# Existence of conserved quantities and their algebra in curved spacetime

Susobhan Mandal

###### Abstract

In General Relativity, finding out the geodesics of a given spacetime manifold
is an important task because it determines which classical processes are
dynamically forbidden. Conserved quantities play an important role in solving
geodesic equations of a general spacetime manifold. Furthermore, knowing all
possible conserved quantities of a system tells about the hidden symmetries of
that system since, conserved quantities are deeply connected with the
symmetries of the system, which are very important in their own right.
Conserved quantities are also useful to capture certain features of spacetime
manifold for an asymptotic observer. In this article, we show the existence of
these conserved charges and their algebra for a class of dynamical systems in
a generic curved spacetime.

##  1 Introduction

For a system any physical quantity $Q$ has, for each kinematically possible
motion, a definite numerical value (which could be vectorial or tensorial in
nature) at each instant of time. If for every dynamically allowed motion of
that system it happens that $\frac{dQ}{dt}=0$, we then say that $Q$ is
conserved (or that $Q$ is a constant of motion) for that particular physical
system. It is often found out that the same quantity $Q$ (or a similar
quantity) is a constant of motion for some broad and interesting class of
systems. We then assert a conservation theorem which is in a way related to
symmetries, characterizing the situations in which $Q$ is conserved.

Conservation theorems are important for various reasons. First of all,
conservation theorems are generic statements about the types of motions that a
dynamical law (or a class of dynamical laws) allows. In particular, they
provide information about certain types of motion which are classically
forbidden. Conservation theorems also give partial information about the
nature of a particular motion, even if the equations are too complicated to
solve analytically [1, 2, 3, 4]. A conserved quantity provides a “first
integral" of the equations of motion (which is second order differential
equation) and sometimes this is sufficient to essentially solve the problem;
other times it can be used to decouple a set of coupled differential
equations.

In General Relativity (GR), dynamics of a system is often talked w.r.t affine
parameter $\lambda$ which in certain cases can be chosen as coordinate time
$t$. In an arbitrary spacetime manifold one does not expect existence of any
conserved quantities however, existence of Killing symmetries in special class
of spacetime manifolds, makes it possible to define certain conserved
quantities easily. Furthermore, if the metric of chosen spacetime manifold is
asymptotically flat then it is straightforward to assign physical meaning to
such quantities (for example angular momentum, energy of a particle etc.)
Another context, in which existence of conserved quantities plays an important
role is to find out integrability condition of a system which basically says a
2n-dimensional hamiltonian system of ordinary differential equations (ODE) is
integrable [5, 6, 7] if it has n (functionally) independent constants of the
motion that are "in involution" where functionally independent means none of
them can be written as a function of the others and "in involution" means that
their Poisson Brackets all vanish.

##  2 Algebra of Killing vectors

Given a spacetime manifold a general diffeomorphism is characterized by a
vector field $\xi$, defined over that spacetime manifold such that under
following

$x^{\prime}=x+\xi\implies x^{\prime\mu}=x^{\mu}+\xi^{\mu}(x)$ (2.1)

co-ordinate transformation,

$\delta
g_{\alpha\beta}=\mathcal{L}(\xi)g_{\alpha\beta}=\nabla_{\alpha}\xi_{\beta}+\nabla_{\beta}\xi_{\alpha}=0$
(2.2)

Denoting $H_{\mu\nu}\equiv\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}$ and
using Killing equation, we obtain

$\begin{split}0&=\nabla_{\mu}H_{\nu\rho}+\nabla_{\rho}H_{\nu\mu}-\nabla_{\nu}H_{\rho\mu}\\\
&=\nabla_{\mu}\nabla_{\nu}\xi_{\rho}+\nabla_{\mu}\nabla_{\rho}\xi_{\nu}+\nabla_{\rho}\nabla_{\nu}\xi_{\mu}+\nabla_{\rho}\nabla_{\mu}\xi_{\nu}\\\
&-\nabla_{\nu}\nabla_{\rho}\xi_{\mu}-\nabla_{\nu}\nabla_{\mu}\xi_{\rho}\\\
&=[\nabla_{\mu},\nabla_{\nu}]\xi_{\rho}+[\nabla_{\rho},\nabla_{\nu}]\xi_{\mu}+\\{\nabla_{\mu},\nabla_{\rho}\\}\xi_{\nu}\\\
&=\mathcal{R}_{\rho\mu\nu\lambda}\xi^{\lambda}+\mathcal{R}_{\mu\rho\nu\lambda}\xi^{\lambda}+2\nabla_{\mu}\nabla_{\rho}\xi_{\nu}+\mathcal{R}_{\nu\rho\mu\lambda}\xi^{\lambda}\\\
\implies&\nabla_{\mu}\nabla_{\rho}\xi_{\nu}=-\frac{1}{2}\mathcal{R}_{\nu\rho\mu\lambda}\xi^{\lambda}\end{split}$
(2.3)

Using this relation, it can be shown that all the isometries of a given
spacetime form a Lie algebra. We show that the commutator of two isometries is
another isometry

$\begin{split}\nabla_{\alpha}[\xi_{a},\xi_{b}]_{\beta}&+\nabla_{\beta}[\xi_{a},\xi_{b}]_{\alpha}=\nabla_{\alpha}(\xi_{a}^{\mu}\nabla_{\mu}\xi_{\beta}^{b}-\xi_{b}^{\mu}\nabla_{\mu}\xi_{\beta}^{a})\\\
&+\nabla_{\beta}(\xi_{a}^{\lambda}\nabla_{\lambda}\xi_{\alpha}^{b}-\xi_{b}^{\lambda}\nabla_{\lambda}\xi_{\alpha}^{a})\\\
&=\xi_{a}^{\mu}\nabla_{\alpha}\nabla_{\mu}\xi_{\beta}^{b}-\xi_{b}^{\mu}\nabla_{\alpha}\nabla_{\mu}\xi_{\beta}^{a}+\xi_{a}^{\mu}\nabla_{\beta}\nabla_{\mu}\xi_{\alpha}^{b}-\xi_{b}^{\lambda}\nabla_{\beta}\nabla_{\lambda}\xi_{\alpha}^{a}\\\
&+\nabla_{\alpha}\xi_{a}^{\mu}\nabla_{\mu}\xi_{\beta}^{b}-\nabla_{\alpha}\xi_{b}^{\mu}\nabla_{\mu}\xi_{\beta}^{a}+\nabla_{\beta}\xi_{a}^{\lambda}\nabla_{\lambda}\xi_{\alpha}^{b}-\nabla_{\beta}\xi_{b}^{\lambda}\nabla_{\lambda}\xi_{\alpha}^{a}\\\
&=-\xi_{a}^{\mu}\nabla_{\alpha}\nabla_{\beta}\xi_{\mu}^{b}+\xi_{b}^{\lambda}\nabla_{\alpha}\nabla_{\beta}\xi_{\lambda}^{a}-\xi_{a}^{\mu}\nabla_{\beta}\nabla_{\alpha}\xi_{\mu}^{b}+\xi_{b}^{\lambda}\nabla_{\beta}\nabla_{\alpha}\xi_{\lambda}^{a}\\\
&=\frac{1}{2}\xi_{a}^{\mu}\xi_{b}^{\lambda}(\mathcal{R}_{\mu\beta\alpha\lambda}-\mathcal{R}_{\lambda\beta\alpha\mu}+\mathcal{R}_{\mu\alpha\beta\lambda}-\mathcal{R}_{\lambda\alpha\beta\mu})=0\end{split}$
(2.4)

Hence, this indicates that the mentioned commutator is a linear combination of
other Killing vector with constant coefficients, known as structure constants
of the isometry algebra. In general, if we consider $\\{\xi_{a},\
a=1,\ldots,r\\}$ represents a basis of linear space of Killing fields of a
spacetime manifold, the following mathematical statement is true

$[\xi_{a},\xi_{b}]=\mathcal{C}_{ab}^{c}\xi_{c}$ (2.5)

with $\mathcal{C}_{ab}^{c}=-\mathcal{C}_{ba}^{c}$. The isometry group is
simply transitive if all Killing fields are linearly independent, otherwise
the group is multiply transitive.

##  3 Canonical formulation

Here we discuss the Poisson bracket formalism in hamiltonian mechanics in
curved spacetime. We provide here a covariant formulation of Poisson bracket
which makes it easy to understand dynamics of a given object in curved
spacetime. Further, notion of conserved charges associated to Killing
symmetries is revisited and closure of their algebra is also shown in this
section.

###  3.1 Hamiltonian mechanics

Most often to study dynamics of a particle on a curved spacetime one would
start with following action

$S=-\int\sqrt{-g_{\mu\nu}dx^{\mu}dx^{\nu}}=-\int\sqrt{-g_{\mu\nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}}d\lambda$
(3.1)

where $\lambda$ is an affine parameter and $\dot{}$ represents derivative
w.r.t $\lambda$. Extremizing this action is equivalent to extremizing line-
element which gives geodesic equations of that manifold.

There is another alternative choice of action which also leads to geodesic
equations but characterestically of different form

$S=\frac{1}{2}\int g_{\mu\nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}d\lambda$ (3.2)

but both the action is equivalent from ‘einbin’ formalism point of view.
Thereore, we start with second action whose correspondence hamiltonian is

$H=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}$ (3.3)

which can be obtained through Legendre transformation and where
$p_{\mu}(\lambda)$ is the canonical conjugate momentum variable defined by

$p_{\mu}(\lambda)=g_{\mu\nu}(x(\lambda))\dot{x}^{\nu}$ (3.4)

which translates geodesic equations in following form

$\begin{split}\frac{dp_{\mu}}{d\lambda}&=\partial_{\rho}g_{\mu\nu}\frac{dx^{\rho}}{d\lambda}\frac{dx^{\nu}}{d\lambda}+g_{\mu\nu}(x)\frac{d^{2}x^{\nu}}{d\lambda^{2}}\\\
&=\partial_{\rho}g_{\mu\sigma}\frac{dx^{\rho}}{d\lambda}\frac{dx^{\sigma}}{d\lambda}-\Gamma_{\mu\rho\sigma}(x)\frac{dx^{\rho}}{d\lambda}\frac{dx^{\sigma}}{d\lambda}\\\
&=\frac{dx^{\rho}}{d\lambda}\frac{dx^{\sigma}}{d\lambda}\left(\partial_{\rho}g_{\mu\sigma}-\frac{1}{2}\partial_{\rho}g_{\mu\sigma}-\frac{1}{2}\partial_{\sigma}g_{\mu\rho}+\frac{1}{2}\partial_{\mu}g_{\rho\sigma}\right)\\\
&=\Gamma_{\sigma\mu\rho}\frac{dx^{\rho}}{d\lambda}\frac{dx^{\sigma}}{d\lambda}=\Gamma_{\
\mu}^{\sigma\ \ \rho}p_{\rho}p_{\sigma}\end{split}$ (3.5)

The equations (3.4) and (3.5) constitute a pair of first order ODEs which is
equivalent to second order geodesic equations. Hamilton’s equations are the
set of powerful results which shows how to derive above two equations from
hamiltonian $H(\lambda)$ itself

$\frac{dx^{\mu}}{d\lambda}=\frac{\partial H}{\partial p_{\mu}},\
\frac{dp_{\mu}}{d\lambda}=-\frac{\partial H}{\partial x^{\mu}}$ (3.6)

where $H(\lambda)=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}$.

###  3.2 Poisson bracket

With the definitions (3.6), one can show that any function
$F(x^{\mu},p_{\nu})$ defined over phase space spanned by canonical conjugate
variables, changes along geodesic according to

$\begin{split}\frac{dF}{d\lambda}&=\frac{dx^{\mu}}{d\lambda}\frac{\partial
F}{\partial x^{\mu}}+\frac{dp_{\mu}}{d\lambda}\frac{\partial F}{\partial
p_{\mu}}\\\ &=\frac{\partial H}{\partial p_{\mu}}\frac{\partial F}{\partial
x^{\mu}}-\frac{\partial H}{\partial x^{\mu}}\frac{\partial F}{\partial
p_{\mu}}\\\ &=\\{F,H\\}\end{split}$ (3.7)

where Poisson bracket of two functions of phase space variable is defined as
follows

$\\{F,G\\}=-\\{G,F\\}=\frac{\partial F}{\partial x^{\mu}}\frac{\partial
G}{\partial p_{\mu}}-\frac{\partial G}{\partial x^{\mu}}\frac{\partial
F}{\partial p_{\mu}}$ (3.8)

Anti-symmetry of Poisson bracket automatically ensures that time-independent
hamiltonian (which does not have explicit time dependence) is a conserved
quantity since

$\\{H,H\\}=0$ (3.9)

Poisson bracket also satisfies linearity in both of its arguments

$\\{F,\alpha_{1}G_{1}+\alpha_{2}G_{2}\\}=\alpha_{1}\\{F,G_{1}\\}+\alpha_{2}\\{F,G_{2}\\},\
\forall\alpha,\beta\in\mathbb{C}$ (3.10)

Another important property that it satisfies is the Jacobi identity

$\\{F,\\{G,K\\}\\}+\\{G,\\{K,F\\}\\}+\\{K,\\{F,G\\}\\}=0$ (3.11)

These properties also imply that the Poisson bracket of any two conserved
quantities of geodesic flow is also a conserved quantity of geodesic flow. If
$F,G$ satisfies $\\{F,H\\}=0=\\{G,H\\}$ then

$\\{\\{F,G\\},H\\}=\\{F,\\{G,H\\}\\}-\\{G,\\{F,H\\}\\}=0$ (3.12)

As $H$ is itself a constant of motion on geodesics the above result are enough
to establish that constants of motion form a Lie-algebra, with Poisson bracket
as Lie bracket. (We will alternatively use the terms constant of motion and
conserved quantity which are equivalent.)

Note that definition of Poisson bracket mentioned above is nor manifestly
covariant, which at first sight, seems to destroy the in buit covariance
nature of GR. Since, the connetion used in GR is symmetric [8] in its lower
two indices, it helps us to extend the definition of Poisson bracket,
maintaining the general covariance

$\\{F,G\\}_{*}\equiv\mathcal{D}_{\mu}F\frac{\partial G}{\partial
p_{\mu}}-\frac{\partial F}{\partial p_{\mu}}\mathcal{D}_{\mu}G$ (3.13)

where for functions over phase-space variables (which are scalar by nature),
we define covariant derivative as

$\mathcal{D}_{\mu}F=\partial_{\mu}F+\Gamma_{\
\mu\nu}^{\lambda}p_{\lambda}\frac{\partial F}{\partial p_{\nu}}$ (3.14)

Now note that for any two arbitrary phase-space functions

$\begin{split}\\{F,G\\}_{*}&=\partial_{\mu}F\frac{\partial G}{\partial
p_{\mu}}-\frac{\partial F}{\partial p_{\mu}}\partial{\mu}G\\\ &+\Gamma_{\
\mu\nu}^{\lambda}p_{\lambda}\left(\frac{\partial F}{\partial
p_{\nu}}\frac{\partial G}{\partial p_{\mu}}-\frac{\partial G}{\partial
p_{\nu}}\frac{\partial F}{\partial p_{\mu}}\right)\\\
&=\partial_{\mu}F\frac{\partial G}{\partial p_{\mu}}-\frac{\partial
F}{\partial p_{\mu}}\partial_{\mu}G=\\{F,G\\}\end{split}$ (3.15)

whose shows both the definition of Poisson bracket numerically gives same
value hence, for sake of convenience we use the first definition from now
onwards.

The second definition also preserves covariance manifestly for scalar
functions which are of the form $J(x,p)=J^{\mu}(x)p_{\mu}$:

$\begin{split}\mathcal{D}_{\mu}J&=p_{\nu}\partial_{\mu}J^{\nu}+\Gamma_{\
\mu\nu}^{\lambda}p_{\lambda}J^{\nu}\\\ &=(\partial_{\mu}J^{\nu}+\Gamma_{\
\mu\lambda}^{\nu})p_{\nu}\\\ &=(\nabla_{\mu}J^{\nu})p_{\nu}\end{split}$ (3.16)

The manifestly covariant nature of Poisson bracket can also be extended to
completely symmetric tensors of rank-n by contracting all indices with
$p_{\mu}$’s which yields

$T(x,p)=\frac{1}{n!}T^{\mu_{1}\ldots\mu_{n}}p_{\mu_{1}}\ldots p_{\mu_{n}}$
(3.17)

A similar result can be obtained for antisymmetric tensors using Grassmann
algebra.

###  3.3 Conserved charges

Symmetry of geodesic structure generated by diffeomorphisms associated with
Killing vector fields implies conservation of certain quantity under geodesic
flow. That quantity is obtained through contraction of Killing vector field
with momentum

$J[\xi]=\xi^{\mu}p_{\mu}$ (3.18)

which is also a generator of symmetry transformations.

Since this is a scalar quantity, it is invariant under coordinate
transformations. It generates a coordinate transformation to which we are
familiar with

$\delta x^{\mu}=\\{J[\xi],x^{\mu}\\}=-\xi^{\mu}(x)$ (3.19)

which is nothing but the infinitesimal diffeomorphism. Similarly we can also
find out the variation of momentum under this generator which is

$\delta p_{\mu}=\\{J[\xi],p_{\mu}\\}=\partial_{\mu}\xi^{\nu}p_{\nu}$ (3.20)

which is also a coordinate dependent transformation. It can be explicitly
checked that under these two transformation, hamiltonian in (3.3) remains
invariant which means

$\frac{dJ[\xi]}{d\lambda}=\\{J[\xi],H\\}=0$ (3.21)

This result was expected since hamiltonian depends on the coordinates only
through metric which is invariant under diffeomorphism [9] associated with
Killing vector fields.

As we have seen that Killing equations are linear in nature and hence, it
forms a linear vector space. Let choose dimension of the vector space to be
$r$, then any arbitrary Killing vector field on this vector space can be
written in terms of linear combination of basis vectors $\\{e_{i}(x),\
i=1,\ldots,r\\}$

$\xi(\alpha_{i})=\alpha_{1}e_{1}(x)+\ldots\alpha_{r}e_{r}(x)$ (3.22)

Hence, each such diffeomorphism under which metric remains invariant can be
written as a linear combination of $r$ numbers of conserved generators/charges
whose coefficients will also be $\\{\alpha_{i}\\}$s

$J[\xi]=\alpha_{1}J_{1}+\ldots+\alpha_{r}J_{r}$ (3.23)

where $J_{i}(x,p)=e_{i}^{\mu}(x)p_{\mu}$. Now, we show that these generators
form a Lie-algebra under Poisson bracket:

$\begin{split}\\{J_{i},J_{j}\\}&=\\{J[e_{i}],J[e_{j}]\\}\\\
&=\\{e_{i}^{\mu}p_{\mu},e_{j}^{\nu}p_{\nu}\\}\\\
&=e_{i}^{\mu}\\{p_{\mu},e_{j}^{\nu}p_{\nu}\\}+\\{e_{i}^{\mu},e_{j}^{\nu}p_{\nu}\\}p_{\mu}\\\
&=-e_{i}^{\mu}\partial_{\mu}e_{j}^{\nu}p_{\nu}+e_{j}^{\nu}\partial_{\nu}e_{i}^{\mu}p_{\mu}\\\
&=(e_{j}^{\nu}\partial_{\nu}e_{i}^{\mu}-e_{i}^{\nu}\partial_{\nu}e_{j}^{\mu})p_{\mu}\end{split}$
(3.24)

In (2.5) it is shown that the Killing vectors form a Lie-algebra, therefore,

$\begin{split}[e_{i},e_{j}]&=-(e_{j}^{\nu}\nabla_{\nu}e_{i}^{\mu}-e_{i}^{\nu}\nabla_{\nu}e_{j}^{\mu})\\\
&=-(e_{j}^{\nu}\partial_{\nu}e_{i}^{\mu}-e_{i}^{\nu}\partial_{\nu}e_{j}^{\mu})+e_{i}^{\nu}e_{j}^{\lambda}\Gamma_{\
\nu\lambda}^{\mu}-e_{j}^{\nu}e_{i}^{\lambda}\Gamma_{\ \nu\lambda}^{\mu}\\\
&=-(e_{j}^{\nu}\partial_{\nu}e_{i}^{\mu}-e_{i}^{\nu}\partial_{\nu}e_{j}^{\mu})\\\
&=\mathcal{C}_{ij}^{k}e_{k}\end{split}$ (3.25)

which shows

$\\{J_{i},J_{j}\\}=-\mathcal{C}_{ij}^{k}e_{k}^{\mu}p_{\mu}=f_{ij}^{k}J_{k}$
(3.26)

where we denote $f_{ij}^{k}=-\mathcal{C}_{ij}^{k}$.

Since, $\\{J_{i}\\}$ forms a linear vector space with a closed algebra under a
Lie bracket (which is Poisson bracket in this case) which is anti-symmetric
and bilinear in its arguments hence, it is proven that $\\{J_{i}\\}$ form a
Lie-algebra with structure constant $f_{ij}^{k}$.

###  3.4 Conservation laws and algebra of conserved charges

It was established in previous section that associated with each continuous
symmetries generated by Killing vector fields, one can define conservation
laws or conserved quantities (denoted by generators $J$) along geodesics. But
the reverse statement is not true or in other words for each conserved
quantity along geodesic flow there might not be a Killing symmetry, and one
such example is hamiltonian of the system. This raises the next question - are
there conserved quantities, higher order in momenta?

Let define a general phase-space function $J(x,p)$ on geodesic which is non-
singular in momentum variables that means it is possible to express the
quantity in following manner

$J(x,p)=\sum_{k=0}^{\infty}\frac{1}{k!}J^{(k)\mu_{1},\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}}$ (3.27)

where the expansion coefficients $J^{(k)\mu_{1},\ldots,\mu_{k}}$ are
completely symmetric in the interchange of indices. Now we ask under what
circumstatnces

$\frac{dJ}{d\lambda}=\\{J,H\\}=0$ (3.28)

In the leading order, it implies

$\frac{\partial J^{(0)}}{\partial x^{\mu}}=0$ (3.29)

which shows $J^{(0)}$ to a constant which can be redefined and make it zero.

At first order one would find back the Killing equation

$\nabla_{\mu}J_{\nu}^{(1)}+\nabla_{\nu}J_{\mu}^{(1)}=0$ (3.30)

because the term is linear in momentum. For the $k^{\text{th}}$ order
following condition needs to be satisfied

$\begin{split}0&=[J^{(k)\mu_{1},\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}},g^{\mu\nu}p_{\mu}p_{\nu}]\\\
&=-\sum_{i=1}^{k}[J^{(k)\mu_{1},\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}p^{\mu_{i+1}}\ldots
p_{\mu_{k}}\partial_{\mu_{i}}g^{\mu\nu}p_{\mu}p_{\nu}]\\\
&+\partial_{\mu}J^{(k)\mu_{1},\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}}g^{\mu\nu}p_{\nu}+\partial_{\nu}J^{(k)\mu_{1},\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}}g^{\mu\nu}p_{\mu}\\\
&=-\frac{1}{2}\sum_{i=1}^{k}[J_{\mu_{1},\ldots,\mu_{k}}^{(k)\ \mu_{i}\ \
}p^{\mu_{1}}\ldots p^{\mu_{i-1}}p^{\mu_{i+1}}\ldots
p^{\mu_{k}}\partial_{\mu_{i}}g^{\mu\nu}p_{\mu}p_{\nu}]\\\
&+\partial_{\mu}J^{(k)\mu_{1},\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}}g^{\mu\nu}p_{\nu}\\\
&=\sum_{i=1}^{k}[J_{\mu_{1},\ldots,\mu_{k}}^{(k)\ \mu_{i}\ \
}p^{\mu_{1}}\ldots p^{\mu_{i-1}}p^{\mu_{i+1}}\ldots
p^{\mu_{k}}\Gamma_{\alpha\beta\mu_{i}}g^{\alpha\mu}g^{\beta\nu}p_{\mu}p_{\nu}]\\\
&+\partial_{\mu}J^{(k)\mu_{1},\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}}g^{\mu\nu}p_{\nu}\end{split}$ (3.31)

which can be written as follows

$\begin{split}0&=\sum_{i=1}^{k}[J_{\mu_{1},\ldots,\mu_{k}}^{(k)\ \beta\ \
}p^{\mu_{1}}\ldots p^{\mu_{i-1}}p_{\mu_{i+1}}\ldots p^{\mu_{k}}\Gamma_{\
\mu\beta}^{\mu_{i}}g^{\mu\nu}p_{\mu_{i}}p_{\nu}]\\\
&+\partial_{\mu}J^{(k)\mu_{1},\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}}g^{\mu\nu}p_{\nu}\\\ &=p^{\mu}p_{\mu_{1}}\ldots
p_{\mu_{k}}\left(\partial_{\mu}J^{(k)\mu_{1},\ldots,\mu_{k}}+\sum_{i}\Gamma_{\
\mu\beta}^{\mu_{i}}J^{(k)\mu_{1},\ldots,\beta,\ldots,\mu_{k}}\right)\\\
&=p^{\mu}p_{\mu_{1}}\ldots
p_{\mu_{k}}\nabla_{\mu}J^{(k)\mu_{1}\ldots\mu_{k}}\\\ \implies
0&=\nabla_{(\mu}J_{\mu_{1}\ldots\mu_{k})}^{(k)}\end{split}$ (3.32)

These are nothing but the generalized Killing equations and solutions of these
equations are called Killing tensors. Just like Killing vectors, Killing
tensors also act as a generator of transformations but on phase space
variables. Under action of this $x^{\mu}$ changes by

$\delta
x^{\mu}=\\{J(x,p),x^{\mu}\\}=-J^{(1)\mu}-\sum_{k=2}^{\infty}\frac{1}{(k-1)!}J^{(k)\mu\mu_{2}\ldots\mu_{k}}p_{\mu_{2}}\ldots
p_{\mu_{k}}$ (3.33)

Note that these transformations are in general velocity dependent
(specifically momentum dependent) for higher order generators/charges.

Closure of the algebra between charges/generators (which are constants along
geodesic flow) of different order, follows from Jacobi identity of Poisson
bracket

$\begin{split}\\{J^{(k)},J^{(l)}\\}&=\\{J^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}},J^{(l)\nu_{1}\ldots\nu_{l}}p_{\nu_{1}}\ldots p_{\nu_{l}}\\}\\\
&=-\sum_{i=1}^{k}J^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\frac{\partial J^{(l)\nu_{1}\ldots\nu_{l}}}{\partial
x^{\mu_{i}}}p_{\mu_{i+1}}\ldots p_{\mu_{k}}p_{\nu_{1}}\ldots p_{\nu_{l}}\\\
&+\sum_{j=1}^{l}p_{\mu_{1}}\ldots
p_{\mu_{k}}J^{(l)\nu_{1}\ldots\nu_{l}}p_{\nu_{1}}\ldots
p_{\nu_{j-1}}\frac{\partial J^{(k)\mu_{1}\ldots\mu_{k}}}{\partial
x^{\nu_{j}}}p_{\nu_{j+1}}\ldots p_{\nu_{l}}\\\
&\equiv\tilde{J}^{(k+l-1)\sigma_{1}\ldots\sigma_{k+l-1}}p_{\mu_{1}}\ldots
p_{\sigma_{k+l-1}}\\\ \implies\\{J^{(k)},J^{(l)}\\}&\sim
J^{(k+l-1)}\end{split}$ (3.34)

Note that for $l=1$ above equation tell us

$\\{J^{(k)},J^{(1)}\\}\sim J^{(k)}$ (3.35)

which means algebra of charge/generator of any order with charge/generator of
order one (linear in momentum) is closed under Poisson bracket but it is not
true for arbitrary values of $k$ and $l$.

Note also that the algebra shown in (3.34) is similar to Virasoro algebra [10,
11] which is the algebra of infinite dimensional Killing vector space for
2-dimensional conformal field theories [12, 13, 14].

##  4 Hamiltonians linear in momentum

###  4.1 Introduction

A hamiltonian quadratic in momentum naturally follows from Lagrangian which is
quadratic in coordinate velocities through Legendre transformations. But in
certain cases one can approximate a Hamiltonian quadratic in momentum such a
way that only the liner momentum piece contained in it survives ultimately,
one such example is shown here.

Consider a charged particle of charge $q$ on 2-dimensional space in a magnetic
field in z-direction $\vec{B}=B_{0}\hat{z}$ and for sake of simplicity we
consider non-relativistic system. In that case, hamiltonian is given by

$H=\frac{1}{2m}\left(\vec{p}+\frac{q}{c}\vec{A}\right)^{2}=\frac{\vec{p}^{2}}{2m}+\frac{q}{mc}\vec{A}.\vec{p}+\frac{q^{2}}{2mc^{2}}\vec{A}.\vec{A}$
(4.1)

Now we choose circular gauge in which $\vec{A}=\frac{B_{0}}{2}(-y,x)$ which
yields

$H=\frac{\vec{p}^{2}}{2m}+\frac{qB_{0}}{2mc}(xp_{y}-yp_{x})+\frac{q^{2}B_{0}^{2}}{8mc^{2}}(x^{2}+y^{2})$
(4.2)

If we consider the quantity $\frac{qB_{0}}{c}$ which is inverse of some time
scale, is high in magnitude then we can drop the first term (which is the
kinetic term) in the hamiltonian which is valid for heavily charged
astrophysical object moving around Neutron stars or in accretion disk then we
obtain an effective hamiltonian which is of following form

$H_{eff}=\omega_{c}(xp_{y}-yp_{x})+\frac{m\omega_{c}^{2}}{2}(x^{2}+y^{2}),\
\omega_{c}=\frac{qB_{0}}{2mc}$ (4.3)

Note that the Hamilton’s equation for coordinate velocity becomes

$\dot{x}=-\omega_{c}y,\ \dot{y}=\omega_{c}x$ (4.4)

which leads to two simple harmonic oscillator equations

$\ddot{x}=-\omega_{c}^{2}x,\ \ddot{y}=-\omega_{c}^{2}y$ (4.5)

which shows that particle is trapped in closed orbit [15, 16, 17].

If $\frac{p_{i}}{m\omega_{c}x^{i}}\gg 1$ then we can write the effective
hamiltonian as follows

$H_{eff}=\omega_{c}(xp_{y}-yp_{x})$ (4.6)

###  4.2 Conserved charges and their algebra

In this section we start with a general hamiltonian linear in momentum which
is of following form

$H=\zeta^{\mu}p_{\mu}$ (4.7)

where $\zeta^{\mu}$ is a vector field defined over spacetime manifold and it’s
a dimensionful quantity.

As earlier let define a general phase-space function $\mathcal{K}(x,p)$ on
geodesic which is non-singular in momentum variables that means it is possible
to express the quantity in following manner

$\mathcal{K}(x,p)=\sum_{k=0}^{\infty}\frac{1}{k!}\mathcal{K}^{(k)\mu_{1},\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}}$ (4.8)

where the expansion coefficients $\mathcal{K}^{(k)\mu_{1},\ldots,\mu_{k}}$ are
completely symmetric in the interchange of indices. Now we ask under what
circumstatnces

$\frac{d\mathcal{K}}{d\lambda}=\\{\mathcal{K},H\\}=0$ (4.9)

Hence, existence of a conserved charge of rank-k implies

$\begin{split}0&=\\{\mathcal{K}^{(k)},H\\}=\\{\mathcal{K}^{(k)\mu_{1},\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}},\zeta^{\mu}p_{\mu}\\}\\\
&=-\sum_{i=1}^{k}\mathcal{K}^{(k)\mu_{1},\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\frac{\partial\zeta^{\nu}}{\partial
x^{\mu_{i}}}p_{\nu}p_{\mu_{i+1}}\ldots p_{\mu_{k}}\\\
&+\frac{\partial\mathcal{K}^{(k)\mu_{1},\ldots,\mu_{k}}}{\partial
x^{\nu}}\zeta^{\nu}p_{\mu_{1}}\ldots p_{\mu_{k}}\\\
&=-\sum_{i=1}^{k}\mathcal{K}^{(k)\mu_{1},\ldots\nu,\ldots,\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\frac{\partial\zeta^{\mu_{i}}}{\partial
x^{\nu}}p_{\mu_{i}}p_{\mu_{i+1}}\ldots p_{\mu_{k}}\\\
&+\frac{\partial\mathcal{K}^{(k)\mu_{1},\ldots,\mu_{k}}}{\partial
x^{\nu}}\zeta^{\nu}p_{\mu_{1}}\ldots p_{\mu_{k}}\\\ &=p_{\mu_{1}}\ldots
p_{\mu_{k}}\left(\zeta^{\nu}\frac{\partial\mathcal{K}^{(k)\mu_{1},\ldots,\mu_{k}}}{\partial
x^{\nu}}-\sum_{i=1}^{k}\mathcal{K}^{(k)\mu_{1},\ldots\nu,\ldots,\mu_{k}}\frac{\partial\zeta^{\mu_{i}}}{\partial
x^{\nu}}\right)\\\
&=\mathcal{L}_{\zeta}\mathcal{K}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}}\\\
\implies&\mathcal{L}_{\zeta}\mathcal{K}^{(k)\mu_{1}\ldots\mu_{k}}=0\end{split}$
(4.10)

which says mathematically that in order to be a conserved charge/generator
Lie-derivative of $\mathcal{K}^{(k)\mu_{1}\ldots\mu_{k}}$ must be zero along
the vector field $\zeta^{\mu}$. These set of conditions are completely
different from the earlier set of conditions in (3.32). Hence, depending on
the existence of solutions of above set of tensorial equation one can generate
conserved charges of different rank. One of the simplest quantities to look at
are the vector fields $\\{\xi^{(i)}\\}$ for which the Lie-bracket
$[\zeta,\xi^{(i)}]=0$ which gives set of conserved quantities
$\\{\xi^{(i)\mu}p_{\mu}\\}$s.

Note that the algebra of charges of different rank remains same as we have
shown earlier, which is

$\\{\mathcal{K}^{(k)},\mathcal{K}^{(l)}\\}\sim\mathcal{K}^{(k+l-1)}$ (4.11)

##  5 Spinning objects

###  5.1 Introduction

The dynamics of angular momentum and spin [18, 19, 20, 21, 22, 23, 24] of
astrophysical objects plays an important role in the understanding of binary
mergers and recent discovery of gravitational waves [25, 26, 27, 28, 29] makes
it possible to measure these properties in curved spacetime. Here, the
discussion is started by mentioning the Poisson algebra between phase-space
variables for point-like objects [30]. This is an idealization of a compact
body since it neglects details of the internal structure by assigning the
point-like object with overall fixed position, momentum and spin, known as the
spinning-particle approximation. A large variety of models for spinning
particles is found in the literature [31, 32, 33].

###  5.2 Covariant phase-space structure

In order to specify a hamiltonian dynamical system, three ingredients are
required which are the phase space, identifying the dynamical degrees of
freedom, the Poisson brackets between these dynamical degrees of freedoom
defining a symplectic structure [34, 35, 36]. The hamiltonian generates the
evolution of the system with given initial conditions by specifying a curve in
the phase space passing through the initial point. The parametrization of
phase-space is not unique, since changes in the parametrization can be
compensated by redefining the Poisson brackets and the hamiltonian.

We start by defining spin-degrees of freedom, described by an anti-symmetric
tensor $\Sigma^{\mu\nu}$

$S^{\mu}=\frac{1}{2\sqrt{-g}}\varepsilon^{\mu\nu\kappa\lambda}u_{\nu}\Sigma_{\kappa\lambda},\
Z^{\mu}=\Sigma^{\mu\nu}u_{\nu}$ (5.1)

where $u^{\mu}$ is a time-like unit vector satisfies $u^{\mu}u_{\mu}=-1$. By
construction above two quantities satisfy following two conditions

$Z^{\mu}u_{\mu}=0,\ \ S^{\mu}u_{\mu}=0$ (5.2)

which means they are space-like in nature.

The full set of phase-space co-ordinates of a spinning particle can be
constructed using position co-ordinate $x^{\mu}$, the covariant momentum
$p_{\mu}$ and the spin tensor $\Sigma_{\mu\nu}$, with antisymmetric Poisson
brackets

$\begin{split}\\{x^{\mu},p_{\nu}\\}=\delta_{\nu}^{\mu},&\ \
\\{p_{\mu},p_{\nu}\\}=\frac{1}{2}\Sigma^{\kappa\lambda}\mathcal{R}_{\kappa\lambda\mu\nu}\\\
\\{\Sigma^{\mu\nu},p_{\lambda}\\}&=\Gamma_{\
\lambda\kappa}^{\mu}\Sigma^{\nu\kappa}-\Gamma_{\
\lambda\kappa}^{\nu}\Sigma^{\mu\kappa}\\\
\\{\Sigma^{\mu\nu},\Sigma^{\kappa\lambda}\\}&=g^{\mu\kappa}\Sigma^{\nu\lambda}-g^{\mu\lambda}\Sigma^{\nu\kappa}-g^{\nu\kappa}\Sigma^{\mu\lambda}+g^{\nu\lambda}\Sigma^{\mu\kappa}\end{split}$
(5.3)

Note that second Poisson bracket becomes trivial in a limit in which spin of
the system vanishes which is consistent.

It is quite simple task to check that these brackets are indeed closed in the
sense that they satisfy the Jacobi identities, hence a consistent symplectic
structure is defined on the phase space. To have a well-defined dynamical
system we need to complete the phase-space structure with a hamiltonian that
generates proper-time evolution of the system. Here, we choose free-particle
hamiltonian in (3.3).

It can be shown explicitly that the chosen hamiltonian generates the following
set of proper-time evolution equations [30]

$\begin{split}\dot{x}^{\mu}=\\{x^{\mu},H\\}&\implies
p_{\mu}=g_{\mu\nu}\dot{x}^{\nu}\\\ \dot{p}_{\mu}=\\{p_{\mu},H\\}&=\Gamma_{\
\lambda\mu}^{\nu}p_{\nu}\dot{x}^{\lambda}+\frac{1}{2}\Sigma^{\kappa\lambda}\mathcal{R}_{\kappa\lambda\mu}^{\
\ \ \nu}p_{\nu}\\\
\dot{\Sigma}^{\mu\nu}=\\{\Sigma^{\mu\nu},H\\}&\implies\dot{\Sigma}^{\mu\nu}+\Gamma_{\
\lambda\kappa}^{\mu}\dot{x}^{\lambda}\Sigma^{\kappa\nu}+\Gamma_{\
\lambda\kappa}^{\nu}\dot{x}^{\lambda}\Sigma^{\mu\kappa}=0\end{split}$ (5.4)

Substituting first equation into second equation one would obtain

$\ddot{x}^{\mu}+\Gamma_{\
\nu\lambda}^{\mu}\dot{x}^{\nu}\dot{x}^{\lambda}=\frac{1}{2}\Sigma^{\kappa\lambda}R_{\kappa\lambda\
\nu}^{\ \ \mu}\dot{x}^{\nu}$ (5.5)

which reduces to familiar geodesic equation in $\Sigma=0$ limit.

###  5.3 Conserved charges and their algebra

The previous construction of conserved charges does not hold here since, the
presence of $\Sigma^{\mu\nu}$ tensor makes $\\{p_{\mu},p_{\nu}\\}\neq 0$. Here
we start by looking at conditions in order to construct conserved charges
which are of following form

$\mathcal{J}=\sum_{n=0}^{\infty}\mathcal{J}^{(2n)\mu_{1}\nu_{1}\mu_{2}\nu_{2}\ldots\mu_{n}\nu_{n}}(x)\Sigma_{\mu_{1}\nu_{1}}\ldots\Sigma_{\mu_{n}\nu_{n}}$
(5.6)

In order to construct conserved charges of rank-2k, following condition must
be satisfied

$\begin{split}0&=\\{\mathcal{J}_{(2k)\mu_{1}\nu_{1}\mu_{2}\nu_{2}\ldots\mu_{k}\nu_{k}}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}},g^{\mu\nu}p_{\mu}p_{\nu}\\}\\\
&=\frac{\partial\mathcal{J}_{(2k)\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}}{\partial
x^{\mu}}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}}g^{\mu\nu}p_{\nu}\\\
&+\sum_{i=1}^{k}\mathcal{J}_{(2k)\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{i-1}\nu_{i-1}}\\{\Sigma^{\mu_{i}\nu_{i}},p_{\mu}\\}\Sigma^{\mu_{i+1}\nu_{i+1}}\ldots\Sigma^{\mu_{k}\nu_{k}}g^{\mu\nu}p_{\nu}\\\
&=\frac{\partial\mathcal{J}_{(2k)\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}}{\partial
x^{\mu}}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}}g^{\mu\nu}p_{\nu}\\\
&-\sum_{i=1}^{k}\mathcal{J}_{(2k)\mu_{1}\nu_{1}\ldots\mu_{i-1}\nu_{i-1}\lambda\nu_{i}\ldots\mu_{k}\nu_{k}}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{i-1}\nu_{i-1}}\Gamma_{\
\mu\mu_{i}}^{\lambda}\Sigma^{\mu_{i}\nu_{i}}\Sigma^{\mu_{i+1}\nu_{i+1}}\ldots\Sigma^{\mu_{k}\nu_{k}}g^{\mu\nu}p_{\nu}\\\
&-\sum_{i=1}^{k}\mathcal{J}_{(2k)\mu_{1}\nu_{1}\ldots\mu_{i-1}\nu_{i-1}\mu_{i}\lambda\ldots\mu_{k}\nu_{k}}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{i-1}\nu_{i-1}}\Gamma_{\
\mu\nu_{i}}^{\lambda}\Sigma^{\mu_{i}\nu_{i}}\Sigma^{\mu_{i+1}\nu_{i+1}}\ldots\Sigma^{\mu_{k}\nu_{k}}g^{\mu\nu}p_{\nu}\\\
&=\Bigg{[}\frac{\partial\mathcal{J}_{(2k)\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}}{\partial
x^{\mu}}-\sum_{i=1}^{k}\left(\mathcal{J}_{(2k)\mu_{1}\nu_{1}\ldots\mu_{i-1}\nu_{i-1}\lambda\nu_{i}\ldots\mu_{k}\nu_{k}}\Gamma_{\
\mu\mu_{i}}^{\lambda}+\mathcal{J}_{(2k)\mu_{1}\nu_{1}\ldots\mu_{i-1}\nu_{i-1}\mu_{i}\lambda\ldots\mu_{k}\nu_{k}}\Gamma_{\
\mu\nu_{i}}^{\lambda}\right)\Bigg{]}\\\
&\times\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{i-1}\nu_{i-1}}\Sigma^{\mu_{i}\nu_{i}}\Sigma^{\mu_{i+1}\nu_{i+1}}\ldots\Sigma^{\mu_{k}\nu_{k}}g^{\mu\nu}p_{\nu}\\\
\implies&\nabla_{\mu}\mathcal{J}_{(2k)\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}=0\end{split}$
(5.7)

where the transpositions of pairs $(\mu_{i}\nu_{i})$ are symmetric for
$\mathcal{J}_{(2k)}$ and it is anti-symmetric in permutation between
$\mu_{i}\leftrightarrow\nu_{i},\ \forall i$.

Like previous case, here also

$\\{\mathcal{J}_{(2k)},\mathcal{J}_{(2l)}\\}\sim\mathcal{J}_{(2k+2l-2)}$ (5.8)

which follows from the algebra

$\\{\Sigma^{\mu\nu},\Sigma^{\kappa\lambda}\\}=g^{\mu\kappa}\Sigma^{\nu\lambda}-g^{\mu\lambda}\Sigma^{\nu\kappa}-g^{\nu\kappa}\Sigma^{\mu\lambda}+g^{\nu\lambda}\Sigma^{\mu\kappa}$
(5.9)

Like previous case, here also putting $l=1$ makes the algebra closed but not
for any other arbitrary $l$ value which require inclusion of infinite number
higher-rank conserved charges

$\\{\mathcal{J}_{(2k)},\mathcal{J}_{(2)}\\}\sim\mathcal{J}_{(2k)}$ (5.10)

Next, we look at the conserved charges which are of the form
$\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots p_{\mu_{k}}$ which is
rank-k and satisfy

$\begin{split}0&=\\{\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}},g^{\mu\nu}p_{\mu}p_{\nu}\\}\\\
&=\frac{1}{(n+1)!}\nabla_{(\mu}\mathcal{Q}_{(k)\mu_{1}\ldots\mu_{k})}p^{\mu_{1}}\ldots
p^{\mu_{k}}p^{\mu}\\\
&+\sum_{i=1}^{k}\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\\{p_{\mu_{i}},p_{\mu}\\}p_{\mu_{i+1}}\ldots
p_{\mu_{k}}g^{\mu\nu}p_{\nu}\\\
&=\frac{1}{(n+1)!}\nabla_{(\mu}\mathcal{Q}_{(k)\mu_{1}\ldots\mu_{k})}p^{\mu_{1}}\ldots
p^{\mu_{k}}p^{\mu}\\\
&+\frac{1}{2}\sum_{i=1}^{k}\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\Sigma^{\kappa\lambda}\mathcal{R}_{\kappa\lambda\mu_{i}\mu}p_{\mu_{i+1}}\ldots
p_{\mu_{k}}g^{\mu\nu}p_{\nu}\\\
&=\frac{1}{(n+1)!}\nabla_{(\mu}\mathcal{Q}_{(k)\mu_{1}\ldots\mu_{k})}p^{\mu_{1}}\ldots
p^{\mu_{k}}p^{\mu}\\\
&+\frac{1}{2}\sum_{i=1}^{k}\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{i-1}\mu\mu_{i+1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\Sigma^{\kappa\lambda}\mathcal{R}_{\kappa\lambda\mu\mu_{i}}p_{\mu_{i+1}}\ldots
p_{\mu_{k}}g^{\mu_{i}\nu}p_{\nu}\\\
&=\frac{1}{(n+1)!}\nabla_{(\mu}\mathcal{Q}_{(k)\mu_{1}\ldots\mu_{k})}p^{\mu_{1}}\ldots
p^{\mu_{k}}p^{\mu}\\\
&+\frac{1}{2}\sum_{i=1}^{k}\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{i-1}\mu\mu_{i+1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}p_{\mu_{i}}\Sigma^{\kappa\lambda}\mathcal{R}_{\kappa\lambda\mu}^{\
\ \ \mu_{i}}p_{\mu_{i+1}}\ldots p_{\mu_{k}}\\\
\implies&\nabla_{(\mu}\mathcal{Q}_{(k)\mu_{1}\ldots\mu_{k})}=0,\ \
\sum_{i=1}^{k}\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{i-1}\mu\mu_{i+1}\ldots\mu_{k}}\mathcal{R}_{\kappa\lambda\mu}^{\
\ \ \mu_{i}}=0\end{split}$ (5.11)

where the second condition in the last line implies

$\sum_{i=1}^{k}\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{i-1}\mu\mu_{i+1}\ldots\mu_{k}}\mathcal{R}_{\kappa\lambda\mu}^{\
\ \
\mu_{i}}=[\nabla_{\kappa},\nabla_{\lambda}]\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}=0$
(5.12)

As we can see that the inclusion of non-zero spin to the system in curved
spacetime adds further condition on $\mathcal{Q}_{(k)}$ in order to make it
conserved quantity.

This is an important point to notice that unlike previous case here

$\\{\mathcal{Q}^{(k)},\mathcal{Q}^{(l)}\\}\nsim\mathcal{Q}^{(k+l-1)}$ (5.13)

since in presence of non-zero spin

$\\{p_{\mu},p_{\nu}\\}=\frac{1}{2}\Sigma^{\kappa\lambda}\mathcal{R}_{\kappa\lambda\mu\nu}$
(5.14)

which makes the Poisson-bracket between these conserved quantities not closed.

Now we are looking conserved quantities which are mixed both in
$\Sigma^{\mu\nu}$s and $p_{\lambda}$s in following form

$\mathcal{C}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k,l)\lambda_{1}\ldots\lambda_{l}}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}}p_{\lambda_{1}}\ldots
p_{\lambda_{l}}$ (5.15)

which is of rank-$(2k,l)$.

Conservation of above quantities put following conditions on these quantities

$\begin{split}0&=\\{\mathcal{C}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k,l)\lambda_{1}\ldots\lambda_{l}}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}}p_{\lambda_{1}}\ldots
p_{\lambda_{l}},g^{\mu\nu}p_{\mu}p_{\nu}\\}\\\
\implies&\nabla_{(\mu}\mathcal{C}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k};|\lambda_{1}\ldots\lambda_{l})}^{(2k,l)}=0,\
\sum_{j=1}^{l}\mathcal{C}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k,l)\lambda_{1}\ldots\lambda_{j-1}\mu\lambda_{j+1}\ldots\lambda_{l}}\mathcal{R}_{\lambda\kappa\mu}^{\
\ \ \lambda_{i}}=0\\\
&\nabla_{\mu}\mathcal{C}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k,l)\lambda_{1}\ldots\lambda_{l}}=0\end{split}$
(5.16)

In the above set of conditions, first equation refers to covariant derivative
acting on $\\{\lambda_{i}\\}$ indices and third equation refers to covariant
derivative acts on lower indices pairwise $(\mu_{i},\nu_{i})$.

Note that this suggests

$\begin{split}\\{\mathcal{J}_{(2k)},\mathcal{J}_{(2l)}\\}&\sim\mathcal{J}_{(2k+2l-2)}\\\
\\{\mathcal{Q}^{(k)},\mathcal{Q}^{(l)}\\}&\sim\mathcal{Q}^{(k+l-1)}+\mathcal{C}^{(2,k+l-2)}\\\
\\{\mathcal{C}^{(2k,l)},\mathcal{C}^{(2m,n)}\\}&\sim\mathcal{C}^{(2k+2m,l+n-1)}+\mathcal{C}^{(2k+2m-2,l+n)}+\mathcal{C}^{(2k+2m+2,l+n-2)}\end{split}$
(5.17)

which form a closed algebra under Poisson bracket.

Whole procedure can be repeated again in principle for analyzing the case for
hamiltonians linear in momentum.

Importance of finding such constants of motion or conserved quantities is that
they are helpful in the analysis of spinning particle dynamics. An obvious
such conserved quantity is the total spin

$I=\frac{1}{2}g_{\mu\kappa}g_{\nu\lambda}\Sigma^{\mu\nu}\Sigma^{\kappa\lambda}=S_{\mu}S^{\mu}+Z_{\mu}Z^{\mu}$
(5.18)

##  6 Spin coupled to curved spacetime

In this section, we show the conserved quantities for a hamiltonian of a
moving object whose spin is dynamically coupled to curvature of spacetime
(suggested in [30]) in following way

$\begin{split}H&=H_{0}+H_{\Sigma}\\\
H_{0}&=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}\\\
H_{\Sigma}&=\frac{\kappa}{4}\mathcal{R}_{\mu\nu\kappa\lambda}\Sigma^{\mu\nu}\Sigma^{\kappa\lambda}\end{split}$
(6.1)

where $\kappa$ is a dimensionful quantity and strength of it is comparably
small which also measures geodesic deviations of the object in the curved
spacetime. In this case, conditions need to be satisfied by quantities in
order to be conserved charges, become different because additional term of the
hamiltonian which captures spin-curvature coupling.

Now let’s start finding out the conditions for the conserved quantities of
different form in this case. First, start with following kind of quantities

$\begin{split}0&=\\{\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}},H_{0}+H_{\Sigma}\\}\\\
&=p^{\mu}\nabla_{\mu}\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}}\\\
&+\\{\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}},H_{\Sigma}\\}\end{split}$
(6.2)

Now we compute the second term in the above equation, since first term is
already derived earlier.

$\begin{split}\\{\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}&\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}},H_{\Sigma}\\}\\\
&=\frac{\kappa}{2}\\{\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}},\Sigma^{\mu\nu}\\}\mathcal{R}_{\mu\nu\kappa\lambda}\Sigma^{\kappa\lambda}\\\
&=\frac{\kappa}{2}\sum_{i=1}^{k}\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{i-1}\nu_{i-1}}\\{\Sigma^{\mu_{i}\nu_{i}},\Sigma^{\mu\nu}\\}\Sigma^{\mu_{i+1}\nu_{i+1}}\ldots\Sigma^{\mu_{k}\nu_{k}}\mathcal{R}_{\mu\nu\kappa\lambda}\Sigma^{\kappa\lambda}\\\
&=\frac{\kappa}{2}\sum_{i=1}^{k}\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{i-1}\nu_{i-1}}(g^{\mu_{i}\mu}\Sigma^{\nu_{i}\nu}-g^{\mu_{i}\nu}\Sigma^{\nu_{i}\mu}-g^{\nu_{i}\mu}\Sigma^{\mu_{i}\nu}+g^{\nu_{i}\nu}\Sigma^{\mu_{i}\mu})\\\
&\times\Sigma^{\mu_{i+1}\nu_{i+1}}\ldots\Sigma^{\mu_{k}\nu_{k}}\mathcal{R}_{\mu\nu\kappa\lambda}\Sigma^{\kappa\lambda}\\\
&=\frac{\kappa}{2}\sum_{i=1}^{k}\Bigg{[}\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}\mathcal{R}_{\
\nu\kappa\lambda}^{\mu_{i}}\Sigma^{\nu_{i}\nu}+\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}\mathcal{R}_{\
\mu\kappa\lambda}^{\mu_{i}}\Sigma^{\nu_{i}\mu}-\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}\mathcal{R}_{\
\nu\kappa\lambda}^{\nu_{i}}\Sigma^{\mu_{i}\nu}\\\
&-\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}\mathcal{R}_{\
\mu\kappa\lambda}^{\nu_{i}}\Sigma^{\mu_{i}\mu}\Bigg{]}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{i-1}\nu_{i-1}}\Sigma^{\mu_{i+1}\nu_{i+1}}\ldots\Sigma^{\mu_{k}\nu_{k}}\Sigma^{\kappa\lambda}\\\
&=\kappa\sum_{i=1}^{k}\Bigg{[}-\mathcal{J}_{\mu_{1}\nu_{1}\ldots\rho\nu_{i}\ldots\mu_{k}\nu_{k}}\mathcal{R}_{\
\mu_{i}\kappa\lambda}^{\rho}-\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{i}\rho\ldots\mu_{k}\nu_{k}}\mathcal{R}_{\
\nu_{i}\kappa\lambda}^{\rho}\Bigg{]}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{i}\nu_{i}}\ldots\Sigma^{\mu_{k}\nu_{k}}\\\
&=-\kappa\Big{[}[\nabla_{\kappa},\nabla_{\lambda}]\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}\Big{]}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}}\Sigma^{\kappa\lambda}\end{split}$
(6.3)

Hence, in a nutshell the condition put in (6.2) becomes

$\begin{split}p^{\mu}\nabla_{\mu}&\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}-\kappa\Sigma^{\kappa\lambda}\Big{[}[\nabla_{\kappa},\nabla_{\lambda}]\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}\Big{]}=0,\
\forall\ p_{\mu},\ \Sigma^{\kappa\lambda}\\\
\implies&\nabla_{\mu}\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}=0=[\nabla_{\kappa},\nabla_{\lambda}]\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}\end{split}$
(6.4)

Therefore, above two conditions need to be satisfied at same time in order
$\mathcal{J}_{\mu_{1}\nu_{1}\ldots\mu_{k}\nu_{k}}^{(2k)}\Sigma^{\mu_{1}\nu_{1}}\ldots\Sigma^{\mu_{k}\nu_{k}}$
to be conserved charge.

There exist another class of quantities for which, the conditions need to be
imposed in order to make them conserved charges, shown below.

$\begin{split}0&=\\{\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}},H_{0}+H_{\Sigma}\\}\\\
&=\frac{1}{(n+1)!}p^{\mu}\nabla_{(\mu}\mathcal{Q}_{\mu_{1}\ldots\mu_{k})}^{(k)}p^{\mu_{1}}\ldots
p^{\mu_{k}}+\frac{1}{2}\Sigma^{\kappa\lambda}[\nabla_{\kappa},\nabla_{\lambda}]\mathcal{Q}_{\mu_{1}\ldots\mu_{k}}^{(k)}p^{\mu_{1}}\ldots
p^{\mu_{k}}\\\ &+\\{\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}},H_{\Sigma}\\}\end{split}$ (6.5)

Here, we only need to evaluate the following quantity

$\begin{split}\\{\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}&p_{\mu_{1}}\ldots
p_{\mu_{k}},H_{\Sigma}\\}=\frac{\kappa}{2}\\{\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}},\Sigma^{\mu\nu}\\}\mathcal{R}_{\mu\nu\kappa\lambda}\Sigma^{\kappa\lambda}\\\
&=\frac{\kappa}{2}\sum_{i=1}^{k}\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\\{p_{\mu_{i}},\Sigma^{\mu\nu}\\}p_{\mu_{i+1}}\ldots
p_{\mu_{k}}\mathcal{R}_{\mu\nu\kappa\lambda}\Sigma^{\kappa\lambda}\\\
&-\frac{\kappa}{2}\sum_{i=1}^{k}\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\frac{\partial\mathcal{R}_{\mu\nu\lambda\kappa}}{\partial
x^{\mu_{i}}}p_{\mu_{i+1}}\ldots
p_{\mu_{k}}\Sigma^{\mu\nu}\Sigma^{\kappa\lambda}\\\
&=-\frac{\kappa}{2}\sum_{i=1}^{k}\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}(\Gamma_{\ \mu_{i}\rho}^{\mu}\Sigma^{\nu\rho}-\Gamma_{\
\mu_{i}\rho}^{\nu}\Sigma^{\mu\rho})p_{\mu_{i+1}}\ldots
p_{\mu_{k}}\mathcal{R}_{\mu\nu\kappa\lambda}\Sigma^{\kappa\lambda}\\\
&-\frac{\kappa}{2}\sum_{i=1}^{k}\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\frac{\partial\mathcal{R}_{\mu\nu\lambda\kappa}}{\partial
x^{\mu_{i}}}p_{\mu_{i+1}}\ldots
p_{\mu_{k}}\Sigma^{\kappa\lambda}\Sigma^{\mu\nu}\\\
=-\frac{\kappa}{2}\sum_{i=1}^{k}&\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\left(\Sigma^{\mu\nu}\frac{\partial\mathcal{R}_{\mu\nu\lambda\kappa}}{\partial
x^{\mu_{i}}}+\Gamma_{\
\mu_{i}\rho}^{\mu}\Sigma^{\nu\rho}\mathcal{R}_{\mu\nu\kappa\lambda}-\Gamma_{\
\mu_{i}\rho}^{\nu}\Sigma^{\mu\rho}\mathcal{R}_{\mu\nu\kappa\lambda}\right)p_{\mu_{i+1}}\ldots
p_{\mu_{k}}\Sigma^{\kappa\lambda}\\\
=-\frac{\kappa}{2}\sum_{i=1}^{k}\Bigg{[}&\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\left(\frac{\partial(\Sigma^{\mu\nu}\mathcal{R}_{\mu\nu\lambda\kappa})}{\partial
x^{\mu_{i}}}-\Gamma_{\
\mu_{i}\rho}^{\mu}\Sigma^{\rho\nu}\mathcal{R}_{\mu\nu\kappa\lambda}-\Gamma_{\
\mu_{i}\rho}^{\nu}\Sigma^{\mu\rho}\mathcal{R}_{\mu\nu\kappa\lambda}\right)\\\
&\times p_{\mu_{i+1}}\ldots
p_{\mu_{k}}\Sigma^{\kappa\lambda}\Bigg{]}\end{split}$ (6.6)

Let’s define an anti-symmetric rank-2 tensor
$\tilde{\Sigma}_{\kappa\lambda}\equiv\mathcal{R}_{\mu\nu\kappa\lambda}\Sigma^{\mu\nu}$,
then

$\begin{split}\implies\\{\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}&p_{\mu_{1}}\ldots
p_{\mu_{k}},H_{\Sigma}\\}\\\
=-\frac{\kappa}{2}\sum_{i=1}^{k}\Bigg{[}&\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}\left(\Sigma^{\kappa\lambda}\frac{\partial\tilde{\Sigma}_{\kappa\lambda}}{\partial
x^{\mu_{i}}}-\Gamma_{\
\mu_{i}\rho}^{\mu}\Sigma^{\rho\nu}\tilde{\Sigma}_{\mu\nu}-\Gamma_{\
\mu_{i}\rho}^{\nu}\Sigma^{\mu\rho}\tilde{\Sigma}_{\mu\nu}\right)p_{\mu_{i+1}}\ldots
p_{\mu_{k}}\Bigg{]}\\\
=-\frac{\kappa}{2}\sum_{i=1}^{k}\Bigg{[}&\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{i-1}}(\Sigma^{\mu\nu}\nabla_{\mu_{i}}\tilde{\Sigma}_{\mu\nu})p_{\mu_{i+1}}\ldots
p_{\mu_{k}}\Bigg{]}\\\ =-\frac{\kappa}{2}p_{\mu_{1}}\ldots
p_{\mu_{k-1}}&\Bigg{[}\mathcal{Q}^{(k)\lambda\mu_{1}\ldots\mu_{k-1}}\Sigma^{\mu\nu}\nabla_{\lambda}\tilde{\Sigma}_{\mu\nu}+\ldots+\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k-1}\lambda}\Sigma^{\mu\nu}\nabla_{\lambda}\tilde{\Sigma}_{\mu\nu}\Bigg{]}\\\
=-\frac{\kappa}{2}p_{\mu_{1}}\ldots
p_{\mu_{k-1}}&\left(\mathcal{Q}^{(k)\lambda\mu_{1}\ldots\mu_{k-1}}+\ldots+\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k-1}\lambda}\right)\Sigma^{\mu\nu}\nabla_{\lambda}\tilde{\Sigma}_{\mu\nu}\end{split}$
(6.7)

Therefore, altogether we have 3 conditions, need to be imposed

$\begin{split}\nabla_{(\mu}\mathcal{Q}_{\mu_{1}\ldots\mu_{k})}^{(k)}=0,&\
[\nabla_{\kappa},\nabla_{\lambda}]\mathcal{Q}_{\mu_{1}\ldots\mu_{k}}^{(k)}=0\\\
\mathcal{Q}^{(k)(\lambda\mu_{1}\ldots\mu_{k-1})}&\nabla_{\lambda}\tilde{\Sigma}_{\mu\nu}\Sigma^{\mu\nu}=0\end{split}$
(6.8)

in order to $\mathcal{Q}^{(k)\mu_{1}\ldots\mu_{k}}p_{\mu_{1}}\ldots
p_{\mu_{k}}$ be conserved charge. There is another kind of conserved charges
which are mixed in momentum and $\Sigma^{\mu\nu}$. Those can also be obtained
in similar way using the above mathematical procedure.

##  7 Discussion

In Newtonian mechanics, an isolated system of two point particles interacting
under gravity is exactly solvable and the resulting motion is periodic. The
energy and angular momentum are represented by two conserved integrals of
motion which is a celebrated result in physics community. The dynamics of
binaries are nonlinear in nature in General Relativity due to its inherent
construction and hence, it is important to find the conserved quantities in
order to predict the dynamics or to extract information about dynamics in
curved spacetime. Conserved quantities also play a huge role in numerical
relativity in terms of generating new solutions for example, by adding a
specified amount of angular momentum or spin to a solution of the vacuum
Einstein equations, producing a new solution with specified angular momentum
or spin but with only slightly perturbed energy-momentum vector for asymptotic
flat spacetimes.

In this article, we have shown the existence of the conserved quantities in
terms of well-posed partial differential equations in a generic curved
spacetime even with no Killing vectors. Conserved charges discussed in this
article, are not associated with diffeomophism invariance of spacetime or in
other words Killing vector fields. Existence of these conserved charges are
independent of asymptotic structure of spacetime manifold. However, like
Killing equations, there also exist spacetime manifolds for which solutions of
those partial differential equations do not exist and as a result, these
conserved charges don’t exist for dynamical systems in those spacetime
manifolds. Furthermore, we also discuss the closed algebra between these
conserved charges under the Poisson bracket both in free-particle case and
also with non-minimal spin-curvature coupling. It is shown that these
conserved charges for a given dynamical system in curved spacetime, form an
infinite-dimensional Lie-algebra similar to the Virasoro algebra.

##  8 Acknowledgement

SM wants to thank CSIR to support this work through doctoral fellowship.

## References

  * [1] Elizabeth Tito and Vadim Pavlov.  Relativistic motion of stars near rotating black holes.  Galaxies, 6(2):61, 2018.
  * [2] Andrey A Grib and Yuri V Pavlov.  Comparison of particle properties in kerr metric and in rotating coordinates.  General Relativity and Gravitation, 49(6):78, 2017.
  * [3] Chen-Yu Liu, Da-Shin Lee, and Chi-Yong Lin.  Geodesic motion of neutral particles around a kerr–newman black hole.  Classical and Quantum Gravity, 34(23):235008, 2017.
  * [4] Jerome Gariel, MAH MacCallum, G Marcilhacy, and NO Santos.  Kerr geodesics, the penrose process and jet collimation by a black hole.  Astronomy & Astrophysics, 515:A15, 2010.
  * [5] Alessandro Torrielli.  Classical integrability.  Journal of Physics A: Mathematical and Theoretical, 49(32):323001, 2016.
  * [6] Olivier Babelon, Denis Bernard, and Michel Talon.  Introduction to classical integrable systems.  Cambridge University Press, 2003.
  * [7] Cewen Cao.  Classical integrable systems.  In Soliton Theory and Its Applications, pages 152–191. Springer, 1995.
  * [8] Robert M Wald.  General relativity.  Chicago Univ. Press, Chicago, IL, 1984.
  * [9] Natalia Kiriushcheva, SV Kuzmin, C Racknor, and SR Valluri.  Diffeomorphism invariance in the hamiltonian formulation of general relativity.  Physics Letters A, 372(31):5101–5105, 2008.
  * [10] Douglas Lundholm.  The virasoro algebra and its representations in physics.  Report for the course “Lie algebras and quantum groups” at KTH, 2005.
  * [11] Valter Moretti and Nicola Pinamonti.  Quantum virasoro algebra with central charge c= 1 on the horizon of a 2d-rindler spacetime.  arXiv preprint hep-th/0307021, 2003.
  * [12] Joshua D Qualls.  Lectures on conformal field theory.  arXiv preprint arXiv:1511.04074, 2015.
  * [13] Paul Ginsparg.  Applied conformal field theory.  arXiv preprint hep-th/9108028, 1988.
  * [14] Ralph Blumenhagen and Erik Plauschinn.  Introduction to conformal field theory: with applications to string theory, volume 779.  Springer Science & Business Media, 2009.
  * [15] Woojin Kwon, Ian Stephens, John Tobin, Leslie Looney, Zhi-Yun Li, Floris van der Tak, and Richard Crutcher.  Saving early disk formation of young stellar objects from the magnetic braking catastrophe.  arXiv preprint arXiv:1805.07348, 2018.
  * [16] Aziz Rabhi and Constança Providência.  Dense stellar matter with trapped neutrinos under strong magnetic fields.  Journal of Physics G: Nuclear and Particle Physics, 37(7):075102, 2010.
  * [17] A Chashkina and SB Popov.  Magnetic field estimates for accreting neutron stars in massive binary systems and models of magnetic field decay.  New Astronomy, 17(6):594–602, 2012.
  * [18] Achille Papapetrou.  Spinning test-particles in general relativity. i.  In Nonlinear Gravitodynamics: The Lense-Thirring Effect, pages 393–403. World Scientific, 2003.
  * [19] William G Dixon.  Dynamics of extended bodies in general relativity. i. momentum and angular momentum.  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 314(1519):499–527, 1970.
  * [20] Robert Wald.  Gravitational spin interaction.  Physical Review D, 6(2):406, 1972.
  * [21] Bahram Mashhoon.  Particles with spin in a gravitational field.  Journal of Mathematical Physics, 12(7):1075–1077, 1971.
  * [22] O Semerák.  Spinning test particles in a kerr field—i.  Monthly Notices of the Royal Astronomical Society, 308(3):863–875, 1999.
  * [23] Andrew J Hanson and Tullio Regge.  The relativistic spherical top.  Annals of Physics, 87(2):498–566, 1974.
  * [24] K Kyrian and O Semerák.  Spinning test particles in a kerr field–ii.  Monthly Notices of the Royal Astronomical Society, 382(4):1922–1932, 2007.
  * [25] Benjamin P Abbott, Richard Abbott, TD Abbott, MR Abernathy, Fausto Acernese, Kendall Ackley, Carl Adams, Thomas Adams, Paolo Addesso, RX Adhikari, et al.  Observation of gravitational waves from a binary black hole merger.  Physical review letters, 116(6):061102, 2016.
  * [26] Benjamin P Abbott, R Abbott, TD Abbott, MR Abernathy, F Acernese, K Ackley, C Adams, T Adams, P Addesso, RX Adhikari, et al.  Gw151226: observation of gravitational waves from a 22-solar-mass binary black hole coalescence.  Physical review letters, 116(24):241103, 2016.
  * [27] LIGO Scientific, BP Abbott, R Abbott, TD Abbott, F Acernese, K Ackley, C Adams, T Adams, P Addesso, RX Adhikari, et al.  Gw170104: observation of a 50-solar-mass binary black hole coalescence at redshift 0.2.  Physical Review Letters, 118(22):221101, 2017.
  * [28] Benjamin P Abbott, Rich Abbott, TD Abbott, Fausto Acernese, Kendall Ackley, Carl Adams, Thomas Adams, Paolo Addesso, RX Adhikari, VB Adya, et al.  Gw170817: observation of gravitational waves from a binary neutron star inspiral.  Physical Review Letters, 119(16):161101, 2017.
  * [29] Benjamin P Abbott, R Abbott, TD Abbott, F Acernese, K Ackley, C Adams, T Adams, P Addesso, RX Adhikari, VB Adya, et al.  Gw170814: a three-detector observation of gravitational waves from a binary black hole coalescence.  Physical review letters, 119(14):141101, 2017.
  * [30] G d’Ambrosi, S Satish Kumar, and JW van Holten.  Covariant hamiltonian spin dynamics in curved space–time.  Physics Letters B, 743:478–483, 2015.
  * [31] IB Khriplovich and AA Pomeransky.  Gravitational interaction of spinning bodies, center-of-mass coordinate and radiation of compact binary systems.  Physics Letters A, 216(1-5):7–14, 1996.
  * [32] IB Khriplovich and AA Pomeransky.  Equations of motion of spinning relativistic particle in external fields.  In Nonlinear Gravitodynamics: The Lense-Thirring Effect, pages 46–69. World Scientific, 2003.
  * [33] Enrico Barausse, Etienne Racine, and Alessandra Buonanno.  Hamiltonian of a spinning test particle in curved spacetime.  Physical Review D, 80(10):104025, 2009.
  * [34] Giuseppe Marmo and Gaetano Vilasi.  Symplectic structures and quantum mechanics.  Modern Physics Letters B, 10(12):545–553, 1996.
  * [35] AR Bosco de Magalhães, CH d’Ávila Fonseca, and MC Nemes.  Classical and quantum coupled oscillators: symplectic structure.  Physica Scripta, 74(4):472, 2006.
  * [36] Andrea Carosso.  Geometric quantization.  arXiv preprint arXiv:1801.02307, 2018.


#  Solving Major Problems
Using Vector Affine Quantization

John R<EMAIL_ADDRESS>
Department of Physics and Department of Mathematics
University of Florida, Gainesville, FL 32611-8440

###### Abstract

Affine quantization is a parallel procedure to canonical quantization, which
is ideally suited to deal with special problems. Vector affine quantization
introduces multiple degrees of freedom which find that working together create
novel tools suitable to eliminate typical difficulties encountered in more
conventional approaches.

##  1 An Introduction to Affine Quantization

###  1.1 Basics of canonical and affine quantization

The usual variables of classical physics are $p$ and $q$, and a common
Hamiltonian is $H(p,q)=\textstyle{\frac{1}{2}}p^{2}+V(q)$. If, for some reason
$q^{2}>0$, which is an example of our interest, then we prefer to use the
dilation $d=pq$ and $q$. For them, the classical Hamiltonian becomes
§$H^{\prime}(d,q)=\textstyle{\frac{1}{2}}d^{2}/q^{2}+V(q)$. However, canonical
quantization (CQ) requires that $-\infty<p\,\&\,q<\infty$, and promotes its
proper variables to operators, $p\rightarrow P\,(=P^{\dagger})$ and
$q\rightarrow Q\,(=Q^{\dagger})$ with $[Q,P]=i\hbar 1\\!\\!1$. Affine
quantization (AQ) chooses to promote proper variables to operators,
$d=pq\rightarrow D=(P^{\dagger}Q+QP)/2\,(=D^{\dagger})$, $q\rightarrow Q\neq
0$, and $[Q,D]=i\hbar Q$. The reason that $Q\neq 0$ is because if $Q=0$ then
$D=0$ and $P$ can not help. The fact that $Q\neq 0$ leads to either $Q>0$,
$Q<0$, or simply both, $Q\neq 0$. Because $Q$ is ‘incomplete’, the operator
$P^{\dagger}\neq P$, and so $P^{\dagger}$ was used for $D$ so that
$D^{\dagger}=D$ However, since $P^{\dagger}Q=PQ$, we find that $D=(PQ+QP)/2$
as well.

###  1.2 An example where affine quantization succeeds

As an example of why affine variables are useful consider the well known
harmonic operator Hamiltonian, ${\cal
H}_{cq}=(P^{2}+Q^{2})/2=[-\hbar^{2}(d^{2}/dx^{2})+x^{2}]/2$, which
successfully uses CQ. However the half-harmonic oscillator, for which $Q>0$,
requires AQ. Since $P^{\dagger}\neq P$, the former Hamiltonian operator fails.
Using CQ, we are led to ${\cal H}_{0}=(PP^{\dagger}+Q^{2})/2)$ AND ${\cal
H}_{1}=(P^{\dagger}P+Q^{2})/2$.222Additional wrong candidate operators include
$[(P^{\dagger}+P)^{2}/4+Q^{2}]/2$,
$[(2P^{\dagger}-P)(2P-P^{\dagger})+Q^{2}]/2$ , $[(P^{\dagger
4}/P^{2}+P^{4}/P^{\dagger 2})/2+Q^{2}]/2,$ etc., all having the same classical
Hamiltonian! These are TWO different, unequal, solutions, both of which came
from the same classical Hamiltonian, $(p^{2}+q^{2})/2$. This is not acceptable
quantization!

Instead, if the problem uses AQ, and we insist that $q>0$, the classical
Hamiltonian is $(p^{2}+q^{2})/2=(d^{2}/q^{2}+q^{2})/2$, and the new, and
different, quantum Hamiltonian becomes

$\displaystyle{\cal H}_{aq}=(D\,Q^{-2}D+Q^{2})/2$ $\displaystyle\hskip
20.00003pt=[-\hbar^{2}\\{[x(d/dx)+(d/dx)x]\,x^{-2}\,[x(d/dx)+(d/dx)x]\\}/4+x^{2}]/2$
$\displaystyle\hskip
20.00003pt=[-\hbar^{2}(d^{2}/dx^{2})+(3/4)\hbar^{2}/x^{2}+x^{2}]/2\;,$ (1)

which has the very same classical Hamiltonian [1], Sec. 1.5. While the full
harmonic oscillator has equally spaced eigenvalues $\hbar[(0,1,2,..)+1/2))$,
the eigenvalues of the half-harmonic oscillator are $2\,\hbar[(0,1,2,..)+1]$,
which shows a close connection of these two operators [2].

It should be clear that the mathematics of this example would permit us to not
only solve the case for $x>0$, but also for $x<0$, and therefore display two
opposing or different solutions at the same time!

This story is an example of how AQ has solved a problem that CQ cannot solve.
The full harmonic oscillator is solved by CQ, and it fails using AQ.
Evidently, each of these procedures can solve two different sets of problems.

Let us now introduce vector affine quantization (VAQ) as a lead-up to an
illustration of a really difficult problem that VAQ can tackle.

##  2 A First Look at Vector Affine Quantization

Individual vectors have multiple similar elements such as $\overrightarrow{P}$
and $\overrightarrow{Q}$ for CQ systems. For AQ systems there is
$D\equiv\textstyle{\frac{1}{2}}(\overrightarrow{P}^{\dagger}\cdot\overrightarrow{Q}+\overrightarrow{Q}\cdot\overrightarrow{P})\;(=D^{\dagger})$
and $\overrightarrow{Q}\;(=\overrightarrow{Q}^{\dagger})\neq 0$. For the same
reason as before,
$D=\textstyle{\frac{1}{2}}(\overrightarrow{P}\cdot\overrightarrow{Q}+\overrightarrow{Q}\cdot\overrightarrow{P})$
is also correct. This particular set of basic operators is designed to force
$\overrightarrow{Q}\neq 0$. This requirement forces every component of
$Q_{j}=0$, for $j=\\{1,2,3,...N\\}$ for a vector with $N$ components. But.
suppose we choose a different restriction such as
$(\overrightarrow{Q}\cdot\overrightarrow{Q}-1)\neq 0$, which targets an
$N$-dimensional sphere. This requires a different version of D. Since D terms
can be different, let us agree that a d-item pertains to a classical issue,
and a D-item pertains to a quantum issue. So far we have only introduced
D-items.

We now introduce the general scalar D-item to use that aspect along with some
${\cal F}(\overrightarrow{Q})\neq 0$, a scalar partner, such as
$(\overrightarrow{Q}\cdot\overrightarrow{Q}-1)\,((\overrightarrow{Q}\cdot\overrightarrow{Q})^{2}-4)^{2}\neq
0$. Such examples choose to be part of a D-item leading to

$\displaystyle\hskip-30.00005pt{D}=\textstyle{\frac{1}{2}}[\,|\overrightarrow{P}|\,(\overrightarrow{Q}\cdot\overrightarrow{Q}-1)\,((\overrightarrow{Q}\cdot\overrightarrow{Q})^{2}-4)^{2}$
$\displaystyle\hskip
40.00006pt+\,((\overrightarrow{Q}\cdot\overrightarrow{Q})^{2}-4)^{2}\,(\overrightarrow{Q}\cdot\overrightarrow{Q})-1)\,|\overrightarrow{P}|\,]\;,$
(2)

where
$|\overrightarrow{P}|^{2}=\overrightarrow{P}^{\dagger}\cdot\overrightarrow{P}$.
This example offers two spheres that have a common center with one sphere
inside the other one. For this example, there are three distinct regions of
interest: (1) outside of the largest sphere, (2) the region between the two
spheres, and (3) inside the smallest sphere. As there were different options
in Sec. 1.1, we can choose any one region, or any two regions, or all three
regions to use as contributions to the quantization.

We now turn to vector field theory models that can handle even bizarre vector
affine quantization of such examples.

##  3 Vector Affine Quantization of Complicated Field Theory Models

An absurd vector model field theory, which is ripe for nonrenormalizability
using CQ, will be found formally soluble using VAQ. The purpose of studying
this model is to show that this ‘toy model’ can be soluble, then surely, less
complex problems, that are more physically relevant, can surely enjoy even
simpler solubility.

At first we introduce classical vector fields $\overrightarrow{\pi}(x)$ and
$\overrightarrow{\varphi}(x)$ and propose an unusual classical Hamiltonian
model. To make things more simple let us agree that
$\overrightarrow{\pi}(x)^{2}\equiv\overrightarrow{\pi}(x)\cdot\overrightarrow{\pi}(x)$
and
$\overrightarrow{\varphi}(x)^{2}\equiv\overrightarrow{\varphi}(x)\cdot\overrightarrow{\varphi}(x)$.
Just to be crazy, let us choose $p$, the power of the interaction term as
$10<p<\infty$, the number of spatial dimensions $s$ as $7<s<\infty$, and the
coupling constant $g\geq 0$, all as part of the classical Hamiltonian,

$\displaystyle
H(\overrightarrow{\pi},\overrightarrow{\varphi})={\textstyle\int}\\{\;\textstyle{\frac{1}{2}}[\overrightarrow{\pi}(x)^{2}+(\nabla{\overrightarrow{\varphi}}(x)^{2})+m^{2}\,\overrightarrow{\varphi}(x)^{2}]$
$\displaystyle\hskip
80.00012pt+g\,|(\overrightarrow{\varphi}(x)^{2}-1)\,(\overrightarrow{\varphi}(x)^{4}-16)|^{p}\;\\}\;d^{s}\\!x\;.$
(3)

While CQ could solve this problem if $g=0$, it would be impossible to solve it
when $g>0$. Let us embark on our procedures using VAQ.

First we note an important feature about quantization of all such field theory
problems that divergences do not arise because
$\overrightarrow{\varphi}(x)=\infty$. Instead, the field
$\overrightarrow{\varphi}(x)$ appears in some denominator that briefly
vanishes, and with large enough power, i.e., large $p$, integrations – think
path-integration quantization – can yield divergencies. Our first job is to
introduce a d-item, using
$|\overrightarrow{\pi}(x)|^{2}=\overrightarrow{\pi}(x)^{2}$, and given by

$\displaystyle
d(x)=|\overrightarrow{\pi}(x)|\,(\overrightarrow{\varphi}(x)^{2}-1)\,(\overrightarrow{\varphi}(x)^{4}-16)\;,$
(4)

which is a well-designed d-item. To protect $d(x)$ this process requires that
both $(\overrightarrow{\varphi}(x)^{2}-1)\neq 0$ and
$(\overrightarrow{\varphi}(x)^{4}-16)\neq 0$. These two equations define two
spheres with the same center, and, in the present case, physics tells us to
adopt all three sections of open regions.

Next, we modify the classical Hamiltonian to become

$\displaystyle\hskip-30.00005ptH^{\prime}(d,\overrightarrow{\varphi})={\textstyle\int}\\{\textstyle{\frac{1}{2}}[\\![\,d(x)^{2}\,[(\overrightarrow{\varphi}(x)^{2}-1)\,(\overrightarrow{\varphi}(x)^{4}-16)]^{-2}$
$\displaystyle+(\nabla{\overrightarrow{\varphi}}(x)^{2})+m^{2}\,\overrightarrow{\varphi}(x)^{2}]\\!]+g\,|(\overrightarrow{\varphi}(x)^{2}-1)\,(\overrightarrow{\varphi}(x)^{4}-16)|^{p}\\}\;d^{s}\\!x\;,$
(5)

which now requires that the interaction term
$|(\overrightarrow{\varphi}(x)^{2}-1)\,(\overrightarrow{\varphi}(x)^{4}-16)|^{p}\neq
0$. So far, this discussion has focussed on the classical story. Now we turn
to the quantum story.

The basic operators involved, again with
$|\hat{\overrightarrow{\pi}}(x)|^{2}=\hat{\overrightarrow{\pi}}(x)\cdot\hat{\overrightarrow{\pi}}(x)$,
are

$\displaystyle\hskip-20.00003pt\hat{D}(x)=\textstyle{\frac{1}{2}}\,\\{|\hat{\overrightarrow{\pi}}(x)|\,(\hat{\overrightarrow{\varphi}}(x)^{2}-1)(\hat{\overrightarrow{\varphi}}(x)^{4}-16)$
$\displaystyle\hskip
30.00005pt+(\hat{\overrightarrow{\varphi}}(x)^{4}-16)(\hat{\overrightarrow{\varphi}}(x)^{2}-1)\,|\hat{\overrightarrow{\pi}}(x)|\,\\}$
(6)

and $\hat{\overrightarrow{\varphi}}(x)$ has its several restrictions, namely
$\hat{\overrightarrow{\varphi}}(x)^{2}\neq 1$ and
$\hat{\overrightarrow{\varphi}}(x)^{2}\neq 4$.. Note that $\hat{D}(x)$ can be
zero thanks to $|\hat{\overrightarrow{\pi}}(x)|=0$.

The quantum Hamiltonian is given by

$\displaystyle\hskip 0.0pt{\cal
H}^{\prime}(\hat{D},\hat{\overrightarrow{\varphi}})={\textstyle\int}\\{\,\textstyle{\frac{1}{2}}[\\![\hat{D}(x)\,[(\hat{\overrightarrow{\varphi}}(x)^{2}-1)(\hat{\overrightarrow{\varphi}}(x)^{4}-16)]^{-2}\,\hat{D}(x)$
(7) . $\displaystyle\hskip
30.00005pt+(\nabla{\hat{\overrightarrow{\varphi}}}(x)^{2})+m^{2}\,\hat{\overrightarrow{\varphi}}(x)^{2}]\\!]+g\,|(\hat{\overrightarrow{\varphi}}(x)^{2}-1)\,(\hat{\overrightarrow{\varphi}}(x)^{4}-16)|^{p}\\}\;d^{s}\\!x\;.$

First, it follows that

$\displaystyle
0<\\{|(\hat{\overrightarrow{\varphi}}(x)^{2}-1)|\times|(\hat{\overrightarrow{\varphi}}(x)^{4}-16)|\\}^{-2}<\infty\;,$
(8)

and, for $1<p<\infty$, then it also follows that

$\displaystyle
0<\\{\,|(\hat{\overrightarrow{\varphi}}(x)^{2}-1)|\times|(\hat{\overrightarrow{\varphi}}(x)^{4}-16)|\,\\}^{p}<\infty\;,$
(9)

a fact which guarantees that the quantization, although naturally quite
complicated, would lead to a valid result, as it did for the half-harmonic
oscillator.

##  4 A Homework Problem to
Sharpen Your Understanding

Following the procedures above, please test your understanding by rendering a
more natural field theory given by

$\displaystyle
H(\pi,\varphi)={\textstyle\int}\\{\,\textstyle{\frac{1}{2}}[\,\overrightarrow{\pi}(x)^{2}+(\nabla{\overrightarrow{\varphi}}(x))^{2}+m^{2}\,\overrightarrow{\varphi}(x)^{2}\,]+g\,(\overrightarrow{\varphi}(x)^{2})^{p}\,\\}\;d^{s}\\!x$
(10)

for all $p\geq 8$ and $s\geq 6$. This analysis points toward finding quantum
operators that are designed to formulate a physically correct quantization.

Other efforts will be needed to provide desired solutions of the appropriate
equations. To give you hope, reread the story of the half-harmonic oscillator
in Sec. 1.2, which found its correct formulation and correct solution using
AQ.

##  5 Why These Procedures Have Affine Roots

Equation (4) can be reexamined as

$\displaystyle
d(x)=|\overrightarrow{\pi}(x)|\,(\overrightarrow{\varphi}(x)^{2}-1)\,(\overrightarrow{\varphi}(x)^{4}-16)\equiv|\overrightarrow{\pi}(x)|\,w(x)\;,$
(11)

in which $w(x)=0$ implies that
$(\overrightarrow{\varphi}(x)^{2}-1)\,(\overrightarrow{\varphi}(x)^{4}-16)\;=0$.333It
is somewhat extraordinary to imagine that removing a single point in one
continuous function it automatically removes two whole spheres in another
continuous function. The pair of classical variables,
$d(x)=|\hat{\overrightarrow{\pi}}(x)|\,w(x)$ and $w(x)$, for each $x$, are
simply like $d=pq$ and $q$.

While classical variables for CQ, namely $p\,\&\,q$, are promoted to quantum
operators, it is required that they be Cartesian variables, specifically that
$d\sigma^{2}=A^{-1}dp^{2}+A\,dq^{2}$, in order to get physically correct
operators [3]. On the other hand, to get physically correct quantum operators,
the classical variables for AQ, namely $d=pq$ and $q\neq 0$, are required to
be from a constant negative curvature, such as
$d\sigma^{2}=B^{-1}q^{2}\,dp^{2}+B\,q^{-2}\,dq^{2}$, where the positive
constant $B$ determines the magnitude of the curvature [1], Sec. 1.4. This
relation requires that $q^{2}>0$, which then applies to either $q>0$, $q<0$,
or both, $q\neq 0$.

For our principal topic, the classical variables,
$d(x)=|\overrightarrow{\pi}(x)|\,w(x)$ and $w(x)\neq 0$ must belong to
constant negative curvatures to promote physically correct quantum operators,
which arises from the metric

$\displaystyle
d\sigma^{2}={\textstyle\int}\\{C(x)^{-1}\;w(x)^{2}\;d|\overrightarrow{\pi}(x)|^{2}+C(x)\;w(x)^{-2}\;dw(x)^{2}\;\\}\;d^{s}\\!x\;,$
(12)

in which $C(x)>0$ determines the magnitude of the constant negative curvatures
for each $x$, and all of $w(x)>0$ and $w(x)<0$ are included in this specific
analysis. The fact that our model has
$(\nabla{\hat{\overrightarrow{\varphi}}}(x))^{2}$ forces an effective
continuity of $w(x)$ along spheres.

##  6 Summary

Canonical quantization tackles many field theory problems with a fixed
procedure. On the other hand, this version of vector affine quantization
allows for a carefully-designed operator retooling in order to tackle a large
variety of Hamiltonians to ensure quantization succeeds as already was the
case in properly quantizing the half-harmonic oscillator. Additional articles
featuring affine quantization can be found in [4, 5, 6].

Having seen what procedures were used, the reader is welcome to carry out a
VAQ of their own favorite, unsolved, quantum field theory problem.

## References

  * [1] J. Klauder, “Quantum Gravity Made Easy”. Journal of High Energy Physics, Gravitation and Cosmology 6, 90-102 (2020); DOI:10.4236/jhepgc.2020.61009.
  * [2] L. Gouba, “Affine Quantization on the Half Line”, Journal of High Energy Physics, Gravitation and Cosmology 7, 352-365 (2021); DOI:10.4236/jhepgc.2021.71019.
  * [3] P.A.M. Dirac, The Principles of Quantum Mechanics, (Claredon Press, Oxford, 1958), page 114, in a footnote.
  * [4] J. Klauder, “The Benefits of Affine Quantization”, Journal of High Energy Physics, Gravitation and Cosmology 6, 175-185 (2020); DOI;10.4236/jhepgc.2020.62014.
  * [5] J. Klauder, “Using Affine Quantization to Analyze Non-Renormalizable Scalar Fields and the Quantization of Einstein’s Gravity”, Journal of High Energy Physics, Gravitation and Cosmology 6, 802-816 (2020); DOI:10.4236/jhepgc.2020.64053.
  * [6] J. Klauder, “Using Coherent States to Make Physically Correct Classical-to-Quantum Procedures That Help Resolve Nonrenomalizable Fields Including Einstein’s Gravity”, Journal of High Energy Physics, Gravitation and Cosmology 7, 1019-1026 (2021); DOI:10.4236/jhepgc.2021.73060.