Derek McKenzie
Apr7-04, 09:02 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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\n\n\nThe standard Hilbert Space formalism of QM takes for granted the\nexistence of a space-time description which we use to describe our\nexperimental procedures and results. That is, we assume at the outset\nthat rulers and clocks exist which enable observers to set up\ndescriptions of experiments in a space-time language. Furthermore, we\nspecify an unambiguous transformation rule for relating the\nexperiences of different observers; namely by means of Projective\nUnitary Representations of the Galilei or Poincare group. For\nconvenience we will assume here our clocks and rulers to be\n\'Lorentzian\'.\n\nWith this framework in place we proceed to \'place lumps of stuff in\nsome box\' and attempt to describe its properties (or, more carefully,\nthe marks this stuff causes to appear on our registration devices)\nusing an appropriate set of Hilbert Space constructs (operators, the\ntaking of traces, etc). As we all know, this all works remarkably well\nand we find out lots of interesting things about the stuff of the\nuniverse.\n\nNow as we look around for more bits of stuff to analyse we get a bit\nadventurous and place one of our rulers in the box just to see what\nhappens. Since a ruler is just another lump of stuff we argue that we\nshould be able to describe its properties with self-adjoint operators.\nIn particular, we attempt to describe the marks it makes against the\nother \'fixed\' rulers of our framework (i.e. we measure it\'s various\nco-ordinates). We find that we fail miserably in the following sense:\nno self-adjoint operators can be found that enable us to interpret the\nruler as a set of localised particles (corresponding to ruler marks)\nwhich transform in a consistent way with those of our background\nrulers! At least, this is what we find if we assume Lorentzian rulers.\nIf we use Galilean rulers, then we can describe the behaviour of a\nruler adequately, but this is unacceptable since we know our\nbackground rulers are *actually* Lorentzian and not *approximately*\nLorentzian.\n\nGiven the elegant Galilean solution, however, we try to convince\nourselves that QM describes our rulers successfully (although only\napproximately), and we move on to the task of validating our clocks.\nThat is, we throw one of our background clocks into the box and make\nmeasurements of its \'ticks\' against those of our background clocks.\nThis time we find ourselves in an even more unacceptable situation. No\nself-adjoint operators exist that make sense of the \'ticks\' as a QM\nobservable. That is, no self-adjoint operator exists which can be\nconsidered to constitute a temporal property for the clock.\n\nThis situation is clearly very serious for it seems to say the\nfollowing: Although rulers and clocks exist (and I know this because I\nhave some of them on my desk at work!), their key properties cannot be\ndescribed within the framework of QM. But this in turn suggests that\neither QM is incomplete, or we have made some incorrect assumptions in\ntrying to derive self-adjoint operators with the \'right\' properties to\njustify the terms \'location-measurement\' and \'time-measurement\'.\n\nNow this problem is of course not new - it amounts to the\nnon-existence of Lorentz-localization Operators and Time Operators in\nQM - however the trouble I find myself facing here is that the problem\nis equally serious as the infamous Measurement Problem since it faces\nus with phenomena which, on the one hand, QM should obviously\ndescribe, but which on the other hand, the mathematics says it can\'t!\nInterestingly, whilst virtually all textbooks give the author\'s ...\ner... position ... on Schrodinger\'s Cat, the Measurement Problem, the\nEPR Paradox, etc, very little, if any, attention is usually paid to\nthis fundamental issue.\n\nSo I guess my summarizing question is: how do others \'find a home\' for\nthis dilemma? Does QFT offer us a consistent explanation for the\nexistence of rulers and clocks? If so, how? If not, how do we justify\ntreating clocks and rulers as being any different to other lumps of\nmatter in the universe? After all, the chipped rock you hold in your\nhand could equally well be called a ruler by me.\n\nDerek McKenzie\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 standard Hilbert Space formalism of QM takes for granted the
existence of a space-time description which we use to describe our
experimental procedures and results. That is, we assume at the outset
that rulers and clocks exist which enable observers to set up
descriptions of experiments in a space-time language. Furthermore, we
specify an unambiguous transformation rule for relating the
experiences of different observers; namely by means of Projective
Unitary Representations of the Galilei or Poincare group. For
convenience we will assume here our clocks and rulers to be
'Lorentzian'.
With this framework in place we proceed to 'place lumps of stuff in
some box' and attempt to describe its properties (or, more carefully,
the marks this stuff causes to appear on our registration devices)
using an appropriate set of Hilbert Space constructs (operators, the
taking of traces, etc). As we all know, this all works remarkably well
and we find out lots of interesting things about the stuff of the
universe.
Now as we look around for more bits of stuff to analyse we get a bit
adventurous and place one of our rulers in the box just to see what
happens. Since a ruler is just another lump of stuff we argue that we
should be able to describe its properties with self-adjoint operators.
In particular, we attempt to describe the marks it makes against the
other 'fixed' rulers of our framework (i.e. we measure it's various
co-ordinates). We find that we fail miserably in the following sense:
no self-adjoint operators can be found that enable us to interpret the
ruler as a set of localised particles (corresponding to ruler marks)
which transform in a consistent way with those of our background
rulers! At least, this is what we find if we assume Lorentzian rulers.
If we use Galilean rulers, then we can describe the behaviour of a
ruler adequately, but this is unacceptable since we know our
background rulers are *actually* Lorentzian and not *approximately*
Lorentzian.
Given the elegant Galilean solution, however, we try to convince
ourselves that QM describes our rulers successfully (although only
approximately), and we move on to the task of validating our clocks.
That is, we throw one of our background clocks into the box and make
measurements of its 'ticks' against those of our background clocks.
This time we find ourselves in an even more unacceptable situation. No
self-adjoint operators exist that make sense of the 'ticks' as a QM
observable. That is, no self-adjoint operator exists which can be
considered to constitute a temporal property for the clock.
This situation is clearly very serious for it seems to say the
following: Although rulers and clocks exist (and I know this because I
have some of them on my desk at work!), their key properties cannot be
described within the framework of QM. But this in turn suggests that
either QM is incomplete, or we have made some incorrect assumptions in
trying to derive self-adjoint operators with the 'right' properties to
justify the terms 'location-measurement' and 'time-measurement'.
Now this problem is of course not new - it amounts to the
non-existence of Lorentz-localization Operators and Time Operators in
QM - however the trouble I find myself facing here is that the problem
is equally serious as the infamous Measurement Problem since it faces
us with phenomena which, on the one hand, QM should obviously
describe, but which on the other hand, the mathematics says it can't!
Interestingly, whilst virtually all textbooks give the author's ...
er... position ... on Schrodinger's Cat, the Measurement Problem, the
EPR Paradox, etc, very little, if any, attention is usually paid to
this fundamental issue.
So I guess my summarizing question is: how do others 'find a home' for
this dilemma? Does QFT offer us a consistent explanation for the
existence of rulers and clocks? If so, how? If not, how do we justify
treating clocks and rulers as being any different to other lumps of
matter in the universe? After all, the chipped rock you hold in your
hand could equally well be called a ruler by me.
Derek McKenzie
existence of a space-time description which we use to describe our
experimental procedures and results. That is, we assume at the outset
that rulers and clocks exist which enable observers to set up
descriptions of experiments in a space-time language. Furthermore, we
specify an unambiguous transformation rule for relating the
experiences of different observers; namely by means of Projective
Unitary Representations of the Galilei or Poincare group. For
convenience we will assume here our clocks and rulers to be
'Lorentzian'.
With this framework in place we proceed to 'place lumps of stuff in
some box' and attempt to describe its properties (or, more carefully,
the marks this stuff causes to appear on our registration devices)
using an appropriate set of Hilbert Space constructs (operators, the
taking of traces, etc). As we all know, this all works remarkably well
and we find out lots of interesting things about the stuff of the
universe.
Now as we look around for more bits of stuff to analyse we get a bit
adventurous and place one of our rulers in the box just to see what
happens. Since a ruler is just another lump of stuff we argue that we
should be able to describe its properties with self-adjoint operators.
In particular, we attempt to describe the marks it makes against the
other 'fixed' rulers of our framework (i.e. we measure it's various
co-ordinates). We find that we fail miserably in the following sense:
no self-adjoint operators can be found that enable us to interpret the
ruler as a set of localised particles (corresponding to ruler marks)
which transform in a consistent way with those of our background
rulers! At least, this is what we find if we assume Lorentzian rulers.
If we use Galilean rulers, then we can describe the behaviour of a
ruler adequately, but this is unacceptable since we know our
background rulers are *actually* Lorentzian and not *approximately*
Lorentzian.
Given the elegant Galilean solution, however, we try to convince
ourselves that QM describes our rulers successfully (although only
approximately), and we move on to the task of validating our clocks.
That is, we throw one of our background clocks into the box and make
measurements of its 'ticks' against those of our background clocks.
This time we find ourselves in an even more unacceptable situation. No
self-adjoint operators exist that make sense of the 'ticks' as a QM
observable. That is, no self-adjoint operator exists which can be
considered to constitute a temporal property for the clock.
This situation is clearly very serious for it seems to say the
following: Although rulers and clocks exist (and I know this because I
have some of them on my desk at work!), their key properties cannot be
described within the framework of QM. But this in turn suggests that
either QM is incomplete, or we have made some incorrect assumptions in
trying to derive self-adjoint operators with the 'right' properties to
justify the terms 'location-measurement' and 'time-measurement'.
Now this problem is of course not new - it amounts to the
non-existence of Lorentz-localization Operators and Time Operators in
QM - however the trouble I find myself facing here is that the problem
is equally serious as the infamous Measurement Problem since it faces
us with phenomena which, on the one hand, QM should obviously
describe, but which on the other hand, the mathematics says it can't!
Interestingly, whilst virtually all textbooks give the author's ...
er... position ... on Schrodinger's Cat, the Measurement Problem, the
EPR Paradox, etc, very little, if any, attention is usually paid to
this fundamental issue.
So I guess my summarizing question is: how do others 'find a home' for
this dilemma? Does QFT offer us a consistent explanation for the
existence of rulers and clocks? If so, how? If not, how do we justify
treating clocks and rulers as being any different to other lumps of
matter in the universe? After all, the chipped rock you hold in your
hand could equally well be called a ruler by me.
Derek McKenzie