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In quantum field theory, **multiplicative quantum numbers** are conserved quantum numbers of a special kind. A given quantum number *q* is said to be **additive** if in a particle reaction the sum of the *q*-values of the interacting particles is the same before and after the reaction. Most conserved quantum numbers are additive in this sense; the electric charge is one example. A **multiplicative** quantum number *q* is one for which the corresponding product, rather than the sum, is preserved.

Any conserved quantum number is a symmetry of the Hamiltonian of the system (see Noether's theorem). Symmetry groups which are examples of the abstract group called **Z _{2}** give rise to multiplicative quantum numbers. This group consists of an operation,

**P**, whose square is the identity,

**P**. Thus, all symmetries which are mathematically similar to parity (physics) give rise to multiplicative quantum numbers.

^{2}= 1In principle, multiplicative quantum numbers can be defined for any abelian group. An example would be to trade the electric charge, **Q**, (related to the abelian group U(1) of electromagnetism), for the new quantum number **exp(2 iπ Q)**. Then this becomes a multiplicative quantum number by virtue of the charge being an additive quantum number. However, this route is usually followed only for discrete subgroups of U(1), of which

**Z**finds the widest possible use.

_{2}## See also[edit]

- Parity, C-symmetry, T-symmetry and G-parity

## References[edit]

*Group theory and its applications to physical problems, by M. Hamermesh*(Dover publications, 1990) ISBN 0-486-66181-4