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The bounds of the integral have no dependence on any of the variables, and hence we may move the differential operator into the integrand, $$\frac{\delta}{\delta \eta (z)} \int \mathrm{d}^4 y \, S_F (z-y) \eta(y) = \int \mathrm{d}^4 y \, S_F (z-y) \delta^{(4)}(z-y)$$ Evaluating the integral using the standard delta distribution identity, we obtain your ...

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The existence of "bosons" is already a consequence of QM -- the notion of indistinguishable particles and the resulting Bose-Einstein (as opposed to Maxwell-Boltzmann) statistics is manifestly not a classical phenomena. Classical particles are always distinguishable, since "that particle there" has a complete set of observables that classically commute with ...

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Not sure. Completely lost about the equations. However, have you seen the experiment in which electrons that pass through one slit are rotated by 2 pi. The interference pattern shifts. The min and maximum flip. I'm not exactly sure why an electron that is in different state would show any kind of interference pattern at all. In any case the experiment might ...

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Let's speak about 1D particles for simplicity. What should be understood first of all is that for indistinguishable particles configuration space isn't the same as for distinguishable ones. For two distinguishable spinless 1D particles configuration space is a square: one side is for $x_1$, another for $x_2$. But if the particles appear indistinguishable, ...

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There is a lot to be said about Majorana and Dirac neutrino masses, but I will try to address only the specific point you raise. When you write a Majorana mass term, for example, $$\mathcal{L}_{\text{Majorana}} = -\frac{m_M}{2} \left[\bar\nu_L(\nu_L)^C + \overline{\nu^C_L}\nu_L\right]$$ you are not assuming that $(\nu_L)^C = \nu_R$, but you can write that ...

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