Multiplicative quantum number
Encyclopedia
In quantum field theory
, multiplicative quantum numbers are conserved quantum number
s 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 Z2 give rise to multiplicative quantum numbers. This group consists of an operation, P, whose square is the identity, P2 = 1. Thus, all symmetries which are mathematically similar to parity (physics)
give rise to multiplicative quantum numbers.
In 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(2iπ 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 Z2 finds the widest possible use.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, multiplicative quantum numbers are conserved quantum number
Quantum number
Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...
s 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
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
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
Noether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
). Symmetry groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
which are examples of the abstract group called Z2 give rise to multiplicative quantum numbers. This group consists of an operation, P, whose square is the identity, P2 = 1. Thus, all symmetries which are mathematically similar to parity (physics)
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
give rise to multiplicative quantum numbers.
In principle, multiplicative quantum numbers can be defined for any Abelian
Abelian
In mathematics, Abelian refers to any of number of different mathematical concepts named after Niels Henrik Abel:- Group theory :*Abelian group, a group in which the binary operation is commutative...
group. An example would be to trade the electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
, Q, (related to the Abelian group U(1) of electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
), for the new quantum number exp(2iπ 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 Z2 finds the widest possible use.