where and 4>\ ' are the positive and negative frequency parts of the field 4>lt as defined in Eq. (1.15), and to obtain the last line use the facts that 0(+)|O) = 0, (0|</>(_) = 0, and the commutator [<Ai+\ </>2is a c-number, so that it is equal to its ground state expectation value. Hence the normal ordered product of operators which satisfy commutation relations like (1.17) can be obtained simply by reordering any terms in which creation operators are on the right and the annihilation operators are on the left, so that all the terms have either two annihilation operators, two creation operators, or a creation operator on the left and an annihilation operator on the right.
Using this definition, the total momentum operator of the one-dimensional string is
The total momentum assumes a simple, clearly interpretable form when expressed in terms of the a's. To obtain it, substitute (1.15) into (1.24), honoring the normal ordered definition (1.23):
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