# Things Change – Even in Hamiltonian General Relativity!

by J. Brian Pitts.

J. Brian Pitts is a Senior Research Associate, Faculty of Philosophy, University of Cambridge.

Observables and the Problem of Time

Mixing gravity and quantum mechanics is hard. Many approaches start with a classical theory and apply the magic of quantization, so it is important to have the classical theory sorted out well first. But the “problem of time” in Hamiltonian General Relativity looms: change seems missing in the canonical formulation.

Are Hamiltonian and Lagrangian forms of a theory equivalent? It’s not so obvious for Maxwell’s electromagnetism or Einstein’s GR, for which the Legendre transformation from the Lagrangian to the Hamiltonian doesn’t exist. It was necessary to reinvent the Hamiltonian formalism: constrained Hamiltonian dynamics. Rosenfeld’s 1930 work was forgotten until after Dirac and (independently) Bergmann’s Syracuse group had reinvented the subject by 1950. Recently a commentary and translation were published by Salisbury and Sundermeyer.

As canonical quantum gravity grew in the 1950s, it seemed less crucial for the Hamiltonian formalism to be mathematically equivalent to the Lagrangian formalism, as long as they were equivalent for the physically real “observables.” But what are observables? Surely the formalism, not postulation, should tell us. Already in the mid-1950s change seemed missing in the Hamiltonian observables. Yet similar conclusions are not drawn about the Lagrangian formalism. How can the two formalisms fail to be equivalent?

Around the early 1980s various authors revived the mathematical equivalence of the Hamiltonian to the Lagrangian formalism. A key difference between the two views involves gauge transformations. Are they the transformations generated by the 1930/1951/1980s+ Rosenfeld-Anderson-Bergmann… “gauge generator” G in which the (first-class) constraints work together as a team and the transformations are essentially familiar Lagrangian expressions? Or is a gauge transformation anything generated by a (first-class) constraint alone? One can find change (essential time dependence) in Hamiltonian GR just where one finds it in the 4-dimensional geometric/Lagrangian formalism: where there is no time-like Killing vector field (in the vacuum case).

But what about the changelessness of observables? The notion of observables has been controversial. Definitions by Bergmann and by Dirac are famous, but Bergmann’s views are diverse on closer inspection. The conclusion that observables do not vary with time has also drawn criticisms from Kuchař and Smolin. Pons, Salisbury and Sundermeyer have proposed a definition that treats (first-class) constraints as a team rather than individuals.

How might one test definitions of observables? By using two equivalent formulations of one theory, assuming that the answer is certain in one formulation. For massive electromagnetism naturally there is no gauge freedom, so everything is observable. Using the “Stueckelberg trick” (useful for massive QED), one can add gauge freedom with an extra gauge compensation field. If observables can be observed (a natural condition for propriety of the term), then the two formulations of electromagnetism must have equivalent observables. Then one can test definitions of observables by direct calculation: a good definition must imply that the two equivalent forms of massive electromagnetism have equivalent observables. It was shown recently that this requirement picks out the Pons-Salisbury-Sundermeyer G team definition.

Of course observables in GR are the really interesting question. Fortunately, one can attempt the same kind of modification of GR that de Broglie and Proca did to electromagnetism, thus arriving at a massive gravity theory. After impressive work in the 1960s (partly reinvented in the 2010s), massive gravity suffered a seemingly fatal dilemma in the early 1970s. In the new millennium both arguments behind the dilemma were questioned. Fortunately, using massive gravity to test definitions of observables works regardless of those subtleties, because neither empirical viability nor suitably for quantization is relevant. One can install artificial gauge freedom in massive gravity (“clock fields” XA, also called “parametrization”) much like the Stueckelberg trick.

Which definition gives equivalent observables? It turns out that one needs a new definition that distinguishes between “external” symmetries (such as coordinate transformations in gravity) and “internal” symmetries (as in electromagnetism). In the new definition, observables, instead of being unchanged (changed by 0) by the team G of first-class constraints generating an external symmetry, can change by a Lie derivative, like scalar fields, vector fields, tensors, etc. Because parametrized massive gravity is so much like GR with 4 scalar fields, the same definition should apply to GR also. Thus observables in GR are basically 4-dimensional tensor calculus all over again, making use as needed of the relations $\dot{q}=\partial H/\partial p$ and even other Hamilton equations. So Hamiltonian observables are local fields that change even in GR. Thus at the classical level, there isn’t a problem of time even for observables. For more information and references, please see the paper in Classical and Quantum Gravity [1].

How, if at all, does such work affect canonical quantum gravity? Should something usually thought to be 0 be non-zero (but still restricted to enforce gauge freedom) instead? That remains to be seen. And what of supergravity’s mixed internal-external transformations?