In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, 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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
should be equal to zero.
For example, in electromagnetism, the equation for the
Gauss' law
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
:
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
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
, 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
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
.
See also
*
First class constraints
A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanis ...
References
Quantum mechanics
Quantum field theory
String theory
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