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“It Ain’t Necessarily So”
Interpretations and Misinterpretations of
Quantum Theory
John Stachel
Frontiers of Fundamental Physics 14
Faculty of Sciences (AMU)
Marseille, 15-18 July 2014
It Ain't Necessarily So
by George Gershwin
It ain't necessarily so
It ain't necessarily so
The t'ings dat yo' li'ble
To read in de Bible,
It ain't necessarily so.
……………………………
I'm preachin' dis sermon to show,
It ain't nece-ain't nece
Ain't nece-ain't nece
Ain't necessarily ... so !
My Apologies in
Advance
Time limits require brevity and brevity is
the mother of dogmatism.
None of my statements should be
interpreted dogmatically– they are all
meant to stimulate critical thinking and
further discussion.
For a copy of my PowerPoint just email
[email protected]
Examples of Misinterpretations
from Two Widely Praised 2013
Books
1) What is the Copenhagen
interpretation?
2) Are Duality and
Complementarity the same?
Princeton University Press, 2013
Einstein and the
Quantum
Stone attacks
“the Copenhagen interpretation,”
focusing on
“Born’s probabilistic interpretation of
the wave-function, Heisenberg’s
uncertainty principle and Bohr’s
mysterious complementarity
principle.” (p. 281)
Einstein and the
Quantum
“Einstein’s later critiques of quantum theory
focused less on its indeterminacy and more
on its strange epistemological status. In
quantum mechanics the actual act of
measurement is part of the theory; these
magic coins just mentioned exist in a state of
(heads, tails)-(tails, heads) uncertainty until
they are measured, and then they are forced
to ‘decide’ which state they are in.”
Heisenberg’s Copenhagen
Interpretation
Stone does not seem to be aware that he is
giving Heisenberg’s interpretation of
quantum mechanics, which is quite different
from Bohr’s interpretation.
You don’t have to take my word for this:
“Nine formulations of quantum mechanics,”
Daniel F. Styer et al, Am. J. Phys. 70 (2002):
pp. 288-297
[O]f the two primary architects of the
Copenhagen interpretation, Werner
Heisenberg maintained that ‘observation of
the position will alter the momentum by an
unknown and undeterminable amount,’
whereas Niels Bohr ‘warned specifically
against phrases, often found in the physical
literature, such as ‘disturbing of phenomena
by observation.’
“Nine formulations of quantum mechanics,”
Daniel F. Styer et al, Am. J. Phys. 70 (2002):
pp. 288-297
The wave function should be regarded as a
mathematical tool for calculating the
outcomes of observations, not as a
physically present entity existing in space
such a football, or a nitrogen molecule, or
even an electric field.
Examples of Misinterpretations
from Two Widely Praised 2013
Books
1) What is the Copenhagen
interpretation?
2) Are Duality and Complementarity
the same?
Pegasus Books, 2013
Farewell to Reality
Danish physicist Niels Bohr and German
Werner Heisenberg argued that particles
and waves are merely the shadowy
projections of an unfathomable reality into
our empirical world of measurement and
perception. …. This approach to quantum
theory became known as the Copenhagen
interpretation….At the heart of this
interpretation lies Bohr’s notion of
complementarity, a fundamental duality of
wave and particle behavior.
But According to Bohr They
Are Not
Since Bohr introduced and
developed the concept of
complementarity in quantum
mechanics, on this one I’ll let Bohr
speak for himself:
Niels Bohr 1885-1962
The Causality Problem in Atomic Physics
(1938)
It is true that the duality between the
undulatory and corpuscular conceptions
exists for matter as well as for light, but this
is only one aspect of a symbolical formalism
and its interpretation must be found in the
classical conceptions. Just as the mass and
charge of the electron can only be defined
classically, the description of the phenomena of radiation cannot dispense with the
idea of the electromagnetic wave field.
The Causality Problem in Atomic
Physics (1938)
The concepts of the photon and the material
wave are on the contrary purely abstract
methods of considering the general nature
of complementarity that exists, by reason of
the individuality of the quantum of action,
between the spatio-temporal representation
and the principle of conservation of
momentum and energy.
DUALITY
CLASSICAL (h=0)
Radiation
Matter
Waves
Particles
MATHEMATICAL REPRESENTATION
Characteristics
Trajectories
(wave fronts)
(world lines, characteristic strips)
DUALS
Bicharacteristics
Ensemble of trajectories
(rays)
(characteristic function)
QUANTUM MECHANICAL (h>0)
photon
wave function
COMPLEMENTARITY
SPACE-TIME DESCRIPTION
(x,t)
CONSERVATION OF
ENERGY- MOMENTUM
(E, p)
CLASSICAL (h=0)
Both can be defined and measured for an individual system
Either can be chosen to define a complete ensemble
QUANTUM MECHANICAL (h>0)
Only open systems can be treated– One must choose between
them to define and measure an individual system
Outline of the Talk:
1) Some background
information on my
approach
Traditional View
A theory is a conceptual
framework providing
predictions .
The results of experiments or
observations decide whether
the theory is right or wrong
Gaston Bachelard (1884-1962)
The Formation of the Scientific
Spirit (1938)
In order to include new experimental
tests, it is necessary to deform the
original concepts, study their
conditions of applicability, and above
all incorporate the conditions of
applicability of a concept into the
very meaning of the concept.
The New Scientific Spirit
(translation 1934).
[P]henomena must ... be carefully
selected, filtered and purified; they must
be cast in the mold of scientific
instruments and produced at the level of
these instruments. Now instruments are
just materialized theories. The
phenomena that come out of them bear
on all sides the mark of theory
The Lesson From
Bachelard
Don’t separate meaning and
measurement:
Incorporate the conditions of
applicability of a concept
into the very meaning of
the concept!
Outline of the Talk:
2) Measurability
Analysis
Measurability Analysis
Measurability analysis identifies those
concepts that a theory defines as
meaningful within some context and
investigates to what extent the values
associated with these concepts are ideally
measurable in the defining context (e.g.
concepts of hardness and viscosity in the
context of fluid and solid states of matter
in classical thermodynamics).
Peter G. Bergmann
Collaborator of Einstein
Pioneer in study of
quantization of
“generally covariant”
theories, including GR
Bergmann and Smith 1982
Measurability Analysis for the Linearized
Gravitational Field
“Measurability analysis identifies those
dynamic field variables that are
susceptible to observation and
measurement (“observables”), and
investigates to what extent limitations
inherent in their experimental
determination are consistent with the
uncertainties predicted by the formal
theory.”
Prolegomena to any future Quantum
Gravity (Stachel 2007)
‘[M]easurability analysis’… is based on
‘the relation between formalism and
observation’; its aim is to shed light on the
physical implications of any formalism: the
possibility of formally defining any
physically significant quantity should
coincide with the possibility of measuring
it in principle; i.e., by means of some
idealized measurement procedure that is
consistent with that formalism.
Warning!
This is not operationalism–
It’s not real because it’s
measurable, it must be
measurable because it’s
real!
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.
Outline of the Talk:
3)What quantization
is and is not
What is NOT Being Claimed
Quantization only makes sense
when applied to “fundamental”
structures or entities.
The Mystique Surrounding
Quantum Mechanics
“Anything touched by this formalism
thereby seems to be elevated– or should it
be lowered?– to a fundamental
ontological status. The very words
‘quantum mechanics’ conjure up visions of
electrons, photons, baryons, mesons,
neutrinos, quarks and other exotic
building blocks of the universe.”
The Mystique Surrounding
Quantum Mechanics (cont’d)
“But the scope of the quantum mechanical
formalism is by no means limited to such
(presumed) fundamental particles. There
is no restriction of principle on its
application to any physical system. One
could apply the formalism to sewing
machines if there were any reason to do
so!” (Stachel 1986)
What IS Quantization?
Quantization is just a way 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
Some Non-fundamental Quanta
1) quasi-particles: particle-like entities
arising in certain systems of interacting
particles, such as phonons and rotons in
hydrodynamics (Landau 1941)
2) phenomenological photons: quantized
electromagnetic waves in a homogeneous,
isotropic dielectric (Ginzburg 1940)
Two Kinds of Relations
There are relations, in which the things
are primary and their relation is
secondary: “relations between
things”
There are relations, in which the
relation is primary while the things
are secondary: “things between
relations”
Particles, Field Quanta
The particles of non-relativistic QM and the quanta
of special-relativistic Quantum Field Theory lack
inherent individuality
They are only individuated (to the extent that they
are) by some process (Feynman’s word) or
phenomenon (Bohr’s word), in which they are
involved.
Bosons and Fermions can be arbitrarily permuted
without changing the probability amplitude for
any process, and so are “things between
relations.”
Successful Quantization
Successful quantization of some classical
formalism does not mean that one has
achieved a deeper understanding of
reality– or better, an understanding of a
deeper level of reality. It means that one
has successfully understood the effects of
the quantum of action on the phenomena
(processes) described by the formalism
“In my Fathers house are many
mansions”-- John 14:2
Having passed beyond the quantum
mystique, one is free to explore how to
apply quantization techniques to various
formulations of a theory without the need
to single one out as the unique “right”
one. One might say: “Let a hundred
flowers blossom, let a hundred schools
contend” (Mao 1956)
Three Morals of This Tale
1) Relation Between Qantiz’ns
If two such quantizations at different levels
are carried out, one may then investigate
the relation between them
Example: Crenshaw demonstrates: “A
limited equivalence between microscopic
and macroscopic quantizations of the
electromagntic field in a dielectric”
[Phys. Rev. A 67 033805 (2003)]
Three Morals of This Tale
1) (cont’d)
If two such quantizations at the same level
are carried out, one may also investigate
the relation between them
Example: the relation between loop
quantization and field quantization of the
electromagnetic field: If you “thicken” the
loops, they are equivalent (Ashtekar and
Rovelli 1992)
Three Morals of This Tale
2) Don’t Go “Fundamental”
The search for a method of quantizing
space-time structures associated with
the Einstein equations is quite
distinct from:
The search for an underlying theory of
all “fundamental” interactions
Three Morals of This Tale
3) Don’t go “Exclusive”
An attempt to quantize one set of space-
time structures does not negate, and
need not replace, attempts to
quantize another set of space-time
structures. Everything depends on the
utility of the results in explaining
some physical processes.
The Causality Problem in Atomic Physics,
I.I.I.C., Warsaw 1938
The essential lesson of the analysis of measurements in quantum theory is thus the emphasis on
the necessity, in the account of the phenomena, of
taking the whole experimental arrangement into
consideration, in complete conformity with the fact
that all unambiguous interpretation of the quantum
mechanical formalism involves the fixation of the
external conditions, defining the initial state of the
atomic system concerned and the character of the
possible predictions as regards subsequent
observable properties of that system (Niels Bohr).
“A Well-defined Phenomenon”
Any measurement in quantum theory can in
fact only refer either to a fixation of the
initial state or to the test of such predictions,
and it is first the combination of
measurements of both kinds which
constitutes a well-defined phenomenon.
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.
Definability and Measurability
One must always establish a qualitative
and quantitative consonance between the
concept of an entity, for which physical
significance is claimed, and an ideal
measurement procedure for that entity.
If it is a quantum concept, h (the quantum
of action) must enter both definition and
measurement procedure
Bohr: 1931 Maxwell Centenary
Talk
It must not be forgotten that only the
classical ideas of material particles
and electromagnetic waves have a
field of unambiguous application,
whereas the concepts of photon and
electron waves have not.
Bohr: 1931 Maxwell Centenary
Talk
Their applicability is essentially limited to
cases in which, on account of the
existence of the quantum of action, it is
not possible to consider the phenomenon
observed as independent of the apparatus
utilized for their observation. …. [T]he
photon idea … is essentially one of
enumeration of elementary processes…,
Science: Confusion in Warsaw
Time Magazine ,13 June 1938
No remarkable new contributions to physical theory
came out of Warsaw, Poland last week, and none
was expected. Nevertheless, an International
Conference on New Theories in Physics, sponsored
by the League of Nations International Institute of
Intellectual Cooperation, was in session there,
attended by some 30 giants of theoretical physics.
On hand were Denmark's Niels Bohr and France's
Louis de Broglie.
Science: Confusion in Warsaw
The physicists' talk was lively and brilliant. But
they spent most of their time trying to find
some way to mend the painful gap between
Relativity and Quantum Mechanics, bickering
politely about the validity and application of
physical theories, asking themselves what
physical reality is after all. Bohr criticized de
Broglie and almost everyone present criticized
Sir Arthur Eddington. Altogether they gave the
impression of giants wallowing in a quagmire.
Paul Langevin 1872-1946
The Positivistic and the Realistic Trends
in the Philosophy of Physics
This idea [Laplacian determinism] is inhuman not only because it
fixes an ideal that is impossible to attain, but because it excludes the
observer from the system observed, because it separates the mind
from the matter which it tries to penetrate. …
In quantum mechanics it is the wave function that describes a
system and which allows us to calculate the probability depending
both upon the system and upon our information about it; it
introduces both the observer and the observed, the subject and the
obect, and every time we obtain new information, the wave
function appears to change. There are therefore as many wave
functions as observers. ….
Already, in this its first form, quantum mechanics closely unites the
subject with the object, the observer and the observed …
Bohr on Langevin 1)
[I]n order to avoid any misunderstanding
concerning the significance of the word
“indeterminism”, … recall that in quantum
effects we were not dealing with behaviour
independent of the objects, but that the
observable phenomena essentially depend
upon the interaction of these objects with
the measuring instruments which fix the
conditions for the experiment.
Bohr on Langevin 1)
That is the reason why we find ourselves
faced by quite a new situation in physics in
which the traditional conceptions of
determinism or indeterminism are not
univocally applicable. It is really wonderful
that in spite of this we can, with the help of
mathematical abstractions, put so much
order into a domain so vast and so rich with
experience, in a way that is entirely rational
and excludes all mysticism.
Langevin’s Response to Bohr 1)
…. He thought … that the use of the word
“corpuscle” [particle], weighed down by
old associations was sometimes a source
of confusion and difficulty. … [T]here is a
kind of intermediate picture that would
suit the corpuscle better than that of an
individual object taken over from classical
mechanics.
Bohr on Langevin 2)
Professor Bohr …wished, with regard to the
use of the corpuscle idea, to draw attention
to the danger there would be in confusing
the problem of the individuality of the
photon, which is entirely quantic, with the
corpuscular properties of the electron,
which can be related to an entirely classical
description.
Bohr on Langevin 2)
It is true that the duality between the
undulatory and corpuscular conceptions
exists for matter as well as for light, but this
is only one aspect of a symbolical formalism
and its interpretation must be found in the
classical conceptions. Just as the mass and
charge of the electron can only be defined
classically, the description of the phenomena of radiation cannot dispense with the
idea of the electromagnetic wave field.
Bohr on Langevin 2)
The concepts of the photon and the material
wave are on the contrary purely abstract
methods of considering the general nature
of complementarity that exists, by reason of
the individuality of the quantum of action,
between the spatio-temporal representation
and the principle of conservation of
momentum and energy.
Bohr on Langevin 2)
In fact we might say that from this point of
view the difference between matter and
light is as fundamental in quantum theory as
in the classical one.
Einstein to Paul Bonofield, September
18, 1939
“I do not believe that the light-quanta have
reality in the same immediate sense as the
corpuscles of electricity [i.e., electrons].
Likewise I do not believe that the particlewaves have reality in the same sense as the
particles themselves. The wave-character of
particles and the particle-character of light
will-- in my opinion-- be understood in a
more indirect way, not as immediate physical
reality."
DUALITY
CLASSICAL (h=0)
Radiation
Matter
Waves
Particles
MATHEMATICAL REPRESENTATION
Characteristics
Trajectories
(wave fronts)
(world lines, characteristic strips)
DUALS
Bicharacteristics
Ensemble of trajectories
(rays)
(characteristic function)
QUANTUM MECHANICAL (h>0)
photon
wave function
COMPLEMENTARITY
SPACE-TIME DESCRIPTION CONSERVATION OF
(x,t)
ENERGY- MOMENTUM (E, p)
Going Beyond Bohr
UNDERLYING SPACE-TIME STRUCTURES
Metric tensor
Affine connection
compatibility conditions
ROLE OF MASS
M= 0→Conformal structure
M>0 →Projective structure
For Some Discussion of This Question,
See FFP12
Photons: Their Emission and
Detection
Under what conditions will an (ideal)
device be able to register?
the emission of a photon (must be
non-destructive);
the detection of a photon (may be
destructive)
Photons: Their Emission and
Detection
The device used must contain a system
with a series of discrete energy and/or
momentum levels, the differences
between which are proportional to h so
that it can emit or absorb photons of
energy hν and momentum h/λ, linked to a
system sufficiently complex that it is able
to record an irreversible change when
such photons are emitted or absorbed.
The Moral of This Tale
Without interaction with a massive system having discrete quantum levels, neither the electromagnetic nor the gravitational
field will ever manifest their
discrete, particulate aspect
(“photons” and “gravitons”)
Outline of the Talk
4) Processes are
Primary, States are
Secondary
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.
Primacy of Process
Phrases such as "at any moment of
time", "at any given time” are
appropriate 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.
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).
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
instantaneous measurements.
Primacy of Process
But sometimes we most definitely
cannot, as Bohr and Rosenfeld
demonstrated for QFT, where the basic
quantities defined by the theory are
space-time averages. Their critique of
Landau and Peierls shows what happens
if you forget this!
"Extension of the principle of indeterminateness for
the relativistic quantum theory"
L. Landau and R. Peierls, Z. Phys. 69, 56 (1931).
Rudolf Peierls
Lev Davidovich Landau
“Indeterminacy in Measurements by
Charged Particles,” Jens Lindhard
“Indeterminacy in Measurements by
Charged Particles”
In 1931, Landau and Peierls raised doubts
about the consistency of the quantum theory
of electromagnetic fields, doubts which, if
true, were expected to deprive the theory of
any physical basis. They maintained that,
due to quantal uncertainty relations, it was
not possible to measure electromagnetic
radiation fields by means of charged
particles.
“Indeterminacy in Measurements by
Charged Particles”
Soon after, Bohr and Rosenfeld
criticized this derivation and went on to
show that electromagnetic fields could
indeed be measured if the point-like
particles of Landau and Peierls were
replaced by spatially extended charge
distributions [and the measurement
extended over a finite time interval- JS].
“Zur Frage der Messbarkeit der elektromagnetischen Feldgrössen,” Bohr & Rosenfeld 1933
Niels Bohr
Leon Rosenfeld
“On the Measurability of Electromagnetic
Field Magnitudes” (Bohr-Rosenfeld 1933)
In their analysis of the co-measurability of
electric and magnetic field components,
rather than Landau-Peierls’ test point
particles, to get finite results they had to
use averages over test bodies occupying
finite space-time regions, paralleling their
similar averaging of the commutation
relations between field components.
To Sum It Up:
A quantum process involves
three stages: preparation,
interaction, registration.
Big question: How does h figure
in the preparation and/or
registration procedures?
Outline of the Talk
5) Commutation
Relations in Quantum Mechanics
Commutation Relations
One central method of taking into account
the quantum of action is by means of
introducing commutation relations
between various particle (non-rel QM) or
field (SR QFT) quantities (“observables”)
into the formalism.
But these commutation relations have
more than a purely formal significance
Some Measurement Problems in
Quantum Gravity – JS
Within quantum mechanics, the uncertainty
relations-- or better, using a direct
translation of the German term
Unbestimmheit, the indeterminacy relations- assert that there is a limit to the
simultaneous measurability of a pair of
classical canonically conjugate variables such
as the position and momentum of a system.
Some Measurement Problems in
Quantum Gravity – JS
And, as Heisenberg was at great pains to
demonstrate in his little book "Physical
Principles of Quantum Theory," the limit
set by the theory on the simultaneous
measurement of any pair of canonically
conjugate variables agrees perfectly with
the limits set by any idealized measurement procedure that takes into account
the finite size of the quantum of action.
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.
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.
Outline of the Talk
6) Commutation
Relations in Quantum Field Theory
Measurement of the space-time interval between
two events … (Amelino-Camelia and JS 2007)
We share the point of view emphasized by
Heisenberg and Bohr and Rosenfeld, that
the limits of definability of a quantity
within any formalism should coincide with
the limits of measurability of that quantity
for all conceivable (ideal) measurement
procedures. For well-established theories,
this criterion can be tested. For example,
in spite of a serious challenge [by Landau
and Peierls], source-free quantum electrodynamics was shown to pass this test.
Olivier Darrigol
The problem of the measurability of
quantum fields (JS transl’n)
The discussion of these fundamental
difficulties at the 1930 Solvay Congress
was dominated by Bohr’s viewpoint …
[T]he scope of these problems and the
nature of their solutions had to be
uncovered by a critique of the basic
concepts of the threatened theories,
The problem of the measurability of
quantum fields
by an evaluation of the possibilities of
definition and of observation within them.
Bohr’s main message: “On can only judge
the coherence of the symbolic method by
examining the limits of observability in the
usual sense”
New Heisenberg Relations?
Heisenberg had been the first to consider the
problem of field measurements in his Chicago
lectures of spring 1929. … [I]n his analysis of the
x-ray microscope, he tended to privilege the
corpuscular and discontinuous viewpoint above
that of the wave viewpoint. Bohr had
succeeded, not without difficulty, in convincing
him that the evaluation of the limits of the
corpus-cular viewpoint necessarily involved
calling upon the wave theory.
New Heisenberg Relations?
But then it became important for Heisenberg to show that, reciprocally, the
domain of applicability of the electromagnetic field concept must be limited by
the existence of corpuscular aspects. He
provided new uncertainty relations
ΔExΔHy ≥ hc/(δl)
for the averages of the electric field Ex and
magnetic field Hy over the same domain of
extension …
New Heisenberg Relations?
Heisenberg, Bohr wrote … should
have taken account not only of the
spatial extension of the field
measurements, but also of their
duration, essential for the estimation
of the role of quantum fluctuations of
the field.
New Heisenberg Relations?
[C]ontrary to the initial arguments of
Heisenberg the Bohr-Rosenfeld article
contains the rigorous proof that
ΔExΔHy = 0
if Ex and Hy are measured in the same
[four-dimensional] domain.
David Kaiser’s article in Conceptual
Foundations of Quantum Field Theory
By 1960 … on the quantum field theory side,
theorists taught their students to proceed
along four steps:
1. Specify an interaction Hamiltonian in
terms of quantum fields.
2. Derive propagators for these fields from
the fields’ equations of motion and
commutation relations
Why Hamiltonians and EqualTime Commutation
Relations?
Tradition!!!
The Peierls Bracket (1999)
Bryce DeWitt
The Peierls Bracket
When expounding the fundamentals of
quantum field theory physicists almost
universally fail to apply the lessons that
relativity theory taught them early in the
twentieth century. Although they usually
carry out their calculations in a covariant
way, in deriving their calculational rules they
seem unable to wean themselves from
canonical methods and Hamiltonians,
The Peierls Bracket
which are holdovers from the nineteenth
century, and are tied to the cumbersome
(3+1)-dimensional baggage of conjugate
momenta, bigger-than-physical Hilbert
spaces and constraints. … One of the
unfortunate results … is that physicists, over
the years, have almost totally neglected the
beautiful covariant replacement for the
canonical Poisson bracket that Peierls
invented in 1952.
Pierre Cartier & Cécile DeWittMorette, (Cambridge 2006)
A Legacy, by Pierre Cartier
Bryce DeWitt constructs the operator
formalism of quantum physics from the
Peierls bracket which leads to the Schwinger
variational principle and to functional
integral representations. The bracket
invented by Peierls in 1952 is a beautiful, but
often neglected, covariant replacement for
the canonical Poisson bracket, or its
generalizations, used in canonical
quantization.
The Pursuit of Quantum Gravity
Memoirs of Bryce DeWitt from 1946 to 2004
The Pursuit of Quantum Gravity,
The remarkable thing about the Peierls’
brackets is that they do not depend for their
definition on the introduction of a canonical
formalism. They are completely determined
by the laws of propagation of Jacobi fields,
and their definition emphasizes the global
spacetime view of dynamics.
When I first realized that Bohr and Rosenfeld
were dealing with Peierls brackets, I became
quite excited.
The Pursuit of Quantum Gravity,
[T]he Peierls bracket is the appropriate
concept for analyzing the quantum
mechanical limitations on measurement
accuracy. This analysis says that
measurements can, in principle, always be
made to an accuracy equal to but no better
than that allowed by the a priori
uncertainties implied by the quantum
mechanical formalism.
The Problem of Quantum
Gravity
We need a theory that can
somehow encompass the
achievements of both Quantum
Field Theory (backgrounddependent) & General relativity
(background-independent)
But That is Another Story!