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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

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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 !

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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 john.stachel@gmail.com

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Examples of Misinterpretations from Two Widely Praised 2013 Books 1)What is the Copenhagen interpretation? 2)Are Duality and Complementarity the same?

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Princeton University Press, 2013

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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)

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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.”

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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:

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“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.’

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“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.

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Examples of Misinterpretations from Two Widely Praised 2013 Books 1)What is the Copenhagen interpretation? 2)Are Duality and Complementarity the same?

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Pegasus Books, 2013

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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.

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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:

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Niels Bohr 1885-1962

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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 pheno- mena of radiation cannot dispense with the idea of the electromagnetic wave field.

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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.

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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

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COMPLEMENTARITY SPACE-TIME DESCRIPTION CONSERVATION OF (x,t) 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

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Outline of the Talk: 1) Some background information on my approach

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Traditional View A theory is a conceptual framework providing predictions. The results of experiments or observations decide whether the theory is right or wrong

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Gaston Bachelard (1884-1962)

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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.

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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

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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!

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Outline of the Talk: 2) Measurability Analysis

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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).

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Peter G. Bergmann Collaborator of Einstein Pioneer in study of quantization of “generally covariant” theories, including GR

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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.”

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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.

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Warning! This is not operationalism– It’s not real because it’s measurable, it must be measurable because it’s real!

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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.

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Outline of the Talk: 3)What quantization is and is not

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What is NOT Being Claimed Quantization only makes sense when applied to “fundamental” structures or entities.

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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.”

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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)

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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

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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)

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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”

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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.”

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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

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“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)

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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)]

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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)

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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

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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.

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The Causality Problem in Atomic Physics, I.I.I.C., Warsaw 1938 The essential lesson of the analysis of measure- ments 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).

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“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.

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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

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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.

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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

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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.

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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…,

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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.

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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.

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Paul Langevin 1872-1946

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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 …

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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.

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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.

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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.

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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.

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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 pheno- mena of radiation cannot dispense with the idea of the electromagnetic wave field.

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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.

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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.

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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 particle- waves 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."

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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

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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

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For Some Discussion of This Question, See FFP12

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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)

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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.

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The Moral of This Tale Without interaction with a mass- ive system having discrete quan- tum levels, neither the electro- magnetic nor the gravitational field will ever manifest their discrete, particulate aspect (“photons” and “gravitons”)

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Outline of the Talk 4) Processes are Primary, States are Secondary

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Lee Smolin

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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.

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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.

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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.

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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).

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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.

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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!

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"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

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“Indeterminacy in Measurements by Charged Particles,” Jens Lindhard

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“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.

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“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].

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“Zur Frage der Messbarkeit der elektromag- netischen Feldgrössen,” Bohr & Rosenfeld 1933 Niels Bohr Leon Rosenfeld

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“On the Measurability of Electromagnetic Field Magnitudes” (Bohr-Rosenfeld 1933) I n 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.

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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?

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Outline of the Talk 5) Commutation Relations in Quan- tum Mechanics

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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

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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.

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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 measure- ment procedure that takes into account the finite size of the quantum of action.

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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.

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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.

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Outline of the Talk 6) Commutation Relations in Quan- tum Field Theory

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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 electro- dynamics was shown to pass this test.

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Olivier Darrigol

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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,

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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”

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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.

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New Heisenberg Relations? But then it became important for Heisen- berg to show that, reciprocally, the domain of applicability of the electro- magnetic field concept must be limited by the existence of corpuscular aspects. He provided new uncertainty relations ΔE x ΔH y ≥ hc/(δl) for the averages of the electric field E x and magnetic field H y over the same domain of extension …

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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.

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New Heisenberg Relations? [C]ontrary to the initial arguments of Heisenberg the Bohr-Rosenfeld article contains the rigorous proof that ΔE x ΔH y = 0 if E x and H y are measured in the same [four-dimensional] domain.

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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

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Why Hamiltonians and Equal- Time Commutation Relations? Tradition!!!

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The Peierls Bracket (1999) Bryce DeWitt

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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,

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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.

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Pierre Cartier & Cécile DeWitt- Morette, (Cambridge 2006)

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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.

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The Pursuit of Quantum Gravity Memoirs of Bryce DeWitt from 1946 to 2004

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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.

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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.

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The Problem of Quantum Gravity We need a theory that can somehow encompass the achievements of both Quantum Field Theory (background- dependent) & General relativity (background-independent)

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But That is Another Story!

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