By Maurice A. de Gosson
This publication 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. hence, Born–Jordan quantization presents the single bodily constant quantization scheme, instead of the Weyl quantization familiar through physicists. during this booklet we boost Born–Jordan quantization from an operator-theoretical viewpoint, and study extensive the conceptual modifications among the 2 schemes. We talk about a number of bodily encouraged methods, particularly the Feynman-integral standpoint. One vital and fascinating function of Born-Jordan quantization is that it's not one-to-one: there are infinitely many classical observables whose quantization is zero.
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Extra resources for Born-Jordan Quantization: Theory and Applications
45, Footnote 76). Let H be a real-valued function; we assume for convenience that C ∞ (R2n × R), but most of what follows remains true if we assume less stringent differentiability conditions (see Abraham and Marsden  for a discussion of sufficient smoothness requirements). A. 1007/978-3-319-27902-2_4 37 38 4 Basic Hamiltonian Mechanics dx j ∂H = (x1 , . . , xn , p1 , . . , pn , t) dt ∂ pj dp j ∂H =− (x1 , . . , xn , p1 , . . 2) where j = 1, . . 3) where ∇z is the gradient in the 2n variables x1 , .
31) we get S2 (x, x , t, t ) = m m 2 ω2 (x − x )2 − (x + x )2 t 2 t 8 and we have here S(x, x , t, t ) − S2 (x, x , t, t ) = O( t). The “mid-point rules” are thus incorrect even to first order, and should therefore be avoided in any rigorous argument. 2 A Correct Short-Time Approximation Assume that H is a classical Hamiltonian function of the physical type n H (x, p, t) = j=1 p 2j 2m j + V (x, t). 34) 48 4 Basic Hamiltonian Mechanics where V (x, x , t ) is the average 1 V (x, x , t ) = V (τ x + (1 − τ )x , t )dτ .
By inserting this expression in the Hamilton– Jacobi equation ∂W ∂t 1 ∂W + ∂t 2m 2 + V = 0. This leads to the equation W1 + (x − x ) ∂W1 +V =0 ∂x and to similar equations for the lower order terms W2 , W3 . . One next observes that the only smooth solution of the equation above is x W1 (x, x , t ) = − 1 x−x V (x , t )d x = −V (x, x , t ) x where the second equality follows from the first by making the change of variables x = τ x + (1 − τ )x . Example 13 Consider again the one-dimensional oscillator Hamiltonian H (x, p) = 1 ( p 2 + m 2 ω 2 x 2 ).