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behaves as a scalar density S(x). Finally, the behavior of AIl(X) and P(x) under parity justifies the denominations pseudovector current and pseudo scalar density respectively. All phase arbitrariness has disappeared from table (3-199). In this section we have discussed the most frequent discrete symmetries and constructed the corresponding unitary (or antiunitary) operators in terms of the fundamental free fields. The necessity of such a construction is clear, for only then can we be sure that the transformed states do exist. It is a different matter to inquire whether the dynamics is left invariant when interactions are taken into account. This requires us to examine whether U HUt = H. When we consider such interacting fields, we shall try to define the symmetry operators acting on the states and the fields according to the above rules. The question will then be: Is the invariance implemented at the dynamical level A fundamental property of local quantum theory first discussed by Pauli, Zumino, and Schwinger states that, in any case, 0 remains an in variance of the theory. This is the famous peT theorem. We sketch here a proof restricted to lagrangian field theory, referring the rigorous-minded reader to the works quoted in the notes for a more general treatment. Let a local quantum field theory be described by an action principle involving an hermitian Lorentz-invariant lagrangian. This is a combination of local scalar densities expressed in terms of the
158 QUANTUM FIELD THEORY
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 particle-antiparticle 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)
(3-201)
We should have added that for a scalar field in general
0cp(x)0 t
(3-202)
If (3-201) 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 (3-199) 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 (3-199). As a consequence, if the vacuum is invariant under 0, so will be the dynamics.
It is interesting to check that equal-time 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)
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