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f!J>All (x)f!J>t em in VS .NET
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basic fields, eventually normal ordered. The quantization respects the spin statistics relation. The PCT theorem then states that even if P, C, and T are not separately invariances, their combination is. This implies, in particular, the existence of antiparticles for charged fields (with masses and spins equal to those of the corresponding particles) and the particleantiparticle identification for neutral fields. In summary we want to show that the lagrangian '(x) behaves under 0 as a neutral scalar field 0 '(x)0 t
= '(  x) = cpt( x) (3201) We should have added that for a scalar field in general
0cp(x)0 t
(3202) If (3201) is satisfied the action will remain invariant We shall be satisfied here with a lagrangian depending on scalar (cp), vector (All)' and spinor (1/1) fields possibly carrying internal indices. The theorem remains tJ;ue for higher spin fields. It is clear that ' can be constructed using All' 1/1, If/ and only hermitian scalar fields combining if necessary the charged ones into CP1 = (cp + cpt)/fi and CP2 = (cp  cpt)/ifi (0cpi(X)0 t = cpi(X)). We can then summarize the action of 0 as follows: 1. It changes the arguments of fields from x to  x. 2. As a consequence derivatives change sign, all +  Ow 3. A vector field AIl(x) gets an extra minus sign, i.e., behaves as the gradient of a scalar field. 4. Any quadratic form in (If/,I/I) gets a sign (1t where P denotes the number of Lorentz indices carried by y matrices or derivatives. This follows from table (3199) and rule 2. 5. Finally, constants are complex conjugated. Since ' is a scalar, each term in a monomial expansion is obtained by contracting an even number of Lorentz indices. The minus signs coming from vector fields, derivatives, or quadratic forms in (iii, 1/1) are cancelled. The net effect is thus to change '(x) into 't(  x) = '(  x), since the order of operators is irrelevant under normal ordering as vector and scalar fields commute and the anticommutation of the 1/1 fields has already been taken into account in (3199). As a consequence, if the vacuum is invariant under 0, so will be the dynamics. It is interesting to check that equaltime commutation rules are invariant under 0. This involves some subtlety since interactions may modify the expression of conjugate variables. Such is the case, for instance, for electromagnetic interactions of charged bosons and more generally when the interaction part of f ' involves derivatives. We may also verify that the hamiltonian density behaves as 0YC'(O, x)0 t YC'(O, x)

