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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: $$D_F(p^2) = \frac{i}{p^2-m^2+i\epsilon} + \int^{\infty}_{4m^2}ds\rho(s)*\frac{i}{p^2-s+i\epsilon} \tag{1}$$ ...
0
votes
1answer
65 views

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{\...
3
votes
1answer
144 views

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 ...
2
votes
0answers
83 views

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
3
votes
2answers
76 views

Multiplying Distributions in finite-temperature Keldysh/Thermo-field field theory

In the real-time finite temperature formalisms (Schwinger-Keldysh or Thermo-field), the free propagators are often defined with terms like: $$ \mathrm{Dirac\ Delta}\ \times \ \mathrm{Thermal\ ...
3
votes
1answer
218 views

Massless $m=0$ 4D Fourier transform of $(p^2 + i \epsilon)^{-2}$

This question is related to this one. I'm assuming that we're in or on the the light-cone $s \leq 0$ in what follows. Suppose I'm interested in computing the following Fourier transform, in the ...
4
votes
2answers
702 views

Bulk-to-Boundary propagator

How can I show that the bulk-to-boundary propagator $$ K(z,x;x')~=~\frac{z^{\Delta}}{[z^2+(x-x')^2]^{\Delta}} \tag{1} $$ goes as a delta function near the boundary $$ K(z,x;x')~\sim ~z^{d-\Delta}\...
2
votes
1answer
2k views

Why is the propagator the Green's function for Schrodinger equation? [duplicate]

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 }{\...
11
votes
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, $$u(x) = \int dx' G(x;x')\rho(x'),$$ ...
2
votes
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\...
18
votes
2answers
8k views

How exactly is the propagator a Green's function for the Schrodinger equation

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)\...