Search results
Results From The WOW.Com Content Network
In a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There are other general examples, as well: it occurs in the theory of Lie algebras , where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this ...
The contravariant four-momentum of a particle with relativistic energy E and three-momentum p = (p x, p y, p z) = γmv, where v is the particle's three-velocity and γ the Lorentz factor, is = (,,,) = (,,,). The quantity mv of above is the ordinary non-relativistic momentum of the particle and m its rest mass. The four-momentum is useful in ...
Another useful result is Poisson's theorem, which states that if two quantities and are constants of motion, so is their Poisson bracket {,}. A system with n degrees of freedom, and n constants of motion, such that the Poisson bracket of any pair of constants of motion vanishes, is known as a completely integrable system .
Since the Dirac bracket respects the constraints, one need not be careful about evaluating all brackets before using any weak equations, as is the case with the Poisson bracket. Note that while the Poisson bracket of bosonic (Grassmann even) variables with itself must vanish, the Poisson bracket of fermions represented as a Grassmann variables ...
The three-dimensional momentum vector p is associated with a four-dimensional vector on a three-dimensional unit sphere = + ^ + + = ^ +, where ^ is the unit vector along the new w axis. The transformation mapping p to η can be uniquely inverted; for example, the x component of the momentum equals p x = p 0 η x 1 − η w , {\displaystyle p_{x ...
In 1946, Hip Groenewold demonstrated that a general systematic correspondence between quantum commutators and Poisson brackets could not hold consistently. [4] [5] However, he further appreciated that such a systematic correspondence does, in fact, exist between the quantum commutator and a deformation of the Poisson bracket, today called the ...
The symplectic structure induces a Poisson bracket. The Poisson bracket gives the space of functions on the manifold the structure of a Lie algebra. If F and G are smooth functions on M then the smooth function ω(J(dF), J(dG)) is properly defined; it is called a Poisson bracket of functions F and G and is denoted {F, G}. The Poisson bracket ...
The subleading terms are all encoded in the Moyal bracket, the suitable quantum deformation of the Poisson bracket.) In general, for the quantities (observables) involved, and providing the arguments of such brackets, ħ -deformations are highly nonunique—quantization is an "art", and is specified by the physical context.