Scalar Field Theory-Vacuum Expectation Value

[Mick83]

Homework Statement

I am given an equation for a quantized, neutral scalar field expanded in creation and destruction operators, and need to find the vacuum expectation value of a defined average field operator, squared. See attached pdf.

Homework Equations

Everything is attached, but I can include more.

The Attempt at a Solution

I've solved part (a) (I think- if someone could check my work it would be appreciated), so I need some idea on how to do part (b). The question clearly asks for the expectation of the squared value, but I don't have a clue how to solve the integral in the first place, which must then be squared to find the expectation value.

 

[Thaakisfox]

switch to spherical coordinates... put your z axis toward the direction of k i.e. measure your \theta angle from k. don't forget to use the correct integral measure...

 

[Mick83]

So use $$k_{\mu}=\bigg(\mathbf{k},\frac{i\omega}{c}\bigg),~~ x_{\mu}=(\mathbf{x},ict),~~ \mathbf{k}\cdot\mathbf{x}=k_{\mu}x_{\mu}-\omega t=k_1x_1+k_2x_2+k_3x_3-\omega t =kr(\sin\theta \cos\phi +\sin\theta \sin\phi +\cos\theta)-\omega t$$? Regardless of the θ or ∅ dependence, $$\int^{\infty}_0 dr~r^2~ e^{r^2/2b}~e^{ikr}$$ gives a horrible result, even when squared.

 

[Thaakisfox]

nono that is not correct.
The kx in the power of the exponential is already only the 3-part of them. the expression is not in a "covariant" form, but the usual separate spatial and time parts.

So If you put your z axis towards the direction of k, and measure the angle \theta from there your integral will be:

$$\int_0^{2\pi}d\phi\int_0^{\infty}dr \, r^2 e^{-r^2/2b^2} \int_{-1}^{1}d\cos{\theta} e^{ikr\cos\theta}$$The theta part can be integrated easily.

U will get the kind of integral wt you wrote:

$$\int_0^{\infty}dr\, r^2 e^{-r^2/2b^2}e^{\pm ikr}$$

now try bringing the integrand into the form:$r^2e^{-ax^2}$ i.e. complete the square on the exponentials. this kind of integral can be simply expressed with gamma functions.

 

[paranormal]

I would first clarify the context of this question and the significance of finding the vacuum expectation value in the given scalar field theory. I would also check the work done for part (a) and ensure its accuracy before moving on to part (b).

To solve part (b), I would approach it by first understanding the integral and its physical interpretation. The integral represents the probability of finding a particular state of the system, in this case, the squared value of the average field operator. Therefore, I would use the appropriate mathematical tools and techniques to evaluate the integral and find the expectation value.

One approach could be to use the creation and destruction operators to express the average field operator in terms of these operators, and then use their properties to simplify the integral. Another approach could be to use the commutation relations of the operators to simplify the expression before integrating.

I would also consult relevant literature and resources to gain a better understanding of the concept and techniques involved in finding the vacuum expectation value in scalar field theory. Additionally, I would seek guidance from my peers and mentors to ensure the accuracy of my approach and solution.

Overall, finding the vacuum expectation value in scalar field theory is a complex task that requires a thorough understanding of the theory and its mathematical tools. I would approach it with a systematic and analytical mindset, seeking to understand the underlying principles and applying appropriate techniques to solve the problem accurately.