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### How to get the imaginary part from the Källén-Lehmann propagator

During field theory course the Källén-Lehmann propagator was defined as follows: $$D_F(p^2) = \frac{i}{p^2-m^2+i\epsilon} + \int^{\infty}_{4m^2}ds\rho(s)*\frac{i}{p^2-s+i\epsilon} \tag{1}$$ ...
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### Derivation of the QFT Propagator

I don't understand how we get from the RHS to the last line. \begin{eqnarray} \left[ \hat{H}_x - i \frac{\partial}{\partial t_x} \right] G^+(x,t_x,y,t_y) &=& -i \delta (t_x - t_y) \sum_n{\...
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### How do you write the Wightman function $\langle\phi(t_1)\phi(t_2)\rangle$ for a massive scalar field in position space?

For a free real scalar field $\phi(t,\mathbf{x})$, we define the Wightman function as: $$W(t_1,t_2) \equiv \langle 0 | \phi(t_1,\mathbf{x}_1) \phi(t_2,\mathbf{x}_2) | 0 \rangle$$ I'm suppressing the ...
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### How can the propagator be written in the below integral form?

I’am finding it difficult to understand as to how the delta function is written as a product of many delta functions with the integral
76 views

Sakurai says that the propagator is simply the Green's function for the time-dependent wave equation satisfying $$\left [ -\frac{\hbar^2}{2m} \triangledown ''^2+V(\mathbf{x''})-ih\frac{\partial }{\... 1answer 875 views ### Dirac Delta in definition of Green function For a inhomogeneous differential equation of the following form$$\hat{L}u(x) = \rho(x) ,$$the general solution may be written in terms of the Green function,$$u(x) = \int dx' G(x;x')\rho(x'),$$... 2answers 641 views ### Klein-Gordon Green's function: derivative of delta distribution? In Peskin/Schroeder there is an explicit calculation showing that the retarded Green's function of the real Klein-Gordon field$$D_R(x-y) ~\equiv~ \theta(x^0 - y^0) \langle 0 | [\phi(x), \phi(y)] |0\...
Sakurai mentions that the propagator is a Green's function for the Schrodinger equation because it solves \left(H-i\hbar\frac{\partial}{\partial t}\right)K(x,t,x_0,t_0) = -i\hbar\delta^3(x-x_0)\...