Deriving the Quantum Properties of a General Configuration Space

[siddhartha04]

$$The requirements, posed on a system given by the configuration coordinates q(t) = (q^1(t), q^2(t), ...) that (1) they be subject to 2nd-order equations of motion: q'(t) = v(t), v'(t) = a(q(t),v(t)) and (2) have a classical configuration space at each time: [q^i(t), q^j(t)] = is nearly enough, alone, to derive the key properties of quantum mechanics, such as the Heisenberg Uncertainty Principle and Heisenberg equations of motion. This feature was first discovered in the early 1990's, where it was shown that if the matrix W^{ij} = [q^i, v^j]/(i h-bar) approaches a non-singular matrix as h-bar -> 0, then the equations of motion must be so constrained that the equations of motion yield a Hamiltonian system in the classical limit, with W^{ij} being the inverse mass matrix (i.e., the hessian d^{2H}/d(p_i)d(p_j)). If the W's, instead, are assumed to be c-numbers, allowing the matrix to be singular, then the result is that the system splits into the direct sum of a classical sector, given by c-number coordinates and velocities, and a quantum sector which is canonically quantized with respect to a Hamiltonian which is constrained to be of a form as a polynomial of order 2 in the conjugate momenta, reducible to the form: H = sum (1/2 W^{ij}(q) p_i p_j) + U(q). The requirement that (1) and (2) be compatible with one another is actually quite strong. For general functions A(q), B(q), ... of the configuration coordinates, define W^{AB} = [A, dB/dt]/(i h-bar) S^{AB} = [dA/dt, dB/dt]/(i h-bar) note then that S^{AB} = -S^{BA}. For general coordinate functions, given the commutativity of the q's, it also follows that [A,B] = . Consistency with time derivatives already implies >From d/dt [A,B]: W^{AB} = W^{BA} >From d/dt [A,B']: i h-bar dW^{AB}/dt = 1/2 ([A,B''] + [B,A'']) i h-bar S^{AB} = 1/2 ([B,A''] - [A,B'']) >From d/dt [A',B']: i h-bar dS^{AB}/dt = [A',B''] - [B',A''], using primes to denote time derivatives. The Jacobi identities imply: >From [q,[q,q]]: Nothing new >From [q,[q,v]]: [A,W^{BC}] = [B,W^{AC}] >From [q,[v,v]]: [A,S^{BC}] = [B',W^{AC}] - [C',W^{AB}] >From [v,[v,v]]: [A',S^{BC}] + [B',S^{CA}] + [C',S^{AB}] = . So, with these preliminaries, we'll show how the result follows. For functions A(q), B(q), ... over configuration space, define the following: A is a classical coordinate if [A,A'] = A is a quantum coordinate if [A,A'] is not . A is canonical if [A,A'] is a c-number. A classical sector S is a linear space of functions over Q whose members are all classical. S is called a quantum sector if all of its members are quantum. It is called canonical, they are all canonical. Since the sector S is to be closed under linear combinations, then consider the case of the combination (A + zB) with A, B in S. If S is classical, one has = [A+zB,A'+z'B+zB'] = z (W^{AB} + W^{BA}). Taking z = 1/2, noting that W^{BA} = W^{AB}, it follows that W^{AB} = . The W matrix is over a classical sector. If S is quantum, or canonical, then by similar arguments it follows that W is respectively non-singular over S or comprises a matrix of c-numbers over S. Finally, a sector S is called closed if its coordinates have accelerations given as functions of the other members of S. For the case of a finite dimensional sector S with basis (A1,...,An), the functions would be of the form: A'' = a^{A}(A1,...,An,A1',...,An'). The result is: a closed canonical sector splits up into a classical sector and a quantum sector with the latter canonically quantized with respect to a Hamiltonian that is a polynomial of order 2 in the conjugate momenta. --------- First, consider the effect of an invertible linear transformation on the coordinates Q^a = sum Z^{a_i} q^i. We'll adopt the summation convention here and below and write this more simply, also in matrix form, as: Q = Z q. Then V = Z v + Z' qV' = Z a(q,v) + 2 Z' v + Z'' q = A(Q,V) where A(Q,V) = Z a(Z^{-1}Q,Z^{-1}V)+ 2 Z' Z^{-1} V+ (Z'' Z^{-1} - 2 Z' Z^{-1} Z' Z^{-1}) Q Writing the commutators in matrix form, we get: [Q,Q] = [Zq,Zq] = Z [q,q] Z^T = W -> [Q,V] = [Zq, Zv + Z'q] = Z W Z^TS -> [V,V] = [Zv + Z'q, Zv + Z'q] = Z S Z^T + (Z' W Z^T - Z W Z'^T) using ()^T to denote transpose. A closed sector thus transforms linearly to a closed sector, with the W's behaving as 2nd order tensors under the transformation. --------- For canonical sectors, since one has: [A,W^{BC}] == [A',W^{BC}], then the Jacobi conditions substantially reduce to the form: [A,S^{BC}] = . and differentiating: [A',S^{BC}] = -[A,S^{BC}']. Additionally, one has (after differentiating): [A',W^{BC}] + [A,W^{BC}'] = -> [A,W^{BC}'] = and, if the sector is closed: [A'',W^{BC}] + [A',W^{BC}'] = -> [A',W^{BC}'] = . Consider the general case, now, where the coordinates themselves (q^1,...,q^n) form a closed canonical sector, with equations of motion as given above. We'll see how this works out in detail in the remainder of the discussion, which will follow in a later article.$$

 

[selfAdjoint]

You have the tex box around the text of your message instead of just around the formulas. And this message doesn't belong in this subforum anyway.

 

[mattmns]

You can use preview post to see how your post is going to look, and if your latex images are going to come out correctly.