Linear space of all constraints on a Hilbert space
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
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
In more general theories, the constraint algebra may be a noncommutative algebra.
^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. arXiv:gr-qc/9710018. Bibcode:1998IJMPD...7...97G. doi:10.1142/S0218271898000103. ISSN 0218-2718. S2CID 3072598.
^Thiemann, Thomas (2006-03-14). "Quantum spin dynamics: VIII. The master constraint". Classical and Quantum Gravity. 23 (7): 2249–2265. arXiv:gr-qc/0510011. Bibcode:2006CQGra..23.2249T. doi:10.1088/0264-9381/23/7/003. hdl:11858/00-001M-0000-0013-4B4E-7. ISSN 0264-9381. S2CID 29095312.
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