Rotational Invariance, The Spin-Statistics Connection And The TCP Theorem
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Quantum Field Theory formulated in terms of hermitian fields automatically leads to a spin-statistics connection when invariance under rotations is required. In three (or more) dimensions of space this implies Bose statistics for integer spin fields and Fermi statistics for half-integer spin fields. One should recall that spin-1/2 fields in three dimensions have two nonhermitian or four hermitian components. This automatic doubling of the number of components enables one to define a pseudoscalar matrix, and this in turn allows one to prove the TCP theorem for rotationally invariant field theories. In two space dimensions one obtains anyon statistics independent of the >spin>. For the quantum mechanics of identical particles we obtain only the possibility of either statistics for either spin as long as the spatial dimension is three (or highs). For two space dimensions we get anyon statistics. This difference is due to the contractibility of closed loops in three or more dimensions. The relation to the arguments of Broyles, of Bacry and of Berry and Robbins is discussed.