Constraint algebra

{{Short description|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.{{Cite journal |last1=Gambini |first1=Rodolfo |last2=Lewandowski |first2=Jerzy |last3=Marolf |first3=Donald |last4=Pullin |first4=Jorge |date=1998-02-01 |title=On the consistency of the constraint algebra in spin network quantum gravity |url=https://www.worldscientific.com/doi/abs/10.1142/S0218271898000103 |journal=International Journal of Modern Physics D |volume=07 |issue=1 |pages=97–109 |doi=10.1142/S0218271898000103 |arxiv=gr-qc/9710018 |bibcode=1998IJMPD...7...97G |s2cid=3072598 |issn=0218-2718}}{{Cite journal |last=Thiemann |first=Thomas |date=2006-03-14 |title=Quantum spin dynamics: VIII. The master constraint |url=https://doi.org/10.1088/0264-9381/23/7/003 |journal=Classical and Quantum Gravity |language=en |volume=23 |issue=7 |pages=2249–2265 |doi=10.1088/0264-9381/23/7/003 |arxiv=gr-qc/0510011 |bibcode=2006CQGra..23.2249T |hdl=11858/00-001M-0000-0013-4B4E-7 |s2cid=29095312 |issn=0264-9381}}

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

:$\backslash nabla\backslash cdot\; \backslash vec\; E\; =\; \backslash rho$

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

:$(\backslash nabla\backslash cdot\; \backslash vec\; E(x)\; -\; \backslash rho(x))\; |\backslash psi\backslash rangle\; =\; 0.$

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