whopkins@csd.uwm.edu
Dec19-04, 06:52 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>Charles Torre wrote:\n> Anybody want to bring us up to date on the status of\n> quantization without a Lagrangian?\n>\n> Charles Torre\n\nBasically, what I do in the 2 articles "Ab Initio Derivation of\nQuantum Mechanics" is show that the d/dt-compatibility and\nJacobi-compatibility conditions that arise from putting together\nthe equations of motion dq/dt = v, dv/dt = a(q,v); with the\nequal-time commutators [q(t),q(t)] (matrix form) = 0 for all time\nt is the non-commutative analogue of the Helmholz conditions that\nin classical dynamics define the existence of a Lagrangian. This\ntakes the 1991 JMP paper "No Lagrangian? No Quantization!" one\nstep further, where the authors there had to resort to an explicit\nmention of a classical limit to get their results. Likewise, I\'m\nrestricting attention to c-number [q,v] commutators, because the\nmore general case requires much deeper analysis using Lie algebras.\nA less restricted subset coming from relaxing the assumptions\neither to (a) q\'s have infinite number of degrees of freedom (then\nthe critical use of a Schur\'s Lemma is blocked) or (b) [q,v]\'s\nallowed to vary with q (which captures n-body systems in GR),\nin which case you need to consider the centralizer of the\nZ_<q>(<q,v>) of the q\'s in the larger algebra generated by the\nq\'s and v\'s. In general dim(Z_<q>(<q,v>)) > dim(<q>), and this\nspace will factor into <q> x <infinite-mass mode subspace>.\n\nThere may be a paper coming out of these deliberations ("On the\nQuantum Dynamics of Moving Bodies" :))\n\nThe singular Lagrangian gives you a singular matrix for [q,v]^{-1}\nwhich are the 0-mass modes, interpreted as constraints. This,\nin contrast, gives you in general a "singular Hamiltonian",\nwith a singular [q,v], and classical "infinite mass" modes. These\nextra modes need not be suject to any Lagrangian and their evolution\ncan\'t make reference to any of the quantum coordinates, except those\n(in the case of a system of infinite number of degrees of freedom)\nthat slip in under Schur\'s Lemma. So, the classical subsystem is,\nin effect, an external system.\n\nThe interpretation of the generalized result\n(2nd-order system + [q,q] = 0) <-> (classical + quantum)\nin which a split occurs is basically a formalization of the\nHeisenberg Cut, itself. A hybrid classical-quantum dynamics\nmay enable one to put the final missing piece into place to\nsolve the problem of how to implement the Heisenberg Cut.\nInstead of the usual approach of trying to explain it away by\nsome mechanism (decoherence, consistent histories, Bohm, Everett,\netc.) which only get you 9/10\'s of the way there, leaving behind\nthe irreducible unexplained core of (improper mixture ---\ncollapse to --> proper mixture), it may be explainable away by\nsimply accepting the Cut at face value.\n\nThis may be necessary in principle. In an infinite universe, there\nare no closed systems. A coarse graining of the environment is\ninevitable in practice. But even more, if the classification\nproblem for 3-metrics modulo diffeomorphism proves to be\nunsolveable, then the prospect of even having a configuration\nspace: (3-metric)/diffeos, in principle is out the window, too,\nwhich means: no universal pure state, mixtures are fundamental\nand the Cut is inevitable in principle.\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>Charles Torre wrote:
> Anybody want to bring us up to date on the status of
> quantization without a Lagrangian?
>
> Charles Torre
Basically, what I do in the 2 articles "Ab Initio Derivation of
Quantum Mechanics" is show that the d/dt-compatibility and
Jacobi-compatibility conditions that arise from putting together
the equations of motion dq/dt = v, dv/dt = a(q,v); with the
equal-time commutators [q(t),q(t)] (matrix form) = for all time
t is the non-commutative analogue of the Helmholz conditions that
in classical dynamics define the existence of a Lagrangian. This
takes the 1991 JMP paper "No Lagrangian? No Quantization!" one
step further, where the authors there had to resort to an explicit
mention of a classical limit to get their results. Likewise, I'm
restricting attention to c-number [q,v] commutators, because the
more general case requires much deeper analysis using Lie algebras.
A less restricted subset coming from relaxing the assumptions
either to (a) q's have infinite number of degrees of freedom (then
the critical use of a Schur's Lemma is blocked) or (b) [q,v]'s
allowed to vary with q (which captures n-body systems in GR),
in which case you need to consider the centralizer of the
Z_<q>(<q,v>) of the q's in the larger algebra generated by the
q's and v's. In general dim(Z_<q>(<q,v>)) > dim(<q>), and this
space will factor into <q> x <infinite-mass mode subspace>.
There may be a paper coming out of these deliberations ("On the
Quantum Dynamics of Moving Bodies" :))
The singular Lagrangian gives you a singular matrix for [q,v]^{-1}
which are the 0-mass modes, interpreted as constraints. This,
in contrast, gives you in general a "singular Hamiltonian",
with a singular [q,v], and classical "infinite mass" modes. These
extra modes need not be suject to any Lagrangian and their evolution
can't make reference to any of the quantum coordinates, except those
(in the case of a system of infinite number of degrees of freedom)
that slip in under Schur's Lemma. So, the classical subsystem is,
in effect, an external system.
The interpretation of the generalized result
(2nd-order system + [q,q] = 0) <-> (classical + quantum)
in which a split occurs is basically a formalization of the
Heisenberg Cut, itself. A hybrid classical-quantum dynamics
may enable one to put the final missing piece into place to
solve the problem of how to implement the Heisenberg Cut.
Instead of the usual approach of trying to explain it away by
some mechanism (decoherence, consistent histories, Bohm, Everett,
etc.) which only get you 9/10's of the way there, leaving behind
the irreducible unexplained core of (improper mixture ---
collapse to --> proper mixture), it may be explainable away by
simply accepting the Cut at face value.
This may be necessary in principle. In an infinite universe, there
are no closed systems. A coarse graining of the environment is
inevitable in practice. But even more, if the classification
problem for 3-metrics modulo diffeomorphism proves to be
unsolveable, then the prospect of even having a configuration
space: (3-metric)/diffeos, in principle is out the window, too,
which means: no universal pure state, mixtures are fundamental
and the Cut is inevitable in principle.
> Anybody want to bring us up to date on the status of
> quantization without a Lagrangian?
>
> Charles Torre
Basically, what I do in the 2 articles "Ab Initio Derivation of
Quantum Mechanics" is show that the d/dt-compatibility and
Jacobi-compatibility conditions that arise from putting together
the equations of motion dq/dt = v, dv/dt = a(q,v); with the
equal-time commutators [q(t),q(t)] (matrix form) = for all time
t is the non-commutative analogue of the Helmholz conditions that
in classical dynamics define the existence of a Lagrangian. This
takes the 1991 JMP paper "No Lagrangian? No Quantization!" one
step further, where the authors there had to resort to an explicit
mention of a classical limit to get their results. Likewise, I'm
restricting attention to c-number [q,v] commutators, because the
more general case requires much deeper analysis using Lie algebras.
A less restricted subset coming from relaxing the assumptions
either to (a) q's have infinite number of degrees of freedom (then
the critical use of a Schur's Lemma is blocked) or (b) [q,v]'s
allowed to vary with q (which captures n-body systems in GR),
in which case you need to consider the centralizer of the
Z_<q>(<q,v>) of the q's in the larger algebra generated by the
q's and v's. In general dim(Z_<q>(<q,v>)) > dim(<q>), and this
space will factor into <q> x <infinite-mass mode subspace>.
There may be a paper coming out of these deliberations ("On the
Quantum Dynamics of Moving Bodies" :))
The singular Lagrangian gives you a singular matrix for [q,v]^{-1}
which are the 0-mass modes, interpreted as constraints. This,
in contrast, gives you in general a "singular Hamiltonian",
with a singular [q,v], and classical "infinite mass" modes. These
extra modes need not be suject to any Lagrangian and their evolution
can't make reference to any of the quantum coordinates, except those
(in the case of a system of infinite number of degrees of freedom)
that slip in under Schur's Lemma. So, the classical subsystem is,
in effect, an external system.
The interpretation of the generalized result
(2nd-order system + [q,q] = 0) <-> (classical + quantum)
in which a split occurs is basically a formalization of the
Heisenberg Cut, itself. A hybrid classical-quantum dynamics
may enable one to put the final missing piece into place to
solve the problem of how to implement the Heisenberg Cut.
Instead of the usual approach of trying to explain it away by
some mechanism (decoherence, consistent histories, Bohm, Everett,
etc.) which only get you 9/10's of the way there, leaving behind
the irreducible unexplained core of (improper mixture ---
collapse to --> proper mixture), it may be explainable away by
simply accepting the Cut at face value.
This may be necessary in principle. In an infinite universe, there
are no closed systems. A coarse graining of the environment is
inevitable in practice. But even more, if the classification
problem for 3-metrics modulo diffeomorphism proves to be
unsolveable, then the prospect of even having a configuration
space: (3-metric)/diffeos, in principle is out the window, too,
which means: no universal pure state, mixtures are fundamental
and the Cut is inevitable in principle.