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1

It looks like the sum appears because of the completeness relation $$\mathbf{1} = \sum_{\alpha} \left| \alpha \right> \left< \alpha \right|$$ taken over all the $\{ \alpha_j \}$ to give $$\mathbf{1} = \sum_{\{ \alpha_j \}} \left| \alpha_j \right> \left< \alpha_j \right|$$ Starting with the first expression, which I believe you corrected in ...

0

The exponential factor is a phase factor, where a change in the exponent represents a rotation in the complex plane. Because $\xi^2$ multiplies a very large number (both $\epsilon$ and $\hbar$ being very small), when $\xi$ is not small any $d\xi$ makes the rotation "fast" compared to the corresponding change in the other factor (the phase is proportional to ...

2

Okay, let's give it a try. $SU(2)$ sector of Standard Model Lagrangian is rather involved, so we will take a look at something simpler. Neutron-proton interaction comes to my mind. In low energy limit it is mediated by a massive scalar particle — a pion. We will be very qualitative about this, in reality there are a lot of details. Lagrangian will look ...

5

While the trace is invariant under a transform to another basis, you need to take into account here that the coherent state basis is not an orthogonal basis and it is overcomplete. We can evaluate the trace of an operator $A$ by inserting identity operators in front of and after the operator and then using resolution of identity in terms of the coherent ...

2

I) Apart from a full proof of the Gutzwiller's formula in the context of the Feynman path integral (FPI), then OP is essentially asking if the FPI knows about the metaplectic correction/Maslov index and caustics? The physics lore is that when the FPI is set up and interpreted properly, it does contain these semiclassical phase factors. II) In practice, let ...

2

No, you don't need to work in the basis where the Hamiltonian is diagonal. It's a fact of linear algebra that the sum of the diagonal elements of a matrix is the same no matter what basis you're in, so you can easily evaluate the trace in whichever basis is convenient.

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In modelling elementary particle interactions, Feynman diagrams are used to represent the scattering amplitude which will give the crossection for the interaction. This is a diagram for calculating the first order contribution to the elastic scattering ( taking the x axis as time, ) of an incoming e+ e- pair to an outgoing e+ e- pair. The exchanged ...

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