Do Quanta Need a New Logic?
John Stachel
Frontiers of Fundamental Physics 14
Epistemology and Philosophy
Marseille, 17 July 2014
Do Quanta Need a New Logic?
Three Aspects of a Logic
Syntax:
A formalism, or set of symbols and rules for
well-formed formulas using these symbols
Semantics:
An interpretation of the formalism: Its meaning
as a description of some aspect of the world
Pragmatics:
Its application to that aspect of the world
Logic-Language-World
Three steps:
Logic is about Language,
Language is about The World.
Panlogism
– The attempt to “short circuit” this process by
identifying the real object and the “concrete-inthought” leads to the assertion:
Logic is about The World
Aron Gurwitsch
Leibniz: Philosophie des
Panlogismus
“Things are realizations of concepts of reason.
It is not sufficient to maintain that the logical
and the ontological viewpoints can never be
fully distinguished from each other, or that
no separation, no abyss exists between
reason and reality. One seems most faithful
to the situation, if one speaks of an identity,
or better of an equivalence of the logical and
the ontological”
Panlogism Reborn!
“ By panlogism I mean the philosophical
tendency to obliterate the distinction
between logical and ontological principles”
JS, “Do Quanta Need a New Logic?” (1986)
I have been combating this viewpoint for
over 35 years:
”The ‘Logic’ of Quantum Logic” in PSA 1974 (Dordrecht:
Reidel 1976), pp. 515-526.
Logic(s) For Quantum Mechanics
If we abandon panlogism, and
the search for The Logic of
Quantum Mechanics, we shall
see that quantum logics (note
the plural!) are just different
ways of reformulating the same
content of quantum theory.
The Danger:
If we accept one formulation of
The Logic of Quantum Mechanics
as providing the answers, it
prevents us from confronting the
real questions about the nature
of quantum mechanics.
Chris Isham
Is it True; or is it False; or Some-where In
Between? The Logic of Quantum Theory
"A key feature of classical physics is that, at any
given time, the system has a definite state, and
this state determines-- and is uniquely
determined by-- the values of all the physical
quantities associated with the system.“
Realism is "the philosophical view that each
physical quantity has a value for any given state
of the system.“
In a Letter to Chris, I Raised Two
Problems:
1) Conditional Properties:
“each physical quantity has a
value”
2) The Primacy of Process:
“for any given state of the
system”
First Problem:
1) Conditional Properties:
“each physical quantity has a
value”
2) The Primacy of Process:
“for any given state of the
system.”
1) Conditional Properties
This is just not true of conditional properties, as
discussed in detail in Do Quanta Need a New
Logic? The example I use concerns the
properties "hardness h" and "viscosity v": Given
a system defined by its chemical composition,
the property "hardness" will only apply-- let
alone have a numerical value on Moh's scale-- if
the system is in its solid state; while "viscosity"
will only apply if the system is in a fluid (liquid
or gaseous) state.
Simple Classical Example
Hardness and Viscosity can be
applied to any substance, but
not simultaneously. If it is in
solid state, hardness applies; if
it is in a fluid state viscosity
applies.
1) Conditional Properties
With the definition of the logical 'negation'
operator, [logic] has already gotten as
complicated as it gets. … [M]y article …
discusses the difference between choice
negation (-) and exclusion negation (∼ ) in
general, and the inevitability of choice
negation for a conditional predicate … if one
wants to derive other predicates from it, and
what follows from this choice even before
getting to the special case of QM.
Non-Standard Logic Needed
Semantics
If we allow elementary propositions of the form:
“System S has hardness h,”
“System S has viscosity v”
Then a non-standard logic is needed:
negation (“not”), conjunction (“and”) and
disjunction (“or”) cannot all follow the laws of
classical logic
Which Negation?
How shall we interpret the proposition:
“System S does not have hardness h1”
It could mean:
“System S has some hardness h2≠ h1”
or it could mean:
“System S is not in the solid state, so the
concept of hardness does not apply”
Which Negation?
The law of the excluded middle:
“p or not p is always true” (p∨∼p=I)
If we choose
“System S has some hardness h2≠ h1”
Then the law of the excluded middle Is not valid for this
variant of intuitionistic logic.
To keep the law, we must choose the disjunction:
“System S has some hardness h2≠ h1 or it is not in
the solid state so the concept of hardness does
not apply”
This choice also leads to a non-standard logic!
Simplest Example
Only Two States, Solid and Liquid:
Solid State, only two values h1, h2 of hardness
Liquid State, only two values v1, v2 of viscosity
Then there are only four elementary propositions,
symbolized by h1, h2, v1, v2
We also need symbols for:
Negation (∼)
The identically false proposition ϕ
The dentically true proposition I
Simplest Example
If we plot all possible combinations of
propositions p, starting from the
identically false proposition ϕ and ending
with the identically true proposition I, and
use dotted lines to represent logical
implication, then we get the following
diagram of a propositional lattice
Propositional Lattice
.
h1 (∼h2)
.
I (∼ϕ)
.
.
.
.
. .
.
.
.
v2 (∼v1) v1 (∼v2)
.
.
.
. .
.
.
.
.
ϕ (∼I )
.
h2(∼h1)
.
David Ritz Finkelstein
The Physics of Logic
(1969)
Finkelstein uses the same diagram to
describe the polarization of photons,
and states:
“The system is the simplest
quantum-like lattice and exhibits
nondistributivity and coherence.”
Enter Pragmatics
But such propositions cannot be tested without
the additional specification of the conditions
C under which system S is being observed
(e.g., temperature T, pressure p).
So if we add these conditions to the form of the
proposition:
“System S under conditions C has hardness h”
Classical Logic Returns!
Then, either:
The proposition is not well-formed, if system
S is not in the solid state under conditions C,
or:
The proposition is well-formed, if system S is
in the solid state under conditions C, and
Classical logic holds for all well-formed
propositions!
Objection
Well and good for this example, but there
is no analogue of the superposition
principle in it. We cannot superpose two
states S1 and S2 of different hardness or a
state of hardness and and a state of
viscosity to get a new state S
Answer:
Let’s look at another classical example:
What Are Colors?
Colors are literally in the mind of the beholder:
The human eye and brain combine to interpret
all electromagnetic waves within a certain
range of frequencies that impinge on the
retina in terms of three dimensions, e.g.:
Brightness, hue and saturation or
Three primary colors
So colors are conditional properties of an open
system, forming a 3-dimensional vector space
Color Logic
Three primary colors:
Red
Green
Blue
Additive Color
They can be mixed (color
superposition) to get:
Magenta Yellow
Cyan
Complementation
Two colors are complementary if, when
superposed (mixed) they produce white.
Red Cyan
Green Magenta
Blue Yellow
Enter Logic
Define elementary propositions:
“Object O has color c”
Interpret negation (“not”) as color
complementation
Interpret conjunction (“and”) as color
addition
And you have a non-standard logic with
superposition of colors!
What Is a Quantum System?
So neither conditional properties nor
superposition are unique to quantum
systems.
Then what makes a quantum system?
As we shall see, it is the role of h, the
quantum of action.
What is Quantization?
Quantization is just a way of accounting for
the effects of h, the quantum of action, on
any process involving some system,–
or rather on theoretical models of such a
system-- “fundamental” or “composite”, in
which the collective behavior of a set of
more fundamental entities is quantized
“Atoms and Human Knowledge”
--Niels Bohr 1957
“..an element of wholeness, so to
speak, in the physical processes, a
feature going far beyond the old
doctrine of the restricted divisibility
of matter. This element is called the
universal quantum of action. It was
discovered by Max Planck in the first
year of this century and came to
inaugurate a whole new epoch in
physics and natural philosophy.
“Atoms and Human Knowledge”
-- (cont’d)
We came to understand that the ordinary
laws of physics, i.e., classical mechanics
and electrodynamics, are idealizations that
can only be applied in the analysis of
phenomena in which the action involved
at every stage is so large compared to the
quantum that the latter can be completely
disregarded.
The Second Problem
1) Conditional Properties:
“each physical quantity has a
value”
2) The Primacy of Process:
“for any given state of the
system”
Lee Smolin
Three Roads to Quantum
Gravity
“[R]elativity theory and quantum
theory each ... tell us-- no, better,
they scream at us-- that our world is a
history of processes. Motion and
change are primary. Nothing is,
except in a very approximate and
temporary sense. How something is,
or what its state is, is an illusion.
Three Roads to Quantum Gravity
It may be a useful illusion for some
purposes, but if we want to think
fundamentally we must not lose sight of
the essential fact that 'is' is an illusion.
So to speak the language of the new
physics we must learn a vocabulary in
which process is more important than,
and prior to, stasis.”
2) Primacy of Process
Phrases such as "at any moment of
time", "at any given time” may be
applied in Newtonian-Galileian physics,
which is based on a global absolute
time. But from SR on to GR, this phrase
involves a convention defining a global
time., and nothing physical can depend
on the choice of convention!
2) Primacy of Process
The only convention-invariant things are
processes, each involving a space-time
region. This suggests-- as do many
other considerations-- that the
fundamental entities in quantum theory
are the transition amplitudes, and that
states should be taken in the c.g.s.
system (cum grano salis).
2) Primacy of Process
And this is true of our measurements as
well: any measurement involves a finite
time interval and a finite 3-dimensional
spatial region. Sometimes, we can get
away with neglecting this, and talking,
for example in NR QM, about ideal
instantaneous measurements.
2) Primacy of Process
But sometimes we most definitely
cannot, as Bohr and Rosenfeld
demonstrated for E-M QFT, where the
basic quantities defined by the theory
(and therefore measurable-- I am not an
operationalist!) are space-time
averages. Their critique of Heisenberg
shows what happens if you forget this!
Closed versus Open Systems
System
Key Concept
Closed
Determinism
Open
Causality
Determinism means fatalism: nothing can
change what happens
Causality means control: by manipulating the
causes, one can change the outcome
“Determinism is really an article of philosophical
faith, not a scientific result” (JS 1968).
The Dogma of Closure
When classical physics treated open systems,
it was tacitly assumed (as an article of faith)
that, by suitable enlargement of the system,
it could always be included in closed system
of a deterministic type. … The contrast
between open and closed should not be
taken as identical with the contrast between
‘phenomenological’ and ‘fundamental’ …
(JS: “Comments on ‘Causality Requirements and the
Theory of Relativity,” 1968)
Do We Really Want Global?
The systems we actually model are finite
processes, and all finite processes are open.
A finite process is a bounded region in spacetime: Its boundary is where new data
(information) can be fed into the system and
the resulting data can be extracted from it.
Example: Asymptotically free in- and outstates in a scattering process.
Properties:
Intrinsic and Extrinsic
Intrinsic properties: Define the nature of the
system
Example: The mass, charge, and spin of an
electron, proton, neutron, etc.
Extrinsic properties: Depend on the relation
of the system to its surroundings
Example: The position, momentum, angular
momentum of a system relative to some
inertial frame of reference.
Heisenberg Physics and Philosophy
Introduction by Paul Davies
One is used to uncertainty in many
physical processes – for example, in the
stock market or in thermodynamics – but
in these cases the uncertainty is due to
missing information rather than to any
fundamental limitation in what may be
known about these systems.
Processes
An open system can undergo a process:
Preparation of the system,
Interaction with its environment,
Registration of some result.
Experiment vs Observation
Experiment: The preparation
result is fixed, predict the result
of the registration (laboratory).
Observation: The registration
result is fixed, retrodict the
result of preparation that led to
it (astronomy, cosmology).
Classical System:
Complete information allows us
to predict/retrodiction with
certainty.
Incomplete information leads to
compute the probability of a
prediction/retrodiction.
What is Probability
Ensemble interpretation:
Large number of copies of the process.
Probability is the ratio of number of
copies giving the predicted outcome to
the total number
Propensity interpretation:
Probability is the propensity of the
process to give the predicted outcome
Quantum Logic (J.S.)
The conditional probability for a system initially
prepared with position qi at time ti to be found in an
interval dq around q at time t is given by:
P(qi , ti; q, t) dq,
where the probability density P(qi , ti; q, t) is
proportional to the Van Vleck determinant of
Hamilton’s principal function.
This probability may be given a propensity
interpretation for a virtual ensemble associated with a
single system, or a frequency interpretation for an
ensemble of identically prepared systems.
Heisenberg Physics and Philosophy
Introduction by Paul Davies
It is essential to appreciate that this
uncertainty is inherent in nature and not
merely the result of technological
limitations in measurement. It is not that
the experimenter is merely too clumsy to
measure position and momentum
simultaneously. The particle simply does
not possess simultaneously precise values
of these two attributes.
Bohr: Atomic Physics and
Human Knowledge
On the lines of objective description, it is
indeed more appropriate to use the word
phenomenon to refer only to observations
obtained under circumstances whose
description includes an account of the whole
experimental arrangement. In such
terminology, the observational problem in
quantum physics is deprived of any special
intricacy,
Atomic Physics and Human
Knowledge
and we are, moreover, directly reminded that
every atomic phenomenon is closed in the
sense that its observation is based on
registrations obtained by means of suitable
amplification devices with irreversible
functioning such as, for example, permanent
marks on a photographic plate, caused by the
penetration of electrons into the emulsion.
What is a Quantum Process?
A quantum process involves
three stages: preparation,
interaction, registration.
Big question: How does h figure
in the preparation and
registration procedures?
Quantum System:
The effects of the quantum of action
h on the process cannot be
neglected.
In general, only probabilities 0 p1
for such processes can be calculated
(this does not exclude the
occasional p = 0 or 1)
The Big Difference:
If in some process, different interaction
paths can lead from preparation to
registration; then to get the total probability
for that process, if the paths are:
Distinguishable: add the probabilities
for each path (classical)
Indistinguishable: add the probability
amplitudes for each path (quantum).
What is a Quantum-Mechanical
Proposition?
Much of the debate over quantum logic(s)
hinges (implicitly if not explicitly) on the
question of the quantum-mechanical
analogue of the classical-mechanical
propositions. Propositions about a classical
system can refer to only the properties of the
system itself, considered as closed-- that is,
not interacting with anything outside the
system.
What is a Quantum-Mechanical
Proposition?
But, as Bohr emphasized, the existence
of the quantum of action h prevents
such a complete separation between a
quantum-mechanical system and its
macroscopic surroundings. Quantum
Mechanics can only deal with open
systems. Two major consequences are:
What is a Quantum-Mechanical
Proposition?
(i) A full description of a quantum-mechanical
phenomenon (see Bohr 1958) or process (see
Feynman 1968– the word “process” will be
used hereafter) must include a specification
of the result of an initial preparation of the
system, an account of the type of interactions
it undergoes subsequently, and of the result
of some act of registration (“measurement”)
to which the system is finally subjected.
What is a Quantum-Mechanical
Proposition?
(ii) A maximal quantum-mechanical
preparation or registration only specifies
“half” the data about a system that would be
specifiable classically. For example, while one
could in principle prepare or register a
classical-mechanical system with a
determinate position and momentum, one
can only prepare or register a quantummechanical system with either a determinate
position or momentum.
What is a Quantum-Mechanical
Proposition?
A typical proposition about a process
involving an electron might read:
“At time t1 the electron was prepared with
momentum p0, subsequently passed
through a certain electric field E, and at a
later time t2 was registered at position q0.”
Quantum mechanics assigns a probability to
such a proposition.
Conditional Probabilities
Note that, as in the classical case, all the
quantum mechanical probabilities for a
process are conditional; one does not ask for
the probability of a final value b tout court,
but its probability given an initial prepared
value a. Once computed, these conditional
probabilities obey the laws of the classical
probability calculus based on classical logic.
Conditional Probabilities
In contrast to the classical case, quantummechanical conditional probabilities cannot
be attributed to ignorance but are
fundamental. Without altering the physical
conditions defining the process considered
(thus producing a different process), it is
impossible to further subdivide a quantummechanical ensemble into sub-ensembles–
let alone into individual trajectories–
Conditional Probabilities
for which probabilities (as opposed to
probability amplitudes) can be defined. For
example, a sample of undecayed radioactive
nuclei all with the same average half-life
cannot be subdivided it into sub-ensembles
each with different predicted average lifetimes. (Of course, retroactively, after some or
all have decayed, it is easy to do so.).
What is a Quantum-Mechanical
Proposition?
Classical propositional logic is all that is
needed to handle such propositions,
describing a complete quantum-mechanical
process.
Logical problems begin when such
propositions are truncated by omission of
reference to the preparation and/or
registration.
What is a Quantum-Mechanical
Proposition?
Attention is focused on “the state of the
system at time t,” to which one tries to attach
a significance similar to that of the state in
the classical case.
The maximal goal of such an approach is to
attribute a complete set of classical
properties, e.g., both position and
momentum, to a quantum system in a given
state.
What is a Quantum-Mechanical
Proposition?
These elementary propositions are then
compounded by suitably-defined operations
of conjunction (“and”) and disjunction (“or”)
in an attempt to give meaning to the
resulting compound propositions, even when
the properties referred to in such compound
propositions are incompatible quantum
mechanically.
What is a Quantum-Mechanical
Proposition?
I shall not enter into further detail about the
various possible non-standard logics that
have been proposed for quantum mechanics
(see my article Quantum Logic), but end with
two reminders:
What is a Quantum-Mechanical
Proposition?
1) If you accept panlogism (“The Logic of …”),
this philosophical choice has nothing to do
with quantum theory.
2) The choice of a logic (“Logics for … “) is
just a matter of selecting one of various
different ways to express the same content.
Final Word on Quantum Logic
(Niels Bohr, 1939)
The question of the logical forms
which are best adapted to quantum
theory is in fact a practical problem,
concerned with the most
convenient manner in which to
express the new situation that
arises in this domain.