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Yes there is link.When the massive charged particles fully express themselves in quantum fields they would behave like-bosons but are not exclusively bosons. It is the property of both the field and the intrinsic property of the charged particles. So matter can behave like wave in quantum fields and away from the quantum fields they become matter. Therefore ...

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Taking the Hermitian conjugate reverses the order of the $\psi$'s. You have $$L_M^\dagger = \left( \bar{\psi}\psi\right)^\dagger = \left( \psi^\dagger \gamma^0\psi\right)^\dagger = \psi^\dagger{\gamma^0}^\dagger \psi = \psi^\dagger\gamma^0\psi = \bar{\psi}\psi = L_M \ ,$$ where we use that $\gamma^0$ is Hermitian.

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It's because first homotopy group for configuration space of one-particle states in 3D space is permutations group. In two words: scalar product $\langle \mathbf p_{1}, \mathbf p_{2}, ...| \mathbf k_{1}, \mathbf k_{2} , ...\rangle$ is invariant under simultaneous permutations $\mathbf p$ and $\mathbf k$ for different particles. For identical particles, ...

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A muon and a positron are different species so the wavefunction need not be symmetric or antisymmetric under interchange of the positions of the two different particles. That's good since the momentum operator takes the derivative in that particles direction and then scales by that particles mass, so it would be weird if they swapped. So you can have an ...

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What is the fundamental reason of the fermion doubling? Answer: there is no fundamental reason of the fermion doubling. Adding proper lattice interaction can always get rid of the doublers, and it works for both abelian and non-abelian gauge groups, as long as the resulting chiral theory is free of all anomalies. (See Xiao-Gang Wen arXiv:1305.1045 and ...

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Comments to the question (v3): A Grassmann-odd number is not a complex number. It is a complex supernumber $z=x+iy$, which can be decomposed in real and imaginary supernumbers, cf. e.g. this and this Phys.SE posts. The Berezin integral $\int\! d\theta~f(\theta)$ over supernumbers is an ordinary complex number $c=a+ib\in\mathbb{C}$, which can be ...

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