# Fiber bundles, Yang–Mills theory, and general relativity

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## Abstract

I articulate and discuss a geometrical interpretation of Yang–Mills theory. Analogies and disanalogies between Yang–Mills theory and general relativity are also considered.

## Keywords

Yang–Mills theory General relativity Fiber bundle interpretation Holonomy interpretation## Notes

### Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. 1331126. Special thanks are due to participants in my 2012 seminar on Gauge Theories, and especially Ben Feintzeig and Sarita Rosenstock for their many discussions on these topics. I am also particularly indebted to Dick Palais and Bob Geroch for enlightening correspondences on the geometrical foundations of Yang–Mills theory. Helpful conversations and correspondence with Dave Baker, Jeff Barrett, Gordon Belot, Erik Curiel, Katherine Brading, Richard Healey, David Malament, Oliver Pooley, Chris Smeenk, Bob Wald, David Wallace, and Chris Wüthrich have also contributed to the development of the ideas presented here. Erik Curiel, Sam Fletcher, and David Malament read the manuscript carefully and noted several slips (though remaining errors are, of course, my own!). Versions of this paper were presented to the Southern California Philosophy of Physics Group and at the Munich Center for Mathematical Philosophy; I am grateful to the participants there for discussion and comments.

## References

- Anandan, J. (1993). Remarks concerning the geometries of gravity and gauge fields. In B. L. Hu, M. P. Ryan, & C. V. Vishveshwara (Eds.),
*Directions in general relativity*(pp. 10–20). New York: Cambridge University Press.CrossRefGoogle Scholar - Arntzenius, F. (2012).
*Space, time, and stuff*. New York: Oxford University Press.CrossRefGoogle Scholar - Baez, J., & Munian, J. (1994).
*Gauge fields, knots and gravity*. River Edge, NJ: World Scientific.CrossRefGoogle Scholar - Barrett, T. (2014). On the structure of classical mechanics.
*The British Journal of Philosophy of Science*. doi: 10.1093/bjps/axu005. - Belot, G. (1998). Understanding electromagnetism.
*The British Journal for Philosophy of Science*,*49*(4), 531–555.CrossRefGoogle Scholar - Belot, G. (2003). Symmetry and gauge freedom.
*Studies in History and Philosophy of Modern Physics*,*34*(2), 189–225.CrossRefGoogle Scholar - Belot, G. (2007). The representation of time and change in mechanics. In J. Butterfield & J. Earman (Eds.),
*Philosophy of physics*(pp. 133–228). Amsterdam: Elsevier.CrossRefGoogle Scholar - Bleecker, D. (1981). Gauge theory and variational principles. Reading, MA: Addison-Wesley. (Reprinted by Dover Publications in 2005).Google Scholar
- Brown, H. (2005).
*Physical relativity*. New York: Oxford University Press.CrossRefGoogle Scholar - Butterfield, J. (2007). On symplectic reduction in classical mechanics. In J. Butterfield & J. Earman (Eds.),
*Philosophy of physics*(pp. 1–132). Amsterdam: Elsevier.CrossRefGoogle Scholar - Catren, G. (2008). Geometrical foundation of classical Yang-Mills theory.
*Studies in History and Philosophy of Modern Physics*,*39*(3), 511–531.CrossRefGoogle Scholar - Curiel, E. (2013). Classical mechanics is Lagrangian; it is not Hamiltonian.
*The British Journal for Philosophy of Science*,*65*(2), 269–321.CrossRefGoogle Scholar - Earman, J. (1995).
*Bangs, crunches, whimpers, and shrieks*. New York: Oxford University Press.Google Scholar - Feintzeig, B., & Weatherall, J. O. (2014). The geometry of the ‘gauge argument’, unpublished manuscript.Google Scholar
- Friedman, M. (1983).
*Foundations of space-time theories*. Princeton, NJ: Princeton University Press.Google Scholar - Geroch, R. (1996). Partial differential equations of physics. In G. S. Hall & J. R. Pulham (Eds.),
*General relativity: Proceedings of the forty sixth Scottish Universities summer school in physics*(pp. 19–60). Edinburgh: SUSSP Publications.Google Scholar - Healey, R. (2001). On the reality of gauge potentials.
*Philosophy of Science*,*68*(4), 432–455.CrossRefGoogle Scholar - Healey, R. (2004). Gauge theories and holisms.
*Studies in History and Philosophy of Modern Physics*,*35*(4), 643–666.CrossRefGoogle Scholar - Healey, R. (2007).
*Gauging what’s real: The conceptual foundations of contemporary gauge theories*. New York: Oxford University Press.CrossRefGoogle Scholar - Kobayashi, S., & Nomizu, K. (1963).
*Foundations of differential geometry*(Vol. 1). New York: Interscience Publishers.Google Scholar - Kolář, I., Michor, P. W., & Slovák, J. (1993).
*Natural operations in differential geometry*. New York: Springer.CrossRefGoogle Scholar - Lee, J. M. (2009).
*Manifolds and differential geometry*. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar - Leeds, S. (1999). Gauges: Aharanov, Bohm, Yang, Healey.
*Philosophy of Science*,*66*(4), 606–627.CrossRefGoogle Scholar - Lyre, H. (2004). Holism and structuralism in \(U(1)\) gauge theory.
*Studies in History and Philosophy of Modern Physics*,*35*(4), 643–670.CrossRefGoogle Scholar - Malament, D. (2012).
*Topics in the foundations of general relativity and Newtonian gravitation theory*. Chicago: University of Chicago Press.CrossRefGoogle Scholar - Maudlin, T. (2007).
*The metaphysics within physics*(pp. 78–103, Ch. 3). New York: Oxford University Press.Google Scholar - Michor, P. (2009).
*Topics in differential geometry*. Providence, RI: American Mathematical Society.Google Scholar - Myrvold, W. C. (2011). Nonseparability, classical, and quantum.
*The British Journal for the Philosophy of Science*,*62*(2), 417–432.CrossRefGoogle Scholar - Nakahara, M. (1990).
*Geometry topology and physics*. Philadelphia: Institute of Physics Publishing.CrossRefGoogle Scholar - North, J. (2009). The ‘structure’ of physics: A case study.
*Journal of Philosophy*,*106*(2), 57–88.CrossRefGoogle Scholar - Palais, R. S. (1981). The geometrization of physics. Institute of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, accessed from http://vmm.math.uci.edu/.
- Penrose, R., & Rindler, W. (1984).
*Spinors and space-time*. New York: Cambridge University Press.CrossRefGoogle Scholar - Rosenstock, S., & Weatherall, J. O. (2015a). A categorical equivalence between generalized holonomy maps on a connected manifold and principal connections on bundles over that manifold. arXiv:1504.02401 [math-ph].
- Rosenstock, S., & Weatherall, J. O. (2015b). On holonomy and fiber bundle interpretations of Yang–Mills theory, unpublished manuscript.Google Scholar
- Swanson, N., & Halvorson, H. (2012). On North’s ‘the structure of physics’. http://philsci-archive.pitt.edu/9314/.
- Taubes, C. H. (2011).
*Differential geometry: Bundles, connections, metrics and curvature*. New York: Oxford University Press.CrossRefGoogle Scholar - Trautman, A. (1965). Foundations and current problem of general relativity. In S. Deser & K. W. Ford (Eds.),
*Lectures on general relativity*(pp. 1–248). Englewood Cliffs, NJ: Prentice-Hall.Google Scholar - Trautman, A. (1980). Fiber bundles, gauge fields, and gravitation. In A. Held (Ed.),
*General relativity and gravitation*(pp. 287–308). New York: Plenum Press.Google Scholar - Wald, R. (1984).
*General relativity*. Chicago: University of Chicago Press.CrossRefGoogle Scholar - Weatherall, J. O. (2015). Regarding the ‘Hole Argument’.
*The British Journal for Philosophy of Science*(Forthcoming). arXiv:1412.0303. - Wu, T. T., & Yang, C. N. (1975). Concept of nonintegrable phase factors and global formulation of gauge fields.
*Physical Review D*,*12*(12), 3845–3857.CrossRefGoogle Scholar