Does decoherence explain phase space emergence from Hilbert spaces? [Archive]

Kanwarpreet Grewal

Sep17-04, 05:32 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>\nHi,\n\nI am thinking about classical-quantum correspondence and the decoherence\ntheory. I think that any theory that shows how classical emerges from\nthe quantum must explain the following:\n\n1) Why are objects around us not in a superposition of states?\n\n2) How phase space emerges from infinite dimentional Hilbert spaces\n\n3) How to derive classical Hamilton\'s equations from Schrodinger\'s\nequation.\n\n\nIt seems to me that decoherence only tries to solve the first problem.\nOr has there been work in decoherence theory about 2) and 3) above?\nWhat are the theories that explain 2) and 3) and what is their\nrelationship with decoherence theory?\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,

I am thinking about classical-quantum correspondence and the decoherence
theory. I think that any theory that shows how classical emerges from
the quantum must explain the following:

1) Why are objects around us not in a superposition of states?

2) How phase space emerges from infinite dimentional Hilbert spaces

3) How to derive classical Hamilton's equations from Schrodinger's
equation.

It seems to me that decoherence only tries to solve the first problem.
Or has there been work in decoherence theory about 2) and 3) above?
What are the theories that explain 2) and 3) and what is their
relationship with decoherence theory?

regards,
Kanwar

Igor

Sep19-04, 06:56 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>\nKanwarpreet Grewal <kanwar@cadence.com> wrote in message news:<4149914C.FAC80FAA@cadence.com>...\n> Hi,\n>\n> I am thinking about classical-quantum correspondence and the decoherence\n> theory. I think that any theory that shows how classical emerges from\n> the quantum must explain the following:\n>\n> 1) Why are objects around us not in a superposition of states?\n>\n> 2) How phase space emerges from infinite dimentional Hilbert spaces\n>\n> 3) How to derive classical Hamilton\'s equations from Schrodinger\'s\n> equation.\n>\n>\n> It seems to me that decoherence only tries to solve the first problem.\n> Or has there been work in decoherence theory about 2) and 3) above?\n> What are the theories that explain 2) and 3) and what is their\n> relationship with decoherence theory?\n>\n> regards,\n> Kanwar\n\n\nFirst of all, I don\'t think the idea is to have classical phase space\nemerge from Hilbert space. In a sense, the notions of classical phase\nspace are already present in QM if you use Wigner\'s phase space\ndistribution approach. Due to the commutivity rules of QM, however,\nthis can be somewhat different from the standard classical theory.\nFor instance, we cannot specify actual points in phase space, but only\nfinite closed regions dictated by the uncertainty principle. Thus, QM\nphase spaces are fundamental examples of what is referred to as\nnon-commutative geometry. But interestingly, most of the fundamental\nideas are still there in one form or another. If you are not familiar\nwith Wigner\'s ideas, you would definitely benefit from looking them\nup. Googling on "Wigner" and "phase space" would be good place to\nbegin.\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 <kanwar@cadence.com> wrote in message news:<4149914C.FAC80FAA@cadence.com>...
> Hi,
>
> I am thinking about classical-quantum correspondence and the decoherence
> theory. I think that any theory that shows how classical emerges from
> the quantum must explain the following:
>
> 1) Why are objects around us not in a superposition of states?
>
> 2) How phase space emerges from infinite dimentional Hilbert spaces
>
> 3) How to derive classical Hamilton's equations from Schrodinger's
> equation.
>
>
> It seems to me that decoherence only tries to solve the first problem.
> Or has there been work in decoherence theory about 2) and 3) above?
> What are the theories that explain 2) and 3) and what is their
> relationship with decoherence theory?
>
> regards,
> Kanwar

First of all, I don't think the idea is to have classical phase space
emerge from Hilbert space. In a sense, the notions of classical phase
space are already present in QM if you use Wigner's phase space
distribution approach. Due to the commutivity rules of QM, however,
this can be somewhat different from the standard classical theory.
For instance, we cannot specify actual points in phase space, but only
finite closed regions dictated by the uncertainty principle. Thus, QM
phase spaces are fundamental examples of what is referred to as
non-commutative geometry. But interestingly, most of the fundamental
ideas are still there in one form or another. If you are not familiar
with Wigner's ideas, you would definitely benefit from looking them
up. Googling on "Wigner" and "phase space" would be good place to
begin.

Arnold Neumaier

Sep24-04, 08:08 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>\n> I am thinking about classical-quantum correspondence and the decoherence\n> theory. I think that any theory that shows how classical emerges from\n> the quantum must explain the following:\n>\n> 1) Why are objects around us not in a superposition of states?\n>\n> 2) How phase space emerges from infinite dimentional Hilbert spaces\n>\n> 3) How to derive classical Hamilton\'s equations from Schrodinger\'s\n> equation.\n>\n>\n> It seems to me that decoherence only tries to solve the first problem.\n> Or has there been work in decoherence theory about 2) and 3) above?\n> What are the theories that explain 2) and 3) and what is their\n> relationship with decoherence theory?\n\n2): Coherent states.\n3): Nelson\'s stochastic mechanics\n\nIn many cases, arbitrary states decohere into coherent states.\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,
>
> I am thinking about classical-quantum correspondence and the decoherence
> theory. I think that any theory that shows how classical emerges from
> the quantum must explain the following:
>
> 1) Why are objects around us not in a superposition of states?
>
> 2) How phase space emerges from infinite dimentional Hilbert spaces
>
> 3) How to derive classical Hamilton's equations from Schrodinger's
> equation.
>
>
> It seems to me that decoherence only tries to solve the first problem.
> Or has there been work in decoherence theory about 2) and 3) above?
> What are the theories that explain 2) and 3) and what is their
> relationship with decoherence theory?

2): Coherent states.
3): Nelson's stochastic mechanics

In many cases, arbitrary states decohere into coherent states.

Arnold Neumaier

Arnold Neumaier

Sep24-04, 08:08 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>Igor wrote:\n> Kanwarpreet Grewal <kanwar@cadence.com> wrote in message news:<4149914C.FAC80FAA@cadence.com>...\n\n>>I am thinking about classical-quantum correspondence and the decoherence\n>>theory. I think that any theory that shows how classical emerges from\n>>the quantum must explain the following:\n>>\n>>1) Why are objects around us not in a superposition of states?\n>>2) How phase space emerges from infinite dimentional Hilbert spaces\n>>3) How to derive classical Hamilton\'s equations from Schrodinger\'s\n>>equation.\n>>\n>> It seems to me that decoherence only tries to solve the first problem.\n>>Or has there been work in decoherence theory about 2) and 3) above?\n>> What are the theories that explain 2) and 3) and what is their\n>>relationship with decoherence theory?\n>\n> First of all, I don\'t think the idea is to have classical phase space\n> emerge from Hilbert space. In a sense, the notions of classical phase\n> space are already present in QM if you use Wigner\'s phase space\n> distribution approach. Due to the commutivity rules of QM, however,\n> this can be somewhat different from the standard classical theory.\n\nIn the classical limit hbar to 0, this difference disappears.\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>Igor wrote:
> Kanwarpreet Grewal <kanwar@cadence.com> wrote in message news:<4149914C.FAC80FAA@cadence.com>...

>>I am thinking about classical-quantum correspondence and the decoherence
>>theory. I think that any theory that shows how classical emerges from
>>the quantum must explain the following:
>>
>>1) Why are objects around us not in a superposition of states?
>>2) How phase space emerges from infinite dimentional Hilbert spaces
>>3) How to derive classical Hamilton's equations from Schrodinger's
>>equation.
>>
>> It seems to me that decoherence only tries to solve the first problem.
>>Or has there been work in decoherence theory about 2) and 3) above?
>> What are the theories that explain 2) and 3) and what is their
>>relationship with decoherence theory?
>
> First of all, I don't think the idea is to have classical phase space
> emerge from Hilbert space. In a sense, the notions of classical phase
> space are already present in QM if you use Wigner's phase space
> distribution approach. Due to the commutivity rules of QM, however,
> this can be somewhat different from the standard classical theory.

In the classical limit \hbar to 0, this difference disappears.

Arnold Neumaier

Kanwarpreet Grewal

Sep28-04, 11:51 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,\n\nThanks to everyone who answered my question. It made me think more.\nWigner distribution seems to be an attempt to study quantum mechanics in\nterms of phase space which is already assumed to exist. And coherent\nstates seems to be the closest quantum approximation of a body with a\ndefinite position and momentum. My point here is that using these we\ncannot hope to get back phase space. This is because phase space was the\ninput to both these concepts and getting them back is no surprise.\n\nMy question is : if we start with a purely quantum universe with only\nHilbert spaces then how does position and momentum become the basis of\nour existance. Are these the preferred basis and does this basis emerge\nafter selection by the envirioment ?\n\nWould an imaginary "animal" with atomic dimentions know anything about\nposition and momentum? If yes then would these concepts be a part of a\nlarge number of others with no special status. And if not then how do we\nemerge with an obsession with these concepts?\n\nregards\nKanwar\n\n\n\nArnold Neumaier wrote:\n\n> Igor wrote:\n> > Kanwarpreet Grewal <kanwar@cadence.com> wrote in message news:<4149914C.FAC80FAA@cadence.com>...\n>\n> >>I am thinking about classical-quantum correspondence and the decoherence\n> >>theory. I think that any theory that shows how classical emerges from\n> >>the quantum must explain the following:\n> >>\n> >>1) Why are objects around us not in a superposition of states?\n> >>2) How phase space emerges from infinite dimentional Hilbert spaces\n> >>3) How to derive classical Hamilton\'s equations from Schrodinger\'s\n> >>equation.\n> >>\n> >> It seems to me that decoherence only tries to solve the first problem.\n> >>Or has there been work in decoherence theory about 2) and 3) above?\n> >> What are the theories that explain 2) and 3) and what is their\n> >>relationship with decoherence theory?\n> >\n> > First of all, I don\'t think the idea is to have classical phase space\n> > emerge from Hilbert space. In a sense, the notions of classical phase\n> > space are already present in QM if you use Wigner\'s phase space\n> > distribution approach. Due to the commutivity rules of QM, however,\n> > this can be somewhat different from the standard classical theory.\n>\n> In the classical limit hbar to 0, this difference disappears.\n>\n> Arnold 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>Hi,

Thanks to everyone who answered my question. It made me think more.
Wigner distribution seems to be an attempt to study quantum mechanics in
terms of phase space which is already assumed to exist. And coherent
states seems to be the closest quantum approximation of a body with a
definite position and momentum. My point here is that using these we
cannot hope to get back phase space. This is because phase space was the
input to both these concepts and getting them back is no surprise.

My question is : if we start with a purely quantum universe with only
Hilbert spaces then how does position and momentum become the basis of
our existance. Are these the preferred basis and does this basis emerge
after selection by the envirioment ?

Would an imaginary "animal" with atomic dimentions know anything about
position and momentum? If yes then would these concepts be a part of a
large number of others with no special status. And if not then how do we
emerge with an obsession with these concepts?

regards
Kanwar

Arnold Neumaier wrote:

> Igor wrote:
> > Kanwarpreet Grewal <kanwar@cadence.com> wrote in message news:<4149914C.FAC80FAA@cadence.com>...
>
> >>I am thinking about classical-quantum correspondence and the decoherence
> >>theory. I think that any theory that shows how classical emerges from
> >>the quantum must explain the following:
> >>
> >>1) Why are objects around us not in a superposition of states?
> >>2) How phase space emerges from infinite dimentional Hilbert spaces
> >>3) How to derive classical Hamilton's equations from Schrodinger's
> >>equation.
> >>
> >> It seems to me that decoherence only tries to solve the first problem.
> >>Or has there been work in decoherence theory about 2) and 3) above?
> >> What are the theories that explain 2) and 3) and what is their
> >>relationship with decoherence theory?
> >
> > First of all, I don't think the idea is to have classical phase space
> > emerge from Hilbert space. In a sense, the notions of classical phase
> > space are already present in QM if you use Wigner's phase space
> > distribution approach. Due to the commutivity rules of QM, however,
> > this can be somewhat different from the standard classical theory.
>
> In the classical limit \hbar to 0, this difference disappears.
>
> Arnold Neumaier

Igor

Sep29-04, 08:29 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 <kanwar@cadence.com> wrote in message news:<41594009.AC560396@cadence.com>...\n> Hi,\n>\n> Thanks to everyone who answered my question. It made me think more.\n> Wigner distribution seems to be an attempt to study quantum mechanics in\n> terms of phase space which is already assumed to exist. And coherent\n> states seems to be the closest quantum approximation of a body with a\n> definite position and momentum. My point here is that using these we\n> cannot hope to get back phase space. This is because phase space was the\n> input to both these concepts and getting them back is no surprise.\n>\n> My question is : if we start with a purely quantum universe with only\n> Hilbert spaces then how does position and momentum become the basis of\n> our existance. Are these the preferred basis and does this basis emerge\n> after selection by the envirioment ?\n\n\nMy own take on this is that this is an arbitrary basis that we\'ve only\nchosen since it goes over rather nicely to quantities that we can\nmeasure with no real problem as h approaches zero. Frankly, I\'m not\nreally sure what other basis would be available to us, but since the\nwave function evolves in a completely deterministic way, the elements\nof the Hilbert space as well as their time derivatives may be a better\nchoice, although these are technically nonobservable. Something along\nthe line of the Bohm analysis may be in order here.\n\n\n\n> Would an imaginary "animal" with atomic dimentions know anything about\n> position and momentum? If yes then would these concepts be a part of a\n> large number of others with no special status. And if not then how do we\n> emerge with an obsession with these concepts?\n\n\nI doubt that such a creature would have any understanding of our\nconcepts of mechanics at all. I do wonder whether one would be able\nto "see" the wave function at that level. If so, that would probably\nbe the basis of their own mechanical universe. If not, there may not\nbe any real answer to that question.\nAgain, I think the only reason we insist on talking about position and\nmomentum is because they are fundamental quantities of classical\nmechanics and we prefer to understand the quantum world in terms of\nthings we already know. But this may not be the best approach,\nhowever.\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 <kanwar@cadence.com> wrote in message news:<41594009.AC560396@cadence.com>...
> Hi,
>
> Thanks to everyone who answered my question. It made me think more.
> Wigner distribution seems to be an attempt to study quantum mechanics in
> terms of phase space which is already assumed to exist. And coherent
> states seems to be the closest quantum approximation of a body with a
> definite position and momentum. My point here is that using these we
> cannot hope to get back phase space. This is because phase space was the
> input to both these concepts and getting them back is no surprise.
>
> My question is : if we start with a purely quantum universe with only
> Hilbert spaces then how does position and momentum become the basis of
> our existance. Are these the preferred basis and does this basis emerge
> after selection by the envirioment ?

My own take on this is that this is an arbitrary basis that we've only
chosen since it goes over rather nicely to quantities that we can
measure with no real problem as h approaches zero. Frankly, I'm not
really sure what other basis would be available to us, but since the
wave function evolves in a completely deterministic way, the elements
of the Hilbert space as well as their time derivatives may be a better
choice, although these are technically nonobservable. Something along
the line of the Bohm analysis may be in order here.

> Would an imaginary "animal" with atomic dimentions know anything about
> position and momentum? If yes then would these concepts be a part of a
> large number of others with no special status. And if not then how do we
> emerge with an obsession with these concepts?

I doubt that such a creature would have any understanding of our
concepts of mechanics at all. I do wonder whether one would be able
to "see" the wave function at that level. If so, that would probably
be the basis of their own mechanical universe. If not, there may not
be any real answer to that question.
Again, I think the only reason we insist on talking about position and
momentum is because they are fundamental quantities of classical
mechanics and we prefer to understand the quantum world in terms of
things we already know. But this may not be the best approach,
however.

Arnold Neumaier

Sep29-04, 12:18 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>\n\nIgor wrote:\n> Kanwarpreet Grewal <kanwar@cadence.com> wrote in message news:<41594009.AC560396@cadence.com>...\n>\n>>Woul d an imaginary "animal" with atomic dimentions know anything about\n>>position and momentum? If yes then would these concepts be a part of a\n>>large number of others with no special status. And if not then how do we\n>>emerge with an obsession with these concepts?\n>\n> I doubt that such a creature would have any understanding of our\n> concepts of mechanics at all. I do wonder whether one would be able\n> to "see" the wave function at that level.\n\nSomeone competent wondered already in 1940, and turned it into a bunch\nof nice stories:\n\nMr Tompkins in Paperback :\nComprising \'Mr Tompkins in Wonderland\' and \'Mr Tompkins Explores the Atom\'\nby George Gamow\nhttp://www.amazon.com/exec/obidos/tg/detail/-/0521447712/\n\nHappy reading!\n\n\nArnold Neumaier\n\n\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>Igor wrote:
> Kanwarpreet Grewal <kanwar@cadence.com> wrote in message news:<41594009.AC560396@cadence.com>...
>
>>Would an imaginary "animal" with atomic dimentions know anything about
>>position and momentum? If yes then would these concepts be a part of a
>>large number of others with no special status. And if not then how do we
>>emerge with an obsession with these concepts?
>
> I doubt that such a creature would have any understanding of our
> concepts of mechanics at all. I do wonder whether one would be able
> to "see" the wave function at that level.

Someone competent wondered already in 1940, and turned it into a bunch
of nice stories:

Mr Tompkins in Paperback :
Comprising 'Mr Tompkins in Wonderland' and 'Mr Tompkins Explores the Atom'
by George Gamow
http://www.amazon.com/exec/obidos/tg/detail/-/0521447712/

Happy reading!

Arnold Neumaier

Arnold Neumaier

Sep29-04, 12:18 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>\n> Thanks to everyone who answered my question. It made me think more.\n> Wigner distribution seems to be an attempt to study quantum mechanics in\n> terms of phase space which is already assumed to exist.\n\nNo. No classical input is needed.\nOne can get Wigner distributions from quantum theories which\nnever mention the classical theory. Wigner distributions don\'t even relate\ndirectly to phase space since they are not probability distributions\non phase space. Once you have canonical commutation relations, you get\nfrom it a symplectic structure and associated Wigner distributions.\nA classical phase space picture only arises in the limit hbar to 0.\n\n\n> My question is : if we start with a purely quantum universe with only\n> Hilbert spaces then how does position and momentum become the basis of\n> our existance. Are these the preferred basis and does this basis emerge\n> after selection by the envirioment ?\n\nPosition and momentum are not a \'preferred basis\' in any technical sense,\nin particular not in the sense of decoherence theory.\n\nThe form of the interaction with the environment decides upon whether\nthere is a preferred set of states; sometimes it is momentum states,\nsometimes it is position states, sometimes 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:
>
> Thanks to everyone who answered my question. It made me think more.
> Wigner distribution seems to be an attempt to study quantum mechanics in
> terms of phase space which is already assumed to exist.

No. No classical input is needed.
One can get Wigner distributions from quantum theories which
never mention the classical theory. Wigner distributions don't even relate
directly to phase space since they are not probability distributions
on phase space. Once you have canonical commutation relations, you get
from it a symplectic structure and associated Wigner distributions.
A classical phase space picture only arises in the limit \hbar to .

> My question is : if we start with a purely quantum universe with only
> Hilbert spaces then how does position and momentum become the basis of
> our existance. Are these the preferred basis and does this basis emerge
> after selection by the envirioment ?

Position and momentum are not a 'preferred basis' in any technical sense,
in particular not in the sense of decoherence theory.

The form of the interaction with the environment decides upon whether
there is a preferred set of states; sometimes it is momentum states,
sometimes it is position states, sometimes coherent states.

Arnold Neumaier

p.kinsler@imperial.ac.uk

Oct26-04, 12:56 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>Kanwarpreet Grewal <kanwar@cadence.com> wrote:\n> Thanks to everyone who answered my question. It made me think more.\n> Wigner distribution seems to be an attempt to study quantum mechanics in\n> terms of phase space which is already assumed to exist. And coherent\n> states seems to be the closest quantum approximation of a body with a\n> definite position and momentum.\n\nIt\'s worth noting that there are other sorts of distribution\nalong the lines of the Wigner -- notably the (Glauber-Sudarshan) P\nand (Husimi) Q ones. Further, there are versions of these with\nexpanded domains, as e.g. the complex-P or positive-P distributions.\n\nThe P distributions are very useful in optics -- they use a coherent\nstate basis, so a coherent state is represented (in all its intrinsic\nquantum uncertainty) by a delta function; the positive-P is nice since\nit is guaranteed to be positive definite, so you can treat it\nlike a probablility distribution (unlike e.g. the Wigner or P), and\ngenerate stochastic equations for numerical solutions. You can also\ngo even further, adding "gauges" to optimise a model according to what\nyou are tring to calculate.\n\nPeter Drummond at UQ has done a lot of this stuff.\n\n--\n---------------------------------+---------------------------------\nDr. Paul Kinsler\nBlackett Laboratory (QOLS) (ph) +44-20-759-47520 (fax) 47714\nImperial College London, Dr.Paul.Kinsler@physics.org\nSW7 2BW, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/\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 <kanwar@cadence.com> wrote:
> Thanks to everyone who answered my question. It made me think more.
> Wigner distribution seems to be an attempt to study quantum mechanics in
> terms of phase space which is already assumed to exist. And coherent
> states seems to be the closest quantum approximation of a body with a
> definite position and momentum.

It's worth noting that there are other sorts of distribution
along the lines of the Wigner -- notably the (Glauber-Sudarshan) P
and (Husimi) Q ones. Further, there are versions of these with
expanded domains, as e.g. the complex-P or positive-P distributions.

The P distributions are very useful in optics -- they use a coherent
state basis, so a coherent state is represented (in all its intrinsic
quantum uncertainty) by a \delta function; the positive-P is nice since
it is guaranteed to be positive definite, so you can treat it
like a probablility distribution (unlike e.g. the Wigner or P), and
generate stochastic equations for numerical solutions. You can also
go even further, adding "gauges" to optimise a model according to what
you are tring to calculate.

Peter Drummond at UQ has done a lot of this stuff.

--
---------------------------------+---------------------------------
Dr. Paul Kinsler
Blackett Laboratory (QOLS) (ph) +44-20-759-47520 (fax) 47714
Imperial College London, Dr.Paul.Kinsler@physics.org
SW7 2BW, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/