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Unifying Geometrical Representations of Gauge Theory

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Abstract

We unify three approaches within the vast body of gauge-theory research that have independently developed distinct representations of a geometrical surface-like structure underlying the vector-potential. The three approaches that we unify are: those who use the compactified dimensions of Kaluza–Klein theory, those who use Grassmannian models (also called gauge theory embedding or \(CP^{N-1}\) models) to represent gauge fields, and those who use a hidden spatial metric to replace the gauge fields. In this paper we identify a correspondence between the geometrical representations of the three schools. Each school was mostly independently developed, does not compete with other schools, and attempts to isolate the gauge-invariant geometrical surface-like structures that are responsible for the resulting physics. By providing a mapping between geometrical representations, we hope physicists can now isolate representation-dependent physics from gauge-invariant physical results and share results between each school. We provide visual examples of the geometrical relationships between each school for \(U(1)\) electric and magnetic fields. We highlight a first new result: in all three representations a static electric field (electric field from a fixed ring of charge or a sphere of charge) has a hidden gauge-invariant time dependent surface that is underlying the vector potential.

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Notes

  1. We have chosen to work in units where \(\hbar = c = 1\) and where we absorb the electron’s charge \(e\) into the definition of \(A_\mu \).

  2. Throughout this paper we use the convention that lower-case Latin letters near the beginning of the alphabet \(a,b,\ldots \) will be gauge-theory color indices, Greek letters \(\mu , \nu ,\ldots \) will be space-time coordinates, upper-case Latin letters \(A,B,\ldots \) will be used for Kaluza–Klein metric indices, and lower-case Latin letters towards the middle of the alphabet \(i,j,\ldots \) will be used for the variables corresponding to subspaces of space-time and the embedding dimensions, where context will keep them distinct. The Kaluza–Klein index values 0 through 3 are the usual space-time coordinates \(t,x,y,z\) and the index value 5 is the fifth dimension coordinate \(x^5\), which is used to parameterize the tiny compact dimension. The appendix provides a summary.

  3. Some specific many-to-one mappings will be provided in Sect. 5 in Eqs. (39) and (40).

  4. The distinction between lower and upper indices are dropped in the epsilon term for convenience (see Weinberg [50, Chap. 15, Appendix]).

  5. We have reused the variable name \(X\) to parametrize each immersion. This is not not same immersion as Eq. (12) nor the same as in the other examples.

  6. We have reused the variable name X to parametrize each immersion. This is not not same immersion as Eq. (12) nor the same as in the other examples.

  7. We have reused the variable name X to parametrize each immersion. This is not not same immersion as Eq. (12) nor the same as in the other examples.

  8. We have reused the variable name X to parametrize each immersion. This is not not same immersion as Eq. (12) nor the same as in the other examples.

  9. We have reused the variable name X to parametrize each immersion. This is not not same immersion as Eq. (12) nor the same as in the other examples.

  10. The authors would like to thank Ricardo Schiappa for highlighting these research directions.

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Acknowledgments

The authors would like to thank Laura Serna, Kevin Cahill, Richard Cook, Matt Robinson, Christian Wohlwend, Ricardo Schiappa, and Yang–Hui He for helpful comments after reviewing the manuscript. We would also like to thank the reviewers for helpful contributions increasing the quality of the final paper. The views expressed in this paper are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the US Government. Distribution A: Approved for public release. Distribution unlimited.

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    Scott Alsid & Mario Serna

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Appendix: Variable Definitions Reference

Appendix: Variable Definitions Reference

\(\mathbf {e}_a\) or \(e_a^j\)

The basis vector for the Grassmannian school. The \(a\) coordinate is an internal ‘color’ index. If there is a Latin index like \(j\), it refers to the embedding space. Forms a rectangular matrix

\(\mathbf {t}_\mu \) or \(t_\mu ^j\)

Coordinate tangent vector for the coordinate \(x^\mu \). Used in defining a metric. If there is a Latin index like \(j\), it refers to the embedding-space dimension. Forms a rectangular matrix

\(u^a_j\)

The tetrad of the hidden-spatial-geometry school. Notice the \(a\) index specifies the ‘frame’ in color space and \(j\) is a ‘frame’ in a slice of space-time. This maps the color index \(a\) to the space-time coordinate tangent vector \(j\) of a spatial metric which represents the gauge field and corresponding electric and magnetic fields. Must be a square matrix

\(\mathbf {X}\) or \(X^j\)

Is the generic vector used to denote an explicit isometric embedding which will be used to induce a metric. The Latin index \(j\) refers to the embedding space

\(\phi ^a\)

Coefficients of the basis element \(\mathbf {e}_a\) which specify a vector in color-space. \(\phi ^a\) changes with a gauge transformation but the vector \( \mathbf {\phi } = \phi ^a \mathbf {e}_a = \phi '^{\,b} \mathbf {e}'_b\) is gauge invariant

\( a, b, c\)

Lower-case Latin letters near the beginning of the alphabet will be gauge-theory color indices

\(\mu , \nu ,\ldots \)

Greek letters will be space-time coordinates

\(A,B,\ldots \)

Upper-case Latin letters will be used for Kaluza–Klein metric indices. Kaluza–Klein index values 0 through 3 are the usual space-time coordinates \(t,x,y,z\) and the index value 5 is the fifth dimension coordinate \(x^5\), which is used to parameterize the tiny compact dimension

\(i,j,\ldots \)

Variables corresponding to subspaces of space-time and the embedding dimensions, where context will keep them distinct

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Alsid, S., Serna, M. Unifying Geometrical Representations of Gauge Theory. Found Phys 45, 75–103 (2015). https://doi.org/10.1007/s10701-014-9841-x

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