Constraint algebra explained

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

\nabla ⋅ \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(\nabla ⋅ \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

Notes and References

1. Gambini . Rodolfo . Lewandowski . Jerzy . Marolf . Donald . Pullin . Jorge . 1998-02-01 . On the consistency of the constraint algebra in spin network quantum gravity . International Journal of Modern Physics D . 07 . 1 . 97–109 . 10.1142/S0218271898000103 . gr-qc/9710018 . 1998IJMPD...7...97G . 3072598 . 0218-2718.
2. Thiemann . Thomas . 2006-03-14 . Quantum spin dynamics: VIII. The master constraint . Classical and Quantum Gravity . en . 23 . 7 . 2249–2265 . 10.1088/0264-9381/23/7/003 . gr-qc/0510011 . 2006CQGra..23.2249T . 11858/00-001M-0000-0013-4B4E-7 . 29095312 . 0264-9381.