View Full Version : New thread : Probability density in spacetime
Jagmeet Singh
Sep29-04, 08:30 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>I propose the following idea (which I am not at all\nsure,but is an appealing idea all the same).Let me\nhave your views on it.\n\nN.B. (For the moderators:-I started this thread using\nphysics forums web-site 3-4 months back,but it was\nsomehow deleted).\n\nJust as the wavefunction in non-relativistic quantum\nmechanics gives probability density in space,the\nwavefunction in relativistic quantum mechanics should\ngive the probability density in space-time---because\nspace and time are at par in relativity.So what this\nreally means is that a particle has some probability\nof going a tiny time,centred around the present,into\nthe past as well as the future.In such a scenario the\ncontinuity equation will not be satisfied(as is the\ncase in say the Klein Gordon equation)----because\nthere are sources and sinks(particles are appearing\nand\ndisappearing).What is the wavefunction or field(which\nis a function of time t,that we understand) that we\nuse in say the K.G. equation?----it would be the\naverage over time t\' of the real wavefunction(which is\na function of time t\',centred around t,going a little\ninto the past and future).So the wavefunction or field\nthat we see is really the average over a bit of past &\nfuture of the real wavefunction.If this is so,perhaps,\nwe could design an experiment to see some interference\nin time kind of effects.\n\nJagmeet\n\n\n\n\n_____________________ _____________\nDo you Yahoo!?\nNew and Improved Yahoo! Mail - 100MB free storage!\nhttp://promotions.yahoo.com/new_mail\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>I propose the following idea (which I am not at all
sure,but is an appealing idea all the same).Let me
have your views on it.
N.B. (For the moderators:-I started this thread using
physics forums web-site 3-4 months back,but it was
somehow deleted).
Just as the wavefunction in non-relativistic quantum
mechanics gives probability density in space,the
wavefunction in relativistic quantum mechanics should
give the probability density in space-time---because
space and time are at par in relativity.So what this
really means is that a particle has some probability
of going a tiny time,centred around the present,into
the past as well as the future.In such a scenario the
continuity equation will not be satisfied(as is the
case in say the Klein Gordon equation)----because
there are sources and sinks(particles are appearing
and
disappearing).What is the wavefunction or field(which
is a function of time t,that we understand) that we
use in say the K.G. equation?----it would be the
average over time t' of the real wavefunction(which is
a function of time t',centred around t,going a little
into the past and future).So the wavefunction or field
that we see is really the average over a bit of past &
future of the real wavefunction.If this is so,perhaps,
we could design an experiment to see some interference
in time kind of effects.
Jagmeet
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Arnold Neumaier
Sep30-04, 01:46 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>\nJagmeet Singh wrote:\n> I propose the following idea (which I am not at all\n> sure,but is an appealing idea all the same).Let me\n> have your views on it.\n>\n> N.B. (For the moderators:-I started this thread using\n> physics forums web-site 3-4 months back,but it was\n> somehow deleted).\n>\n> Just as the wavefunction in non-relativistic quantum\n> mechanics gives probability density in space,the\n> wavefunction in relativistic quantum mechanics should\n> give the probability density in space-time---because\n> space and time are at par in relativity.So what this\n> really means is that a particle has some probability\n> of going a tiny time,centred around the present,into\n> the past as well as the future.\n\nThis was explored by Bloch, Phys. Z. U.S.S.R., 5 (1943) 301,\nbut is not completely consistent, since the probability\ndensity must integrate to 1 over the 3-space defined by t=const\nand _not_ over the 4-space, since the interpretation must\nreduce to the standard one in the nonrelativistic limit.\nThus one would need at least a notion of probability density\nfor being at a point on a spacelike hyperplane, for any fixed\nhyperplane. Or even on any spacelike hypersurface.\n\nThis leads to Tomonaga\'s form of renormalized QED,\nfor which he (together with Schwinger and Feynman)\nreceived the 1965 Nobel prize. His Nobel lecture at\nhttp://nobelprize.org/physics/laureates/1965/tomonaga-lecture.html\nis interesting to read, though it says much more about the early history\nof renormalization than about the probability interpretation.\n\nTomonaga\'s form of QED is now almost forgotten since everything is\ndone with Feynman diagrams. But it may still contain a key to\nthe dynamical aspects of QED, which have been almost completely\nignored in the last 50 years.\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>Jagmeet Singh wrote:
> I propose the following idea (which I am not at all
> sure,but is an appealing idea all the same).Let me
> have your views on it.
>
> N.B. (For the moderators:-I started this thread using
> physics forums web-site 3-4 months back,but it was
> somehow deleted).
>
> Just as the wavefunction in non-relativistic quantum
> mechanics gives probability density in space,the
> wavefunction in relativistic quantum mechanics should
> give the probability density in space-time---because
> space and time are at par in relativity.So what this
> really means is that a particle has some probability
> of going a tiny time,centred around the present,into
> the past as well as the future.
This was explored by Bloch, Phys. Z. U.S.S.R., 5 (1943) 301,
but is not completely consistent, since the probability
density must integrate to 1 over the 3-space defined by t=const
and _not_ over the 4-space, since the interpretation must
reduce to the standard one in the nonrelativistic limit.
Thus one would need at least a notion of probability density
for being at a point on a spacelike hyperplane, for any fixed
hyperplane. Or even on any spacelike hypersurface.
This leads to Tomonaga's form of renormalized QED,
for which he (together with Schwinger and Feynman)
received the 1965 Nobel prize. His Nobel lecture at
http://nobelprize.org/physics/laureates/1965/tomonaga-lecture.html
is interesting to read, though it says much more about the early history
of renormalization than about the probability interpretation.
Tomonaga's form of QED is now almost forgotten since everything is
done with Feynman diagrams. But it may still contain a key to
the dynamical aspects of QED, which have been almost completely
ignored in the last 50 years.
Arnold Neumaier
Charles J. Quarra
Oct1-04, 09:43 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>\n\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<415B045B.6050804@univie.ac.at>...\n>\n> This was explored by Bloch, Phys. Z. U.S.S.R., 5 (1943) 301,\n> but is not completely consistent, since the probability\n> density must integrate to 1 over the 3-space defined by t=const\n> and _not_ over the 4-space, since the interpretation must\n> reduce to the standard one in the nonrelativistic limit.\n> Thus one would need at least a notion of probability density\n> for being at a point on a spacelike hyperplane, for any fixed\n> hyperplane. Or even on any spacelike hypersurface.\n\na question about the many-time interpretation: what would be the\nequivalent of partial traces?\n\n\nint ( Psi( x_0, t_0 , x_1 , t_1 ) * Psi*( x_0\' , t_0\' , x_1 , t_1 ) ,\nx_1=R^3 , t_1=-inf..inf) ?\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<415B045B.6050804@univie.ac.at>...
>
> This was explored by Bloch, Phys. Z. U.S.S.R., 5 (1943) 301,
> but is not completely consistent, since the probability
> density must integrate to 1 over the 3-space defined by t=const
> and _not_ over the 4-space, since the interpretation must
> reduce to the standard one in the nonrelativistic limit.
> Thus one would need at least a notion of probability density
> for being at a point on a spacelike hyperplane, for any fixed
> hyperplane. Or even on any spacelike hypersurface.
a question about the many-time interpretation: what would be the
equivalent of partial traces?
\int ( \Psi( x_0, t_0 , x_1 , t_1 ) * \Psi*( x_0' , t_0' , x_1 , t_1 ) ,x_1=R^3 , t_1=-inf..inf) ?
Arnold Neumaier
Oct3-04, 03:49 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>\nCharles J. Quarra wrote:\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<415B045B.6050804@univie.ac.at>...\n>\n>>This was explored by Bloch, Phys. Z. U.S.S.R., 5 (1943) 301,\n>>but is not completely consistent, since the probability\n>>density must integrate to 1 over the 3-space defined by t=const\n>>and _not_ over the 4-space, since the interpretation must\n>>reduce to the standard one in the nonrelativistic limit.\n>>Thus one would need at least a notion of probability density\n>>for being at a point on a spacelike hyperplane, for any fixed\n>>hyperplane. Or even on any spacelike hypersurface.\n>\n>\n> a question about the many-time interpretation: what would be the\n> equivalent of partial traces?\n>\n>\n> int ( Psi( x_0, t_0 , x_1 , t_1 ) * Psi*( x_0\' , t_0\' , x_1 , t_1 ) ,\n> x_1=R^3 , t_1=-inf..inf) ?\n\nI don\'t understand your question.\nThis expression does not seem to have a useful interpretation.\n\nThe multiple times serve to allow looking at arbitrary spacelike\nhypersurfaces. But probabilities are always relative to two such\nhypersurfaces representing initial time and final time, not between\nmultiple times.\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>Charles J. Quarra wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<415B045B.6050804@univie.ac.at>...
>
>>This was explored by Bloch, Phys. Z. U.S.S.R., 5 (1943) 301,
>>but is not completely consistent, since the probability
>>density must integrate to 1 over the 3-space defined by t=const
>>and _not_ over the 4-space, since the interpretation must
>>reduce to the standard one in the nonrelativistic limit.
>>Thus one would need at least a notion of probability density
>>for being at a point on a spacelike hyperplane, for any fixed
>>hyperplane. Or even on any spacelike hypersurface.
>
>
> a question about the many-time interpretation: what would be the
> equivalent of partial traces?
>
>
> \int ( \Psi( x_0, t_0 , x_1 , t_1 ) * \Psi*( x_0' , t_0' , x_1 , t_1 ) ,
> x_1=R^3 , t_1=-inf..inf) ?
I don't understand your question.
This expression does not seem to have a useful interpretation.
The multiple times serve to allow looking at arbitrary spacelike
hypersurfaces. But probabilities are always relative to two such
hypersurfaces representing initial time and final time, not between
multiple times.
Arnold Neumaier
<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>Arnold Neumaier wrote:\n\n> This was explored by Bloch, Phys. Z. U.S.S.R., 5 (1943) 301,\n> but is not completely consistent, since the probability\n> density must integrate to 1 over the 3-space defined by t=const\n> and _not_ over the 4-space, since the interpretation must\n> reduce to the standard one in the nonrelativistic limit.\n> Thus one would need at least a notion of probability density\n> for being at a point on a spacelike hyperplane, for any fixed\n> hyperplane. Or even on any spacelike hypersurface.\n\n\nWhen we are talking of the K.G.equation,we are not worried\nabout the probability density not integrating to 1 in 3-space.In the\nnon-relativistic limit,the Schrodinger equation is, in any case,\nsatisfied which guarantees that the continuity equation is\nsatisfied(and hence the probabilty\ndensity integrates to 1) no matter what shady underhand interpretations\nI give to the wavefunction.\n\nBesides, I\'ve found the following paper on google which seems to be\ndiscussing ideas similar to \'prob. density in spacetime\' :\nhttp://arxiv.org/abs/quant-ph/0306007\n\nHave a look at it.\n\n------------------------------------------------------------------------\nThis post submitted through the LaTeX-enabled physicsforums.com\nTo view this post with LaTeX images:\nhttp://www.physicsforums.com/showthread.php?t=45264#post329561\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>Arnold Neumaier wrote:
> This was explored by Bloch, Phys. Z. U.S.S.R., 5 (1943) 301,
> but is not completely consistent, since the probability
> density must integrate to 1 over the 3-space defined by t=const
> and _not_ over the 4-space, since the interpretation must
> reduce to the standard one in the nonrelativistic limit.
> Thus one would need at least a notion of probability density
> for being at a point on a spacelike hyperplane, for any fixed
> hyperplane. Or even on any spacelike hypersurface.
When we are talking of the K.G.equation,we are not worried
about the probability density not integrating to 1 in 3-space.In the
non-relativistic limit,the Schrodinger equation is, in any case,
satisfied which guarantees that the continuity equation is
satisfied(and hence the probabilty
density integrates to 1) no matter what shady underhand interpretations
I give to the wavefunction.
Besides, I've found the following paper on google which seems to be
discussing ideas similar to 'prob. density in spacetime' :
http://arxiv.org/abs/http://www.arxiv.org/abs/quant-ph/0306007
Have a look at it.
------------------------------------------------------------------------
This post submitted through the LaTeX-enabled physicsforums.com
To view this post with LaTeX images:
http://www.physicsforums.com/showthread.php?t=45264#post329561
Charles J. Quarra
Oct5-04, 06:42 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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<415DBA44.7020903@univie.ac.at>...\n> Charles J. Quarra wrote:\n> > a question about the many-time interpretation: what would be the\n> > equivalent of partial traces?\n> >\n> >\n> > int ( Psi( x_0, t_0 , x_1 , t_1 ) * Psi*( x_0\' , t_0\' , x_1 , t_1 ) ,\n> > x_1=R^3 , t_1=-inf..inf) ?\n>\n> I don\'t understand your question.\n> This expression does not seem to have a useful interpretation.\n>\n> The multiple times serve to allow looking at arbitrary spacelike\n> hypersurfaces. But probabilities are always relative to two such\n> hypersurfaces representing initial time and final time, not between\n> multiple times.\n\n\nNote the first argument against using 3+1 wavefunctions is that one\nwants that its normalized in space-like sheets so when one _measures_\nit, the probability is one of being somewhere in that sheet. But since\nmeasurement is an more or less ad-hoc element in the formulation one\nshould take with a grain of salt the argument of normalization on\nspace-like sheets.\n\nOn a side note, Thinking in a wavefunction psi(x0,t0, x1,t1, ...xn,tn)\nof n particles then that has an inmediate interpretation as the\namplitude of particle 0 being at x0 at time t0, while ...particle n\nbeing at xn at time tn. What i understand of this formulation that is\nnon-trivial and that is not present in single-time formulations is\nthat there can exists amplitudes correlations between events at\ndifferent time directly, without relying on hamiltonian evolution. In\nschrödinger single-time wavefunction, the Hillbert tensor product of\nparticle states spaces are a basis of |X0> (x) |X1> , where Xn is a\nthree-vector. Hence if a ket in the spanned space is projected to a\nsubspace of a given state in Particle-0 state, the resulting factor\nstate in P-1 is spanned by three-vector states of the _same_ time. The\nsubletie is that for different evolution operators this factor state\nwill evolve to different things at different times. _not so in the\nDirac scheme_. In some way the Dirac many-time wavefunction entangles\nevents at different times independently of particular evolution\noperators.\n\n\nWhen i say that there is a probability of finding the particle at\ntime Tc in a wide region after being in the state |Xb,Tb> and evolving\nwitht U(T,Tb) im computing it as\n\nint ( || <Xs,T|U(T,Tb)|Xb,Tb> ||^2 , Xs=region , T=Tc )\n\nWhile in fact, no experimental setup detects events occuring in a\ngiven time instant! instead one detects events in a thick time slice\nof T=Tc..Tc + Delta_T, where Delta_T is a time in which the X-ray film\nor whatever is exposed to potential events.\nSo actually the probability should be something like\n\nint ( || <Xs,T|U(T,Tb)|Xb,Tb> ||^2 , Xs=region, T=Tc..Tc + Delta_T )\n\nFrom what i see, normalizing in all 3+1 space is consistent as long\nas i do no physical measurement. Is that true in general?\n\nFurthermore, imagine i have a state of one particle described by\na|Xa,Ta> + b|Xb,Tb>. U is the evolution operator, so i have at time\nTc, Tc > Ta > Tc\n\naU(Tc,Ta)|Xa,Ta> + bU(Tc,Tb)|Xb,Tb>\n\nif The rough probability density | <Xs,Tc|U(Tc,Tb)|Xb,Tb> |^2 happens\nto be zero, the a measurement resulting in detecting the particle at\nthe |Xs,Tc> state could be interpreted in this formulation in a\nprojection _not_ on the state |Xs,Tc>, but actually in the state\n|Xa,Ta> with probability |a|^2, which happens at a past time. Do you\nknow an argument why this scheme to interpret measurement would not\nwork?\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<415DBA44.7020903@univie.ac.at>...
> Charles J. Quarra wrote:
> > a question about the many-time interpretation: what would be the
> > equivalent of partial traces?
> >
> >
> > \int ( \Psi( x_0, t_0 , x_1 , t_1 ) * \Psi*( x_0' , t_0' , x_1 , t_1 ) ,
> > x_1=R^3 , t_1=-inf..inf) ?
>
> I don't understand your question.
> This expression does not seem to have a useful interpretation.
>
> The multiple times serve to allow looking at arbitrary spacelike
> hypersurfaces. But probabilities are always relative to two such
> hypersurfaces representing initial time and final time, not between
> multiple times.
Note the first argument against using 3+1 wavefunctions is that one
wants that its normalized in space-like sheets so when one _measures_
it, the probability is one of being somewhere in that sheet. But since
measurement is an more or less ad-hoc element in the formulation one
should take with a grain of salt the argument of normalization on
space-like sheets.
On a side note, Thinking in a wavefunction \psi(x0,t0, x1,t1, ...xn,tn)
of n particles then that has an inmediate interpretation as the
amplitude of particle being at x0 at time t0, while ...particle n
being at xn at time tn. What i understand of this formulation that is
non-trivial and that is not present in single-time formulations is
that there can exists amplitudes correlations between events at
different time directly, without relying on hamiltonian evolution. In
schrödinger single-time wavefunction, the Hillbert tensor product of
particle states spaces are a basis of |X0> (x) |X1> , where Xn is a
three-vector. Hence if a ket in the spanned space is projected to a
subspace of a given state in Particle-0 state, the resulting factor
state in P-1 is spanned by three-vector states of the _same_ time. The
subletie is that for different evolution operators this factor state
will evolve to different things at different times. _not so in the
Dirac scheme_. In some way the Dirac many-time wavefunction entangles
events at different times independently of particular evolution
operators.
When i say that there is a probability of finding the particle at
time Tc in a wide region after being in the state |Xb,Tb> and evolving
witht U(T,Tb) im computing it as
\int ( || <Xs,T|U(T,Tb)|Xb,Tb> ||^2 ,[/itex] Xs=region , T=Tc )
While in fact, no experimental setup detects events occuring in a
given time instant! instead one detects events in a thick time slice
of T=Tc..Tc + \Delta_T, where \Delta_T is a time in which the X-ray film
or whatever is exposed to potential events.
So actually the probability should be something like
\int ( || <Xs,T|U(T,Tb)|Xb,Tb> ||^2 , Xs=region, T=Tc.[itex].Tc + \Delta_T )
From what i see, normalizing in all 3+1 space is consistent as long
as i do no physical measurement. Is that true in general?
Furthermore, imagine i have a state of one particle described by
a|Xa,Ta> + b|Xb,Tb>. U is the evolution operator, so i have at time
Tc, Tc > Ta > Tc
aU(Tc,Ta)|Xa,Ta> + bU(Tc,Tb)|Xb,Tb>
if The rough probability density | <Xs,Tc|U(Tc,Tb)|Xb,Tb> |^2 happens
to be zero, the a measurement resulting in detecting the particle at
the |Xs,Tc> state could be interpreted in this formulation in a
projection _not_ on the state |Xs,Tc>, but actually in the state
|Xa,Ta> with probability |a|^2, which happens at a past time. Do you
know an argument why this scheme to interpret measurement would not
work?
Arnold Neumaier
Oct5-04, 06:44 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>gptejms wrote:\n> Arnold Neumaier wrote:\n>\n>\n>>This was explored by Bloch, Phys. Z. U.S.S.R., 5 (1943) 301,\n>>but is not completely consistent, since the probability\n>>density must integrate to 1 over the 3-space defined by t=const\n>>and _not_ over the 4-space, since the interpretation must\n>>reduce to the standard one in the nonrelativistic limit.\n>>Thus one would need at least a notion of probability density\n>>for being at a point on a spacelike hyperplane, for any fixed\n>>hyperplane. Or even on any spacelike hypersurface.\n>\n>\n>\n> When we are talking of the K.G.equation,we are not worried\n> about the probability density not integrating to 1 in 3-space. In the\n> non-relativistic limit,the Schrodinger equation is, in any case,\n> satisfied which guarantees that the continuity equation is\n> satisfied(and hence the probabilty\n> density integrates to 1)\n\nOnly the probability density at _fixed_ time. If you also integrate\nover time, the integral diverges - precisely this was my point.\nIt implies that the \'probability at a spacetime point\' makes no sense\nwithout referring to a specific spacelike hypersurface. Spacetime\nbehavior must be handled in terms of correlation functions.\n\n\n> Besides, I\'ve found the following paper on google which seems to be\n> discussing ideas similar to \'prob. density in spacetime\' :\n> http://arxiv.org/abs/quant-ph/0306007\n\nHe assumes at least that the probabilities integrate to 1 at an\ninitial spacelike hypersurface and a final spacelike hypersurface,\nsee (39). This is acceptable if it is assumed that nothing can be measured\nin between.\n\nUsing dynamics between two arbitrary spacelike hypersurfaces\nis close to the view of Tomonaga, mentioned earlier in this thread.\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>gptejms wrote:
> Arnold Neumaier wrote:
>
>
>>This was explored by Bloch, Phys. Z. U.S.S.R., 5 (1943) 301,
>>but is not completely consistent, since the probability
>>density must integrate to 1 over the 3-space defined by t=const
>>and _not_ over the 4-space, since the interpretation must
>>reduce to the standard one in the nonrelativistic limit.
>>Thus one would need at least a notion of probability density
>>for being at a point on a spacelike hyperplane, for any fixed
>>hyperplane. Or even on any spacelike hypersurface.
>
>
>
> When we are talking of the K.G.equation,we are not worried
> about the probability density not integrating to 1 in 3-space. In the
> non-relativistic limit,the Schrodinger equation is, in any case,
> satisfied which guarantees that the continuity equation is
> satisfied(and hence the probabilty
> density integrates to 1)
Only the probability density at _fixed_ time. If you also integrate
over time, the integral diverges - precisely this was my point.
It implies that the 'probability at a spacetime point' makes no sense
without referring to a specific spacelike hypersurface. Spacetime
behavior must be handled in terms of correlation functions.
> Besides, I've found the following paper on google which seems to be
> discussing ideas similar to 'prob. density in spacetime' :
> http://arxiv.org/abs/http://www.arxiv.org/abs/quant-ph/0306007
He assumes at least that the probabilities integrate to 1 at an
initial spacelike hypersurface and a final spacelike hypersurface,
see (39). This is acceptable if it is assumed that nothing can be measured
in between.
Using dynamics between two arbitrary spacelike hypersurfaces
is close to the view of Tomonaga, mentioned earlier in this thread.
Arnold Neumaier
Arnold Neumaier
Oct6-04, 08:03 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 J. Quarra wrote:\n\n> When i say that there is a probability of finding the particle at\n> time Tc in a wide region after being in the state |Xb,Tb> and evolving\n> witht U(T,Tb) im computing it as\n>\n> int ( || <Xs,T|U(T,Tb)|Xb,Tb> ||^2 , Xs=region , T=Tc )\n>\n> While in fact, no experimental setup detects events occuring in a\n> given time instant!\n\nIn classical physics this is accounted for by averaging values\nwith suitable weights over a time window, and in QM one can use the\nsame principle.\n\nWe know from QM the conditional probabilities Pr(X|t=T),\nso if we know the probability density P(T) for measuring at\nan uncertain time T in [T1,T2] we obtain\nPr(t in [T1,T2]) = integral_{T1,T2} dT P(T) Pr(X|t=T)\nNo multitime wave function is needed for that.\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>Charles J. Quarra wrote:
> When i say that there is a probability of finding the particle at
> time Tc in a wide region after being in the state |Xb,Tb> and evolving
> witht U(T,Tb) im computing it as
>
> \int ( || <Xs,T|U(T,Tb)|Xb,Tb> ||^2 , Xs=region , T=Tc )
>
> While in fact, no experimental setup detects events occuring in a
> given time instant!
In classical physics this is accounted for by averaging values
with suitable weights over a time window, and in QM one can use the
same principle.
We know from QM the conditional probabilities Pr(X|t=T),
so if we know the probability density P(T) for measuring at
an uncertain time T in [T1,T2] we obtain
Pr(t in [T1,T2]) = integral_{T1,T2} dT P(T) Pr(X|t=T)
No multitime wave function is needed for that.
Arnold Neumaier
Jagmeet Singh
Oct7-04, 07:05 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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<cju1eh\\$tt4\\$1@lfa222122.richmond.edu>...\ n\n> He assumes at least that the probabilities integrate to 1 at an\n> initial spacelike hypersurface and a final spacelike hypersurface,\n> see (39). This is acceptable if it is assumed that nothing can be measured\n> in between.\n\n\nHave looked at eq.(39) but fail to understand one thing.If probability\ndensity integrates to 1 at end-points,and in between there is unitary\nevolution via Schrodinger equation ,why shouldn\'t the probability\ndensity integrate to 1 in between?I can understand that for\nrelativistic quantum mechanics,prob. density integrating to one at\nendpoints doesen\'t guarantee the same for intermediate points.\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<cju1eh$tt4$1@lfa222122.richmond.edu>...
> He assumes at least that the probabilities integrate to 1 at an
> initial spacelike hypersurface and a final spacelike hypersurface,
> see (39). This is acceptable if it is assumed that nothing can be measured
> in between.
Have looked at eq.(39) but fail to understand one thing.If probability
density integrates to 1 at end-points,and in between there is unitary
evolution via Schrodinger equation ,why shouldn't the probability
density integrate to 1 in between?I can understand that for
relativistic quantum mechanics,prob. density integrating to one at
endpoints doesen't guarantee the same for intermediate points.
Arnold Neumaier
Oct8-04, 06:19 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nJagmeet Singh wrote:\n> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<cju1eh\\$tt4\\$1@lfa222122.richmond.edu>...\ n>\n>>He assumes at least that the probabilities integrate to 1 at an\n>>initial spacelike hypersurface and a final spacelike hypersurface,\n>>see (39). This is acceptable if it is assumed that nothing can be measured\n>>in between.\n>\n> Have looked at eq.(39) but fail to understand one thing.If probability\n> density integrates to 1 at end-points,and in between there is unitary\n> evolution via Schrodinger equation, why shouldn\'t the probability\n> density integrate to 1 in between?\n\nIn relativistic theories, there is no well-defined \'in-between\'.\nOf course, one expects the p.d. to integrate to 1 over arbitrary\nmaximal spacelike hypersurfaces.\n\nThe point of my remark was, however, that one cannot expect the p.d.\nto integrate to 1 if one also integrates over time...\n\n\nArnold Neumaier\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>Jagmeet Singh wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<cju1eh$tt4$1@lfa222122.richmond.edu>...
>
>>He assumes at least that the probabilities integrate to 1 at an
>>initial spacelike hypersurface and a final spacelike hypersurface,
>>see (39). This is acceptable if it is assumed that nothing can be measured
>>in between.
>
> Have looked at eq.(39) but fail to understand one thing.If probability
> density integrates to 1 at end-points,and in between there is unitary
> evolution via Schrodinger equation, why shouldn't the probability
> density integrate to 1 in between?
In relativistic theories, there is no well-defined 'in-between'.
Of course, one expects the p.d. to integrate to 1 over arbitrary
maximal spacelike hypersurfaces.
The point of my remark was, however, that one cannot expect the p.d.
to integrate to 1 if one also integrates over time...
Arnold Neumaier
Jagmeet Singh
Oct11-04, 03:52 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>\n\nArnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4165580E.5000603@univie.ac.at>...\n\n> In relativistic theories, there is no well-defined \'in-between\'.\n> Of course, one expects the p.d. to integrate to 1 over arbitrary\n> maximal spacelike hypersurfaces.\n>\n> The point of my remark was, however, that one cannot expect the p.d.\n> to integrate to 1 if one also integrates over time...\n>\n>\n> Arnold Neumaier\n\nAt non-relativistic speeds, \\delta t over which you integrate the\n4-dimensional wavefunction is expected to be very small,so that\nSchrodinger equation is satisfied in the lab. time T(see paper)--so\none expects the prob. density to integrate to one at all lab times.\nOne could expect to see interesting effects if one probes at finer\ntime\nscales---one such experiment is proposed in the\npaper(Stern Gerlach experiment in time).If you insist on prob. density\nintegrating to 1 at all time scales(finer or gross) in the\nnon-relativistic limit,then I can say that \\delta t is zero in this\nlimit.\nIt (i.e. \\delta t) begins to depart from zero as you pick up speed\ni.e. relativity comes into the picture.\n\nJagmeet Singh\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>Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4165580E.5000603@univie.ac.at>...
> In relativistic theories, there is no well-defined 'in-between'.
> Of course, one expects the p.d. to integrate to 1 over arbitrary
> maximal spacelike hypersurfaces.
>
> The point of my remark was, however, that one cannot expect the p.d.
> to integrate to 1 if one also integrates over time...
>
>
> Arnold Neumaier
At non-relativistic speeds, \delta t over which you integrate the
4-dimensional wavefunction is expected to be very small,so that
Schrodinger equation is satisfied in the lab. time T(see paper)--so
one expects the prob. density to integrate to one at all lab times.
One could expect to see interesting effects if one probes at finer
time
scales---one such experiment is proposed in the
paper(Stern Gerlach experiment in time).If you insist on prob. density
integrating to 1 at all time scales(finer or gross) in the
non-relativistic limit,then I can say that \delta t is zero in this
limit.
It (i.e. \delta t) begins to depart from zero as you pick up speed
i.e. relativity comes into the picture.
Jagmeet Singh
gptejms
Oct12-04, 10:50 AM
In relativistic theories, there is no well-defined 'in-between'.
Of course, one expects the p.d. to integrate to 1 over arbitrary
maximal spacelike hypersurfaces.
The point of my remark was, however, that one cannot expect the p.d.
to integrate to 1 if one also integrates over time...
Arnold Neumaier
At non-relativistic speeds, \delta t over which you integrate the
4-dimensional wavefunction is expected to be very small,so that
Schrodinger equation is satisfied in the lab. time T(see paper)--so
one expects the prob. density to integrate to one at all lab times.
One could expect to see interesting effects if one probes at finer time
scales---one such experiment is proposed in the
paper(Stern Gerlach experiment in time).If you insist on prob. density integrating to 1 at all time scales(finer or gross) in the non-relativistic limit,then I can say that \delta t is zero in this limit.
It (i.e. \delta t) begins to depart from zero as you pick up speed
i.e. relativity comes into the picture.
Jagmeet Singh
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