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For a complex scalar field $\Phi$, the field has the expansion $$ \Phi(x^0,\mathbf{x}) = \int \frac{d^{3}\mathbf{p}}{\sqrt{ 2 E_{\mathbf{p}} (2\pi)^3 } }\ \bigg[ e^{- i E_{\mathbf{p}}x^0 + i \mathbf{p …

Is the following statement true? $$ \lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x) $$ where $\mathscr{P}$ is the Cauchy principal value. The above i …

I'm reading through "H. Kleinert and V. Schulte-Frohlinde" notes for "Critical Properties of $\phi^{4}$-Theories", and I've reached this point in the lecture notes: $\ $ $\ $ The trace property …

we have the equality: $$ \iint_{X\times Y} f(x,y) d(x,y) = \int_X \left[ \int_Y f(x,y) dy \right] dx = \int_Y \left[ \int_X f(x,y) dx \right] dy $$ So this tells us: We can safely switch the order of integration … Furthermore, what if the variable upon which we need to place our cutoff is different depending on the order of integration? This seems very concerning! …

The point of the question is to ask what is the function given by the following integral: $$ H(x,y) \ \equiv \ \int \frac{d^{4}p}{(2\pi)^{4}} \frac{e^{-i p \cdot (x-y)}}{(p^{2}+m^{2}-i\epsilon)^{2}} $ …

I'm following Chapter 3 of Kapusta and Gale's Finite-Temperature Field Theory here. I'm considering the following integral (the unrenormalized self-energy evaluated at zero-four momentum): $$ \mathca …

In Equation (8) of this paper by Groote et. al., we are given the following Euclidean identity: $$ \int \frac{d^{4}\mathbf{p}_{\mathrm{E}}}{(2\pi)^{4}} \frac{e^{ i \mathbf{p}_{\mathrm{E}} \cdot \mathb …

When writing integrals that look like $$ \int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}} \ = \int \frac{d^4p}{(2\pi)^4}\ 2\pi\ \delta(p^2+m^2)\Theta(p^0) $$ it is often said …

This is somewhat related to an earlier question I asked about the following diagram in $\phi^{4}$ theory: I've been following these lecture notes by H. Kleinert and V. Schulte-Frohlinde. Saying we …

Consider the following loop diagram: If $k$ is the incoming/outgoing momentum and we're integrating over momentum $p$, the above diagram corresponds to: $$ - \lambda \frac{1}{k^{2} + m^{2}} \int \f …

In a real scalar massive $\phi^4$-interacting theory consider the amputated sunset diagram. This is the integral out of Kleinert and Schulte-Frohlinde Critical Properties of $\phi^4$-Theories: The a …