CP3-21-45

Noether Symmetries, Dynamical Constants of Motion,

and Spectrum Generating Algebras

Daddy Balondo Iyelaa and Jan Govaertsb,c,111Fellow of the Stellenbosch
Institute for Advanced Study (STIAS), Stellenbosch, Republic of South Africa

aDépartement de Physique, Université de Kinshasa (UNIKIN),
B.P. 190 Kinshasa XI, Democratic Republic of Congo
E-mail<EMAIL_ADDRESS>
ORCID: http://orcid.org/0000-0001-8282-7914
bCentre for Cosmology, Particle Physics and Phenomenology (CP3),
Institut de Recherche en Mathématique et Physique (IRMP),
Université catholique de Louvain (UCLouvain),
2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium
E-mail<EMAIL_ADDRESS>
ORCID: http://orcid.org/0000-0002-8430-5180
cInternational Chair in Mathematical Physics and Applications (ICMPA–UNESCO
Chair)
University of Abomey-Calavi, 072 B.P. 50, Cotonou, Republic of Benin

When discussing consequences of symmetries of dynamical systems based on
Noether’s first theorem, most standard textbooks on classical or quantum
mechanics present a conclusion stating that a global continuous Lie symmetry
implies the existence of a time independent conserved Noether charge which is
the generator of the action on phase space of that symmetry, and which
necessarily must as well commute with the Hamiltonian. However this need not
be so, nor does that statement do justice to the complete scope and reach of
Noether’s first theorem. Rather a much less restrictive statement applies,
namely that the corresponding Noether charge as an observable over phase space
may in fact possess an explicit time dependency, and yet define a constant of
the motion by having a commutator with the Hamiltonian which is nonvanishing,
thus indeed defining a dynamical conserved quantity. Furthermore, and this
certainly within the Hamiltonian formulation, the converse statement is valid
as well, namely that any dynamical constant of motion is necessarily the
Noether charge of some symmetry leaving the system’s action invariant up to
some total time derivative contribution. The present contribution revisits
these different points and their consequences, straightaway within the
Hamiltonian formulation which is the most appropriate for such issues.
Explicit illustrations are also provided through three general but simple
enough classes of systems.

##  1 Introduction

The first Noether theorem, relevant to global continuous Lie symmetries of a
dynamical system, is standard textbook material (such discussions seldom
include the second and third Noether theorems as well, relevant to gauge or
local continuous Lie symmetries; see for instance Ref.[1]). Usually, in
physics textbooks, the presentation proceeds from the variational principle
considered within the Lagrangian formulation for the system’s dynamics, with a
regular Lagrange function $L(q^{\alpha},\dot{q}^{\alpha})$ which is defined
over configuration space spanned by the coordinates $q^{\alpha}(t)$, with of
course $\dot{q}^{\alpha}(t)=dq^{\alpha}(t)/dt$. Often this Lagrangian is taken
not to have any explicit time dependency in the time evolution coordinate $t$,
and is assumed to be such that the differential
$(dt\,L(q^{\alpha},\dot{q}^{\alpha}))$ be exactly invariant under some class
of global continuous transformations in the variables $q^{\alpha}$ (or
separately in $t$). Consequently such transformations correspond to some form
of geometrical symmetry in the space $q^{\alpha}$ under which the
configuration space Euler-Lagrange equations of motion are covariant, thus
indeed inducing a mapping between solutions to these equations, namely a
symmetry of these equations leaving invariant the set of their solutions.

As is well known, Noether’s first theorem then implies the existence of
conserved Noether charges or constants of the motion, in one-to-one
correspondence with each of the independent constant symmetry transformation
parameters. These Noether charges are defined over the associated phase space
formulation of the same dynamics spanned by the coordinates
$(q^{\alpha},p_{\alpha})$, $p_{\alpha}$ being the momenta canonically
conjugate to $q^{\alpha}$. While under such specific circumstances, these
charges are such that they commute with the system’s canonical Hamiltonian
$H(q^{\alpha},p_{\alpha})$, namely possess vanishing Poisson brackets with
that Hamiltonian for the classical dynamics, and vanishing quantum commutators
with the quantum Hamiltonian for the quantum dynamics (provided the symmetry
is not broken by quantum anomalies incurred through quantisation). Presumably
this is also the reason why in quantum mechanics textbooks more often than not
a symmetry is taken to be associated to a quantum observable which necessarily
commutes with the quantum Hamiltonian.

However this need not be so. Indeed there are known instances of dynamical
constants of the motion, which even though conserved through time evolution,
do not commute with the Hamiltonian (see for instance Ref.[2]). Their
conservation is then made possible by having an explicit time dependency for
the corresponding Noether charges, $Q(q^{\alpha},p_{\alpha};t)$, such that
their Hamiltonian equation of motion implies the following relation between
that time dependency and, say in the classical context, their Poisson bracket
with the Hamiltonian,

$\frac{\partial Q}{\partial
t}=-\left\\{Q,H\right\\},\qquad\frac{dQ}{dt}=\frac{\partial Q}{\partial
t}\,+\,\left\\{Q,H\right\\}=0.$ (1)

The reach of Noether’s first theorem is thus more general still than under the
specific restricted conditions recalled above. First a set of equations of
motion may possess symmetries beyond geometrical symmetries in configuration
space, the latter thus acting on the configuration space coordinates, on the
one hand, and separately on the conjugate momenta only through an induced
action, on the other hand. By extending the analysis based on the variational
principle to the first-order Hamiltonian phase space action for the dynamics,
symmetries more general than such geometrical transformations may be brought
to the fore, with their own conserved Noether charges on account of Noether’s
first theorem which is operative now over phase space, while such symmetries
then mix all phase space configuration space and conjugate momenta coordinates
into one another and no longer separately as do geometrical symmetries. Famous
examples of such symmetries of the Hamiltonian formulation which are not
manifest geometrical symmetries of the Lagrangian formulation are, the
dynamical SO(4) symmetry of the Coulomb-Kepler problem in 3 dimensional
euclidean space, and the dynamical SU($d$) symmetry of the spherical symmetric
harmonic oscillator in $d$ dimensional euclidean space. The conserved charges
of these examples and symmetries however, do commute with the Hamiltonian, and
do not possess an explicit time dependency in phase space.

Furthermore since equations of motion follow from the variational principle up
to terms in the varied action that are total time derivatives (i.e., surface
terms in time), symmetry transformations under which the equations of motion
are covariant may leave the action invariant but only up to some total time
derivative contribution which is induced by the transformation. Thus in actual
fact the (Lagrangian or Hamiltonian) action need not be exactly invariant
under a symmetry transformation, but only up to a total differential in time
(as is the case, for instance, for the dynamics of the free non-relativistic
massive particle in $d$ dimensional euclidean space under Galilean boost
transformations which are certainly symmetries of that system, as part of the
full Galilean symmetry group of Newton’s mechanics in inertial frames)[1, 2].
Of course Noether’s first theorem applies to such circumstances as well,
leading again to constants of the motion as conserved Noether charges.

The scope of Noether’s original theorems is thus wider than what is usually
presented in most textbooks. In particular, when the global continuous
symmetry transformations of which the parameters are constants involve
(configuration space or phase space) functions which carry an explicit time
dependency (think of Galilean boosts, for instance), then necessarily as phase
space functions the associated Noether charges also possess an explicit time
dependency, and thus do not commute with the Hamiltonian — hence the name of
dynamical constants of motion. Nevertheless, even under such circumstances any
element of the enveloping algebra of all conserved charges is again a
conserved charge, while the set of conserved charges is closed under Poisson
brackets or commutation relations, whether these are dynamical conserved
charges or not.

In the context of the quantised system, these observations have the following
consequences. As a subset of this Lie algebra of all constants of the motion,
the Lie sub-algebra of those symmetry conserved charges which do not possess
any explicit time dependency has vanishing commutators with the quantum
Hamiltonian. Hence the spectrum of energy eigenstates and their degeneracies
is organised into irreducible representations of that Lie sub-algebra, namely
more specifically of its primary generators. This is the situation most often
considered in the context of quantum mechanics, which is such that symmetry
transformations associated to time independent conserved charges act by
mapping into one another those degenerate energy eigenstates that belong to a
same energy level. Note that these symmetries may extend beyond the Lagrangian
geometrical ones, as do for instance the SO(4) or SU($d$) dynamical symmetries
of the two examples mentioned above. On the other hand, for dynamical
conserved charges with their explicit time dependency which therefore do not
commute with the quantum Hamiltonian, necessarily the action of the associated
symmetries maps between (linear combinations of) energy eigenstates that
belong to different energy levels, thereby generating part, if not all, of the
complete Hilbert space of quantum energy eigenstates for the system. In other
words, the enveloping Lie algebra of all (dynamical and non-dynamical)
conserved charges of a system is a spectrum generating Lie algebra for that
system. The Lie sub-algebra of the primary generators of the enveloping Lie
algebra of all conserved charges is often referred to as a spectrum generating
algebra or a dynamical algebra of the system (even when no explicit time
dependency is involved, which may be sometimes source of some confusion).

Thus given Noether’s first theorem, and this now in full generality, whenever
the first-order Hamiltonian action of a dynamical system is invariant, up to
some total time derivative contribution, under any global continuous
transformation acting on phase space, this transformation defines a symmetry
of that dynamics such that its equations of motion are covariant under that
transformation, while that transformation itself, in linearised form, is
generated through the Poisson brackets of the phase space coordinates with a
conserved Noether charge which, even when displaying an explicit time
dependency, determines a (then dynamical) constant of the motion.

But then what about the status of the converse statement? Namely, could there
exist dynamical constants of the motion which, through their Poisson brackets
generate covariant symmetries of the Hamiltonian equations of motion, and yet
would not leave the first-order Hamiltonian action invariant up to some total
time derivative? In other words, could there exist non-Noetherian dynamical
constants of the motion beyond those whose existence is a consequence of
Noether’s first theorem? In spite of statements to that effect in the
literature claiming the possibility of non-Noetherian symmetries, it will
shown hereafter that provided one addresses the issue within the Hamiltonian
formulation of the dynamics — which from a number of points of view may be
considered to be more fundamental than the Lagrangian one — any dynamical
constant of the motion indeed generates through its Poisson brackets a global
continuous symmetry of the dynamics and its equations of motion, and
corresponds to the Noether charge associated to that symmetry which indeed
leaves the Hamiltonian action invariant up to a total time derivative. Any
dynamical constant of the motion is a Noether charge for some global
continuous symmetry acting on phase space, under which the system’s
Hamiltonian equations of motion are covariant (for another discussion of such
issues, see Refs.[3, 4]).

One of the purposes of the present contribution is to (re)visit the different
points addressed above, specifically within the Hamiltonian formulation, and
in particular for the converse statement to the first Noether theorem. As far
as the present authors are aware, such a general discussion is not readily
available in such a form in the literature, and in particular for the latter
point. Furthermore and with the same pedagogical aims in mind, the general
analysis is then illustrated with three explicit examples as well, for which
all dynamical constants of the motion and their spectrum generating algebras
are identified.

In brief, this paper is organised as follows. In view of the above comments
Section 2 revisits Noether’s first theorem, and its converse statement, in a
streamlined presentation directly within the Hamiltonian formulation and by
assuming the least restrictive and most general possible conditions under
which dynamical constants of motion are indeed always Noether symmetry
generators and spectrum generating observables, even when these charges do not
commute with the Hamiltonian. Section 3 provides a first and simple
illustration of the general discussion for a system with a purely harmonic
dynamics. This is extended to the free relativistic scalar field in Minkowski
spacetime in Section 4. Section 5 details the analysis of the classical and
quantum dynamical constants of motion which generate spatial translations and
Galilean boosts in euclidean space in the case of a nonrelativistic particle
subjected to a constant force field, to highlight through an explicit
illustration the consistent role and relevance of dynamical conserved
observables as symmetry generators and spectrum generating operators. Finally
some concluding comments are presented in Section 6.

##  2 Symmetries and Conserved Hamiltonian Observables

###  2.1 Noether symmetries and conserved observables

Given the context outlined in the Introduction, let us consider the
Hamiltonian formulation of an arbitrary dynamical system, whose phase space is
spanned by local canonical coordinates $(q^{\alpha},p_{\alpha})$ and its time
dynamics governed by a Hamiltonian $H(q^{\alpha},p_{\alpha})$ (taken here to
be time independent), where $\alpha$ denotes a — finite or infinite — discrete
and/or continuous set of possibly multi-indices that label the system’s
degrees of freedom. Such a dynamics is characterised by a first-order
Hamiltonian action principle given by

$S[q^{\alpha},p_{\alpha}]=\int
dt\,\left[\lambda\dot{q}^{\alpha}p_{\alpha}-(1-\lambda)q^{\alpha}\dot{p}_{\alpha}-H(q^{\alpha},p_{\alpha})\right],$
(2)

where the summation convention is in place, while $\lambda\in\mathbb{R}$ is an
arbitrary real constant parameter (which accounts for the freedom available in
redefining the action up to a total time derivative contribution without
modifying the canonical character of the phase space coordinates
$(q^{\alpha},p_{\alpha})$; usual choices for a value of $\lambda$ are
$\lambda=1,1/2,0$). The variational principle implies the canonical
Hamiltonian equations of motion,

$\dot{q}^{\alpha}=\frac{\partial H}{\partial
p_{\alpha}},\qquad\dot{p}_{\alpha}=-\frac{\partial H}{\partial q^{\alpha}},$
(3)

which also follow from the Poisson brackets
$\dot{q}^{\alpha}=\\{q^{\alpha},H\\}$ and
$\dot{p}_{\alpha}=\\{p_{\alpha},H\\}$ given the canonical Poisson brackets

$\\{q^{\alpha},p_{\beta}\\}=\delta^{\alpha}_{\beta}.$ (4)

More generally the time evolution of any phase space observable
$F(q^{\alpha},p_{\alpha};t)$ — even when complex valued — having possibly an
explicit time dependency, is determined from the classical Hamiltonian
equation of motion,

$\frac{dF}{dt}=\frac{\partial F}{\partial t}+\\{F,H\\}.$ (5)

Its solutions $F(q^{\alpha}(t),p_{\alpha}(t);t)$ are uniquely determined once
an initial value has been specified, namely
$F_{0}=F(q^{\alpha}_{0},p_{\alpha,0};t_{0})$ where
$q^{\alpha}_{0}=q^{\alpha}(t_{0})$ and $p_{\alpha,0}=p_{\alpha}(t_{0})$.

According to Noether’s first theorem, if the action is left invariant possibly
up to a total time derivative contribution under a one-parameter continuous
group of transformations then corresponding to a Lie symmetry of the system
and its dynamics (which may be a subgroup of a many parameter symmetry Lie
group), there exists a combination of the equations of motion that reduces to
the total time derivative of a specific quantity — namely the corresponding
Noether charge — which therefore is itself a conserved observable for any
solution to the system’s equations of motion, whose constant value is function
of the specific parameters that characterise that solution.

Note well that the present analysis is restricted to one-parameter Lie
symmetries, and to dynamics generated by a Hamiltonian function which is not
explicitly time dependent (and thus related through a Legendre transformation
to a Lagrange function which itself is time independent). A more general
discussion addressing the possibility of time dependent Hamiltonians (and
Lagrangians), and the non-abelian Lie algebra structure itself of the full
ensemble of dynamical Noether constants of motion, is not included in the
present work.

Consequently, by having trivially a vanishing Poisson bracket with itself the
Hamiltonian identifies a first conserved observable as a Noether charge, which
generates the global continuous symmetry of the dynamics under constant and
finite translations in the time evolution parameter. If $t$ parametrises
physical time, this Noether charge measures the total energy of the system.
For this reason, let us henceforth restrict to transformations that leave
invariant the time evolution coordinate $t$.

Specifically given the above Hamiltonian formulation let us thus consider the
most general possible form of possibly time dependent global continuous
transformations of the phase space variables, $q^{\alpha}$ and $p_{\alpha}$,
dependent on a single constant parameter $\epsilon$, which would leave the
first-order Hamiltonian action invariant up to a total time derivative
contribution. Let us represent such transformations in terms of the following
expressions,

$\tilde{q}^{\alpha}=\tilde{q}^{\alpha}(q^{\alpha},p_{\alpha};t|\epsilon),\qquad\tilde{p}_{\alpha}=\tilde{p}_{\alpha}(q^{\alpha},p_{\alpha};t|\epsilon),$
(6)

such that

$S[\tilde{q}^{\alpha},\tilde{p}_{\alpha}]=S[q^{\alpha},p_{\alpha}]\,+\,\int
dt\,\frac{d}{dt}\left(\Lambda(q^{\alpha},p_{\alpha};t|\epsilon)\right),$ (7)

where $\Lambda(q^{\alpha},p_{\alpha};t|\epsilon)$ is identified implicitly
through the variation of the Hamiltonian action under the transformation and
is defined only up to an arbitrary additive constant contribution. Since under
such conditions the variation of $S[\tilde{q},\tilde{p}_{\alpha}]$ relative to
$(\tilde{q}^{\alpha},\tilde{p}_{\alpha})$ differs from the variation of
$S[q^{\alpha},p_{\alpha}]$ relative to $(q^{\alpha},p_{\alpha})$ only by a
total time derivative contribution, the Hamiltonian equations of motion of the
system are form invariant under the transformations under consideration. In
other words these transformations map solutions into one another, leaving the
set of solutions invariant. They thus define a symmetry of the dynamics.

When considering the linearised form of these transformations, to first order
in the global symmetry transformation parameter $\epsilon$, one establishes
the following first Noether identity,

$\delta S=S[q^{\alpha}+\delta q^{\alpha},p_{\alpha}+\delta
p_{\alpha}]-S[q^{\alpha},p_{\alpha}]=\int
dt\,\frac{d}{dt}\left(\epsilon\,\Lambda(q^{\alpha},p_{\alpha};t)\right),$ (8)

with $\tilde{q}^{\alpha}=q^{\alpha}+\delta q^{\alpha}$ and
$\tilde{p}_{\alpha}=p_{\alpha}+\delta p_{\alpha}$, while

$\delta
q^{\alpha}=\epsilon\,\phi^{\alpha}(q^{\alpha},p_{\alpha};t),\qquad\delta
p_{\alpha}=\epsilon\,\chi_{\alpha}(q^{\alpha},p_{\alpha};t),\qquad\Lambda(q^{\alpha},p_{\alpha};t|\epsilon)=\epsilon\,\Lambda(q^{\alpha},p_{\alpha};t).$
(9)

Here $\phi^{\alpha}(q^{\alpha},p_{\alpha};t)$,
$\chi_{\alpha}(q^{\alpha},p_{\alpha};t)$ and
$\Lambda(q^{\alpha},p_{\alpha};t)$ characterise the possibly time dependent
action of the linearised symmetry transformation on the phase space variables
$(q^{\alpha},p_{\alpha})$. Note well that even though the functions
$\phi^{\alpha}(q^{\alpha},p_{\alpha};t)$,
$\chi_{\alpha}(q^{\alpha},p_{\alpha};t)$ and
$\Lambda(q^{\alpha},p_{\alpha};t)$ may possess an explicit time dependency
which follows from the nature of the symmetry, such a transformation still
corresponds to (the linearised form of) a global symmetry (as opposed to a
local or gauge symmetry) since the symmetry parameter $\epsilon$ is (a
(space)time independent) constant. Furthermore the implicitly identified
surface term in time, $\Lambda(q^{\alpha},p_{\alpha};t)$, remains itself
determined only up to an arbitrary additive constant contribution.

A direct analysis of the above Noether identity for $\delta S$, thus valid for
any symmetry transformation leaving the Hamiltonian equations of motion form
invariant, namely the following explicit expression valid to first order in
$\epsilon$,

$\displaystyle\delta S$ $\displaystyle=$ $\displaystyle\int
dt\,\left[\lambda\left(\frac{d}{dt}\delta
q^{\alpha}\right)p_{\alpha}+\lambda\left(\frac{dq^{\alpha}}{dt}\right)\,\delta
p_{\alpha}-(1-\lambda)\delta
q^{\alpha}\left(\frac{dp_{\alpha}}{dt}\right)-(1-\lambda)q^{\alpha}\left(\frac{d\delta
p_{\alpha}}{dt}\right)\right.-$ (10)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.-\delta
q^{\alpha}\frac{\partial H}{\partial q^{\alpha}}-\delta
p_{\alpha}\frac{\partial H}{\partial p_{\alpha}}\right]=\int
dt\,\frac{d}{dt}\left[\epsilon\,\Lambda\right],$

which also reduces to the following identity for the variations $\delta
q^{\alpha}$ and $\delta p_{\alpha}$ that are determined by the symmetry
transformation,

$0=\int dt\,\left\\{\frac{d}{dt}\left[\lambda\,\delta
q^{\alpha}p_{\alpha}-(1-\lambda)\,q^{\alpha}\delta
p_{\alpha}-\epsilon\,\Lambda\right]\,-\,\delta
q^{\alpha}\,\left(\frac{dp_{\alpha}}{dt}+\frac{\partial H}{\partial
q^{\alpha}}\right)\,+\,\delta
p_{\alpha}\,\left(\frac{dq^{\alpha}}{dt}-\frac{\partial H}{\partial
p_{\alpha}}\right)\right\\},$ (11)

then readily leads to the statement of the first Noether theorem within the
Hamiltonian formulation, namely,

$\frac{dQ}{dt}=-\left(\frac{dq^{\alpha}}{dt}-\frac{\partial H}{\partial
p_{\alpha}}\right)\chi_{\alpha}\,+\,\left(\frac{dp_{\alpha}}{dt}+\frac{\partial
H}{\partial q^{\alpha}}\right)\phi^{\alpha},$ (12)

with the Noether charge given by the following phase space observable,

$Q(q^{\alpha},p_{\alpha};t)=\lambda\,\phi^{\alpha}(q^{\alpha},p_{\alpha};t)\,p_{\alpha}\,-\,(1-\lambda)\,q^{\alpha}\,\chi_{\alpha}(q^{\alpha},p_{\alpha};t)\,-\,\Lambda(q^{\alpha},p_{\alpha};t).$
(13)

Note that the Noether charge is defined up to the same additive constant that
remains arbitrary in the identification of the time surface contribution in
$\Lambda$, a fact which of course is fully consistent with the status of a
conserved observable for the Noether charge.

Even though the time surface term $\Lambda$ is defined implicitly through the
variation of the Hamiltonian action under the symmetry transformation up to an
additive constant contribution, it is possible to establish a series of
conditions in the form of differential equations that the function
$\Lambda(q^{\alpha},p_{\alpha};t)$ has to satisfy in each of its variables. By
making explicit in (10) all contributions linear in $\dot{q}^{\alpha}$, in
$\dot{p}_{\alpha}$, and independent of these, one identifies the following
necessary consistency conditions, respectively,

$\displaystyle\frac{\partial\Lambda}{\partial q^{\alpha}}$ $\displaystyle=$
$\displaystyle\lambda\frac{\partial\phi^{\beta}}{\partial
q^{\alpha}}p_{\beta}+\lambda\chi_{\alpha}-(1-\lambda)q^{\beta}\frac{\partial\chi_{\beta}}{\partial
q^{\alpha}},$ $\displaystyle\frac{\partial\Lambda}{\partial p_{\alpha}}$
$\displaystyle=$ $\displaystyle\lambda\frac{\partial\phi^{\beta}}{\partial
p_{\alpha}}p_{\beta}-(1-\lambda)\phi^{\alpha}-(1-\lambda)q^{\beta}\frac{\partial\chi_{\beta}}{\partial
p_{\alpha}},$ (14) $\displaystyle\frac{\partial\Lambda}{\partial t}$
$\displaystyle=$ $\displaystyle\lambda\frac{\partial\phi^{\alpha}}{\partial
t}p_{\alpha}-(1-\lambda)q^{\alpha}\frac{\partial\chi_{\alpha}}{\partial
t}-\phi^{\alpha}\frac{\partial H}{\partial
q^{\alpha}}-\chi_{\alpha}\frac{\partial H}{\partial p_{\alpha}}.$

Based on these properties one may then establish through explicit calculation
that,

$\frac{\partial Q}{\partial t}=\phi^{\alpha}\frac{\partial H}{\partial
q^{\alpha}}+\chi_{\alpha}\frac{\partial H}{\partial
p_{\alpha}},\qquad\left\\{Q,H\right\\}=-\phi^{\alpha}\frac{\partial
H}{\partial q^{\alpha}}-\chi_{\alpha}\frac{\partial H}{\partial p_{\alpha}},$
(15)

so that the Hamiltonian equation of motion for the Noether charge reads,

$\frac{dQ}{dt}=\frac{\partial Q}{\partial t}+\left\\{Q,H\right\\}=0,$ (16)

which is indeed the Hamiltonian equation of motion for a conserved observable
which carries an explicit time dependency in addition to its phase space
dependency, with in particular then a nonvanishing Poisson bracket with the
system’s Hamiltonian. Furthermore from the identities (2.1) it readily follows
that

$\left\\{q^{\alpha},\epsilon\,Q\right\\}=\epsilon\,\phi^{\alpha}=\delta
q^{\alpha},\qquad\left\\{p_{\alpha},\epsilon\,Q\right\\}=\epsilon\,\chi_{\alpha}=\delta
p_{\alpha},$ (17)

establishing that the Noether charge $Q$ is indeed the observable that
generates the linearised transformations of the corresponding symmetry group’s
action on the system’s phase space.

###  2.2 Conserved observables and Noether symmetries

Let us also explicitly establish the converse result to Noether’s first
theorem. Namely that any conserved observable, even when carrying an explicit
time dependency over phase space, is the generator of a one-parameter symmetry
group of the dynamics which also leaves the first-order Hamiltonian action
invariant up to a total time derivative contribution. Let
$F(q^{\alpha},p_{\alpha};t)$ be such a conserved observable, thus with the
property that

$\frac{dF}{dt}=\frac{\partial F}{\partial t}+\left\\{F,H\right\\}=0.$ (18)

It readily follows from this conservation property in the form of $\partial
F/\partial t=-\left\\{F,H\right\\}$, that,

$\displaystyle\frac{d}{dt}\frac{\partial F}{\partial q^{\alpha}}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial t}\frac{\partial
F}{\partial q^{\alpha}}+\left\\{\frac{\partial F}{\partial
q^{\alpha}},H\right\\}=-\frac{\partial}{\partial
q^{\alpha}}\left\\{F,H\right\\}+\left\\{\frac{\partial F}{\partial
q^{\alpha}},H\right\\}=-\left\\{F,\frac{\partial H}{\partial
q^{\alpha}}\right\\},$ $\displaystyle\frac{d}{dt}\frac{\partial F}{\partial
p_{\alpha}}$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
t}\frac{\partial F}{\partial p_{\alpha}}+\left\\{\frac{\partial F}{\partial
p_{\alpha}},H\right\\}=-\frac{\partial}{\partial
p_{\alpha}}\left\\{F,H\right\\}+\left\\{\frac{\partial F}{\partial
p_{\alpha}},H\right\\}=-\left\\{F,\frac{\partial H}{\partial
p_{\alpha}}\right\\}.$ (19)

Furthermore consider now the linearised phase space transformations generated
by $F$ with constant parameter $\epsilon$,

$\delta
q^{\alpha}=\left\\{q^{\alpha},\epsilon\,F\right\\}=\epsilon\,\frac{\partial
F}{\partial
p_{\alpha}}\equiv\epsilon\,\phi^{\alpha}(q^{\alpha},p_{\alpha};t),\qquad\delta
p_{\alpha}=\left\\{p_{\alpha},\epsilon\,F\right\\}=-\epsilon\,\frac{\partial
F}{\partial
q^{\alpha}}\equiv\epsilon\,\chi_{\alpha}(q^{\alpha},p_{\alpha};t),$ (20)

which define the quantities $\phi^{\alpha}$ and $\chi_{\alpha}$ as relevant to
the present discussion (to be distinguished for now from those introduced in
the previous Subsection 2.1). These transformations then generate the
following linearised variations of the Hamiltonian (note that for any
observable $G(q^{\alpha},p_{\alpha};t)$ one has to linear order for these
transformations, $G(q^{\alpha}+\delta q^{\alpha},p_{\alpha}+\delta
p_{\alpha};t)=G(q^{\alpha},p_{\alpha};t)+\epsilon\left\\{G,F\right\\}$, as may
easily be checked),

$\displaystyle H(q^{\alpha}+\delta q^{\alpha},p_{\alpha}+\delta p_{\alpha})$
$\displaystyle=$ $\displaystyle
H(q^{\alpha},p_{\alpha})-\epsilon\,\left\\{F,H\right\\},$
$\displaystyle\frac{\partial H}{\partial q^{\alpha}}(q^{\alpha}+\delta
q^{\alpha},p_{\alpha}+\delta p_{\alpha})$ $\displaystyle=$
$\displaystyle\frac{\partial H}{\partial
q^{\alpha}}(q^{\alpha},p_{\alpha})-\epsilon\,\left\\{F,\frac{\partial
H}{\partial q^{\alpha}}\right\\},$ (21) $\displaystyle\frac{\partial
H}{\partial p_{\alpha}}(q^{\alpha}+\delta q^{\alpha},p_{\alpha}+\delta
p_{\alpha})$ $\displaystyle=$ $\displaystyle\frac{\partial H}{\partial
p_{\alpha}}(q^{\alpha},p_{\alpha})-\epsilon\,\left\\{F,\frac{\partial
H}{\partial p_{\alpha}}\right\\}.$

Given these series of identities let us now consider a specific but otherwise
arbitrary solution to the Hamiltonian equations of motion,
$(q^{\alpha}(t),p_{\alpha}(t))$, and its linearly transformed configuration
$(\tilde{q}^{\alpha}(t),\tilde{p}_{\alpha}(t))$ with

$\displaystyle\tilde{q}^{\alpha}(t)$ $\displaystyle=$ $\displaystyle
q^{\alpha}(t)+\epsilon\left\\{q^{\alpha},F\right\\}=q^{\alpha}(t)+\epsilon\frac{\partial
F}{\partial
p_{\alpha}}(q^{\alpha}(t),p_{\alpha}(t);t)=q^{\alpha}(t)+\epsilon\,\phi^{\alpha}(q^{\alpha}(t),p_{\alpha}(t);t),$
$\displaystyle\tilde{p}_{\alpha}(t)$ $\displaystyle=$ $\displaystyle
p_{\alpha}(t)+\epsilon\left\\{p_{\alpha},F\right\\}=p_{\alpha}(t)-\epsilon\frac{\partial
F}{\partial
q^{\alpha}}(q^{\alpha}(t),p_{\alpha}(t);t)=p_{\alpha}(t)+\epsilon\,\chi_{\alpha}(q^{\alpha}(t),p_{\alpha}(t);t).\qquad$
(22)

To linear order in $\epsilon$ it then follows simply that,

$\displaystyle\frac{d}{dt}\tilde{q}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\partial H}{\partial
p_{\alpha}}-\epsilon\,\left\\{F,\frac{\partial H}{\partial
p_{\alpha}}\right\\}=\frac{\partial H}{\partial
p_{\alpha}}(\tilde{q}^{\alpha},\tilde{p}_{\alpha}),$
$\displaystyle\frac{d}{dt}\tilde{p}_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{\partial H}{\partial
q^{\alpha}}+\epsilon\,\left\\{F,\frac{\partial H}{\partial
q^{\alpha}}\right\\}=-\frac{\partial H}{\partial
q^{\alpha}}(\tilde{q}^{\alpha},\tilde{p}_{\alpha}).$ (23)

In other words, any solution $(q^{\alpha}(t),p_{\alpha}(t))$ to the
Hamiltonian equations of motion is transformed into another solution to the
same equations of motion under the action on phase space of the transformation
generated by the conserved observable $\epsilon F(q^{\alpha},p_{\alpha};t)$,
however general the latter may be. Any conserved observable $F$ is thus the
generator of a one-parameter Lie symmetry group of the system’s dynamics.

The same conclusion may also be reached based directly on the transformation
of the first-order Hamiltonian action (2) linearised to first order in
$\epsilon$. A direct analysis then finds for that linearised variation,

$\delta S=\epsilon\,\int\,dt\left[\frac{d}{dt}\left(\lambda\frac{\partial
F}{\partial p_{\alpha}}p_{\alpha}+(1-\lambda)q^{\alpha}\frac{\partial
F}{\partial q^{\alpha}}-F\right)\,+\,\left(\frac{\partial F}{\partial
t}+\left\\{F,H\right\\}\right)\right].$ (24)

Hence indeed when $F$ is a conserved observable, through Poisson brackets it
generates phase space transformations that leave the action invariant up to a
total time derivative contribution of a quantity $\Lambda$ given by,

$\displaystyle\Lambda(q^{\alpha},p_{\alpha};t)$ $\displaystyle=$
$\displaystyle\lambda\frac{\partial F}{\partial
p_{\alpha}}p_{\alpha}+(1-\lambda)q^{\alpha}\frac{\partial F}{\partial
q_{\alpha}}-F$ (25) $\displaystyle=$
$\displaystyle\lambda\phi^{\alpha}(q^{\alpha},p_{\alpha};t)p_{\alpha}-(1-\lambda)q^{\alpha}\chi_{\alpha}(q^{\alpha},p_{\alpha};t)-F(q^{\alpha},p_{\alpha};t).$

When compared to the general expression (13) for the Noether charge associated
to the symmetry generated by $F$, it is clear that the conserved observable
$F$ is itself the Noether charge of the symmetry that it generates.

There is thus a one-to-one correspondence between all one-parameter global
continuous Lie symmetries of a Hamiltonian dynamics and the set of all its
conserved observables, including those that possess an explicit time
dependency over phase space and then do not have a vanishing Poisson bracket
(or quantum commutator) with the Hamiltonian. According to Noether’s first
theorem, any such Lie symmetry under which the first-order Hamiltonian action
is invariant up to a total time derivative contribution and thus such that its
Hamiltonian equations of motion are covariant, implies the existence of a
(possibly time dependent) conserved observable which is the generator of that
symmetry over phase space through Poisson brackets. But the converse statement
is thus true as well, namely that any (possibly time dependent) conserved
observable is the generator of a one-parameter Lie symmetry under which the
Hamiltonian equations of motion are covariant and for which the first-order
Hamiltonian action is invariant up to a total time derivative contribution.
Namely any dynamical constant of the motion is the generator of a Noether
symmetry, within the Hamiltonian formulation of a dynamical system. All Lie
symmetries are Noetherian symmetries.

Furthermore the set of all conserved observables is a Lie algebra under the
Poisson bracket structure. Indeed it is obvious that any linear combination
with constant coefficients of conserved observables is again a conserved
observable. The same is true for the product of two conserved observables.
Hence the set of all conserved observables, which generate all the symmetries
of a dynamical system, is a closed linear algebra. In addition, because of the
Jacobi identity obeyed by Poisson brackets, it may readily be shown that the
Poisson bracket of any two conserved observables is again a conserved
observable, whether explicitly time dependent or not. In other words, the
algebra of all conserved observables is a Lie algebra as well, namely the Lie
algebra of all the continuous Lie symmetries of the Hamiltonian dynamics of a
dynamical system. Of course, the Lie algebra of all conserved observables may
possess closed Lie sub-algebras of constants of motion, explicitly time
dependent or not, of a geometrical character or not, of direct physical
relevance and interest.

##  3 Harmonic dynamics

As a first illustration of the general discussion above, let us consider a
system with a phase space parametrised by the canonical variables
$(q^{\alpha},p_{\alpha})$ of which the time evolution is governed by the
Hamiltonian

$H(q^{\alpha},p_{\alpha})=\sum_{\alpha}\left(\frac{1}{2}p^{2}_{\alpha}+\frac{1}{2}\omega^{2}_{\alpha}{q^{\alpha}}^{2}\right),$
(26)

where $\omega_{\alpha}>0$ are positive real valued angular frequencies. For
instance under the rescaling
$(q^{\alpha},p_{\alpha})\rightarrow(\sqrt{m}q^{\alpha},p_{\alpha}/\sqrt{m})$
with $m>0$ being a mass parameter, and if the rescaled coordinates
$q^{\alpha}$ are finite in number and correspond to cartesian coordinates in
some euclidean space $\mathbb{R}^{d}$, this system is that of an anisotropic
harmonic oscillator in that configuration space and represented relative to
its principal normal mode axes.

Of course such a system hides no secrets whatsoever. Its equations of motion
reduce to the harmonic dynamics of each of the degrees of freedom
$q^{\alpha}$,

$\ddot{q}^{\alpha}+\omega^{2}_{\alpha}\,q^{\alpha}=0,\qquad
p_{\alpha}=\dot{q}^{\alpha},$ (27)

with obvious solutions.

As is well known, in such a situation it is most appropriate to use a complex
parametrisation of phase space, corresponding to Fock algebra generators in
the quantum case, defined by,

$a_{\alpha}(q^{\alpha},p_{\alpha})=\sqrt{\frac{\omega_{\alpha}}{2}}\left(q^{\alpha}+\frac{i}{\omega_{\alpha}}p_{\alpha}\right),\qquad
a^{\dagger}(q^{\alpha},p_{\alpha})=\sqrt{\frac{\omega_{\alpha}}{2}}\left(q^{\alpha}-\frac{i}{\omega_{\alpha}}p_{\alpha}\right),$
(28)

which is such that, for the classical system,

$\left\\{a_{\alpha},a^{\dagger}_{\beta}\right\\}=-i\,\delta_{\alpha,\beta},\qquad
H=\sum_{\alpha}\,\omega_{\alpha}a^{\dagger}_{\alpha}a_{\alpha}.$ (29)

Consequently one has the following equations of motion,

$\frac{d}{dt}a_{\alpha}=\left\\{a_{\alpha},H\right\\}=-i\omega_{\alpha}a_{\alpha},\qquad\frac{d}{dt}a^{\dagger}_{\alpha}=\left\\{a^{\dagger}_{\alpha},H\right\\}=i\omega_{\alpha}\,a^{\dagger}_{\alpha},$
(30)

of which the solutions are

$a_{\alpha}(t)=a_{\alpha,0}\,e^{-i\omega_{\alpha}(t-t_{0})},\qquad
a^{\dagger}_{\alpha,0}(t)=a^{\dagger}_{\alpha,0}\,e^{i\omega_{\alpha}(t-t_{0})},$
(31)

with $a_{\alpha,0}$ being complex valued integration constants (and
$a^{\dagger}_{\alpha,0}$ their complex conjugates) specifying the values of
$a_{\alpha}$ (hence of $q^{\alpha}$ and $p_{\alpha}$ as well) at $t=t_{0}$.

Quite obviously the phase space observables
$a_{\alpha}(q^{\alpha},p_{\alpha})$ and
$a^{\dagger}_{\alpha}(q^{\alpha},p_{\alpha})$, which as such have no explicit
time dependency, are not conserved since they have nonvanishing Poisson
brackets (or quantum commutators) with the Hamiltonian $H$. However let us now
introduce the following explicitly time dependent phase space observables,

$A_{\alpha}(t)\equiv
A_{\alpha}(q^{\alpha},p_{\alpha};t)=e^{i\omega_{\alpha}(t-t_{0})}\,a_{\alpha}(q^{\alpha},p_{\alpha}),\quad
A^{\dagger}(t)\equiv
A^{\dagger}_{\alpha}(q^{\alpha},p_{\alpha};t)=e^{-i\omega_{\alpha}(t-t_{0})}\,a^{\dagger}_{\alpha}(q^{\alpha},p_{\alpha}),$
(32)

in terms of which the original basic parametrisation $(q^{\alpha},p_{\alpha})$
of phase space is expressed as,

$\displaystyle q^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\omega_{\alpha}}}\left(e^{-i\omega_{\alpha}(t-t_{0})}\,A_{\alpha}(t)\,+\,e^{i\omega_{\alpha}(t-t_{0})}A^{\dagger}_{\alpha}(t)\right),$
$\displaystyle p_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{i\omega_{\alpha}}{\sqrt{2\omega_{\alpha}}}\left(e^{-i\omega_{\alpha}(t-t_{0})}A_{\alpha}(t)\,-\,e^{i\omega_{\alpha}(t-t_{0})}A^{\dagger}_{\alpha}(t)\right).$
(33)

By construction the observables $A_{\alpha}(t)$ and $A^{\dagger}_{\alpha}(t)$
are all conserved quantities, since,

$\frac{d}{dt}A_{\alpha}=\frac{\partial}{\partial
t}A_{\alpha}+\left\\{A_{\alpha},H\right\\}=0,\qquad\frac{d}{dt}A^{\dagger}_{\alpha}=\frac{\partial}{\partial
t}A^{\dagger}_{\alpha}+\left\\{A^{\dagger}_{\alpha},H\right\\}=0,$ (34)

while their constant values for solutions to the Hamiltonian equations of
motion are given by the integration constants for the time evolution of
$a_{\alpha}$ and $a^{\dagger}_{\alpha}$,

$\displaystyle A_{\alpha}(q^{\alpha}(t),p_{\alpha}(t);t)$ $\displaystyle=$
$\displaystyle
A_{\alpha}(q^{\alpha}(t_{0}),p_{\alpha}(t_{0});t_{0})=a_{\alpha,0},$
$\displaystyle A^{\dagger}_{\alpha}(q^{\alpha}(t),p_{\alpha}(t);t)$
$\displaystyle=$ $\displaystyle
A^{\dagger}_{\alpha}(q^{\alpha}(t_{0}),p_{\alpha}(t_{0});t_{0})=a^{\dagger}_{\alpha,0}.$
(35)

By substitution in the phase space parametrisation (33), one then identifies
the solution to the Hamiltonian equations of motion in phase space. Note that
the Poisson brackets of these dynamical constants of the motion still take the
canonical values,

$\left\\{A_{\alpha}(q^{\alpha},p_{\alpha};t),A^{\dagger}_{\beta}(q^{\alpha},p_{\alpha};t)\right\\}=-i\,\delta_{\alpha,\beta}.$
(36)

Based on this observation it is clear that the complete algebra of all
conserved phase space observables that generate the full dynamical symmetry
Lie algebra of this harmonic dynamics, constructed as composite quantities out
of its phase space variables $(q^{\alpha},p_{\alpha})$, is spanned by the set
of all possible monomials comprised of a product of an arbitrary number of
$A^{\dagger}_{\alpha}(q^{\alpha},p_{\alpha};t)$ factors, followed by the
product of an arbitrary number of $A_{\alpha}(q^{\alpha},p_{\alpha};t)$
factors, written in normal ordered form relative to these Fock algebra
generators in the quantum case, and this for all possible values of $\alpha$.
One example of such a conserved observable being simply, of course, the
Hamiltonian itself, in the form,

$H=\sum_{\alpha}\omega_{\alpha}A^{\dagger}_{\alpha}(q^{\alpha},p_{\alpha};t)\,A_{\alpha}(q^{\alpha},p_{\alpha};t).$
(37)

Given this understanding of the full dynamical symmetry Lie algebra of this
system, it also becomes clear that in the quantum case, and then considered
within the Schrödinger picture, these monomials which are the generators of
this dynamical symmetry, by annihilating and creating arbitrary numbers of
arbitrary energy quanta of the system, span as well the complete spectrum
generating algebra of all energy eigenstates of this dynamics starting with
their action on the Fock vacuum annihilated by
$A_{\alpha}(q^{\alpha},p_{\alpha};t)$, namely $|\Omega\rangle$ such that
$A_{\alpha}(q^{\alpha},p_{\alpha};t)|\Omega\rangle=0$ and
$\langle\Omega|\Omega\rangle=1$. From that point of view let us remark that
the conserved observables $A^{\dagger}_{\alpha}(q^{\alpha},p_{\alpha};t)$ and
$A_{\alpha}(q^{\alpha},p_{\alpha};t)$ are, on their own already, the primary
generators of the full spectrum generating algebra, which is indeed the
enveloping algebra constructed out of these primary conserved observables.

Finally note that by restricting this choice of monomials to those of the
following bi-linear form in the creation and annihilation operators for all
possible values of $\alpha$ and $\beta$, namely
$A^{\dagger}_{\alpha}(t)A_{\beta}(t)$, one generates a closed subalgebra of
conserved observables and Lie symmetry generators of the full dynamical
symmetry Lie algebra, to which the Hamiltonian $H$ belongs. In the quantum
case the action of this subalgebra is also closed in the subspace of Hilbert
space spanned by all single quanta quantum states
$A^{\dagger}_{\alpha}(t)|\Omega\rangle$ for all values of $\alpha$, even
though not all these states are degenerate in energy if the frequencies
$\omega_{\alpha}$ are not all equal.

##  4 Free Scalar Fields in $d$+1 Dimensional Minkowski Spacetime

As a second illustration, let us now consider the relativistic Poincaré
invariant free real Klein-Gordon scalar field of mass $\mu\geq 0$ in a $D=d$+1
dimensional Minkowski spacetime, $\phi(x^{\mu})\in\mathbb{R}$, with its
dynamics determined from the usual spacetime local Lagrangian action (in units
such that $c=1=\hbar$),

$S[\phi]=\int
d^{D}x^{\mu}\left[\frac{1}{2}\left(\partial_{t}\phi\right)^{2}-\frac{1}{2}\left(\vec{\nabla}_{\vec{x}}\,\phi\right)^{2}-\frac{1}{2}\mu^{2}\phi^{2}\right].$
(38)

In Hamiltonian form we thus have,

$S[\phi,\pi]=\int\,d^{D}x^{\mu}\left[\partial_{t}\phi\,\pi\,-\,\left(\frac{1}{2}\pi^{2}+\frac{1}{2}\left(\vec{\nabla}_{\vec{x}}\phi\right)^{2}+\frac{1}{2}\mu^{2}\phi^{2}\right)\right]=\int
d^{D}x^{\mu}\left[\partial_{t}\phi\,\pi\,-\,{\cal H}\right],$ (39)

with $\pi(t,\vec{x})=\partial_{t}\phi(t,\vec{x})$ being the canonical
conjugate momentum to $\phi(t,\vec{x})$, and ${\cal H}$ the canonical
Hamiltonian density. It should be clear, by considering the momentum space
representation of the field in terms of its Fourier transformed modes, that
this system is of the general harmonic form discussed in the previous Section.
The analysis thus readily proceeds accordingly.

Given the original phase space parametrisation in terms of the canonically
conjugate variables $(\phi(\vec{x}),\pi(\vec{x}))$, let us now consider its
complex parametrisation in terms of complex variables “in momentum space”,
$(a(\vec{k}),a^{\dagger}(\vec{k}))$, defined through a Fourier transform such
that,

$\displaystyle\phi(\vec{x})$ $\displaystyle=$
$\displaystyle\int_{(\infty)}\frac{d^{d}\vec{k}}{(2\pi)^{d}\,2\omega(k)}\left(e^{i\vec{k}\cdot\vec{x}}\,a(\vec{k})\,+\,e^{-i\vec{k}\cdot\vec{x}}\,a^{\dagger}(\vec{k})\right),$
$\displaystyle\pi(\vec{x})$ $\displaystyle=$
$\displaystyle\int_{(\infty)}\frac{d^{d}\vec{k}}{(2\pi)^{d}\,2\omega(k)}\left(-i\omega(k)\right)\left(e^{i\vec{k}\cdot\vec{x}}\,a(\vec{k})\,-\,e^{-i\vec{k}\cdot\vec{x}}\,a^{\dagger}(\vec{k})\right),$
(40)

where $\omega(k)=\sqrt{\vec{k}^{2}+\mu^{2}}$ and $k=|\vec{k}|$. Note that the
plane wave factors involved in this mode decomposition of the field, namely
$e^{i\vec{k}\cdot\vec{x}}/(2\pi)^{d/2}$, meet the necessary completeness
relation,

$\int_{(\infty)}\frac{d^{d}\vec{k}}{(2\pi)^{d}}\,e^{-i\vec{k}\cdot\vec{x}}\,e^{i\vec{k}\cdot\vec{y}}=\delta^{(d)}(\vec{x}-\vec{y}\,),$
(41)

while they also determine a complete set of orthonormalised functions in
space,

$\int_{(\infty)}\frac{d^{d}\vec{x}}{(2\pi)^{d}}\,e^{-i\vec{k}\cdot\vec{x}}\,e^{i\vec{\ell}\cdot\vec{x}}=\delta^{(d)}(\vec{k}-\vec{\ell}\,).$
(42)

As is well known, in the context of the quantised field theory already, one
then has the following commutation relations for this Fock algebra
parametrisation,

$\left[a(\vec{k}),a^{\dagger}(\vec{\ell})\right]=(2\pi)^{d}\,2\omega(k)\,\delta^{(d)}(\vec{k}-\vec{\ell}\,)\,\mathbb{I},$
(43)

and the following expressions for the Hamiltonian, $H$, or total conserved
energy, and the total conserved momentum of the quantised field, given the
choice of normal ordering relative to the Fock algebra generators
$(a(\vec{k}),a^{\dagger}(\vec{k}))$ for composite operators, namely,

$H=\int_{(\infty)}\frac{d^{d}\vec{k}}{(2\pi)^{d}\,2\omega(k)}\,a^{\dagger}(\vec{k})\,\omega(k)\,a(\vec{k}),\qquad\vec{P}=\int_{(\infty)}\frac{d^{d}\vec{k}}{(2\pi)^{d}\,2\omega(k)}\,a^{\dagger}(\vec{k})\,\vec{k}\,a(\vec{k}),$
(44)

with indeed,

$\left[H,a^{\dagger}(\vec{k})\right]=\omega(k)\,a^{\dagger}(\vec{k}),\qquad\left[\vec{P},a^{\dagger}(\vec{k})\right]=\vec{k}\,a^{\dagger}(\vec{k}),\qquad\left[\vec{P},H\right]=0.$
(45)

The Heisenberg equations of motion for the Fock algebra operators are,

$\frac{d}{dt}a(\vec{k})=-i\left[a(\vec{k}),H\right]=-i\omega(k)\,a(\vec{k}),\qquad\frac{d}{dt}a^{\dagger}(\vec{k})=-i\left[a^{\dagger}(\vec{k}),H\right]=i\omega(k)\,a^{\dagger}(\vec{k}).$
(46)

These nonconserved observables thus have the following time dependency for
solutions to the quantum dynamics of the system,

$a(t,\vec{k})=e^{-i\omega(k)(t-t_{0})}\,a(t_{0},\vec{k}),\qquad
a^{\dagger}(t,\vec{k})=e^{i\omega(k)(t-t_{0})}\,a^{\dagger}(t_{0},\vec{k}),$
(47)

where $a(t_{0},\vec{k})$ and $a^{\dagger}(t_{0},\vec{k})$ are initial values
for these observables at the reference time $t=t_{0}$.

Thus once again, let us consider the following basic conserved quantum
observables possessing an explicit time dependency in phase space, and related
to the Fock algebra at the reference time $t_{0}$ and considered in the
Schrödinger picture,

$A(\vec{k};t)=e^{i\omega(k)(t-t_{0})}\,a(\vec{k}),\qquad
A^{\dagger}(\vec{k};t)=e^{-i\omega(k)(t-t_{0})}\,a^{\dagger}(\vec{k}),$ (48)

whose commutation relations remain of the Fock algebra type,

$\left[A(\vec{k};t),A^{\dagger}(\vec{\ell};t)\right]=(2\pi)^{d}\,2\omega(k)\,\delta^{(d)}(\vec{k}-\vec{\ell}\,)\,\mathbb{I}.$
(49)

In the Heisenberg picture, these observables are constants of the motion since
$dA_{H}(\vec{k};t|t)/dt=\partial A_{H}(\vec{k};t|t)/\partial
t-i\left[A_{H}(\vec{k};t|t),H\right]=0$ (with the partial time derivative in
the first term in the r.h.s. of this expression acting only on the explicit
time dependency of the phase space observable $A(\vec{k};t)$ in the
Schrödinger picture), namely,

$\displaystyle
A_{H}(\vec{k};t|t)=A_{H}(\vec{k};t_{0}|t_{0})=a(t_{0},\vec{k}),\qquad
A^{\dagger}_{H}(\vec{k};t|t)=A^{\dagger}_{H}(\vec{k};t_{0}|t_{0})=a^{\dagger}(t_{0},\vec{k}).$
(50)

Furthermore the original phase space variables are then expressed as follows
in terms of the phase space parametrisation
$(A_{H}(\vec{k};t|t),A^{\dagger}_{H}(\vec{k};t|t))$ in the Heisenberg picture,

$\displaystyle\phi_{H}(t,\vec{x})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\int_{(\infty)}\frac{d^{d}\vec{k}}{(2\pi)^{d}\,2\omega(k)}\left(e^{-i\omega(k)(t-t_{0})+i\vec{k}\cdot\vec{x}}\,A_{H}(\vec{k};t|t)+e^{i\omega(k)(t-t_{0})-i\vec{k}\cdot\vec{x}}\,A^{\dagger}_{H}(\vec{k};t|t)\right),$
(51) $\displaystyle\pi_{H}(t,\vec{x})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\int_{(\infty)}\frac{d^{d}\vec{k}}{(2\pi)^{d}\,2\omega(k)}\left(-i\omega(k)\right)\left(e^{-i\omega(k)(t-t_{0})+i\vec{k}\cdot\vec{x}}\,A_{H}(\vec{k};t|t)-e^{i\omega(k)(t-t_{0})-i\vec{k}\cdot\vec{x}}\,A^{\dagger}_{H}(\vec{k};t|t)\right),$

thereby leading to the usual expression for the time dependency of the
dynamical quantum scalar field in the Heisenberg picture, in terms of the
integration constants $(a(t_{0},\vec{k}),a^{\dagger}(t_{0},\vec{k}))$. Note
that in the Heisenberg picture the conserved total energy-momentum of the
field is expressed as follows in terms of the conserved Fock algebra
generators,

$H=\int_{(\infty)}\frac{d^{d}\vec{k}}{(2\pi)^{d}\,2\omega(k)}\,A^{\dagger}_{H}(\vec{k};t|t)\,\omega(k)\,A_{H}(\vec{k};t|t),\quad\vec{P}=\int_{(\infty)}\frac{d^{d}\vec{k}}{(2\pi)^{d}\,2\omega(k)}\,A^{\dagger}_{H}(\vec{k};t|t)\,\vec{k}\,A_{H}(\vec{k};t|t).\qquad$
(52)

Obviously the full dynamical symmetry Lie algebra and spectrum generating
algebra of the system is spanned by the full enveloping algebra generated by
the primary conserved Fock space operators, considered now within the
Schrödinger picture, namely $A(\vec{k};t)$ and $A^{\dagger}(\vec{k};t)$. In
particular the sub-algebra of bi-linears in the creation and annihilation
operators, $A^{\dagger}(\vec{k}_{1};t)\,A(\vec{k}_{2};t)$, acts on the
subspace of Hilbert space comprised of all 1-particle quantum states with
definite momentum eigenvalue by changing their momentum and energy values.
Yet, these transformations define global continuous Lie symmetries of the free
field dynamics, even though they do not commute with the quantum Hamiltonian
when $\omega(k_{1})\neq\omega(k_{2})$. Note that comments made to that effect
in the Conclusions of Ref.[5] thus prove to have been misconceived.

##  5 The Nonrelativistic Particle in a Constant Force Field

As a third and last illustration, let us consider a nonrelativistic particle
of mass $m>0$ with position vector $\vec{x}(t)$ relative to some inertial
frame in a $d$-dimensional euclidean space, subjected to a constant and thus
conservative force $\vec{F}$ (see for instance Ref.[2]), for which the
configuration space equation of motion reads,

$m\frac{d^{2}\vec{x}}{dt^{2}}=\vec{F}=-\vec{\nabla}_{\vec{x}}V(\vec{x}\,),\qquad
V(\vec{x}\,)=-\vec{x}\cdot\vec{F}.$ (53)

Geometrical global continuous symmetries of this dynamics include constant
translations in time, constant translations in space, constant rotations in
space that leave invariant the force vector $\vec{F}$, and constant velocity
Galilean boosts in space. To each class of these transformations there
correspond conserved Noether charges which are the generators of these
transformations once the phase space parametrisation of the system is
considered. Knowing that the Hamiltonian is the Noether charge for constant
time translations, hereafter the dynamical conserved charges associated to
constant space translations and constant velocity Galilean boosts are being
considered specifically, leaving aside the generators for spatial rotations in
those planes perpendicular to $\vec{F}$. Denoting by $\vec{a}$ the constant
spatial translation vector, and by $\vec{v}$ the constant Galilean boost
vector (not to be confused with the particle’s velocity, $\dot{\vec{x}}(t)$),
the action of these transformations on the degrees of freedom $\vec{x}(t)$
(namely $x^{i}(t)$ with $i=1,2,\cdots,d$) and the velocity momentum
$\vec{p}(t)=m\dot{\vec{x}}(t)$ are represented as follows,

$\vec{x}\,^{\prime}(t)=\vec{x}(t)\,+\,\vec{a}\,+\,t\,\vec{v},\qquad\vec{p}\,^{\prime}(t)=\vec{p}(t)\,+\,m\,\vec{v}.$
(54)

###  5.1 Classical dynamics and dynamical constants of the motion

The above equation of motion (53) is of course the Euler-Lagrange equation
given the following choice of Lagrangian action for this system,

$S_{\rm L}[\vec{x}]=\int
dt\,\left[\frac{1}{2}m\dot{\vec{x}}^{\,2}\,+\,\vec{x}\cdot\vec{F}\right].$
(55)

Its associated first-order Hamiltonian action is given as, by explicitly using
as well the free parameter $\lambda\in\mathbb{R}$ as in the general discussion
of Sect.2,

$S[\vec{x},\vec{p}\,]=\int
dt\left[\,\lambda\,\dot{\vec{x}}\cdot\vec{p}\,-\,(1-\lambda)\,\vec{x}\cdot\dot{\vec{p}}\,-\,H(\vec{x},\vec{p}\,)\right],\qquad
H(\vec{x},\vec{p}\,)=\frac{1}{2m}\vec{p}\,^{2}\,-\,\vec{x}\cdot\vec{F},$ (56)

where the canonical conjugate momentum coincides with the velocity momentum,
$\vec{p}=m\dot{\vec{x}}$.

Clearly whether in the Lagrangian or the Hamiltonian formulation, none of
these actions are exactly invariant under the considered symmetry
transformations, but they are indeed invariant up to a total time derivative
contribution which is a necessary and sufficient condition for a Noether
symmetry. In the case of the Lagrangian action one has,

$S_{\rm L}[\vec{x}\,^{\prime}]=S_{\rm L}[\vec{x}]+\int
dt\frac{d}{dt}\left[m\vec{v}\cdot\vec{x}+\frac{1}{2}t\,m\vec{v}\,^{2}+t\,\vec{a}\cdot\vec{F}+\frac{1}{2}t^{2}\,\vec{v}\cdot\vec{F}\right],$
(57)

while the first-order Hamiltonian action is such that

$S[\vec{x}\,^{\prime},\vec{p}\,^{\prime}]=S[\vec{x},\vec{p}\,]+\int
dt\frac{d}{dt}\Lambda(\vec{x},\vec{p};t|\vec{a},\vec{v}),$ (58)

where

$\Lambda(\vec{x},\vec{p};t|\vec{a},\vec{v})=\lambda\,m\vec{v}\cdot\vec{x}+\lambda
t\,m\vec{v}\,^{2}-(1-\lambda)\,t\,\vec{v}\cdot\vec{p}-\frac{1}{2}t\,m\vec{v}\,^{2}+\frac{1}{2}t^{2}\,\vec{v}\cdot\vec{F}-(1-\lambda)\,\vec{a}\cdot\vec{p}+t\,\vec{a}\cdot\vec{F}.$
(59)

Therefore in linearised form to first order in $\vec{a}$ and $\vec{v}$ one
obtains,

$\Lambda(\vec{x},\vec{p};t|\vec{a},\vec{v})\simeq\vec{a}\cdot\left[-(1-\lambda)\,\vec{p}+t\,\vec{F}\right]\,+\,\vec{v}\cdot\left[\lambda\,m\vec{x}\,-\,(1-\lambda)\,t\,\vec{p}\,+\,m\,\frac{1}{2}t^{2}\,\vec{F}\right].$
(60)

On the other hand, since in the notations of Sect.2.1 one has for the
transformed phase space coordinates,

$\delta\vec{x}=\vec{a}+t\,\vec{v},\qquad\delta\vec{p}=m\vec{v},$ (61)

and given the identification (13) for Noether charges, one readily identifies
the Noether charge for spatial translations to read,

$\vec{T}(\vec{x},\vec{p};t)=\vec{p}\,-\,t\,\vec{F},$ (62)

and that for Galilean boosts,

$\vec{\gamma}(\vec{x},\vec{p};t)=-m\vec{x}+t\,\vec{p}\,-\,\frac{1}{2}t^{2}\,\vec{F},$
(63)

both of which are thus dynamical constants of motion inclusive of their
explicit time dependency as phase space observables. Explicitly one finds for
their Hamiltonian equations of motion,

$\frac{\partial\vec{T}}{\partial
t}=-\vec{F},\quad\left\\{\vec{T},H\right\\}=\vec{F},\qquad\frac{\partial\vec{\gamma}}{\partial
t}=\vec{p}-t\vec{F}=\vec{T},\quad\left\\{\vec{\gamma},H\right\\}=-\vec{p}+t\vec{F}=-\vec{T},$
(64)

so that indeed,

$\frac{d\vec{T}}{dt}=\frac{\partial\vec{T}}{\partial
t}+\left\\{\vec{T},H\right\\}=\vec{0},\qquad\frac{d\vec{\gamma}}{dt}=\frac{\partial\vec{\gamma}}{\partial
t}+\left\\{\vec{\gamma},H\right\\}=\vec{0}.$ (65)

Furthermore the Poisson bracket algebra of these Noether charges is given as,

$\left\\{T_{i},T_{j}\right\\}=0,\quad\left\\{\gamma_{i},\gamma_{j}\right\\}=0,\quad\left\\{T_{i},\gamma_{j}\right\\}=m\delta_{ij},\quad\left\\{\vec{T},H\right\\}=\vec{F},\quad\left\\{\vec{\gamma},H\right\\}=-\vec{T},$
(66)

showing that the mass parameter $m$ determines a classical central extension
of the algebra of the Galilean group[6], with a value independent of the
external force $\vec{F}$. In particular note how the dynamical observables
$T_{i}(\vec{x},\vec{p};t)$ and $\gamma_{i}(\vec{x},\vec{p};t)/m$ are
canonically conjugate variables in phase space for all values of $t$, with the
same Poisson brackets as the basic phase space coordinates $x_{i}$ and
$p_{i}$. In addition, since one finds,

$\left\\{\vec{x},\vec{a}\cdot\vec{T}\right\\}=\vec{a},\quad\left\\{\vec{p},\vec{a}\cdot\vec{T}\right\\}=\vec{0},\quad\left\\{\vec{x},\vec{v}\cdot\vec{\gamma}\right\\}=t\,\vec{v},\quad\left\\{\vec{p},\vec{v}\cdot\vec{\gamma}\right\\}=m\,\vec{v},$
(67)

the conserved quantities $\vec{T}$ and $\vec{\gamma}$ are indeed the phase
space generators of spatial translations and of Galilean boosts, respectively,
themselves transforming as,

$\left\\{\vec{T},\vec{a}\cdot\vec{T}\right\\}=\vec{0},\quad\left\\{\vec{\gamma},\vec{a}\cdot\vec{T}\right\\}=-m\vec{a},\quad\left\\{\vec{T},\vec{v}\cdot\vec{\gamma}\right\\}=m\vec{v},\quad\left\\{\vec{\gamma},\vec{v}\cdot\vec{\gamma}\right\\}=\vec{0}.$
(68)

The $(\vec{x},\vec{p}\,)$ phase space parametrisation may also be expressed in
terms of these Noether charges, as,

$\vec{p}=\vec{T}(\vec{x},\vec{p};t)+t\,\vec{F},\qquad\vec{x}=\frac{1}{m}\left(-\vec{\gamma}(\vec{x},\vec{p};t)+t\,\vec{T}(\vec{x},\vec{p};t)+\frac{1}{2}t^{2}\,\vec{F}\right).$
(69)

In particular, since the solution to the Hamiltonian equations of motion for
the Noether charges is trivial and in the form of,

$\vec{T}(\vec{x}(t),\vec{p}(t);t)=\vec{T}(\vec{x}_{0},\vec{p}_{0};0)=\vec{p}_{0},\qquad\vec{\gamma}(\vec{x}(t),\vec{p}(t);t)=\vec{\gamma}(\vec{x}_{0},\vec{p}_{0};0)=-m\vec{x}_{0},$
(70)

where $\vec{x}_{0}$ and $\vec{p}_{0}$ are the initial values for $\vec{x}(t)$
and $\vec{p}(t)$ at $t=0$, respectively, the solution to the Hamiltonian
equations of motion for the latter degrees of freedom is readily given as,
directly from their representation above in terms of the Noether charges,

$\vec{x}(t)=\vec{x}_{0}+\frac{t}{m}\vec{p}_{0}+\frac{t^{2}}{2m}\vec{F},\qquad\vec{p}(t)=\vec{p}_{0}+t\vec{F},$
(71)

of course as it should.

Consequently the full dynamical symmetry Lie algebra of the system is simply
the enveloping algebra of the dynamical conserved Noether charges $\vec{T}$
and $\vec{\gamma}$, which are the primary generators of that algebra also as a
spectrum generating algebra. For instance in terms of these constants of the
motion the Hamiltonian of the particle reads,

$H(\vec{x},\vec{p}\,)=\frac{1}{2m}\vec{T}\,^{2}(\vec{x},\vec{p};t)+\frac{1}{m}\vec{F}\cdot\vec{\gamma}(\vec{x},\vec{p};t),$
(72)

with the following constant value for the solution determined by the initial
conditions $\vec{x}_{0}$ and $\vec{p}_{0}$,

$H(\vec{x}(t),\vec{p}(t))=\frac{1}{2m}\vec{p}_{0}\,^{2}-\vec{x}_{0}\cdot\vec{F}.$
(73)

This observation is also consistent with the variation of the Hamiltonian
under finite transformations generated by these two Noether charges, given as,

$H(\vec{x}+\vec{a}+t\vec{v},\vec{p}+m\vec{v}\,)\,-\,H(\vec{x},\vec{p}\,)=-\vec{a}\cdot\vec{F}+\vec{v}\cdot\vec{T}+\frac{1}{2}m\vec{v}\,^{2}=-\vec{a}\cdot\vec{F}+\vec{v}\cdot\left(\vec{p}-t\vec{F}\right)+\frac{1}{2}m\vec{v}\,^{2},$
(74)

a result which coincides with the finite second order recursive action of the
Noether charges through Poisson brackets, namely,

$\displaystyle
H(\vec{x}+\vec{a}+t\vec{v},\vec{p}+m\vec{v}\,)=H(\vec{x},\vec{p}\,)+\left\\{H(\vec{x},\vec{p}\,),\vec{a}\cdot\vec{T}(\vec{x},\vec{p};t)+\vec{v}\cdot\vec{\gamma}(\vec{x},\vec{p};t)\right\\}+$
$\displaystyle\qquad\qquad+\frac{1}{2}\left\\{\left\\{H(\vec{x},\vec{p}\,),\vec{a}\cdot\vec{T}(\vec{x},\vec{p};t)+\vec{v}\cdot\vec{\gamma}(\vec{x},\vec{p};t)\right\\},\vec{a}\cdot\vec{T}(\vec{x},\vec{p};t)+\vec{v}\cdot\vec{\gamma}(\vec{x},\vec{p};t)\right\\}.$
(75)

Thus spatial translations and Galilean boosts, which are global continuous Lie
symmetries of the dynamics, indeed map between solutions with different energy
values, since their generators do not commute with the Hamiltonian even though
these observables constitute the primary generators of all dynamical conserved
quantities of the system.

Incidentally note that the Noether charges of the remaining geometrical
symmetry not included in the present discussion, namely SO($d-1$) spatial
rotations in all those planes which are perpendicular to $\vec{F}$, already
belong to the enveloping algebra generated by the Noether dynamical charges
$\vec{T}$ and $\vec{\gamma}$. Indeed if the components of $\vec{x}$ and
$\vec{p}$ that are perpendicular to $\vec{F}$ are denoted by $\vec{x}_{\bot}$
and $\vec{p}_{\bot}$, respectively, and likewise for those of $\vec{T}$ and
$\vec{\gamma}$, the Noether charges for these spatial rotations are given in
the following form,

$L_{ij}=x_{\bot,i}\,p_{\bot,j}-x_{\bot,j}\,p_{\bot,i}=-\frac{1}{m}\left(\gamma_{\bot,i}\,T_{\bot,j}\,-\,\gamma_{\bot,j}\,T_{\bot,i}\right).$
(76)

###  5.2 Quantising the dynamics — at $t=0$

Given the above canonical Hamiltonian formulation of the system, let us
consider its quantisation relative to the reference time $t=0$. The space of
quantum states is a representation of the commutation relations for the basic
phase space observables at $t=0$, promoted now to quantum operators
$(\hat{\vec{x}}_{0}\equiv\hat{\vec{x}}(t=0),\hat{\vec{p}}_{0}\equiv\hat{\vec{p}}(t=0))$
which span a tensor product of Heisenberg algebras, namely (only the
nonvanishing commutators are given),

$\left[\hat{x}_{0,i},\hat{p}_{0,j}\right]=i\hbar\,\delta_{ij}\,\mathbb{I},\qquad\hat{\vec{x}}_{0}\,^{\dagger}=\hat{\vec{x}}_{0},\qquad\hat{\vec{p}}_{0}\,^{\dagger}=\hat{\vec{p}}_{0}.$
(77)

The quantum Hamiltonian is then expressed as,

$\hat{H}=\frac{1}{2m}\hat{\vec{p}}_{0}^{\,2}\,-\,\hat{\vec{x}}_{0}\cdot\vec{F},$
(78)

which is trivially such that

$\frac{\partial\hat{H}}{\partial t}=0,\qquad\left[\hat{H},\hat{H}\right]=0,$
(79)

implying that, from the point of view of the Heisenberg equation of motion in
the Heisenberg picture of time evolution, it is a conserved quantity,
$\hat{H}(t)=\hat{H}(t=0)=\hat{H}$, without any explicit time dependency.

Within the Schrödinger picture for time evolution, the Noether charges for
spatial translations and Galilean boosts are given as,

$\hat{\vec{T}}(t)=\hat{\vec{p}}_{0}\,-\,t\,\vec{F}\,\mathbb{I},\quad\hat{\vec{\gamma}}(t)=-m\hat{\vec{x}}_{0}+t\,\hat{\vec{p}}_{0}-\frac{1}{2}t^{2}\,\vec{F}\,\mathbb{I},\qquad\hat{\vec{T}}(t)^{\,\dagger}=\hat{\vec{T}}(t),\quad\hat{\vec{\gamma}}(t)^{\,\dagger}=\hat{\vec{\gamma}}(t).$
(80)

Note that $\hat{\vec{T}}(t=0)=\hat{\vec{p}}_{0}$ and
$\hat{\vec{\gamma}}(t=0)=-m\hat{\vec{x}}_{0}$. These dynamical conserved
quantum observables are such that,

$\left[\hat{\vec{T}}(t),\hat{H}\right]=i\hbar\,\vec{F}\,\mathbb{I},\qquad\left[\hat{\vec{\gamma}}(t),\hat{H}\right]=-i\hbar\,\hat{\vec{T}}(t),$
(81)

while their algebra is given as,

$\left[\hat{T}_{i}(t),\hat{T}_{j}(t)\right]=0,\qquad\left[\hat{T}_{i}(t),\hat{\gamma}_{i}(t)\right]=i\hbar\,m\,\delta_{ij}\,\mathbb{I},\qquad\left[\hat{\gamma}_{i}(t),\hat{\gamma}_{j}(t)\right]=0.$
(82)

###  5.3 Time evolution in the Heisenberg picture

Within the Heisenberg picture the time evolution of a quantum observable
$\hat{A}_{H}(t)$ is governed by the Heisenberg equation of motion (which is
direct correspondence with its classical Hamiltonian equation of motion),

$i\hbar\frac{d\hat{A}_{H}(t)}{dt}=i\hbar\frac{\partial\hat{A}_{H}(t)}{\partial
t}+\left[\hat{A}_{H}(t),\hat{H}\right],$ (83)

in which the time independency of the Hamiltonian operator within the
Heisenberg picture is already accounted for (namely $\hat{H}_{H}(t)=\hat{H}$),
while the partial time derivative term on the r.h.s. of this relation only
acts on the possible explicit time dependency of the observable $\hat{A}(t)$
in the Schrödinger picture. If $\hat{A}(t)$ denotes that observable in the
Schrödinger picture as obtained through quantisation at $t=0$ of the classical
dynamics as discussed in Sect.5.2, the general solution for the time evolution
of the observable in the Heisenberg picture is thus given by the following
unitary transformation involving the generator for finite translations in
time,

$\hat{A}_{H}(t)=e^{\frac{i}{\hbar}t\,\hat{H}}\,\hat{A}(t)\,e^{-\frac{i}{\hbar}t\,\hat{H}}.$
(84)

In particular by applying the relevant Baker-Campbell-Hausdorff formula (for
$e^{A}Be^{-A}$), one readily finds for the time evolution of the basic phase
space observables in the Heisenberg picture,

$\hat{\vec{x}}_{H}(t)=\hat{\vec{x}}_{0}+\frac{t}{m}\hat{\vec{p}}_{0}+\frac{t^{2}}{2m}\vec{F}\,\mathbb{I},\qquad\hat{\vec{p}}_{H}(t)=\hat{\vec{p}}_{0}+t\,\vec{F}\,\mathbb{I},$
(85)

in direct correspondence with their classical solutions (on account of the
linearity of the classical and quantum equations of motion of the system).
Similarly one finds for the Noether charges in the Heisenberg picture,

$\hat{\vec{T}}_{H}(t)=\hat{\vec{p}}_{H}(t)-t\,\vec{F}\,\mathbb{I}=\hat{\vec{p}}_{0},\qquad\hat{\vec{\gamma}}_{H}(t)=-m\hat{\vec{x}}_{H}(t)+t\,\hat{\vec{p}}_{H}(t)-\frac{1}{2}t^{2}\vec{F}\,\mathbb{I}=-m\hat{\vec{x}}_{0},$
(86)

establishing that these are indeed constants of the motion, in spite of the
fact that they do not commute with the Hamiltonian, with
$\left[\hat{\vec{T}}_{H}(t),\hat{H}\right]=i\hbar\vec{F}\,\mathbb{I}$ and
$\left[\hat{\vec{\gamma}}_{H}(t),\hat{H}\right]=-i\hbar\,\hat{\vec{T}}_{H}(t)$,
while they determine a Heisenberg algebra with
$\left[\hat{T}_{H,i}(t),\hat{\gamma}_{H,j}(t)/m\right]=i\hbar\delta_{ij}\,\mathbb{I}$.

In a likewise manner finite symmetry transformations of quantum observables in
the Heisenberg picture generated by these Noether charges are represented by
the following unitary transformations,

$\hat{A}_{H}(t|\vec{a},\vec{v})=e^{\frac{i}{\hbar}\left(\vec{a}\cdot\hat{\vec{T}}_{H}(t)+\vec{v}\cdot\hat{\vec{\gamma}}_{H}(t)\right)}\,\hat{A}_{H}(t)\,e^{-\frac{i}{\hbar}\left(\vec{a}\cdot\hat{\vec{T}}_{H}(t)+\vec{v}\cdot\hat{\vec{\gamma}}_{H}(t)\right)}.$
(87)

In particular one then finds,

$\hat{\vec{x}}_{H}(t|\vec{a},\vec{v})=\hat{\vec{x}}_{H}(t)+\left(\vec{a}+t\,\vec{v}\right)\mathbb{I},\qquad\hat{\vec{p}}_{H}(t|\vec{a},\vec{v})=\hat{\vec{p}}_{H}(t)+m\vec{v}\,\mathbb{I},$
(88)

as well as,

$\hat{H}(\vec{a},\vec{v}\,)=\hat{H}-\vec{a}\cdot\vec{F}\,\mathbb{I}+\vec{v}\cdot\hat{\vec{T}}_{H}(t)+\frac{1}{2}m\vec{v}\,^{2}\,\mathbb{I}=\frac{1}{2m}\left(\hat{\vec{p}}_{0}+m\vec{v}\,\mathbb{I}\right)^{2}-\left(\hat{\vec{x}}_{0}+\vec{a}\,\mathbb{I}\right)\cdot\vec{F},$
(89)

indeed as should be expected given the corresponding expressions for the
classical dynamics.

To conclude, let us point out that the full dynamical symmetry Lie algebra of
constants of motion is thus spanned by the enveloping algebra of the Noether
charges $(\hat{\vec{T}}_{H}(t),\hat{\vec{\gamma}}_{H}(t))$ as primary
generators which, as dynamical conserved charges, also determine a generating
algebra for the entire energy spectrum of the system, to which we now turn.

###  5.4 Time evolution in the Schrödinger picture

Within the Schrödinger picture the time evolution of a quantum state
$|\psi,t\rangle$ is governed by the Schrödinger equation,

$i\hbar\frac{d|\psi,t\rangle}{dt}=\hat{H}\,|\psi,t\rangle,$ (90)

while quantum observables $\hat{A}(t)$ evolve in time only provided they
possess an explicit time dependency (which is thus not generated by any action
of the Hamiltonian operator), as is the case for instance for the Noether
charges $(\hat{\vec{T}}(t),\hat{\vec{\gamma}}(t))$ in the Schrödinger picture.

The space of quantum states provides a representation of the Heisenberg
algebra spanned by the basic phase space operators
$(\hat{\vec{x}}_{0},\hat{\vec{p}}_{0})$, leading to the well known
configuration and momentum space wave function representations of that algebra
using normalised position and momentum eigenstates such that,

$\hat{\vec{x}}_{0}|\vec{x}\,\rangle=\vec{x}\,|\vec{x}\,\rangle,\quad\hat{\vec{p}}_{0}|\vec{p}\,\rangle=\vec{p}\,|\vec{p}\,\rangle,\quad\langle\vec{x}_{1}|\vec{x}_{2}\rangle=\delta^{(d)}(\vec{x}_{1}-\vec{x}_{2}),\quad\langle\vec{p}_{1}|\vec{p}_{2}\rangle=\delta^{(d)}(\vec{p}_{1}-\vec{p}_{2}).$
(91)

Consequently,

$\int_{(\infty)}d^{d}\vec{x}\,|\vec{x}\,\rangle\,\langle\vec{x}\,|=\mathbb{I}=\int_{(\infty)}d^{d}\vec{p}\,|\vec{p}\,\rangle\,\langle\vec{p}\,|,\qquad|\varphi\rangle=\int_{(\infty)}d^{d}\vec{x}\,|\vec{x}\,\rangle\,\varphi(\vec{x})=\int_{(\infty)}d^{d}\vec{p}\,|\vec{p}\,\rangle\,\tilde{\varphi}(\vec{p}\,),$
(92)

with the configuration and momentum space wave functions representing the
abstract state $|\varphi\rangle$ defined by
$\varphi(\vec{x}\,)=\langle\vec{x}\,|\varphi\rangle$ and
$\tilde{\varphi}(\vec{p}\,)=\langle\vec{p}\,|\varphi\rangle$, respectively.
Finally the local phase of the position and momentum eigenstates may be chosen
such that the operators $(\hat{\vec{x}}_{0},\hat{\vec{p}}_{0})$ possess the
following wave function representations[7],

$\langle\vec{x}\,|\hat{\vec{x}}_{0}|\varphi\rangle=\vec{x}\,\varphi(\vec{x}\,),\quad\langle\vec{x}\,|\hat{\vec{p}}_{0}|\varphi\rangle=-i\hbar\vec{\nabla}_{\vec{x}}\,\varphi(\vec{x}\,),\quad\langle\vec{p}\,|\hat{\vec{x}}_{0}|\varphi\rangle=i\hbar\vec{\nabla}_{\vec{p}}\,\tilde{\varphi}(\vec{p}\,),\quad\langle\vec{p}\,|\hat{\vec{p}}_{0}|\varphi\rangle=\vec{p}\,\tilde{\varphi}(\vec{p}\,),\quad$
(93)

which in particular implies the following change of basis overlaps and Fourier
transformations relating the wave functions $\varphi(\vec{x}\,)$ and
$\tilde{\varphi}(\vec{p}\,)$, in which a final relative global phase choice is
implicit,

$\langle\vec{x}\,|\vec{p}\,\rangle=\frac{1}{(2\pi\hbar)^{d/2}}\,e^{\frac{i}{\hbar}\vec{x}\cdot\vec{p}},\qquad\langle\vec{p}\,|\vec{x}\,\rangle=\frac{1}{(2\pi\hbar)^{d/2}}\,e^{-\frac{i}{\hbar}\vec{x}\cdot\vec{p}}.$
(94)

Consequently the Noether charges have the following wave function
representations, namely in configuration space,

$\hat{\vec{T}}(t)\ :\quad-i\hbar\vec{\nabla}_{\vec{x}}\,-\,t\,\vec{F}\
;\qquad\hat{\vec{\gamma}}(t)\
:\quad-m\vec{x}\,-\,i\hbar\,t\,\vec{\nabla}_{\vec{x}}\,-\,\frac{1}{2}t^{2}\,\vec{F},$
(95)

and in momentum space,

$\hat{\vec{T}}(t)\ :\quad\vec{p}\,-\,t\,\vec{F}\ ;\qquad\hat{\vec{\gamma}}(t)\
:\quad-i\hbar\,m\,\vec{\nabla}_{\vec{p}}\,+\,t\,\vec{p}\,-\,\frac{1}{2}t^{2}\,\vec{F}.$
(96)

However since the Noether charges themselves define what is essentially
another Heisenberg algebra, so that $\hat{\vec{T}}(t)$ on the one hand, and
$\hat{\vec{\gamma}}(t)$ on the other hand, determine a complete set of
commuting observables, one may consider in a likewise manner two further bases
in Hilbert space spanned by the normalised eigenstates of each of these
Noether charges, with time independent eigenvalues since these are dynamical
conserved observables. Hence let us consider the following eigenstates,

$\hat{\vec{T}}(t)\,|\vec{T},t\rangle=\vec{T}\,|\vec{T},t\rangle,\qquad\hat{\vec{\gamma}}(t)\,|\vec{\gamma},t\rangle=\vec{\gamma}\,|\vec{\gamma},t\rangle,$
(97)

normalised such that

$\langle\vec{T}_{1},t|\vec{T}_{2},t\rangle=\delta^{(d)}(\vec{T}_{1}-\vec{T}_{2}),\qquad\langle\gamma_{1},t|\vec{\gamma}_{2},t\rangle=\delta^{(d)}(\vec{\gamma}_{1}-\gamma_{2}).$
(98)

Consequently

$\int_{(\infty)}d^{d}\vec{T}\,|\vec{T},t\rangle\,\langle\vec{T},t|=\mathbb{I}=\int_{(\infty)}d^{d}\vec{\gamma}\,|\vec{\gamma},t\rangle\,\langle\vec{\gamma},t|,$
(99)

thereby leading to the $\vec{T}$\- and $\vec{\gamma}$-wave function
representations of quantum states,

$|\varphi\rangle=\int_{(\infty)}d^{d}\vec{T}\,|\vec{T},t\rangle\,\varphi_{\vec{T}}(\vec{T},t)=\int_{(\infty)}d^{d}\vec{\gamma}\,|\vec{\gamma},t\rangle\,\varphi_{\vec{\gamma}}(\vec{\gamma},t),$
(100)

where

$\varphi_{\vec{T}}(\vec{T},t)=\langle\vec{T},t|\varphi\rangle,\qquad\varphi_{\vec{\gamma}}(\vec{\gamma},t)=\langle\vec{\gamma},t|\varphi\rangle.$
(101)

Finally the local phase of the $\hat{\vec{T}}$\- and
$\hat{\vec{\gamma}}$-eigenstates may be chosen in such a way that the Noether
charges have the following $\vec{T}$\- and $\vec{\gamma}$-wave function
representations, namely in the $\vec{T}$-representation,

$\hat{\vec{T}}(t)\ :\quad\vec{T}\ ;\qquad\hat{\vec{\gamma}}(t)\
:\quad-i\hbar\,m\,\vec{\nabla}_{\vec{T}},$ (102)

and in the $\vec{\gamma}$-representation,

$\hat{\vec{T}}(t)\ :\quad i\hbar\,m\,\vec{\nabla}_{\vec{\gamma}}\
;\qquad\hat{\vec{\gamma}}(t)\ :\quad\vec{\gamma}.$ (103)

Quite obviously one then has as well, up to some phase factor
$\Phi(\vec{T},\vec{\gamma},t)$ still to be specified or identified at least
implicitly (as will be done hereafter),

$\langle\vec{T},t|\vec{\gamma},t\rangle=\frac{1}{(2\pi
m\hbar)^{d/2}}\,e^{\frac{i}{m\hbar}\vec{T}\cdot\vec{\gamma}}\,e^{i\Phi(\vec{T},\vec{\gamma},t)},\quad\langle\vec{\gamma},t|\vec{T},t\rangle=\frac{1}{(2\pi
m\hbar)^{d/2}}\,e^{-\frac{i}{m\hbar}\vec{T}\cdot\vec{\gamma}}\,e^{-i\Phi(\vec{T},\vec{\gamma},t)}.$
(104)

####  5.4.1 Energy eigenstates

Before determining the $\vec{x}$\- and $\vec{p}$-representations of the
$\hat{\vec{T}}$\- and $\hat{\vec{\gamma}}$-eigenstates, let us turn to the
identification of the energy eigenstates of the system, in the form

$|\psi_{E},t\rangle=e^{-\frac{i}{\hbar}t\,E}\,|E\rangle,\qquad\hat{H}|E\rangle=E\,|E\rangle.$
(105)

Given that the potential $V(\vec{x}\,)=-\vec{x}\cdot\vec{F}$ is unbounded
below (and above), the system does not possess a ground state, a feature which
in itself is not a physical inconsistency. Provided it remains decoupled from
any external interaction other than the constant force field, the system
displays stability under time evolution in any of its possible energy
configurations, of which the continuum spectrum covers the entire real line,
$-\infty<E<+\infty$. Consequently energy eigenstates are to be normalised
according to

$\langle E_{1}|E_{2}\rangle=\delta(E_{1}-E_{2}),\qquad
E_{1},E_{2}\in\mathbb{R}.$ (106)

Note well however that energy eigen-levels will (for $d\geq 3$) possess a
degeneracy related to the SO($d-1$) symmetry of the dynamics under rotations
in all those planes perpendicular to the constant force vector $\vec{F}$.

For later convenience let us introduce the unit vector $\hat{F}$ in the
direction of $\vec{F}$,

$\hat{F}=\frac{1}{F}\vec{F},\qquad F=|\vec{F}|>0,\qquad\vec{F}=F\,\hat{F}.$
(107)

Any vector, say $\vec{x}$ of $\vec{p}$, is then to be decomposed as follows in
terms of components parallel and perpendicular relative to $\hat{F}$,

$\vec{x}=x_{\parallel}\,\hat{F}\,+\,\vec{x}_{\bot},\quad\vec{p}=p_{\parallel}\,\hat{F}\,+\,\vec{p}_{\bot},\qquad
x_{\parallel}=\vec{x}\cdot\hat{F},\quad
p_{\parallel}=\vec{p}\cdot\hat{F},\qquad\vec{x}_{\bot}\cdot\hat{F}=0=\vec{p}_{\bot}\cdot\hat{F}.$
(108)

In order to solve the stationary Schrödinger equation it is best to work in
the $\vec{p}$-representation, leading to,

$\left(\frac{1}{2m}\vec{p}\,^{2}-i\hbar\,\vec{F}\cdot\vec{\nabla}_{\vec{p}}\right)\,\tilde{\varphi}_{E}(\vec{p}\,)=E\,\tilde{\varphi}_{E}(\vec{p}\,),\qquad\tilde{\varphi}_{E}(\vec{p}\,)=\langle\vec{p}\,|E\rangle,$
(109)

or equivalently,

$-i\hbar\,F\,\frac{\partial}{\partial
p_{\parallel}}\,\tilde{\varphi}_{E}(p_{\parallel},\vec{p}_{\bot})=\left(E-\frac{1}{2m}\vec{p}_{\bot}^{\,2}-\frac{1}{2m}p_{\parallel}^{2}\right)\,\tilde{\varphi}_{E}(p_{\parallel},\vec{p}_{\bot}).$
(110)

The general solution, properly normalised, is then of the form,

$\tilde{\varphi}_{E}(\vec{p}\,)=\langle\vec{p}\,|E\rangle=\frac{1}{\sqrt{2\pi\hbar
F}}\,\tilde{\varphi}_{\bot}(\vec{p}_{\bot})\,{\rm exp}\left\\{\frac{i}{\hbar
F}\left[\left(E-\frac{1}{2m}\vec{p}_{\bot}^{\,2}\right)\,p_{\parallel}\,-\,\frac{1}{6m}\,p^{3}_{\parallel}\right]\right\\},\quad
E\in\mathbb{R},$ (111)

where $\tilde{\varphi}_{\bot}(\vec{p}_{\bot})$ is some arbitrarily chosen
function parametrising the degeneracy of the energy levels under SO($d-1$)
rotations and normalised such that

$\int_{(\infty)}d^{d-1}\vec{p}_{\bot}\,|\tilde{\varphi}_{\bot}(\vec{p}_{\bot})|^{2}=1.$
(112)

If need be, a basis for these functions
$\tilde{\varphi}_{\bot}(\vec{p}_{\bot})$ may be constructed in terms of Bessel
functions and SO($d-1$) hyperspherical harmonics in the space
$\vec{p}_{\bot}$.

Through inverse Fourier transformations the $\vec{x}$-representation of these
states is obtained from,

$\displaystyle\varphi_{E}(x_{\parallel},\vec{x}_{\bot})=\varphi_{E}(\vec{x}\,)=\langle\vec{x}\,|E\rangle=\int_{(\infty)}\frac{d^{d}\vec{p}}{(2\pi\hbar)^{d/2}}\,e^{\frac{i}{\hbar}\vec{x}\cdot\vec{p}}\,\tilde{\varphi}_{E}(\vec{p}\,)=$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2\pi\hbar
F}}\int_{-\infty}^{\infty}\frac{dp_{\parallel}}{\sqrt{2\pi\hbar}}\,e^{\frac{i}{\hbar}x_{\parallel}p_{\parallel}}\,e^{\frac{i}{\hbar
F}\left[\left(E-\frac{1}{2m}\vec{p}_{\bot}^{\,2}\right)\,p_{\parallel}\,-\,\frac{1}{6m}\,p^{3}_{\parallel}\right]}\times\,\int_{(\infty)}\frac{d^{d-1}\vec{p}_{\bot}}{(2\pi\hbar)^{(d-1)/2}}\,e^{\frac{i}{\hbar}\vec{x}_{\bot}\cdot\vec{p}_{\bot}}\,\tilde{\varphi}_{\bot}(\vec{p}_{\bot}),$

and this in particular in terms of the Airy function of the first kind for
what the dependency on the cartesian coordinate $x_{\parallel}$ aligned with
the constant force field is concerned[2] (which results from the integral over
$p_{\parallel}$).

####  5.4.2 $\hat{\vec{T}}(t)$-eigenstates

Working once again in the $\vec{p}$-representation,
$\hat{\vec{T}}(t)$-eigenstates $|\vec{T},t\rangle$ are characterised by the
following eigenvalue equation,

$\left(\vec{p}\,-\,t\,\vec{F}\right)\,\tilde{\varphi}_{\vec{T}}(\vec{p},t)=\vec{T}\,\tilde{\varphi}_{\vec{T}}(\vec{p},t),\qquad\tilde{\varphi}_{\vec{T}}(\vec{p},t)=\langle\vec{p}\,|\vec{T},t\rangle.$
(114)

Therefore necessarily

$\tilde{\varphi}_{\vec{T}}(\vec{p},t)=\delta^{(d)}\left(\vec{p}-t\,\vec{F}-\vec{T}\right)\,\tilde{\chi}_{\vec{T}}(t),$
(115)

where $\tilde{\chi}_{\vec{T}}(t)$ is a still to be determined function of $t$
only, such that the states $|\vec{T},t\rangle$ obey the Schrödinger equation,
namely,

$\left(\frac{1}{2m}\vec{p}^{\,2}-i\hbar\vec{F}\cdot\vec{\nabla}_{\vec{p}}\right)\,\tilde{\varphi}_{\vec{T}}(\vec{p},t)=i\hbar\frac{\partial}{\partial
t}\,\tilde{\varphi}_{\vec{T}}(\vec{p},t).$ (116)

As a result one must have,

$i\hbar\,\frac{d}{dt}\tilde{\chi}_{\vec{T}}(t)=\frac{1}{2m}\left(\vec{T}_{\bot}^{2}+(T_{\parallel}+tF)^{2}\right)\,\tilde{\chi}_{\vec{T}}(t)=\frac{1}{2m}\left(\vec{T}^{2}+2t\,\vec{T}\cdot\vec{F}+t^{2}\vec{F}^{2}\right)\,\tilde{\chi}_{\vec{T}}(t).$
(117)

By choosing appropriately the hitherto unspecified local phase of the states
$|\vec{T},t\rangle$, and imposing the normalisation of the states
$|\vec{T},t\rangle$ as indicated above, one thus finds,

$\langle\vec{p}\,|\vec{T},t\rangle=\tilde{\varphi}_{\vec{T}}(\vec{p},t)=\delta^{(d)}\left(\vec{p}-t\vec{F}-\vec{T}\right)\,e^{-\frac{i}{\hbar}\frac{t}{2m}\left(\vec{T}^{2}+t\,\vec{T}\cdot\vec{F}+\frac{t^{2}}{3}\,\vec{F}^{2}\right)},$
(118)

as well as in the $\vec{x}$-representation,

$\langle\vec{x}\,|\vec{T},t\rangle=\frac{1}{(2\pi\hbar)^{d/2}}\,e^{\frac{i}{\hbar}\vec{x}\cdot(\vec{T}+t\vec{F})}\,e^{-\frac{i}{\hbar}\frac{t}{2m}\left(\vec{T}^{2}+t\,\vec{T}\cdot\vec{F}+\frac{t^{2}}{3}\,\vec{F}^{2}\right)}.$
(119)

Incidentally note that as a matter of fact the $\hat{\vec{T}}(t)$-eigenstates
coincide essentially with the momentum eigenstates but with a time dependency
in their momentum value and in their phase factor, in the form,

$|\vec{T},t\rangle=e^{-\frac{i}{\hbar}\frac{t}{2m}\left(\vec{T}^{2}+t\,\vec{T}\cdot\vec{F}+\frac{t^{2}}{3}\,\vec{F}^{2}\right)}\,|\vec{p}=\vec{T}+t\vec{F}\rangle.$
(120)

####  5.4.3 $\hat{\vec{\gamma}}(t)$-eigenstates

Working once again in the $\vec{p}$-representation,
$\hat{\vec{\gamma}}(t)$-eigenstates $|\vec{\gamma},t\rangle$ are characterised
by the following eigenvalue equation,

$\left(-i\hbar\,m\,\vec{\nabla}_{\vec{p}}+t\,\vec{p}-\frac{1}{2}t^{2}\,\vec{F}\right)\,\tilde{\varphi}_{\vec{\gamma}}(\vec{p},t)=\vec{\gamma}\,\tilde{\varphi}_{\vec{\gamma}}(\vec{p},t),\qquad\tilde{\varphi}_{\vec{\gamma}}(\vec{p},t)=\langle\vec{p}\,|\vec{\gamma},t\rangle.$
(121)

Therefore necessarily

$\tilde{\varphi}_{\vec{\gamma}}(\vec{p},t)=e^{\frac{i}{\hbar
m}\left(\vec{\gamma}\cdot\vec{p}+\frac{t^{2}}{2}\vec{F}\cdot\vec{p}-\frac{t}{2}\vec{p}^{\,2}\right)}\,\tilde{\chi}_{\vec{\gamma}}(t),$
(122)

where $\tilde{\chi}_{\vec{\gamma}}(t)$ is a still to be determined function of
$t$ only, such that the states $|\vec{\gamma},t\rangle$ obey the Schrödinger
equation, namely,

$\left(\frac{1}{2m}\vec{p}^{\,2}-i\hbar\vec{F}\cdot\vec{\nabla}_{\vec{p}}\right)\,\tilde{\varphi}_{\vec{\gamma}}(\vec{p},t)=i\hbar\frac{\partial}{\partial
t}\,\tilde{\varphi}_{\vec{\gamma}}(\vec{p},t).$ (123)

As a result one must have,

$i\hbar\,\frac{d}{dt}\tilde{\chi}_{\vec{\gamma}}(t)=\frac{1}{m}\left(\vec{F}\cdot\vec{\gamma}+\frac{t^{2}}{2}\vec{F}^{2}\right)\,\tilde{\chi}_{\vec{\gamma}}(t).$
(124)

By choosing appropriately the hitherto unspecified local phase of the states
$|\vec{\gamma},t\rangle$, and imposing the normalisation of the states
$|\vec{\gamma},t\rangle$ as indicated above, one thus finds,

$\langle\vec{p}\,|\vec{\gamma},t\rangle=\tilde{\varphi}_{\vec{\gamma}}(\vec{p},t)=\frac{1}{(2\pi\hbar\,m)^{d/2}}\,e^{-\frac{i}{\hbar}\frac{t}{m}\left(\vec{F}\cdot\vec{\gamma}+\frac{t^{2}}{6}\vec{F}^{2}\right)}\,e^{\frac{i}{\hbar
m}\left(\vec{\gamma}\cdot\vec{p}+\frac{t^{2}}{2}\vec{F}\cdot\vec{p}-\frac{t}{2}\vec{p}^{\,2}\right)},$
(125)

as well as in the $\vec{x}$-representation,

$\langle\vec{x}\,|\vec{\gamma},t\rangle=\frac{1}{(2\pi\hbar\,it)^{d/2}}\,e^{\frac{i}{\hbar}\frac{1}{2mt}\left(m\vec{x}+\vec{\gamma}+\frac{t^{2}}{2}\vec{F}\right)^{2}}\,e^{-\frac{i}{\hbar}\frac{t}{m}\left(\vec{F}\cdot\vec{\gamma}+\frac{t^{2}}{6}\vec{F}^{2}\right)}.$
(126)

The knowledge of the configuration and momentum wave function representations
of the $\hat{\vec{T}}(t)$\- and $\hat{\vec{\gamma}}(t)$-eigenstates, enables
one to also compute their overlap, in view of the earlier remark that these
operators essentially define a Heisenberg algebra. Completing the relevant
momentum integrations for the product
$\langle\vec{T},t|\vec{p}\,\rangle\langle\vec{p}\,|\vec{\gamma},t\rangle$, one
finds,

$\langle\vec{T},t|\vec{\gamma},t\rangle=\frac{1}{(2\pi\hbar\,m)^{d/2}}\,e^{\frac{i}{\hbar}\frac{1}{m}\vec{T}\cdot\vec{\gamma}}\,e^{\frac{i}{\hbar}\frac{t}{m}\left(\vec{T}+t\vec{F}\right)^{2}},$
(127)

a result which is certainly perfectly in line with the Heisenberg nature of
the algebra defined by these two Noether charges, and finally determines the
phase factor introduced in (104) as being given by
$\Phi(\vec{T},\vec{\gamma},t)=t(\vec{T}+t\vec{F})^{2}/(\hbar m)$.

####  5.4.4 Finite quantum spatial translations and Galilean boosts

Given the Noether charges in the Schrödinger picture expressed as

$\hat{\vec{T}}(t)=\hat{\vec{p}}_{0}-t\vec{F}\,\mathbb{I},\qquad\hat{\vec{\gamma}}(t)=-m\hat{\vec{x}}_{0}+t\hat{\vec{p}}_{0}-\frac{1}{2}t^{2}\vec{F}\,\mathbb{I},$
(128)

and using the different wave function representations constructed above, it is
a simple matter to check explicitly that indeed the states $|\vec{T},t\rangle$
and $|\vec{\gamma},t\rangle$ are eigenstates of these operators with time
independent and thus conserved eigenvalues, which obey as well the time
dependent Schrödinger equation, namely,

$\hat{\vec{T}}(t)|\vec{T},t\rangle=\vec{T}\,|\vec{T},t\rangle,\quad\hat{\vec{\gamma}}(t)|\vec{\gamma},t\rangle=\vec{\gamma}\,|\vec{\gamma},t\rangle,\quad\hat{H}|\vec{T},t\rangle=i\hbar\frac{d}{dt}|\vec{T},t\rangle,\quad\hat{H}|\vec{\gamma},t\rangle=i\hbar\frac{d}{dt}|\vec{\gamma},t\rangle.$
(129)

Even though dynamical with an explicit time dependency within the Schrödinger
picture, these Noether charges are nevertheless constants of the motion, in
full agreement with Noether’s first theorem within this quantum context.

Let us now consider finite transformations generated by these charges, within
the Schrödinger picture, beginning with spatial translations. Acting on the
basic phase space observables, a straightforward calculation finds,

$e^{\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{T}}(t)}\,\hat{\vec{x}}_{0}\,e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{T}}(t)}=\hat{\vec{x}}_{0}\,+\,\vec{a}\,\mathbb{I},\qquad
e^{\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{T}}(t)}\,\hat{\vec{p}}_{0}\,e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{T}}(t)}=\hat{\vec{p}}_{0},$
(130)

as it should be. Given that the $\hat{\vec{T}}(t)$ eigenstates are essentially
momentum eigenstates with a time dependency in their momentum eigenvalue and
in their phase, it readily follows that finite spatial translations transform
momentum eigenstates according to,

$e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{T}}(t)}\,|\vec{p}\,\rangle=e^{-\frac{i}{\hbar}\,\vec{a}\cdot(\vec{p}-t\vec{F})}\,|\vec{p}\,\rangle.$
(131)

Given the change of basis overlaps $\langle\vec{p}\,|\vec{x}\,\rangle$, from
this result one then establishes as well that,

$e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{T}}(t)}\,|\vec{x}\,\rangle=e^{\frac{i}{\hbar}\,t\,\vec{a}\cdot\vec{F}}\,|\vec{x}+\vec{a}\,\rangle.$
(132)

Considering now finite quantum Galilean boost transformations, one finds, thus
within the Schrödinger picture,

$e^{\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\,\hat{\vec{x}}_{0}\,e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}=\hat{\vec{x}}_{0}\,+\,t\,\vec{v}\,\mathbb{I},\qquad
e^{\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\,\hat{\vec{p}}_{0}\,e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}=\hat{\vec{p}}_{0}\,+\,m\vec{v}\,\mathbb{I},$
(133)

as it should be. Furthermore using the change of basis
$\langle\vec{\gamma},t|\vec{p}\,\rangle$, an explicit calculation establishes
that under Galilean boosts momentum eigenstates transform according to,

$e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\,|\vec{p}\,\rangle=e^{\frac{i}{\hbar}\left(\frac{t^{2}}{2}\vec{v}\cdot\vec{F}-t\,\vec{v}\cdot\vec{p}-\frac{t}{2}m\vec{v}^{\,2}\right)}\,|\vec{p}+m\vec{v}\,\rangle,$
(134)

from which it follows as well that, or else using the overlaps
$\langle\vec{\gamma},t|\vec{x}\,\rangle$,

$e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\,|\vec{x}\,\rangle=e^{\frac{i}{\hbar}\left(m\vec{v}\cdot\vec{x}+\frac{1}{2}mt\,\vec{v}^{\,2}+\frac{t^{2}}{2}\vec{v}\cdot\vec{F}\right)}\,|\vec{x}+t\,\vec{v}\,\rangle.$
(135)

Finally let us address the action of these symmetry transformations on the
energy eigenstates. First it is straightforward to establish that one has,
within the Schrödinger picture,

$e^{\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{T}}(t)}\,\hat{H}\,e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{T}}(t)}=\hat{H}-\vec{a}\cdot\vec{F}\,\mathbb{I},\qquad
e^{\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\,\hat{H}\,e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}=\hat{H}+\vec{v}\cdot\hat{\vec{T}}(t)+\frac{1}{2}m\vec{v}^{\,2}\,\mathbb{I},$
(136)

indeed as ought to be the case in view of the corresponding results for the
classical dynamics. Consider now an arbitrary energy eigenstate,

$|\psi_{E},t\rangle=e^{-\frac{i}{\hbar}t\,E}\,|E\rangle,\qquad\hat{H}|E\rangle=E\,|E\rangle,\qquad\hat{H}|\psi_{E},t\rangle=E\,|\psi_{E},t\rangle=i\hbar\frac{d}{dt}|\psi_{E},t\rangle.$
(137)

Within the Schrödinger picture and under a spatial translation this state is
transformed into

$|\psi_{E},t;\vec{a}\,\rangle=e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{T}}(t)}\,|\psi_{E},t\rangle=e^{-\frac{i}{\hbar}t\,E}\,e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{T}}(t)}\,|E\rangle=e^{-\frac{i}{\hbar}t(E-\vec{a}\cdot\vec{F})}\,e^{-\frac{i}{\hbar}\vec{a}\cdot\hat{\vec{p}}_{0}}\,|E\rangle.$
(138)

Applying the identity established above for the transformed Hamiltonian
itself, it readily follows that

$\hat{H}|\psi_{E},t;\vec{a}\,\rangle=\left(E-\vec{a}\cdot\vec{F}\right)\,|\psi_{E},t;\vec{a}\,\rangle=i\hbar\frac{d}{dt}|\psi_{E},t;\vec{a}\,\rangle.$
(139)

The transformed state $|\psi_{E},t;\vec{a}\,\rangle$ is thus some energy
eigenstate of energy $(E-\vec{a}\cdot\vec{F})$ which solves the Schrödinger
equation. The dynamical conserved Noether charge for spatial translations,
$\hat{\vec{T}}(t)$, is indeed a spectrum generating symmetry operator mapping
between energy eigenstates and between general solutions to the quantum
dynamics.

Consider now the Galilean boost transformed state,

$|\psi_{E},t;\vec{v}\,\rangle=e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\,|\psi_{E},t\rangle=e^{-\frac{i}{\hbar}t\,E}\,e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\,|E\rangle=e^{-\frac{i}{\hbar}t\,E}\,e^{\frac{i}{\hbar}\left(\frac{t^{2}}{2}\vec{v}\cdot\vec{F}+m\vec{v}\cdot\hat{\vec{x}}_{0}-t\,\vec{v}\cdot\hat{\vec{p}}_{0}\right)}\,|E\rangle.$
(140)

In this case one finds,

$\hat{H}\,|\psi_{E},t;\vec{v}\,\rangle=\left(E+\frac{1}{2}m\vec{v}^{\,2}\right)\,|\psi_{E},t;\vec{v}\,\rangle\,+\,e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\,\vec{v}\cdot\hat{\vec{T}}(t)\,|\psi_{E},t\rangle.$
(141)

Given the identity,

$e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\,\hat{\vec{T}}(t)\,e^{\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}=\hat{\vec{T}}(t)-m\vec{v}\,\mathbb{I},$
(142)

one may write as well,

$\hat{H}\,|\psi_{E},t;\vec{v}\,\rangle=\left(E+\frac{1}{2}m\vec{v}^{\,2}\right)\,|\psi_{E},t;\vec{v}\,\rangle\,+\,\left(\vec{v}\cdot\hat{\vec{T}}(t)-m\vec{v}^{\,2}\right)\,|\psi_{E},t;\vec{v}\,\rangle.$
(143)

On the other hand when considering the Schrödinger equation, one has for the
transformed state,

$\displaystyle i\hbar\frac{d}{dt}|\psi_{E},t;\vec{v}\,\rangle$
$\displaystyle=$ $\displaystyle
e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\,\left[i\hbar\frac{d}{dt}+e^{\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\left(i\hbar\frac{d}{dt}e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\right)\right]\,|\psi_{E},t\rangle$
(144) $\displaystyle=$ $\displaystyle
E\,|\psi_{E},t;\vec{v}\,\rangle+e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\left[e^{\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\left(i\hbar\frac{d}{dt}e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\right)\right]\,|\psi_{E},t\rangle.$

Using the following Baker-Campbell-Hausdorff identity valid for any two
operators $A$ and $B$ which commute with their commutator $[A,B]$,

$e^{A+B}=e^{-\frac{1}{2}[A,B]}\,e^{A}\,e^{B},$ (145)

it may be established that one has,

$e^{\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\left(i\hbar\frac{d}{dt}e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\right)=\frac{1}{2}m\vec{v}^{\,2}\,\mathbb{I}-t\,\vec{v}\cdot\vec{F}\,\mathbb{I}+\vec{v}\cdot\hat{\vec{p}}_{0}=\frac{1}{2}m\vec{v}^{\,2}\,\mathbb{I}+\vec{v}\cdot\hat{\vec{T}}(t).$
(146)

Consequently the time evolution of the transformed state is such that,

$i\hbar\frac{d}{dt}|\psi_{E},t;\vec{v}\,\rangle=\left(E+\frac{1}{2}m\vec{v}^{\,2}\right)\,|\psi_{E},t;\vec{v}\,\rangle\,+\,e^{-\frac{i}{\hbar}\vec{v}\cdot\hat{\vec{\gamma}}(t)}\,\vec{v}\cdot\hat{\vec{T}}(t)\,|\psi_{E},t\rangle=\hat{H}\,|\psi_{E},t;\vec{v}\,\rangle,$
(147)

namely it indeed obeys the Schrödinger equation.

In other words, and in contradistinction to spatial translations, under a
Galilean boost an energy eigenstate is not mapped into some other well defined
energy eigenstate (that could have possessed the corresponding classical
energy value, $E+m\vec{v}^{\,2}/2+\vec{v}\cdot\vec{T}$, had it not been for
the quantum non-commutativity of $\hat{\vec{\gamma}}(t)$ and
$\hat{\vec{T}}(t)$ as quantum operators, while the states
$|\psi_{E},t;\vec{v}\,\rangle$ are not sharp $\hat{\vec{T}}(t)$-eigenstates),
but rather into a specific quantum superposition of all energy eigenstates
with a certain distribution characterised by the action of the operator
$\vec{v}\cdot\hat{\vec{T}}(t)$ onto the transformed state
$|\psi_{E},t;\vec{v}\,\rangle$ (see (143)). Nevertheless Galilean boosts do
map between general solutions to the Schrödinger equation. Therefore these
observations confirm the status of the dynamical conserved Noether charge
$\hat{\vec{\gamma}}(t)$ as being a spectrum generator symmetry operator that
does indeed map between solutions to the quantum dynamics, even though it does
not map energy eigenstates into energy eigenstates but rather into quantum
superpositions thereof.

That Galilean boosts map energy eigenstates into distributions of energy
eigenstates rather than into another energy eigenstate of specific energy may
also be confirmed by considering the mean energy distribution for transformed
energy eigenstates of values $E_{1}$ and $E_{2}$, namely,

$\displaystyle\langle\psi_{E_{1}},t;\vec{v}\,|\hat{H}|\psi_{E_{2}},t;\vec{v}\,\rangle$
$\displaystyle=$
$\displaystyle\langle\psi_{E_{1}},t|\left(\hat{H}+\vec{v}\cdot\hat{\vec{T}}(t)+\frac{1}{2}m\vec{v}^{\,2}\,\mathbb{I}\right)|\psi_{E_{2}},t\rangle$
(148) $\displaystyle=$
$\displaystyle\left(E_{1}+\frac{1}{2}m\vec{v}^{\,2}\right)\,\delta(E_{1}-E_{2})\,+\,\langle
E_{1}|\vec{v}\cdot\hat{\vec{T}}(t)|E_{2}\rangle\,e^{\frac{i}{\hbar}t(E_{1}-E_{2})}.\qquad$

This expression illustrates explicitly that it is the quantum Noether charge
for spatial translations, $\hat{\vec{T}}(t)$, which maps energy eigenstates
into different energy eigenstates, that is responsible for the spreading in
energy of the action on energy eigenstates of the quantum Noether charge for
Galilean boosts, $\hat{\vec{\gamma}}(t)$. This feature is a direct consequence
of the fact that these two Noether charges do not commute and generate a
Heisenberg algebra on their own. Nevertheless their status as symmetry
generators and dynamical constants of the motion remains perfectly consistent
within the quantum context as well, in spite of their explicit time dependency
as operators in the Schrödinger picture for the quantum dynamics of this
system.

##  6 Conclusions

Symmetries of a dynamical system are transformations of the set of all its
possible configurations that map any solution to its equations of motion to
some other solution, thereby leaving invariant indeed the ensemble of all
these solutions. Most standard textbooks dealing with the properties following
from the existence of symmetries set the analysis within the Lagrangian
variational principle and of course rely on Noether’s first theorem in the
case of continuous Lie symmetries. However, they present these matters in such
a manner that it leads one to believe, and learn, that such Lie symmetries are
generated by conserved Noether charges that necessarily are time independent
phase space observables, hence that these constants of the motion necessarily
commute with the Hamiltonian, and that necessarily symmetries must leave the
Lagrange function exactly invariant. However, this need not be so. Such
statements are too restrictive and specific; neither do they do full justice
to the true and complete content of Noether’s first theorem, nor to the full
reach and meaning of it.

In complete generality however, and as a matter of fact, there do exist Lie
symmetries generated by conserved Noether charges that are dynamical, namely
which possess an explicit time dependency as observables over phase space and
thus do not commute with the Hamiltonian and yet define constants of the
motion, and potentially define spectrum generating Lie algebras of
observables; and there exist as well symmetries that leave the system’s action
invariant but only up to some total time derivative contribution. Furthermore,
these features apply whether to the Lagrangian or to the Hamiltonian
formulation of the dynamics, the latter however, offering the advantage of
identifying then further dynamical symmetries beyond those of the Lagrangian
formulation with its symmetries of a geometrical character in configuration
space only, namely symmetry transformations that mix all phase space variables
indifferently, whether the configuration space coordinates or their canonical
conjugate momenta.

In addition, within the Hamiltonian formulation, the converse statements are
true as well, namely that any dynamical constant of the motion is necessarily
the Noether charge for some symmetry of the first-order Hamiltonian action and
its dynamics, which remains invariant up to a total time derivative
contribution under the action of that symmetry on phase space. All Lie
symmetries are indeed so-called Noetherian symmetries, certainly within the
Hamiltonian setting; there are no non-Noetherian Lie symmetries (in spite of
statements to the contrary in the literature). This remains valid upon (a
complete or partial) Hamiltonian reduction (of all or some of the canonically
conjugate momenta to generalised velocities in configuration space) to a
Lagrangian formulation, provided those Lie symmetry transformations under
considerations act by mixing the configuration coordinates and their
generalised velocities indifferently (as they do within the Hamiltonian
formulation for these coordinates and their canonically conjugate momenta).

This less restricted and richer understanding of the possible consequences of
continuous Lie symmetries in dynamical systems, directly rooted in the
complete content of Noether’s first theorem, is not as widely known and
integrated within textbooks as it would deserve to be, in particular certainly
when it comes to the Hamiltonian approach. In a discussion which aims to be
pedagogical as well, the first part of this work (re)visits these matters
within the Hamiltonian setting in as general and streamlined a way as possible
(the sole restrictions being that the Hamiltonian does not involve any
explicit time dependency, and that the system is regular, without first- or
second-class constraints, for ease of discussion). In particular it includes a
detailed presentation and justification of the statement converse to Noether’s
first theorem, establishing a one-to-one correspondence between Noether Lie
symmetries (leaving the action invariant only up to some total time derivative
contribution) and constants of the motion.

To illustrate the general discussion in all its different aspects, three
different relatively simple and straightforward classes of systems are
considered as well. Namely, first the harmonic dynamics of a collection of
coupled harmonic oscillators in the basis of their normal modes, with its full
algebra of dynamical Noether constants of motion. Next, the extension of such
dynamics to that of a free relativistic scalar field in Minkowski spacetime,
with the identification of all its dynamical conserved Noether charges
constructed out of its time dependent observables that create and annihilate
single quanta of each of the field’s oscillating modes. And finally, the
dynamics of a nonrelativistic particle in euclidean space subjected to a
constant force, and its symmetries under spatial translations and Galilean
boosts, with a detailed analysis specifically of its quantum dynamics and
quantum conserved Noether charges whether in the Heisenberg or in the
Schrödinger picture for quantum time evolution, as they may be identified from
these two classes of global Lie symmetries of that system and be shown to be
indeed spectrum generating operators.

One last general consideration may be worth pointing out, to again confirm the
complete consistency of having dynamical quantum constants of the motion with
their explicit time dependency as conserved Noether charges which generate
continuous symmetries and yet do not commute with the quantum Hamiltonian, and
now in the case when even the Hamiltonian itself possesses an explicit time
dependency and does not define generally itself a conserved quantity, namely
$H(t)\equiv H(q^{\alpha},p_{\alpha};t)$.

Consider the quantum system to have been quantised at time $t=t_{0}$, with in
particular (thus in the Schrödinger picture) its quantum Hamiltonian
$\hat{H}(t)$, some arbitrary quantum observable $\hat{A}(t)$, and some
dynamical Noether charge $\hat{Q}(t)$ with its value
$\hat{Q}_{0}=\hat{Q}(t_{0})$. In the Schrödinger picture time evolution is
determined from the Schrödinger equation,

$i\hbar\frac{d}{dt}|\psi,t\rangle=\hat{H}(t)\,|\psi,t\rangle,$ (149)

of which the solution is given in the form,

$|\psi,t\rangle=U(t,t_{0})\,|\psi,t_{0}\rangle,$ (150)

where $|\psi,t_{0}\rangle$ is the initial quantum state at $t=t_{0}$, and
$U(t,t_{0})$ is the unitary quantum evolution operator such that
$U^{\dagger}(t_{2},t_{1})=U^{-1}(t_{2},t_{1})=U(t_{1},t_{2})$,
$U(t,t)=\mathbb{I}$, and defined by the time-ordered exponential,

$t_{2}>t_{1}\,:\qquad U(t_{2},t_{1})={\rm
T}\,e^{-\frac{i}{\hbar}\int_{t_{1}}^{t_{2}}dt\,\hat{H}(t)}.$ (151)

In particular, for an arbitrary normalised quantum state,
$\langle\psi,t|\psi,t\rangle=\langle\psi,t_{0}|\psi,t_{0}\rangle=1$, the
quantum expectation value for the arbitrary observable $\hat{A}(t)$ is given
by,

$\langle\hat{A}\rangle(t)=\frac{\langle\psi,t|\hat{A}(t)|\psi,t\rangle}{\langle\psi,t|\psi,t\rangle}=\langle\psi,t_{0}|U^{\dagger}(t,t_{0})\,\hat{A}(t)\,U(t,t_{0})|\psi,t_{0}\rangle.$
(152)

Within the Heisenberg picture time evolution of quantum observables is defined
to be given by, say for the arbitrary observable $\hat{A}(t)$,

$\hat{A}_{H}(t)=U^{\dagger}\,(t,t_{0})\,\hat{A}(t)\,U(t,t_{0}),$ (153)

with thus in particular,

$\hat{H}_{H}(t)=U^{\dagger}(t,t_{0})\,\hat{H}(t)\,U(t,t_{0}).$ (154)

These operators are the solutions to the Heisenberg equation of motion in the
Heisenberg picture, namely,

$i\hbar\frac{d}{dt}\hat{A}_{H}(t)=i\hbar\frac{\partial}{\partial
t}\hat{A}_{H}(t)\,+\,\left[\hat{A}_{H}(t),\hat{H}_{H}(t)\right],$ (155)

where explicitly,

$\frac{\partial}{\partial t}\hat{A}_{H}(t)\equiv
U^{\dagger}(t,t_{0})\,\frac{\partial\hat{A}(t)}{\partial t}\,U(t,t_{0}).$
(156)

In particular in the case of a quantum dynamical Noether charge one has,

$i\hbar\frac{d}{dt}\hat{Q}_{H}(t)=i\hbar\frac{\partial}{\partial
t}\hat{Q}_{H}(t)\,+\,\left[\hat{Q}_{H}(t),\hat{H}_{H}(t)\right]=0,\qquad\hat{Q}_{H}(t)=\hat{Q}_{H}(t_{0})=\hat{Q}(t_{0})=\hat{Q}_{0}.$
(157)

Consequently the quantum expectation value of a dynamical Noether charge for
an arbitrary (normalised) quantum state is simply obtained in the form,

$\langle\hat{Q}\rangle(t)=\langle\psi,t|\hat{Q}(t)|\psi,t\rangle=\langle\psi,t_{0}|\hat{Q}_{H}(t)|\psi,t_{0}\rangle=\langle\psi,t_{0}|\hat{Q}_{0}|\psi,t_{0}\rangle.$
(158)

In other words, under such general conditions, such a quantum expectation
value indeed readily defines a time independent conserved quantity whose value
is set by the choice of initial conditions, on account of Noether’s first
theorem.

Quite many other possible choices of dynamical systems than the few simple
illustrations presented herein, for which similar considerations could be
applied and developed to lead to further insight and understanding into the
classical and quantum dynamics easily come to one’s mind. The celebrated
Landau problem is certainly a case in point. However quantum field theories,
coupled to specific background spacetime geometries and with their infinite
number of degrees of freedom and potentially huge dynamical symmetries[5],
offer great potential in that respect, as witnessed by the recent resurgence
of interest into so-called Bondi-Metzner-Sachs (BMS) symmetries (see for
instance Ref.[8]).

## Acknowledgements

Data sharing is not applicable to this article as no new data were created or
analysed in this study.

DBI acknowledges the support of an “extraordinary” postdoctoral Fellowship of
the Académie de Recherche et d’Enseignement Supérieur (ARES) of the Wallonia-
Brussels Federation of Belgium towards a six months stay at CP3 in 2019 during
which the present work was initiated. The work of JG is supported in part by
the Institut Interuniversitaire des Sciences Nucléaires (IISN, Belgium).

## References

  * [1] J. Govaerts, Hamiltonian Quantisation and Constrained Dynamics, Leuven University Press, Leuven, 1991.
  * [2] J. Govaerts, J. Phys. Conf. Ser. 1208 (2019) 012011 (arXiv:1907.00154 [hep-th]).
  * [3] P.G.L Leach and G.P. Flessas, Journal of Nonlinear Mathematical Physics 15 (2008) 9-21 (https://doi.org/10.2991/jnmp.2008.15.1.2).
  * [4] P.G.L. Leach, Appl. Anal. Discrete Math. 6 (2012) 238-246
(https://doi.org/10.2298/AADM120625015L).

  * [5] D. Balondo Iyela and J. Govaerts, J. Math. Phys. 60 (2019) 092302.
  * [6] A. Cabo, J.-L. Lucio-Martinez and M. Napsuciale, Ann. Phys. 244 (1995) 1.
  * [7] J. Govaerts and V.M. Villanueva, Int. J. Mod. Phys. A15 (2000) 4903.
  * [8] G. Compère, Advanced Lectures in General Relativity, Lect. Notes Phys. 952 (Springer Nature Switzerland AG 2019) (arXiv:1801.07064 [hep-th]).