\sqrt{k^2+m^2}##
From this, I find
$$\bra{P'} \phi^4 \ket{P} = \int \frac {d^3 k_1 d^3 k_2 d^3 k_3 d^3 k_4} {16 \omega_{k_1}\omega_{k_2}\omega_{k_3}\omega_{k_4} (2\pi)^{12} }\bra{0} a_{P'}(a_{k_1} a_{k_2} a_{k_3}a_{k_4}e^{i(k_1+k_2+k_3+k_4)x}+...)a^{\dagger}_{P}\ket{0}$$ (a total of 16 different terms)
Right now, I am trying to figure out how to calculate terms like
## \bra{0}a_{P'}a_{k_1}a^{\dagger}_{k_2}a_{k_3}a^{\dagger}_{k_4}a^{\dagger}_{P}\ket{0} ##. Any examples would be greatly appreciated.
 
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Commute an annihilation or creation operator to the right or left of the product to render the expectation value zero, repeat this with the new terms generated by the commutator. Although to show that this quantity is divergent you don't have to do this, just look at how the integrand scales with ##k##. After doing all the commutators you will find that the integrand scales as ##\sim k^{-2}##, which means that even with one integration measure ##d^3 k## the integral will still be divergent since ##\int \frac{d^3 k}{k^2} \sim \Lambda##.
 
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Related to Calculating Terms in $\bra{P'}\phi^4\ket{P}$

1. How do you calculate the terms in $\bra{P'}\phi^4\ket{P}$?

To calculate the terms in $\bra{P'}\phi^4\ket{P}$, you first need to expand the equation using the Wick theorem. This will give you a series of contractions between the field operators. Then, you need to use the rules for evaluating these contractions, which involve using the propagator and the delta function. Finally, you can simplify the expression and solve for the terms.

2. What is the significance of calculating terms in $\bra{P'}\phi^4\ket{P}$?

Calculating terms in $\bra{P'}\phi^4\ket{P}$ is important in quantum field theory as it allows us to calculate the probability amplitude for a specific process to occur. This is useful in understanding the behavior of particles and predicting the outcomes of experiments.

3. What is the difference between calculating terms in $\bra{P'}\phi^4\ket{P}$ and $\bra{P}\phi^4\ket{P}$?

The main difference between calculating terms in $\bra{P'}\phi^4\ket{P}$ and $\bra{P}\phi^4\ket{P}$ is the initial and final states of the particles involved. In $\bra{P'}\phi^4\ket{P}$, the initial and final states are different, while in $\bra{P}\phi^4\ket{P}$, they are the same. This difference affects the number and type of contractions that need to be evaluated.

4. What are some common challenges in calculating terms in $\bra{P'}\phi^4\ket{P}$?

One common challenge in calculating terms in $\bra{P'}\phi^4\ket{P}$ is dealing with divergent integrals. This can arise due to the infinite number of possible contractions between field operators. Another challenge is keeping track of the different contractions and simplifying the expression to obtain a final result.

5. How can calculating terms in $\bra{P'}\phi^4\ket{P}$ be applied in practical situations?

Calculating terms in $\bra{P'}\phi^4\ket{P}$ has practical applications in particle physics and cosmology. It can be used to calculate the scattering amplitudes of particles in high-energy experiments, as well as to study the interactions between particles in the early universe. This can provide insights into the fundamental forces and laws of nature.

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