Abstract
The world appears to be well described by gauge theories; why? I suggest that gauge is more than mathematical redundancy. Gauge-dependent quantities can not be predicted, but there is a sense in which they can be measured. They describe “handles” though which systems couple: they represent real relational structures to which the experimentalist has access in measurement by supplying one of the relata in the measurement procedure itself. This observation leads to a physical interpretation for the ubiquity of gauge: it is a consequence of a relational structure of physical quantities.
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Notes
For an interesting philosophical discussion of the problem, and general references, see [6].
This interpretation has gained support from recent results on the possibility of computing amplitudes using methods that make gauge look just like a complication [7], and by AdS/CFT suggestions that gravitational physics can be coded on the asymptotic boundary. In the quantum-gravity community, it grounds the common idea that the space of solutions of the hamiltonian constraint exhausts the physics of a system.
A related argument is Feynman’s suggestion ([8] II.15.3-4) that we should take the Maxwell potential (a gauge-dependent quantity) seriously on the ground that it determines the energy of a current (it appears in a coupling).
Notation in this note is standard and can be found in all common theoretical-physics textbooks.
Plus additional global quantities if the topology is non-trivial.
In fact, all the variables of the system (9) are of the same form. The system can be rewritten in the form
$$\begin{aligned} L = \frac{1}{2} \sum _{n=1}^{N+M-1} \left( \dot{z}_{n+1} - \dot{z}_{n}\right) , \end{aligned}$$(12)where \(x_n=z_n\) and \(y_n=z_{n+N}\), which in turn can be split into two subsystems at any value of \(n\), showing that any of its gauge invariant variables
$$\begin{aligned} d_n= z_{n+1} - z_{n} \quad n=1, \dots , N+M-1 \end{aligned}$$(13)is a relative observable, in the sense above. Thus, all gauge-invariant degrees of freedom of this system have such a relational character.
Of course the Yang–Mills connection connects the internal frame orientations of two points of spacetime and the metric expresses the distance between two points of spacetime.
I have always found this story a bit mysterious. Entities in Nature are not there to realize purposes: they are there and have properties. The Yang–Mills field is there and has symmetries; it is not there for the purpose of implementing a symmetry. We should distinguish our understanding of Nature from the path of discovery. If we understand that large wild mammals have disappeared from America because of human hunting, we should not conclude that humans exists for the purpose of exterminating mammals.
Physics is about relations and this is implemented in the standard dynamics, which codes relations between values of physical variables –including \(t\)– or, equivalently, the evolution of all variables in a single variable (\(t\)) taken as reference. A detailed discussion of this point is in Section 3.2.4 of [11].
This interpretation of the relation between gauge and time evolution is reinforced by the analysis of the coupling of general-covariant systems developed in [12]. When coupling general-covariant systems, one finds systems of hamiltonian constraints generating multi-dimensional orbits. If the physics singles out a preferred time variable, the others independent variables appear as gauges. See [12].
The fact that a gauge variable can be measured even though it cannot be predicted is not a paradox and it does not imply that a gauge theory is useless because it is unable to predict the results of possible measurements. Any use of a physical theory, in fact, includes predictable quantities and unpredictable ones. The last can be identified with those determination the measurement location. The main example is time: we look at the clock to read what time is it, and then we may predict the value of some variable at that time. A theory can be useful and predictive even if it does not predict at which time we are going to look at the clock.
This raises an intriguing question in the quantum context. In quantum theory we compute probabilities for outcomes when a system \(S\) interacts with an “apparatus” \(O\). According to the preferred interpretation of quantum theory, we view such interactions as a measurement by a classical apparatus (Copenhagen); a generic physical interaction (relational interpretation); the establishing of an entangled state with an external system (many-world); possibly with effective suppression of interference terms (decoherence); a physical event not yet described by standard theory (GRW, Penrose); or an interaction revealing the underlying hidden variables dynamics (DeBroglie-Bohm) ...Are we forced to include the detector into the quantum system, in order to compute transition probabilities, if the interaction between \(S\) and \(O\) is a gauge invariant interaction between gauge variables? I think the answer is negative [16], but I leave it open in this paper, which is confined to classical theory.
There is also a weaker sense in which measures are relational: we measure a quantity by comparing it to some standard: that’s why quantities are assigned units on some scale. This is not related to gauge invariance, although the idea of gauge originated from Weyl’s considerations about the relativity of scale.
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Acknowledgments
Thanks to Eugenio Bianchi and Hal Haggard for a critical discussion of the paper and suggestions, and to Alejandro Perez for a lifelong discussion on this issue.