Jacobi Action Research Articles

Classical and Quantum Descriptions of the Geometrodynamics of Black Holes

Analytical aspects of the classical and quantum geometrodynamics of charged black hole are considered. We start with a reduced action for a spherically symmetric of Maxwell-Einstein system written in characteristic variables. The feature of these configurations is that they admit two motion integrals, the total mass and charge. Momenta and constraint are introduced. Using the conservation laws and the Hamiltonian constraint, the momenta as functions of configuration variables are found. It turns out that the system of equations relating momenta and functional derivatives of an action on a configuration space (CS) is integrable. This allows us to obtain the action functional, as a solution of the Einstein-Hamilton-Jacobi equation in functional derivatives. Variations of the action functional with respect to mass and charge of the configuration lead to the motion trajectories in the CS. Further, the space-temporal action is transformed into an action in the configuration space similar to the Jacobi action of classical mechanics. This induces a metric in the CS. Thus, the metric on the CS is introduced and its geometry is studied. The field variables transformation is obtained which brings the CS metric to the "quasi-Lorentzian" form. On this basis, quantization is considered. Taking into account the structure of the CS, the momentum operators, DeWitt equations, mass and charge operators are constructed. Further, for comparison, consider the reduced CBH model defined in the T-region. In this simplified formulation, the T-model equations are integrated and lead to CBH with a continuous spectrum of mass and charge.

Reassessing the problem of time of quantum gravity

In this paper I raise a worry about the most extended resolutions of the problem of time of canonical quantizations of general relativity. The reason for this is that these resolutions are based on analogies with deparametrizable models for which the problem can be solved, while I argue in this paper that there are good reasons for doubting about these resolutions when the theory is not deparametrizable, which is the case of general relativity. I introduce an example of a non-deparametrizable model, a double harmonic oscillator system expressed by its Jacobi action, and argue that the problem of time for this model is not solvable, in the sense that its canonical quantization doesn’t lead to the quantum theory of two harmonic oscillators and the standard resolutions of the problem of time don’t work for this case. I argue that as general relativity is strongly analogous to this model, one should take seriously the view that the canonical quantization of general relativity doesn’t lead to a meaningful quantum theory. Finally, I comment that this has an impact on the foundations of different approaches to quantum gravity.

Quantum Chern–Simons Theories on Cylinders: BV-BFV Partition Functions

We compute partition functions of Chern–Simons type theories for cylindrical spacetimes I times Sigma , with I an interval and dim Sigma = 4l+2, in the BV-BFV formalism (a refinement of the Batalin–Vilkovisky formalism adapted to manifolds with boundary and cutting–gluing). The case dim Sigma = 0 is considered as a toy example. We show that one can identify—for certain choices of residual fields—the “physical part” (restriction to degree zero fields) of the BV-BFV effective action with the Hamilton–Jacobi action computed in the companion paper (Cattaneo et al., Constrained systems, generalized Hamilton–Jacobi actions, and quantization, arXiv:2012.13270), without any quantum corrections. This Hamilton–Jacobi action is the action functional of a conformal field theory on Sigma . For dim Sigma = 2, this implies a version of the CS-WZW correspondence. For dim Sigma = 6, using a particular polarization on one end of the cylinder, the Chern–Simons partition function is related to Kodaira–Spencer gravity (a.k.a. BCOV theory); this provides a BV-BFV quantum perspective on the semiclassical result by Gerasimov and Shatashvili.

On Time

This note describes the restoration of time in one-dimensional parameterization-invariant (hence timeless) models, namely the classically-equivalent Jacobi action and gravity coupled to matter. It also serves as a timely introduction by examples to the classical and quantum BV-BFV formalism as well as to the AKSZ method.

On quantum and relativistic mechanical analogues in mean-field spin models

Conceptual analogies among statistical mechanics and classical or quantum mechanics have often appeared in the literature. For classical two-body mean-field models, such an analogy is based on the identification between the free energy of Curie–Weiss-type magnetic models and the Hamilton–Jacobi action for a one-dimensional mechanical system. Similarly, the partition function plays the role of the wave function in quantum mechanics and satisfies the heat equation that plays, in this context, the role of the Schrödinger equation. We show that this identification can be remarkably extended to include a wider family of magnetic models that are classified by normal forms of suitable real algebraic dispersion curves . In all these cases, the model turns out to be completely solvable as the free energy as well as the order parameter are obtained as solutions of an integrable nonlinear PDE of Hamilton–Jacobi type. We observe that the mechanical analogue of these models can be viewed as the relativistic analogue of the Curie–Weiss model and this helps to clarify the connection between generalized self-averaging in statistical thermodynamics and the semiclassical dynamics of viscous conservation laws.

On Eshelby tensors in the context of the thermodynamics of open systems: Application to volumetric growth

The connections between the notion of Eshelby tensor and the variation of Hamiltonian like action integrals are investigated, in connection with the thermodynamics of continuous open bodies exchanging mass, heat and work with their surrounding. Considering first a homogeneous representative volume element (RVE), it is shown that a possible choice of the Lagrangian density is the material derivative of a suitable thermodynamic potential. The Euler equations of the so built action integral are the state laws written in rate form. As the consequence of the optimality conditions of the resulting Jacobi action, the vanishing of the surface contribution resulting from the general variation of this Hamiltonian action leads to the well-known Gibbs–Duhem condition. A general three-field variational principle describing the equilibrium of heterogeneous systems is next written, based on the zero potential, the stationarity of which delivers a balance law for a generalized Eshelby tensor in a thermodynamic context. Adopting the rate of the grand potential as the lagrangian density, a generalized Gibbs–Duhem condition is obtained as the transversality condition of the thermodynamic action integral, considering a solid body with a movable boundary. The stationnarity condition of the surface part of the thermodynamic action traduces a relationship between the virtual work of the field variables and the virtual work of the material forces at the moving boundary. This framework is applied to the volumetric growth of spherical tissue elements due to the diffusion of nutrients, whereby a growth model relating the growth velocity gradient to a growth like Eshelby stress built from the grand potential is set up.

Jacobi’s principle and the disappearance of time

Jacobi's action principle is known to lead to a problem of time. For example, the timelessness of the Wheeler-DeWitt equation can be seen as resulting from using Jacobi's principle to define the dynamics of 3-geometries through superspace. In addition, using Jacobi's principle for non-relativistic particles is equivalent classically to Newton's theory but leads to a time-independent Schrodinger equation upon Dirac quantization. In this paper, we study the mechanism for the disappearance of time as a result of using Jacobi's principle in these simple particle models. We find that the path integral quantization very clearly elucidates the physical mechanism for the timeless of the quantum theory as well as the emergence of duration at the classical level. Physically, this is the result of a superposition of clocks which occurs in the quantum theory due to a sum over all histories. Mathematically, the timelessness is related to how the gauge fixing functions impose the boundary conditions in the path integral.

Relational motivation for conformal operator ordering in quantum cosmology

Operator ordering in quantum cosmology is a major as-yet unsettled ambiguity with not only formal but also physical consequences. We determine the Lagrangian origin of the conformal invariance that underlies the conformal operator-ordering choice in quantum cosmology. This arises particularly naturally and simply from relationalist product-type actions (such as the Jacobi action for mechanics or Baierlein–Sharp–Wheeler-type actions for general relativity), for which all that is required is for the kinetic and potential factors to rescale in compensation to each other. These actions themselves mathematically sharply implement philosophical principles relevant to whole-universe modelling, so that the motivation for conformal operator ordering in quantum cosmology is thereby substantially strengthened. Relationalist product-type actions also give emergent times which amount to recovering Newtonian, proper and cosmic time in various contexts. The conformal scaling of these actions directly tells us how emergent time scales; if one follows suit with the Newtonian time or the lapse in the more commonly used difference-type Euler–Lagrange or Arnowitt–Deser–Misner-type actions, one sees how these too obey a more complicated conformal invariance. Moreover, our discovery of the conformal scaling of the emergent time permits relating how this simplifies equations of motion with how affine parametrization simplifies geodesics.

Jacobi's action and the recovery of time in general relativity.

We argue that the usual action principle of general relativity, applied to spacetimes with closed spatial geometries, should be regarded as analogous to Jacobi's form of the principle of stationary action, in which the energy rather than a physical time is fixed. Following the paradigm of quantization based on Jacobi's action for a nonrelativistic particle, we show that the Wheeler-DeWitt equation corresponds to a time-independent Schr\odinger equation. The relationship between Jacobi's and Hamilton's action principles then allows us to derive a time-dependent Wheeler-DeWitt equation of the Schr\odinger type. In this equation, the role of energy is played by the cosmological constant and that of physical time by the four-volume of spacetime.

Jacobi's action and the recovery of time in general relativity

Jacobi's action and the recovery of time in general relativity