Klauder coherent states and Quantum-Classical correspondence [Archive]

Kanwarpreet Grewal

Oct28-04, 01:29 PM

<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>\nHi,\n\nSee J.R Klauder\'s papers:\n\nThe Current State of Coherent States : quant-ph/0110108\nCoherent States for Discrete Spectrum Dynamics : quant-ph/9810044\n\nIn these papers Klauder tries to generalize the definition of\ncoherent states.\nHe selects 4 conditions for coherent states:\n\n1) Resolution of Unity\n2) Temporal Stability\n3) Overcompleteness\n4) Action Identity\n\nI have no problems with 1 - 3 . However I dont understand 4).\nKlauder insists that\nfor a coherent state |J,Y> to be physically meaningful J and Y have to\nbe classical action\nangle variables. So my questions are:\n\n1) Why the insistance that (J,Y) should be classical action angle\nvariables?\n2) How are action-angle variables important in quantum-classical\ncorrespondence\n3) Is Klauder trying to form a theory of how classical emerges from\nquantum?\n\nKanwar\n\n--\n|=============================================== =======|\nGo, wondrous creature! mount where Science guides:\nGo, measure earth, weigh air, and state the tides:\nInstruct the planets in what orbs to run,\nCorrect old time and regulate the Sun;\n|=========================================== ===========|\n\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>Hi,

See J.R Klauder's papers:

The Current State of Coherent States : http://www.arxiv.org/abs/quant-ph/0110108
Coherent States for Discrete Spectrum Dynamics : http://www.arxiv.org/abs/quant-ph/9810044

In these papers Klauder tries to generalize the definition of
coherent states.
He selects 4 conditions for coherent states:

1) Resolution of Unity
2) Temporal Stability
3) Overcompleteness
4) Action Identity

I have no problems with 1 - 3 . However I dont understand 4).
Klauder insists that
for a coherent state |J,Y> to be physically meaningful J and Y have to
be classical action
angle variables. So my questions are:

1) Why the insistance that (J,Y) should be classical action angle
variables?
2) How are action-angle variables important in quantum-classical
correspondence
3) Is Klauder trying to form a theory of how classical emerges from
quantum?

Kanwar

--
|================================================= =====|
Go, wondrous creature! mount where Science guides:
Go, measure earth, weigh air, and state the tides:
Instruct the planets in what orbs to run,
Correct old time and regulate the Sun;
|================================================= =====|

Arnold Neumaier

Oct29-04, 02:07 PM

<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>\nKanwarpreet Grewal wrote:\n> Hi,\n>\n> See J.R Klauder\'s papers:\n>\n> The Current State of Coherent States : quant-ph/0110108\n> Coherent States for Discrete Spectrum Dynamics : quant-ph/9810044\n>\n> In these papers Klauder tries to generalize the definition of\n> coherent states.\n> He selects 4 conditions for coherent states:\n>\n> 1) Resolution of Unity\n> 2) Temporal Stability\n> 3) Overcompleteness\n> 4) Action Identity\n>\n> I have no problems with 1 - 3 . However I dont understand 4).\n> Klauder insists that\n> for a coherent state |J,Y> to be physically meaningful J and Y have to\n> be classical action\n> angle variables. So my questions are:\n>\n> 1) Why the insistance that (J,Y) should be classical action angle\n> variables?\n\nHe gives the reason on p.6 of quant-ph/0110108:\n\'\'We now seek an additional physical criterion that picks out a single\nset of weights {rho_n} for a given Hamiltonian, thereby reducing the\nvast family of coherent states down to a single set.\'\'\nThe action principle makes classical mechanics come naturally from\nthe Dirac-Frenkel variational principle applied to the set of coherent\nstates instead of the whole state space.\n\n\n> 2) How are action-angle variables important in quantum-classical\n> correspondence\n\nThe action angle variables are relevant classically to characterize\ncompletely integrable systems. They are the natural coordinate system\nof phase space of such a system since the dynamics is trivialized.\nThis makes for very easy calculations...\n\n\n> 3) Is Klauder trying to form a theory of how classical emerges from\n> quantum?\n\nIt is known for a long time that classical mechanics is essentially\nthe approximation of quantum mechanics obtained by restricting\nstates to coherent states.\n\n\nArnold Neumaier\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>Kanwarpreet Grewal wrote:
> Hi,
>
> See J.R Klauder's papers:
>
> The Current State of Coherent States : http://www.arxiv.org/abs/quant-ph/0110108
> Coherent States for Discrete Spectrum Dynamics : http://www.arxiv.org/abs/quant-ph/9810044
>
> In these papers Klauder tries to generalize the definition of
> coherent states.
> He selects 4 conditions for coherent states:
>
> 1) Resolution of Unity
> 2) Temporal Stability
> 3) Overcompleteness
> 4) Action Identity
>
> I have no problems with 1 - 3 . However I dont understand 4).
> Klauder insists that
> for a coherent state |J,Y> to be physically meaningful J and Y have to
> be classical action
> angle variables. So my questions are:
>
> 1) Why the insistance that (J,Y) should be classical action angle
> variables?

He gives the reason on p.6 of http://www.arxiv.org/abs/quant-ph/0110108:
''We now seek an additional physical criterion that picks out a single
set of weights {\rho_n} for a given Hamiltonian, thereby reducing the
vast family of coherent states down to a single set.''
The action principle makes classical mechanics come naturally from
the Dirac-Frenkel variational principle applied to the set of coherent
states instead of the whole state space.

> 2) How are action-angle variables important in quantum-classical
> correspondence

The action angle variables are relevant classically to characterize
completely integrable systems. They are the natural coordinate system
of phase space of such a system since the dynamics is trivialized.
This makes for very easy calculations...

> 3) Is Klauder trying to form a theory of how classical emerges from
> quantum?

It is known for a long time that classical mechanics is essentially
the approximation of quantum mechanics obtained by restricting
states to coherent states.

Arnold Neumaier

kanwarpreet Grewal

Dec10-04, 10:18 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>Hi,\nThanks for your reply. It helped me understand these papers much better.\nHowever I have more questions now:\n\n1) Klauder says that classical mechanics is what remains of QM if we restrict\n\nvariations alone coherent states only.\nSo is this an alternate theory to decoherence theory?\n\n2) How does decoherence theory and this theory fit together? I know that\nin decoherence theory coherent states are preferred over others(say squeezed\nstates) by interactions with the environment.\n\n3) In decoherence theory coherent states are the effect of selections. But in\nKlauder\'s theory coherent states are the cause of classicality.\n\nI dont understand these theories together. Can you please help.\n\nregards\nKanwar\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>Hi,
Thanks for your reply. It helped me understand these papers much better.
However I have more questions now:

1) Klauder says that classical mechanics is what remains of QM if we restrict

variations alone coherent states only.
So is this an alternate theory to decoherence theory?

2) How does decoherence theory and this theory fit together? I know that
in decoherence theory coherent states are preferred over others(say squeezed
states) by interactions with the environment.

3) In decoherence theory coherent states are the effect of selections. But in
Klauder's theory coherent states are the cause of classicality.

I dont understand these theories together. Can you please help.

regards
Kanwar

Arnold Neumaier

Dec11-04, 03:04 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>kanwarpreet Grewal wrote:\n> Hi,\n> Thanks for your reply. It helped me understand these papers much better.\n> However I have more questions now:\n>\n> 1) Klauder says that classical mechanics is what remains of QM if we restrict\n>\n> variations alone coherent states only.\n> So is this an alternate theory to decoherence theory?\n\nNo. Coherent states are kinematical objects, that have nothing directly\nto do with dynamics. As pure states they are mailny useful for\nconservative dynamics. Decoherence theory tells what happens to systems\nin a dissipative environment, and how that environment selects preferred\nstates. Sometimes they are position or momentum eigenstates, sometimes\ncoherent states - depending on the interactions.\n\n\n> 2) How does decoherence theory and this theory fit together? I know that\n> in decoherence theory coherent states are preferred over others(say squeezed\n> states) by interactions with the environment.\n\nWhat is preferred depends on the interaction. Coherent states are a\ntool, not a theory. Decoherence is a dynamical phenomenon, not a theory.\nSometimes the two fit, namely in circumstances where we expect the\nsystem to behave almost classically.\n\n\n> 3) In decoherence theory coherent states are the effect of selections. But in\n> Klauder\'s theory coherent states are the cause of classicality.\n\nKlauder does not make any dynamical statements. All he says is: If you\napproximate all your states to coherent states you get a restricted\nvariational principle that reproduces classical mechanics. Thus he gives\nsome intuition why there is a classical approximation to quantum\nmechanics. And a method of computation (developed by Heller and others)\nwhen this approximation is good enough. But no discussion about when\nthis is the case.\n\nOn the other hand, decoherence is about showing that the reduced\ndynamics of a system interacting with a typical environment must be\nessentially classical. If this is the case, a description by coherent\nstates is appropriate and accurate.\n\n\nArnold Neumaier\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>kanwarpreet Grewal wrote:
> Hi,
> Thanks for your reply. It helped me understand these papers much better.
> However I have more questions now:
>
> 1) Klauder says that classical mechanics is what remains of QM if we restrict
>
> variations alone coherent states only.
> So is this an alternate theory to decoherence theory?

No. Coherent states are kinematical objects, that have nothing directly
to do with dynamics. As pure states they are mailny useful for
conservative dynamics. Decoherence theory tells what happens to systems
in a dissipative environment, and how that environment selects preferred
states. Sometimes they are position or momentum eigenstates, sometimes
coherent states - depending on the interactions.

> 2) How does decoherence theory and this theory fit together? I know that
> in decoherence theory coherent states are preferred over others(say squeezed
> states) by interactions with the environment.

What is preferred depends on the interaction. Coherent states are a
tool, not a theory. Decoherence is a dynamical phenomenon, not a theory.
Sometimes the two fit, namely in circumstances where we expect the
system to behave almost classically.

> 3) In decoherence theory coherent states are the effect of selections. But in
> Klauder's theory coherent states are the cause of classicality.

Klauder does not make any dynamical statements. All he says is: If you
approximate all your states to coherent states you get a restricted
variational principle that reproduces classical mechanics. Thus he gives
some intuition why there is a classical approximation to quantum
mechanics. And a method of computation (developed by Heller and others)
when this approximation is good enough. But no discussion about when
this is the case.

On the other hand, decoherence is about showing that the reduced
dynamics of a system interacting with a typical environment must be
essentially classical. If this is the case, a description by coherent
states is appropriate and accurate.

Arnold Neumaier

Arnold Neumaier

Dec13-04, 10:10 PM

<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>easily obscured\nby propaganda and fancy talk.\n\n139. And note this important difference: It is conceivable that our\nenvironmental problems (for example) may some day be settled through a\nrational, comprehensive plan, but if this happens it will be only\nbecause it is in the long-term interest of the system to solve these\nproblems. But it is NOT in the interest of the system to preserve\nfreedom or small-group autonomy. On the contrary, it is in the\ninterest of the system to bring human behavior under control to the\ngreatest possible extent. Thus, while practical considerations may\neventually force the system to take a rational, prudent approach to\nenvironmental problems, equally practical considerations will force\nthe system to regulate human behavior ever more closely (preferably by\nindirect means that will disguise the encroachment on freedom.) This\nisn\'t just our opinion. Eminent social scientists (e.g. James Q.\nWilson) have stressed the importance of "socializing" people more\neffectively.\n\nREVOLUTION IS EASIER THAN REFORM\n\n\n\n140. We hope we have convinced the reader that the system cannot be\nreformed in a such a way as to reconcile freedom with technology. The\nonly way out is to dispense with the industrial-technological system\naltogether. This implies revolution, not necessarily an armed\nuprising, but certainly a radical and fundamental change in the nature\nof society.\n\n\n\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>easily obscured
by propaganda and fancy talk.

139. And note this important difference: It is conceivable that our
environmental problems (for example) may some day be settled through a
rational, comprehensive plan, but if this happens it will be only
because it is in the long-term interest of the system to solve these
problems. But it is NOT in the interest of the system to preserve
freedom or small-group autonomy. On the contrary, it is in the
interest of the system to bring human behavior under control to the
greatest possible extent. Thus, while practical considerations may
eventually force the system to take a rational, prudent approach to
environmental problems, equally practical considerations will force
the system to regulate human behavior ever more closely (preferably by
indirect means that will disguise the encroachment on freedom.) This
isn't just our opinion. Eminent social scientists (e.g. James Q.
Wilson) have stressed the importance of "socializing" people more
effectively.

REVOLUTION IS EASIER THAN REFORM

140. We hope we have convinced the reader that the system cannot be
reformed in a such a way as to reconcile freedom with technology. The
only way out is to dispense with the industrial-technological system
altogether. This implies revolution, not necessarily an armed
uprising, but certainly a radical and fundamental change in the nature
of society.

kanwarpreet Grewal

Dec14-04, 12:24 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>this will not solve all problems. Industrial society\nhas already done tremendous damage to nature and it will take a very\nlong time for the scars to heal. Besides, even pre-industrial\nsocieties can do significant damage to nature. Nevertheless, getting\nrid of industrial society will accomplish a great deal. It will\nrelieve the worst of the pressure on nature so that the scars can\nbegin to heal. It will remove the capacity of organized society to\nkeep increasing its control over nature (including human nature).\nWhatever kind of society may exist after the demise of the industrial\nsystem, it is certain that most people will live close to nature,\nbecause in the absence of advanced technology there is not other way\nthat people CAN live. To feed themselves they must be peasants or\nherdsmen or fishermen or hunter, etc., And, generally speaking, local\nautonomy should tend to increase, because lack of advanced technology\nand rapid communications will limit the capacity of governments or\nother large organizations to control local communities.\n\n185. As for the negative consequences of eliminating industrial\nsociety -- well, you can\'t eat your cake and have it too. To gain one\nthing you have to sacrifice another.\n\n186. Most people hate psychological conflict. For this reason they\navoid doing any serious thinking about difficult social issues, and\nthey like to have such issues presented to them in simple,\nblack-and-white terms: THIS is all good and THAT is all bad. The\nrevolutionary ideology should therefore be developed on two levels.\n\n187. On the more sophisticated level the ideology should address\nitself to people who are intelligent, thoughtful and rational. The\nobject should be to create a core of people who will be opposed to the\nindustrial system on a rational, thought-out basis, with full\nappreciation of t\n\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>this will not solve all problems. Industrial society
has already done tremendous damage to nature and it will take a very
long time for the scars to heal. Besides, even pre-industrial
societies can do significant damage to nature. Nevertheless, getting
rid of industrial society will accomplish a great deal. It will
relieve the worst of the pressure on nature so that the scars can
begin to heal. It will remove the capacity of organized society to
keep increasing its control over nature (including human nature).
Whatever kind of society may exist after the demise of the industrial
system, it is certain that most people will live close to nature,
because in the absence of advanced technology there is not other way
that people CAN live. To feed themselves they must be peasants or
herdsmen or fishermen or hunter, etc., And, generally speaking, local
autonomy should tend to increase, because lack of advanced technology
and rapid communications will limit the capacity of governments or
other large organizations to control local communities.

185. As for the negative consequences of eliminating industrial
society -- well, you can't eat your cake and have it too. To gain one
thing you have to sacrifice another.

186. Most people hate psychological conflict. For this reason they
avoid doing any serious thinking about difficult social issues, and
they like to have such issues presented to them in simple,
black-and-white terms: THIS is all good and THAT is all bad. The
revolutionary ideology should therefore be developed on two levels.

187. On the more sophisticated level the ideology should address
itself to people who are intelligent, thoughtful and rational. The
object should be to create a core of people who will be opposed to the
industrial system on a rational, thought-out basis, with full
appreciation of t