# Physical Significance of Noether Symmetries

Asghar Qadir  Abdus Salam School of Mathematical Sciences (ASSMS) Government
College University, 68B New Muslim Town, Lahore, Pakistan
<EMAIL_ADDRESS>Ugur Camci  Department of Chemistry and Physics, Roger
Williams University, One Old Ferry Road, Bristol, Rhode Island 02809, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we will trace the development of the use of symmetry in
discussing the theory of motion initiated by Emmy Noether in 1918. Though it
started with its use in Classical Mechanics, and has been heavily used in
engineering applications of Mechanics, it came into its own in Relativity, and
Quantum Theory and their applications in Particle Physics and Field Theory. It
will be beyond the scope of this article to explain the Quantum Field Theory
applications in any detail, but the base for understanding it will be provided
here. We will also go on to discuss an insight from some more mathematical
developments related to Noether symmetry.

###### pacs:

04.30, 04.30.Nk, 04.50.+h, 98.70.Vc

##  I Introduction

Physics started as a study of motion in Greek times and was formalized by
Aristotle (c. 350 BC) by a set of “laws”, which he declared as “self-evident
truths”, based on his view of the Universe as it was then visualized. The law
for motion on the Earth was based on the nature of the object moving (how much
of Earth, Water, Air or Fire there is in it), and for motion in the heavens on
the “truth” that heavenly objects are made of the perfect element, aether, and
hence move along perfect circles, unless they are contaminated by proximity to
the Earth, in which case epicycles (perfect circles about a point moving in a
perfect circle, or further repetitions thereof) develop. The closer the object
to the Earth, the more epicycles it will have. Using this law the motion of
the then known “planets”: the Sun, Moon, Venus, Mars, Mercury, Jupiter and
Saturn, were supposedly explained using 127 epicycles. The number was reduced
to 57 by Ptolemy (c. 150 AD) by extending from motion in a plane to motion in
three-dimensional space and using spheres in place of circles. The Muslim
scholars followed the same way of thinking, but Al-Zarkali (c. 1050 AD) was
ready to use ellipses instead of perfect circles. Later this work, and the
astronomical data of Ulugh Beg (c. 1420 AD), led Nicholas Copernicus to
heliocentric planetary orbits nc  in 1543 AD, with the Earth replacing the Sun
as a planet. Johannes Kepler jk  replaced Copernicus’ circular orbits by
ellipses in 1619, which finally led Isaac Newton to his laws of motion and
universal gravitation in  in 1687. While Newton’s law is supposedly universal,
his methods work well only for two bodies and become unwieldy for several
bodies. The method was extended by Joseph Louis Lagrange in 1778 and 1779,
jll1  and later used by William Rowan Hamilton wrh ; wrh2  in 1834 and 1835,
for systems of particles. Newton had used Calculus for his purpose and
Lagrange used the Calculus of Variations that Leonhard Euler le  had fully
developed by 1773. It was this formulation that Emmy Noether had used, which
completed the classical view and led to the modern view of Mechanics.

Emmy Noether (1882-1935) was a German female - and in those days German
females could not enter academics. However, she had remarkable mathematical
capability and was able to get the support of David Hilbert to work in the
field unofficially. Finally she had to emigrate to America to achieve her true
potential. She had numerous contributions in various branches of Mathematics,
but we are concerned with her contribution to the use of symmetry in
Mechanics.

In normal parlance “symmetry” is used in an aesthetic sense to express balance
and harmony, as in William Blake’s poem, The Tyger:
“Tyger, tyger burning bright;
In the forests of the night;
What immortal hand or eye
Could frame thy fearful symmetry?”
Almost as common is its use in the geometric sense of leaving a figure or
shape invariant under some transformation, such as reflection or rotation. It
is odd, however, that the common use has such a strong hold that many do not
realise that one can count the symmetries of objects. (AQ has even known an
expert in Differential Topology to objet to the idea of counting symmetries.)

The geometrical concept of symmetry had come from the Greeks. However, while
considering solutions of polynomial equations of degree 5 or more in 1771,
Lagrange extended the concept to invariance of polynomials under permutations
of its elements jll2 . While this led to many other developments in Algebra, I
am here concerned with its use by Abel nha  and Galois eg  to invent Group
Theory, so as to prove that there could be no solution of quintic or higher
degree polynomial equations by means of radicals. In particular, Abel’s work
led to the Abelian Group and Galois’ to the Galois Group. This inspired Sophus
Lie (1842-1899) to try to emulate the success of Abel and Galois for
differential equations in 1883 sl1 . Note the leap over all other algebraic
equations to reach out to all differential equations. This was overambitious
and Lie never managed to complete the attempt. Nevertheless, it led to
enormous developments in the solution of nonlinear differential equations.
This will be discussed in the next section.

One might think that Geometry and Dynamics had no contact till the time that
Albert Einstein and Marcel Grossmann used Geometry to generalize the
Restricted Theory of Relativity aq1 , but that is not the case at all. As
pointed out by Julian Barbour jbb , starting in antiquity and going through
Copernicus, Kepler, Galileo and Newton, Kinematics and then Dynamics, have
been inextricably entwined with Geometry. At the base of the link between them
is the idea of symmetry. Aristotle insisted on perfect circles because he
perceived them as the most symmetric figures possible. Ellipses, to the
contrary, were perceived as imperfect, and hence not to be used for celestial
motion. It took a deeper, hidden, symmetry for the ellipses to be perceived as
“beautiful” by Kepler. We will see how the hidden symmetry was uncovered by
Noether. This symmetry could not have been understood till the advent of
calculus and it was the Geometry that used Calculus, Differential Geometry,
that Einstein and Grossmann introduced into considerations of Dynamics. We
will be concerned with the importance of the not-so-obvious symmetries that
have become all important in Modern Physics.

The plan of the paper is as follows. In the next section we will briefly
review Lie’s Symmetry Analysis and go on, in the subsequent sections three and
four, to review Euler’s variational principle for particles and fields,
respectively, and its use by Lagrange and Hamilton in the principle of least
action and the equations of motion of Hamilton and of Euler and Lagrange. A
geometrical application is given in section five, to generalize the concept of
a straight line in flat space to curved spaces. The applications of the
Noether’s theorem in Classical Mechanics, Economics, Classical Field Theory,
Relativistic Field Theory, will be discussed in the section six. Some
extensions in obtaining Noether symmetries and Noether invariants will also be
given in this section. Some applications of Quantum Field theory will be given
in the next section. Complex Lie and Noether symmetries will be considered in
the section eight, and a discussion and conclusion presented in the last
section.

##  II Lie Symmetry Analysis

Before Lie, the usual method to solve a differential equation (DE) was by ad-
hoc approaches or by approximating it by a linear DE and solving that. In
general the approximation will work well enough in some domain and become
arbitrarily bad in others. Thus one would need to prove the existence of a
solution and to determine the domain in which the approximation is good
enough. Since these will be different for each DE, one is reduced to solving
one DE at a time and cannot rely on any method for whole classes of DEs. Among
the methods that had been used for solving nonlinear DEs was the
transformation of independent and dependent variables. Analogous to Abel and
Galois, Lie looked for invariance of the DE under such transformations sl1 ;
sl2 ; sl3 ; sl4 , so that it could be determined when the DEs could be solved,
or their order reduced, by transformation, and then proceed with the
transformation. Lie used not only the groups of symmetries, but the algebra of
the corresponding infinitesimal symmetry generators. The DEs are not
necessarily single (scalar) but could be systems of (vector) DEs. Further, he
did not restrict the domain of the DEs to be real, but took it to be complex.

For completeness we start with basic definitions so as to present the notation
used. If there are $l$ independent variables represented as a vector ${\bf x}$
and $m$ dependent variables represented by ${\bf y}$, a Lie point symmetry
generator is the operator

${\bf X}={\bf A}({\bf x},{\bf y})\cdot{\bf\nabla}_{{\bf x}}+{\bf B}({\bf
x},{\bf y})\cdot{\bf\nabla}_{{\bf y}}\,,$ (1)

or using indices $a$ for the independent variables and $i$ for the dependent
variables

${\bf X}=A^{a}(x^{b},y^{i})\frac{\partial}{\partial
x^{a}}+B^{i}(x^{b},y^{j})\frac{\partial}{\partial
y^{i}}\,,\,(a,b,...=1,...,l;i,j,...=1,...m)\,,$ (2)

where the Einstein summation convention, that repeated indices are summed
over, has been used. Further, if the DE is of order $n$, one must prolong the
space and the generators to incorporate all the derivatives of the dependent
variables with respect to the independent variables. For ordinary differential
equations (ODEs)

${\bf X}^{[n]}=A(x,y^{j})\frac{\partial}{\partial
x}+B^{i}(x,y^{j})\frac{\partial}{\partial
y^{i}}+B^{i\,[1]}(x,y^{j},{y^{j}}^{\prime})\frac{\partial}{\partial
y^{i}}+...\,,\,\,(a=1,...,l;i=1,...m)\,,$ (3)

where

$B^{i\,[k]}=D_{x}B^{i\,[k-1]}-{y^{i}}^{\prime}D_{x}A\,,\quad(k=1,...,m)\,,$
(4)

$B^{i\,[0]}$ simply being $B^{i}$, and $D_{x}$ is the total derivative in the
prolonged space,

$D_{x}=\frac{\partial}{\partial x}+{y^{i}}^{\prime}\frac{\partial}{\partial
y^{i}}+...+y^{i(k)}\frac{\partial}{\partial y^{i(k-1)}}\,.$ (5)

For partial differential equations (PDEs), $A$ has to be replaced by ${\bf A}$
and $\partial/\partial x$ by $\nabla_{\bf x}$ in Eq. (3). While the former is
easily converted to index notation as $A^{a}$ and $\partial/\partial x^{a}$ in
Eq. (3), for the latter one has to write $y^{i\,[k]}$, which is the partial
derivative with respect to all $x^{a}$ to all orders up to $k$.

The set of all prolonged symmetry generators, ${\bf X}_{q}\,(q=1,...,p)$,
forms a $p$-dimensional Lie algebra, which determines what reduction there can
be of the DE. It is a Lie algebra if the commutators of the symmetry
generators satisfy $[{\bf X}_{r},{\bf X}_{s}]=C^{q}_{rs}{\bf X}_{q}$ where
$C^{q}_{rs}$ are constants, called structure constants, that determine the
structure of the Lie algebra. The generators must also satisfy the Jacobi
identities,

$[[{\bf X}_{[i},X_{j}],X_{k]}]=0~{}.$ (6)

The square brackets in the subscript denote a skew linear combination,
signifying that the terms with the suffices in an even permutation are
positive and those in an odd permutation negative, so that interchanging any
two indices reverses the sign of the expression. The total expression is
divided by the factorial of the number of indices involved. A system of $m$
ODEs of order $n$,

$E^{i}(x,y^{j},{y^{j}}^{\prime},...,y^{j\,[n]})=0,$

is said to be symmetric under the transformation generated by $\bf{X}$, if
${\bf X}^{[n]}E^{i}=0$, when restricted to the solutions of $E^{i}=0$, which
is denoted by ${\bf X}^{[n]}E^{i}=0|_{{\bf E}=0}$. The generalization to PDEs
is as before, with the corresponding complications.

It is worth pondering the prolonged space. For a scalar ODE there are only two
variables, the independent and dependent, and so the manifold considered is
only two-dimensional. However, for a DE we have to treat the derivative as
unknown, for if it were known the DE would already be reduced or solved. Hence
it must be treated as another dependent variable, leading to a three-
dimensional manifold. If the DE is of second order, we would need to also
treat the second derivative as an independent variable. Thus, for an $n^{\rm
th}$ order ODE, we need to use an $(n+2)-$ dimensional space. This is the
prolonged or extended space, also called a jet space. For an $m$-dimensional
system of $n^{\rm th}$ order ODEs we need an $(nm+m+1)-$dimensional space. For
a PDE of $l$ independent variables the dimension is $(nml+m+l)$. Since the
largest group acting on an $n-$dimensional manifold is $GL(n)$, that would be
the upper limit for the algebra. In fact, since the algebra of infinitesimal
generators must leave the group identity out, and only add to it, the relevant
group would be $SL(n)$ and the corresponding algebra $sl(n)$, which has
$n(n-1)/2$ generators. Thus the dimension for the general PDE mentioned is
$(mnl+m+l)(mnl+m+l-1)/2$. Obviously, the dimension of the manifold rapidly
becomes unwieldy with increase in order, and the number of independent and
dependent variables. Even for a second order, two dimensional PDE of two
variables the dimension is $66$, and this ignores other complications arising
for PDEs, which will be discussed later. This is what makes it necessary to
use algebraic computing programmes.

##  III The Variational Principle for Particles

The laws of refraction had been explained by assuming that light takes minimum
time to go from a point in one medium to a point in the other medium, taking
into account the change in its speed in the media. This principle was used by
Lagrange jll1  for his extension of Newton’s mechanics. He wrote Newton’s
mechanics for a system of particles as if it were a single particle in a
higher dimensional space. For $N$ particles, Newton’s laws would be written in
terms of their positions, ${\bf r}_{i}(t)$, and velocities
${\bf\dot{r}}_{i}(t)$ ($i=1,...,N$). Lagrange wrote them as
$(q^{a}(t),\dot{q}^{a}(t)),(a=1,...,3N)$. This change makes it possible to
also incorporate m constraints between the generalized coordinates, so that
the dimension of the space for the single particle is $n=3N-m$. He then
required that the free energy, i.e. the difference between the kinetic and
potential energy, be minimum, over the entire motion. This is called the
principle of least action.

Lagrange’s methods have been used in areas far removed from Mechanics, like
Economics. The function to be minimized (with or without constraints) is
called the Lagrangian, $L[t,q^{i}(t),\dot{q}^{i}(t)]$, and the action is
defined as the total of the minimized quantity over the time period from the
initial time, $t_{i}$, to the final time, $t_{f}$ :

$S\left[q^{i}(t),\dot{q}^{i}(t)\right]=\int_{t_{i}}^{t_{f}}\mathcal{L}\left[t,q^{i}(t),\dot{q}^{i}(t)\right]dt\qquad(i=1,...,n).$
(7)

In Economics, the $q^{i}(t)$ would represent the quantity of a commodity and
its time rate of change would be directly related to the price of that
commodity in the market. The Lagrangian would be the cost for all things
bought at the time, and the action would be the total money spent. It is
useful to think of this analogy for Mechanics. In that case the money spent at
one instant is the free-energy and the total money spent is the total energy
spent. The object “wants” to spend the least energy to get from the start to
the end, and “chooses” the path that will do this.

The original mechanical application of Lagrange’s formalism regarded the
Lagrangian as the difference between the kinetic and potential energy

$\mathcal{L}\left[t,q^{i}(t),\dot{q}^{i}(t)\right]=T(\dot{q}^{i}(t))-V(q^{i}(t))\,,$
(8)

where the explicit time-dependence is absent. This was because in the system
of the Sun and planets the gravitational potential remained constant, and
there was no meaning to the kinetic energy changing with time. The Economic
analogy replaces the kinetic energy by the profit and the potential energy by
the loss. In Economics, we are familiar with money “evaporating” due to
inflation. Similarly, the potential can be made time-dependent and energy can
be lost to friction, or be radiated away. Hence there should be explicit time
dependence. Not only may the system lose energy, it could gain energy. The
corresponding phenomenon in Economics is “deflation”, where the money
appreciates in value. One might think that would be desirable, but it leads to
the Economy slowing down. In Physics, one gets more energy, but it is unusable
as it is thermalized; it would lead to a “heat death”.

For the action functional, $S$, to be minimal when we vary the functions that
are its arguments by $\delta q^{i},\delta\dot{q}^{i}$, it must be unchanged,
i.e. $\delta S=0$. Writing this explicitly,

$\delta S=\int_{t_{i}}^{t_{f}}\left[\frac{\partial\mathcal{L}}{\partial
q^{i}}\delta
q^{i}+\frac{\partial\mathcal{L}}{\partial\dot{q}^{i}}\delta\dot{q}^{i}+\frac{\partial\mathcal{L}}{\partial
t}\delta t\right]dt=0\,.$ (9)

If we require that the initial and final positions of the system of particles
are fixed, then $\delta q^{i}(t_{i})=\delta q^{i}(t_{f})=0$. Writing the
variations in the integrand in terms of $\delta q^{i}$ and integrating the
$\delta\dot{q}^{i}$ by parts using the above boundary conditions,

$0=\int_{t_{i}}^{t_{f}}\left[\frac{\partial\mathcal{L}}{\partial
q^{i}}-\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot{q}^{i}}\right)+\frac{1}{\dot{q}^{i}}\frac{\partial\mathcal{L}}{\partial
t}\right]\delta q^{i}dt\,.$ (10)

Since the $\delta q^{i}$ in the integrand is arbitrary and the integral is
zero the rest of the integrand must be zero. This gives the time-dependent
Euler-Lagrange (EL) equations for a system of particles

$\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot{q}^{i}}\right)=\frac{\partial\mathcal{L}}{\partial
q^{i}}+\frac{1}{\dot{q}^{i}}\frac{\partial\mathcal{L}}{\partial t}\,.$ (11)

It is easily checked that by the EL equations,
$(\dot{q}^{i}\partial\mathcal{L}/\partial\dot{q}^{i}-\mathcal{L})^{{}^{.}}=\partial\mathcal{L}/\partial
t$. Assuming that there is no explicit time dependence in the Lagrangian,
$(\dot{q}^{i}\partial{{\mathcal{L}}}/\partial\dot{q}^{i}-{\mathcal{L}})$ is a
conserved quantity, which is the energy, and is called the Hamiltonian.

So far only a system of particles has been considered. In earlier times a
fluid was generally regarded as distinct from particles, though people like
Democritus (c. 460-370 BC) argued that even water must finally be particulate.
With Robert Boyle (1627-1691 AD) and John Dalton (1766-1844 AD) the idea was
lifted out of the realms of Philosophy to a scientific base in Chemistry.
Later, Daniel Bernoulli db  treated the flow of fluids mathematically as a
nearly infinite system of particles. The term “molecule” was coined much later
by Amedeo Avogadro aa  and defines our modern view of material fluids. He (and
his brother Johann, who claimed that he had written the work prior to Daniel)
sent their work to Leonhard Euler.

As the third term comes from the explicit time-dependence of the Lagrangian,
if it is positive it corresponds to dissipation of energy in Physics, such as
the velocity dependent friction. For the Universe as a whole, i.e. in the
cosmological context, it corresponds to the energy getting absorbed into the
expanding spacetimes, like water into an expanding sponge. In Economics it
would be the money “evaporating” by inflation. To the contrary, if it is
negative, it corresponds to absorption of energy from the environment in
Physics and to energy getting “squeezed out” of the spacetime, as from a
sponge. In Economics it corresponds to deflation. If the analogy holds, we can
expect the cosmological deflation to “squeeze out” thermal radiation. Instead
of merely being crushed to death, the Universe would be broiled and crushed to
death.

##  IV The Variational Principle for Fields

Even if all material objects, including fluids like water and air, are made of
particles, there are so many particles that one might as well take them to be
infinitely many. Even that would not provide us with the full power of
calculus, so it was worthwhile to use the continuum limit. Further, with
Michael Faraday mf  and James Clerk Maxwell’s jcm  theory of electricity and
magnetism, they had to be treated as continuous fields. As such, it became
necessary to extend the Lagrange formalism to a continuum in space and in
time. Faraday and Maxwell’s electric and magnetic fields, ${\bf E}(t,{\bf x})$
and ${\bf H}(t,{\bf x})$, pervade all of space at all times. Let us briefly
review the developments.

Faraday’s law relates changes in the magnetic field over time to changes in
the electric field over space. Maxwell believed that the converse effect
should also hold and modified Ampere’s law relating the current density, ${\bf
j}$ to a magnetic field varying over space, to include a change of the
electric field over time. The Gauss equations use his divergence theorem to
relate the divergence of the electric and magnetic fields to the strengths of
their sources. Since there is no “magnetic charge”, the divergence of the
magnetic field is zero. Since electric causes have magnetic effects and vice
versa, Maxwell unified the theories of electricity and magnetism to
electromagnetism. The theory was formulated mathematically in a set of four
PDEs, written in the cumbersome formalism of the time. In more modern notation
they are:

$\displaystyle\nabla\cdot{\bf E}$ $\displaystyle=$
$\displaystyle\frac{\rho}{\epsilon}\qquad\qquad\qquad\qquad\qquad\qquad\,{\rm(Gauss^{\prime}s\,\,law)};$
(12) $\displaystyle\nabla\cdot{\bf H}$ $\displaystyle=$ $\displaystyle
0\qquad\qquad\qquad\qquad\qquad\qquad\,\,{\rm(Gauss^{\prime}s\,\,law)};$ (13)
$\displaystyle\nabla\times{\bf H}$ $\displaystyle=$
$\displaystyle\epsilon\frac{\partial{\bf E}}{\partial t}+{\bf
j}\qquad\qquad{\rm(Modified\,\,Ampere^{\prime}s\,\,law)};$ (14)
$\displaystyle\nabla\times{\bf E}$ $\displaystyle=$
$\displaystyle-\mu\frac{\partial{\bf H}}{\partial t}+{\bf
j}\qquad\qquad\qquad\quad{\rm(Faraday^{\prime}s\,\,law)};$ (15)

in electrostatic units, where $\rho$ is the charge density, $\epsilon$ is the
dielectric constant of the material and $\mu$ its magnetic susceptibility
(see, for example aq2 ). The vacuum has a dielectric constant and magnetic
susceptibility, denoted by a subscript zero. Thus the spacetime that carries
the field takes the place of the aether - not a mechanical aether but an
electromagnetic aether - which supports the field.

Notice the symmetry in these equations, in that two deal only with fields and
two have material sources involved (the charge and current density). Even in
the absence of these sources at some point in space at some time, the fields
can still exist. There is also a symmetry between space and time in the
equations. However, in Maxwell’s theory there is a scalar potential, $\phi$
for the electric force and a vector potential, ${\bf A}$, for the magnetic
field, so that ${\bf E}=-\nabla\phi$ and $\mu\,{\bf H}=\nabla\times{\bf A}$.
While we can restore the symmetry of the Maxwell equations by considering the
source-free case, how can we obtain a symmetry in the potentials? In his
seminal paper on Special Relativity, Albert Einstein ae1  brought out the
symmetry by unifying space and time into a single spacetime, so that a point
in spacetime is given by a four-vector, $x^{\mu}=(x^{0},x^{i})=(ct,{\bf x})$.
In the same way the electromagnetic potential is a four vector,
$A^{\mu}=(\phi/c,-{\bf A})$. Though the first field theory was for fluids,
that was artificial, as it dealt with discrete particles as if they formed a
continuum. In this case the fields are continua. In four-vectors we can write
the Maxwell fields in terms of a tensor, $F_{\mu\nu}=A_{\nu,\mu}-A_{\mu,\nu}$
or, in the language of forms ${\bf F}={\bf d}\wedge{\bf A}$. The corresponding
Maxwell equations are

$F^{\mu\nu}_{\,\,\,;\nu}=j^{\mu}\,,\qquad
F_{\mu\nu,\rho}+F_{\rho\mu,\nu}+F_{\nu\rho,\mu}=0\,;$ (16) ${\bf d}\cdot{\bf
F}={\bf j}\,,\qquad{\bf d}\wedge{\bf F}=0\,.$ (17)

In fact, inserting the definition of the Maxwell field as the skew derivative
(generalized curl) of the four-vector potential, the second equations are
easily seen to be identities, ${\bf d}\wedge{\bf d}\wedge{\bf A}=0$, as the
exterior derivative, ${\bf d}\wedge$, is associative. In other words the
definition of the four-vector potential makes the magnetic Gauss law and
Faraday’s law into identities, while the physical content now resides in the
equations with sources: Gauss’ electric law, and Ampere’s modified law. The
difference between “;” and “,” is explained in the next section.

It is easily shown (see for example aq1 ) that in the absence of any source
the electric and magnetic fields satisfy the wave equation and the speed of
the electromagnetic wave, $1/\sqrt{\epsilon\mu}$, so that the speed of light
in vacuum is $c=1/\sqrt{\epsilon_{0}\mu_{0}}$. Since the dielectric constant
and magnetic susceptibility of the vacuum are less than for any material
medium, the speed of the electromagnetic wave in vacuum is the maximum speed
of these waves. Maxwell had already noted in his work jcm , that this is the
speed of light. Einstein had pointed out that it is the maximum attainable
speed for any form of matter or energy ae1 ; ae2 ; ae3 ; ae4 ; ae5 .

We need to repeat the variational procedure for fields in $n$-dimensional
spaces and four-dimensional spacetime, as slightly new features arise both
times. Let us keep in mind three-dimensional spaces first, but be ready to
extend the three to $n$. A scalar field, $\phi$, is an explicit function of
the independent variable, $t$ and of the three spatial variables $x^{i}(t)$.
Though this is not necessary, we limit the Lagrangian to be a function of the
field and its first total $t$-derivative,
${\mathcal{L}}[\phi(t,x^{i}(t)),\dot{\phi}]$, where
$\dot{\phi}=\phi_{t}+\dot{x}^{i}\phi_{i}$. The problem now is what is meant by
the derivative of the functional ${\mathcal{L}}$ with respect to a function
$\phi$, rather than a continuous variable like $t$? For the latter we just
take the limit as $\delta t$ tends to $0$. A function can tend to zero in
infinitely many ways. As such, we need a measure of the function, let us say
the root mean square norm, and then let that tend to zero. To distinguish
between the two concepts of derivative, we use $\delta$ in place of
$\partial$. Now repeating the previous variational procedure, yields the EL
equations for scalar fields

$\frac{\delta{\mathcal{L}}}{\delta\phi}=\frac{\partial}{\partial
t}\left(\frac{\delta{\mathcal{L}}}{\delta\phi_{t}}\right)+\left(\frac{\delta{\mathcal{L}}}{\delta\phi_{i}}\right)_{;i}~{}\,,$
(18)

where $\phi_{t}$ and $\phi_{i}$ stand for the partial derivative of $\phi$
relative to $t$ and $x^{i}$, respectively. For use in Mechanics we can take
$i=1,2,3$, but in Economics we have to allow for all the commodities. It would
be worth exploring the use of fields in Economics. If, instead of a scalar
field, there is a vector field, $A_{r}(t,x^{i}(t))$, the EL equations become

$\frac{\delta{\mathcal{L}}}{\delta A_{r}}=\frac{\partial}{\partial
t}\left(\frac{\delta{\mathcal{L}}}{\delta
A_{r,t}}\right)+\left(\frac{\delta{\mathcal{L}}}{\delta
A_{r,i}}\right)_{;i}~{}\,,$ (19)

You might have expected that the examples of the electric and magnetic fields
would be given here, but the fact is that a field theory of one without the
other would be incomplete. We need to use the relativistic electromagnetic
field, $A_{\mu}$. However, there is a complication. In relativity, time is
like any other coordinate, so we do not have a separate independent variable,
but four independent variables, and the field is $A_{\mu}(x^{\nu})$. Now the
variation is with respect to each function of each variable. The EL equations
for a relativistic vector field become,

$\frac{\delta{\mathcal{L}}}{\delta
A_{\mu}}=\left(\frac{\delta{\mathcal{L}}}{\delta
A_{\mu,\nu}}\right)_{;\nu}~{}.$ (20)

Using the electromagnetic Lagrangian,

${\mathcal{L}}=A_{\mu}j^{\mu}+\frac{1}{4}F_{\mu\nu}F^{\mu\nu},$

in the above EL equations gives the first of the Maxwell equations (16). The
other is, of course, an identity.

##  V A Geometrical Application of the Lagrangian

The geometrical Lagrangian is obviously for a continuum, but not so obviously
for a field theory. If we limit our discussion to three-dimensional space, or
surfaces, or even to Minkowski space, it is not for a field. As we saw,
electromagnetism, which is the epitome of a field theory, has a vector field,
$A_{\mu}$. Geometrically, the arc length square is given by

$ds^{2}=g_{ij}(x^{k})dx^{i}dx^{j},$

where $g_{ij}$ is the matrix representation of the metric tensor, ${\bf g}$,
in some $n$-dimensional coordinate system in index notation and $dx^{i}$ are
infinitesimal changes of the position vector in those coordinates aq2 . Thus
geometry needs a (second rank) tensor field, $g_{ij}$. Note that not only the
metric coefficients, but the metric tensor itself, varies from point to point.

The shortest path between two points, P and Q, called a geodesic, is obtained
by minimizing the integral of the arc length, $ds$, along the path from one to
the other,

$s_{PQ}=\int_{P}^{Q}ds=\int_{P}^{Q}\ell[x^{i}(s),\dot{x}^{j}(s)]ds=\int_{P}^{Q}g_{ij}(x^{k})\dot{x}^{i}\dot{x}^{j}ds~{}\,,$
(21)

where $\ell[x^{i}(s),\dot{x}^{j}(s)]$ is the Lagrangian, which has a constant
value as afunction of $s$, but as a functional it depends on the position and
velocity vectors. Using the EL equations, (11), without explicit dependence on
the parameter $s$, we obtain

$\frac{d}{ds}(g_{ij}\delta^{i}_{k}\dot{x}^{j}+g_{ij}\delta^{j}_{k}\dot{x}^{i})=g_{ij,k}\dot{x}^{i}\dot{x}^{j}~{}.$
(22)

The total derivative of $\dot{x}^{i}$ is $\ddot{x}^{i}$ and, since $g_{ij}$ is
an explicit function of $x^{i}$ but not of $s$,
$\frac{d}{ds}g_{ij}=g_{ij,l}\dot{x}^{l}$. Hence

$g_{kj}\ddot{x}^{j}+g_{ik}\ddot{x}^{i}+g_{kj,l}\dot{x}^{j}\dot{x}^{l}+g_{ik,l}\dot{x}^{i}\dot{x}^{l}=g_{ij,k}\dot{x}^{i}\dot{x}^{j}~{}.$
(23)

As the metric tensor defines the length of a vector, so it and its inverse
must exist at every point. In index notation we write the inverse as $g^{ik}$
such that $g_{ij}g^{ik}=\delta^{j}_{k}$, the Kronecker delta, which is the
identity matrix in index notation. Multiplying (23) through by half the
inverse metric, relabeling dummy indices and transposing the term on the right
side, we obtain the geodesic equation

$\ddot{x}^{i}+\Gamma^{i}_{jk}\dot{x}^{j}\dot{x}^{k}=0~{}\,,$ (24)

where $\Gamma^{i}_{jk}$ is the Christoffel symbol, defined by

$\Gamma^{i}_{jk}=\frac{1}{2}g^{il}\left(g_{jl,k}+g_{kl,j}-g_{jk,l}\right)~{}\,,$
(25)

which gives the difference between the covariant derivative denoted by “$;k$”
and the partial derivative denoted by “$,k$”, namely for any $A^{i}$,

$A^{i}_{;k}=A^{i}_{,k}+\Gamma^{i}_{jk}A^{j}~{}.$ (26)

One of Euclid’s theorems for a plane says that, the shortest path between two
points is the straight line joining them. Of course, there is no straight line
in a curved space (like the surface of the Earth, where straight lines in
three dimensions are excluded). The straightest available path in an
$n$-dimensional space, is the curve whose unit tangent vector, ${\bf t}$, does
not change direction along it, i.e. $\frac{d}{ds}{\bf t}=0$. Now, we could
write the tangent vector in the local coordinates discussed, with the basis
vectors, ${\bf e}_{i}$, so that ${\bf t}=\dot{x}^{i}{\bf e}_{i}$. Since every
vector can be written as a linear combination of basis vectors, so the partial
derivative along along any direction can be so written. Hence, ${\bf
e}_{i;j}=\Gamma^{k}_{ij}{\bf e}_{k}~{}$, As such the Christoffel symbol arises
from the differentiation of the basis vectors. It is shown in Differential
Geometry (see for example aq2 ) that this set of coefficients is given by
(25). Thus the generalization of Euclid’s theorem to curved spaces is: “the
shortest available path between two points, is the straightest available
curve.” In a flat space in Cartesian coordinates the basis vectors are
constant, but in general, like in Gauss’ theory of surfaces, they vary. The
simplest examples are of the polar basis vector in a plane in polar
coordinates, and all basis vectors on a sphere.

##  VI Symmetries of Fields and Lagrangians

What would be meant by the symmetry of a field? For one thing, the field may
not depend on some independent variable(s), such as time or position. Writing
the field, which may be a scalar, vector or tensor, as ${\bf A}$, then
$\partial{\bf A}/\partial t=0$, or the equivalent for one or more position
variable. In that case we say that ${\bf X}=\partial/\partial t$, or the
equivalent for some position variable is a translation symmetry of the field.
On the other hand, it could be that the field itself “inflates” (or
“deflates”) with time, or along some spatial direction. In that case
$t\partial/\partial t$ or $x\partial/\partial x$ say, is called a scaling
symmetry of the field. Even if the field depends on the independent variables,
it could be that some physically (or economically) relevant quantities do not.
For example it may be that the energy and momentum re-scale in such a way that
the difference between the square of the energy and a constant times the
square of the other is constant, as in Special Relativity, where
$E^{2}-p^{2}c^{2}=m^{2}c^{4}$. In this case we say that the mass, $m$, is an
invariant and the corresponding infinitesimal symmetry generator is
$p^{\mu}=\partial/\partial x^{\mu}$, which gives the energy momentum four-
vector. Notice that this says that energy-momentum is collectively but not
separately, conserved. The conserved quantity is the Hamiltonian, $mc^{2}$. In
Physics one would call the former conservation a “conservation law”, but
people in Symmetry Analysis call the latter by that name. There can be other
symmetry generators, like a combined scaling symmetry such as
$x\partial/\partial x+y\partial/\partial y$, along the line $y=x$, or
rotations given for example by $y\partial/\partial x-x\partial/\partial y$ for
rotation about the $z-$axis. In Relativity, the Lorentz transformation for
uniform linear motion in the $x$-direction is $ct\partial/\partial
x+x\partial/\partial(ct)$. Thus in Special Relativity this would be a
conservation law. There can also be symmetries of combinations of the field,
or of it and the first derivative of the field, but not of the field itself.

Of special relevance are symmetries of the Lagrangian, because it gives the
dynamics arising from the variational principle for the Lagrangian, while also
allowing a reduction of the number of variables that are involved in the DE,
or reducing its order. This double reduction makes these symmetries especially
useful. They are called Noether symmetries. Noether’s theorem says that to
each symmetry of the Lagrangian, there corresponds a conserved scalar quantity
(called a Noether charge). Each conserved quantity is a first integral of some
part of the equations of motion. At the same time, when we use that quantity
as a new “variable”, it reduces the equation by trivializing that part of it.
The most obvious invariant is the Lagrangian itself by definition. This is not
useful for any reduction, as it is tautologically true. In fact, scaling it by
a constant cannot change the equations of motion, since they are linear and
homogeneous in the Lagrangian. The great thing about using Geometry for
kinematics or dynamics is that in that case every geometrical symmetry will be
a non-trivial Noether symmetry and provide a double reduction for the
equations of the theory. The charges give us physical information and help to
reduce the order of the equations. Before continuing with the applications, we
need to present the definition of Noether symmetries and explain how they are
determined.

###  VI.1  Noether Symmetries

Noether’s theorem noether  is applicable to a dynamical system of ordinary or
partial differential equations obtained from a variational principle olver ;
bluman-anco ; ibragimov ; bluman2010 . Let $x^{a}$ be $\ell$ independent
variables and $q^{i}$ be $m$ dependent variables which are arbitrary
(sufficiently smooth) functions of independent variables. The total derivative
operator $D_{x}$ given in (5) can be recast into the form

$D_{a}=\frac{\partial}{\partial
x^{a}}+q^{i}_{a}\frac{\partial}{\partial\,q^{i}}+q^{i}_{ab}\frac{\partial}{\partial\,q^{i}_{b}}+...\,,$
(27)

where the derivatives of $q^{i}$ with respect to $x^{a}$ are represented by
$q^{i}_{a}=D_{a}q^{i}\,,q^{i}_{ab}=D_{b}D_{a}q^{i}$, and so on. Then, the
Euler-Lagrange operator, for each $i$, is defined by

$\frac{\delta}{\delta q^{i}}=\frac{\partial}{\partial q^{i}}+\sum_{N\geq
1}\left(-1\right)^{N}D_{a_{1}}...D_{a_{N}}\frac{\partial}{\partial\,q^{i}_{a_{1}..a_{N}}}\,.$
(28)

We now consider the EL equations of motion

$U^{k}\left(x^{a},q^{i},q^{i}_{(1)},...,q^{i}_{(N)}\right)=0\,,$ (29)

which is an ${\it N}^{th}$-order system of $m(\geq 1)$ PDEs or, the ODEs if
$\ell=1$. Eq. (29) is assumed to be of maximal rank and locally solvable,
where the collection of Nth-order derivatives is denoted by $q^{i}_{(N)}$. If
there exists a function $\mathcal{L}(x^{a},q^{i},...,q^{i}_{(M)})$, $M<N$,
such that (29) is equivalent to

$\frac{\delta\mathcal{L}}{\delta q^{i}}=0\,,$ (30)

then $\mathcal{L}$ is called a Lagrangian of (29). Here the ${\it M}^{th}$
prolonged operator has the form ibr1998

$\displaystyle{\bf X}^{[M]}={\bf X}+\eta^{\prime
i}_{\,a}\frac{\partial}{\partial\,q^{i}_{a}}+\eta^{\prime
i}_{\,ab}\frac{\partial}{\partial\,q^{i}_{ab}}+...+\eta^{\prime
i}_{\,a_{1}...a_{M}}\frac{\partial}{\partial\,q^{i}_{a_{1}...a_{M}}}\,,$ (31)

where the operator ${\bf X}$ is of the form

${\bf X}=\xi^{a}(x^{b},q^{j})\frac{\partial}{\partial
x^{a}}+\eta^{i}(x^{b},q^{j})\frac{\partial}{\partial\,q^{i}}\,,$ (32)

and $\eta^{\prime i}_{\,a}$ and $\eta^{\prime i}_{\,ab}$ are defined by

$\eta^{\prime i}_{\,a}=D_{a}\eta^{i}-q^{i}_{b}D_{a}\xi^{b}\,,\quad\eta^{\prime
i}_{\,ab}=D_{a}D_{b}\eta^{i}-q^{i}_{bc}D_{a}\xi^{c}-q^{i}_{ac}D_{b}\xi^{c}-q^{i}_{c}D_{a}D_{b}\xi^{c}\,.$
(33)

A Noether symmetry generator corresponding to a Lagrangian
$\mathcal{L}(x^{a},q^{i},...,q^{i}_{(M)})$ is the operator ${\bf X}$ in (32)
if there exists a vector ${\bf K}=(K^{1},...,K^{\ell})$, or a function
$K(x^{a},q^{i})$ if there is only one independent variable (i.e., $\ell=1$),
such that

${\bf X}^{[M]}(\mathcal{L})+\mathcal{L}\,D_{a}(\xi^{a})=D_{a}(K^{a})\,,$ (34)

where $M<N$. Further, the operator ${\bf X}^{[M]}$, which is also called the
Lie-Bäcklund symmetry, is a Noether symmetry of $\mathcal{L}$ corresponding to
an Euler-Lagrange equations (30) if and only if the Lie characteristic
function $W^{i}=\eta^{i}-\xi^{b}\,q^{i}_{b}$ of ${\bf X}^{[M]}$ is also the
characteristic of the conservation law

$D_{a}(I^{a})=0\,,$ (35)

where $I^{a}$ has the form

$I^{a}=N^{a}(\mathcal{L})-K^{a}\,,$ (36)

and the Noether operator associated with the operator ${\bf X}^{[M]}$ is
defined by Ibragimov ibr1979  as follows:

$\displaystyle N^{a}=\xi^{a}+W^{i}\frac{\delta}{\delta\,q^{i}_{a}}+\sum_{M\geq
1}D_{a_{1}}D_{a_{2}}...D_{a_{M}}\left(W^{i}\right)\frac{\partial}{\partial\,q^{i}_{a\,a_{1}...a_{M}}}\,.$
(37)

Here ${\bf I}=(I^{1},...,I^{\ell})$ will be called a conserved vector of the
EL equations (29), or a conserved quantity (or first integral) if $\ell=1$.
Notice that for variational problems with Lagrangian functions depending on
higher-order derivatives, the main conservation theorems are valid, but the
conserved vector (36) for a Lagrangian function depending on any order
derivatives has a different form.

###  VI.2 Classical Mechanics

Time-translational invariance of the Lagrangian in Classical Mechanics reduces
the number of variables in the equations of motion from four to three and it
implies energy conservation. The former property makes it easier to solve the
ODEs involved, but the latter allows one to know the answer without solving
the equations. For a simple harmonic oscillator, one can use the further
spatial symmetry to reduce them to a single second-order ODE with constant
coefficients, but the momentum conservation gives planar motion and the law of
energy conservation says that a frictionless oscillator will continue its
motion forever, and that the greater the friction the faster the motion will
die out. The formal aspects of the calculations of Noether symmetries in this
case are explained below.

Consider a first order Lagrangian for only one independent variable, so that
the EL equations of motion are given by (11). Then the energy functional
associated with $\mathcal{L}$ is defined by

$E_{\mathcal{L}}={\dot{q}}^{i}\frac{\partial\mathcal{L}}{\partial{\dot{q}}^{i}}-\mathcal{L},$
(38)

which is also the Hamiltonian of the system. This yields the first integral as
a system of ODEs of the form

${\ddot{q}}^{i}=w^{i}(\tau,q^{k},{\dot{q}}^{k}).$ (39)

The Noether symmetry generator for this Lagrangian is given by

${\bf X}=\xi(t,q^{i})\frac{\partial}{\partial
t}+\eta^{i}(t,q^{i})\frac{\partial}{\partial\,q^{i}}\,,$ (40)

if there exists a function $K(t,q^{k})$ and the Noether symmetry condition

${\bf X}^{[1]}(\mathcal{L})+\mathcal{L}\,D_{t}(\xi)=D_{t}K\,,$ (41)

is satisfied. Here ${\bf X}^{[1]}$ is the first prolongation of Noether
symmetry generator ${\bf X}$, i.e.,

$\displaystyle{\bf X}^{[1]}={\bf
X}+{\eta^{\prime}}^{i}(t,q^{k},{\dot{q}}^{k})\frac{\partial}{\partial{\dot{q}}^{i}}\,,$
(42)

where
${\eta^{\prime}}^{i}(t,q^{k},{\dot{q}}^{k})=D_{t}\eta^{i}-{\dot{q}}^{i}D_{t}\xi$.
For every Noether symmetry generator ${\bf X}$, the conservation law (36)
becomes $D_{\tau}(I)=0$, where $I$ is the corresponding Noether flow, and it
has the expression

$I=-\xi
E_{\mathcal{L}}+\eta^{i}\frac{\partial\mathcal{L}}{\partial{\dot{q}}^{i}}-K\,,$
(43)

which is a conserved quantity of the system of equations (39).

There is a second way of finding symmetries of a Lagrangian $\mathcal{L}$ for
a given dynamical system, called the strict Noether symmetry approach that
yields $\pounds_{\bf X}\mathcal{L}=0$, where $\pounds_{\bf X}$ is the Lie
derivative operator along ${\bf X}$ capo1996 . The only difference is that
here $K$ vanishes. Both approaches are useful in a variety of problems arising
from physics and applied mathematics, and lead to first integrals. The
important thing is that which approach yields a conserved quantity. The
classical Noether symmetries have the advantage of yielding conserved
quantities or conservation laws, directly ibragimov . Cyclic variables are
also used and are related to Noether symmetries, but there is ambiguity in
their choice. For more details see capo2007 .

Another example is of a particle that would normally undergo geodesic motion,
but is forced off it, like a charged particle in an EM field. For the present
purpose take a scalar potential $V(s,x^{k})$, then the geodesic Lagrangian
describing the motion of the massive or massless (i.e., lightlike) particles
can be written as

$\mathcal{L}(s,x^{k},\dot{x}^{k})=\frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}-V(s,x^{k})\,,$
(44)

which gives rise to the forced geodesic equations of motion

$\ddot{x}^{i}+\Gamma^{i}_{jk}\dot{x}^{j}\dot{x}^{k}=F^{i}~{}\,,$ (45)

where $F^{i}=g^{ij}V_{,j}$ is the conservative force field. For every Noether
symmetry, there is a a first integral for the system of equations (45) of the
form

$I=-\xi E_{\mathcal{L}}+g_{ij}\eta^{i}\dot{x}^{j}-K\,,$ (46)

where the energy functional (38) for the geodesic Lagrangian is given by

$E_{\mathcal{L}}=\frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}+V(s,x^{k})\,,$ (47)

which is the Hamiltonian of the system.

###  VI.3 Economics

Shifting from Mechanics to Economics, time-translational invariance of the
Lagrangian implies that prices will stay constant over time and the total
amount in the Economy stays constant. “Amount of what?”, you ask. This hides
an aspect of Economics with no Classical Mechanics analogue. Two distinct
quantities may be taken: wealth; or money. Wealth refers to the actual goods
and services in the Economy, while money is an arbitrary measure for that
wealth, called a “numeraire”. Being arbitrary, we can change the amount of
wealth to a unit of money. The Government may do this to count in wealth that
would be there in the Economy, but not as yet registered in the accounting; or
to appear to be richer. In the former case, there is a growing economy, and
the “value of money” remains constant. In the latter case the value of money
will decline while the Economy remains stagnant. The Government could even
anticipate the wealth yet to be generated, before it has been, so as to
accelerate economic growth. As such, it would “borrow from the future”. This
is called “credit creation”. It can lead to runaway inflation leading to a
credit crunch, as has been seen.

Already we have seen economic insights provided by the invariant beyond the
benefit of reducing the number of variables in the EL-equations. More follow.
In 1945 John von Neumann demonstrated jvn  that the rate of interest equals
the rate of growth for an optimally growing Economy. In other words, for
optimal growth, the amount of money in the Economy must grow at the same rate
as the amount of wealth. It was then shown that aq3 ; aq32  that there must be
inflation in a growing economy. The excess money behaves in much the same way
as the entropy in Physics corresponding to the shortfall of efficiency of a
heat engine. It should be noted that the Noether invariant in both
applications (Physics and Economics) has significance far beyond its use for
reducing the number of variables and order of the governing ODEs.

Though there is no analogue of the two ways of considering the “amount” of
something in Classical Mechanics, there is in Relativistic Mechanics. The
measures of length and time vary from observer to observer, since they are
arbitrary measures of the invariant length and duration. Consequently, the
invariants play a far more significant role in Relativity than they do in
Classical Mechanics. To explain this will need to provide a quick review of
geometrical symmetries.

###  VI.4 Geometrical Symmetries

There are two generalizations of the derivative for a manifold. One is to
first map an open set on the manifold to a coordinate frame, $\mathbb{R}^{n}$,
take the derivative along a vector in the usual way and then map the
derivative back to the manifold. The second is to directly take the rate of
change by infinitesimal movement along a curve on the manifold. The former is
called the intrinsic derivative and the latter the Lie derivative. The former
procedure obviously incorporates the derivatives of the basis vectors, while
the latter does not. The intrinsic derivative, left in the coordinate system,
is called the covariant derivative and denoted by “;”, as we saw. The Lie
derivative along a vector field, ${\bf t}$, will be denoted by $\pounds_{\bf
t}$. If the intrinsic derivative is used in a Taylor series to move from one
point on the manifold to another, it is called parallel transport, as a curve
traced out by transporting a vector field, ${\bf p}$, along a curve with
tangent vector field ${\bf t}$, will appear parallel as seen in the coordinate
system. However, it will not be parallel on the manifold. If the Lie
derivative is used for moving on the manifold it is called Lie transport.

This is most easily seen by taking a sphere as the manifold and the base curve
to be a line of latitude. Transporting a unit North pointing vector traces out
the next line of latitude. At the equator it is not so obvious, but near the
North pole, as we know from our school Geography, on the map of the Earth it
does not look parallel, and the square on the map does not look like a square
on the globe. The reason is that the lengths of the upper and lower sides of
the “square” on the globe are unequal, but the angles subtended by them are
the same. Thus if ${\bf p}$ takes point $P$ to point $Q$ on one line of
latitude, and ${\bf t}$ takes $P$ to $R$ on the same line of longitude and $Q$
to $S$ on the next one, by definition of subtending the same angle, ${\bf p}$
must take $R$ to $S$, so ${\bf p}$ is Lie transported along ${\bf t}$ and the
square on the globe closes. However, since the lengths of the upper and lower
${\bf p}$ are unequal, it cannot close on the map.

Let us get a bit more technical. If ${\bf p}$ is to be Lie transported from
$P$ to $Q$ by ${\bf t}$, then ${\bf p}|_{Q}=exp(\pounds_{\bf t}){\bf p}|_{P}$.
Now, if ${\bf p}$ is invariant under this Lie transport, then clearly
$\pounds_{\bf t}{\bf p}=0$. In this case the “square” mentioned above, or more
generally the “rectangle”, closes and so ${\bf t}{\bf p}={\bf p}{\bf t}$.
Taken into the coordinate system, $t^{a}p^{b}_{;a}=p^{a}t^{b}_{;a}$. Now,
since the Christoffel symbol is symmetric the terms involving it on both
sides, $p^{a}t^{c}\Gamma^{b}_{ac}$ and $t^{a}p^{c}\Gamma^{b}_{ac}$, cancel and
so we have $t^{a}p^{b}_{,a}=p^{a}t^{b}_{,a}$. This is the way that the
derivative of the basis vector is removed from the Lie derivative.

###  VI.5 General Relativity

First let us introduce General Relativity (GR). It requires that all observers
be equally good. “Observers” are conceived as disembodied persons — massless
points endowed with a clock attached to a spatial frame of reference — who
move on geodesics. “Equally good” means that physical laws are no simpler in
one frame than another. This does not necessarily mean that one cannot
distinguish between different frames. For example, one could feel
acceleration, so a frame of zero acceleration could be determined, but the
presence or absence of acceleration would not be relevant for the statement of
a physical law. As such, those laws must be defined on manifolds. Special
Relativity (SR) does assume that there is no way to distinguish between two
un-accelerated frames. However, this must again be seen in a special context
of two observers communicating with each other, without reference to a third
aq2 . By choosing the frame of reference in which the cosmic microwave
radiation is isotropic (up to statistical fluctuations) we can determine the
rest-frame of the Universe. As such, GR entails using Lie derivatives and Lie
transport.

As in the earlier example, let us take ${\bf t}$ to be the unit tangent vector
to the geodesic of an observer, $O$ and ${\bf p}$ be the position vector of
another observer $O^{\prime}$ relative to $O$. Then ${\bf t}({\bf
p})=\dot{{\bf p}}$ and ${\bf t}[{\bf t}({\bf p})]=\ddot{{\bf p}}$, i.e. the
velocity and the acceleration of $O^{\prime}$ as seen by $O$. In Geometry, the
latter is called geodesic deviation and will be denoted by ${\bf\mathcal{A}}$.
In index notation

${\mathcal{A}}^{\mu}=t^{\alpha}(t^{\nu}p^{\mu}_{;\nu})_{;\alpha}~{}.$ (48)

Using the Lie transport requirement we can interchange the $t$ and $p$ inside
the bracket and, on expanding the bracket use the geodesic condition on one
term. What remains is

${\mathcal{A}}^{\mu}=t^{\alpha}t^{\nu}[p^{\mu}_{;\nu;\alpha}-p^{\mu}_{;\alpha;\nu}]~{}.$
(49)

Now, by definition, the Riemann curvature tensor is defined mtw  by the skewed
second derivative of a vector, so that we get

${\mathcal{A}}^{\mu}=R^{\mu}_{~{}\alpha\nu\beta}t^{\nu}p^{\alpha}t^{\beta}~{}.$
(50)

This means that acceleration is related to the curvature of the spacetime
manifold. Physically, acceleration is caused by matter and energy. We saw how
energy conservation arose for a system of particles in Classical Mechanics. It
would be useful to generalize that concept to other physical fields in a four-
dimensional spacetime.

On account of mass-energy equivalence we no longer have conservation of mass
and energy separately, but only of the matter-energy tensor, also called the
stress-energy tensor, (see mtw , Ch.4). It includes the momentum 4-vector and
spatial fluid stress tensor $\sigma^{ij}=dF^{i}/dS_{j}$, where $dF^{i}$ is the
force acting on an area element of the fluid, $dS_{j}$. If it is an
irrotational perfect fluid, the stress energy tensor density is

$T^{\mu\nu}=\rho
c^{2}\delta^{\mu}_{0}\delta^{\nu}_{0}+\sigma^{ij}\delta^{\mu}_{i}\delta^{\nu}_{j}~{},$
(51)

and the usual conservation laws for relativistic fluids, derived by the
requirement that the flux of fluid and energy crossing a closed surface is
conserved. Using Gauss’ divergence theorem, it gives

$T^{\mu\nu}_{\quad;\nu}=0~{}.$ (52)

For a field with a vector-valued potential $A_{r}$, for an irrotational
perfect fluid, the stress energy tensor density is

$T^{\mu\nu}=g^{\mu\alpha}A_{r,\alpha}\frac{\delta{\mathcal{L}}}{\delta
A_{r,\nu}}-g^{\mu\nu}{\mathcal{L}}~{}.$ (53)

For $r=1$ we get a scalar field, for $r=\mu$ we get a scalar field, and for
$r=\mu\nu$ we get a tensor field. For any number of fields, we simply have to
add the stress energy tensor for each one to get the total stress-energy
tensor.

Returning to GR, we need that ${\bf T}$ be related to the Riemann tensor,
${\bf R}$, by a second rank tensor function, which is divergence-free. Since
the function is to be second rank and the Riemann tensor is fourth rank, we
need to take the trace of the Riemann tensor, namely the Ricci tensor. Using
the Bianchi identities that are satisfied by the Riemann tensor, we obtain the
linear combination of the Ricci tensor and scalar that is divergence-free,
${\mathcal{E}}_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$, called the
Einstein tensor, yielding the simplest non-trivial relation, called the
Einstein Field Equations, (EFEs)

$\kappa T_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}~{},$
(54)

where $\kappa=8\pi G/c^{4}$, gives the coupling of gravity with matter, $G$
being Newton’s constant, $\Lambda$ is a constant of integration, called the
“cosmological constant”. In their full generality they are a system of ten
non-homogeneous, second order nonlinear PDEs for ten functions (metric
coefficients) of four variables. Even given the source, it would be impossible
to solve them generally.

Let us now take a quick look at Noether symmetries for the geodesic Lagrangian
(44) in GR for some well-known spacetimes which have been used ofr
classifiation according to their symmetry generators. This has been done for
static plane, static spherical, and static cylindrically symmetric spacetimes
by Feroze and her collaborators feroze1 ; feroze2 ; feroze3 ; feroze4 ;
feroze5 ; feroze6 . The complete classification of non-static plane and non-
static spherically symmetric spacetimes via Noether symmetry worked by Jamil
et al. jamil1 ; jamil2 . The Lie and Noether symmetries of geodesic equations
have been studied for the Friedmann metrics by Tsamparlis and Paliathanasis
tsamparlis1 . These symmetries have also been obtained for some of the
Bianchi-type spacetimes in tsamparlis2 ; ptm2015 ; ah2016 ; hy2017 . The
complete analysis of Noether symmetries for Gödel-type and pp-wave spacetimes
studied by Camci et al. ug2014 ; ug2015 ; uy2015 .

####  VI.5.1 Spacetime Symmetries

This is where the need to use symmetries comes in. Given enough symmetries we
can reduce the number of independent variables. Thus, if we have the maximum
possible symmetry of a flat space, i.e. Minkowski space, the functions are
fully given directly, being constant in Cartesian coordinates and only
trivially dependent on the independent variables in other coordinates. This
brings out a problem of determining whether the dependence is trivial or not.
Can a change of the independent variables remove the apparent dependence on
them? For this purpose we need an invariant characterization of the symmetry
involved. Since the metric tensor is the (tensor) potential, we need that the
symmetry direction be one along which ${\bf g}$ is Lie transported. If the
symmetry vector is denoted by ${\bf k}$, we need that $\pounds_{\bf k}{\bf
g}=0$. In index notation this reduces to the Killing equation,

$k_{(\mu;\nu)}=0=k^{\alpha}_{,\mu}g_{\alpha\nu}+k^{\alpha}_{,\nu}g_{\alpha\mu}+k^{\alpha}g_{\mu\nu,\alpha}~{}.$
(55)

A vector, ${\bf k}$, satisfying this equation is called a Killing vector (KV),
or an isometry. If ${\bf k}$ is a unit timelike isometry then we can choose
coordinates such that $k^{\mu}=\delta^{\mu}_{0}$ and use it to define the time
coordinate and the metric coefficients will be time-independent in these
coordinates. Similarly, if there are three unit spacelike isometries, ${\bf
k}_{i}$, we can use them to define three spatial coordinates by
$k^{\mu}=\delta^{\mu}_{i}$, for $(i=1,2,3)$. In the former case the metric
coefficients will be independent of time, and in the latter of space. As such,
we will have time translation invariance in the former case and space
translation invariance in the latter case. The Noether invariants
corresponding to them will be the energy and momentum. If the spacelike
vectors generate the Lie algebra $SO(3)$, there can only be trivial dependence
of the metric coefficients on the three coordinates and there will be
rotational invariance. The corresponding conserved quantity will be the
angular momentum. If the timelike and spacelike vectors in pairs, generate an
$SO(1,1)$ there will also be invariance under Lorentz transformations and the
corresponding conserved quantity will be the spin angular momentum. If all
(six) of these “rotational” symmetries exist, the Lie algebra will be
$SO(1,3)$. If the four translations also exist the spacetime must be flat and
one can choose Cartesian coordinates, in which the metric tensor is diagonal,
being $1$ in the time component and $-1$ in each spatial component.

In a curved spacetime the symmetry cannot be higher dimensional, but can have
the same number of dimensions if the curvature is a non-zero constant. Thus
the number of symmetry generators will be ten and the associated group will be
the de Sitter (dS) group, $SO(1,4)$ for positive curvature and the anti-de
Sitter, (AdS) group, $SO(2,3)$, both of which contain $SO(1,3)$ as a subgroup.
Since the metric has a timelike KV, there is still energy conservation.
However, the distinction between linear and angular momentum disappears. The
point is that the geodesic on which the linear momentum was being conserved,
now bends around and closes on itself for dS and bends hyperbolically away for
AdS, so that it becomes rotational motion as well. In all other cases some
conservation laws will be lost.

To be concrete, the dS metric is:

$ds^{2}=e^{\nu(r)}dt^{2}-e^{-\nu(r)}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)~{}\,,$ (56)

where distances are measured in light seconds and

$e^{\nu(r)}=1-\frac{r^{2}}{R^{2}}\,,$ (57)

$R$ being a constant. If, instead,

$e^{\nu(r)}=1+\frac{r^{2}}{R^{2}}~{}\,,$ (58)

we have the AdS metric. In these cases the stress-energy tensor is
proportional to the metric tensor and so, by the EFEs, the Ricci tensor must
also be proportional to the metric tensor. Such spaces are called Einstein
spaces. In the former case the stress-energy tensor has a positive trace, and
in the latter a negative trace. Were we to interpret that trace as the energy,
which would be done for a fluid, the former would have a positive energy and
the latter a negative energy. The difference this makes may be best visualized
by conceiving of a very large space city, in which some live on the outside
and others on the inside. Those on the outside would see a horizon below which
there is nothing, albeit smaller than they would see on Earth. The latter
would see everything collecting up at the horizon and above that a sky that
would look just like the city around them. The former is what one gets with
positive energy and the latter what arises with negative energy.

The gravitational field for a point particle of mass $m$ and charge $Q$ is
given by

$e^{\nu(r)}=1-\frac{2m}{c^{2}r}+\frac{Q^{2}}{r^{2}}~{}\,,$ (59)

in units with Newton’s gravitational constant chosen to be unity, and yields a
traceless Ricci tensor. If there is no charge, $Q=0$, the spacetime is Ricci-
flat. The former is called the Reissner-Nordström metric, and the latter the
Schwarzschild metric. In both cases we are only left with time translational
and rotational invariance, or energy and angular momentum conservation. The
fact that momentum conservation is lost is easily seen by considering a test
particle left near a gravitational source, in which case it will fall towards
the source. That spin angular momentum conservation is lost has the
consequence that precession can be generated or lost in a gravitational field,
and can be tested. For gravitational waves time translation is also lost and
so energy is not conserved.

Some more conservation laws are found in an Einstein Universe, which has the
symmetry group, $SO(4)\bigotimes\mathbb{R}$, for which the coefficient of the
time metric coefficient in Eq. (56), is unity and the coefficient of the
radial coefficient is given by Eq. (57). This is an Einstein space and we
recover the usual translation invariance as well, leading to linear momentum
conservation. If Eq. (58) is satisfied instead of Eq. (57), it is the anti-
Einstein Universe and the usual angular momentum conservation is replaced by a
spin angular momentum conservation, as we get an $SO(1,2)$ instead of an
$SO(3)$, symmetry group. The total number of conserved quantities is seven.

There are other metrics with $6$ isometries, corresponding to spaces that have
a constant coefficient for the solid angle element and they have the
corresponding Noether invariants, but their physical significance is not quite
that obvious. There are no spherically symmetric metrics with only $5$
isometries as was shown in a complete classification of spacetimes by their
isometries aq4 . There are many other spherically symmetric spacetimes with
$4$ KVs and any number with only the $3$ of angular momentum, that define
spherical symmetry.

Dispensing with spherical symmetry, the Kerr metric represents a spinning
point mass, $m$, with angular momentum per unit mass, $a$,

$\displaystyle
ds^{2}=Adt^{2}-\frac{\rho^{2}}{\Delta}dr^{2}-\rho^{2}d\theta^{2}-B^{2}d\phi^{2}-Cdtd\phi~{}\,,$
$\displaystyle\rho^{2}=r^{2}+a^{2}\cos^{2}\theta~{},\quad\Delta=r^{2}-2mr+a^{2}~{},\quad~{}A=1-\frac{2mr}{\rho^{2}}~{}\,,$
(60) $\displaystyle
B=(r^{2}+a^{2})\sin^{2}\theta+\frac{2mr}{\rho^{2}}a^{2}\sin^{4}\theta~{},\quad~{}C=\frac{2mr}{\rho^{2}}a\sin^{2}\theta\,,$

which has two KVs, for time translation and axial rotation. However, it has
three Noether symmetries, two of which correspond to energy and angular
momentum and an additional one coming from a Killing tensor (see, for example,
hs ). It has been shown aq5  that this invariant corresponds to the total
angular momentum squared. It is interesting to note that these are the same
quantities that are conserved in Quantum Mechanics and that the angular
momentum vector is not conserved there either aq6 .

####  VI.5.2 Conformal Symmetries

In School Geometry one first learns of congruent triangles and then of similar
triangles. The congruent triangles give invariance of the figure under
translation, but the similar triangles give it under translation and scaling,
which is provided by changing lengths while leaving angles invariant. This is
achieved by scaling the metric tensor, ${\bf g}\rightarrow\tilde{{\bf
g}}=\Omega^{2}({\bf x}){\bf g}$, which is called a conformal transformation.
In this case there will be a conformal Killing vector, (cKV), or conformal
isometry. Thus, if there is a timelike cKV, though energy conservation is
lost, a re-scaled energy is conserved. This applies, for example, to the
Friedmann metrics

$ds^{2}=dt^{2}-a^{2}(t)\left[d\chi^{2}+f_{k}^{2}(\chi)(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right]~{},$ (61)

where $k$ gives the normalized constant curvature, being $+1$ for a sphere,
$0$ for a plane and $-1$ for a hyperbola; the corresponding $f_{k}(\chi)$ are
$\sin\chi$, $\chi$ and $\sinh\chi$. For the first case the range of $\chi$ is
$2\pi$, for the other two it is semi-infinite. In this case there are only 6
isometries as the timelike one is lost, and so energy is not conserved.
However, there is a cKV, $a^{-1}(t)\delta^{\mu}_{0}$, and so energy is
conserved up to scaling. The scaling comes from the expansion factor for the
Universe, $a(t)$, and the re-scaled conserved quantity is the number of
particles in the expanding volume. In terms of the Lagrangian for the metric
tensor, the Lagrangian is not conserved but it is scaled. This yields a
conformal Noether invariant, which is the energy.

####  VI.5.3 Symmetries of the Electromagnetic Field

In the Lagrangian for the electromagnetic field, given immediately after Eq.
(20), the first term gives the source and the second represents the pure
electromagnetic field. The corresponding stress-energy tensor for the pure
electromagnetic field, given by Eq. (53) is

$T^{\mu}_{\nu}=F^{\mu\alpha}F_{\nu\alpha}-\frac{1}{4}\delta^{\mu}_{\nu}F^{\alpha\beta}F_{\alpha\beta}~{}.$
(62)

Writing this in terms of the electric and magnetic fields, the Lagrangian is
$(E^{2}-H^{2})/8\pi$, for the pure time component of the stress-energy tensor,
i.e. the Hamiltonian or energy, we get $(E^{2}+H^{2})/8\pi$, the space-time
part is the momentum vector, which gives the Poynting vector, ${\bf E\times
H}/4\pi$, and the spatial part gives the Maxwell stress tensor

$\sigma^{i}_{j}=E^{i}E_{j}+B^{i}B_{j}-\frac{1}{2}(E^{2}+B^{2})\delta^{i}_{j}~{},$
(63)

which signifies the symmetric stresses in a shear-free “fluid”.

Now the 4-gradient of a scalar function can be added to the 4-vector
potential, ${\bf A}$ without changing the Maxwell tensor, ${\bf F}$, since the
latter is the 4-curl of the former, and the curl-grad vanishes in all
dimensions, i.e. if ${\bf A}\rightarrow{\bf\tilde{A}}={\bf A}+{\bf d}f$ then
${\bf F}\rightarrow{\bf\tilde{F}}$. This non-uniqueness of the field is called
“gauge-freedom” and we say that A is invariant under gauge transformations.
This freedom must have a group associated with it, and consequently a “Noether
charge”. A conservation associated with it comes from the observation that the
second divergence of the Maxwell tensor must be zero, as the second derivative
is symmetric but the Maxwell tensor is skew. Hence, the divergence of the
first of the Maxwell equations, (16), implies that $j^{\mu}_{\,\,;\mu}=0$,
which amounts to $(\rho v^{\mu})_{;\mu}=0$, which says that the total time
derivative of the charge density is zero, i.e. the electric charge is
conserved.

For the transformation ${\bf\tilde{A}}=e^{\iota f({\bf x})}{\bf A}$,
$\nabla{\bf\tilde{A}}=e^{\iota f({\bf x})}[\nabla{\bf A}+\iota(\nabla f){\bf
A}]$. If we now define the so-called “covariant derivative”,
$\tilde{\nabla}=\nabla-\iota\nabla$ then the covariant curl gives
${\bf\tilde{F}}={\bf F}$, so that we recover gauge invariance by multiplying
by a position-dependent phase. The multiplication by a phase can be understood
geometrically by considering $e^{\iota\,\theta}$ acting on a position vector
in the complex plane, i.e. the number $z=x+\iota y$. This transforms to the
new point $(x\cos\theta-y\sin\theta)+\iota(y\cos\theta+x\sin\theta)$, which is
just a rotation through the angle $\theta$. If $\theta$ is a constant we say
that it is a global gauge transformation and if it is variable a local gauge
transformation. Thus, for the electromagnetic field we have local gauge
invariance. The physically relevant symmetry is the local gauge symmetry which
yields the conservation of electric charge. Written as the addition of the
gradient of a scalar function does not indicate what the associated group is,
but in the complex form employed above, this becomes the unitary group in one
dimension, $U(1)$. In other words, we can regard Maxwell’s theory as the
consequence of a $U(1)$ local gauge symmetry, called $U_{em}(1)$. Local gauge
theories really came into their own in Quantum Theory (QT), which we shall be
seeing in the next section.

##  VII Symmetries in Quantum Theory

In Quantum Theory (QT) discrete symmetries play a significant role. Since the
Lagrangian is left invariant under their action, they are Noether symmetries
and will give some conserved quantity. Since they are discrete, they will not
reduce the number of variables or order of the equations, but will limit the
range. Thus a reflection symmetry halves the range and a global rotational
symmetry limits the variable to a semi-closed interval that can be chosen to
be $[0,2\pi)$. If there is reflection symmetry, the number of solutions is
more than halved (as they must all be even functions; or if there is
antisymmetry they must all be odd). This break-up applies to translational
shifts, and hence to momentum. However, for angular momentum, we get ${\bf
r}\rightarrow-{\bf r}$ and ${\bf p}\rightarrow-{\bf p}$, so ${\bf r}\times{\bf
p}\rightarrow{\bf r}\times{\bf p}$. This symmetry is called parity. Thus,
while in a mirror left is converted to right the angular momentum arrow
remains unchanged. If a quantity remains unchanged, it is said to have
positive parity and if it reverses direction, it is said to have negative
parity. It was believed that all fundamental particles have either one or the
other (with no cases of neither) and the total parity in any interaction is
conserved.

Salam proposed that parity is violated in weak nuclear interactions (due to
which there is radioactive decay), and he sent his draft paper for comments to
Wolfgang Pauli, a leader of QT and especially of the spin quantum number,
which would be reversed if parity is violated. Pauli sent him back the
message, “Tell my young friend Salam to think of something better.” He added,
“Parity is conserved — I can feel it in my bones,” punning on the use of the
phrase “feel in my bones”, as the phosphorous in the bones would decay if
parity is not conserved. The fact is that it is violated and the phosphorous
in the bones does decay, but the rate of decay is orders of magnitude less
than the biological degradation of the bones and could not be noted. Pauli’s
reasoning was specious and spurious. When Yang and Mills received the Nobel
Prize for independently discovering the same principle, Pauli apologised to
Salam, but that did not restore Salam’s claim to priority.

There is also time reversal symmetry or asymmetry in any process, in that a
film of the process run in reverse would be indistinguishable from the
original, e.g. the collision of two billiard balls on a billiards table, or
the swing of an ideal pendulum. However, if the surface of the table is rough,
or the pendulum is damped, one can distinguish the forward direction from the
reverse. Though non-frictional dynamics obeys time reversal invariance, for
fundamental particles it was taken for granted that it is conserved, with
positive or negative values. Another property of fundamental particles is
electric charge. It is found that corresponding to each fundamental particle,
there is an otherwise identical particle with opposite charge, called an
antiparticle, even though one may be far more common than the other (like
electrons and positrons or like protons and antiprotons). Writing the parity
reversal operator as P, time-reversal as T and charge conjugation as C, though
each one may be separately violated it has been proved that their product,
CPT, is conserved in all fundamental processes, (the CPT Theorem).

More relevant for recent developments, are continuous symmetries. Recall that
QT uses a complex “wave function”, satisfying the Schrödinger, Klein-Gordon,
or Dirac equations. It gives a complex amplitude whose magnitude square
represents the probability of finding a quantum entity at some place at some
time. (In fact, even for the classical electromagnetic field theory, it is
convenient to use complex variables, see e.g. ll ; jdj .) When the classical
electromagnetic field is “quantized” by Dirac’s procedure (see e.g. jjs ), the
4-vector field corresponds to a spin-one particle, called the photon. If the
gravitational field could be quantized, it would correspond to a spin-two
field that people call the graviton. More generally, a field represented by a
tensor of rank $n$ corresponds to spin-$n$ quantized field a. These have a
real representation but, Dirac showed that there would be half-integer spin
fields as well, such as the electron, represented by a 4-dimensional complex
vector of the representation space of $U_{em}(1)$, called a Dirac spinor.
Spin-$n/2$ has an $n$-index spinor. The usual vector corresponds to a two
index spinor, or a single index tensor. Note that $U(1)$ is locally isomorphic
to $SO(2)$, as they both give rotations, but the former is a double covering
of the latter because of the complex representation.

While rotations in two dimensions commute, and so $SO(2)$ and $U(1)$ are
Abelian, they do not in general and so are non-Abelian. As
$U(n)=SU(n)\bigotimes U(1)$, the next simplest unitary group is $SU(2)$. Like
the transpose of an orthogonal matrix is its inverse, the Hermitian conjugate
of a unitary matrix is its inverse. Since a $2\times 2$ matrix has four
complex entries subject to four real constraints, $U(2)$ has four independent
parameters and hence $SU(2)$ has three. As 3-d rotations are also three,
$SU(2)$ is locally isomorphic to $SO(3)$, again with a double covering. This
is the symmetry group for weak interactions at high enough energy $\sim
150~{}GeV$, as shown by Glashow, Salam and Weinberg as . In fact, the weak and
electromagnetic forces are unified at energies $\sim 150GeV$ but the
electroweak (EW) symmetry group, $SU_{W}(2)\bigotimes U_{Y}(1)$, breaks down
at lower energies to the usual $U_{em}(1)$ below that. The “$Y$” is the
conserved hypercharge, which mixes the charge of electromagnetism with that of
the weak nuclear force, on account of which the photon of electromagnetism is
a mixture of the bare $U_{em}(1)$ and the neutral component of the
$SU_{Y}(2)$, called a neutral current (the other two being charged currents),
above the unification energy. Murray Gell-Mann mgm  had proposed an $SU(3)$
gauge group for three quarks, for the strong nuclear force. Originally an
energy-dependent term not respecting the symmetry, was inserted in the
Lagrangian to break the symmetry, which is negligible at lower energies, but
dominates at higher energies, and so the symmetry breaks — gradually. Peter
Higgs pwh  suggested a mechanism whereby the symmetry breaks spontaneously at
a critical energy, due to a field with a vacuum expectation value at higher
energies that acquires a mass and becomes a physical spin zero particle,
called a “Higgs boson”, as the energy drops below its mass. This mechanism was
used by Salam to develop the “electroweak” unified theory, and was then used
for $SU(3)$-breaking. The new conserved quantum number in this case was called
“colour”, and the resulting theory called quantum chromo-dynamics (QCD),
denoted by $SU_{c}(3)$. It has eight generators and that yields eight gauge
bosons called “gluons”. The standard model of particle physics is, then,
$SU_{C}(3)\bigotimes SU_{W}(2)\bigotimes U_{Y}(1)$.

###  VII.1 Gauge Grand Unification Symmetry

The critical energy of the standard model is the same as the EW theory, but it
consists of three forces with very different strengths: (a) weak $\sim
10^{-8}$; (b) electromagnetism $\sim 10^{-2}$; and (c) strong $\sim 1$. As the
interaction energy increases the strengths change; the weak and
electromagnetic becoming stronger at relative rates that go inversely as the
relative strengths and the strong gets weaker. At first it appeared that all
three should meet at somewhere around $10^{12}-10^{15}GeV$. Since the Universe
is cooling as it expands (like the gas in a refrigerator), this was taken to
indicate that there may have been a time in the early stages of the Universe
when all three were unified and this “grand unified theory” (GUT) broke at the
critical energy to yield the Universe as we see it now. In that case we would
need a bigger group that would break down to the groups of the standard model.

In 1971 Jogesh Pati and Abdus Salam proposed an $SU_{C}(4)\bigotimes
SU_{f}(4)$ GUT (published in 1974 ps ), with the weakly interacting particles
(called leptons) as a fourth colour prior to symmetry breaking, and with four
“flavours” of the leptons/quarks, $(u,d,e,\nu_{e})$ giving the the usual
protons, neutrons and electrons and a more massive set $(s,c,\mu,\nu_{\mu})$,
that was known at the time. Whereas Gell-Mann used fractional charges for his
quarks: $+2/3$ for $u$; $-1/3$ for $d$; $-3/3$ for $e$; and $0/3$ for
$\nu_{e}$, Pati and Salam used integer charges. but it has an enormous $225$
generators! The smallest simple group containing the full standard model is
$SU(5)$, which could break into the standard model at some critical energy,
taken to be $\sim 10^{15}GeV$ and there would be a $\sim 10^{15}GeV$ Higgs
boson associated with that unification. This was proposed in 1974 by Howard
Georgi and Sheldon Glashow gg . The number of generators would be 24 in this
case. By this time there was reason to believe that there were three sets of
the basic four particles, and the two heavier sets were not needed for the
unified theory, but came as redundant copies. The one set had three colours
for the quarks and only one each for the leptons. However, the theory took
left-handed and right-handed spins for the quarks and the electron, but only a
left-handed version of the neutrino, $\nu_{e}$. This unaesthetic break-up was
put in a cumbersome way into two representations of the group. To avoid this
problem in 1975 Harald Fritzsch and Peter Minkowski fm  proposed an $SO(10)$
theory that put all the particles into a single multiplet, but at the expense
of expanding the number of generators to 45. While the $SU(5)$ breaks in a
single step, $SO(10)$ can break into a left-right symmetric model, which then
breaks down into the standard model.

More precise experiments showed that the three strengths do not come together
at a single energy, with the extrapolations forming a triangle. As such, the
basic raison d’etre was lost and the whole unification enterprise seemed to be
in serious jeopardy. Further, since the quarks and leptons could inter-convert
at sufficiently high energies, the proton would be unstable. The predictions
for proton decay of $SU_{C}(4)\bigotimes SU_{f}(4)$ and $SU(5)$ were
experimentally violated and the only reason $SO(10)$ escaped was that it did
not have a definite prediction — which is hardly a recommendation. Something
more was needed to save GUTs.

###  VII.2 Supersymmetry and Unification

Quantum Field Theory (QFT) had problems since its inception. Taken beyond the
lowest level, the calculations for any interaction yield infinite
probabilities, called “divergences”. Since the wave functions are unit norm
vectors in a Hilbert space, it is argued that if they seem to become infinite
they should be “renormalized”. However, it turns out that all theories,
especially gravity, cannot be renormalized. Renormalizing infinity seems
suspect to many in any case. A finite theory, which can then be renormalized
in a meaningful way is needed. A method came from a novel proposal of treating
spinors and tensors as different representations of a unified symmetry, but
using commutators for the products of tensors and anti-commutators for the
products of spinors (or multi-spinors), which complicates the Lie algebra to
what is called a “super-algebra”. This supersymmetry (SUSY) was proposed by
Julius Wess and Bruno Zumino wz . Their cumbersome formalism was put into a
more usable form by Abdus Salam and his student, John Strathdee ss .

Tensors correspond to fields with integer spins in units of Planck’s constant
(divided by $2\pi$), $\hbar$, while spinors correspond to half-integer spins.
The thermodynamic distributions, giving the speeds or energies of the gas
particles for the former are called bosons and for the latter fermions, are
different at low temperatures but behave in much the same way at higher
temperatures. In no way could this be taken to be the unification talked of.
However, there is no spontaneous symmetry breaking mechanism for it either.
Nevertheless, assuming that the symmetry does apply in the sense of a
fundamental theory at some higher energy and then breaks down, it would modify
the standard model significantly by introducing an extra parameter of the
symmetry-breaking energy. In the standard model the number of Higgs bosons is
not constrained. For definiteness one takes the minimal number of Higgs
fields, and this is called the minimal standard model (MSM). With SUSY one
gets an MSSM. It was found that the three constants came together at
$10^{16}GeV$ if SUSY is assumed to be broken at $1TeV=10^{3}Gev$. Thus, if
SUSY is to save GUTs, it must be seen to break at this energy, which was
reached long ago at the Large Hadron Collider at CERN. It has not been seen so
far, and people are jumping through hoops to keep SUSY alive, but she is on
life-support.

Could a higher unification of all forces save the day? The divergences of the
bosons seemed to be canceled at the lowest non-trivial level of calculation of
interaction cross-sections, by those of its fermionic super-partner, and vice
versa. Gravity is a non-renormalizable theory. With SUSY it is called
supergravity (SUGRA). The super-partner of the hypothetical “graviton” was a
spin 3/2 field called a gravitino. In SUGRA it turned out that at the next
non-trivial level the divergences did not cancel. This problem was resolved by
using an extended SUGRA, in which a second gravitino was inserted. This
mechanism worked by introducing a new gravitino at the next level up to the
eighth level, because of the extra parameter inserted, but by the same token
it will not work beyond that pvn . One also needed to go to higher dimensions
of spacetime and strings or membranes instead of point particles sssm .

###  VII.3 Twistor Quantization and Unification

A totally different approach to combine GR and QT was proposed by Roger
Penrose. As he saw it, the problem with combining the two is that since the
field to be quantized is the metric tensor, which defines the distance between
two points, once the quantization is done the spacetime points will no longer
form a continuum, but will be discrete. Thus, we would no longer be able to
use Calculus for Geometry. He pointed out that the quantity that we know is
quantized, and we know how to deal with, is angular momentum or spin angular
momentum. Regarding particles as just a collection of spins, one could only
know of the existence of the other by exchanging a unit of spin, $\hbar/2$. He
wrote a paper entitled Spin Networks in 1967 or 1968, which he gave to one of
us (AQ), when he was suggesting possible lines for AQ’s PhD research (when he
gave one of us, AQ, an unpublished paper on spin networks to see if he wanted
to work on the idea for his PhD). It was later published in 1971 rp1 . He
regarded the bundles of spin as moving at the speed of light, so that he could
express them in terms of spinors (for which he had given a geometrical
visualization rp2 ).

Spinors can be thought of as vectors of the representation space of the
symplectic group, $Sp(2n)$, over $\mathbb{R}$. This group leaves invariant the
null structure, so that all vectors have zero magnitude. Penrose was dealing
with two-component spinors, so that the symplectic structure is provided by
the Levi-Civita symbol, $\epsilon_{AB}~{}(A,B=0,1)$, which is 0 when $A=B$, 1
when $A=0,B=1$ and $-1$ when $A=1,B=0$. Using the set of four Pauli spin
matrices $\sigma_{\mu}^{AA^{\prime}}$, a vector, $x^{\mu}$ can then be written
as a complex matrix, $x^{AA^{\prime}}=x^{\mu}\sigma_{\mu}^{AA^{\prime}}$. If
the vector is null then we can write
$x^{AA^{\prime}}=\xi^{A}\overline{\xi}^{A^{\prime}}$. Here $\xi^{A}$ is a two-
component spinor (for spin vector) and can be visualised as a flagpole lying
along the null cone with a half-plane element stuck on it like a pennant. The
symmetry group for it is $SU(2)$. Multiplying the spinor by a complex number
scales the flagpole by the magnitude of the factor and the pennant is rotated
through twice the argument of the factor. This explains why $SU(2)$ is a
double covering of $SO(3)$, since the scaling by $e^{\iota\pi}$ will leave the
vector unchanged but will reverse the direction of the spinor.

The spinor representation of the covariant derivative operator $\nabla_{\mu}$
is $\nabla_{AA^{\prime}}$ , and the Killing equation can be written for a
spinor as $\nabla^{AC^{\prime}}\xi^{B}+\nabla^{BC^{\prime}}\xi^{A}=0$, which
is called the twistor equation rp3 . The twistor contains the information of
the spinor, $\xi^{A}$ and an $\eta^{A^{\prime}}=\iota x^{AA^{\prime}}\xi_{A}$.
Thus the twistor is given by the pair of spinors,
$Z^{\alpha}=(\xi^{A},\eta^{A^{\prime}}),~{}(\alpha=0,...,3)$. Single or multi-
index twistors are solutions of the zero rest-mass field equations rp4 , with
the number of indices corresponding to the spin of the field. A twistor has
four complex components, corresponding to eight real components, but the
relevant information does not depend on the overall magnitude, as the position
can go on sliding up the flagpole out to infinity. Thus the twistor is an
entire null ray. As such one only needs the projective space of twistors,
which is three complex, or six real, dimensional. The $x^{\mu}$ in the twistor
will generally be complex and the twist in the congruence of geodesics (given
by the relevant spin coefficient, see pr ) is either positive or negative. The
corresponding twistors are said to belong to $C^{+}$ or $C^{-}$. However, when
$x^{\mu}$ is real, there is one extra constraint and there are only five real
components left. Such twistors are called null twistors. Hence there is a
five-dimensional hypersurface, $N$, separating the positive and negative
projective twistor spaces, $C^{+}$ and $C^{-}$. Elements of $N$ correspond to
entire real null rays in real Minkowski space, and the congruence of such rays
is also five dimensional. The Penrose transform between the twistor space and
complexified Minkowski space signifies a duality between the two.

We can generate solutions of the zero rest-mass field equations by using
certain “intertwining integrals” aq7 ; aq8 . This enables us to do contour
integration over $N$, for twistor fields, which will start in one of the six-
dimensional spaces, pass through $N$ and then go back into the original one.
This procedure yields scattering amplitudes, and hence probabilities, for the
various scattering processes in the high energy limit rp5 ; aq9 . Whereas
standard QFT yields infinite probabilities, and SUSY, SUGRA, superstrings and
supermembranes hope to achieve cancellations of the infinities, the twistor
approach yields finite answers automatically, and the renormalization involved
is only division by a magnitude obtained by summing a convergent series of
finite magnitude terms.

##  VIII Complex Lie and Noether Symmetries

In his work, Lie had considered the transformation of the independent and
dependent variable by differentiable transformations, called point
transformations, as the space of the variables can be thought of as two-
dimensional, one for the independent and one for the dependent, with the
specific values of the variables represented by a point. When the variables
are transformed the points get shifted. Thus
$(x,y)\rightarrow(\tilde{x}(x,y),\tilde{y}(x,y))$ is a point transformation.
It is particularly useful to make the differentiability explicit, by using the
transformations in infinitesimal form, so that one sees a smooth path traced
out by the moving point. Thus we can write

$\tilde{x}=x+\epsilon\xi(x,y)+O(\epsilon^{2})~{},~{}\tilde{y}=y+\epsilon\eta(x,y)+O(\epsilon^{2})~{},$
(64)

where $\epsilon>0$ is an infinitesimal. The infinitesimal generator of the
transformation is then

${\bf X}=\xi(x,y)\frac{\partial}{\partial
x}+\eta(x,y))\frac{\partial}{\partial y}~{}.$ (65)

A scalar ODE is said to be symmetric or invariant under a transformation if
the graph of its solution is preserved by the transformation. He provided
various methods to check whether a given system of differential equations is
invariant under a given point transformation and others were developed later,
see e.g. hs . In particular, he discussed when the equations could be
converted to linear form by point transformations, called linearization, and
thereby solved comparatively easily. He especially studied the criteria for
scalar second order ODEs to be so transformed sl3 , see e.g bk . Despite the
importance of PDEs in applications (especially in Fluid Dynamics), we restrict
our attention to ODES. The reason is that while the general solution of an ODE
is unique up to (at most) as many arbitrary constants as the order of the
equation, PDEs, have infinitely many solutions. The definite statements
available for ODEs are lost for PDEs.

For second order ODEs, $y^{\prime\prime}=w(x,y,y^{\prime})$ we need to include
the first derivative as if it were an independent variable and use a 3-d
space. For higher order scalar ODEs the space has to be extended (or
prolonged) to include all derivatives up to the next to highest derivative, as
if all are independent of each other. The graph of the solution must then
remain invariant in the projected two-dimensional space. Thus for the second
order scalar ODE the prolonged infinitesimal generator is

${\bf X}^{[1]}=\xi(x,y)\frac{\partial}{\partial
x}+\eta(x,y))\frac{\partial}{\partial
y}+\eta^{[1]}(x,y,y^{\prime}))\frac{\partial}{\partial y^{\prime}}~{},$ (66)

where $\eta^{[1]}$ is given in terms of $\eta$ and the derivative of $\xi$. As
the order increases the generator gets prolonged further by adding another
term involving the derivative with respect to the next order derivative. For
an $n^{th}$ order ODE we need to prolong up to $\eta^{[n-1]}$, where

$\eta^{[n-1]}(x,y,\ldots,y^{(n-1)})=\frac{d}{dx}\eta^{[n-2]}(x,y,\ldots,y^{(n-3)})-y^{(n-1)}\frac{d}{dx}\xi(x,y)~{}.$
(67)

Lie had proved that second order scalar ODEs are linearizable only if they are
at most cubically semilinear and the four coefficients satisfy a set of four
first derivative constraints involving two arbitrary functions. Tresse at
reduced them to two second order constraints without the arbitrary functions.
Note that these equations do not have to be solved but only checked. Methods
were later developed to simply write down the solution of second order
quadratically semilinear linearizable systems mq1  and were later generalized
to the cubically semilinear case mq2 ; mq22 .

Lie used complex functions of complex variables for his analysis of
differential equations. Of course, he had to take the functions to be not only
continuous but differentiable. Now complex differentiability implies
analyticity. He did not explicitly use this fact in his analysis. However, it
obviously has significant consequences, as the dependent and independent
variables will be constrained by the Cauchy-Riemann (CR) equations. Thus, even
a scalar first order ODE, split into its real and imaginary parts, is actually
a system of four first order PDEs. This point was noted and exploited by Ali,
Mahomed and Qadir amq1 ; amq12 , who called it complex symmetry analysis
(CSA). If the dependent variable is $w=u+\iota v$ and independent variable
$z=x+\iota y$, one function of one variable yields four functions of two
variables, and so a system of PDEs. For the resulting system to be of ODEs,
one must restrict the independent variable to the real part, $x$, only. Thus
the complex scalar ODE will now correspond to a system of two ODEs along with
a set of CR equations. A first order ODE will then split into the two first
order ODEs, $u^{\prime}=f(x,u,v)$ and $v^{\prime}=g(x,u,v)$, by taking the
real and imaginary parts of the equation. The CR equations for the
corresponding system will be $f_{u}=g_{v}~{},~{}f_{v}=-g_{u}$. It is this
system that would now be analysed. The formalism is easily extended to higher
order ODEs.

The complex generator ${\bf W}$ splits into its real and imaginary parts, as

$\displaystyle{\bf W}$ $\displaystyle=$
$\displaystyle\xi(x,w)\frac{\partial}{\partial
x}+\eta(x,w))\frac{\partial}{\partial w}$ (68) $\displaystyle=$
$\displaystyle\xi^{r}(x,u,v)\frac{\partial}{\partial x}$ $\displaystyle+$
$\displaystyle\frac{1}{2}[(\eta^{r}(x,u,v)\frac{\partial}{\partial
u}+\eta^{i}(x,u,v)\frac{\partial}{\partial
v})+\iota(\eta^{i}(x,u,v)\frac{\partial}{\partial
u}-\eta^{r}(x,u,v)\frac{\partial}{\partial v})]$ $\displaystyle=$
$\displaystyle{\bf X}+\iota{\bf Y}~{},$

the half coming from the requirement that ${\bf W}w=1$. The prolongation is,
of course, still more cumbersome to write, but easy to obtain. Ali, Mahomed
and Qadir applied their CSA to the linearization of second order scalar ODEs
amq2 , and found that they could linearize the complex scalar second order ODE
corresponding to a 2-d non-linearizable system. In fact, it was found that
systems with less than the eight generators required for a single scalar
second order ODE may correspond to a complex scalar linearizable ODE. Further,
systems without enough infinitesimal symmetry generators to be solvable by
symmetry methods, there was an example of a second order 2-d system that has
no generator, but corresponds to a complex scalar second order ODE, and its
solution was obtained sqa ; sqa1 ; sqa2 .

Extending CSA to Lagrangians to be able to use it for Noether symmetries is
non-trivial. The reason is that the Lagrangian is necessarily defined for the
real domain. The extension presents us with two basic problems that may be
thought of as two faces of the same coin. The Lagrangian is the kernel of a
functional, and functionals map functions into $\mathbb{R}$ not $\mathbb{C}$.
The whole purpose of defining it is to find the form of the dependent variable
for which the functional takes a minimum value. As such the image space for it
has to be an ordered set like $\mathbb{R}$, and not a partially ordered set
like $\mathbb{C}$. The first problem is simply dealt with by fiat, redefining
the range of the functional to be $\mathbb{C}$. The associated problem is
dealt with by using the magnitude of the functional to be minimal, rather the
functional itself amq3 .

It turned out that invariants could be obtained for the complex Lagrangians
and provide new insights into the physical significance of the Noether
invariant faq . In particular, when a complex scalar harmonic oscillator
equation is split, it gives a system of coupled harmonic oscillators and the
Noether invariant gives the energy in each oscillator and identifiably
separately in the field between them. This was expressed as “seeing the energy
in the field through complex glasses”. It is easy to extend to the time
dependent harmonic oscillator and see how the energy in the coupled
oscillators and in the field are transferred about. It turns out that the
complex Noether symmetries do not provide any new invariants, but they do
provide them more easily and put them into different combinations that may be
more insightful sqf .

##  IX Concluding Remarks

Mathematically, Noether symmetries provide double reduction of the order
and/or the number of variables in a differential equation. One of the methods
of achieving a reduction is by determining an invariant combination of the
dependent and independent variables and their derivatives. Writing the
invariant as an arbitrary constant, it can be used to write, say, the highest
derivative in the combination, in terms of the other variables. It is
particularly needed for PDEs, as “solving them” without the boundary
conditions is not very meaningful, and the invariants can incorporate the
boundary conditions. While other methods could be used for the same
mathematical purpose, they cannot provide the physical insights of the
“Noether charge” that the invariants do. In this paper the physical insights
obtained from Lie and Noether symmetries and invariants was reviewed. Worth
special mention is the use of Noether symmetries in Quantum Theory, which has
not received enough attention from those working in Symmetry Analysis. Since
one of the most important outstanding problems in fundamental Physics is the
unification of Quantum Theory and General Relativity, the introduction of new
methods may lead in solving it. One such method is the explicit use of complex
analyticity of Lie groups that was also briefly mentioned, but there are many
others.

There is considerable activity in spacetime symmetries gsh , including not
only isometries (also called KVs), but also the scaling symmetry of the metric
tensor, called a homothety (the special case of a constant conformal factor).
There is also interest in the symmetries of the Ricci (Ric), Riemann (Rie) and
Weyl (C) tensors, which are called Ricci, Riemann and Weyl collineations,
given by $\pounds_{\bf k}{\bf Ric}=0$, $\pounds_{\bf k}{\bf Rie}=0$ and
$\pounds_{\bf k}{\bf C}=0$. Much of it is on their physical significance
tm1990 ; bq1993 ; hall1996 ; bqaa ; cb2002 ; bkq ; hqs  and many problems
continue to arise, which would be worth exploring. A different line was
developed of obtaining all metrics with isometry groups containing a minimal
group, which was called a “complete classification” of spacetimes with the
minimal symmetry bq1987 ; qz1995 ; dz1997 . The idea was to be able to pick up
spacetimes that have the desired symmetry and use the metric to obtain the
stress-energy tensor by constructing the Einstein tensor for it. Thus one
manages to “solve the Einstein equations without having to solve them”. This
was initially done for isometries of spherically symmetric, static metrics,
but was then extended to isometry groups of only three dimensions. It was
further extended to homotheties and collineations. Classification by Noether
symmetries feroze2 ; feroze3 ; feroze4 ; feroze5 ; feroze6 ; jamil1 ; jamil2
has yielded solutions of the Einstein equations along with their conserved
quantities. This line is also worth pursuing. A more difficult problem is to
completely classify by a two-dimensional isometry group, like cylindrical
symmetry. If that is done, it may be possible to extend it to homotheties and
collineations.

As mentioned in Section 7, there is great need for workers in Lie symmetry
analysis to enter into QFT, as the work using symmetries is pursued by
physicists who are not generally well-versed in the methods developed by Lie
and subsequent workers. (However, it is not enough for the workers to simply
take an open problem in QFT, without understanding what is entailed in it. It
would be all too easy to get spurious and irrelevant results if the context is
not understood.) In particular, any new solutions of the Newman-Penrose
equations rp2 ; pr  would be extremely useful as they come equipped with their
physical significance. Since they deal with invariants a Noether symmetry
formulation for them may be possible, and would be a major contribution.

Coming to Section 8, the field is wide open. Very powerful complex methods
have been developed, where they can be applied. One direction to go in
following this up, is to check open problems to see if the powerful complex
methods can be used there or not. Unfortunately, one runs the risk of wasting
a lot of time finding that the methods are not applicable to the chosen
problem. One must also bear in mind that while CSA provides solutions of
systems of ODEs that cannot be solved by traditional symmetry methods, Noether
symmetries will not give new invariants, but only new combinations of the old
invariants. As pointed out above, they can provide new insights into the
physical significance of the invariants. A very much more important line of
work to follow in this area, is to find an explanation of why the complex
methods provide answers, where they do. If one can answer that question, one
should be able to formulate criteria for the applicability of complex methods
and thus avoid wasting time on searching for problems where complex methods
can be applied and go directly to those where it can; or perhaps even find
ways to tweak the methods to make them more generally applicable.

## Acknowledgements

We would like to thank unknown referees for their valuable comments and
suggestions.

## References

  * (1) Copernicus, N. Three Copernican Treatises. Dover Publications: Mineola, New York, USA, 2004.
  * (2) Kepler, J. Harmonices Mundi (Harmony of the Worlds), 1619; To be found in Johannes Kepler Gesammelte Werke, KGW Vol. 6, Capar, M., ed., 1940; 2nd ed., 1981; ISBN 3-406-01648-0.
  * (3) Newton, I. Philosophi Naturalis Principia Mathematica; Benjamin Motte, 1687; Eng. Tr. Cajori, F. Sir Isaac Newton’s mathematical principles of natural philosophy and the system of the world; University of California Press, 1947.
  * (4) Lagrange, J. L. Mcanique Analytique, 2nd ed.; Courcier: Paris, France, 1811; Reissued by Cambridge University Press: Cambridge, MA, USA, 2009.
  * (5) Hamilton, W.R. On a general method in dynamics by which the Study of the motions of all free Systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function. Philos. Trans. R. Soc. Lond. 1834, 125, 247-308.
  * (6) Hamilton, W.R. Second essay on a general method in dynamics. Philos. Trans. R. Soc. Lond. 1835, 125, 95-144.
  * (7) Euler, L. Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti. Euler Archive-All Works, 1744. The Latin title translates as a method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense.
  * (8) Lagrange, J.L. Suite des rflexions sur la résolution algébrique des équations, Section troisieme, De la résolution des équations du cinquieme degré $\&$ des degrés ultérieurs. Nouv. Mem. Acad. R. Sci. Berlin 1771, 202-3, 138-254.
  * (9) Abel, N. H. Mémoire sur les équations algébriques, où on démontre l’impossibilité de la resolution de l’équation générale du cinquiéme dégré ; Christiania, De l’imprimerie de Groendahl, 1824; in Sylow, L., Lie, S., Eds.; Cuvres Completes de Niels Henrik Abel (in French), 2nd ed.; Groendahl $\&$ Soen, 1881.
  * (10) Galois, É. Analyse d’un mémoire sur la résolution algébrique des équations. Bull. des Sci. Math. 1830, 13, 171-172.
  * (11) Lie, S. Klassification und Integration von gewönlichen Differentialgleichungen zwischen $x,y$, die eine Gruppe von Transformationen gestaten I, II, III. Arch. Math. 1883, 8, 187-224, 249-288 and 371-458 respectively.
  * (12) Qadir, A. Relativity: An Introduction to the Special Theory; World Scientific: Singapore, 1989.
  * (13) Barbour, J.B.The Discovery of Dynamcis; Oxford University Press: Oxford, New York, USA, 2001.
  * (14) Lie, S. Differential Equations; Chelsea Publ. Co.: New York, USA, 1967.
  * (15) Lie, S. Lectures on Differential Equations with Known Infinitesimal Transformations; Teubner: Leipzig, Germany (in German, Lie’s lectures by Sheffers, G.), 1891.
  * (16) Lie, S. Theorie der Transformationsgruppen, III; Teubner: Leipzig, Germany, 1893.
  * (17) Bernoulli, D. Hydrodynamica, sive de viribus et motibus fluidorum commentarii, (in Latin, source ETH-Bibliothek Zrich, Rar 5503); Johannis Reinholdi Dulseckeri, 1738; doi:10.3931/e-rara3911.
  * (18) Avogadro, A. Masses of the Elementary Molecules of Bodies. J. Phys. (Paris) 1811, 73, 58-76.
  * (19) Faraday, M. Experimental Researches in Electricity-Vols. 1 and 2; Richard, E., John, E., eds.; Richard and John Edward Taylor, 1839; Vol. 3. Richard Taylor and William Francis, 1855.
  * (20) Maxwell, J.C. VII. A dynamical theory of the electromagnetic field. Phil. Trans. R. Soc. Lond. 1865, 155, 459-512.
  * (21) Qadir, A. Einstein’s General Theory of Relativity; Cambridge Scholars Pulishing: UK, 2020.
  * (22) Einstein, A. Zur elektrodynamik bewegter körper. Ann. Phys. (Berl.) 1905, 17, 891-921.
  * (23) Einstein, A.; Grossmann, M. Entwurf einer verallgemeinerten relativitätstheorie und eine theorie der gravitation: I. Physikalischer teil von A. Einstein; II. Mathematischer teil von M. Grossmann. Z. für Math. Phys. 1913, 62, 225-244, 245-261.
  * (24) Einstein, A.;Grossmann, M. Kovarianzeigenschaften der feldgleichungen der auf die verallgemeinerte relativitätstheorie gegrndeten gravitationstheorie. Z. für Math. Phys. 1914, 63, 215-225.
  * (25) Einstein, A. Zur allgemeinen relativitätstheorie. Preuss. Akad. der Wissen. Sitz. 1915 (part 2), 844-847.
  * (26) Einstein, A. Grundlage der allgemeinen relativitätstheorie. Ann. Phys. (Berl.) 1916, 49, 769-822.
  * (27) Noether, E. Invariante Variationsprobleme. Göttingen Math. Phys. Kl. 1918, 2, 235; English translation by Tavel, M.A. Invariante variation problems. Transp. Theory Statist. Phys. 1971, 1, 186-207.
  * (28) Olver, P. Applications of Lie Groups to Differential Equations; Springer-Verlag: New York, USA, 1986.
  * (29) Bluman, G.; Anco, S.C. Symmetries and Integration Methods for Differential Equations; Springer Applied Mathematics Series 154; Springer-Verlag: New York, USA, 2002.
  * (30) Ibragimov, N. H. CRC Handbook of Lie Group Analysis of Differential Equations: Symmetries, Exact Solutions and Conservation Laws; CRC Press: Boca Raton, FL, USA, 1994.
  * (31) Bluman, G.; Cheviakov, A.; Anco, S. C. Applications of Symmetry Methods to Partial Differential Equations: Springer Applied Mathematics Series 168; Springer-Verlag: New York, USA, 2010.
  * (32) Ibragimov, N. H.; Kara, A.H.; Mahomed, F. M. Lie-Bäcklund and Noether Symmetries with Applications. Nonlinear Dyn. 1998, 15, 115-136.
  * (33) Ibragimov, N. H. The Noether identity, in Continuum Dynamics. 1979, 38, 26-32. Publisher: Institute of Hydrodynamics, USSR Acad. Sci., Siberian Branch, Novorsibirsk. (Russian).
  * (34) Capozziello, S.; De Ritis R.; Rubano, C.; Scudellaro, P. Nöther symmetries in cosmology. Riv. Nuovo Cimento 1996 19, 1-114.
  * (35) Capozziello, S.; Stabile, A.; Troisi, A. Spherically symmetric solutions in f(R)-gravity via Noether Symmetry Approach. Class. Quantum Grav. 2007, 24, 2153-2166
  * (36) Neumann, J.v. A Model of General Economic Equilibrium. Rev. Econ. Stud. 1945, 13, 1–9.
  * (37) Qadir, A. Economic implications of Von Neumann’s general equilibrium model, Pak. Ec. & Soc. Rev. 1975, XIII, 213–214.
  * (38) Qadir, A.; Qadir, K. Inflation in a growing economy. Pak. Ec. & Soc. Rev. 1981, XIX, 85–92.
  * (39) Misner, C. W.; Thorne, K. S.; Wheeler, J. A. Gravitation; W.H. Freeman: San Francisco, USA, 1973.
  * (40) Feroze, T.; Mahomed, F. M.; Qadir, A. The connection between isometries and symmetries of geodesic equations of underlying spaces, Nonlinear Dyn. 2006, 45, 65.
  * (41) Feroze, T. New conserved quantities for the spaces of different curvatures, Mod. Phys. Lett. A 2010, 25, 1107.
  * (42) Feroze, T.; Hussain, I. Noether symmetries and conserved quantities for spaces with a section of zero curvature. J. Geom. Phys. 2011, 61, 658.
  * (43) Ali, F.; Feroze, T. Classification of Plane Symmetric Static Space-Times According to Their Noether Symmetries. Int. J. Theor. Phys. 2013, 52, 3329–3342.
  * (44) Ali, F.; Feroze, T.; Ali, S. Complete classification of spherically symmetric static spacetimes via Noether symmetries. Theor. Math. Phys. 2015, 184, 973–985.
  * (45) Ali, F.; Feroze, T. Complete classification of cylindrically symmetric static spacetimes and the corresponding conservation laws. Sigma Math. 2016, 4, 50.
  * (46) Jamil, B.; Feroze, T. Conservation laws corresponding to the Noether symmetries of the geodetic Lagrangian in spherically symmetric spacetimes. Int. J. Mod. Phys. D 2017, 26, 1741006.
  * (47) Jamil, B.; Feroze, T.; Vargas, A. Geometrical/Physical interpretation of the conserved quantities corresponding to Noether symmetries of plane symmetric spacetimes. Adv. Math. Phys. 2017, 2017, 4384093.
  * (48) Tsamparlis, M.; Paliathanasis, A. Lie and Noether symmetries of geodesic equations and collineations. Gen. Relativ. Gravit. 2010, 42, 2957–2980.
  * (49) Tsamparlis, M.; Paliathanasis, A. The geometric nature of Lie and Noether symmetries. Gen. Relativ. Gravit. 2011, 43, 1861-1881.
  * (50) Paliathanasis, A.; Tsamparlis, M.; Mustafa, M.T. Symmetry analysis of the Klein-Gordon equation in Bianchi I spacetimes. Int. J. Geom. Meth. Mod. Phys. 2015, 12, 1550033.
  * (51) Ali, S.; Hussain, I. A study of positive energy condition in Bianchi V spacetimes via Noether symmetries. Eur. Phys. J. C 2016, 76.
  * (52) Hickman, M.; Yazdan, S. Noether symmetries of Bianchi type II spacetimes. Gen. Relativ. Gravit. 2017, 49, 65.
  * (53) Camci, U. Symmetries of geodesic motion in Gödel-type spacetimes. J. Cosmol. Astropart. Phys. 2014, 07, 002.
  * (54) Camci, U. Noether gauge symmetries of geodesic motion in stationary and nonstatic Gödel-type spacetimes. Int. J. Mod. Phys. Conf. Series 2015, 38, 1560072.
  * (55) Camci, U; Yildirim, A. Noether gauge symmetry classes for pp-wave spacetimes. Int. J. Geom. Meth. Mod. Phys. 2015, 12, 1550120.
  * (56) Qadir, A.; Ziad, M. The classification of spherically symmetric spacetimes. Nuovo Cimento B 1995, 110, 317–334.
  * (57) Stephani, H. Differential Equations: Their Solution Using Symmetries; Cambridge University Press: Cambridge, UK, 1989.
  * (58) Qadir, A.; Quamar, J.; Rafique, M. Carter’s fourth invariant of the motion. Phys. Lett. A 1985, 109, 90–92.
  * (59) Qadir, A.; Siddiqui, A.A. Similarities between classical timelike geodesics in a naked Reissner-Nordstrom singularity background and the behaviour of electrons in quantum theory. Int. J. Mod. Phys. D 2007, 15, 25–34.
  * (60) Landau, L.D.; Lifshitz, E.M. The Classical Theory of Fields, 4th ed.; Elsevier: Burlinghton MA, USA, 1975.
  * (61) Jackson, J.D. Classical Electrodynamics, 3rd ed.; John Wiley and Sons: New York, USA, 1998.
  * (62) Sakurai, J.J.; Neapolitino, J. Modern Quantum Mechanics, 3rd ed.; Cambridge University Press: Cambridge, UK, 2021.
  * (63) Salam, A. Gauge unification of fundamental forces. Rev. Mod. Phys. 1980, 52, 525–538.
  * (64) Gell-Mann, M. The eightfold way, in The Eightfold Way; Gell-Mann, M., Ne’eman, Y., Eds.; W.A. Benjamin Inc.: New York, USA, 1964.
  * (65) Higgs, P.W. Broken symmetries and the masses of gauge bosons. Phys. Rev. Lett. 1964, 13, 508–509.
  * (66) Pati, J.C.; Salam, A. Lepton number as the fourth “color”. Phys. Rev. D 1974, 10, 275-289.
  * (67) Georgi, H.; Glashow, S.L. Unity of all elementary particle forces. Phys. Rev. Lett. 1974, 8, 438–441.
  * (68) Fritzsch, H.; Minkowski, P. Unified interactions of leptons and hadrons. Ann. Phys. 1975, 93, 193–266.
  * (69) Wess, J.; Zumino, B. Supergauge transformations in four dimensions. Nucl. Phys. B 1974, 70, 39–50.
  * (70) Salam, A.; Strathdee, J. Super-symmetry and non-Abelian gauges. Phys. Lett. B 1974, 51, 353–355.
  * (71) Van Nieuwenhuizen, P. Supergravity. Phys. Rep. 1981 68, 189–398.
  * (72) Bandos, I.; Pasti, P.; Sorokin, D.; Tonin, M.; Volkov, D. Superstrings and supermembranes in the doubly supersymmetric geometrical approach. Nucl. Phys. B 1995, 446, 79–118.
  * (73) Penrose, R. Angular momentum: an approach to combinatorial spacetime. In Quantum Theory and Beyond; Bastin, T., ed.; Cambridge University Press: Cambridge, UK, 1971; pp.151–180.
  * (74) Penrose, R. An Analysis of the Structure of Space-Time. Adams Prize Essay; Cambridge University: Cambridge, UK, 1967.
  * (75) Penrose, R. Twistor algebra. J. Math. Phys. 1967, 8, 345–366.
  * (76) Penrose, R. Solution of sero rest-mass field equations. J. Math. Phys. 1969, 10, 38–39.
  * (77) Penrose, R.; Rindler, W. Spinors and Space-Time: Vol. 2, Spinor and Twistor Methods in Space-Time Geometry, 1st ed.; Cambridge University Press: New York, USA, 1988.
  * (78) Qadir, A. Twistor Fields. PhD thesis, London University, UK, 1971.
  * (79) Qadir, A. An interesting representation of Lie algebras of linear groups. Int. J. Theor. Phys. 1976, 14, 74–101.
  * (80) Penrose, R., MacCallum, M. A. H. Twistor theory: An approach to the quantisation of fields and space-time. Phys. Rep. 1973, 6, 241–315.
  * (81) Qadir, A. Penrose graphs. Phys. Rep. 1978, 39, 131–167.
  * (82) Bluman, G.W.; Kumei, S. Symmetries and Differential Equations; Springer Verlag: New York, USA, 1989.
  * (83) Tresse, A. Sur les Invariants Différentiels des Groupes Continus de Transformations. Acta Math. 1894, 18, 1–88.
  * (84) Mahomed, F.M.; Qadir, A. Linearization criteria for a system of second order quadratically semi-linear ordinary differential equations. N͡onlinear Dyn. 2007, 48, 417–422.
  * (85) Mahomed, F. M.; Qadir, A. Invariant linearization criteria for systems of cubically semi-linear second order ordinary differential equations. J. Nonlin. Math. Phys. 2009, 16 283–298.
  * (86) Fredericks, E.; Mahomed, F. M.; Momoniat, E.; Qadir, A. Constructing a space from the system of geodesic equations. Comp. Phys. Commun. 2008, 179, 438-442.
  * (87) Ali, S.; Mahomed, F. M.; Qadir, A. Complex Lie symmetries for scalar second-order ordinary differential equations. Nonlinear Anal. Real World Appl. 2009, 10, 3335–3344.
  * (88) Ali, S. Complex Lie Symmetries for Differential Equations. Ph.D. Thesis, National University of Sciences and Technology, Islamabad, Pakistan, 2009.
  * (89) Ali, S.; Mahomed, F. M.; Qadir, A. Linearizability criteria for systems of two second order ordinary differential equations by complex methods. Nonlinear Dyn. 2011, 66, 77–88.
  * (90) Safdar, M.; Qadir, A.; Ali, S. Linearizability of systems of ordinary differential equations obtained by complex symmetry analysis. Math. Probl. Eng. 2011, 2011, 171834.
  * (91) Ali, S.; Qadir, A.; Safdar, M. Inequivalence of classes of linearizable systems of cubically semi-linear ordinary differential equations obtained by real and complex symmetry analysis. Math. Comput. Appl. 2011, 16, 923–934.
  * (92) Ali, S.; Safdar, M.; Qadir, A. Linearization from complex Lie point transformations. J. Appl. Math. 2014, 2014, 793247.
  * (93) Ali, S.; Mahomed, F.M.; Qadir, A. Complex Lie symmetries for variational problems. J. Nonlinear Math. Phys. 2008, 15, 25–35.
  * (94) Farooq, M.U.; Ali, S.; Qadir, A. Invariants of two-dimensional systems via complex Lagrangians with applications. Commun. Nonlinear Sci. Numer. Simul. 2010, 16, 1804–1810.
  * (95) Safdar, M.; Qadir, A.; Farooq, M.U. Comparison of Noether symmetries and first integrals of two-dimensional systems of second order ordinary differential equations by real and complex methods. Symmetry 2019, 11, 1180.
  * (96) Hall, G. S.Symmetries and Curvature Structure in General Relativity; World Scientific: Singapore, 2004.
  * (97) Tsamparlis, M.; Mason, D. P. Ricci collineation vectors in fluid space-times. J. Math. Phys. 1990, 31, 1707–1722.
  * (98) Bokhari, A. H.; Qadir, A. Collineations of the Ricci tensor. J. Math. Phys. 1993, 34, 3543–3552.
  * (99) Hall, G. S.; Roy, I.; Vaz, E. G. L. R. Ricci and matter collineations in space-time. Gen. Relativ. Gravit. 1996, 28, 299–310.
  * (100) Bokhari, A. H.; Qadir, A.; Ahmed, M. S.; Asghar, M. Classification of spherically symmetric static space-times by their curvature collineations. J. Math. Phys. 1997, 38, 3639–3649.
  * (101) Camci, U.; Barnes, A. Ricci collineations in Friedmann-Robertson-Walker spacetimes. Class. Quantum Grav. 2002, 19, 393–404.
  * (102) Bokhari, A.H.; Kashif, A.R.; Qadir, A. A Complete Classification of Curvature Collineations of Cylindrically Symmetric Static Metrics. Gen. Relativ. Gravit. 2003, 35, 1059–1076.
  * (103) Hussain, I.; Qadir, A.; Saifullah, K. Weyl Collineations that are not Curvature Collineations. Int. J. Mod. Phys. D 2005, 14, 1431–1437.
  * (104) Bokhari, A. H.; Qadir, A. Symmetries of static, spherically symmetric space-times. J. Math. Phys. 1987, 28, 1019–1022.
  * (105) Qadir, A.; Ziad, M. The classification of spherically symmetric space-times. Nuovo Cimento B 1995, 110, 317–334.
  * (106) Ahmad, D.; Ziad, M. Homothetic motions of spherically symmetric space-times. J. Math. Phys. 1997, 38, 2547–2552.