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

(Derek McKenzie) writes:

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.

Well, in these theories we have a difference between parameters,
such as time and location, and dynamical variables. The angle of
a clock hand is a dynamical variable, as is the distance
of an object from one end of a ruler. On the other hand, the
supposed time and location of the various events are parameters,
and we consider the dynamical variables which track the putative
parameters as closely as possible. The QM no-go theorems tell us
that we can't have dynamical variables which are in perfect sync
with the parameters.

R.