Gauge symmetry and the arrow of time: How to count what counts (2509.14720v1)

Abstract: This thesis addresses two major problems in the philosophy of physics. The first is how to identify the minimal physical content of a theory; that is, what features of a theory are truly needed to make predictions, and what can be removed without changing its empirical consequences. The second is the problem of time's arrow: why time seems to have a direction, even though the fundamental laws of physics treat the past and future symmetrically. I show that answering the first question leads to insights about the second. In particular, I argue that the overall size of the Universe is not used to make predictions in cosmology, and so should not count as part of the theory's minimal physical content. Describing the Universe without this feature leads to a striking result: the arrow of time becomes a local phenomenon. Observers like us who see a Universe full of matter clumped together to form structures like stars and planets are statistically much more likely to see increasing clumpiness into the future than into the past. This tendency helps explain our experience of time's direction.

• The paper introduces a universal definition of gauge symmetry and applies it to propose a novel mechanism for the arrow of time in cosmology.
• It employs dynamical similarity as a gauge symmetry, reducing dynamics to contact geometry to expose attractor structures and Janus points.
• The work challenges conventional Liouville measure typicality by proposing a new time-dependent measure with significant implications for entropy and cosmological models.

Gauge Symmetry, Dynamical Similarity, and the Arrow of Time: A Technical Analysis

This dissertation presents a comprehensive framework for understanding gauge symmetry in physical theories and leverages this framework to propose a novel explanation for the Arrow of Time (AoT) in cosmology. The work is distinguished by its generalization of Dirac's analysis of gauge symmetry, its explicit representational account, and its extension to non-Hamiltonian systems, particularly those exhibiting dynamical similarity. The implications for cosmology are significant, as the treatment of dynamical similarity as a gauge symmetry leads to a new mechanism for time asymmetry that does not rely on the Past Hypothesis or conventional entropy-based arguments.

Universal Definition of Gauge Symmetry

The author introduces a universal definition of gauge symmetry, motivated by Dirac's identification of gauge structure with representational redundancy underdetermined by the dynamical laws. This definition is formalized through the Principle of Essential and Sufficient Autonomy (PESA), which requires that:

• Gauge symmetry is present when the representational structures of a theory are underdetermined by the phenomena.
• The dynamical equations must be formulated so that their underdetermination matches the underdetermination of the representations by the phenomena.

This approach is operationalized by distinguishing between the full set of representational structures $\mathcal{A}$ and the subset of observable structures $\mathcal{O} \subset \mathcal{A}$ that are both necessary and sufficient for empirical adequacy. Gauge symmetries are then defined as transformations on $\mathcal{A}$ that leave $\mathcal{O}$ invariant, and the theory must be reformulated so that only $\mathcal{O}$ is dynamically determined.

The framework is general enough to encompass standard gauge symmetries (e.g., Yang–Mills, diffeomorphism invariance) and, crucially, extends to cases where the symmetry does not preserve the canonical symplectic structure, such as dynamical similarity.

Dynamical Similarity as a Gauge Symmetry

A central technical innovation is the treatment of dynamical similarity—a scaling symmetry that rescales both configuration and momentum variables, but not their rates of change—as a gauge symmetry in cosmology. The author demonstrates that:

• In cosmological models (e.g., FLRW, Newtonian $N$-body), the overall scale of the universe is empirically unobservable, while scale changes (e.g., the Hubble parameter) are observable.
• Implementing dynamical similarity as a gauge symmetry requires projecting the dynamics onto a reduced state space (contact space) where the scale degree of freedom is eliminated, but its conjugate momentum (e.g., Hubble parameter) is retained.
• The resulting reduced dynamics is governed by contact geometry rather than symplectic geometry, leading to dissipative-like evolution and the possibility of attractor structures.

This approach is technically nontrivial, as it requires generalizing the Dirac constraint analysis and the gauge principle to systems where the symmetry generator does not preserve the symplectic form. The author provides explicit constructions of the reduced (contact) dynamics and analyzes the resulting flow equations, measures, and attractor behavior.

The Janus–Attractor Scenario and the Arrow of Time

The most significant application of this framework is the Janus–Attractor (JA) scenario for the Arrow of Time. The key technical points are:

• Janus points are dynamically privileged states (often of minimal complexity or maximal symmetry) that serve as temporal midpoints in the reduced (gauge-invariant) dynamics.
• Attractors are dynamically stable states toward which the system evolves in the reduced state space, despite the underlying time-reversal invariance of the fundamental laws.
• The existence of both Janus points and attractors in the reduced dynamics allows for the definition of an Arrow of Time: for observers near an attractor, the direction from the Janus point to the attractor is distinguished by a large, monotonic gradient in physically relevant quantities (e.g., complexity, Hubble parameter).

The author demonstrates, with explicit models, that this scenario can explain the observed time asymmetry in cosmology without invoking a Past Hypothesis or special initial conditions. In particular:

• In a self-gravitating Newtonian $N$-body model, gauge-fixing dynamical similarity solves the smoothness problem (the low-entropy, homogeneous early universe).
• In FLRW cosmology, the same procedure solves the red-shift problem (the large, monotonic expansion rate in the past).

The technical analysis shows that the attractor structure and the time-dependent measure on the reduced state space naturally lead to the observed time asymmetries, and that the choice of measure is dictated by the gauge structure rather than by an arbitrary convention.

Implications for Measures and Typicality

A critical technical consequence of treating dynamical similarity as a gauge symmetry is the breakdown of Liouville's theorem in the reduced (contact) dynamics. The standard Liouville measure, which is time-independent in symplectic systems and underpins typicality arguments in statistical mechanics, is not preserved in contact systems. Instead:

• The natural measure on the reduced state space is time-dependent and determined by the contact structure and the "drag" variable (e.g., Hubble parameter).
• This invalidates conventional typicality arguments based on the Liouville measure and requires a new family of measures, consistent with the gauge symmetry, for counting dynamical possibilities.

The author provides a detailed analysis of the construction and transformation properties of these measures, showing that they include those used in modern cosmology and that their time dependence is essential for explaining the Arrow of Time.

Trade-offs, Limitations, and Deployment

Trade-offs and Limitations:

• The approach requires a careful specification of the representational context and the identification of gauge versus observable structures, which may be nontrivial in complex or quantum systems.
• The framework is classical; quantum generalizations are not addressed, though the author suggests the approach may provide a template for future work.
• The explanation of the Arrow of Time is model-dependent; while the mechanism is general, its realization in more realistic or higher-dimensional models may require further technical development.

Deployment and Implementation:

• For practical implementation, the procedure involves:
  • Identifying the dynamical similarity symmetry and its generator.
  • Constructing the reduced (contact) state space by quotienting out the gauge direction.
  • Formulating the reduced dynamics and identifying attractor and Janus point structures.
  • Computing the time-dependent measure on the reduced state space for typicality and probability assignments.
• The approach is computationally tractable in finite-dimensional models (e.g., $N$-body, minisuperspace cosmology) and can be extended to more complex systems with appropriate mathematical tools (e.g., jet bundles, infinite-dimensional contact geometry).

Theoretical and Practical Implications

Theoretical:

• The work provides a rigorous, generalizable definition of gauge symmetry that resolves longstanding ambiguities in the philosophy and foundations of physics.
• It demonstrates that time asymmetry can arise from the interplay of gauge symmetry and dynamical structure, without recourse to special initial conditions or entropy-based arguments.
• The analysis challenges the universality of the Liouville measure and typicality arguments in cosmology, suggesting that the correct measure is dictated by the gauge structure of the theory.

Practical:

• The framework offers a new methodology for constructing and interpreting cosmological models, particularly in the context of the initial conditions problem and the interpretation of cosmological data.
• It provides a principled basis for the choice of measure in cosmological typicality arguments, with direct implications for the interpretation of inflation, the multiverse, and anthropic reasoning.

Future Directions

• Extension to quantum cosmology and quantum gravity, where the identification of gauge and observable structures is even more subtle.
• Application to more realistic cosmological models, including inhomogeneities, anisotropies, and higher-dimensional theories.
• Investigation of the implications for the interpretation of entropy, information, and complexity in cosmology and statistical mechanics.

Conclusion

This dissertation advances the technical understanding of gauge symmetry and its role in the foundations of physics, providing a robust framework for both the identification of gauge structure and the construction of empirically adequate models. By treating dynamical similarity as a gauge symmetry in cosmology, the work offers a novel, technically precise mechanism for the emergence of the Arrow of Time, grounded in the geometry of the reduced state space and the properties of contact dynamics. The implications for cosmology, the philosophy of physics, and the methodology of theoretical modeling are substantial, and the framework sets the stage for further developments in both classical and quantum contexts.