quantum mechanics and curvature of space-time [Archive]

alistair

Sep27-04, 03:31 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>\nThe amplitude of a wavefunction squared is proportional to the\nprobability of finding a particle at a position in space.\nThis squaring of the amplitude is often compared to the intensity\n(energy density) of an electromagnetic wave.\nThe stress energy momentum tensor of general relativity contains\nthe term Too - the energy density.Can Too be related to the\nsquare of the amplitude of the quantum mechanical wavefunction?\nIs Too (and therefore the curvature of spacetime due to energy\ndensity) proportional to the probability of finding a particle at a\nposition in space?\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>The amplitude of a wavefunction squared is proportional to the
probability of finding a particle at a position in space.
This squaring of the amplitude is often compared to the intensity
(energy density) of an electromagnetic wave.
The stress energy momentum tensor of general relativity contains
the term Too - the energy density.Can Too be related to the
square of the amplitude of the quantum mechanical wavefunction?
Is Too (and therefore the curvature of spacetime due to energy
density) proportional to the probability of finding a particle at a
position in space?

Igor Khavkine

Sep28-04, 10:20 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\nalistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0409241312.3e02324@posting.google.c om>...\n> The amplitude of a wavefunction squared is proportional to the\n> probability of finding a particle at a position in space.\n> This squaring of the amplitude is often compared to the intensity\n> (energy density) of an electromagnetic wave.\n> The stress energy momentum tensor of general relativity contains\n> the term Too - the energy density.Can Too be related to the\n> square of the amplitude of the quantum mechanical wavefunction?\n> Is Too (and therefore the curvature of spacetime due to energy\n> density) proportional to the probability of finding a particle at a\n> position in space?\n\nInstead of wondering how T_{00} could be related to the wavefunction,\none can simply calculate it. T_{00} is a function of the dynamical variables\nof the theory (a function of its momentum for a single particle) and as\nsuch is an observable. Observables are represented as hermitian operators\nthat we can calculate expecration values of. What you are looking for\nis <T_{00}> whose relation to the wavefunction is quite obvious.\n\nAs to how <T_{00}>, or in more generality <T_{ij}>, affects the equations\nof motion for the gravitational field, I don\'t really know. The biggest\nproblem is that gravity coupled to matter must be consisten (i.e. the\ninduced curvature induces the right equations of motion for matter)\nand that\'s hard to do when part of the system is treated classically and\nthe other quantum-mechanically. But I do remember many of the details\nsubtleties were hashed out between Arnold Neumaier and Arkadiusz Jadczyk\nin this thread: http://groups.google.ca/groups?th=b7be680c23c06f9d\n\nIgor\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>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0409241312.3e02324@posting.google.com>...
> The amplitude of a wavefunction squared is proportional to the
> probability of finding a particle at a position in space.
> This squaring of the amplitude is often compared to the intensity
> (energy density) of an electromagnetic wave.
> The stress energy momentum tensor of general relativity contains
> the term Too - the energy density.Can Too be related to the
> square of the amplitude of the quantum mechanical wavefunction?
> Is Too (and therefore the curvature of spacetime due to energy
> density) proportional to the probability of finding a particle at a
> position in space?

Instead of wondering how T_{00} could be related to the wavefunction,
one can simply calculate it. T_{00} is a function of the dynamical variables
of the theory (a function of its momentum for a single particle) and as
such is an observable. Observables are represented as hermitian operators
that we can calculate expecration values of. What you are looking for
is <T_{00}> whose relation to the wavefunction is quite obvious.

As to how <T_{00}>, or in more generality <T_{ij}>, affects the equations
of motion for the gravitational field, I don't really know. The biggest
problem is that gravity coupled to matter must be consisten (i.e. the
induced curvature induces the right equations of motion for matter)
and that's hard to do when part of the system is treated classically and
the other quantum-mechanically. But I do remember many of the details
subtleties were hashed out between Arnold Neumaier and Arkadiusz Jadczyk
in this thread: http://groups.google.ca/groups?th=b7be680c23c06f9d

Igor

Arnold Neumaier

Sep28-04, 10:20 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>\nalistair wrote:\n> The amplitude of a wavefunction squared is proportional to the\n> probability of finding a particle at a position in space.\n> This squaring of the amplitude is often compared to the intensity\n> (energy density) of an electromagnetic wave.\n> The stress energy momentum tensor of general relativity contains\n> the term Too - the energy density.Can Too be related to the\n> square of the amplitude of the quantum mechanical wavefunction?\n> Is Too (and therefore the curvature of spacetime due to energy\n> density) proportional to the probability of finding a particle at a\n> position in space?\n\nIn very simple cases (single particle in an external potential),\n<T_00(x)> is proportional to |psi(x)|^2.\n\nIn general, the expectation of the number density, not the energy\ndensity, is proportional to the probability of finding a particle\nat a position in space. For relativistic QFT, the Baryon or\nlepton number can be negative, however, and the interpretation\nin terms of probabilities breaks down.\n\n\nArnold Neumaier\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>alistair wrote:
> The amplitude of a wavefunction squared is proportional to the
> probability of finding a particle at a position in space.
> This squaring of the amplitude is often compared to the intensity
> (energy density) of an electromagnetic wave.
> The stress energy momentum tensor of general relativity contains
> the term Too - the energy density.Can Too be related to the
> square of the amplitude of the quantum mechanical wavefunction?
> Is Too (and therefore the curvature of spacetime due to energy
> density) proportional to the probability of finding a particle at a
> position in space?

In very simple cases (single particle in an external potential),
<T_{00}(x)> is proportional to |\psi(x)|^2.

In general, the expectation of the number density, not the energy
density, is proportional to the probability of finding a particle
at a position in space. For relativistic QFT, the Baryon or
lepton number can be negative, however, and the interpretation
in terms of probabilities breaks down.

Arnold Neumaier

Ken S. Tucker

Sep29-04, 02:25 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\nalistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0409241312.3e02324@posting.google.c om>...\n> The amplitude of a wavefunction squared is proportional to the\n> probability of finding a particle at a position in space.\n> This squaring of the amplitude is often compared to the intensity\n> (energy density) of an electromagnetic wave.\n> The stress energy momentum tensor of general relativity contains\n> the term Too - the energy density.Can Too be related to the\n> square of the amplitude of the quantum mechanical wavefunction?\n> Is Too (and therefore the curvature of spacetime due to energy\n> density) proportional to the probability of finding a particle at a\n> position in space?\n\nOffhand I would say yes but with an unorthodox caveat,\nbut I\'m curious too.\n\nThe usual Einstein Law is G_uv = T_uv (approximately).\n\nWhen solving that equation near the Sun it is conventional\nto set G_uv =0 meaning the energy density is zero and the\npoint where the solution occurs is regarded as a pure vacuum.\n\nBut I ask and question "what is a pure vacuum"?\n\nDo we measure a volume of 1 cubic meter and find no\nparticles, or should we measure 1,000,000 cubic miles\nand find the Sun within that volume and arrive at a\ndifferent answer for density?\n\nSo the G_uv may very well be a geometric representation\nof the probability field, provided we define what a\nvacuum is and over what volume we should use to calculate\nenergy density.\nRegards\nKen S. Tucker\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>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0409241312.3e02324@posting.google.com>...
> The amplitude of a wavefunction squared is proportional to the
> probability of finding a particle at a position in space.
> This squaring of the amplitude is often compared to the intensity
> (energy density) of an electromagnetic wave.
> The stress energy momentum tensor of general relativity contains
> the term Too - the energy density.Can Too be related to the
> square of the amplitude of the quantum mechanical wavefunction?
> Is Too (and therefore the curvature of spacetime due to energy
> density) proportional to the probability of finding a particle at a
> position in space?

Offhand I would say yes but with an unorthodox caveat,
but I'm curious too.

The usual Einstein Law is G_{uv} = T_{uv} (approximately).

When solving that equation near the Sun it is conventional
to set G_{uv} =0 meaning the energy density is zero and the
point where the solution occurs is regarded as a pure vacuum.

But I ask and question "what is a pure vacuum"?

Do we measure a volume of 1 cubic meter and find no
particles, or should we measure 1,000,000 cubic miles
and find the Sun within that volume and arrive at a
different answer for density?

So the G_{uv} may very well be a geometric representation
of the probability field, provided we define what a
vacuum is and over what volume we should use to calculate
energy density.
Regards
Ken S. Tucker

chris h fleming

Oct1-04, 04:22 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\nalistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0409241312.3e02324@posting.google.c om>...\n> The amplitude of a wavefunction squared is proportional to the\n> probability of finding a particle at a position in space.\n> This squaring of the amplitude is often compared to the intensity\n> (energy density) of an electromagnetic wave.\n> The stress energy momentum tensor of general relativity contains\n> the term Too - the energy density.Can Too be related to the\n> square of the amplitude of the quantum mechanical wavefunction?\n> Is Too (and therefore the curvature of spacetime due to energy\n> density) proportional to the probability of finding a particle at a\n> position in space?\n\nYou either need to start with a relativistic formulation of QM or with\nQFT. Take a look at Semiclassical gravity.\n\nhttp://relativity.livingreviews.org/Articles/lrr-2004-3/articlesu2.html#x8-50003.1\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>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0409241312.3e02324@posting.google.com>...
> The amplitude of a wavefunction squared is proportional to the
> probability of finding a particle at a position in space.
> This squaring of the amplitude is often compared to the intensity
> (energy density) of an electromagnetic wave.
> The stress energy momentum tensor of general relativity contains
> the term Too - the energy density.Can Too be related to the
> square of the amplitude of the quantum mechanical wavefunction?
> Is Too (and therefore the curvature of spacetime due to energy
> density) proportional to the probability of finding a particle at a
> position in space?

You either need to start with a relativistic formulation of QM or with
QFT. Take a look at Semiclassical gravity.

http://relativity.livingreviews.org/Articles/lrr-2004-3/articlesu2.html#x8-50003.1