GENERAL COVARIANCE, ARTIFICIAL GAUGE FREEDOM AND EMPIRICAL EQUIVALENCE A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by James Brian Pitts Don Howard, Director Graduate Program in History and Philosophy of Science Notre Dame, Indiana July 2008 c Copyright by James Brian Pitts 2008 All Rights Reserved GENERAL COVARIANCE, ARTIFICIAL GAUGE FREEDOM AND EMPIRICAL EQUIVALENCE Abstract by James Brian Pitts This dissertation updates the debate over the nontriviality of general covari- ance for Einstein’s General Theory of Relativity (GTR) and considers particle physics in the debate over underdetermination and empirical equivalence. Both tasks are tied to the unexplored issue of artificial gauge freedom, a valuable form of descriptive redundancy. Whereas Einstein took general covariance to characterize GTR, Kretschmann thought it merely a formal feature that any theory could have. Anderson and Friedman analyzed substantive general covariance as the lack of absolute objects, fields the same in all models. Some extant counterexamples and a new one involv- ing the electron spinor field are resolved. However, Geroch and Giulini diagnose an absolute object in GTR itself in the metric’s volume element. One might in- stead analyze substantive general covariance as formal general covariance achieved without hiding preferred coordinates as scalar “clock fields,” recalling Einstein’s early views. Theories with no metric or multiple metrics make the age of the uni- verse meaningless or ambiguous, respectively, so the ancient and medieval debate over the eternity of the world should be recast. Particle physics provides case studies for empirical equivalence. Proca’s elec- tromagnetism with some nonzero photon mass constitutes a family of rivals to James Brian Pitts Maxwell’s theory. Whereas any Proca theory can be distinguished empirically from Maxwell’s, the Proca family approaches Maxwell’s for small masses, yield- ing permanent underdetermination with only approximate empirical equivalence. The weak nuclear force also displays a smooth massless limit classically, but not after quantization, recalling the instability of empirical equivalence under change of auxiliary hypotheses. The standard electroweak theory apparently permits a photon mass term and hence underdetermination, but possible further unifica- tion might not. The question of underdetermination regarding massive gravity is unresolved. Physicists often reformulate a theory to install additional fields and new gauge symmetries preserving empirical content. Post-positivist philosophers might judge the result a distinct and inferior theory, but physicists consider it a perhaps supe- rior formulation of the same theory. Evidently a step back from na¨ıve realism to- wards mathematical-empirical equivalence is appropriate for physical ontology and theory individuation. Artificial gauge freedom licenses a generalized Kretschmann objection, but the clock field case suggests a resolution. For Dilkushi, more precious than jewels. ii CONTENTS ACKNOWLEDGMENTS ........................... v CHAPTER1:INTRODUCTION... .. .. .. .. ... .. .. .. .. 1 1.1 Overview................................ 1 1.2 GeneralCovariance .......................... 4 1.3 EmpiricalEquivalence . 10 1.4 ArtificialGaugeFreedom. 14 CHAPTER 2: GENERAL COVARIANCE, ANDERSON-FRIEDMAN AB- SOLUTE OBJECTS, VARIATIONAL PRINCIPLES & CLOCK FIELDS 18 2.1 Absolute Objects, Admissible Coordinates and Groupoids..... 18 2.2 TensorDensities............................ 37 2.3 Confined Objects and Global Space-time Topology . 49 2.4 Jones-Geroch Vector Field Counterexample and Friedman’s Reply 50 2.5 Redefinition with Variational Principle, Maidens’s Worry, and the Rosen-SorkinAnswer .. ... .. .. .. .. ... .. .. .. .. 53 2.6 Eliminating Local Irrelevance Excludes the Geroch-Jones Vector FieldfromDustandOtherFluids . 62 2.7 Torretti’s and Norton’s Constant Curvature Space Counterexam- plesDoHaveAbsoluteObjects . 71 2.8 Geroch-Giulini Scalar Density Counterexample √ g in GTR . 73 − 2.9 Bach-Weyl Gravity Lacks √ g ................... 78 2.10 Implications of Non-Metric or− Multi-Metric Theories for Ancient- MedievalDebate Overthe Eternityof the World . 80 2.11 Which Fields to Test for Absoluteness? Equivalence, Irreducibility, Concomitants, etc. .......................... 83 2.12 Absoluteness and Nonvariationality? . 97 2.13 Tetrad-Spinor Counterexample and Resolution via Nonlinear Geo- metric Objects and the Ogievetsky-Polubarinov Spinor Formalism 103 2.14 Unwanted Vanishing Absolute Objects: Concomitant Irrelevance . 115 2.15 Conclusions about Anderson-Friedman Absolute Objects Program 120 iii 2.16 Parametrization: Preferred Coordinates as Clock Fields and the Einstein-KretschmannDebate . 126 CHAPTER 3: INEXACT EMPIRICAL EQUIVALENCE IN ELECTRO- MAGNETIC, YANG-MILLS, AND GRAVITATIONAL THEORIES . 134 3.1 Typesof InexactEmpiricalEquivalence. 134 3.2 Approximate Empirical Equivalence in Electromagnetism: The Pos- sibilityofMassivePhotons . 142 3.3 No Approximate Equivalence in Yang-Mills Theories? . 146 3.4 Approximate Equivalence in Electroweak Theory: Yang-Mills The- orywithEssentiallyAbelianSector . 149 3.5 Is There Approximate Equivalence for Gravity? General Relativity andItsMassiveVariants . 150 CHAPTER 4: EXACT EMPIRICAL EQUIVALENCE AND ARTIFICIAL GAUGEFREEDOM............................ 153 4.1 Overview................................ 153 4.2 Modified Batalin-Fradkin-Tyutin Procedure Gives Stueckelberg For- mulationDirectly .. .. ... .. .. .. .. ... .. .. .. .. 157 4.3 Gauge Freedom in Massive Gravity: An External Symmetry . 165 4.4 Generalized BFT Conversion, Unconstrained Theories and Parametriza- tion .................................. 174 4.5 Artificial Gauge Freedom and Recent Views on Underdetermina- tion:GlymourandQuine . 177 CHAPTER5:CONCLUSION. 185 5.1 Insights for the Philosophy of Physics . 185 5.2 Insights for the General Philosophy of Science . 190 BIBLIOGRAPHY ............................... 196 iv ACKNOWLEDGMENTS I am very grateful to my advisor Don Howard for his wise guidance and support at all stages of this work. Katherine Brading’s great interest has been most helpful. Ikaros Bigi has served as an oracle in particle physics whenever I have consulted him. In return I can only offer the conclusions that particle physics is important for the philosophy of space-time and the general philosophy of science. Anja Jauernig has helped greatly to extract the philosophical lessons out of technical issues. William McGlinn’s encouragement is appreciated. Robert Geroch’s influence has been enormously beneficial on chapter 2. His counterexample to the Anderson-Friedman program radically transformed an an- cestor of that chapter (Pitts, 2006a). The present work contains in section 2.11 a theorem and proof outline that he donated. Harvey Brown and Oliver Pooley have been helpful in many ways beyond what the text can show. Discussion with John Norton on underdetermination and empirical equivalence has been helpful. Ted Jacobson, Charles Misner and Aron Wall have contributed insightful probing on the material of chapter 2. I thank Roger Jones for discussion about his coun- terexample to the Anderson-Friedman project, David Malament for contributing helpful perspective on the Anderson-Friedman project, and many people with whom I have discussed the matter of spinors in curved space-time. Discussion with William Lane Craig regarding the debate over the eternity of the world is appreciated. All views expressed here are my own. v I thank the University of Notre Dame for a Presidential Fellowship that made the conditions of the production of this work pleasant. The Reilly Center for Science, Technology and Values has provided generous support for conference presentations. The Mustard Seed Foundation’s generous support for 2007-8 gave me considerable liberty for writing. One of the great methodological lessons of this work is the importance of old eastern European particle physics and differential geometry literature, which for- tunately is generally available in English or perhaps French or German. Where my needs exceeded the ample holdings on campus, the Notre Dame libraries gen- erously tracked down materials by interlibrary loan without limit, allowing me to find many gems and ensuring that I did not lack a paper that I might need. The libraries of the University of Maryland kindly gave me a library card during my stay in the Washington, D.C. area. The Joint Institute for Nuclear Research in Dubna, Russia was able and willing to answer my request for preprint R-1890 containing the unpublished appendix of the spinor paper by Ogievetsky and Pol- ubarinov (Ogievetski˘i and Polubarinov, 1965). More personally, I thank my parents for their encouragement and support during my long years of education. My wife Dilkushi arrived late in the writing process, but has transformed my life. vi CHAPTER 1 INTRODUCTION 1.1 Overview This dissertation will address two longstanding issues in the philosophy of physics or the more general philosophy of science, namely, general covariance and empirical equivalence. Both of these issues are strongly tied to the largely untouched conceptual and technical question of artificial gauge freedom. It has been claimed, starting with Einstein in the 1910s and including workers in Loop Quantum Gravity today, that general covariance is the chief novel and distinc- tive feature of the General Theory of Relativity (GTR), the great