Department of History and Philosophy of Science

University of Pittsburgh

This site is http://www.pitt.edu/~jdnorton/teaching/philphys

This document may be updated in the course of term.

Updated: September 1, 2004. (Ising simulation link added November 10, 2004)

Schedule

The Three Principal Problems of Philosophy of Modern Physics

Topics and Readings

Space and Time

Quantum Theory

Statistical Physics

More

Further Literature

The Fine Print

Other course documents

1. Mathematical structures. 2. Physical ontology 3. Appearances

**First Problem**. When do intertransformable mathematical structures represent the same physical ontology?

(Esp. philosophy of space and time.)

- The two ways in which the mathematical structures in question may be related are by symmetry transformations and gauge transformations.

A*symmetry transformation*leaves the mathematical structure unchanged, e.g. a translation or a rotation in a Euclidean space. This is the strongest form of interrelation. A symmetry transformation expresses a homogeneity of space or a relativity principle. e.g. The Lorentz transformation is a symmetry of a Minkowski spacetime. In the case of a symmetry transformation, we can only say that the original and transformed structures represent something different in the ontology by calling on something outside what is described by the mathematical structures of the theory. e.g. In a Minkowski spacetime free of all other material structures, a Lorentz transformation can be construed as shifting one "God's eye" observer's frame to another in relative motion.

A*gauge transformation*alters the mathematical structure but in a way for which we have some reason to believe that the physical ontology represented is unchanged. The simplest example is a real valued field in Euclidean space representing a Newtonian gravitational field. Since differences in potential only have physical meaning, we can always add a real constant to the field without changing the physical ontology, although the numerical value of the potential field will be changed throughout. This addition is a gauge transformation. The principal challenge with a gauge transformation is finding grounds for judging that the distinct mathematical structures represent the same thing physically. Two sorts of grounds have been used: verificationism--there is no observable difference between what the two structures represent; indeterminism--if we insist the two structures represent distinct physical ontologies, we gratuitiously visit indeterminism of the theory.

There is an interesting inversion of the general problem with the existence of unitarily inequivalent representations of the commutation relations in algebraic quantum field theory. These are structures that are not intertransformable, but plausibly should somehow represent the same ontology. Or should they? **Second Problem**. What do the mathematical structures of a successful theory represent in the physical ontology?

(Esp. philosophy of quantum theory)

- With any mathematical theory, we must associate the mathematical structures of the theory with something in the physical ontology. Usually this is not so difficult to achieve. Invariant line elements in a metrical space are associated with physical distances in ordinary space. Or the magnetic field strength vector field of electrodynamics is associated with the magnetic field supported by a magnet. One instance of this problem has proven remarkably recalcitrant:

*Interpretations of quantum theory*Quantum theory has proven enormously successful in dealing with appearances. However the debate rages without evident hope of resolution over just what physical ontology ought to be ascribed to the theory's mathematical structures. The Schroedinger wave of a particle was initially hoped to be some sort of charge distribution; and then perhaps a wave of probability or potentiality, whatever that might mean; and most recently the theory is somehow about "quantum information." The most refractory problems have been presented by superpositions and entangled states. The problems of the latter were paraded in the Einstein-Podolsky-Rosen paper of 1935 and developed later by Bell. The concensus arising from this problem is of an unavoidable non-locality or non-separability in the theory's physical ontology. The problems of the former (superpositions) were emphasized by Schroedinger's "cat" paper also of 1935. Seventy years later, the enormous literature in "interpretations of quantum theory" is largely given over to trying to specifying just what the physical ontology of quantum theory should be.

*The measurement problem*What complicates the problem of interpretation enormously is a possible incompleteness of the theory's formal apparatus. In traditional developments, two types of time development are distinguished: Schroedinger evolution (spreading of the wave packet) and measurement (collapse of the wave packet). Unfortunately we do not have precise rules to decide when one or the other happens. Rather we have pretty good rules of thumb that work well for practical application of the theory. If a wave interacts with something big, it collapses; if it interacts with one or two other particles, or none at all, it spreads. But how are we to handle the intermediate cases?

Some interpretations ("no collapse theories") do away with the measurement problem by seeking to develop the theory entirely from Schroedinger time evolution. Such approaches complicate matters since we no longer have a single fixed theory. Collapse and no-collapse theories differ on just what time evolutions are possible within the mathematical structures. **Third Problem**. How do we reconcile a physical ontology with seemingly incompatible appearances?

(Esp. philosophy of statistical physics)

- In this problem, we face some sort of systematic mismatch between the physical ontology and appearances. Two important instances arise in statistical physics.

*The asymmetry of time.*Excepting weak interactions, all our microphysics is time reversible. The appearances of the macroscopic world are strongly directed in time, distinguishing past from future. How can this asymmetry arise from the microphysics?

*Phase transitions.*The interactions of microphysics are represented by differential equations that admit no discontinuities. The appearance of thermal systems is characterized by discontinuous transformations--phase transitions, such as between solid and gaseous state. How can the appearance of discontinuity arise from a continuous ontololgy?

This problem can arise in many other contexts. e.g. Einstein in 1905 worried about how to reconcile the presence of a preferred ether state of rest in then current electrodynamics with appearances that conformed to the principle of relativity. Some versions of no-collapse theories of quantum mechanics must explain how quantities appear to have sharp values when their version of quantum mechanics says that they really do not.

Newton, Leibniz and the debate over absolute space.

Einstein and special relativity

Einstein and the vexed problem of whether general relativity extends the principle of relativity to acceleration.

Mach's Principle and Machian theories of space and time.

Newtonian cosmology turns out to host a rather unexpected principle of relativity tha governs uniform acceleration. Because of its unfamiliarity, it is a good place to hone our intutions on the principle of relativity.

Norton, J. D. "The Force of Newtonian Cosmology: Acceleration is Relative" Philosophy of Science, 62 (1995), pp.511-22. Download.

The revival of Einstein's "hole argument" was an early and important element in the development of interest in gauge freedoms in philosophy of physics. One aspect of its importance lies in its clear delineation of the two sorts of physical groundings for the judgment that a transformation represents a non-physical gauge freedom: verificationism and determinism.

Norton, J. D. "The Hole Argument,"

A perennial topic of debate is whether the physical effects of special relativity (e.g. length contraction) require some sort of physical explanation that in turn requires some sort of state of rest such as Lorentz envisaged was supplied by his ether.

Balashov, Y. and Janssen, M. "Presentism and relativity,"

Brown, H, R. and Pooley, O. "Minkowski space-time: a glorious non-entity." 2004 Download from philsci-archive.

Is the reality of time a moving instant of "now," a moving "now" that converts potential futures into an existent past, or an eternal coexistence of all times? Does modern physics have anything to add to this old debate? Savitt, S. "Being and Becoming in Modern Physics,"

Earman, J. "Thoroughly Modern McTaggart: Or, What McTaggart Would Have Said if He Had Read the General Theory of Relativity,"

Mauldlin, T. "Thoroughly Muddled McTaggart: Or, How to Abuse Gauge Freedom to Create Metaphysical Monostrosities" With a Response by John Earman

Much of what is said about symmetries and gauge freedoms is of direct relevance to general problems in philosophy of science.

Ismael, J. and Van Fraassen, B. "Symmetry as a Guide to Superfluous Theoretical Structure," Ch. 23 in K. Brading and E. Castellani, eds.,

Norton, J. D. "The Formal Equivalence of Grue and Green and How It Undoes the New Riddle of Induction."

Initially the affect appeared to show that the gauge field of electrodynamics did matter empirically in the quantum mechanical context, so we seemed to have a splendid example of a mathematical construct migrating to the physically real. Later reflection suggested that this is not so and that what has empirical significance is a gauge indepedent holonomy.

Healey, R. "Nonlocality and the Aharonov-Bohm Effect,"

Healey, R. "On the Reality of Gauge Potentials," Philosophy of Science, 68 (2001), pp. 432-55.

Batterman, R. "Falling cats, parallel parking, and polarized light",

Nounou, A. M. "A fourth way to the Aharonov-Bohm Effect." Ch. 10 in K. Brading and E. Castellani, eds.,

Just how should one respond to the discovery of a gauge freedom in a physical theory? Redhead, M. "The Interpretation of Gauge Symmetry," Ch. 7 in K. Brading and E. Castellani, eds.,

Earman, J. Gauge matters.

Earman, J. "Tracking Down Gauge: An Ode to the Constrained Hamiltonian Formalism," Ch. 8 in K. Brading and E. Castellani, eds.,

Two distinct strategies emerged in the nineteenth century. One could overdescribe the physical system and then use group theoretic methods to deprive some parts of the description of physical significance; or one would strive to introduce just as much mathematical structure as already had physical significance.

Norton, J. D. "Geometries in Collision: Einstein, Klein and Riemann." in J. Gray, ed.,

Bohmian Mechanics

Collapse models: von Nuemann's two rules and the GRW (Ghirardi, Rimini, Weber) proposal

Consistent histories

Copenhagen Interpretation: Bohr's influential vision

Decoherence

Everett Relative-State formulation/Many worlds interpretation

Modal Interpretations

This is a recent version of the idea popularly rendered as the notion that a superposition in quantum mechanics is to be associated with many worlds existing in parallel.

Wallace, D., "Worlds in the Everett interpretation,"

Wallace, D. "Everett and structure,"

Wallace, D., "Everettian rationality: defending Deutsch's approach to probability in the Everett interpretation,"

Quantum mechanics tells us that, in Schroedinger's famous set up, we create a cat in a superposition of live and dead states. The proposal is that this superposed cat rapidly gets into interaction with many other systems and that, through this chaotic interaction, reverts to the more famililar cat that definitely either alive or dead.

Bacciagaluppi, G "The Role of Decoherence in Quantum Theory"

Zurek, W.H., "Decoherence, einselection and the existential interpretation (The Rough Guide)."

Zurek, W.H., "Decoherence, einselection and the quantum origins of the classical."

A number of characteristically quantum effects can be described in terms of information processing. This has led to the notion that what is distinctive about quantum theory is it harbors a new notion of information, quantum information. The effects are all expressions of entanglement and include quantum teleportation, quantum cryptography and quantum computation

Rae, A. "Quantum Information," Ch. 12 in

Bub, J. "Quantum Entanglement and Information,"

Whether quantum theory requires a new notion of information, as opposed to that already captured in Shannon's classical theory has been debated.

Brukner, C. and Zeilinger, A., Operationally Invariant Information in Quantum Measurements,

Timpson, C. G. "On a Supposed Conceptual Inadequacy of the Shannon Information in Quantum Mechanics,"

Duwell, A. "Quantum Information Does not Exist,"

This theorem expresses the fundamental limits on which properties may be possessed simultaneously by a quantum system. It is one of the most important of the "no go" theorems that preclude quantum craziness arising merely as an artifact of our ignorance of a hidden but classically well behaved microphysics.

Redhead, M. "The Kochen Specker Paradox," Ch. 4 in

Held, Carsten "The Kochen-Specker Theorem,"

Hrushovski, E. and Pitowsky, I "Generalizations of Kochen and Specker's theorem and the effectiveness of Gleason's theorem"

Within the constraints of the Kochen Specker theorem, just how many possessed states may be ascribed to a quantum system? Jeff Bub and Rob Clifton found a nice geometrical answer.

Dickson, M. [Section 3] of "The Modal Interpretations of Quantum Theory,"

Bub, J. and Clifton, R. "A Uniqueness Theorem for 'No Collapse' Interpretations of Quantum Mechanics,"

Bub, J. "The problem of interpretation," Ch. 4 in

Brown, H. R. and Uffink, J. "The Origins of Time-Asymmetry in Thermodynamics: The Minus First Law,"

The temporal asymmetry of familiar processes now is often traced back to the low entropy state of the universe near the big bang. Is this explanation cogent or does it in turn raise problems in need of further explanation?

Price, H. "On the Origins of the Arrow of Time: Why There is Still a Puzzle About the Low Entropy Past,"

Callender, C. "There is No Puzzle About the Low Entropy Past,"

Ch.11 and 12 in C. Hitchcock, ed.,

Parker, D. (2004) "Thermodynamic Irreversibility: Does the Big Bang Explain what it Purports to Explain?"

The best known examples of how discontinuous phase transitions may be wrestled from an apparently continuous microphysics are spin lattices. Here is a brief introduction to how this works. Reading to be determined.

A nice simulation of a 2D Ising model.

A short introduction to notions that now dominate statistical physics and have enabled considerable clarification of the nature of phase transitions. Phase transitions are recovered by taking limits of infinitely many components. The renormalization group enables us to map out the behavior of systems at their critical points.

"Asymptotic Explanation," Chapter 4 in R. W. Batterman,

Should we dispense with the idea that phase transitions involve discontinuities in thermodynamic quantities so that we need not take the thermodynamic limit of infinitely many particles?

Callender, C. "Taking Thermodynamics Too Seriously,"

Batterman, Robert (2004) Critical Phenomena and Breaking Drops: Infinite Idealizations in Physics. Download from philsci-archive

A long standing tradition maintains that proper attention to information theory explains why Maxwell's demon must fail to reverse the second law of thermodynamics. John Earman and I (Norton) have long argued that this explanation depends on question beginning. More recently I've pointed to systematic misapplications of statistical physics that seriously compromise Landauer's principle, the notion that memory erasure must be accompanied by an entropy cost.

Earman, J. and Norton, J. D. "Exorcist XIV: The Wrath of Maxwell's Demon."

Norton, J. D. "Eaters of the Lotus: Landauer's Principle and the Return of Maxwell's Demon." Prepared for New Directions in the Foundations of Physics, American Institute of Physics, College Park, MD. April 30-May 2, 2004. Download from http://www.pitt.edu/~jdnorton/homepage/cv.html

Is there some kind of unreasonable miracle is the immense fertility of mathematics in physics?

Wigner, E., "The Unreasonable Effectiveness of Mathematics in the Natural Sciences",

Davey, K. "Unresaonable Effectiveness: Historical/Aesthetic" and "Unreasonable Effectiveness: Descriptive" Chs. 7-8 in

Zeno thought that the idea of completing an infinity of operations was deeply problematic. It is now becoming clear that the completion of an infinity of actions is not just a logical possibility but is licensed by quite a few respectable physical theories and with some curious outcomes. But is the license really licit?

Alper, J. S., Mark Bridger, M. , Earman, J. and Norton, J. D. , "What is a Newtonian System? The Failure of Energy Conservation and Determinism in Supertasks,"

Albert, D., 1992,

Barrett, J.

Brading, K. and Castellani, E. , eds.,

Bub, J.

Cushing, J. T. and McMullin, E., eds.,

Hughes, R. I. G.

Norton, J. "Introduction to the Philosophy of Space and Time, " in

Preskill, J.,

Rae, A.

Redhead, M.

Sklar, L.

Sklar, L.

Stanford Encyclopedia of Philosophy http://plato.stanford.edu

Torretti, R.

Bub, J. Appendix to

Clifton, Rob, Introductory Notes on the Mathematics Needed for Quantum Theory. 1996. Download from philsci-archive

Ismael, Jenann, "Quantum Mechanics,"

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