From: Mitchell.Porter@launchpad.unc.edu (Mitchell Porter)
Newsgroups: sci.physics,alt.sci.physics.new-theories,sci.philosophy.meta
Subject: Bell's paper on QFT - Part 2
Date: 12 Jul 1994 09:56:41 -0400
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Summary: ... or wave function collapse, or anything like that.
Xref: columba.udac.uu.se sci.physics:14592 alt.sci.physics.new-theories:1907 sci.philosophy.meta:1978
Part Two of a transcription of Chapter 19 of:
_Speakable and unspeakable in quantum mechanics_ by J.S. Bell
3. Dynamics
For the time evolution of the state vector we retain the ordinary Schroedinger
equation,
d/dt |t> = - iH|t> (4)
where H is the ordinary Hamiltonian operator.
For the fermion number configuration we prescribe a stochastic development.
In a small time interval dt configuration m jumps to configuration n with
probability
dt Tnm (5)
where
Tnm = Jnm/Dm (6)
Jnm = SUM 2 Re (7)
qp
Dm = SUM || ^2 (8)
q
provided Jnm > 0, but
Tnm = 0 if Jnm <= 0 (9)
From (5) the evolution of a probability distribution Pn over configurations
n is given by
d/dt Pn = SUM (Tnm Pm - Tmn Pn) (10)
m
Compare this with a mathematical consequence of the Schroedinger equation (4):
d/dt ||^2 = SUM 2 Re
mp
or
d/dt Dn = SUM Jnm = SUM (Tnm Dm - Tmn Dn) (11)
m m
If we assume that at some initial time
Pn(0) = Dn(0) (12)
then from (11) the solution of (10) is
Pn(t) = Dn(t) (13)
Envisage then the following situation. In the beginning God chose 3-space
and 1-time, a Hamiltonian H, and a state vector |0>. Then She chose a fermion
configuration n(0). This She chose at random from an ensemble of possibilities
with distribution D(0) related to the already chosen state vector |0>. Then
She left the world alone to evolve according to (4) and (5).
It is notable that although the probability distribution P in (13) is
governed by D and so by |t>, the latter is not to be thought of as just a way
of expressing the probability distribution. For us |t> is an independent
beable of the theory. Otherwise its appearance in the transition probabilities
(5) would be quite unintelligible.
The stochastic transition probabilities (5) replace here the deterministic
guiding equation of the de Broglie-Bohm "pilot wave" theory. The introduction
of a stochastic element, for beables with discrete spectra, is unwelcome, for
the reversibility [14] of the Schroedinger equation strongly suggests that
quantum mechanics is not fundamentally stochastic in nature. However I suspect
that the stochastic element introduced here goes away in some sense in the
continuum limit.
4. OQFT and BQFT
OQFT is "ordinary" "orthodox" "observable" quantum field theory, whatever
that may mean. BQFT is de Broglie-Bohm beable quantum field theory. To what
extent do they agree? The main difficulty with the question is the absence of
any sharp formulation of OQFT. We will consdier two different ways of reducing
the ambiguity.
In OQFT1 the world is considered as one big experiment. God prepared it at
the initial time t = 0, and let it run. At some much later time T She will
return to judge the outcome. In particular She will observe the contents of
all the physics journals. This will include of course the records of our own
little experiments - as distributions of ink on paper, and so of fermion
number. From (13) the OQFT1 probabilty D that God will observe one
configuration rather than another is identical with the BQFT probability P
that the configuration _is_ then one thing rather than another. In this sense
there is complete agreement between OQFT1 and BQFT on the result of God's big
experiment - including the results of our little ones.
OQFT1, in contrast with BQFT, says nothing about events in the system in
between preparation and observation. However adequate this may be from an
Olympian point of view, it is rather unsatisfactory for us. We live in between
creation and last judgement - and imagine that we experience events. In this
respect another version of OQFT is more appealing. In OQFT2, whenever the
state can be resolved into a sum of two (or more) terms
|t> = |t,1> + |t,2> (14)
which are "macroscopically different", then in disregard for the Schroedinger
equation the state "collapses" somehow into one term or the other:
|t> -> (N1)^(-1/2) |t,1> with probability N1 (15)
|t> -> (N2)^(-1/2) |t,1> with probability N2
where
N1 = || N2 = || (16)
In this way the state is always, or nearly always, macroscopically unambiguous
and defines a macroscopically definite history for the world. The words
"macroscopic" and "collapse" are terribly vague. Nevertheless this version of
OQFT is probably the nearest approach to a rational formulation of how we use
quantum theory in practice.
Will OQFT2 agree with OQFT1 and BQFT at the final time T? This is the main
issue in what is usually called "the Quantum Measurement Problem". Many
authors, analyzing many models, have convinced themselves that the state
vector collapse of OQFT2 is consistent with the Schroedinger equation of OQFT1
"for all practical purposes".[15] The idea is that even when we retain both
components in (13), evolving as required by the Schroedinger equation, they
remain so different as not to interfere in the calculation of anything of
interest. The following sharper form of this hypothesis seems plausible to me:
the macroscopically distinct components remain so different, for a very long
time, as not to interfere in the calculation of D and J.[5] In so far as this
is true, the trajectories of OQFT2 and BQFT will agree macroscopically.
5. Concluding remarks
We have seen that BQFT is in complete accord with OQFT1 as regards the final
outcome. It is plausibly consistent with OQFT2 in so far as the latter is
unambiguous. BQFT already has the advantage over OQFT1 of being relevant at
all times, and not just at the final time. It is superior to OQFT2 in being
completely formulated in terms of unambiguous equations.
Yet even BQFT does not inspire complete happiness. For one thing there is
nothing unique about the choice of fermion number density as basic local
beable. We could have others instead, or in addition. For example the Higgs
field of contemporary gauge theories could serve very well to define "the
positions of things". Other possibilities have been considered by K.
Baumann.[4] I do not see how this choice can be made experimentally
significant, so long as the final results of experiments are defined so
grossly as by the positions of instrument pointers, or of ink on paper.
And the status of Lorenz invariance is very curious. BQFT agrees with OQFT
on the result of the Michelson-Morley experiment, and so on. But the
formulation of BQFT relies heavily upon a particular division of space-time
into space and time. Could this be avoided?
There is indeed a trivial way of imposing Lorentz invariance.[4] We can
imagine the world to differ from vacuum only over a limited region of infinite
Euclidean space (we forget general relativity here). Then an overall
centre of mass system is defined. We simply assert that our equations hold
in this centre of mass system. Our scheme is then Lorentz invariant. Many
others could be made Lorentz invariant in the same way ... for example
Newtonian mechanics. But such Lorentz invariance would not imply a null
result for the Michelson-Morley experiment ... which could detect motion
relative to the cosmic mass centre. To be predictive, Lorentz invariance
must be supplemented by some kind of locality, or separability,
consideration. Only then, in the case of a more or less isolated object,
can motion relative to the world as a whole be deemed more or less irrelevant.
I do not know of a good general formulation of such a locality requirement.
In classical field theory, part of the requirement could be formulation in
terms of differential (as distinct from integral) equations in 3+1 dimensional
space-time. But it seems clear that quantum mechanics requires a much bigger
configuration space. One can formulate a locality requirement by permitting
arbitrary external fields, and requiring that variation thereof have
consequences only in their future light cones. In that case the fields could
be used to set measuring instruments, and one comes into difficulty with
quantum predictions for correlations related to those of Einstein, Podolsky
and Rosen.[18] But the introduction of external fields is questionable. So I
am unable to prove, or even formulate clearly, the proposition that a sharp
formulation of quantum field theory. such as that set out here, must
disrespect serious Lorentz invariance. But it seems to me that this is
probably so.
As with relativity before Einstein, there is then a preferred frame in the
formulation of the theory ... but it is experimentally indistinguishable.
[20,21,22] It seems an eccentric way to make a world.
Notes and references
1. D. Bohm, _Phys.Rev._ 85, 166 (1952).
2. D. Bohm, _Phys.Rev._ 85, 180 (1952).
3. D. Bohm and B. Hiley, _Found.Phys._ 14, 270 (1984).
4. K. Baumann, preprint, Graz (1984).
5. J.S. Bell, _Phys.Rep._ 137, 49-54 (1986).
6. T.H.R. Skyrme, _Proc.Roy.Soc._ A260, 127 (1961). A.S. Goldhaber,
_Phys.Rev.Lett._ 36, 1122 (1976). F. Wilczek and A. Zee, _Phys.Rev.Lett._ 51,
2250 (1983).
7. L. de Broglie, _Tentative d'Interpretation Causale et Nonlineaire de la
Mechanique Ondulatoire_. Gauthier-Villars, Paris (1956).
8. J.S. Bell, _Rev.Mod.Phys._ 38, 447 (1966).
9. J.S. Bell, in _Quantum Gravity_, p.611. Edited by Isham, Penrose and
Sciama, Oxford (1981). (Originally TH. 1424-CERN, 1971 Oct 27).
10. J.S. Bell, _Found.Phys._ 12, 989 (1982).
11. A. Shimony, _Epistemological Letters_, Jan 1978, 1.
12. B. Zumino, private communication.
13. B. d'Espagnat, _Phys.Rep._ 110, 202-63 (1984).
14. I ignore here the small violation of time reversibility that has shown up
in elementary particle physics. It could be of "spontaneous" origin. Moreover
PCT remains good.
15. This is touched on in Refs. 9, 16, and 17, and in many papers inn the
anthology of Wheeler and Zurek Ref.19.
16. J.S. Bell, _Helv.Phys.Acta_ 48 (93) (1975).
17. J.S. Bell, _Int.J.Quant.Chem._: Quantum Chemistry Symposium 14, p155
(1980).
18. J.S. Bell, _Journal de Physique_, Colloque C2, suppl. au no 3, Tome 42,
pC2-41, mars 1981.
19. J.A. Wheeler and W.H. Zurek (editors), _Quantum Theory and Measurement_.
Princeton University Press, Princeton 1983.
20. J.S. Bell, in _Determinism, Causality, and Particles_, p17. Edited by
M. Flato et al. Dordrecht-Holland, D. Rediel (1976).
21. P.H. Eberhardt, _Nuovo Cimento_ 46B, 392 (1978).
22. K. Popper, _Found.Phys._ 12, 971 (1982).
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Subject: (fwd) Bell's Paper on QFT, Part 1
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From: Mitchell.Porter@launchpad.unc.edu (Mitchell Porter)
Newsgroups: sci.physics,alt.sci.physics.new-theories,sci.philosophy.meta
Subject: Bell's Paper on QFT, Part 1
Date: 12 Jul 1994 09:06:20 -0400
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Summary: ... or measurements, or systems, or apparatus, ...
Xref: columba.udac.uu.se sci.physics:14589 alt.sci.physics.new-theories:1905 sci.philosophy.meta:1977
Part One of a transcription of Chapter 19 of:
_Speakable and unspeakable in quantum mechanics_ by J.S. Bell
Reproduced without permission
"Beables for Quantum Field Theory" J.S. Bell
Dedicated to Professor D. Bohm
1. Introduction
Bohm's 1952 papers [1,2] on quantum mechanics were for me a revelation. The
elimination of indeterminism was very striking. But more important, it seemed
to me, was the elimination of any need for a vague division of the world into
"system" on the one hand, and "apparatus" or "observer" on the other. I have
always felt since that people who have not grasped the ideas of those papers
... and unfortunately they remain the majority ... are handicapped in any
discussion of the meaning of quantum mechanics.
When the cogency of Bohm's reasoning is admitted, a final protest is often
this: it is all nonrelativistic. This is to ignore that Bohm himself, in an
appendix to one of the 1952 papers[2], already applied his scheme to the
electromagnetic field. And application to scalar fields is straightforward[3].
However until recently[4,5], to my knowledge, no extension covering Fermi
fields had been made. Such an extension will be sketched here. The need for
Fermi fields might be questioned. Fermions might be composite structures of
some kind[6]. But also they might not be, or not all. The present exercise
will not only include Fermi fields, but even give them a central role. The
dependence on the ideas of de Broglie[7] and Bohm[1,2], and also on my own
simplified extension to cover spin[8,9,10], will be manifest to those
familiar with these things. However no such familiarity will be assumed.
A preliminary account of these notions[5] was entitled "Quantum field theory
without observers, or observables, or measurements, or systems, or apparatus,
or wavefunction collapse, or anything like that". This could suggest to some
that the issue in question is a philosophical one. But I insist that my
concern is strictly profession. I think that conventional formulations of
quantum theory, and of quantum field theory in particular, are
unprofessionally vague and ambiguous. Professional theoretical physicists
ought to be able to do better. Bohm has shown us a way.
It will be seen that all the essential results of ordinary quantum field
theory are recovered. But it will also be seen that the very sharpness of the
reformulation brings into focus some awkward questions. The construction of
the scheme is not at all unique. And Lorentz invariance plays a strange,
perhaps incredible, role.
2. Local beables
The usual approach, centred on the notion of "observable", divides the world
somehow into parts: "system" and "apparatus". The "apparatus" interacts from
time to time with the "system", "measuring" "observables". During
"measurement" the linear Schroedinger evolution is suspended, and an
ill-defined "wave-function collapse" takes over. There is nothing in the
mathematics to tell what is "system" and what is "apparatus", nothing to tell
which natural processes have the status of "measurements". Discretion and good
taste, born of experience, allow us to use quantum theory with marvelous
success, despite the ambiguity of the concepts named above in quotation marks.
But it seems clear that in a serious fundamental formulation such concepts
much be excluded.
In particular we will exclude the notion of "observable" in favour of that
of "_be_able". The _be_ables of the theory are those elements which might
correspond to elements of reality, to things which exist. Their existence does
not depend on "observation". Indeed observation and observer must be made out
of beables.
I use the term "_be_able" instead of some more committed term like "being"
[11] or "beer" [12] to recall the essentially tentative nature of any physical
theory. Such a theory is at best a _candidate_ for the description of nature.
Terms like "being", "beer", "existent" [11-13], etc., would seem to me lacking
in humility. In fact "beable" is short for "maybe-able".
Let us try to promote some of the usual "observables" to the status of
beables. Consider the conventional axiom:
the probability of observables (A, B,...) (1)
if observed at time t
being observed to be (a, b,...)
is
SUM || ^2
q
where q denotes additional quantum numbers which
together with the eigenvalues (a, b,..)
form a complete set.
This we replace by
the probability of beables (A, B,...) (2)
at time t
being (a, b,...)
is
SUM || ^2
q
where q denotes additional quantum numbers which
together with the eigenvallues (a, b,...)
form a complete set.
Not all "observables" can be given beable status, for they do not all have
simultaneous eigenvalues, i.e. do not all commute. It is important to realize
therefore that most of these "observables" are entirely redundant. What is
essential is to be able to define the positions of things, including the
positions of instrument pointers or (the modern equivalent) of ink on computer
output.
In making precise the notion "positions of things" the energy density
T00(x) comes to mind. However the commutator
[T00(x),T00(y)]
is not zero, but proportional to derivatives of delta functions. So the T00(x)
do not have simultaneous eigenvalues for all x. We would have to devise some
new way of specifying a joint probability distribution.
We fall back then on a second choice - fermion number density. The
distribution of fermion number in the world certainly includes the positions
of instruments, instrument pointers, ink on paper... and much much more.
For simplicity we replace the three-space-continuum by a dense lattice,
keeping time t continuous (and real!). Let the lattice points be
enumerated by
l = 1,2,...L
where L is very large. Define lattice point fermion number operators
Psi+(l)Psi(l)
where summation over Dirac indices and over all Dirac fields is understood.
The corresponding eigenvalues are integers
F(l) = 1,2,...4N
where N is the number of Dirac fields. The fermion number configuration of the
world is a list of such integers
n = (F(1),F(2),...F(L))
We suppose the world to have a definite such configuration at every time t:
n(t)
The lattice fermion numbers are the local beables of the theory, being
associated with definite positions in space. The state vector |t> also we
consider as a beable, although not a local one. The complete specification of
our world at time t is then a combination
(|t>, n(t))
It remains to specify the time evolution of such a combination.
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