- #1

siddhartha04

- 1

- 0

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.[/tex]