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siddhartha04
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The requirements, posed on a system given<br />
by the configuration coordinates<br />
q(t) = (q^1(t), q^2(t), ...)<br />
that<br />
(1) they be subject to 2nd-order equations of motion:<br />
q'(t) = v(t), v'(t) = a(q(t),v(t))<br />
and<br />
(2) have a classical configuration space at each time:<br />
[q^i(t), q^j(t)] =<br />
is nearly enough, alone, to derive the key properties of<br />
quantum mechanics, such as the Heisenberg Uncertainty<br />
Principle and Heisenberg equations of motion.<br />
<br />
This feature was first discovered in the early 1990's,<br />
where it was shown that if the matrix<br />
W^{ij} = [q^i, v^j]/(i h-bar)<br />
approaches a non-singular matrix as h-bar -> 0, then<br />
the equations of motion must be so constrained that<br />
the equations of motion yield a Hamiltonian system in<br />
the classical limit, with W^{ij} being the inverse<br />
mass matrix (i.e., the hessian d^{2H}/d(p_i)d(p_j)).<br />
<br />
If the W's, instead, are assumed to be c-numbers,<br />
allowing the matrix to be singular, then the result<br />
is that the system splits into the direct sum of a<br />
classical sector, given by c-number coordinates and<br />
velocities, and a quantum sector which is canonically<br />
quantized with respect to a Hamiltonian which is<br />
constrained to be of a form as a polynomial of order 2<br />
in the conjugate momenta, reducible to the form:<br />
H = sum (1/2 W^{ij}(q) p_i p_j) + U(q).<br />
<br />
The requirement that (1) and (2) be compatible with one<br />
another is actually quite strong. For general<br />
functions A(q), B(q), ... of the configuration coordinates,<br />
define<br />
W^{AB} = [A, dB/dt]/(i h-bar)<br />
S^{AB} = [dA/dt, dB/dt]/(i h-bar)<br />
note then that<br />
S^{AB} = -S^{BA}.<br />
For general coordinate functions, given the commutativity<br />
of the q's, it also follows that [A,B] = .<br />
<br />
Consistency with time derivatives already implies<br />
>From d/dt [A,B]: W^{AB} = W^{BA}<br />
>From d/dt [A,B']:<br />
i h-bar dW^{AB}/dt = 1/2 ([A,B''] + [B,A''])<br />
i h-bar S^{AB} = 1/2 ([B,A''] - [A,B''])<br />
>From d/dt [A',B']:<br />
i h-bar dS^{AB}/dt = [A',B''] - [B',A''],<br />
using primes to denote time derivatives.<br />
<br />
The Jacobi identities imply:<br />
>From [q,[q,q]]: Nothing new<br />
>From [q,[q,v]]: [A,W^{BC}] = [B,W^{AC}]<br />
>From [q,[v,v]]: [A,S^{BC}] = [B',W^{AC}] - [C',W^{AB}]<br />
>From [v,[v,v]]: [A',S^{BC}] + [B',S^{CA}] + [C',S^{AB}] = .<br />
<br />
So, with these preliminaries, we'll show how the result<br />
follows.<br />
<br />
For functions A(q), B(q), ... over configuration space,<br />
define the following:<br />
<br />
A is a classical coordinate if [A,A'] =<br />
A is a quantum coordinate if [A,A'] is not .<br />
A is canonical if [A,A'] is a c-number.<br />
<br />
A classical sector S is a linear space of functions over<br />
Q whose members are all classical. S is called a quantum<br />
sector if all of its members are quantum. It is called<br />
canonical, they are all canonical.<br />
<br />
Since the sector S is to be closed under linear<br />
combinations, then consider the case of the combination<br />
(A + zB) with A, B in S. If S is classical, one has<br />
= [A+zB,A'+z'B+zB'] = z (W^{AB} + W^{BA}).<br />
Taking z = 1/2, noting that W^{BA} = W^{AB}, it follows<br />
that W^{AB} = . The W matrix is over a classical<br />
sector.<br />
<br />
If S is quantum, or canonical, then by similar arguments<br />
it follows that W is respectively non-singular over S<br />
or comprises a matrix of c-numbers over S.<br />
<br />
Finally, a sector S is called closed if its coordinates<br />
have accelerations given as functions of the other<br />
members of S. For the case of a finite dimensional<br />
sector S with basis (A1,...,An), the functions would<br />
be of the form:<br />
A'' = a^{A}(A1,...,An,A1',...,An').<br />
<br />
The result is: a closed canonical sector splits up into<br />
a classical sector and a quantum sector with the latter<br />
canonically quantized with respect to a Hamiltonian that<br />
is a polynomial of order 2 in the conjugate momenta.<br />
<br />
---------<br />
<br />
First, consider the effect of an invertible linear<br />
transformation on the coordinates<br />
Q^a = sum Z^{a_i} q^i.<br />
We'll adopt the summation convention here and below and<br />
write this more simply, also in matrix form, as:<br />
Q = Z q.<br />
Then<br />
V = Z v + Z' qV' = Z a(q,v) + 2 Z' v + Z'' q = A(Q,V)<br />
where<br />
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<br />
Writing the commutators in matrix form, we get:<br />
[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]<br />
= Z S Z^T + (Z' W Z^T - Z W Z'^T)<br />
using ()^T to denote transpose.<br />
<br />
A closed sector thus transforms linearly to a closed<br />
sector, with the W's behaving as 2nd order tensors<br />
under the transformation.<br />
<br />
---------<br />
<br />
For canonical sectors, since one has:<br />
[A,W^{BC}] == [A',W^{BC}],<br />
then the Jacobi conditions substantially reduce to the<br />
form:<br />
[A,S^{BC}] = .<br />
and differentiating:<br />
[A',S^{BC}] = -[A,S^{BC}'].<br />
Additionally, one has (after differentiating):<br />
[A',W^{BC}] + [A,W^{BC}'] = -> [A,W^{BC}'] =<br />
and, if the sector is closed:<br />
[A'',W^{BC}] + [A',W^{BC}'] = -> [A',W^{BC}'] = .<br />
<br />
Consider the general case, now, where the coordinates<br />
themselves (q^1,...,q^n) form a closed canonical sector,<br />
with equations of motion as given above.<br />
<br />
We'll see how this works out in detail in the remainder<br />
of the discussion, which will follow in a later article.
