**Remarks on Griffiths' "Consistent Quantum Theory".**

Griffiths considers quantum theory from the point of view of "logic",
rather than on the basis of "measurements and actual knowledge".

The mathematical and quantum mechanical foundations of Griffiths' work
are securely rooted in von Neumann's work, which, as he developed it,
was erected upon the ideas of the measurement-based Copenhagen
interpretation. However, von Neumann altered those ideas by shifting the
Heisenberg boundary between the measured and measuring parts of nature
so that the entire physical world, including the bodies and brains of
the human observers, became described in terms of the quantum (Hilbert
Space) mathematics. This move left open the embarrassing question of
what became of the "measuring system", which formerly had played the
essential role of determining which question, from among a continuum of
possibilities, would be posed, and when it would be posed. In the
Copenhagen approach that measuring system was conceived to be a
"physical" system that included both the human observers and their
instruments, which, however, in spite of containing many atoms and
molecules, and other quantum physical elements, were empirically
described in terms of the concepts of classical physics. Von Neumann
shifted the apparatus and the brains and bodies of the measuring system
into the physically/mathematically described quantum world\-\--where
they most naturally belonged\-\--and thereby eliminated the awkward
separation in the physically unified world into two differently
described parts. But after this shift there was no physical agency
around to make the essential choices of the 'what and when' of the
probing questions. Von Neumann alluded to the "abstract ego" as the
needed agency of choice, but this introduction into the dynamics of a
non-physical agency is an anathema to materialist-minded quantum
physicists because it shifts the dynamical function of the
observer/participant, which is essential to the proper working of the
theory, out of the physically described world.

Griffiths' move points to a solution of this problem. His central idea
is to regard properties that *could be actual* properties of a physical
system as elements in a space of logical possibilities, in a way
somewhat reminiscent of the shift of mathematics to logic. In the case
of physics the aspects shifted to logical space were the Yes/No bits of
information associated with the two eigenvalues, one and zero, of the
projection operators of quantum theory. This move focuses on "possibly
actual" properties at the expense of the (mere) potentialities residing
in the evolving state of the universe between "actual" events, and at
the expense of any consideration about how the selection of the basis
associated with an event is made, and about how the actual outcome
becomes known to someone, or to something. The Yes-or-No bits are
shifted into a purely logical space of true or false propositions. This
is a space where it can be more naturally operated upon, and become part
of, the mental or conceptual aspects of the observer/participant, which
von Neumann referred to as the "abstract ego". In this way the treatment
becomes akin to classical physics, where the issue of how properties
become known is not considered.

Logic deals in useful ways with propositions that are either true or
false. Correspondingly, the quantum description of the "actual" is built
around propositions that are treated as being either true or false,
e.g., "The Geiger Counter Clicked!" This discreteness aspect was
described by Bohr, as "The essential element of wholeness characterized
by the quantum of action, and completely foreign to classical physical
principles." This feature of properties, as being either true of false,
is characteristic also of classical physics, but in classical physics
one can subdivide without limit, whereas in quantum physics there is a
limitation imposed by Planck's unit of action, and there are also
[potentialities]{.underline}, which are nonfactual realities of a kind
that have no place in classical physics. In the traditional
conceptualization of quantum theory we imagine such True-or-False
statements to be statements about the status of measuring devices, but
Griffiths construes them as statements of Yes/No properties
independently of the possibility of their being measured or known.

The fundamental logical structure of quantum theory expresses the
information that describes the *actual* state of any system S large or
small, in terms of the plus one or zero eigenvalues of a set of
projection operators in the Hilbert Space associated with that system S!
These projection operators must be orthogonal (P~i~ P ~j~ = P~i~ δ~ij~ )
and sum to unity: (i.e., be a decomposition of unity.) For any such set
there is basis in the Hilbert space such that each of the projectors in
this set is represented by a matrix that is filled with zeros and ones,
with all the ones located only on the diagonal. Different choices of
basis allow different "mutually consistent" sets of P's to be defined.
But the key point is that only sets of P's that are all mutually
orthogonal give allowed "mutually consistent" sets of Yes/No bits of
information. Two projection operators that cannot be expressed in terms
of a set of ones lying on the diagonal *in some one common basis* are
not mutually compatible: no [actual]{.underline} state can be specified
by giving, simultaneously, their eigenvalues.

Griffiths' point is that this characterization the description of the
actual state of a quantum system S at an instant of time is a feature of
the quantum structure the system S itself, for any S of any size, large
or small, without ever considering some *other* physical system, a
measuring apparatus, that has performed, or cann perform, a measurement
upon this system S. That Copenhagen approach leads to the further
question of the state of the measuring apparatus, and how information
gets conveyed to it. The chain stops only at mind, which is conceived to
operate with Yes/No bits of information. But one need not embark upon
this chain: the description of the state of the system S pertains to
that system S itself. And the various possible descriptions define a
logical space.

But a normal requirement in logic is that one has a set of mutually
consistent propositions: propositions that can be manipulated by the
actions in the logic without logical contradictions arising. So
Griffiths naturally focuses on sets of properties that are mutually
consistent. If one restricts attention to consistent properties then
they can be dealt with logically, without fear of contradiction,

This conceptualization of quantum theory aids von Neumann's formulation,
because it removes the need to think that there must be a "physical
classically described measuring device" intervening between the observed
system and the observing/manipulating mind. The actual properties of the
brain are simply properties of the quantum mechanically described brain
itself. No classically described observing physical system is required
to specify its state!

This understanding does not, however, resolve the "basis problem", or,
more specifically, the problem of "what chooses the set of mutually
consistent of projection operators whose eigenvalues specify the
[actual]{.underline} properties of the brain".

.

One can carry this "logical" approach one step further, and consider not
just the actual state of S at a single time t, but the actual state of
the expanded system consisting of the n states of the original system at
a sequence of n different times {t~1~,...t~n~}. Griffiths' "consistent
histories" condition specifies the conditions needed to make the actual
state of this multi-time system specifiable in terms of Yes/No bits of
information that can be manipulated by the normal rules of logic.
"Actuality" in physics becomes thereby treatable in ordinary logical
terms. These "consistency" conditions are simply that the transformation
from initial time t~1~ to final time t~n~ can be carried through a
time-ordered sequence of n decompositions of the Hilbert Space unit
operator [only]{.underline} by set of paths (histories), each defined by
a sequence of n projectors, such that no (non-identical) pair of paths
in this set can *interfere* (i.e., connect some initial projector at
time t~1~ to some final projector at time t~n~.). Thus the alternative
paths in the set behave like classically distinct paths. A set of paths
satisfying these conditions is called a set of "consistent histories".
Strong interaction with the environment tends to make path "decohere"
(become non-interfering).

An important point is that "macroscopic" systems have a special
property. It is assumed that "macroscopic" systems are imbedded in an
environment that links the different projection operators associated
with these macro systems to environmental states in a way that converts
an appropriate coordinate-space reduced density matrix for the
macroscopic projectors to (nearly) diagonal form. This makes the
macroscopic properties (obtained by tracing over environmental variable)
automatically (nearly) commuting, which makes histories formed
exclusively out of these macroscopic projectors automatically
consistent, or nearly so.

Given any such set of consistent set of (macroscopically characterized)
histories, one can consider a finer analysis obtained by including more
times and non-macroscopic projectors, in order to be able to discuss,
logically, ideas about what is, in some sense, going on between the
macroscopic events.

Making things"logical" can perhaps be helpful in the resolution of
conceptual paradoxes. But the problem remains: "What chooses the
questions to be put to nature, and the times at which the questions are
put?" It appears that "mind" has a big and important job to do. .

Griffiths summarizes his conclusions on the penultimate page of text:

1.  Measurements play no fundamental role in quantum theory, just as
    they play no role in classical mechanics. In both cases, measurement
    apparatus and the process of measurement are described using the
    same basic mechanical principles which apply to all other physical
    objects and processes.

2.  Quantum mechanics is a local theory in the sense that it can be
    understood without supposing that there are mysterious influences
    which propagate over long distances more rapidly than the speed of
    light. \...The idea that the quantum world is permeated by
    superluminal influences has come about because of an inadequate
    understanding of quantum measurements\-\--in particular, the
    assumption that wave function collapse is a physical process\-\--or
    by assuming the existence of hidden- variables...or by employing
    counterfactual arguments that do not satisfy the single-framework
    rule.

3.  Both quantum mechanics and classical mechanics are consistent with
    an *independent reality*, a real world whose properties and
    fundamental laws do no depend upon what human beings happen to
    believe, desire, or think.

As regards point 1, Griffiths gives a special role to "pointers"\-\-- to
the "pointer basis"\-\--for identifying properties that are treated
differently from the others. On pages 132-135 he brings in detectors and
pointer basis to distinguish the dead and alive states of the cat from
their superposition. But, in general, all macroscopic states are given a
preferred status, which arises, however, not from a "measurement being
performed on the system by another system" but from the dynamics of the
interaction with the environment: macroscopic properties. It causes the
density matrix of any macroscopic state to become (nearly) diagonal in
appropriate coordinate space variables, which mean that they (nearly)
commute, which means that any history composed only of macro-properties
is consistent. Then one can contemplate adding non-macro properties at
intervening times, in order to supply a consistent set of answers to
abstract questions. But the answers will depend, in general, on which
refined framework of consistent histories is used

The consideration of these non-macroscopic properties is a rather
sterile endeavour, because there is no connection to macro observables.
And if one uses only macroscopic properties, then any single
framework\-\--defined by any single set of sequence of macroscopic
projectors\-\--are essentially consistent. The crucial question left
open is: what chooses the timings of the actual events, and the form of
the process-1 actions? That is where Whitehead's ideas\-\--about the
process that makes these decisions\-\--naturally enters.

As regards point 2, it is of course true that all proofs of the need for
superluminal transfers of information require the use of counterfactual
elements. Indeed, I believe it was I who originally emphasized this
point, and ardently pursued it. But Griffiths' purpose is not to reject,
wholesale, all counterfactual arguments, but rather to reject proposed
nonlocality arguments on the basis of his basic principle, which is that
no logically acceptable argument can use several different frameworks.
(A "framework" is one single n-fold decomposition of the history
unity.). However, my proof of the logical need in QM for superluminal
transfer of information uses only one framework, and hence conforms to
Griffiths' logical principle. (See detailed discussion below)

Nor is the idea that collapse of the wave function is "a physical
process" successfully evaded. The wave function is a tool for making
statistical predictions on the basis of the known facts, and as such it
generally changes when new facts are introduced. Which new facts come
into existence depend importantly upon which probing actions are chosen
and performed. The entire utility of quantum theory rests on the fact
that we can compute in advance probabilities of outcomes for various
alternative future choices, and the fact that our doing these
calculations can influence which action we will take. So, given the
causal gap pertaining to the choice of basis, it cannot be asserted that
the extension of a set of consistent histories into the future is
uniquely specified. Different extensions of the set of consistent
histories will lead to different potentialities for the future. Thus the
potentialities corresponding to a surviving cat are different from those
corresponding to a dead cat, and from what they would have been if the
experiment had been set up differently. If these choices are a
reflection of our values, desires etc., then these psychologically
described things can influence the course of our lives.

Potentialities can be a part of the quantum reality that is not subsumed
by the "actual". The fact the wave function is used to make predictions,
does not mean that it has no *real* existence. Griffiths calls the wave
function a "pre-probability", thereby making it appear purely subjective
and imaginary. But Heisenberg suggested that it is the counterpart in
our theory of a real "objective tendency", existing in nature: a
"potentia". These tendencies do follow the quantum rules, and they
change when the quantum state changes, when a collapse occurs, when an
addition is made to the growing body of actual facts. The fact that
Griffiths focuses on "actualities" does not entail that potentialities
cannot exist.

As regards point 3, it must be allowed that the existence of the causal
gap make it possible for the choices of the basis, and of the timings of
events, to be influenced by conceptual (idea-like) realities, and hence
for our actions to depend upon our desires. Shifting the processing of
information from brain to brain-mind makes it [easier]{.underline} for
things in the domain of mind, such as our desires, to affect our
actions.

I now return to the issue of my proof of the need for superluminal
information transfer.

My proof uses an initial state that was used previously by Hardy to
derive a certain paradox, a logical contradiction. Griffiths' analyzes
Hardy's argument and finds fault with it. Griffiths' most direct
argument is that Hardy's argument leads to a contradiction, hence must
be incorrect. Griffiths' "logical" approach is supposed to guarantee the
applicability of the laws of logic. Hardy argues that the obtaining of
the contradiction arose not from a failure of the locality assumption,
but rather from a violation of his (Griffiths") central principle,
namely that a logically valid argument cannot employ several different
"frameworks".

Griffiths argument pertaining to Hardy's (second) paradox (p.343-348) is
not quite as systematic and thorough-going as his earlier arguments. But
it probably does succeed in showing that Hardy's argument cannot be
carried through using only a single framework.

But my proof (see the final Appendix in my new book, "The Mindful
Universe") differs from Hardy's in that it involves no contradiction,
but is instead a direct logically consistent argument that uses only one
framework.

One of the primary virtues of these EPR-Bell-type arguments is that they
refer only to the macroscopic visible properties. Thus there should be
no reference to the unseen intermediaries, such as the **c** and **d**
(c-bar and d-bar) that appear in (25.33) of Griffiths' book.

Then the proof of consistency is essentially trivial, since I use only
one time ordering in my argument, and use only macro variables, which
as, Griffiths notes, effectively migrate to the later times (see p.345
second full paragraph)

The appropriate framework is, using Griffiths' notation (See his page
270 for his

pivet points notation ( **o** ) in connection with counterfactuals: I
also use boldface for the barred symbols in his diagram. (25..23), and
equations (25.29) --(25.32)).

**OE**

O F **o Always: If O and E then if instead B then F**

**BF**

**Ψ~o~ o**

**OE**

B E **o Sometimes: If O and E then if instead B then E**

**BE**

Because the symbols all represent macro-properties they all migrate to
the left and the four states on the left are all mutually orthogonal
(See Griffiths p. 345) and there is no interference between paths:
backing up to a pivet point and then moving again to the right merely
retraces the original path: there are no closed loops! But all four
lines used in my argument are represented in this family of consistent
histories.