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Here is one tricky way to show that the Dirac representation is invariant under C,P,T-transformations. The free Dirac theory refers to the direct sum $\left( 0 , \frac{1}{2}\right) \oplus \left( \frac{1}{2}, 0 \right)$ of the Lorentz group irreducible spinor representations. As it can be shown, the representation $\left(\frac{n}{2}, \frac{m}{2}\right) \oplus ...


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Keeping it simple, let's asume that $\psi(a)$ creates a particle in the state $a$ (i.e., characterized by some collection of quantum numbers that we call $a$), $$ \psi(a)|0\rangle=|a\rangle .$$ and $\psi(b)$ does the same for $b$. We can create a state with two particles: $$ \psi(b)\psi(a)|0\rangle = \psi(b)|a\rangle = |a;b\rangle $$ $$ ...


2

In order to show this, figure out how the transformations $C, P, T$ act on each element in the equation individually. Pay special attention to $C$. That's a tricky one! For a start on C you can check out this physics SE post.


0

A useful reference is http://arxiv.org/pdf/0906.1663.pdf, by Peschel and Eisler. A common approach is to make use of the fact that the two point function you calculated is independent of whether one uses the full density matrix or the reduced density matrix, provided one looks at operators that are local to the region that one is not tracing over. If one ...


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Anomalies (not anamolies) are a whole subject whose basics are covered by one or several chapters of almost any good enough quantum field theory textbook so it's counterproductive to retype this whole chapter here. But generally, in quantum field theory, anomalies are quantum mechanical effects breaking symmetries that exist in the classical theory – ...


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Let's look to the expression for field with mass $m$ and spin $s$ (for massless case following statements exist in similar form): $$ \tag 1 \hat {\psi}_{a}(x) = \sum_{\sigma = -s}^{s}\int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi )^{3} 2E_{\mathbf p}}}\left( u^{\sigma}_{a}(\mathbf p )e^{-ipx}\hat{a}_{\sigma}(\mathbf p ) + v^{\sigma}_{a}(\mathbf p ...


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What you refer to is probably to ground state of a infinite potential wall. If this is the case, then there, in the ground state, the particles are not localized. You can find the solution of the orthogonal ground-states here: https://en.wikipedia.org/wiki/Infinite_potential_well We can reduce the Problem to a one dimensional case, that doesn't change the ...


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Comments to the question (v2): The canonical expansion of the two-fermion wave function seems more related to the canonical form of antisymmetric real matrices in the framework of vector spaces and linear algebra. The Darboux' theorem in the framework of manifolds and differential geometry (which is a surprisingly potent result) is overkill for the ...



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