by R. F. Streater

This page contains some remarks about research topics in physics which seem to me not to be suitable for students. Sometimes I form this view because the topic is too difficult, and sometimes because it has passed its do-by date. Some of the topics, for one reason or another, have not made any convincing progress.

- I. Hidden Variables
- II. Nelson's Stochastic Mechanics
- III. Quantum Logic
- IV. Trivalent Logic
- V. The Scalar Wightman Theory in 4 Space-Time Dimensions
- VI. Jordan Algebras, Octonions, p-adics, Quaternions
- VII. Physics from Fisher Information
- VIII. Phase, the Operator Conjugate to the Number Operator
- IX. Rigorous Feynman Path Integrals
- X. Euclidean Gravity
- XI. Bohmian Mechanics
- XII. The Many-Worlds Interpretation of QM
- XIII. Non-self-adjoint observables
- XIV. Analytic S-matrix bootstrap
- XV. Wolfram's physics as a computer programme
- XVI. Dirac's programme of quantisation
- XVII. Converting R. Penrose to the Copenhagen view
- XVIII. Mach's principle
- IXX. Spheralons
- XX. Hilbert space as classical phase space
- XXI. Spin-statistics `theorems' in non-relativistic quantum mechanics
- XXII.
*Diff M*as a gauge group - XXIII. First causes in physics
- XXIV. Stapp's quantum theory of the brain
- XXV. The New Scientist
- XXVI. Other lists of hopeless theories
- For more links, see Frieder Kleefeld.

This subject has been thoroughly worked out and is now understood. A thesis on this topic, even a correct one, will not get you a job.

Quantum theory is a generalisation of probability theory, and quantum mechanics is a generalisation of a stochastic process. The theory of a stochastic process {X(t)} was first put on a sound mathematical footing by Kolmogorov in his book of 1933. By a Kolmogorovian theory, we shall mean a set-up that conforms to the axioms of this book. Later in life, Kolmogorov played with the ideas of von Mises, who tried to found probability theory on a frequentist philosophy. This programme was made completely rigorous by van Lambalgen, who, according to Gill, had to replace the axiom of choice by a new notion. We do not refer to this late-life conversion as Kolmogorovian in this article. Indeed, we call the frequentist approach "preKolmogorovian". Perhaps a better word is "contextual", as will be explained.

The key to Kolmogorov's theory of a stochastic process {X(t)} is
the proof that there exists a single sample space, on which all the
X(t) in the process are
random variables. This sample space is the space of paths: at each time,
X(t) has a definite value (for each sample), so X(t) is a random version
of an "element of reality" in the words of Einstein, Podolski and Rosen.
They say that, in any theory, a concept (such as that given by a symbol)
arising in the theory
is an element of reality if it takes a real value. Surely, they meant to
allow that it could take different values in different states. We may then
take the set of states envisaged by EPR as the sample space Omega, and an
element of
reality is then a real-valued function on Omega. If we add the technical
requirement that the function has to be measurable, then the set of
elements of reality is a subset of the set of random variables on Omega,
and we can use Kolmogorov's theory.
Bell's inequalities, which hold in any Kolmogorovian theory, do not
hold in quantum mechanics. This shows that
there are some predictions of quantum theory
that cannot be obtained from **any** Kolmogorovian theory. This has
been put to the
test; the work of Aspect, Dalibrand and Roger (1982) shows that for some
systems of two correlated photons, the best experimental estimates for
certain spin correlations
violate the Bell inequalities, and so cannot be explained by
classical probability theory, with or without hidden variables. The
statistics observed in the experiments do however agree very well with the
quantum predictions. Bohm is said (by some) to have found a hidden
assumption, that of "locality", in Bell's theorem. In earlier work, Bohm
had certainly proposed a theory with a quantum potential which has
instantaneous influence over arbitrarily large distances. It is true that
Bell made the assumption, universal among probabilists since Kolmogorov's
book, that there exists
a sample space on which all the non-hidden variables are random
variables. Bohm's new suggestion was to allow yourself the freedom to
choose to represent a given observable X by different random variables (on
different spaces), when the experimental context is different. This is
similar to, but more definite than, the "Copenhagen" view of
Niels Bohr, that one's view of reality depends on what is being
measured. Bohm
allowed the context to be deemed to be different for observer Alice=A in
the following set-up. The system is in a given quantum state. A is
measuring the given observable X, while another observer Bob=B, a long way
from A, is measuring either an observable Y, or an observable Z
complementary to Y. We are told that (X,Y) is a compatible set of
observables, and also that (X,Z) is a compatible set. By allowing that the
random variable chosen by A to model X if B measures Y to be a different
random variable from the one she chooses if Bob measures Z, one can
reproduce the quantum correlations exactly. Each pair, (X,Y) and
(X,Z), generates a commutative subalgebra of the algebra of
observables, and these two subalgebras are not
equal. PreKolmogorovians, and some modern
statisticians would regard (X,Y) as one statistical theory, and
(X,Z) as another. Since the correlation between Y and Z cannot be given
(they are incompatible experiments) we cannot reconstruct, from the data
given, a
single sample space on which all three observables are random
variables. One can reproduce the observed quantum correlations
exactly by allowing X in the theory of the pair (X,Y), to be a different
random variable
from the X used in the study of the pair (X,Z). The need for this freedom
of choice is bizarrely
called non-locality by Bohm (it is nothing to do with locality in
space-time). The modern word is "contextual": the choice of statistical
model depends on the context: what variable is B measuring? To set up
the two models, consider the quantum mechanical state of the system; this
defines states on two abelian algebras, A(X,Y) and A(X,Z), that generated
by X,Y and that generated by X,Z, respectively. Then the restriction
of the given quantum state to one or other of these subalgebras gives
us a state on each algebra, and from this, we get two
statistical models. The
representation of the observables as random variables is, in each case,
then an
immediate consequence of Gelfand's general construction of a
representation of a commutative C* sub-algebra by bounded random
variables defined on the spectrum of the subalgebra. We see that the
random variable used by A to represent observable X depends on which
subalgebra one chooses. Even the sample space is different. Care is
needed in contemplating this; it is not true
that the measurement by B influences A instantaneously. Alice
cannot even know what experiment, Y or Z, B is performing, until she gets
a message (contrary to the claim of R. Penrose, in *The Emperor's New
Mind*). She cannot set up a classical description of the joint
measurement of X and Y (or X and Z) until later, when Alice has heard
from Bob, as to what he is measuring. See
*Classical and Quantum Probability*,
Journal of Math. Phys., **41**, 3556-3603, June 2000. In this article,
I describe Bohm's suggestion as preKolmogorovian probability, as it uses
the frequentist philosophy, and uses many sample spaces for the same
observable in different contexts. In
this sense, it fails to have any "elements of reality", not even
random ones. Frequentist probability has no predictive capability, and is
merely
a noting of the experimental statistics. For the abstract, see this page, and the full article (early
version) is in the archive,
math-ph/0002049.
Below, I describe the original (1952) attempt by Bohm at a purely
classical interpretation of Schrodinger's equation.

Bell's theorem, together with the experiments of Aspect et al., shows that the theoretical idea to use hidden classical variables to replace quantum theory is certainly a lost cause, and has been for forty years. This conclusion should not be taken to mean that entangled states are fully understood; there is still exciting work to be done in the experimental study of some of the quantum predictions related to quantum information theory and quantum communication.

For an account of the EPR `paradox' in the Copenhagen interpretation, see this page.

For a stimulating discussion about quantum mechanics and quantum gravity, and a new model theory, see the article by Prugovecki.

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Nelson invented stochastic mechanics as a dynamical theory of a particle
governed by a classical stochastic process, which gives the same answers
as quantum mechanics for the time-evolution of the probability
distribution of the particle's position.
He later abandonned it as not correct physics. See Nelson's
contribution to "Field Theory and the
Future of Stochastic Mechanics", in **Stochastic Processes in Classical
and Quantum Systems**, Lect. Notes in Phys **262**, p 438,
1986. Springer-Verlag. This has left a stranded group
of enthusiasts, who continue to study the theory. Nelson discusses the
extension of the theory to encompass two or more
particles; this has been taken up by L. M. Morato
and N. C. Petroni, J. Phys A, **33**, 5833-5848, 2000. It seems to me
that once we have
two independent particles, both with an underlying classical stochastic
dynamics, then it is possible to find a statistical prediction of the
theory that is not the same as
that of quantum theory. We just need to set up four pairs of pairwise
commuting observables, some of which are complementary as in Bell's
examples. This would give us a Bell inequality for the Nelson theory, or
ANY other description by a classical stochastic process, but
not for the quantum theory. Therefore, there will be observable statistics
in the classical theory that differ from those predicted by quantum
theory. This convinces me that "it is not
correct physics".
Chris Weed has remarked that the
recent preprint "Quantum Theory from Quantum Gravity", by F. Markopoulou
and Lee Smolin, seems to suffer from the same problem. The authors
claim in the title and
abstract that starting from a classical deterministic spin dynamics, coupled to
a heat bath, one can perhaps get quantum mechanics. They try to avoid
Bell's theorem
by saying "The non-local hidden variables required to satisfy
the conditions of Bell's theorem are the links ... in the graph". The authors
seem to be trying to construct a non-local quantum dynamics. There is no need to,
since relativistic quantum mechanics is, or should be, local.
The authors final theory is not full quantum mechanics. It is a Nelson theory
in which there is a wave function satisfying a version of Schrodinger's equation.
The mod-squared wave function is the probability density that the particle is present
at the time in question. However, as in Bohmian theory,
the velocity is not an observable incompatible with the position, but is
the average of Nelson's forward and backward
velocities. More, the positions of the particles at different
times are random variables on the same sample space, to wit, the sample space of the
driving noise of the stochastic dynamics. They have a good, real correlations, unlike in
quantum mechanics, in which the correlation between position at different times
is complex. Since the dimension of the Hilbert space is at least four,
there are pairs of compatible observables of Bell type which do not give the
same results as quantum mechanics. The authors are not saved by any non-locality
of the dynamics.

I believe that Nelson's reason for giving the theory up
was the difficulty in preventing action at a distance and the transmission
of information faster than light. Doubts were expressed in the beautiful book,
**Quantum Fluctuations**, by Edward Nelson, Princeton Series in Physics,
1985. Morato and Petroni surmise (page
5833) that it was the study of entangled states which turned Nelson
against it. Certainly, entangled states feature in Bell's inequality, and
the study of them would have been enough for me to abandon the theory,
stochastic mechanics, even if I had invented it. It is not true, however,
that any information is transmitted faster than light in quantum field
theory, even in the presence of entangled states: there is correlation
between distant states, but no signal is transmitted. This answers a
question,
posed to me after the experiment of Aspect et al., of whether Wightman and
I planned
a new edition of our book, in which we abandon
the locality axiom: NO.

For a resolution of the EPR `paradox', see this page

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This subject was invented by von Neumann, Jordan and Wigner. They devised
an algebraic set of axioms for propositional logic, different from the
Boolean algebra of classical logic. The Boolean algebra of a set A is the
collection of its subsets, including the whole and the empty set; the
collection of subsets is called the *power set* of the given set A.
The power set is furnished with a *sum*, the union of two subsets
minus their intersection,
and a *product*, the intersection of two subsets. This structure
forms a ring in modern parlance; it obeys the distributive property. It is
an algebra only in the formal sense
that any ring is an algebra over the trivial field containing 0 and 1.
In contrast, the
quantum logic of von Neumann and friends is a non-distributive lattice.
There is really only one result in the subject: Piron's theorem, which
says that (subject to the covering property, and some regularity
assumptions like measurability)
any quantum logic is isomorphic to the lattice of subspaces of a Hilbert
space over some field. This leads to the idea of replacing von
Neumann's complex Hilbert space with real quantum mechanics,
when the field is **R**, or quaternion quantum mechanics, when the field is
the non-commutative field of quaternions. One might even try p-adic
fields. So far, nothing of physical importance has arisen from these
attempts, except possibly the quaternionic case (see below). The same can
be said about attempts to generalise the logic
even more, in an attempt to avoid Piron's theorem.

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Quantum logic is criticised for not being a logic in the book, Modern Logic and Quantum Mechanics, by Rachel Wallace Garden (Adam Hilger, 1984). She points out that the propositions of a (classical) logic should admit valuations, which is an assignment of truth or falsehood to each proposition. The lattice approach leads to a structure for which valuations might not be meaningful, and for which the distributive property fails. The book describes R. Garden's approach; she allows a three-valued valuation, true, false, undecided. But she maintains the distributive property, and so remains close to a classical theory. Quantum theory fits into the general scheme; for her, a "state" is, in physicists' language, a pure state, the simultaneous eigenstate of a maximal abelian subalgebra of the observables. This abelian algebra defines the measurement, M, being done. Such a state defines a probability space, giving the same distributions as predicted by quantum theory. The sample space of this is contextual, in that a different choice of M gives a different sample space. The construction of the sample space is via the Stone space of the Boolean `algebra' defined by M. In finite dimensions, this is isomorphic to the space constructed using the Gelfand isomorphism in my work, Classical and Quantum Probability, mentioned above. Her conclusion, that the theory leads to a Kolmogorovian probability theory, is bizarre; the same observable is realised by various different random variables; even the sample space depends on which (compatible) further observables are being measured. It is clearly a frequentist theory, very prekolmogorovian: apart from the introduction of various sample spaces for the same observable in different contexts, there is no general requirement taking the place of Kolmogorov's consistency condition. For the examples of classical and quantum probability, this requirement is satisfied in her set-up. However, it is difficult to see how any other case of the general formalism could be constructed obeying the consistency condition. For example, it is not required that a given observable have the same distribution when it is regarded as a random variable on different spaces.

Experience shows that it is more productive to use classical logic, but to
change the
probability theory from classical to quantum. Then we may replace the
algebra of all bounded operators on a given Hilbert space by a more
general C*-algebra, and still be able to do physics. See Haag's book *Local
Quantum Physics*. Very little physics has resulted from quantum logic
or trivalent logic.

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Although it is nearly proved that there are no solutions except the free or quasifree fields in four space-time dimensions to the construction of a scalar Wightman field via the lattice approximation, some people still hold out hope that a clever trick will be found to avoid the nearly proved fact that the only fixed point of the renormalisation procedure is a trivial field. So this is not a good field for a research student. It would be a good thing if someone could rule out a non-trivial solution completely; the trouble is, that triviality can only be proved if we agree to follow some specific constructive procedure, like starting with free-fields on the lattice with periodic boundary conditions, and an ultraviolet cut-off to the perturbation, and then taking limits. The optimists would always be able to say "I would not start from there". It is very demoralizing to be a research student working on a theory which will probably lead to a trivial theory, if it leads to anything. This is not to say that the techniques of constructive quantum field theory should not be studied. Glimm and Jaffe, Frohlich, Albeverio and others have laid the foundations for the construction of gauge fields and other geometric systems. Seneor, Magnen and Rivasseau have even shown that pure QCD (without quarks) makes sense in a box with periodic B. C. in the continuum limit, in four dimensional Euclidean space-time. The asymptotic freedom allows a convergent limit to the continuum to be made, as predicted by K. Wilson. The infrared limit, and quark confinement, is not yet a tractable problem; nevertheless, gauge theory is more promising than the scalar theory.

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As if quantum theory were not general enough, people have attempted a more
general algebraic system. One such is to replace the C*-algebra of complex
quantum mechanics by a real Jordan algebra, which is not
associative, which makes their study difficult. There is
a Jordan relation, which allows some manipulations to be used. A special
type of Jordan algebra is the Hermitian part of a C*-algebra, with the
symmetric product (XY+YX)/2 as the product of X and Y. This is clearly
Hermitian, even if X and Y do not commute. Such a Jordan algebra J is
called a *special* Jordan algebra. It does not go beyond quantum
theory, as we can reconstruct the C*-algebra from J in a natural way, just
by complexifying it. Any other closed topological Jordan algebra contains
the matrix algebra of 3 by 3 matrices of octonions. The octonions form a
non-associative field. This is useless for physics, since no one knows how
to form the tensor product of general Jordan algebras; so we do not know
how to describe two independent systems from more elementary ones. I
conjecture that this is impossible unless both Jordan algebras were
special. In abstract terms, Jordan algebras do not form a tensor category.

Quaternions and octonions enter in the classification of Lie groups. See the page of John Baez for some leads.

A different sort of change to quantum mechanics is to replace the complex numbers by p-adic numbers as the field of the Hilbert space. This is a smaller change, as this possibility is allowed by Piron's theorem if we replace real numbers by p-adics in the definition of probability. Tensor structures are then possible. This activity has, however, failed to add anything to physics (so far), and should be avoided by students.

This does not apply to the idea, arising from Piron's thesis, to
replace
the complex numbers by the quaternions. These are associative, but not
commutative. This is no problem in quantum mechanics! Early work by Jauch,
and the thesis of Emch, seemed to show that
quaternions were not really
interesting, because, for elementary problems, no real difference
between quaternion quantum mechanics and complex quantum mechanics shows
up. However, as S. L. Adler has recently pointed
out, this very fact is a
plus for quaternion quantum mechanics: it would be wrong if it gave
completely different answers for the hydrogen atom, for instance. So it
remains an intriguing question, whether e.g. gauge field theory is more
naturally quantised using quaternion Hilbert space, *H*. I have argued,
and so has David Finkelstein that to be
able to form tensor products of quaternion Hilbert spaces, the vector
space should be a right as well as a left module over the field; then by using
the left and right products, we
furnish the Hilbert space *H* with an action
psi--->q psi q^{-1}, for every vector psi in *H*. This structure is
not clearly discussed in Adler's book. It was treated much earlier, in 1979, by
L. P. Horwitz and L. C. Biedenharn, Jour. Math. Phys **20**, 269. See also
L. P. Horwitz and A. Razen, Acta Applic. Math. **24**, 141- and 179-, (1991).
Finkelstein has pointed out that this bimodule structure
furnishes *H* with a real structure: real vectors are those that
commute with all quaternions q. Thus, the only interesting quaternion
quantum mechanics is a real one, with the quaternion group as a
superselection symmetry. This means that in a theory with a complex
Hilbert space and SU(2) as a gauge group, we might as well start with a
real *H*. If the
SU(2) non-abelian gauge theory is true physics, it
finds its natural explanation in the quaternion theory: only if the
theory has SU(2) symmetry will it allow tensor products.

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The claims made for the book of this name, written by Roy Frieden and published by Cambridge University Press, $80$, do not stand up to examination. Let us just look at the claim that quantum mechanics can be derived from the foundational idea that in any measurement, the information deficit K=I-J, is stationary. The author tentatively identifies I with the kinetic energy, and J with the potential energy (which he says is equal to the kinetic energy, a wild claim). In his next step, he postulates the formalism of quantum mechanics, and in particular the identification of the mean energy W in a normalised state |psi) as the scalar product (psi|(T+V)psi) in the usual Hilbert space of wave functions, where T+V is the Schrodinger operator. Far from deriving quantum mechanics from information theory, Frieden assumes quantum mechanics in the form of Schrodinger's stationary equation. By definition,

W:=(psi|(T+V)psi)/(psi|psi) . . . . . . .(1)

He then finds the turning points
as we vary |psi), of (psi|(T+V)psi)-(psi|psi)W, which of course by
eq. (1) is zero for all |psi).
The functional derivative of this relative to |psi) is therefore zero for
all |psi); in calculating it, he erroneously forgets the fact that W
depends on |psi). So what he is really doing is finding which states
make the quantum-mechanical mean energy (psi|(T+V)psi) stationary, subject
to (psi|psi)=constant. Then W appears as a Lagrange multiplier. His solution
is that |psi) must be an
eigenstate of the Schrodinger operator, and W must be an eigenvalue;
he then claims to have derived quantum mechanics from information
theory (forgetting that he started with Schrodinger's equation). One can
more soberly argue that he has proved that every measured state is an
eigenstate of energy. The trouble with this idea is that the form chosen
for the Fisher information, T, is argued earlier in the book to be that
which one gets when measuring the
*position* of the particle, not its energy. Measuring the position of
a particle does not send it into an eigenstate of energy. As it is,
his result is hardly new, being the basis of the Rayleigh-Ritz
method of finding the eigenvalues, known since the nineteenth century. The
author fails to mention that in the rest of the book, it is K=I-J=T-V, the
"information deficit", that is
supposed to be a minimum, and therefore stationary, not T+V; but what is a
minus sign between friends?

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It is unwise to embark on a study of the phase operator, if the only tools known to you are those in Dirac's book. For, we quickly arrive at the following paradox.

Theorem: let *h* be any positive number; then *h* = 0.

"PROOF".

Consider the self-adjoint operator *N* defined on the Hilbert space
of square-integrable functions on [0,2 pi]; *N* can be defined by
Stone's theorem as the generator of the continuous group of translations

{*U(a/h)psi*}*(x)* := *psi(x - a mod(2 pi)).*

Then if the wave-function is differentiable, with square integrable derivative, we have

* N = (h/i)(d/dx)*.

Thus, the number operator is well defined; its eigenfunctions are
*exp(inx)*, having integer eigenvalues *n = ... -1,0,1,2...*, and
are well known to form a complete normalisable orthogonal system.

In particular, the constant function *psi_0 = 1* is an
eigenfunction with eigenvalue zero, also known as the ground state:

*N psi_0 = 0*.

Now define the phase operator *X* as the operator of multiplication
by *x*:

*[X psi](x) := x psi(x)*.

Then *X* is a bounded, everywhere defined operator, and on the
dense set of differentiable functions we have the Heisenberg commutation
relations

[*N, X*] = *h/i*,

since if psi(x) is differentiable, so is x psi(x).

Now compute the matrix element of this commutator in the state
*psi_0*; this is zero, as *N* hits its ground state either to the
left or to the right:

*0 = (psi_0* |[*N, X*]| *psi_0)* = *h/i(psi_0*
|*psi_0) = 2 pi h/i*.

Therefore *h*, any positive number, must be zero. Think about it.
A brief hint where to find the mistake in the argument is to be found here.

A related paradox is seen from the uncertainty relations; if
*qp-pq = ihI* on a dense domain, then the product of the standard
deviations of p and q is greater than h/2 (or equal to it for a special
wavefunction). So if *N, X* obey Heisenberg's commutation relations
then there can be no eigenstates of *N* for which the variance of
*X* is finite; this is contradicted by the eigenstates exp(*inx*), and
the fact that *X*, the phase, is bounded. Where is the mistake in the
book proof of the uncertainty relations, when applied to N, X?

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The use of Feynman path integrals in physics has been an extremely useful
heuristic. From time to time, people ask, whether these integrals can be
made rigorous, in the sense that they are defined by an underlying measure
on path-space. The answer is that, even in the Gaussian case, there is no
such measure. This is known as Cameron's theorem. It means that the usual
manipulations of integral calculus,
such as changing the order of integrations (Fubini's theorem),
or differentiation under the integral sign, *might* give wrong
answers. Often, the answer given by a Gaussian Feynman integral is
correct. In such a case, one might find a justification by introducing
concepts such the *promeasures* of Cecile
deWitt-Morette. A more general method, which works in quantum field
theories of space-time dimension 3, is to adopt a euclidean approach, in
which functions are analytically continued to imaginary time, as in Glimm and
Jaffe. This converts the Feynman integral to a
Wiener integral (which does
exist); the integrals can then be defined; physics then has to be
recovered by analytic continuation back to
real space-time, and the proof that this is possible is one of the harder
parts of the analysis.

A rigorous meaning can be given directly to the Feynman integral if
white noise analysis is used; we arrive at a distribution, which is
not a measure. See the work
of Streit and of Albeverio. Thus, the
integral is regarded as a continuous linear
functional, on a topological space *X* of functions of infinitely
many variables. *X* is
a dense subset of Fock space, *F*. The topology of *X* is
determined by some seminorms chosen to fit
the problem at hand. The dual space of *X* is, as usual, the set
of continuous complex linear functionals on *X*. The triple
*(X, F,X**) make up what is known
as a Gelfand triple. This has allowed a path-integral treatment of some
singular interactions in quantum Field theory to be solved. So, while
difficult and mathematical, this is a promising variation of the problem,
whose original version was very lost.

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The idea of working in imaginary time has been applied to the problem of quantising gravity. The most ardent advocate of this programme is Stephen Hawking. I am reminded of a TV programme about Hawking that was shown a few years ago; we see the Oxford relativist, P. K. Tod, beginning a lecture at Cambridge. After the metric, g_{mu,nu}, appears on the board, Stephen asks, what metric are you using, Lorentzian of Euclidean? Tod replies that being from Oxford, he was of course using the Lorentz metric. To this Stephen replies, "Oh! we gave that up years ago!". Nowadays, the boot is rather on the other foot. The analytic continuation to complex time was probably first introduced into quantum field theory by G. C. Wick in perturbation theory. By replacing t by it, the (non-compact) Lorentz group is converted into the (compact) rotation group, and the Feynman integral into the Wiener integral, which exists. So the mathematics looks promising. The general idea was made rigorous in flat space-time in the early fifties by A. S. Wightman. One is to replace the Fourier transform of the vacuum expectation values by the Laplace transform. To make sense, the energy must be bounded below. Energy is the self-adjoint operator that generates the symmetry group of time translations. Unfortunately, a general space-time, used in general relativity, is not flat and the symmetry group does not seem to have a generator, let alone a positive one, according to Bernard Kay. So, the first step of the Euclidean programme fails. Even so, the path integrals of quantum gravity can be formulated in imaginary time by optimists. However, the (necessary) step, to continue back to real time, has little prospect of being implemented. Most research groups in quantum gravity have voted with their feet, leaving Euclidean gravity as a lost cause.

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Bohmian mechanics was developed by David Bohm [A suggested interpretation
of the quantum theory in terms of hidden variables, I, II, Physical
Review, **85**, 166-179, and 180-193, 1952;]

This subject was assessed by the NSF of the USA as follows [Cushing,
J. T., review
of Bohm, D., and Hiley, B., *The Undivided Universe*, Foundations of
Physics, **25**, 507, 1995.]
"...The causal interpretation [of Bohm] is inconsistent with experiments
which
test Bell's inequalities. Consequently...funding...a research programme in
this area would be unwise". I agree with this recommendation; however, it
is more accurate to say that the Bohm model
without spin gives predictions that differ from those of quantum
mechanics. After the Aspect experiments on photon spins, Bohm accepted
that spin must be treated as in quantum mechanics, and this can be
incorporated into the model.

The starting point is an idea of Madelung, that the quantum probability current has many of the properties of the current of a fluid. To get the full Schrodinger equation, the quantum phase must also be given a dynamics. Bohm found that the phase obeys a sort of Hamilton-Jacobi equation, modified by an extra term involving Planck's constant. He called this extra term the quantum potential. It was hoped that QM can be reformulated as a "realistic" theory in the sense of EPR.

This subject is not likely to succeed. In the first place, the theory predicts effects that move faster than the speed of light, and this is due to the quantum potential, which is non-local. Secondly, the theory without contextuality cannot predict the same results as quantum mechanics, contrary to the claims made by Bohm. He shows that for any time t, his theory predicts the same distribution for the position X(t) of a particle at time t as does the quantum theory of Schrodinger, and also any function of X(t) is also distributed as in quantum theory. In Bohmian mechanics, X(t) is a random variable on the initial space of positions, and the only uncertainty is in the initial configuration of the system. Thus, the probability theory is classical, ie obeys the axioms of Kolmogorov. By Bell's theorem, the theory predicts the Bell inequalities for certain pairs of compatible variables. Thus, the correlations for these pairs of observables are not predicted to have the same values as in quantum theory, which violates the Bell inequalities. The original version, and some second thoughts, is described by D. Durr, S. Goldstein and N. Zanghi, 21-44, in: Bohmian Mechanics and Quantum Theory (ed J. T. Cushing. A. Fine, and S. Goldstein, Kluwer, 1996). It is argued that all observables ultimately come down to position measurements of the particle. It is not clear, however, whether the supporters of the theory expect all the position measurements needed to measure say momentum can be done at the same time. As Fine remarks in the same book, page 235, a classical theory predicts the form of the joint distribution of observables that do not commute in the quantum theory, and so have no joint distribution in QM. In quantum theory, the positions of the particle at different times do not commute, whereas Bohm's theory predicts their joint distribution. This is an indication that there are observables whose distribution is not correctly given by Bohm's theory. Indeed there are. This easy fact is denied by most of the authors in Durr et al.

The classical velocity of the Bohmian particle is not related to the measured values of the velocity in quantum theory. In any real psi, the Bohmian velocity is zero, whereas the measurements (as predicted by quantum theory) is a random variable with distribution given by |phi|^2, where phi the Fourier transform of psi. The ground state of the hydrogen atom is real for all time, and so in Bohmian physics, nothing moves. This is not a contradiction, according to Fine (p. 245): "...reality...is significantly different when observed than when unobserved". Squires (p 133) uses the equality of the quantum and Bohmian distributions of position (which does not, in fact hold for the joint distributions of positions at two different times) to argue that the incorrect Bohmian value of the velocity is no problem. He claims that we can measure the momentum by measuring the positions of certain probes. These "will be correct". However, one can imagine measuring the momentum of a particle by measuring its position at two different times, (and dividing the distance by the time interval); in Bohmian theory this will give a measure of the average Bohmian velocity over that time interval. Squires is saying that he does not believe that this velocity is what will be observed in any experiment. So much for the reality of the model.

The theory is not consistent in its interpretation. It is set up as
a probability theory; but a measurement of the position of the particle
does not result in a conditioning of the probability distribution of the
position. This is at the insistence of J. S. Bell, who championed the Bohm
theory when it was more or less discredited. Bell said "No-one can
understand this theory until he is willing to see psi as a real objective
field rather than just a probability amplitude" [The Speakable and
Unspeakable in Quantum Mechanics, Cambridge University Press, page
128, 1987]. This allows believers to
claim that there is no measurement problem, as a measurement just reveals
what the position *is*. The usual rules of probability would lead to
the replacement of psi by a conditional probability, after a measurement
has given us more info as to the whereabouts of the particle. However,
this conditioning is exactly the collapse of the wave packet, and this is
a dirty word among Bohmists: to allow it here might weaken the case
against quantum mechanics. /p>

The failure to condition after measurement leads to peculiar results
even for positions. Y. Aharonov and L. Vaidman (pp 141-154) in "About
position measurements", give many examples of this, similar to those in
Englert, B.-G., Scully, M. O., Sussmann, G., and Walthier,
H. : Surrealistic
Bohm Trajectories, Zeitsschrift fur Naturforschung, **47a**, 1175-1186,
1992. Ahahonov and Vaidman admit: "We worked hard, but in vain, searching
for an error in
our and Englert et al arguments"[sic]. They conclude "The proponents of
the Bohm theory do not see the phenomena we describe here as difficulties
of the theory", and quote a riposte of the Bohmists [Durr, D., Fusseder,
W., Goldstein, S., Zanghi, N., * Comment on Surrealistic Bohm
Trajectories*, Z. Fur Natur, **48a**, 1261-1262, 1993.

After the results of the experiment by Aspect et al. were known to be inconsistent with a classical model with hidden variables, Bohm called a press conference. He said that "contrary to the expectations of most physicists, the experiments show a violation of Bell's inequalities". On the contrary, the experiments, agreeing with the quantum predictions, were expected by the vast majority of physicists. Squires (q.v.) expresses a more up-to-date view of the Bohmists: "Any `completion' of quantum theory which is consistent with all its predictions must be non-local....most physicists have come to accept this non-locality in some form or other". WRONG! most physicists accept the Copenhagen interpretation, in which quantum probability does not obey Einstein's concept of reality, but is both local and non-contextual. For an account of the Copenhagen interpretation, applied to the EPR experiment, see this page.

Einstein has given the following definition: In any
theory, a concept is called an *element of reality* if is assigned a
[real] value. He goes on to say that if an attribute can be measured
without in
any way altering the system, then it should be represented by an element
of reality [in any good theory]. The famous EPR experiment, in which the
momentum of a particle is 100% anti-correlated with another at a far
distance, shows that by measuring the momentum of the far particle we can
find that of the first. Naturally, EPR thought that this does not change
the first particle in any way. So they concluded that momentum should be
classed as an element of reality [in any good theory]. Bohm concocted a
similar EPR pair, in which the spins are 100% anti-correlated. Thus we
might conclude that Bohmists regard both the momentum of a particle, and
its spin, as elements of reality. Not a bit of it! According to Durr et
al, (pp 21-44), spin, momentum, and energy, in Bohmian mechanics are
not elements of reality; they must be considered as
operators, whose measured values are given by their eigenvalues, and
whose probability distribution is given by the expectation value of
positive-operator-valued measures in the state psi. But this is quantum
probability, as formulated by E. B. Davies and J. T. Lewis ("An
operational approach to quantum probability",
Commun. Math. Phys., **17**, 239-260, 1970) and
expounded in the book "Quantum Theory of Open Systems", by
E. B. Davies, (Academic Press, 1978). To make sure the reader
does not use it for position measurements as well, it is denigrated in
Durr et al. as a mere technicality. It is "even more abstract than operators",
(p 32), but is computed "by elementary functional analysis...[and] is a
near mathematical triviality". But it is used by recent Bohmians
for the theory of spin measurements when explaining the
violation of the Bell inequalities as found experimentally by Aspect,
Roger et al.

The definition of "element of reality" in EPR needs elaboration to
meet modern standards of mathematics. Since a system can be in various
configurations or states, the real number assigned to the element of
reality cannot
always be the same. So the definition should have said,
"In any theory, a concept is called an *element of reality* if, in a
given configuration of the system, it is assigned a number." With this
definition, we may call Gamma = *phase
space*, the set of pure states, and then an element of reality is a map
from Gamma to the
reals. If we add the technical requirement of measurability, this is what
is called an observable in modern classical dynamics.

The EPR definition can be adapted to a theory with randonmess, given by
a measure mu
on Gamma: "In any theory with randomness, a concept is called an
*element of reality* if it is given by a random variable X on Gamma."
This is none other than an element of reality as above: it is assigned a
real value for each omega in Gamma. Although Einstein was against
randomness ("HE does not play dice"), he used it a lot in some of his
best papers. Bohmists take this as the definition of
reality: [Durr et al, bottom of p 35]. For them, energy, momentum and spin
are not elements of reality. However, the claim that position is an
element of reality (in Einstein's sense) at all times, true of Bohmian
mechanics, is not true in
quantum mechanics, contrary to their belief. In QM, positions at different
times do not commute, so there is no representation of them by random
variables on a common sample space: some correlations
referring to positions at different times, fail to satisfy Bell's
inequalities, which they would do if there were such a representation.

We do not wish to define the concept, element of reality, as
above in the manner of Einstein; for then in QM, there are no elements of
reality. We can generalise
the concept from classical to quantum probability as follows, bearing in
mind that self-adjoint operators are the quantum analogues of random
variable:
"In any quantum theory, based on the Hilbert space H, a concept is called
an *element of reality*
if it is a self-adjoint operator X on H. A state is a density operator rho
on H, and the mean of any element of reality in a state rho is Trace(rho
X)." We see here the von Neumann definition of observables in quantum
mechanics; it must be generalised if there are superselection rules.

I do not wish to call "elements of reality" any assignment (of
operators to concepts) which must be changed to another assignment if the
context is different. As an example of a contextual assignment, any model
agreeing with the Aspect experiment, and which describes spin as
a random variable, must use different sample spaces depending on the
context. See some examples of "context" in the above account
of hidden variables, Lost Cause
I. On the contrary, contextuality is not needed in the
quantum theory of
spin. To see this, suppose we have the EPR pair of spins
1/2 particles as in Bohm's modification of EPR. Take *H= C^2 tensor
C^2*, where *C^2* is the two-dimensional Hilbert space. We may
assign the operators
**s tensor I** to the spin of a particle on the left, and
**I tensor s** to the spin on the right, where **s** is the vector
of Pauli matrices. Then these two operators commute elementwise, so the
measurement of one spin does not disturb the other (locality). Moreover,
we do not need to
change **s tensor I** to various other operators depending on what
simultaneous measurement is being done to the other particle, in order to
agree with the Aspect experiment. Thus the assignment is non-contextual.
We conclude that quantum probability allows a realistic, local description
of the Aspect experiments, provided that we modernise Einstein's
concept of reality. This is the view of the vast majority of
physicists, contrary to Squires's statement.

Better steer clear of Bohmians.

BACK to Contents

This subject arose in cosmology; quantum mechanics was introduced in the belief that it must be used in any good model of the universe as a whole. This is misguided: the large scale dynamics of galaxies is just the sort of domain in which the delicate phase relations between components of a quantum state become washed out, and classical mechanics takes over. This is recognised in the recent consensus on the idea of the idea of classical localisation in cosmology, also known as decoherence. It is in astrophysics, the physics of stars, that we must allow for nuclear reactions, particle creation, and so on. The study of a star does not need a detailed description of each and every particle, but should be treated by statistical mechanics. The idea of the wave-function of the universe is meaningless; we do not even know what variables it is supposed to be a function of.

In the original version of many-worlds, in the Ph. D. thesis of H. Everett, the universe is in a pure state, not a more general density operator. The vector in Hilbert space H describes the observer and his apparatus as well as the system being observed. There is no reduction of the wave function. The time-evolution is unitary. One can then list all possible values of every observable (that are consistent in the sense of quantum theory). Since time is continuous, this leads to a very large set of "consistent histories": in modern terms, one first chooses a maximal abelian subalgebra A of B(H) at each time, and a point in the joint spectrum at each time; the set of these choices is the sample space of an elaborate stochastic process. Then one does this for another maximal abelian algebra, and so on, until every possible experimental arrangement that might have been set up over all time has been included. Instead of agreeing that at the present time, up to now, one of these histories has occurred, and the others have not, Everett suggested that every branch actually occurred, not only of each classical stochastic process associated with a fixed maximal abelian algebra, but for every choice of A; each branch is entirely invisible to all the others. In later days, Everett suggested that the many worlds were not actual, but were many views of the same world. But why should different observers see such different things? The idea that the many worlds are actual was then taken up by de Witt.

The reason for this extravagance in reality is that Everett was writing in the physics tradition of probability theory. One needed to have an ensemble in order to discuss probability. With only one universe, physicists thought that the methods of probability would not by justifiable. These days, we simply say that we have the observables, the self-adjoint elements of B(H), and a density operator (more likely than a pure state) giving the expected value of each observable. We can justify the choice of state either by information theory, or by the claim to be modelling. That is, we try a state, make calculations, and compare with experiment. Then try another state...

There is nothing to the many-worlds theory. There are no theorems,
conjectures, experimental predictions or results of any sort, other
than those of Hilbert space. It is not a
*cogent* idea. In fact, it would
need a specification of what observables correspond to each operator
before anything rich enough to be worth studying arises. It is hoped
by its advocates that the existence of so many worlds can be
used to justify the law of large numbers, from which one might give the
theory a probabilistic interpretation. If this were possible, then we just
arrive at quantum mechanics.

The idea that the full details of the observer should be included in the Hilbert space is in violation of the scientific ethos. Science is not the collection of impressions got by watching Nature unfurl. In the most useful phrase from the book by Durr, Goldstein and Zanghi, Bohmian Mechanics and Quantum Theory, "we do not come to the Navier-Stokes equations by admiring water waves". We find the laws of Nature by reproducible experiments. The theory needs a cut, between the observer and the system, and the details of the apparatus should not appear in the theory of the system. The heat capacity of a crystal should not depend on the shape of the calorimeter used to measure it. The result of an experiment should not depend how it is performed (contrary to the claim of Durr et al. q. v. above, p 36, lines 1-2), but on what it is measuring. Moreover, the tentative models of the observers found in papers on many-worlds, to be included in the wave function, are not needed for cosmology: all the experimental atomic and nuclear physicists who ever lived have made a negligible contribution to the reduction of the universe's wave-function.

The contrary ethos, that we should consider the detailed properties of the minds of observers, has been advocated by Matthew J. Donald, in his version of the `many-minds' approach. This is a theory of `no collapse', in which probability is replaced by the idea that the observer is one of an ensemble of minds. This activity takes the subject even further from physics that the original many-worlds picture.

Back to Contents.

It has been suggested that the concept of observable in quantum mechanics
might be extended to include, not only self-adjoint operators, but also
any operator similar to a real diagonal operator. That is, *A* is
observable if there exists a similarity transformation *S*, with
densely defined inverse, such that *SAS^{-1}* is real and diagonal.
The idea comes from the easy fact that such operators, which we shall call
diagons, have real spectrum. If we postulate, with von Neumann, that any
measured value of an observable is an eigenvalue, or more generally, lies
in the spectrum, then allowing such operators will not lead to any complex
measured values. Of course, self-adjoint operators are diagons, so the
proposal is an extension of the set of observables in quantum mechanics.
Is there anything theoretically wrong with it, and if not, why was it not
discovered by the people working in quantum logic and Jordan algebras?

The first thing tried was to expand the allowed dynamics so as to allow
a non-self-adjoint diagon as the energy operator, and to ask for the
consequences. Unless we abandon
the relation between the energy and the generator of time-evolution,
which is the most successful idea in classical as well as
quantum theory, we are then led to a non-unitary evolution. This was the
point of the foundational work of Misra and Prigogine: in a dissipative
system, we need a contraction semigroup, while the observed values of the
energy *H* must be real. Indeed, a semigroup occurs
naturally as the linear
approximation to my own non-linear heat equations, paper [132],
*Stability of a hot Smoluchowski fluid*. Where this work differs from
the scheme of Misra and Prigogine is that, as in any good theory of
non-equilibrium thermodynamics, what is dissipated in [132] is not energy,
or total probability, but information. In [132] total probability is
conserved in time, and energy is simply transferred from
one sort to another,
such as from potential energy to heat. The mean total energy is conserved
(the first law of thermodynamics). A dynamics in which the total energy
leaks away to zero would obviously violate this law. For this reason,
choosing a diagon as Hamiltonian does not seem to be a good idea.
This relates to another question, that of transition
probabilities, which is given by the mod square of the scalar product of
the two states in question. Since the eigenvectors of a diagon *H* in
general are not orthogonal, there will be a non-zero transition
probability that the observed energy of an eigenstate will
be higher or lower than the that of the initial state. This cannot be a
good fundamental theory; it looks more like a system with random noise.

Carl Bender and coworkers were led to study
diagons from analytic
perturbation theory. They have tried to make it into a valid
generalisation of QM. See Bender, C. M., Brody, D., and Jones, H. F.,
*Complex extension of quantum mechanics*, Phys Rev Lett **89**,
270401, 2002. This and possible lines of further work are also
described in *Must
a Hamiltonian be Hermitian?*. Here, the authors consider the usual
Schrodinger operators *p* and *q* acting on wave-functions of
*q*, and build Hamiltonians that are not hermitian as operators on
L^2(**R**), but which are invariant under an antiunitary operator
identified as *PT* (parity times time-reversal). They argue that such
a Hamiltonian not only has real spectrum (which is proved), but is also
very likely to have
a complete set of eigenfunctions (which has been verified to great
accuracy by numerical studies). They do require that the boundary
conditions are carefully chosen in the eigenvalue problem. It is not clear
under what conditions the eigenfunctions are square-integrable. The
eigenfunctions split into pairs, related by the *PT* transformation
and they define a natural indefinite scalar product. This is made positive
definite by changing the sign of the `norm' on half the basis. This
operation is likened to the charge operator *C* and Dirac's trick
with the
positron states: the scalar product uses an analogue of the *PCT*
congugation in place of complex conjugation. Thus they end
up with a Hermitian operator *H* as Hamiltonian
relative to this new scalar product. All the eigenfunctions have finite
norm in the new setting. In the models discussed,
various anharmonic oscillators with complex potentials, neither
*p* nor *q* commutes with *H*. It would appear that they
are not observable, being non-hermitian in the new scalar product. An
interesting problem (but not one for a Ph. D. student) is whether a viable
two-particle theory can be constructed. We need a Hermitian operator
to act as the position of the centre-of-mass of the two particles; this
will generate the symmetry group of Galilean boosts. We need another, the
total momentum operator, to generate translations. Their
commutation relations with *H* are also prescribed by the rules of
the Lie algebra. Then, they must give one of the known reducible
projective representations of the Galilean group, perhaps leading to a
theory in which the two particles do not possess individual observable
positions and momenta. My guess is that it cannot be done.

One of von Neumann's axioms, used to exclude hidden variables
in quantum mechanics, is that the sum of two observables should be an
observable. For example, if one's hidden variables are to be random
variables, and all r.v. are observable, then this axiom is true. Von
Neumann allowed a more general scheme, in which the quantum
self-adjoint observable
could be random; his axiom is also true in this case. Nevertheless, this
axiom has been criticised, almost derided, by Bohmians, as unwarrented: if
two observables are not simultaneously measurable, there is no operational
(that is, experimental) definition of their sum. Their means can be
summed, but there might be no observable of which this is the
mean. Diagons provide a concrete model in which von
Neumann's axiom is violated: if *A* and *B* are diagons, then in
general *A + B* is not a diagon. This reveals the basic problem: the
expectation values of a diagon in all (pure) states are real only if it is
self-adjoint. Thus we would be motivated to assume that a diagon is
observable only in states in which its mean is real, and this is usually
only its eigenstates. Then the question would arise, suppose that we have
an apparatus for measuring the observable, *A*, and we feed it a
state for which the mean is complex: what would our apparatus do?

If we have only one observable in a theory, and it is a diagon, then we can make it self-adjoint by changing the metric in the Hilbert space, so that its eigenvectors are mutually orthogonal. This was remarked by Bender and friends for their class of models. If we have more than one observable, however, we can play this trick only among those observables that mutually commute. The scalar product between two Hilbert space vectors would then depend on what set of mutually compatible observables we are thinking of measuring, and so the transition probability would have no invariant meaning.

In a quantum theory with a diagon as energy, we lose Wigner's theory of symmetry; the observables cease to be a vector space, which means that we are not allowed to perturb a diagon with any old diagon; and the energy will be observable only in some states. Not much is left.

See also Critique of PT-Symmetric Quantum Mechanics, by Ali Mostafazadeh.

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After the success of Mandelstam's double dispersion relations in 1958, Chew and Mandelstam applied it to the pion-nucleon system, in a truncated form. The S-matrix must be unitary, and in the elastic region, this amounts to the parametrisation of the scattering by the partial wave scattering lengths. Suppose we include only the four channels, having angular momentum 1/2 or 3/2, and isotopic spin 1/2 or 3/2. This is described by four scattering lengths, each a function of energy. Each of these obeys a dispersion relation, and so can be expressed in terms of an integral of the corresponding imaginary part over all energies. The imaginary part is related by unitarity to the unknown scattering lengths themselves. Thus four non-linear integral equations in four unknown functions are obtained. The numerical solution of these equations gave the very welcome prediction that there should be a resonance in the (3/2,3/2) channel. That is, states with angular momentum 3/2, and isotopic spin also 3/2, should have a very large crossection, and the predicted energy of this resonance was near to the experimentally observed (3/2,3/2) resonance. The only inputs needed were the pi/proton mass ratio, m say, and the strength, g, of the strong coupling constant, apart from isotopic spin and Lorentz symmetry.

Chew and Frautschi saw their way to do better, and their idea can be schematically explained as follows. We know that if we have N unknowns, and they are to obey N non-linear coupled equations, then typically, we expect there to be a discrete set of solutions, many of which might be ruled out for physical reasons. We might hope that this would lead to a unique solution. Indeed, there is only one world, they argued, so the correct set of equations must lead to it in a unique manner. This is, I should add, an unsound argument, unless we add that the equations we use are complete as well as correct. And completeness can have no other general definition other than that the equations have a unique solution. It follows that we cannot use the principle of uniqueness to find the very equations themselves; for this would tell us (if we omitted a true but needed condition) to reject as incorrect a partial system of equations which is correct. It was noticed that in S-matrix theory we have infinitely many equations, expressing unitarity of S, for infinitely many unknowns, the scattering lengths at each energy.

In the bootstrap method, one picks oneself up by one's own bootlaces as follows: regard the pi/proton mass-ratio m and the coupling constant g as unknowns, as well as the four scattering lengths at all energies; we keep the same equations as were successfully used to predict the (3,3) resonance. We still have the same number of equations as unknowns (infinite), and so there might be a unique solution, or even better, a discrete spectrum of solutions predicting all masses of the baryon resonances. We truncate the infinite set of equations to a finite set, with unknowns listed as m,g, and a finite set of N phase-shifts in a finite range of energy. Together, these make up the unknown vector X. We then keep N+2 of the unitarity equations, which express the vector X of unknowns as a non-linear function F of its components:

X = F(X)

To solve, we use the iteration method: guess an initial value for the unknowns, such as X(1)=(m(1),g(1),0,0...); this is called the input. We put it in the non-linear term, thus getting the output of the first iteration:

X(2) = F(X(1))

Continue, until we get convergence. Unfortunately, the method did not converge very rapidly, if at all. So a better method is to consider the difference,

X - F(X)

as a non-linear expression in the components of X, and find the conditions that its square be minimised, by varying X. Then at least the error (in making the original truncation, perhaps) would be small. This gave a prediction for all the constants m,g etc of the system. Unfortunately, when more energies were included, or three-particle corrections were made, the results were not close to the previous approximation.

When we first heard of this at Imperial College, Paul Matthews said: the bootstrap method consists in making a poor approximation to an exact algebraic identity, and then asking for what values of the fundamental constants make this approximations as least bad as possible?

We who have worked in quantum field theory would have expected that there is more than one theory: in quantum electrodynamics, we can put in any coupling constant, instead of the fine-structure constant alpha, and do the same calculations. We would not expect that general principles alone would determine alpha. In non-relativistic scattering theory, a large number of different potentials lead to a unitary and analytic S-matrix. Thus, we might have thought that the equations of the bootstrap method are algebraic identities for each choice of m,g. What happens when we approximate the equations, and then solve for the parameters, including m and g? We get a numerically unstable system which gets worse the closer we approximate the algebraic identities.

My fellow student, Claud Lovelace, became
a Senior Fellow at CERN, and while there he
proved that the bootstrap equations had no solution. When Euan Squires
gave us a seminar at IC on his research during his stay on the West coast
of USA, I asked him whether he was worried by Claud's result. He looked at
me rather shocked, but saw a way out: "Aha, you mean that the equations
have no *exact* solution. We are physicists, and are looking for *
approximate* solutions!" For non-experts, I should say that there is no
such thing as an approximate solution to an equation with no
solutions; the best you can hope for is a solution of an equation that
approximates the terms of the recalcitrant equation. As such, it is not
a unique concept, since any equation, or system of equations, can be
regarded as the small alpha limit of any number of quite arbitrary
changes to the equations, where the changes are proportional to alpha. In
the case of the
bootstrap equations, the dispersion integral probably diverges, and needs
a subtraction, an adjustment that introduces new unknowns into the
equations. Then one cannot predict all the constants of nature from them.

I wrote out a refutation of the bootstrap method, and submitted it to Phys Rev Lett. But when the referees complained that it was not comprehensible, I withdrew it, rather than rewrite it; I was not an expert of numerical instability.

Indirectly, the bootstrap theory led to the writing of my book, PCT... with Arthur Wightman. I think it was 1961, and Wightman went as usual to the "Rochester" conference on particle physics. In the then recent past, people like him and Lehmann, Kallen, Symanzik, Zimmermann and Res Jost had been welcome at the meeting, to explain the new rigorous results. That year, their slot was occupied by the new bootstrap analytic S-matrix theory. After the talk by Geoffrey Chew, Wightman asked him why he {\em postulated} analyticity, rather than founding it on the axiom of positive energy. Chew said, all physical functions are analytic, because the functions of physics are always smooth. Then what about infinitely differentiable functions that were non-analytic? Chew thought for less than ten seconds, and then said "There aren't any". This so annoyed Wightman that he complained to Bethe, that the rigorous people had been excluded to make way for someone who did not know what an analytic function was. Bethe replied that the rigorous theory was out of reach to most of the audience, and that we, the people concerned, should explain its most important results in a very short book. So this is what we did.

Back to Contents

This deals mainly with classical physics, in that dynamics is given by a
computer algorithm determining a possibly chaotic path in the phase space
of the
system. Quantum dynamics is simulated by invoking the Feynman path
integral, which is lost cause #9. In the introduction
to his *opus
magnum*, Wolfram suggests that quantum uncertainty might be caused by
chaotic dynamics given by an infinite calculation. Who knows? it might be
so; if chaotic dynamics can imitate classical randomless, perhaps an
infinite calculation can imitate quantum randomness. Some sort of cocycle
between the random variables arising in the simulation might lead to
non-commutativity. Perhaps. But there is no hint of this in his article.
Instead, he relies on the belief: `I would not be surprised if most of the
supposed quantum randomness they measure is actually more properly
attributed to intrinsic randomness generation associated with their
internal mechanisms'. He also dismisses the possibility of chaos in the
dynamics of a single particle, since the Schrodinger equation is
linear. He does not mention it, but this is also true of the dynamics of
two particles with spin. So he must wonder how the non-classical
correlations of the *EPR* experiment, performed by Aspect *et al*.,
are generated.

In the early days of quantum theory, Dirac proposed that the quantum
analogue of the Poisson bracket of two classical observables should be the
commutator of the corresponding quantum
observables. This leads to the Dirac quantisation rule: to each classical
observable *A* should be assigned an operator *q(A)*, such that
the map * A --> q(A)* is a homomorphism of Lie algebras. This means
that the classical observables must form an algebra with bracket the
Poisson bracket, and the operators form a Lie algebra with the commutator
as bracket.

It turns out that if one wants an irreducible representation of the
Poisson algebra by operators, then it only works for some sub-algebras of
the set of smooth functions of canonical variables, * p, q *. Of
course, this includes the harmonic oscillator, which must have been the
inspiration for Dirac's idea. Indeed, in the case of the quadratic
algebra, the idea works for any number of degrees of freedom, including
quantum fields (free ones). This leads directly to Irving Segal's
real wave and complex wave realisations of the free field.

Attempts to extend the idea to polynomials of higher degree led to
contradictions with the requirement that the map be a homomorphism; the
quantum operators do not commute, so there is some ambiguity in defining a
polynomial of * p * and * q * in the quantum case. Groenewald
and van Hove showed that there is no ordering convention of the quantum
operators that works for all polynomials. The
most convincing result is that of Lionel Wollenberg: there is an algebraic
property of the Poisson algebra that is invariant under Lie homomorphisms,
but which is not shared by the quantum algebra of operators.
Nevertheless, there are **STILL** books on quantum mechanics being
published which state that Dirac had solved the problem of finding a
quantum theory associated with a given classical theory. Maybe this is
what Bert Schroer is complaining about in his article, in
which he (Schroer) denounces "quantisation": quantum theory has its
its own way of dreaming up models, and should not have to rely on classical
models plus quantisation.

A more lenient approach was invented by J.-M. Souriau, in his brilliant
book, *Structures des systemes dynamiques*, Dunod, 1970.. He relaxed
the requirement that the quantum algebra be irreducible, and was able to
find a class of homomorphic mappings
from the Poisson algebra to the algebra of linear operators on the space
of smooth functions on phase space. Independently, in 1964 Irving Segal
constructed a realisation of the quantum field algebra as
(Gateau) differential operators on the linear space of smooth functionals
on the Hilbert space of one-particle states. Segal did not require that
the map be defined for all polynomial functionals (or tame functions),
and his formula differs from that suggested by an extension of Souriau's
formula to infinite dimensions. This was the subject of my paper [22],
"Canonical Quantisation", CMP **2**, 354-374, 1966, written while I
was Segal's research assistant at MIT. I found that Segal's map
did not preserve the Lie structure, and that his claim that the action of
the Lorentz group on the classical phase-space was mapped into the Lorentz
action on the quantum operators was not true. Segal was kind enough to
give me his hand-written notes of the calculation he had done, which
convinced him of his claim. He had had extraordinary bad luck in this
calculation. I remark on page 370 of [22] that Segal's quantisation
differs from that of Souriau in eq (53) by the omission of the terms
involving the action *S(t)*. In his check that the generators of the
Lorentz group in his quantisation satisfy the correct Lie-algebra
relations, Segal had arrived at an expression that should vanish if his
theory were Lorentz invariant. This expression, which filled a page,
arose exactly from the omitted term in (53), which came in twice.
The second time it entered it was multipled by the sign of -Trace(G),
where G is the Lorentz metric. This is +1 if we use (as Segal had done
from the outset of his calculation) the time-like convention for the
metric. Then we get a non-vanishing result, and Lorentz invariance fails.
But Segal, seeing that all the 30 or so terms would cancel, term by term,
if he took this trace to be positive, forgot that he had a time-like
convention, and happily
cancelled out all Lorentz-violating terms; this he considered such a
remarkable chance that he was convinced that the theory was right.

Souriau theory is now called "prequantisation" rather than geometric quantisation, its early name. As I show in [22], it is not possible to find an invariant subspace of the space of functionals, on which the energy is bounded below, for a non-linear quantum field theory. Kostant has been able to make prequantisation work when the observables form a solvable Lie algebra of type I, by the method of "holomorphic induction". This reduces to the Bargmann-Segal space of holomorphic functions on phase space for the harmonic oscillator. This was the subject of the other paper I wrote while at MIT, [25], "The representations of the oscillator group".

G. G. Emch has been able to identify the reducible representations in some of these constructions as KMS temperature states.

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No chance; I've tried it. For the Copenhagen view, see here; an article from a Bayesian viewpoint is to be found in the article by R. Duvenhage, The nature of information in quantum mechanics. I find it difficult to see how this author's claim that there is no physical collapse of the wave-function can agree with the experimental results of measuring incompatible observables in succession by filters.

Einstein has credited Mach with inspiring him to look for a purely geometric model of gravity, in which the physics is independent of the coordinates chosen for its description. This led Mach to surmise that in a universe with only one body in it, there would be no physical distinction between the cases when the body is rotating, and when it is not: these two situations are tranformed into each other by moving from coordinates in an inertial frame to the rotating frame. He, Mach, predicted that it is only the presence of the fixed stars that allows one to feel centrifugal force, which tells us that we are rotating. Clearly, Newtonian physics does not satisfy this, which is a startling consequence of Mach's general principle. Hochvath showed that Einstein's equations for GR, which possess a solution representing a rotating body in an otherwise empty universe, does not satisfy Mach's principle. He showed that a test-particle on the surface of the body would feel centrifugal force. This inspired Hoyle and Narlikar to develop a more elaborate geometrical theory, which they said obeyed Mach's principle. However, Hochvath also found a solution to their equations, representing a single rotating body, and again violationg Mach's principle. This work did not induce the authors to withdraw their paper, however; they found aspects of the solution to be unphysical. Nevertheless, it is fair to state that their equations do not incorporate Mach's principle.

It is safe to assume that Mach was not correct in his prediction. Indeed, when we change the coordinate system from an inertial frame to an accelerated frame, we must add a suitable gravitational field (an apparent gravitational field in the original coordinates). Mach did not think of that.

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These are turning points of the Lagrangian which, in the Euclidean approach to QFT, correspond to maxima of the Lagrangian instead of the minimum. At first sight, they do not correspond to a normalizable state of the system. On a second look, one finds that they do not make sense at all.

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If one rejects the interpretation of the uncertainty relation as saying that QM is intrinsically a probability theory, one might be tempted to assume that the state vector in Hilbert space could act as a phase space of a deterministic theory. In that case, a mixed state might be described by a probability measure on the unit sphere in Hilbert space. This assumption leads to a change in the statistical mechanics of the system, different from that usually accepted: in particular, the Gibbs state of a particle of spin 1/2 differs from that found in the literature. The difference can be measured, and this has been proposed as a test of this new idea. This would make an interesting Ph. D. in experimental physics: we should test whether there is any physical difference between two states given by the same density operators, but originally formed by mixing different sets of pure states. However, most people would bet that the usual density-operator formalism would be confirmed, so to work on this problem is a high-risk strategy for a student.

A detailed and rigorous mathematical account of such a theory, with many references up to 2000, can be found on the web using Google:

**Extended Quantum Mechanics**

by Pavel Bona, *Acta Physica Slovaca,* **50**, 1-198, 2000.

The author shows that several non-linear theories (such as that of
Kibble) are special cases of the formalism, and discusses the consistency
of any theory which is non-linear in the state. To avoid absurdities, he
points out that there are two kinds of mixing: these
are not dissimilar to the two concepts used in my book, **Statistical
Dynamics**, the every-day randomizing of samples with no physical
contact with each other, and the mixing achieved by
putting all the samples in one box. The first must be described by an
affine mixture, but in the latter case, the system acts
non-linearly on each of its parts. I have no quarrel with non-linearity.
It is the unorthodox feature, that one can distinguish two states with
the same density operator, but different history, that is risky ground for
a student.

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In a recent paper,
* No spin-statistics connection in non-relativistic quantum
mechanics*, R. E. Allen and A. R. Mondragon felt obliged to remind the
reader of simple counter-examples to any `theorem' claiming that spin 1/2
particles must be Fermions, and spin 0 particles must be Bosons, within
non-relativistic quantum mechanics.
They say: "In several recent papers....quantum mechanics is modified so as
to force a spin-statistics connection, but the resulting theory is quite
different from standard physics". I agree. Keep away.

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In quantum cosmology, it is often asserted that all observables are
invariant under the diffeomorphism group of the space-time. This has led
to a fantastic claim by Julian Barbour that one can remove time as a
variable in cosmology. This is countered by Stuart Kauffman and Lee Smolin
in an article,
* A possible solution for the problem of time in quantum cosmology*,
where the authors mention that one possible weakness in Barbour's argument
is that
he is dealing with spaces that are not algorithmically classifiable. It
is possible that they are referring to the result by A. A. Markov,
*Insolubility of the problem of homeomorphy*, pp 300-306,
in Proc. Intern. Congress of Math., 1958, CUP, Cambridge, 1958.
I think that the problem is much simpler: cosmologists have failed to note
the distinction between a gauge group and a mere symmetry group. In
quantum theory, a gauge group is a group of transformations of the algebra of
all operators entering the theory, but which leave invariant every hermitian
operator representing an observable. On the other hand, a symmetry is an
automorphism of the algebra of observables which either commutes with the
Hamiltonian, or is part of a dynamical group like the Lorentz boosts. There
is at least one observable that is changed by it. If two
observers look at the same system, using
different coordinates, they see the same world, but obtain different readings on
their dials. These must be related by a symmetry transformation, called a
passive symmetry by Wigner. Wigner's theorem says that in quantum
mechanics, every passive symmetry defines a unitary or anti-unitary
operator U on the Hilbert space; this can be used to define an active
symmetry: if Phi is a possible state of the system, as seen by an observer
O, then so is U Phi, and it can be seen by O. It will be distinguishable from
Phi, since U is not a gauge transformation.

See this page for other views on the issue.

In classical field theory, a symmetry shows up as an invariance of the
Lagrangian. Thus, the Einstein-Hilbert Lagrangian is invariant under
diffeomorphisms of the manifold M, and *Diff M* is a symmetry group
of GR.

I often hear the argument that, since physics should not depend on the
coordinates used, *Diff M* must be a symmetry. This is a
fallacious argument; in a
coordinate-free formulation of field theory on a manifold, one can
choose a dynamics other than that given by the Einstein-Hilbert Lagrangian, and
*Diff M* might no longer be a symmetry. Indeed, in any universe with
massive particles (like our own), the forces other than gravitational do not
possess *Diff M* as a symmetry group. This can easily be seen; such a
dynamics is not even invariant under constant scaling of space-time.

In any theory with locality, it does not make sense to regard *Diff M*
as a gauge group. An active diffeomorphism might move a state Phi localised
near the observer to one far away, and such a transformed state U Phi is
clearly not the same as Phi. It is no
good saying that the two states are `physically equivalent'. This might mean
that the diffeomorphed observer, moving with the state, and
told to apply the inverse diffeo to his coordinates, would get the same
readings from U Phi as he would have done at his former position on
measuring Phi in his former coordinate system. This bewildered observer
might mistake Phi for U Phi. Whether
they are different states is determined physically: is there any observable
which has a different mean in the two states?. In statistical physics, the
question is, do they both make separate contributions to the partition function
(not contradicted by the
possibility that they contribute the same amount)?. They do, in any theory
in which the concept of local observable is meaningful. Observable
thermodynamic properties like specific heats, rate of thermalisation etc
depend on the multiplicity of the energy-spectrum, and states related by a
geometric symmetry such as rotation have the same energy but are not
the same state in general. Since statistical mechanics is one of the main
tools of cosmology, practitioners of cosmology need to be sure about
multiplicity, as measured by a single observer.

There aren't any. But what about final causes? I hear you cry. None: these are the same as first causes.

Read my review of Stapp's theory.

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In a series of extravagently illustrated
articles on quantum theory, the editors of NEW SCIENTIST are giving credence
to the myth that in quantum communication it is possible to send information
faster than the speed of light. That this is not possible is explained in my
article on the *EPR* experiment.

The Department of Physics and Astronomy, University of Pittsburgh, has two easily-read sites, here and here, giving hints on how to recognise and avoid junk and bunk. These sites also have useful links to courses in maths and physics, as does this one.

Go to my **HOME PAGE** for more links.

© by Ray Streater, 12/Oct/2003.