By Maurice A. de Gosson
This e-book provides a complete mathematical learn of the operators at the back of the Born–Jordan quantization scheme. The Schrödinger and Heisenberg photos of quantum mechanics are identical provided that the Born–Jordan scheme is used. therefore, Born–Jordan quantization presents the one bodily constant quantization scheme, instead of the Weyl quantization generic via physicists. during this e-book we increase Born–Jordan quantization from an operator-theoretical standpoint, and study extensive the conceptual variations among the 2 schemes. We talk about numerous bodily prompted methods, specifically the Feynman-integral perspective. One vital and exciting function of Born-Jordan quantization is that it isn't one-to-one: there are infinitely many classical observables whose quantization is zero.
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Shows that the key to their approach lies in the differentiation rule for products of non-commuting variables. They actually give two definitions, and prove thereafter that both coincide if and only if one makes an essential assumption on the ordering of the quantization of monomials. 27) m=1 (δ r k the Kronecker delta). 26) and form the product of all the following factors, and thereafter the product of the preceding factors (in that order). 28) x r −1− j p s x j . 29) =0 r −1 = 1 j=0 The second definition (explicitly given in formula (3) of ) is similar to that of an ordinary partial derivative: ∂y ∂ yk f (· · ·, yk + α, · · ·) .
27) m=1 (δ r k the Kronecker delta). 26) and form the product of all the following factors, and thereafter the product of the preceding factors (in that order). 28) x r −1− j p s x j . 29) =0 r −1 = 1 j=0 The second definition (explicitly given in formula (3) of ) is similar to that of an ordinary partial derivative: ∂y ∂ yk f (· · ·, yk + α, · · ·) . 29) become ∂ ( ps x r ) ∂p ∂ s r (p x ) ∂x = s p s−1 x r 2 = r p s x r −1 . 2 Their next step consists in identifying both notions of partial derivative; more specifically they want that the quantization H (still to be defined) of a Hamiltonian function satisfies the equalities ∂H ∂p = 1 ∂H ∂p , 2 ∂H ∂x ∂H ∂x = 1 .
It is proven there, among other properties, that every symplectic matrix can be written as a product of exactly two free symplectic matrices. 2 Hamilton’s Two-Point Characteristic Function The notion of generating functions extends to arbitrary free canonical transformations, in particular to Hamiltonian flows. 11) γ H (x , p ) where the integral is calculated along the phase-space trajectory s −→ f s,t joining (x , p ) at time t to (x, p) at time t is a generating function, that is: (x, p) = f t,tH (x , p ) ⇐⇒ p = ∇x W (x, x , t, t ) .