Constraint algebra

In theoretical physics, a constraint algebra is a linear space of all constraints and all of their polynomial functions or functionals whose action on the physical vectors of the Hilbert space should be equal to zero.^[1][2]

For example, in electromagnetism, the equation for the Gauss' law

${\displaystyle \nabla \cdot {\vec {E}}=\rho }$

is an equation of motion that does not include any time derivatives. This is why it is counted as a constraint, not a dynamical equation of motion. In quantum electrodynamics, one first constructs a Hilbert space in which Gauss' law does not hold automatically. The true Hilbert space of physical states is constructed as a subspace of the original Hilbert space of vectors that satisfy

${\displaystyle (\nabla \cdot {\vec {E}}(x)-\rho (x))|\psi \rangle =0.}$

In more general theories, the constraint algebra may be a noncommutative algebra.

See also

• First class constraints