Equivalent Theories and Changing Hamiltonian Observables in General Relativity

Pitts, J. Brian

doi:10.1007/s10701-018-0148-1

Cited by 9 publications

(18 citation statements)

References 56 publications

(93 reference statements)

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“…section 2.2) was established. The will become clear in the next section, where we show how the non-relativistic limit of the gauge-invariant transition amplitude leads precisely to the factorisation (67). We already know a solution to the "background" Hamilton-Jacobi equation (68).…”

Section: The Semiclassical Approach To the Problem Of Time Wkb Timementioning

confidence: 88%

“…Under reparametrisations of the worldline, the components of tensors transform covariantly. In fact, there is no problem in defining physical quantities (observables) to be covariant rather than invariant under worldline reparametrisations (see the discussions in [66][67][68] as well as in [69,70]). Thus, we can define observables to be worldline tensors.…”

Section: Observablesmentioning

confidence: 99%

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Construction of quantum Dirac observables and the emergence of WKB time

Chataignier

2020

Phys. Rev. D

100

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We describe a method of construction of gauge-invariant operators (Dirac observables or "evolving constants of motion") from the knowledge of the eigenstates of the gauge generator of time-reparametrisation invariant mechanical systems. These invariant operators evolve unitarily with respect to an arbitrarily chosen time variable. We emphasise that the dynamics is relational, both in the classical and quantum theories. In this framework, we show how the "emergent WKB time" often employed in quantum cosmology arises from a weak-coupling expansion of invariant transition amplitudes, and we illustrate an example of singularity avoidance in a vacuum Bianchi I (Kasner) model. * Gauge conditions of the form given in (5) are sufficient for our purposes, although more general gauge conditions are possible [12].† This corresponds to the statement that invariant extensions are obtained by writing gauge-fixed quantities in an arbitrary gauge.

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Section: The Semiclassical Approach To the Problem Of Time Wkb Timementioning

confidence: 88%

Section: Observablesmentioning

confidence: 99%

Construction of quantum Dirac observables and the emergence of WKB time

Chataignier

2020

Phys. Rev. D

100

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“…Since c. 1980 a reforming literature has aimed to recover spatio-temporal coordinate freedom in GR and mathematical equivalence between the Hamiltonian and Lagrangian formulations (e.g., [13,[19][20][21][22][23][24][25][26][27][28][29]). Recently it was shown how to bootstrap a definition of observables using equivalent formulations of massive theories, one without gauge freedom (so everything is observable) and one with gauge freedom [17,35]. It turns out that observables are basically tensor fields (or more generally geometric objects [36][37][38] including connections or any set of components with a coordinate transformation rule) that are invariant under any internal gauge symmetries present [39]; thus observables are merely covariant, not invariant, under coordinate transformations.…”

Section: Hamiltonian Change Seems Missing But Lagrangian Change Is Notmentioning

confidence: 99%

“…Using the principle that equivalent theories (or theory formulations) should have equivalent observables, I argued that observables in GR should be invariant under internal gauge symmetries (which is not novel) but only covariant under coordinate transformations, so observables are basically geometric objects, or tensor calculus all over again, such as g and its concomitants [17,35,39]. The nonlinear group realization formalism comes very close to including spinors as part of a geometric object along with the metric (or its conformal part).…”

Section: Observables In Einstein-dirac Theorymentioning

confidence: 99%

Change in Hamiltonian General Relativity with Spinors

Pitts

2021

Found Phys

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In General Relativity in Hamiltonian form, change has seemed to be missing, defined only asymptotically, or otherwise obscured at best, because the Hamiltonian is a sum of first-class constraints and a boundary term and thus supposedly generates gauge transformations. By construing change as essential time dependence (lack of a time-like Killing vector field), one can find change locally in vacuum GR in the Hamiltonian formulation just where it should be. But what if spinors are present? This paper is motivated by the tendency in space-time philosophy tends to slight fermionic/spinorial matter, the tendency in Hamiltonian GR to misplace changes of time coordinate, and the tendency in treatments of the Einstein-Dirac equation to include a gratuitous local Lorentz gauge symmetry along with the physically significant coordinate freedom. Spatial dependence is dropped in most of the paper, both restricting the physical situation and largely fixing the spatial coordinates. In the interest of including all and only the coordinate freedom, the Einstein-Dirac equation is investigated using the Schwinger time gauge and Kibble-Deser symmetric triad condition are employed as a $$3+1$$ 3 + 1 version of the DeWitt-Ogievetsky-Polubarinov nonlinear group realization formalism that dispenses with a tetrad and local Lorentz gauge freedom. Change is the lack of a time-like stronger-than-Killing field for which the Lie derivative of the metric-spinor complex vanishes. An appropriate $$3+1$$ 3 + 1 -friendly form of the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first class-constraints, is shown to change the canonical Lagrangian by a total derivative, implying the preservation of Hamilton’s equations. Given the essential presence of second-class constraints with spinors and their lack of resemblance to a gauge theory (unlike, say, massive photons), it is useful to have an explicit physically interesting example. This gauge generator implements changes of time coordinate for solutions of the equations of motion, showing that the gauge generator makes sense even with spinors.

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“…Recently the author showed that for massive electromagnetism, the requirement that equivalent theories have equivalent observables (in other words, that gauge-fixing/un-fixing doesn't change the observable content) is inconsistent with the separate first-class constraint view but fits perfectly with the gauge generator G [47,48]. Massive electromagnetism approaches massless (Maxwell) as m → 0, whether classically or in quantum field theory [5,24,44].…”

Section: Observables Reformed With the Gauge Generator Gmentioning

confidence: 99%

What Are Observables in Hamiltonian Einstein–Maxwell Theory?

Pitts

2019

Found Phys

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Is change missing in Hamiltonian Einstein-Maxwell theory? Given the most common definition of observables (having weakly vanishing Poisson bracket with each first-class constraint), observables are constants of the motion and nonlocal. Unfortunately this definition also implies that the observables for massive electromagnetism with gauge freedom (Stueckelberg) are inequivalent to those of massive electromagnetism without gauge freedom (Proca). The alternative Pons-Salisbury-Sundermeyer definition of observables, aiming for Hamiltonian-Lagrangian equivalence, uses the gauge generator G, a tuned sum of first-class constraints, rather than each first-class constraint separately, and implies equivalent observables for equivalent massive electromagnetisms.For General Relativity, G generates 4-dimensional Lie derivatives for solutions. The Lie derivative compares different space-time points with the same coordinate value in different coordinate systems, like 1 a.m. summer time vs. 1 a.m. standard time, so a vanishing Lie derivative implies constancy rather than covariance. Requiring equivalent observables for equivalent formulations of massive gravity confirms that G must generate the 4-dimensional Lie derivative (not 0) for observables.These separate results indicate that observables are invariant under internal gauge symmetries but covariant under external gauge symmetries, but can this bifurcated definition work for mixed theories such as Einstein-Maxwell theory? Pons, Salisbury and Shepley have studied G for Einstein-Yang-Mills. For Einstein-Maxwell, both F µν and g µν are invariant under electromagnetic gauge transformations and covariant (changing by a Lie derivative) under 4dimensional coordinate transformations. Using the bifurcated definition, these ⋆ Forthcoming in Foundations of Physics

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Equivalent Theories and Changing Hamiltonian Observables in General Relativity

Abstract: Change and local spatial variation are missing in Hamiltonian general relativity according to the most common definition of observables as having 0 Poisson bracket with all first-class constraints.

Cited by 9 publications

References 56 publications

Construction of quantum Dirac observables and the emergence of WKB time

Construction of quantum Dirac observables and the emergence of WKB time

Change in Hamiltonian General Relativity with Spinors

What Are Observables in Hamiltonian Einstein–Maxwell Theory?

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