whopkins@csd.uwm.edu
Dec16-04, 08:07 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Continuing our discussion, we now consider the case where the\nconfiguration space coordinates q = (q^1,q^2,...) themselves form a\nclosed canonical sector (as per the definitions stated in the previous\narticle). This means that\n\n(1) They satisfy 2nd order equations of motion\ndq/dt = v, dv/dt = a(q,v)\n(2) The [q,q] commutators are all 0\n(3) The [q,v] commutators are all c-numbers,\nso that the [q,[q,v]], [v,[q,v]] commutators 0.\n\nThe matrix W = [q,v]/(i h-bar), as noted previously, is\nsymmetric; and as noted in the previous article, the\nbasic structure of a sector is closed under linear\ntransformations, with the W\'s transforming as a 2nd\norder tensor. Therefore, we may reduce W to the block\ndiagonal form\nW = block-diag(0, m^{-1})\nwhich effects the splitting of the configuration space\ninto a classical and quantum part (as per the definitions\nof the previous article).\n\nFor each coordinate Q of the classical sector, we already\nhave [Q,q^i] = 0, but now also [Q,v^i] = 0 = [V,q^i], where\nV = dQ/dt. Therefore the Q\'s are c-numbers. Differentiating\nthe second relation, we find that\n0 = [V,v^i] + [Q,a^i(q,v)] = [V,v^i]\nso the V\'s are c-numbers as well. Therefore, none of the\nq\'s or v\'s from the classical sector will enter any further\ninto the commutator relations, so that we may focus our\nattention entirely on the coordinates from the quantum\nsector.\n\nAn obvious candidate to try for the conjugate momentum is\nP = mv. In fact, working out the commutation relations,\nnoting that the m\'s are c-numbers, we find that\n[q^a,P_b] = m_{bc} [q^a,v^c]\n= i h-bar m_{bc} W^{ac}\n= i h-bar delta^a_b\nwhich is what we\'re looking for. But for the P\'s we get:\n[P_a,P_b] = m_{ac}m_{bd} [v^c,v^d]\n= i h-bar s_{ab}\nwhere we define\ns_{ab} = m_{ac} m_{bd} S^{cd}.\nThe equations of motion in terms of the q\'s and P\'s are:\nm dq/dt = P, dP/dt = F(q,P)\nwhere\nF(q,P) = m a(q,Wv)\nwhere the functional dependence in terms of the quantum\ncoordinates is made explicit.\n\nThe Jacobi-compatibility conditions now become\n[q^a,s_{bc}] = 0\n[P_a,s_{bc}] + [P_b,s_{ca}] + [P_c,s_{ab}] = 0.\n\nFor polynomials A(q,P), we may argue inductively that\n[q^a,A] = i h-bar dA/dp_a\nand similarly for polynomials in B(q) in q only, that\n[p_a,B] = -i h-bar dB/dq^a.\nAssuming the polynomials are dense in the algebra\ngenerated by the q\'s and v\'s (or q\'s and P\'s) and that\nthe product (and commutator) have suitable continuity\nproperties, this extends by continuity to all A in the\nsubalgebra generated by the q\'s and p\'s, and all B in\nthe subalgebra generated by the q\'s alone.\n\nLikely, this argument will not extend to the case where\nthe q\'s have an infinite number of degrees of freedom,\nsince it would apparently lead to a Stone - von Neumann\ntheorem for the Heisenberg algebra of infinite number\nof degrees of freedom (where we already know no such\nresult exists). So, this argument probably will only\napply for systems of finite number of degrees of\nfreedom. The extension to the general case of infinite\ndegrees of freedom may end up incorporating an additional consideration\nof superselection sectors.\n\nA result of these considerations, since the m\'s commute\nwith the q\'s and P\'s, is that\n0 = [q,m] = i h-bar dm/dP\nso m = m(q), from which it follows that\n0 = [P,m] = -i h-bar dm/dq\nso that the m\'s are independent of all the quantum\ncoordinates.\n\nThe Jacobi-constraints reduce to the form\n[q^a,s_{bc}] = 0\n[P_a,s_{bc}] + [P_b,s_{ca}] + [P_c,s_{ab}] = 0.\n\n>From the first, we get\nds_{bc}/dq^a = 0\nso that s = s(q), functions of q alone. From this,\nit follows that the second relation reduces to\nds_{ab}/dq^c + ds_{bc}/dq^a + ds_{ca}/dq^b = 0\nwhich, in conjunction with the anti-symmetry of s, is\nthe condition required for there to be an A_a(q) such\nthat\ns_{ab} = dA_b/dq^a - dA_a/dq^b.\n\nThis leads directly to a suitable definition for the\nconjugate momenta: p = mv + A. The commutator relations\nfor [q,p] retain the desired form\n[q^a,p_b] = i h-bar delta^a_b\nbut now those for the p\'s reduce to 0:\n[p_a,p_b] = 0,\nthus showing that the quantum sector is canonically\nquantized. The inductive argument outlined above, then,\napplies to both the p\'s and q\'s to yield the correspondence:\n[q,()] = i h-bar d/dp\n[p,()] = -i h-bar d/dq,\nwhich we\'ll freely use below.\n\nRedefining F = F + dA/dt, the equations of motion now\nbecome:\nm dq/dt + A = p, dp/dt = A(q,p)\n\nThe d/dt-compatibility requirements now become\nm_{ab} = m_{ba}\n[q^a,F_b] = -W^{ac}[p_a,A_c]\n[p_a,F_b] = [p_b,F_a].\nThe last two lead to the equations\ndF_b/dp_a = W^{ac} dA_c/dq^a\ndF_b/dq^a = dF_a/dq^b\nwhich are the conditions required for there to be a\nfunction U = U(q) such that\nF_a = -dU/dq^a, A_a = -m_{ab} dU/dp_b.\nSince the A = A(q), then dA/dp = 0, therefore the\nsecond derivatives d^2U/dpdp are all 0. Therefore,\nU is linear in the p\'s with\nU = a(q) + b^a(q) p_a.\nThus, the equations of motion may be cast into the form:\ndp_a/dt = -dU/dq^a\ndq^a/dt = W^{ab}p_b + b^a\n= d/dp_b(U + 1/2 W^{ab}p_a p_b)\nThis yields an appropriate Hamiltonian\nH = a(q) + b^a(q) p_a + 1/2 W^{ab} p_a p_b\nwith respect to which\ndp_a/dt = -dH/dq^a = [p_a,H]/(i h-bar)\ndq^a/dt = dH/dp_a = [q^a,H]/(i h-bar)\nThis extends, inductively, to all polynomials A(q,p) to\nyield\ni h-bar dA/dt = [A,H]\nand, by continuity, to the entire subalgebra generated\nby the p\'s and q\'s.\n\nAs a final note, we point out that for systems of finite\nnumber of degrees of freedom, since a type of Schur\'s\nlemma applies --\n[p,A] = 0 = [q,A] -> dA/dq = 0 = dA/dq\n-> A c-number independent of p, q\nthen for the equations in the classical sector\ndq/dt = v, dv/dt = a(q,v)\nsince the q\'s, v\'s and a\'s are all c-numbers, then the\nfunction a(q,v) depends only on the classical coordinates\nand is independent of the quantum sector; so that this\nsector is closed (using our definition of "closed" from\nthe previous article).\n\nTherefore, the classical sector must be regarded as\nEXTERNAL. However, the Hamiltonian in the quantum\nsector may dependend on the classical coordinates.\nTherefore, if one parametrizes the solution set to\nthe 2nd order system for the classical part, then each\nsolution index will select out a SUPERSELECTION sector\nin the state space for the quantum sector.\n\nIn Newtonian Physics, the coordinates and velocities are\nrelated by a law of inertia of the form\nm dq/dt = p, dp/dt = F(q,p)\nwhere the m\'s are constant. These m\'s will be the inverse\nof the W matrix on the quantized system, so that the matrix\nis constant, as well. So, the considerations above,\nrequiring [q,v]\'s to be c-numbers, apply generally to\nn-body systems in Newtonian dynamics.\n\nLikewise, in relativistic dynamics, one may write down a\nHamiltonian of the form\nH = 1/2 g^{ab}(q) p_a p_b\nwith the functions g = g(q) for special relativity being\nconstant. The W matrix will be a constant multiple of\nthe g matrix, so that the requirement [p,W] = 0 or\ndW/dq = 0 applies there as well. For a general\nrelativistic system, one loses [p,W] = 0 and [v,W] = 0\nand has only [q,W] = 0.\n\nTherefore, a natural extension to the considerations above\nis to enquire what follows if we relax the assumptions to\nthe form:\n(1) Equations of motion:\ndq/dt = v, dv/dt = a(q,v)\n(2) [q,q] = 0\n(3) [q,[q,v]] = 0, but not necessarily [v,[q,v]] = 0.\nThe delineation of the possibilities is left as an open\nissue.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Continuing our discussion, we now consider the case where the
configuration space coordinates q = (q^1,q^2,...) themselves form a
closed canonical sector (as per the definitions stated in the previous
article). This means that
(1) They satisfy 2nd order equations of motion
dq/dt = v, dv/dt = a(q,v)
(2) The [q,q] commutators are all
(3) The [q,v] commutators are all c-numbers,
so that the [q,[q,v]], [v,[q,v]] commutators .
The matrix W = [q,v]/(i h-bar), as noted previously, is
symmetric; and as noted in the previous article, the
basic structure of a sector is closed under linear
transformations, with the W's transforming as a 2nd
order tensor. Therefore, we may reduce W to the block
diagonal form
W = block-diag(0, m^{-1})
which effects the splitting of the configuration space
into a classical and quantum part (as per the definitions
of the previous article).
For each coordinate Q of the classical sector, we already
have [Q,q^i] = 0, but now also [Q,v^i] == [V,q^i], where
V = dQ/dt. Therefore the Q's are c-numbers. Differentiating
the second relation, we find that
= [V,v^i] + [Q,a^i(q,v)] = [V,v^i]
so the V's are c-numbers as well. Therefore, none of the
q's or v's from the classical sector will enter any further
into the commutator relations, so that we may focus our
attention entirely on the coordinates from the quantum
sector.
An obvious candidate to try for the conjugate momentum is
P = mv. In fact, working out the commutation relations,
noting that the m's are c-numbers, we find that
[q^a,P_b] = m_{bc} [q^a,v^c]
= i h-bar m_{bc} W^{ac}
= i h-bar \delta^a_b
which is what we're looking for. But for the P's we get:
[P_a,P_b] = m_{ac}m_{bd} [v^c,v^d]
= i h-bar s_{ab}
where we define
s_{ab} = m_{ac} m_{bd} S^{cd}.
The equations of motion in terms of the q's and P's are:
m dq/dt = P, dP/dt = F(q,P)
where
F(q,P) = m a(q,Wv)
where the functional dependence in terms of the quantum
coordinates is made explicit.
The Jacobi-compatibility conditions now become
[q^a,s_{bc}] =[P_a,s_{bc}] + [P_b,s_{ca}] + [P_c,s_{ab}] = .
For polynomials A(q,P), we may argue inductively that
[q^a,A] = i h-bar dA/dp_a
and similarly for polynomials in B(q) in q only, that
[p_a,B] = -i h-bar dB/dq^a.
Assuming the polynomials are dense in the algebra
generated by the q's and v's (or q's and P's) and that
the product (and commutator) have suitable continuity
properties, this extends by continuity to all A in the
subalgebra generated by the q's and p's, and all B in
the subalgebra generated by the q's alone.
Likely, this argument will not extend to the case where
the q's have an infinite number of degrees of freedom,
since it would apparently lead to a Stone - von Neumann
theorem for the Heisenberg algebra of infinite number
of degrees of freedom (where we already know no such
result exists). So, this argument probably will only
apply for systems of finite number of degrees of
freedom. The extension to the general case of infinite
degrees of freedom may end up incorporating an additional consideration
of superselection sectors.
A result of these considerations, since the m's commute
with the q's and P's, is that
= [q,m] = i h-bar dm/dP
so m = m(q), from which it follows that
= [P,m] = -i h-bar dm/dq
so that the m's are independent of all the quantum
coordinates.
The Jacobi-constraints reduce to the form
[q^a,s_{bc}] =[P_a,s_{bc}] + [P_b,s_{ca}] + [P_c,s_{ab}] = .
>From the first, we get
ds_{bc}/dq^a =
so that s = s(q), functions of q alone. From this,
it follows that the second relation reduces to
ds_{ab}/dq^c + ds_{bc}/dq^a + ds_{ca}/dq^b =
which, in conjunction with the anti-symmetry of s, is
the condition required for there to be an A_a(q) such
that
s_{ab} = dA_b/dq^a - dA_a/dq^b.
This leads directly to a suitable definition for the
conjugate momenta: p = mv + A. The commutator relations
for [q,p] retain the desired form
[q^a,p_b] = i h-bar \delta^a_b
but now those for the p's reduce to 0:
[p_a,p_b] = 0,
thus showing that the quantum sector is canonically
quantized. The inductive argument outlined above, then,
applies to both the p's and q's to yield the correspondence:
[q,()] = i h-bar d/dp[p,()] = -i h-bar d/dq,
which we'll freely use below.
Redefining F = F + dA/dt, the equations of motion now
become:
m dq/dt + A = p, dp/dt = A(q,p)
The d/dt-compatibility requirements now become
m_{ab} = m_{ba}[q^a,F_b] = -W^{ac}[p_a,A_c][p_a,F_b] = [p_b,F_a].
The last two lead to the equations
dF_b/dp_a = W^{ac} dA_c/dq^adF_b/dq^a = dF_a/dq^b
which are the conditions required for there to be a
function U = U(q) such that
F_a = -dU/dq^a, A_a = -m_{ab} dU/dp_b.
Since the A = A(q), then dA/dp = 0, therefore the
second derivatives d^{2U}/dpdp are all . Therefore,
U is linear in the p's with
U = a(q) + b^a(q) p_a.
Thus, the equations of motion may be cast into the form:
dp_a/dt = -dU/dq^adq^a/dt = W^{ab}p_b + b^a= d/dp_b(U + 1/2 W^{ab}p_a p_b)
This yields an appropriate Hamiltonian
H = a(q) + b^a(q) p_a + 1/2 W^{ab} p_a p_b
with respect to which
dp_a/dt = -dH/dq^a = [p_a,H]/(i h-bar)
dq^a/dt = dH/dp_a = [q^a,H]/(i h-bar)
This extends, inductively, to all polynomials A(q,p) to
yield
i h-bar dA/dt = [A,H]
and, by continuity, to the entire subalgebra generated
by the p's and q's.
As a final note, we point out that for systems of finite
number of degrees of freedom, since a type of Schur's
lemma applies --
[p,A] = = [q,A] -> dA/dq == dA/dq-> A c-number independent of p, q
then for the equations in the classical sector
dq/dt = v, dv/dt = a(q,v)
since the q's, v's and a's are all c-numbers, then the
function a(q,v) depends only on the classical coordinates
and is independent of the quantum sector; so that this
sector is closed (using our definition of "closed" from
the previous article).
Therefore, the classical sector must be regarded as
EXTERNAL. However, the Hamiltonian in the quantum
sector may dependend on the classical coordinates.
Therefore, if one parametrizes the solution set to
the 2nd order system for the classical part, then each
solution index will select out a SUPERSELECTION sector
in the state space for the quantum sector.
In Newtonian Physics, the coordinates and velocities are
related by a law of inertia of the form
m dq/dt = p, dp/dt = F(q,p)
where the m's are constant. These m's will be the inverse
of the W matrix on the quantized system, so that the matrix
is constant, as well. So, the considerations above,
requiring [q,v]'s to be c-numbers, apply generally to
n-body systems in Newtonian dynamics.
Likewise, in relativistic dynamics, one may write down a
Hamiltonian of the form
H = 1/2 g^{ab}(q) p_a p_b
with the functions g = g(q) for special relativity being
constant. The W matrix will be a constant multiple of
the g matrix, so that the requirement [p,W] = or
dW/dq = applies there as well. For a general
relativistic system, one loses [p,W] = and [v,W] =
and has only [q,W] = .
Therefore, a natural extension to the considerations above
is to enquire what follows if we relax the assumptions to
the form:
(1) Equations of motion:
dq/dt = v, dv/dt = a(q,v)
(2) [q,q] =
(3) [q,[q,v]] = 0, but not necessarily [v,[q,v]] = .
The delineation of the possibilities is left as an open
issue.
configuration space coordinates q = (q^1,q^2,...) themselves form a
closed canonical sector (as per the definitions stated in the previous
article). This means that
(1) They satisfy 2nd order equations of motion
dq/dt = v, dv/dt = a(q,v)
(2) The [q,q] commutators are all
(3) The [q,v] commutators are all c-numbers,
so that the [q,[q,v]], [v,[q,v]] commutators .
The matrix W = [q,v]/(i h-bar), as noted previously, is
symmetric; and as noted in the previous article, the
basic structure of a sector is closed under linear
transformations, with the W's transforming as a 2nd
order tensor. Therefore, we may reduce W to the block
diagonal form
W = block-diag(0, m^{-1})
which effects the splitting of the configuration space
into a classical and quantum part (as per the definitions
of the previous article).
For each coordinate Q of the classical sector, we already
have [Q,q^i] = 0, but now also [Q,v^i] == [V,q^i], where
V = dQ/dt. Therefore the Q's are c-numbers. Differentiating
the second relation, we find that
= [V,v^i] + [Q,a^i(q,v)] = [V,v^i]
so the V's are c-numbers as well. Therefore, none of the
q's or v's from the classical sector will enter any further
into the commutator relations, so that we may focus our
attention entirely on the coordinates from the quantum
sector.
An obvious candidate to try for the conjugate momentum is
P = mv. In fact, working out the commutation relations,
noting that the m's are c-numbers, we find that
[q^a,P_b] = m_{bc} [q^a,v^c]
= i h-bar m_{bc} W^{ac}
= i h-bar \delta^a_b
which is what we're looking for. But for the P's we get:
[P_a,P_b] = m_{ac}m_{bd} [v^c,v^d]
= i h-bar s_{ab}
where we define
s_{ab} = m_{ac} m_{bd} S^{cd}.
The equations of motion in terms of the q's and P's are:
m dq/dt = P, dP/dt = F(q,P)
where
F(q,P) = m a(q,Wv)
where the functional dependence in terms of the quantum
coordinates is made explicit.
The Jacobi-compatibility conditions now become
[q^a,s_{bc}] =[P_a,s_{bc}] + [P_b,s_{ca}] + [P_c,s_{ab}] = .
For polynomials A(q,P), we may argue inductively that
[q^a,A] = i h-bar dA/dp_a
and similarly for polynomials in B(q) in q only, that
[p_a,B] = -i h-bar dB/dq^a.
Assuming the polynomials are dense in the algebra
generated by the q's and v's (or q's and P's) and that
the product (and commutator) have suitable continuity
properties, this extends by continuity to all A in the
subalgebra generated by the q's and p's, and all B in
the subalgebra generated by the q's alone.
Likely, this argument will not extend to the case where
the q's have an infinite number of degrees of freedom,
since it would apparently lead to a Stone - von Neumann
theorem for the Heisenberg algebra of infinite number
of degrees of freedom (where we already know no such
result exists). So, this argument probably will only
apply for systems of finite number of degrees of
freedom. The extension to the general case of infinite
degrees of freedom may end up incorporating an additional consideration
of superselection sectors.
A result of these considerations, since the m's commute
with the q's and P's, is that
= [q,m] = i h-bar dm/dP
so m = m(q), from which it follows that
= [P,m] = -i h-bar dm/dq
so that the m's are independent of all the quantum
coordinates.
The Jacobi-constraints reduce to the form
[q^a,s_{bc}] =[P_a,s_{bc}] + [P_b,s_{ca}] + [P_c,s_{ab}] = .
>From the first, we get
ds_{bc}/dq^a =
so that s = s(q), functions of q alone. From this,
it follows that the second relation reduces to
ds_{ab}/dq^c + ds_{bc}/dq^a + ds_{ca}/dq^b =
which, in conjunction with the anti-symmetry of s, is
the condition required for there to be an A_a(q) such
that
s_{ab} = dA_b/dq^a - dA_a/dq^b.
This leads directly to a suitable definition for the
conjugate momenta: p = mv + A. The commutator relations
for [q,p] retain the desired form
[q^a,p_b] = i h-bar \delta^a_b
but now those for the p's reduce to 0:
[p_a,p_b] = 0,
thus showing that the quantum sector is canonically
quantized. The inductive argument outlined above, then,
applies to both the p's and q's to yield the correspondence:
[q,()] = i h-bar d/dp[p,()] = -i h-bar d/dq,
which we'll freely use below.
Redefining F = F + dA/dt, the equations of motion now
become:
m dq/dt + A = p, dp/dt = A(q,p)
The d/dt-compatibility requirements now become
m_{ab} = m_{ba}[q^a,F_b] = -W^{ac}[p_a,A_c][p_a,F_b] = [p_b,F_a].
The last two lead to the equations
dF_b/dp_a = W^{ac} dA_c/dq^adF_b/dq^a = dF_a/dq^b
which are the conditions required for there to be a
function U = U(q) such that
F_a = -dU/dq^a, A_a = -m_{ab} dU/dp_b.
Since the A = A(q), then dA/dp = 0, therefore the
second derivatives d^{2U}/dpdp are all . Therefore,
U is linear in the p's with
U = a(q) + b^a(q) p_a.
Thus, the equations of motion may be cast into the form:
dp_a/dt = -dU/dq^adq^a/dt = W^{ab}p_b + b^a= d/dp_b(U + 1/2 W^{ab}p_a p_b)
This yields an appropriate Hamiltonian
H = a(q) + b^a(q) p_a + 1/2 W^{ab} p_a p_b
with respect to which
dp_a/dt = -dH/dq^a = [p_a,H]/(i h-bar)
dq^a/dt = dH/dp_a = [q^a,H]/(i h-bar)
This extends, inductively, to all polynomials A(q,p) to
yield
i h-bar dA/dt = [A,H]
and, by continuity, to the entire subalgebra generated
by the p's and q's.
As a final note, we point out that for systems of finite
number of degrees of freedom, since a type of Schur's
lemma applies --
[p,A] = = [q,A] -> dA/dq == dA/dq-> A c-number independent of p, q
then for the equations in the classical sector
dq/dt = v, dv/dt = a(q,v)
since the q's, v's and a's are all c-numbers, then the
function a(q,v) depends only on the classical coordinates
and is independent of the quantum sector; so that this
sector is closed (using our definition of "closed" from
the previous article).
Therefore, the classical sector must be regarded as
EXTERNAL. However, the Hamiltonian in the quantum
sector may dependend on the classical coordinates.
Therefore, if one parametrizes the solution set to
the 2nd order system for the classical part, then each
solution index will select out a SUPERSELECTION sector
in the state space for the quantum sector.
In Newtonian Physics, the coordinates and velocities are
related by a law of inertia of the form
m dq/dt = p, dp/dt = F(q,p)
where the m's are constant. These m's will be the inverse
of the W matrix on the quantized system, so that the matrix
is constant, as well. So, the considerations above,
requiring [q,v]'s to be c-numbers, apply generally to
n-body systems in Newtonian dynamics.
Likewise, in relativistic dynamics, one may write down a
Hamiltonian of the form
H = 1/2 g^{ab}(q) p_a p_b
with the functions g = g(q) for special relativity being
constant. The W matrix will be a constant multiple of
the g matrix, so that the requirement [p,W] = or
dW/dq = applies there as well. For a general
relativistic system, one loses [p,W] = and [v,W] =
and has only [q,W] = .
Therefore, a natural extension to the considerations above
is to enquire what follows if we relax the assumptions to
the form:
(1) Equations of motion:
dq/dt = v, dv/dt = a(q,v)
(2) [q,q] =
(3) [q,[q,v]] = 0, but not necessarily [v,[q,v]] = .
The delineation of the possibilities is left as an open
issue.