Canonical commutation relation

In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, between the position operator x and momentum operator px in the x direction of a point particle in one dimension, where = x px − px x is the commutator of x and px , i is the imaginary unit, and ℏ is the reduced Planck's constant h/2π . In general, position and momentum are vectors of operators and their commutation relation between different components of position and momentum can be expressed as where δ i j {displaystyle delta _{ij}} is the Kronecker delta. This relation is attributed to Max Born (1925), who called it a 'quantum condition' serving as a postulate of the theory; it was noted by E. Kennard (1927) to imply the Heisenberg uncertainty principle. The Stone–von Neumann theorem gives a uniqueness result for operators satisfying (an exponentiated form of) the canonical commutation relation. By contrast, in classical physics, all observables commute and the commutator would be zero. However, an analogous relation exists, which is obtained by replacing the commutator with the Poisson bracket multiplied by iℏ: This observation led Dirac to propose that the quantum counterparts f̂, ĝ of classical observables f, g satisfy In 1946, Hip Groenewold demonstrated that a general systematic correspondence between quantum commutators and Poisson brackets could not hold consistently. However, he did appreciate that such a systematic correspondence does, in fact, exist between the quantum commutator and a deformation of the Poisson bracket, the Moyal bracket, and, in general, quantum operators and classical observables and distributions in phase space. He thus finally elucidated the correspondence mechanism, the Wigner–Weyl transform, that underlies an alternate equivalent mathematical representation of quantum mechanics known as deformation quantization. The group H 3 ( R ) {displaystyle H_{3}(mathbb {R} )} generated by exponentiation of the 3-dimensional Lie algebra determined by the commutation relation [ x ^ , p ^ ] = i ℏ {displaystyle =ihbar } is called the Heisenberg group. This group can be realized as the group of 3 × 3 {displaystyle 3 imes 3} upper triangular matrices with ones on the diagonal. According to the standard mathematical formulation of quantum mechanics, quantum observables such as x ^ {displaystyle {hat {x}}} and p ^ {displaystyle {hat {p}}} should be represented as self-adjoint operators on some Hilbert space. It is relatively easy to see that two operators satisfying the above canonical commutation relations cannot both be bounded. Certainly, if x ^ {displaystyle {hat {x}}} and p ^ {displaystyle {hat {p}}} were trace class operators, the relation Tr ⁡ ( A B ) = Tr ⁡ ( B A ) {displaystyle operatorname {Tr} (AB)=operatorname {Tr} (BA)} gives a nonzero number on the right and zero on the left.