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Constraint algebra

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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.

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

∇ ⋅ E → = ρ {\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

( ∇ ⋅ E → ( x ) − ρ ( x ) ) | ψ ⟩ = 0. {\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

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. arXiv:gr-qc/9710018. Bibcode:1998IJMPD...7...97G. doi:10.1142/S0218271898000103. ISSN 0218-2718. S2CID 3072598.
  2. 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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