Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-02-05T09:23:43.540Z Has data issue: false hasContentIssue false
Series:   Elements in the Philosophy of Physics

Gauge Theory and the Geometrisation of Physics

Published online by Cambridge University Press:  30 January 2025

Henrique De Andrade Gomes
Henrique De Andrade Gomes
Affiliation:
University of Oxford

Summary

This Element is broadly about the geometrization of physics, but mostly it is about gauge theories. Gauge theories lie at the heart of modern physics: in particular, they constitute the Standard Model of particle physics. At its simplest, the idea of gauge is that nature is best described using a descriptively redundant language; the different descriptions are said to be related by a gauge symmetry. The over-arching question this Element aims to answer is: why is descriptive redundancy fruitful for physics? I will provide three inter-related answers to the question: ``Why gauge theory?'', that is: why introduce redundancies in our models of nature in the first place? The first is pragmatic, or methodological; the second is based on geometrical considerations, and the third is broadly relational.
Get access

Keywords

Gauge theorysymmetryfiber bundlesaffine geometryNoether’s theorems
Type
Element
Information
Series: Elements in the Philosophy of Physics
Online ISBN: 9781009029308
Publisher: Cambridge University Press
Print publication: 30 January 2025

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, J. F. (1996). Lectures on Exceptional Lie Groups. Chicago, IL: University of Chicago Press.Google Scholar
Aharonov, Y., & Bohm, D. (1959). Significance of electromagnetic potentials in the quantum theory. The Physical Review, 115, 485–491. https://link.aps.org/doi/10.1103/PhysRev.115.485.CrossRefGoogle Scholar
Anandan, J. (1977). Gravitational and rotational effects in quantum interference. Physical Review D, 15, 1448–1457. https://link.aps.org/doi/10.1103/PhysRevD.15.1448.CrossRefGoogle Scholar
Atiyah, M. (1957). Complex analytic connections in fibre bundles. Transactions of the American Mathematical Society 85(1), 181–207.CrossRefGoogle Scholar
Avery, S. G., & Schwab, B. U. W. (2016). Noether’s second theorem and ward identities for gauge symmetries. Journal of High Energy Physics, 2016(2). http://dx.doi.org/10.1007/JHEP02(2016)031.CrossRefGoogle Scholar
Baez, J., & Munian, J. (1994). Gauge Fields, Knots and Gravity. Singapore: World Scientific.CrossRefGoogle Scholar
Bamonti, N. (2023). What is a Reference Frame in General Relativity? https://arxiv.org/abs/2307.09338.Google Scholar
Belot, G. (1998). Understanding electromagnetism. The British Journal for the Philosophy of Science, 49(4), 531–555. https://doi.org/10.1093/bjps/49.4.531.CrossRefGoogle Scholar
Berghofer, P., Francois, J., Friederich, S., et al. (2023). Elements in the Foundations of Physics: Gauge Symmetries, Symmetry Breaking, and Gauge-Invariant Approaches. Cambridge, UK Cambridge University Press.CrossRefGoogle Scholar
Bonora, L., & Cotta-Ramusino, P. (1983). Some remarks on BRS transformations, anomalies and the cohomology of the Lie algebra of the group of gauge transformations. Communications in Mathematical Physics, 87(4), 589–603. http://link.springer.com/10.1007/BF01208267.CrossRefGoogle Scholar
Boothby, W. M. (2010). An introduction to differentiable manifolds and Riemannian geometry (Rev. 2. ed., [Nachdr.] ed.). Amsterdam: Academic Press. (Literaturverz. S. 403–409)Google Scholar
Brading, K., & Brown, H. R. (2000). Noether’s theorems and gauge symmetries hep-th/0009058.Google Scholar
Brading, K., & Brown, H. R. (2003). Symmetries and Noether’s theorems. In Brading, K. & Castellani, E. (Eds.), Symmetries in Physics: Philosophical Reflections (pp. 89–109). Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511535369.006.CrossRefGoogle Scholar
Brown, H. (1999). Aspects of objectivity in quantum mechanics. In Butterfield, J. & Pagonis, C. (Eds.), From Physics to Philosophy. Cambridge: Cambridge University Press. http://philsci-archive.pitt.edu/223/.Google Scholar
Brown, H. (2022). Do symmetries ‘explain’ conservation laws? The modern converse Noether theorem vs pragmatism. In Read, J. & Teh, N. J. (Eds.), The Philosophy and Physics of Noether’s Theorems: A Centenary Volume (pp. 144–168). Cambridge: Cambridge University Press. https://doi.org/10.1017/9781108665445.008.CrossRefGoogle Scholar
Carrozza, S., & Höhn, P. A. (2022). Edge modes as reference frames and boundary actions from post-selection. Journal of High Energy Physics, 2022(2). https://doi.org/10.1007/jhep02(2022)172.CrossRefGoogle Scholar
Ciambelli, L., & Leigh, R. G. (2021). Lie Algebroids and the Geometry of Off-Shell BRST. Nuclear Physics B, 972, 115553, https://doi.org/10.1016/j.nuclphysb.2021.115553.CrossRefGoogle Scholar
Coleman, S. (1965). Fun with SU(3). In Seminar on High-Energy Physics and Elementary Particles (pp. 331–352). Boston, MA: MIT press.Google Scholar
de León, M., & Zajac, M. (2020). Hamilton–Jacobi theory for gauge field theories. Journal of Geometry and Physics, 152, 103636. http://dx.doi.org/10.1016/j.geomphys.2020.103636.CrossRefGoogle Scholar
Dougherty, J. (2020). The non-ideal theory of the Aharonov–Bohm effect. Synthese, 198(2021), 12195–12221.CrossRefGoogle Scholar
Dowker, J. S. (1967). A gravitational Aharonov–Bohm effect. Il Nuovo Cimento B (1965–1970), 52, 129–135.CrossRefGoogle Scholar
Earman, J. (1987). Locality, nonlocality and action at a distance: A skeptical review of some philosophical dogmas. In Kargon, R., Achinstein, P., & Kelvin, W. T. (Eds.), Kelvin’s Baltimore Lectures and Modern Theoretical Physics: Historical and Philosophical Perspectives (pp. 449–490). Cambridge, MA: MIT Press. http://d-scholarship.pitt.edu/12972/.Google Scholar
Earman, J. (2002). Gauge matters. Philosophy of Science, 69(S3), S209–S220. https://doi.org/10.1086/341847.CrossRefGoogle Scholar
Earman, J. (2003). Tracking down gauge: An ode to the constrained Hamiltonian formalism. In Brading, K. & Castellani, E. (Eds.), Symmetries in Physics: Philosophical Reflections (pp. 140–162). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Earman, J. (2004). Curie’s principle and spontaneous symmetry breaking. International Studies in the Philosophy of Science, 18(2 & 3), 173–198. https://doi.org/10.1080/0269859042000311299.CrossRefGoogle Scholar
Earman, J. (2019). The role of idealizations in the Aharonov–Bohm effect. Synthese, 196(5), 1991–2019. https://doi.org/10.1007/s11229-017-1522-9.CrossRefGoogle Scholar
Ehrenberg, W., & Siday, R. E. (1949). The refractive index in electron optics and the principles of dynamics. Proceedings of the Physical Society. Section B, 62(1), 8–21. https://doi.org/10.1088/0370-1301/62/1/303.CrossRefGoogle Scholar
Fischer, A. E., & Marsden, J. E. (1979). The initial value problem and the dynamical formulation of general relativity. In Hawking, S. W. (Ed.), General Relativity: An Einstein Centenary Survey. New York: cambridge university press, pp. 138–211.Google Scholar
Ehresmann, Charles. 1950. Les connexions infinitésimales dans un espace fibré différentiable. Séminaire Bourbaki : années 1948/49 – 1949/50 – 1950/51, exposés 1-49, Séminaire Bourbaki, no. 1 (1952), Talk no. 24, 16 p.Google Scholar
Ford, L. H., & Vilenkin, A. (1981). A gravitational analogue of the Aharonov–Bohm effect. Journal of Physics A: Mathematical and General, 14(9), 2353–2357. https://doi.org/10.1088/0305-4470/14/9/030.CrossRefGoogle Scholar
Göckeler, M., & Schücker, T. (1989). Differential Geometry, Gauge Theories, and Gravity. Cambridge: Cambridge University Press.Google Scholar
Gomes, H. (2019). Gauging the boundary in field-space. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. www.sciencedirect.com/science/article/pii/S1355219818302144. https://doi.org/10.1016/j.shpsb.2019.04.002.CrossRefGoogle Scholar
Gomes, H. (2021). Holism as the significance of gauge symmetries. European Journal of Philosophy of Science, 11, 87.CrossRefGoogle Scholar
Gomes, H. (2022). Noether charges, gauge-invariance, and non-separability. In Read, J., and Teh, N. J., (Eds.), The Philosophy and Physics of Noether’s Theorems: A Centenary Volume (pp. 296–321). Cambridge: Cambridge University Press. https://doi.org/10.1017/9781108665445.013.CrossRefGoogle Scholar
Gomes, H. (2024a). Gauge theory as the geometry of internal vector spaces. https://philsci-archive.pitt.edu/id/eprint/24069.Google Scholar
Gomes, H. (2024b). Representational schemes for theories with symmetries. https://philsci-archive.pitt.edu/id/eprint/24070.Google Scholar
Gomes, H., & Butterfield, J. (2022). How to choose a gauge? The case of Hamiltonian electromagnetism. Erkenntnis. 89, 1581–1615. https://doi.org/10.1007/s10670-022-00597-9.CrossRefGoogle Scholar
Gomes, H., Hopfmüller, F., & Riello, A. (2019). A unified geometric framework for boundary charges and dressings: Non-abelian theory and matter. Nuclear Physics B, 941, 249–315. www.sciencedirect.com/science/article/pii/S0550321319300483. https://doi.org/10.1016/j.nuclphysb.2019.02.020.CrossRefGoogle Scholar
Gomes, H., & Riello, A. (2017). The observer’s ghost: Notes on a field space connection. Journal of High Energy Physics (JHEP), 05, 017. https://link.springer.com/article/10.10072FJHEP0528201729017. article number 17.CrossRefGoogle Scholar
Gomes, H., & Riello, A. (2021). The quasilocal degrees of freedom of Yang-Mills theory. SciPost Physics, 10, 130. https://scipost.org/10.21468/SciPostPhys.10.6.130.CrossRefGoogle Scholar
Greaves, H., & Wallace, D. (2014). Empirical consequences of symmetries. British Journal for the Philosophy of Science, 65(1), 59–89.CrossRefGoogle Scholar
Guillemin, V., & Pollack, A. (2010). Differential Topology. London: AMS Chelsea. https://books.google.co.uk/books?id=FdRhAQAAQBAJ.Google Scholar
Healey, R. (2007). Gauging What’s Real: The Conceptual Foundations of Gauge Theories. Oxford: Oxford University Press.CrossRefGoogle Scholar
Healey, R., & Gomes, H. (2021). Holism and nonseparability in physics. In Zalta, E. N. (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2016 ed.). Stanford: Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/spr2016/entries/physics-holism/.Google Scholar
Henneaux, M., & Teitelboim, C. (1992). Quantization of Gauge Systems. Princeton: Princeton University Press.CrossRefGoogle Scholar
Hiley, B. (2013). The early history of the Aharonov–Bohm effect. arXiv preprint arXiv:1304.4736.Google Scholar
Jacobs, C. (2021). Symmetries as a Guide to theStructure of Physical Quantities (Unpublished doctoral dissertation). University of Oxford.Google Scholar
Jacobs, C. (2023). The metaphysics of fibre bundles. Studies in History and Philosophy of Science, 97, 34–43. www.sciencedirect.com/science/article/pii/S0039368122001777. https://doi.org/10.1016/j.shpsa.2022.11.010.CrossRefGoogle ScholarPubMed
Kobayaschi, S. (1957). Theory of connections. Annali di Matematica 43, 119–194. https://doi.org/10.1007/BF02411907.CrossRefGoogle Scholar
Kobayashi, S., & Nomizu, K. (1963). Foundations of Differential Geometry. Vol I. New York: Interscience, a division of John Wiley.Google Scholar
Kolar, I., Michor, P., & Slovak, J. (1993). Natural Operations in Differential Geometry. Berlin: Springer.CrossRefGoogle Scholar
Kosmann-Schwarzbach, Y. (2011). The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century. New York: Springer Science+Business Media. (Translated by Bertram E. Schwarzbach)CrossRefGoogle Scholar
Lovelock, D. (1972). The four-dimensionality of space and the Einstein tensor. Journal of Mathematical Physics, 13, 874–876. https://doi.org/10.1063/1.1666069.CrossRefGoogle Scholar
Mackenzie, K. C. H. (2005). General Theory of Lie Groupoids and Lie Algebroids. Cambridge University Press. https://doi.org/10.1017/CBO9781107325883.CrossRefGoogle Scholar
Martin, C. A. (2002). Gauge principles, gauge arguments and the logic of nature. Philosophy of Science, 69(S3), S221–S234.CrossRefGoogle Scholar
Michor, P. W. (2008). Topics in Differential Geometry (No. volume 93). Providence, RI: American Mathematical Society. (Includes bibliographical references (pages 479–488) and index. Description based on print version record.)Google Scholar
Myrvold, W. C. (2011). Nonseparability, classical, and quantum. The British Journal for the Philosophy of Science, 62(2), 417–432. https://doi.org/10.1093/bjps/axq036.CrossRefGoogle Scholar
Nakahara, M. (2003). Geometry, Topology and Physics. Bristol: Institute of Physics.Google Scholar
Noether, E. (1918). Invariante variationsprobleme. Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse, 7, 235–257. English translation by M. A. Tavel: https://arxiv.org/abs/physics/0503066.Google Scholar
Olver, P. (1986). Applications of Lie Groups to Differential Equations. New York: Springer-Verlag.CrossRefGoogle Scholar
O’Raifertaigh, L. (1997). The Dawning of Gauge Theory. Princeton: Princeton University Press. www.jstor.org/stable/j.ctv10vm2qt.CrossRefGoogle Scholar
Rosenstock, S., & Weatherall, J. O. (2016). A categorical equivalence between generalized holonomy maps on a connected manifold and principal connections on bundles over that manifold. Journal of Mathematical Physics, 57(10), 102902. http://philsci-archive.pitt.edu/11904/.CrossRefGoogle Scholar
Rosenstock, S., & Weatherall, J. O. (2018). Erratum: “A categorical equivalence between generalized holonomy maps on a connected manifold and principal connections on bundles over that manifold”. Journal of Mathematical Physics, 59(2), 029901.CrossRefGoogle Scholar
Rovelli, C. (2014). Why gauge? Foundation of Physics, 44(1), 91–104. https://doi.org/10.1007/s10701-013-9768-7.CrossRefGoogle Scholar
Ryder, L. H. (1996). Quantum Field Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Sardanashvily, G. (2009). Fibre Bundles, Jet Manifolds and Lagrangian Theory: Lectures for Theoreticians. arXiv:0908.1886.Google Scholar
Schutz, B. F. (1980). Geometric Methods of Mathematical Physics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Shech, E. (2018). Idealizations, essential self-adjointness, and minimal model explanation in the Aharonov–Bohm effect. Synthese, 195(11), 4839–4863. https://doi.org/10.1007/s11229-017-1428-6.CrossRefGoogle Scholar
Steenrod, N. 1951. The Topology of Fibre Bundles. Series: Princeton Mathematical Series. Princeton Landmarks in Mathematics and Physics. Princeton University Press.Google Scholar
Struyve, W. (2011). Gauge invariant accounts of the Higgs mechanism. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 42(4), 226–236. www.sciencedirect.com/science/article/pii/S1355219811000384. https://doi.org/10.1016/j.shpsb.2011.06.003.CrossRefGoogle Scholar
Swanson, N. (2019). On the ostrogradski instability, or, why physics really uses second derivatives. The British Journal for the Philosophy of Science. http://philsci-archive.pitt.edu/15932/.Google Scholar
Teller, P. (1997). A metaphysics for contemporary field theories. Studies in History and Philosophy of Modern Physics, 28(4), 507–522.CrossRefGoogle Scholar
Teller, P. (2000). The gauge argument. Philosophy of Science, 67, S466–S481. www.jstor.org/stable/188688.CrossRefGoogle Scholar
Thierry-Mieg, J. (1980). Geometrical reinterpretation of faddeev-popov ghost particles and brs transformations. Journal of Mathematical Physics, 21(12), 2834–2838. https://doi.org/10.1063/1.524385.CrossRefGoogle Scholar
’t Hooft, G. (1980). Gauge theories and the forces between elementary particles. Scientific American, 242, 90–166.CrossRefGoogle Scholar
Wallace, D. (2009). QFT, Antimatter and Symmetry. Unpublished Manuscript, http://arxiv.org/abs/0903.3018.Google Scholar
Wallace, D. (2014). Deflating the Aharonov-Bohm effect. arxiv: 1407.5073.Google Scholar
Wallace, D. (2022). Isolated systems and their symmetries, part II: Local and global symmetries of field theories. Studies in History and Philosophy of Science, 92, 249–259. www.sciencedirect.com/science/article/pii/S0039368122000267. https://doi.org/10.1016/j.shpsa.2022.01.016.CrossRefGoogle ScholarPubMed
Wallace, D. (2024). Gauge Invariance through Gauge Fixing. Studies in History and Philosophy of Science, 108, 38–45.CrossRefGoogle ScholarPubMed
Weatherall, J. (2016). Fiber bundles, Yang–Mills theory, and general relativity. Synthese, 193(8), 2389–2425. http://philsci-archive.pitt.edu/11481/.CrossRefGoogle Scholar
Wells, R. O. N. (1988). The Mathematical Heritage of Hermann Weyl. New York: American Mathematical Society. https://books.google.co.uk/books?id=e0MECAAAQBAJ.CrossRefGoogle Scholar
Weyl, H. (1929). Gravitation and the electron. Proceedings of the National Academy of Sciences of the United States of America, 15(4), 323–334. www.ncbi.nlm.nih.gov/pmc/articles/PMC522457/.CrossRefGoogle ScholarPubMed
Wu, T. T., & Yang, C. N. (1975). Concept of nonintegrable phase factors and global formulation of gauge fields. Physical Review D, 12, 3845–3857. https://link.aps.org/doi/10.1103/PhysRevD.12.3845.CrossRefGoogle Scholar
Yang, C. N. (1983). Articles. In Selected papers (1945–1980) of chen ning yang (pp. 101–567). W. H. Freeman. www.worldscientific.com/doi/abs/10.1142/9789812703354_0002. https://doi.org/10.1142/9789812703354_0002.Google Scholar
Yang, C. N., & Mills, R. L. (1954). Conservation of isotopic spin and isotopic gauge invariance. Physical Review, 96, 191–195. https://link.aps.org/doi/10.1103/PhysRev.96.191.CrossRefGoogle Scholar
Zee, A. (2016). Group Theory in a Nutshell for Physicists. Princeton: Princeton University Press. (Literaturverzeichnis: Seite 581–582. - Hier auch später erschienene, unveränderte Nachdrucke).Google Scholar

Save element to Kindle

To save this element to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Gauge Theory and the Geometrisation of Physics
Available formats
×

Save element to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Gauge Theory and the Geometrisation of Physics
Available formats
×

Save element to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Gauge Theory and the Geometrisation of Physics
Available formats
×