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3

I think the answer is it depends on distance (relative to the size of your system). Another well known example of a boson which is comprised of fermionic components is the helium-4 atom, which has integer spin (both the nucleus and the neutral atom itself). Fermionic or bosonic behavior of a composite particle (or system) is only seen at large (compared ...

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In fact, you have $\{\psi^a(x), \bar \psi_b(y)\} = 0$, as an operator, for a space-like interval $(x-y)^2 <0$ (stricly speaking, this is a distribution, for instance, at $x_0=y_0$, this is the distribution $\delta^a_b \delta^3(\vec x-\vec y))$, together with relations $\{\psi^a(x), \psi_b(y)\} =\{\bar \psi^a(x), \bar \psi_b(y)\} = 0$ Now, if you look ...

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Yes, you can show this using only the fact that the Clifford Algebra has a unique representation up to similarity transformation in any dimension. This is shown in the first few pages of http://arxiv.org/pdf/hep-th/9811101.pdf Then you observe that if $\gamma^\mu$ obeys the clifford algebra, then so does $-(\gamma^\mu)^T$. $\mathcal{C}$ is then defined as ...

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You may write : $(\gamma^\mu D_\mu) (\psi_i)_k = [\delta_i^j \partial_\mu - i g A_\mu^a (t_a)_{i}^j] ~~ (\gamma^\mu)^k_l ~~(\psi_j)_l \tag{1}$ Here $i,j$ are in $1..N$, and $k,l$ are in $1..4$, $(\psi_i)_k$ is the k-th ($1 \leq k \leq 4$) component of the i-th ($1 \leq i \leq N$) spinor. We could use the notation $\psi_{j~l} = (\psi_j)_l$, now we see ...

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In a sense, you are right. It's the same information, put in a different order. Try to do that with a simpler case, for instance two two-dimensional vector space, one with index $\alpha=1,2$, the other with index $a=1,2$. Then any vector of this now four-dimensional vector space can be written either $\psi_{\alpha,a}$ or $\psi'_{a,\alpha}$ or even $v_i$ with ...

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Comments to the question (v2): When working with superobjects (both supernumbers and superoperators), we normally assume that they have definite Grassmann-parity. The Grassmann-parity $|\hat{A}|$ of a Grassmann-even (Grassmann-odd) superoperator is 0 (1) modulo 2, respectively. A supernumber $z$ can be viewed as special case of a superoperator $\hat{A}$ ...

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