Statistical Gravity, ADM Splitting, and AQ <i></i>

Riccardo Fantoni

doi:10.20944/preprints202502.1388.v1

Submitted:

17 February 2025

Posted:

18 February 2025

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Abstract

I propose a possible way to render numerically accessible the path integral Monte Carlo computations required in the Statistical Gravity theory described in a recent publication [Riccardo Fantoni, Quantum Reports, {\bf 6}, 706 (2024)]. This requires the use of the Arnowitt, Deser, and Misner (ADM) splitting and of the Affine Quantization (AQ) method.

Keywords:

General Relativity

;

Einstein-Hilbert action

;

ADM splitting

;

Affine Quantization

;

Statistical Physics

;

Path Integral

;

Monte Carlo

Subject:

Physical Sciences - Theoretical Physics

I. Introduction

The idea to realize a quantum theory of gravity has a long history [1,2]. Recently we proposed a theory for Statistical Gravity [3], the FEBB. Leaving aside the feasible experimental confirmations for it, it is yet important to prove that it gives rise to quantities (observables thermal averages) that are mathematically well defined and can therefore be computed (at least numerically). We are thinking, for example, at the problems that one may encounter in computing a constrained quantum field theory [4,5,6,7,8,9,10], even the simplest one as the scalar (relativistic euclidean). In these cases we could experience how important it was to use the method of Affine Quantization (AQ) (as opposed to the canonical quantization) in order to render the particular theory non trivial. But even before worrying about the renormalizability of the particular quantum field theory it makes sense to worry about the soundness of the place it occupies in the underlying Hilbert space.

With this in mind, in this short paper, following the idea already put forward in Ref. [7] for a construction of a well defined Quantum Gravity, we propose to use the method of AQ also to construct a well defined Statistical Gravity.

In these complex tensorial quantum field theories, even the determination of the relevant semiclassical action can become a formidable task due to the intertwining of the tensorial calculus and the commutation calculus. Here we will not carry out any of this necessary complex calculus explicitly but will just lay down the problem showing that it is a well defined one.

II. Einstein’s Field Equations from a Variational Principle

Sempre caro mi fu quest’ermo colle, e questa siepe, che da tanta parte dell’ultimo orizzonte il guardo esclude.

Giacomo Leopardi L’ Infinito

The Einstein-Hilbert action in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the

(- + + +)

metric signature, the action is given as [11,12]

S = \int (\frac{1}{2 κ} R + L_{F}) \sqrt{- g} d^{4} x .

(2.1)

where

g \equiv det (g_{μ ν})

is the determinant of the metric tensor matrix,

\sqrt{- g}

is the scalar density,

x \equiv (c t, x)

is an event with

t \equiv x^{0} / c

time and

x \equiv (x^{1}, x^{2}, x^{3})

a point in space,

\sqrt{- g} d^{4} x

is the invariant “volume” element, R is the Ricci scalar,

κ = 8 π G c^{- 4}

is the Einstein gravitational constant (G is the gravitational constant and c is the speed of light in vacuum), and

\sqrt{- g} L_{F}

is a Lagrangian density of “interaction” containing the contribution from matter, electromagnetic, or other gauge bosons fields to the action. If it converges, the integral is taken over the whole spacetime. If it does not converge, S is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the Euler-Lagrange equation of the Einstein-Hilbert action. The action was proposed [11] by David Hilbert in 1915 as part of his application of the variational principle

δ S = 0

to a combination of gravity and matter, electromagnetism, or other gauge bosons fields. Note that in the variation of the Ricci scalar one needs to assume that the Gibbons–Hawking–York boundary term [13] gives no contribution to the variation of the action, which is justified at events not in the closure of the boundary, when the variation of the metric vanishes in a neighbourhood of the boundary or when there is no boundary.

The equations of motion coming from the stationary-action principle are then like so (see second section of Ref. [3])

G_{μ ν} \equiv R_{μ ν} - \frac{1}{2} g_{μ ν} R = κ T_{μ ν},

(2.2)

which are the Einstein field equations, where

T_{μ ν} = \frac{- 2}{\sqrt{- g}} \frac{δ (\sqrt{- g} L_{F})}{δ g^{μ ν}} = - 2 \frac{δ L_{F}}{δ g^{μ ν}} + g_{μ ν} L_{F},

(2.3)

is the stress-energy tensor and

κ = 8 π G / c^{4}

has been chosen such that the non-relativistic limit yields the usual form of Newton’s gravity law.

III. ADM 3+1 Foliation of Spacetime

Ma sedendo e mirando, interminati spazi di là da quella, e sovrumani silenzi, e profondissima quiete io nel pensier mi fingo, ove per poco il cor non si spaura.

Giacomo Leopardi L’ Infinito

Arnowitt, Deser and Misner (ADM) proposed in 1962 the following 3+1 foliation of spacetime [14]

d s^{2} = - N^{2} d t^{2} + g_{i j} (d x^{i} + N^{i} d t) (d x^{j} + N^{j} d t),

(3.1)

where now Latin indexes run over the three spatial components

1, 2, 3

. They called N the lapse and

N_{i}

the shift. To split the time component from the 3 spatial components they chose the following

| | g_{μ ν} | | = (\begin{matrix} - (N^{2} - N^{i} N_{i}) & N_{i} \\ N_{i} & g_{i j} \end{matrix}),

(3.2)

| | g^{μ ν} | | = (\begin{matrix} - 1 / N^{2} & N^{i} / N^{2} \\ N^{i} / N^{2} & g^{i j} - N^{i} N^{j} / N^{2} \end{matrix}),

(3.3)

which are inverse by sight. Note also that

\sqrt{- 4 g} = N \sqrt{3 g}

where

3 g = det {g_{i j}}

and

4 g = det {g_{μ ν}}

and we indicate with a presuperscript 4 the full four dimensional tensor and with a presuperscript 3 the spatial

3 \times 3

tensor, when strictly necessary to avoid confusion. Therefore we will raise (or lower) Greek indexes with the full metric tensor

g^{μ ν}

and Latin indexes with the spatial metric tensor

g^{i j}

which also satisfies

g_{i k} g^{k j} = δ_{i}^{j}

.

ADM showed that if one chooses as generalized coordinate

g_{i j}

and conjugated momentum

π^{i j} \equiv \sqrt{- 4 g} ({Γ_{p}}^{0}_{q} - g_{p q} {Γ_{r}}^{0}_{s} g^{r s}) g^{i p} g^{j q},

(3.4)

then the spacetime metric Lagrangian

L \equiv \sqrt{- 4 g} 4 R = - g_{i j} {π^{i j}}_{, 0} - N R^{0} - N_{i} R^{i} - 2 {(π^{i j} N_{j} - \frac{1}{2} π N^{i} + N^{| j} \sqrt{3 g})}_{, i},

(3.5)

where we denote with a semicolon

(;)

the usual covariant derivative in the full spacetime and with a bar

(|)

a spatial covariant derivative, and

R^{0} \equiv - \sqrt{3 g} [3 R + \frac{1}{3 g} (\frac{1}{2} π^{2} - π^{i j} π_{i j})],

(3.6)

R^{i} \equiv - 2 {π^{i j}}_{| j},

(3.7)

π \equiv π_{i}^{i} .

(3.8)

Eq. (9) is the Hamiltonian constraint whereas Eq. (10) the momentum constraint. In fact, since the last term in Eq. (8) only contributes a “surface” term to the metric action

S \propto \int L d^{4} x

, if spacetime extends to infinity it can be taken as giving a negligible contribution.

Upon taking variations with respect to the lapse and shift provides the constraint equation

R^{0} = 0

and

R^{i} = 0

and then the lapse and shift themselves can be freely specified, reflecting the fact that coordinates systems can be freely specified in both space and time.

Since

g_{i j}

is a strictly positive definite tensor, in our recent paper [7] we proposed to use affine variables in place of the canonical variables

g_{i j}

and

π^{i j}

in order to cure such unholonomous constraint. We then introduce a “dilation” conjugate variable

π_{j}^{i} = g_{k j} π^{i k}

. This classical momentric (a name that is the combination of momentum and metric and was invented by John Klauder) tensor and the spatial metric tensor become the new basic canonical affine variables. By doing so and recalling that

{g^{i j}}_{| k} = 0

we reach to the following classical Lagrangian

L = - g_{i j} {π^{i j}}_{, 0} - N R^{0} - N_{i} R^{i},

(3.9)

R^{i} = - 2 g^{i k} {π_{k}}^{j}_{| j},

(3.10)

R^{0} = \frac{1}{\sqrt{3 g}} [π_{j}^{i} π_{i}^{j} - \frac{1}{2} π^{2}] - \sqrt{3 g} 3 R .

(3.11)

where we dropped the gradient term in the Lagrangian since it gives no contribution to the classical action

S = \int_{0} \int_{Ω} {- g_{i j} {π^{i j}}_{, 0} - N R^{0} - N_{i} R^{i}} d (c t) d^{3} x .

(3.12)

where

Ω

is the region of space and time starts from the beginning at

t = 0

.

In Affine Quantization (AQ) we promote the two canonical affine variables

g_{i j}

and

π_{j}^{i}

to operators

{\hat{g}}_{i j}

and

{\hat{π}}_{j}^{i}

and write the corresponding affine semiclassical (including just the terms up to order ℏ in the

ℏ \to 0

limit) Lagrangian

L^{'}

using the commutation relations between the spatial metric operator and the momentric operator (these are given, for example, in Ref. [7] and derived again in the Appendix A).

IV. Path Integral Formulation of Statistical Gravity

E come il vento odo stormir tra queste piante, io quello infinito silenzio a questa voce vo comparando: e mi sovvien l’eterno, e le morte stagioni, e la presente e viva, e il suon di lei.

Giacomo Leopardi L’ Infinito

Then the action for Einstein’s theory of general relativity is one for a particular field theory where the field is the metric tensor

g_{μ ν} (x)

a symmetric tensor with 10 independent components, each of which is a smooth function of 4 variables. We will indicate all these components with the notation

{g} (x)

. We will also work in euclidean time

x^{0} \equiv c t \to i c t

so that the metric signature becomes

(+ + + +)

.

The thermal average of an observable

O [{g} (x)]

will then be given by the following expression [3]

〈 O 〉 = \frac{\int O [{g} (x)] exp (- υ S^{'}) D^{10} {g} (x)}{\int exp (- υ S) D^{10} {g} (x)},

(4.1)

so that

〈 1 〉 = 1

. Here

S^{'}

is the affine action

S^{'} = \int_{0}^{β} \int_{Ω} \{\frac{1}{2 κ} L^{'} + L_{F} N \sqrt{3 g}\} d (c t) d^{3} x,

(4.2)

1 / υ

is a positive constant of dimension of energy times length,

c t \in [0, β [

where

β = 1 / {\tilde{k}}_{B} \tilde{T}

,

{\tilde{k}}_{B}

is a Boltzmann constant of dimensions of one divided by length and by degree Kelvin, and

\tilde{T}

an effective temperature in degree Kelvin (which can be made a field [3],

\tilde{T} (x)

). Since the thermal average involves taking a trace we must have

g_{μ ν} (c t + β, x) = g_{μ ν} (c t, x)

. We will also require periodic spatial boundary conditions on the finite volume

Ω \subset I R^{3}

which is the closest thing to a formal thermodynamic limit. As usual we will use

D^{10} {g} (x) \equiv \prod_{x} d^{10} {g} (x)

and the functional integrals will be calculated on a lattice using the path integral Monte Carlo (PIMC) method [15]. Moreover we will choose

d^{10} {g} (x) \equiv \prod_{μ \leq ν} d g^{μ ν} (x)

where the 10-dimensional space of the 10 independent components of the symmetric metric tensor is assumed to be flat.

The determination of

L^{'}

looks like a formidable task that needs to take care of the commutation relations among the spatial metric and the momentric operators but it seems to be necessary to overcome the numerical singularities that may arise from the geometrical unholonomous constraint of having a strictly positive definite spatial metric. Here we are thinking of the possible loss of ergodicity in the PIMC as its paths wander through and explore the accessible region delimited by the sharp constraints which can be variously intricate. We see AQ as a way to smooth out the geometrical constraints so to recover ergodicity and be able to sample the whole relevant region efficiently.

V. Conclusions

Così tra questa immensità s’annega il pensier mio: e il naufragar m’è dolce in questo mare.

Giacomo Leopardi L’ Infinito

In this short paper we present a plausible representation (realization) of the FEBB defined in Ref. [3]. This requires the use of the ADM 3+1 splitting and the AQ procedure. We just lay down the representation but without finding its explicit form which would require a rather formidable calculus where one needs to deal with commutation relations among tensorial objects. We believe that a Monte Carlo algorithm may lose ergodicity in the presence of sharp constraints which AQ can otherwise smooth out.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgments

I thank Luciano Rezzolla for discussions on the ADM splitting and for sharing his lecture notes and suggesting chapter 7 of his book “Relativistic Hydrodynamics”.

Conflicts of Interest

The author has no conflicts to disclose.

Appendix A. Commutators Between the Spatial Metric and the Momentric

We start from the Poisson brackets (at fixed time) between the two canonical variables

g_{i j}

and

π^{i j}

:

\begin{matrix} {g_{i j} (x), g_{k l} (x^{'})} & = & 0, \\ {g_{i j} (x), π^{k l} (x^{'})} & = & \frac{δ g_{i j} (x)}{δ g_{m n} (x^{''})} \frac{δ π^{k l} (x^{'})}{δ π^{m n} (x^{''})} \\ = & \frac{1}{2} δ^{3} (x - x^{''}) δ^{3} (x^{'} - x^{''}) δ_{m}^{k} δ_{n}^{l} [δ_{i}^{m} δ_{j}^{n} + δ_{j}^{m} δ_{i}^{n}] \end{matrix}

(A1)

\begin{matrix} = & \frac{1}{2} δ^{3} (x - x^{'}) [δ_{i}^{k} δ_{j}^{l} + δ_{i}^{l} δ_{j}^{k}], \end{matrix}

(A2)

\begin{matrix} {π^{i j} (x), π^{k l} (x^{'})} & = & 0, \end{matrix}

(A3)

where in the second equation we used the symmetry of the metric tensor to write

g_{i j} = [g_{i j} + g_{j i}] / 2

and

δ^{3}

is a three dimensional Dirac delta function.

We then find the Poisson brackets between the two canonical affine variables

g_{i j}

and

π_{i}^{j} = g_{i k} π^{k j}

:

\begin{matrix} {g_{i j} (x), π_{k}^{l} (x^{'})} & = & {g_{i j} (x), g_{k n} (x^{'}) π^{n l} (x^{'})} \\ = & g_{k n} (x^{'}) {g_{i j} (x), π^{n l} (x^{'})} \end{matrix}

(A4)

\begin{matrix} = & \frac{1}{2} δ^{3} (x - x^{'}) [δ_{j}^{l} g_{k i} (x) + δ_{i}^{l} g_{k j} (x)], \\ {π_{i}^{j} (x), π_{k}^{l} (x^{'})} & = & {g_{i n} (x) π^{n j} (x), g_{k m} (x^{'}) π^{m l} (x^{'})} \\ = & g_{k m} π^{n j} {g_{i n} (x), π^{m l} (x^{'})} - g_{i n} π^{m l} {g_{k m} (x^{'}), π^{n j} (x)} \end{matrix}

(A5)

\begin{matrix} = & \frac{1}{2} δ^{3} (x - x^{'}) [δ_{i}^{l} π_{k}^{j} (x) - δ_{k}^{j} π_{i}^{l} (x)] . \end{matrix}

(A6)

And in the end we pass to operator commutators, promoted from the Poisson brackets

{\dots, \dots} \to [\dots, \dots] / (i ℏ)

. After being smeared with suitable test functions, the result is that both the metric and the momentric tensors can be made self-adjoint operators (for example choosing for the momentric

({\hat{g}}_{i k} {\hat{π}}^{j k} + {\hat{π}}^{j k} {\hat{g}}_{i k}) / 2

), and the metric operators will satisfy the required positivity requirements.

References

C. W. Misner, Feynman Quantization of General Relativity. Rev. Mod. Phys. 1957, 29, 497. [CrossRef]
J. R. Klauder, A Straight Forward Path to a Path Integration of Einstein’s Gravity. Annals of Physics 2022, 447, 169148. [CrossRef]
R. Fantoni, Statistical Gravity through Affine Quantization, Quantum Rep. 6, 706 (2024a).
R. Fantoni and J. R. Klauder, Monte Carlo evaluation of the continuum limit of the two-point function of the Euclidean free real scalar field subject to affine quantization. J. Stat. Phys. 2021, 184, 28. [CrossRef]
R. Fantoni and J. R. Klauder, Scaled Affine Quantization of φ44 in the Low Temperature Limit, Eur. Phys. J. C 82, 843 (2022a).
R. Fantoni and J. R. Klauder, Scaled Affine Quantization of Ultralocal φ24 a comparative Path Integral Monte Carlo study with scaled Canonical Quantization, Phys. Rev. D 106, 114508 (2022b).
J. R. Klauder and R. Fantoni, The Magnificent Realm of Affine Quantization: valid results for particles, fields, and gravity, Axioms 12, 911 (2023).
R. Fantoni, Static screening in a degenerate electron plasma, The Physics Educator 6, 2420004 (2024b).
R. Fantoni, Continuum limit of the Green function in scaled affine φ44 quantum Euclidean covariant relativistic field theory, Quantum Rep. 6, 134 (2024c).
J. R. Klauder and R. Fantoni, The Secret to Fixing Incorrect Canonical Quantizations, Academia Quantum 1, (2024). [CrossRef]
D. Hilbert, Die Grundlagen der Physik, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen – Mathematisch-Physikalische Klasse 3, 395 (1915).
R. P. Feynman, Feynman Lectures on Gravitation (Addison-Wesley, 1995) p. 136 Eq. (10.1.2).
https://en.wikipedia.org/wiki/Gibbons-Hawking-York_boundary_term (accessed on 2025/02/28 18:34:39).
R. Arnowitt, S. Deser, and C. Misner, The Dynamics of General Relativity, In: Witten, L., Ed., Gravitation: An Introduction to Current Research, Wiley & Sons, New York, 227 (1962), arXiv: gr-qc/0405109.
D. M. Ceperley, Path integrals in the theory of condensed helium, Rev. Mod. Phys. 67, 279 (1995).

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Statistical Gravity, ADM Splitting, and AQ

Abstract

Keywords:

Subject:

I. Introduction

II. Einstein’s Field Equations from a Variational Principle

III. ADM 3+1 Foliation of Spacetime

IV. Path Integral Formulation of Statistical Gravity

V. Conclusions

Data Availability Statement

Acknowledgments

Conflicts of Interest

Appendix A. Commutators Between the Spatial Metric and the Momentric

References

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