Rickey W. Austin
St Claire Scientific Research, Development, and Publishing
The Q-series established a Schrödinger-type compatibility condition for the
transport closure system of scalar--conformal NUVO theory, demonstrating that the
coupled evolution of closure density and transport phase is consistent with a
Schrödinger-type equation in the hydrogenic sector.
The present paper elevates this compatibility result to a general theorem,
establishing the time-dependent Schrödinger equation on the complete Hilbert
space of QM1 as a structural consequence of the scalar--conformal transport
geometry.
Three gaps in the existing series are closed in sequence.
First, the general correspondence between a spatial modulation of the scalar
capacity field and a potential energy function in the
transport closure law is derived explicitly, extending the hydrogenic result of the
Q-series to an admissible class of scalar-conformal configurations.
Second, the Hamiltonian is shown to be self-adjoint
on the second-order Sobolev domain for potentials in the admissible
class, via an application of the Kato--Rellich theorem to the scalar--conformal
transport structure.
Third, Stone's theorem for strongly continuous unitary groups is applied to the
self-adjoint Hamiltonian to derive the unitary time-evolution operator
and the Schrödinger equation
as its infinitesimal generator equation.
With these foundations in place, the Noether-type conservation laws are derived from
transport symmetry: invariance of the transport closure law under time translation,
spatial translation, and spatial rotation yields conservation of the expectation
values of energy, momentum, and angular momentum respectively.
The Ehrenfest theorem is then established as a corollary, identifying classical
transport trajectories as the evolution of expectation values under Schrödinger
dynamics.
No dynamical postulates are introduced.
The Schrödinger equation, the Hamiltonian, the unitary evolution, and the
conservation laws all emerge as structural consequences of the scalar--conformal
transport geometry and the operator framework established in the prior series.
The scalar--conformal NUVO program has developed through a disciplined
sequence of sector papers, each extending the prior work by a single
controlled step.
The M-series fixed the foundational geometry: the scalar capacity field
conformally modulates a reference Lorentzian metric through
the delivery substrate and loop taxonomy were established, and the variational
structure of the scalar--conformal action was derived.
The Q-series developed the exchange sector: closure conditions, holonomic
coherence, quantization from integer winding number, the hydrogenic
correspondence that identifies , and a
Schrödinger-type compatibility result for the hydrogenic transport
closure system.
The QB-series completed the first stage of the quantum-mechanical
development: state representation as a complex encoding of transport closure,
momentum and energy operators derived from transport generators, the
canonical commutation relation, a pre-Hilbert inner product from holonomic
coherence, and the Born frequency law and measurement correspondence as
structural consequences of coherence-gated interaction dynamics.
QM1 then extended this finite-dimensional framework to the complete
separable Hilbert space
normalization was derived as a structural constraint from closure
conservation, the momentum and energy generators were promoted to
essentially self-adjoint operators on their Sobolev domains, and the
spectral theorem together with the resolution of the identity were
established for the transport generators.
Within this sequence, QM4 occupies a specific and load-bearing position.
QM1 provides the Hilbert space and establishes the self-adjointness
of the free momentum generators
QM2 and QM3 were developed using only the algebraic and inner-product
structure of QM1: QM2 derived the superposition principle and interference
from the linearity of the transport closure equations on , and QM3
established the uncertainty relations from the canonical commutation
relation.
Neither QM2 nor QM3 requires a general Hamiltonian or a time-evolution
operator; they depend only on the operator algebra and Hilbert space
structure already in hand.
The situation changes fundamentally at QM5 and beyond.
The angular momentum algebra of QM5 requires a dynamical context in which
the angular momentum generators commute with the Hamiltonian and conservation
is a consequence of rotational symmetry.
The harmonic oscillator of QM6 requires a Hamiltonian with a quadratic
potential, derived from the scalar capacity geometry, and a Schrödinger
evolution from which the energy spectrum is extracted.
The multi-particle dynamics of QM7, the spin dynamics of QM8, the
entanglement structure of QM9, and the scattering analysis of QM10 all
require the time-dependent Schrödinger equation and the unitary
time-evolution operator as their foundational dynamical input.
The present paper, QM4, establishes this dynamical foundation.
Three structural gaps in the existing series, identified by the gap
analysis preceding this work, must be closed before the Schrödinger
equation can be established as a theorem.
The Q-series established Schrödinger-type compatibility for the
hydrogenic sector: the coupled evolution of closure density and transport
phase is consistent with a Schrödinger-type equation for the hydrogenic
Hamiltonian.
It did not establish a general correspondence between the scalar capacity
field and a potential energy function, did not prove that the resulting
Hamiltonian is self-adjoint on , and did not derive the
Schrödinger equation as a theorem valid on the complete Hilbert space.
A fourth gap---the Noether bridge between the geometric transport symmetry
of the M-series and the conservation of quantum-mechanical expectation
values---was also identified: the M-series establishes transport invariance
at the level of the action functional, but the translation of this
invariance into conservation laws for expectation values requires the
Schrödinger dynamics of the present paper.
All four gaps are closed here in sequence, each building on the one before.
The results established in the present paper propagate forward through the
remainder of the QM-series without exception.
The Schrödinger equation, the unitary time-evolution operator, and the
conservation structure established below are foundational inputs to QM5
through QM11.
The covariant extension of the Schrödinger dynamics, which opens the
path to the RQM-series, is prepared in QM11 using the operator framework
and dynamical structure established here.
The central objective of the present paper is to establish the complete
dynamical framework for the scalar--conformal NUVO transport closure system
on the Hilbert space of QM1.
Specifically, the paper aims to establish six related claims in logical order.
A spatial modulation
of the scalar capacity field, treated as a perturbation of the uniform
baseline , generates a real-valued potential
function in the exchange-sector transport closure equation.
The correspondence is derived explicitly from
the scalar--conformal transport law and reduces to the Coulomb potential
in the hydrogenic sector, recovering the Q-series result as a special case.
The potential generated by any spatially localized scalar
capacity modulation belongs to the admissible class of potentials
that are relatively bounded with respect to the kinetic operator
with relative bound strictly less than one.
This places in the hypotheses of the Kato--Rellich theorem.
The Hamiltonian
where denotes multiplication by , is self-adjoint on the
second-order Sobolev domain
for all potentials in the admissible class.
This is established by an application of the Kato--Rellich theorem to
the scalar--conformal transport structure, using the self-adjointness of
on and the relative boundedness established
in claim 2.
Stone's theorem, applied to the self-adjoint operator
on , yields a unique strongly continuous one-parameter
unitary group
on .
This group is the time-evolution operator of the scalar--conformal transport
closure system.
The time-dependent Schrödinger equation
holds on
as the infinitesimal generator equation of , and the solution
is the unique strong solution for each initial datum
The Schrödinger equation is thereby established as a theorem, not
introduced as a postulate.
The invariance of the transport closure law under time translation,
spatial translation, and spatial rotation implies conservation of the
expectation values
respectively, where momentum conservation holds for translationally
invariant potentials and angular momentum conservation holds for
rotationally symmetric potentials.
The Ehrenfest theorem follows as a corollary, identifying the classical
transport trajectory with the evolution of mean position and momentum
under the Schrödinger dynamics.
These six claims form a logically ordered chain in which each depends on
those that precede it.
Claims 1 and 2 establish the physical Hamiltonian and verify its
membership in the class required by the functional analysis of claim 3.
Claim 3 provides the self-adjoint generator required by claim 4.
Claim 4 provides the unitary group whose generator equation is
claim 5.
Claim 5 and the commutation structure recalled later together yield
claim 6.
The present work maintains without modification the interpretive discipline
established in the Q-series and continued through the QB-series and QM1.
The following exclusions are in force throughout.
The Schrödinger equation is not postulated.
In the standard quantum-mechanical formalism,
is introduced as a fundamental dynamical law with no derivation.
In the present framework it is a theorem: the generator equation of the
unitary group constructed from the self-adjoint Hamiltonian
via Stone's theorem.
The three-step derivation---, Kato--Rellich, Stone---is
the content of the potential, Hamiltonian, and Schrödinger dynamics sections,
and none of its steps introduces a dynamical postulate.
The Hamiltonian is not introduced as a primitive object.
In the standard formalism, is chosen to represent the energy of
the physical system under study.
In the present framework
is derived: the kinetic term arises from the free transport
closure system with uniform scalar capacity, and the potential term
arises from the spatial modulation of the scalar capacity field.
The Hamiltonian is therefore not a choice but a consequence.
The potential is not introduced by hand.
Every potential treated in the QM-series---the Coulomb potential of the
hydrogenic sector, the harmonic potential of QM6, the barrier potentials
of QM10---arises from a specific scalar capacity modulation
through the correspondence derived in this paper.
The admissible class of potentials is determined by the scalar--conformal
geometry, not by external physical input.
The conservation laws are not assumed from classical mechanics.
They are derived from the commutation structure of the transport generators
with the Hamiltonian, which is in turn a consequence of the symmetry
properties of the scalar--conformal transport law.
No classical Noether theorem is invoked; the quantum Noether structure
of this paper is self-contained within the Hilbert space framework.
No wavefunction collapse, measurement postulate, or new probabilistic
axiom is introduced.
The Schrödinger dynamics established in the present paper are
deterministic: given the initial closure state
the full time evolution
is uniquely determined.
The statistical interpretation of is inherited from the
Born frequency law of QB6 and the Parseval identity of QM1, and is not
re-derived here.
Section 2 recalls the scalar--conformal geometry, the free transport
structure, the Schrödinger-type compatibility result of the Q-series,
and the operator and norm-preservation results of QM1, and identifies
precisely the four structural gaps that the present paper closes.
Section 3 derives the general correspondence between a scalar capacity
modulation and a potential energy function ,
defines the admissible class of scalar--conformal potentials, and verifies
that the hydrogenic Coulomb potential is recovered as a special case.
Section 4 establishes self-adjointness of the Hamiltonian
on the Sobolev domain via the Kato--Rellich theorem,
records the essential self-adjointness on the Schwartz domain, and
describes the spectral structure of for admissible potentials.
Section 5 applies Stone's theorem to construct the unitary time-evolution
group and derives the Schrödinger equation as its infinitesimal
generator equation, establishes uniqueness of the evolution, and recovers
the Q-series hydrogenic compatibility result as a special case of the
general theorem.
Section 6 establishes unitarity of , records the consistency
of this result with the norm-preservation corollary of QM1, and shows
that the Schrödinger dynamics preserves the full inner product on
.
Section 7 derives the Noether-type conservation laws from the commutation
structure of transport generators, establishing conservation of
, , and
from time-translation, spatial-translation,
and rotational symmetry respectively, and records the Noether bridge as
a theorem.
Section 8 establishes the Ehrenfest theorem as a corollary of the
conservation structure and identifies the classical transport trajectory
with the evolution of mean position and momentum.
Section 9 collects interpretive clarifications, maintains the interpretive
boundary conditions of the prior series, and records the scope of the
present construction.
Section 10 summarizes the results, records their programmatic significance
for the QM-series and the approach to RQM, and prepares the transition to QM5.
The present section collects the results from the M-, Q-, QB-, and
QM1-series that are directly needed for the developments of
Sections 3--8.
Each subsection recalls what has been established, states it in the
form in which it will be used, and provides the relevant citations.
The final subsection identifies the four structural gaps that the
prior series leaves open and that the present paper closes.
The M-series established the scalar--conformal geometric framework
that underlies all subsequent sector developments.
The physical metric is
Equation. Scalar-conformal metric.
where is the scalar capacity field
and is the reference Lorentzian metric.
The baseline scalar level represents the structural
capacity supported by the delivery substrate in the absence of
localized occupation; in the unperturbed state
uniformly.
The dimensionless scalar diagnostic is
Equation. Scalar diagnostic.
so that in the baseline and
wherever the scalar capacity field is modulated by the presence of a
localized structure or exchange process.
The exchange-sector action functional, derived in the M-series and
developed in the Q-series, involves the scalar capacity field through
the conformal factor.
In the non-relativistic limit, spatial gradients of ---
equivalently, spatial gradients of ---appear in the
transport closure equation as restoring terms.
These restoring terms play the role of potential energy in the
transport dynamics: a region where , reduced structural
capacity due to localized occupation, presents an energy barrier to
incoming transport, while a region where the gradient structure of
is attractive acts as a potential well that can support
bound closure modes.
The explicit form of this potential correspondence is derived in
Section 3; the present subsection records that the scalar--conformal
geometry provides the geometric origin of all potential energy in the
NUVO framework.
Remark.
The identification of potential energy with the gradient structure of
the scalar capacity field is not an assumption imported
into the present paper.
It is a consequence of the M-series action functional, which was
derived from the scalar--conformal variational structure without
reference to any external potential.
The present paper makes this identification explicit and derives its
functional form in Section 3.
In the baseline state , uniform scalar
capacity with no spatial modulation, the exchange-sector transport closure
system reduces to the free transport system.
The Q-series and QB-series established that in this limit the transport
generator associated with spatial displacement is the momentum operator
and the energy associated with free transport is the kinetic energy.
The kinetic operator is accordingly
Equation. Kinetic operator.
where
is the spatial Laplacian on and is the structural
mass parameter fixed by the hydrogenic correspondence of the Q-series
through the identification .
The self-adjoint properties of on follow from
QM1.
By QM1 Theorem 5.2, each momentum generator is essentially
self-adjoint on the Schwartz domain with unique
self-adjoint closure having domain .
For the kinetic operator ,
which involves second-order differentiation, the corresponding domain
is the second-order Sobolev space:
Equation. Second-order Sobolev domain.
where is understood in the distributional sense.
The operator is self-adjoint on ; this is a
standard consequence of the Fourier-transform characterization of the
Laplacian on and is not re-derived here.
Remark.
The promotion from the first-order Sobolev domain
of the individual momentum operators to the second-order Sobolev domain
of the kinetic operator is automatic: since
involves two derivatives, the domain condition
requires two distributional derivatives to be square-integrable, which is
precisely the definition of .
The domain will serve as the domain of the full
Hamiltonian
in Section 4, provided the potential does not enlarge it---a condition
verified by the Kato--Rellich theorem.
The Q-series established that the exchange-sector transport closure
system, in the hydrogenic configuration, is compatible with a
Schrödinger-type representation.
Specifically, Q3 showed that the coupled deterministic evolution of
the closure density and the transport phase
can be written in a form consistent with
Equation. Schrödinger-type compatibility.
where is the hydrogenic Hamiltonian operator and
is the QB1 complex encoding.
Q4 then identified the discrete eigenvalues of
with the holonomic closure modes and recovered the hydrogenic energy
spectrum
through the correspondence limit.
It is essential for the present development to state precisely what
the Q-series result does and does not establish.
The Q-series result is a compatibility statement: the transport
closure evolution is consistent with the Schrödinger-type compatibility
equation in the hydrogenic sector.
It is not a derivation of that equation as a theorem on the complete
Hilbert space , for three reasons.
First, the Q-series did not establish that is
self-adjoint on .
Compatibility with the Schrödinger-type form does not require the
operator to be self-adjoint; it requires only that the equation hold
in a formal sense within the finite-dimensional closure mode space.
Self-adjointness on , which requires the Kato--Rellich
analysis of Section 4, was not established.
Second, the Q-series did not establish uniqueness of the evolution.
Without self-adjointness of on and the
application of Stone's theorem, there is no guarantee that the
compatibility relation defines a unique time evolution for all initial
data in .
Third, the Q-series result is confined to the hydrogenic sector.
A general scalar capacity modulation
generates a potential different from the Coulomb form, and the
Q-series provides no general framework for treating such configurations.
The present paper closes all three limitations.
The Q-series compatibility result is recovered as a corollary of the
general theorem in the later hydrogenic recovery proposition.
Four results from QM1 are directly used in the present paper and are
recalled here in the form in which they will be applied.
Norm preservation (QM1 Corollary 3.4).
For any admissible transport evolution with
the norm satisfies
for all .
This result was derived from the divergence-free structure of the
continuity equation and is independent of the Schrödinger dynamics.
In Section 6 it will be shown that the same conclusion follows from
the unitarity of , providing a second, operator-level
derivation that is consistent with and independent of the geometric
transport derivation of QM1.
Hilbert space structure (QM1 Theorem 4.3).
The space
equipped with the closure inner product
is a separable complex Hilbert space.
This is the ambient space on which the Hamiltonian of Section 4 acts
and on which Stone's theorem of Section 5 is applied.
Essential self-adjointness of momentum generators (QM1 Theorem 5.2).
The operators
are essentially self-adjoint on with closure domain
.
This result underlies the self-adjointness of on
recalled above, and is the starting point for the Kato--Rellich analysis
of Section 4.
Canonical commutation relation on (QM1 Proposition 5.4).
On the dense domain
Equation. Canonical commutation relation.
for all .
This relation is used in Section 7 to compute the commutator
and in Section 8 to derive the Ehrenfest
equations.
The prior series leaves four structural gaps that must be closed before
the Schrödinger equation can be established as a theorem on .
They are stated here with the precision required for the reader to
follow the logical sequence of the paper.
Gap (i): The general correspondence.
The Q-series derived the Coulomb potential
from the hydrogenic scalar capacity modulation by direct correspondence
with the hydrogenic spectrum.
This derivation is sector-specific and does not provide a general
framework for determining from an arbitrary spatial modulation
of the scalar capacity field.
Without this general correspondence, the Hamiltonian cannot
be defined for transport sectors beyond the hydrogenic case, and the
QM-series cannot progress to the harmonic oscillator (QM6), barrier
potentials (QM10), or the general central force problem (QM5).
Gap (i) is closed in Section 3.
Gap (ii): Self-adjointness of .
QM1 established that the free kinetic operator
is self-adjoint on .
The addition of a potential operator , multiplication by ,
to a self-adjoint operator does not in general preserve self-adjointness:
if is sufficiently singular, the operator
may fail to be symmetric, or may be symmetric but not self-adjoint, or
may have multiple self-adjoint extensions.
The Kato--Rellich theorem provides a sufficient condition---relative
boundedness of with respect to with relative bound
less than one---that guarantees self-adjointness of on the
original domain .
Verifying this condition for the scalar--conformal potentials of
Gap (i) is the content of Section 4.
Without Gap (ii) closed, the spectral theorem cannot be applied to
and Stone's theorem cannot be invoked.
Gap (iii): The Schrödinger equation as a theorem on .
Stone's theorem states that a self-adjoint operator on
uniquely determines a strongly continuous unitary group
and that this group satisfies the generator equation
for
Applying Stone's theorem with
which requires Gap (ii) to be closed first, yields the Schrödinger equation
as the generator equation of the unitary time-evolution group .
This derivation is carried out in Section 5.
Gap (iv): The Noether bridge.
The M-series established that the scalar--conformal action is invariant
under time translation, spatial translation, and spatial rotation.
At the geometric level, this invariance implies the existence of
classically conserved quantities via the Noether theorem.
However, the quantum-mechanical statement---that the expectation values
are conserved under the Schrödinger evolution---is a separate result
that requires the dynamics of Gap (iii) to be in place.
The bridge from geometric transport symmetry, in the M-series, to
conservation of quantum expectation values, in the present paper, is
constructed in Section 7 using the commutation structure of the
transport generators with .
This bridge has not been explicitly constructed in any prior paper
of the NUVO series.
The four gaps are not independent: Gap (ii) depends on Gap (i),
Gap (iii) depends on Gap (ii), and Gap (iv) depends on Gap (iii).
The present paper closes them in the order (i)--(ii)--(iii)--(iv),
with each section providing the input required by the next.
The fundamental claim of the present section is that every potential
energy function appearing in the QM-series is not introduced as external
physical input but is derived from a spatial modulation of the scalar
capacity field .
The section proceeds in four steps.
First, the exchange-sector transport closure law is expanded to first
order in the scalar capacity modulation, and it is shown that the
first-order correction takes the form of a multiplicative potential.
Second, the explicit correspondence between the modulation field
and the potential function is stated as a definition
and derived from the exchange-sector action structure of the M-series.
Third, the hydrogenic Coulomb potential is recovered as a special case,
providing an internal consistency check on the correspondence.
Fourth, the admissible class of scalar--conformal potentials is defined
and verified to satisfy the hypotheses of the Kato--Rellich theorem
invoked in Section 4.
The baseline scalar capacity , spatially uniform,
corresponds to the free transport sector in which the exchange-sector
Hamiltonian reduces to the pure kinetic operator recalled in
Section 2.
A spatially varying scalar capacity field
Equation. Scalar capacity modulation.
represents a departure from the baseline due to the presence of a
localized structural configuration---a bound holonomic closure mode,
a charged source, or any other exchange-sector occupation that modifies
the local structural capacity.
The condition defines the first-order, or linear
response, regime in which the transport closure law can be expanded in
powers of and the leading correction identified cleanly.
The exchange-sector action functional, derived in M1 from the
scalar--conformal variational structure, takes the general form
Equation. Exchange-sector action.
where is the
exchange-sector Hamiltonian density, which depends on
through the conformal factor.
In the baseline state , the Hamiltonian density
reduces to the free kinetic density:
Equation. Free Hamiltonian density.
For the modulated field above, the Hamiltonian density expands as
Equation. Hamiltonian density expansion.
The first-order variation
is determined by the structure of the M-series exchange-sector action
and is computed below.
The critical structural claim---that this variation is a function of
alone, not a differential operator acting on ---is established by
the following lemma.
Lemma. First-order capacity modulation generates a multiplicative potential.
Let
with
a smooth, compactly supported modulation satisfying
Then the exchange-sector transport closure law, derived from the
action above by variation with respect to , takes the form
Equation. Modulated transport closure law.
where
is a real-valued function of position alone.
Proof.
Varying the exchange-sector action with respect to
and using the Hamiltonian density expansion gives
The first term is the free kinetic operator.
For the second term, the M-series exchange-sector action couples
to the closure state through the local structural
capacity: the conformal modulation modifies the
exchange-sector energy at position by an amount proportional to
the local closure content at that point, with no
derivative of appearing at first order.
This is a consequence of the fact that enters the
M-series action as a background field, not a dynamical variable, that
couples to the closure density
through the structural capacity per unit volume, which is a local,
or contact, interaction.
The first-order variation of with respect
to at fixed is therefore a multiplication
operator:
where is a real-valued function whose explicit form is determined
by the M-series exchange-sector action in the next subsection.
Combining gives the modulated transport closure law.
Remark.
The multiplicative, or contact, character of the first-order potential
is the key structural feature established by the preceding lemma.
It reflects the fact that the scalar capacity modulation
acts as a spatially varying energy offset for the
exchange-sector closure state: at each point , the local departure
from baseline capacity contributes an energy per unit closure content
equal to .
This is consistent with the interpretive framework of the M-series, in
which the scalar capacity field governs the energetic cost of
exchange-sector occupation per unit structural volume.
Higher-order corrections in generate derivative couplings
and non-local terms, which are systematically computable but lie outside
the scope of the present first-order treatment.
The preceding lemma establishes that the first-order correction to
the transport closure law is a multiplicative potential .
The task of the present subsection is to identify explicitly
as a function of the modulation .
From the M-series exchange-sector action, the first-order coupling
between the scalar capacity modulation and the exchange-sector closure
density takes the form
Equation. First-order coupling.
where is a structural coupling constant with dimensions
of energy, determined by the conformal factor of the M-series baseline
geometry and the mass parameter .
The negative sign reflects the convention that a positive modulation
increased capacity, reduces the exchange energy of the closure state,
consistent with the attractive well structure of the hydrogenic sector.
Varying the first-order coupling with respect to yields
the potential operator.
Definition. Scalar-conformal potential.
Given a scalar capacity modulation
satisfying the conditions of the preceding lemma, the associated
scalar-conformal potential is
Equation. Scalar-conformal potential.
where the structural coupling constant is fixed
by the M-series exchange-sector action and satisfies
Equation. Structural coupling constant.
in which is the characteristic length scale of the
hydrogenic scalar capacity modulation established in Q4, and
is the squared charge in SI units.
This expression is established by the hydrogenic specialization below.
Remark.
The structural coupling constant is ultimately a
derived quantity whose precise value follows from the M-series conformal
factor structure and the Q-series exchange-sector action.
The expression above records the value obtained by specializing to the
hydrogenic sector and requiring consistency with the Coulomb potential
established in Q4.
A first-principles derivation of directly from the
M1 action functional, without appeal to the hydrogenic specialization,
is a natural next step within the NUVO program and is deferred as a
separate item.
The results of the present paper require only that
is a well-defined positive constant, which follows from the existence
of the hydrogenic correspondence.
Proposition. Hydrogenic specialization.
Let denote the scalar capacity modulation
field generated by the hydrogenic source configuration, as established
in Q4.
Then the scalar-conformal potential of the preceding definition, evaluated
for
yields
Equation. Coulomb recovery.
thereby recovering the Coulomb potential of the hydrogenic sector as a
special case of the general scalar-conformal correspondence
Proof.
The Q-series established, through the hydrogenic correspondence of Q4,
that the scalar capacity modulation due to the hydrogenic source has the
radial form
Equation. Hydrogenic modulation.
where is the characteristic length set by the structural
parameters of the hydrogenic configuration.
Substituting this into the general correspondence and using the expression
for gives
which is the Coulomb potential.
Remark.
The hydrogenic specialization serves as an internal consistency check
on the scalar-conformal correspondence: the general formula
must reproduce the Coulomb potential in the hydrogenic sector, since
the Q-series derived the hydrogenic spectrum from the Coulomb form.
The proposition confirms this consistency and simultaneously fixes the
coupling constant through the requirement that the
two derivations agree.
Any future first-principles derivation of from M1
must reproduce
as a necessary condition for internal consistency of the NUVO program.
The application of the Kato--Rellich theorem in Section 4 requires the
potential operator , multiplication by , to be relatively
bounded with respect to the kinetic operator with relative
bound strictly less than one.
The present subsection defines this condition precisely and verifies that
the scalar-conformal potentials of the preceding definition satisfy it
for all physically admissible modulations.
Definition. Admissible potential class.
A real-valued function
is admissible if it is relatively bounded with respect to the kinetic
operator
with relative bound : there exist constants
such that
Equation. Relative bound.
for all
The collection of all admissible potentials is denoted
A sufficient condition for membership in is the decomposition
with
and
since such a decomposition implies the relative bound with in the
limit of small-norm approximation.
This decomposition is available for all spatially localized modulation
fields.
Proposition. Scalar-conformal potentials are admissible.
Let
be a spatially localized scalar capacity modulation, and let
be the associated scalar-conformal potential.
Then
In particular, the hydrogenic Coulomb potential
is admissible with relative bound for all values of the coupling.
Proof.
For a general localized
since
and is bounded and square-integrable, decomposes as
where, for any ,
Since is bounded by
and supported within the spatial support of ,
, a bounded function on a bounded set, and
bounded outside the ball of radius .
A standard Sobolev interpolation argument then yields the relative bound
with constants and depending on
Since is bounded in for bounded , one
can choose large enough so that .
For the Coulomb potential
this is a classical result.
The Coulomb potential in is in the Kato class
and satisfies the relative bound with respect to
with for any value of the coupling
Remark.
The admissibility condition above is satisfied by all scalar--conformal
potentials generated by physically localized modulation fields.
In particular, it covers the potentials needed for the physical sectors
of the QM-series: the Coulomb potential of QM5, central force and
hydrogenic; the harmonic potential
of QM6, bounded from below, in but growing
at infinity, which is treated separately via the Kato class for
polynomially bounded potentials; the piecewise constant barrier potentials
of QM10; and multi-particle Coulomb interactions of QM7.
The correspondence
established in the present section is a first-order result in the
perturbative expansion of the exchange-sector transport law in powers
of .
The scope of this result and the directions of its generalization are
recorded here.
Within scope: the correspondence applies to all smooth, spatially localized
modulations
in the first-order regime
it covers time-independent configurations
static potentials; and it produces potentials
for all such modulations, as established above.
Every potential treated in QM5 through QM10 lies within this class.
Outside scope: time-dependent modulations
which generate time-dependent potentials of the form
relevant for the electromagnetic interaction sector, treated in the
RQM-series; strong modulations with
for which the first-order expansion is insufficient and higher-order
terms in contribute to the Hamiltonian; and singular
potentials outside , which would require a separate
self-adjoint extension analysis beyond the scope of the Kato--Rellich
theorem.
Remark.
The harmonic oscillator potential
treated in QM6, grows unboundedly at spatial infinity and does not belong
to the class of the preceding analysis.
Its admissibility with respect to is established separately:
the operator
is essentially self-adjoint on by a direct
argument using the explicit eigenbasis, the Hermite functions, independently
of the Kato--Rellich theorem.
The scalar--conformal origin of the harmonic potential---a quadratically
growing modulation
of the scalar capacity field, corresponding to a confining structural
configuration---is consistent with the general correspondence of the
scalar-conformal potential definition and is developed in QM6.
With the scalar-conformal potential established
in Section 3, the present section constructs the Hamiltonian
as a genuine self-adjoint operator on the
Hilbert space of QM1.
Self-adjointness is not a formal nicety: it is the precise condition
required by Stone's theorem in Section 5 to guarantee that
generates a unique unitary time-evolution group.
Without it, the Schrödinger equation either fails to define a unique
evolution or generates an evolution that does not preserve the
-norm, both of which would be inconsistent with the closure
conservation established in QM1.
The section proceeds in four steps.
The Hamiltonian is first defined on the natural initial domain
, where it is well-defined as a differential
operator.
The Kato--Rellich theorem is then stated as a standalone result so that
its hypotheses can be verified explicitly before the theorem is applied.
Self-adjointness of on the Sobolev domain
is then established as the main theorem of this section.
Finally, essential self-adjointness on is
recorded as a corollary, and the spectral structure of is
described.
The scalar-conformal Hamiltonian is assembled from the two components
whose properties are established in the preceding sections: the kinetic
operator of Section 2 and the potential operator
arising from the scalar capacity modulation of Section 3.
Definition. Formal Hamiltonian.
The scalar-conformal Hamiltonian associated to a potential
is the operator
Equation. Hamiltonian definition.
defined initially on the dense domain
where acts as a multiplication operator and
is the spatial Laplacian.
That is well-defined on follows
immediately from its component structure.
The kinetic operator
maps to itself, since differentiation
and multiplication by polynomials both preserve the Schwartz class.
The potential operator acts on
by pointwise multiplication:
For , the product is in for every
: since decays faster than any
polynomial and grows at most polynomially in the admissible class,
the product is square-integrable.
The operator is therefore a densely defined linear operator
on with initial domain .
Remark.
The formal Hamiltonian is symmetric on
: for any
,
Symmetry of on follows from
the integration-by-parts argument of QM1 Lemma 5.1 applied twice,
since involves two derivatives.
Symmetry of follows from the fact that is real-valued:
Symmetry is a necessary condition for self-adjointness but is not
sufficient; the Kato--Rellich theorem provides the sufficient condition.
The Kato--Rellich theorem is the functional-analytic tool that converts
the relative boundedness established in Section 3 into a self-adjointness
conclusion for the full Hamiltonian.
It is stated here as a standalone theorem so that its hypotheses can be
verified explicitly against the operators at hand before the theorem is
applied.
Theorem. Kato--Rellich.
Let be a self-adjoint operator on with domain
, and let be a symmetric operator satisfying
Suppose is relatively bounded with respect to with relative
bound : there exist constants
such that
for all .
Then is self-adjoint on , and if is
essentially self-adjoint on some core
then is also essentially self-adjoint on .
The proof is a classical result of functional analysis, cited for
orientation.
The condition is sharp: if the conclusion may fail, and
there exist symmetric perturbations with relative bound exactly one
for which is not self-adjoint on .
The requirement is therefore not merely a technical convenience
but the precise condition that distinguishes controllable perturbations
from those that may destroy the self-adjoint domain structure.
Remark.
The relative bound condition
has a direct physical interpretation in the transport closure setting.
The kinetic operator controls the regularity of the closure
state through its second-order derivative structure: states in
have two square-integrable derivatives.
The relative bound condition states that the potential energy
cannot extract more than a fraction of the kinetic energy plus a
fixed -norm contribution.
Physically, this means the potential does not dominate the kinetic term
at short distances: the closure state remains regular on
even in the presence of the potential.
This is the functional-analytic statement of the condition that the
scalar capacity modulation is a controlled perturbation
of the baseline geometry, consistent with the first-order expansion of
Section 3.
The hypotheses of the Kato--Rellich theorem are verified against the
operators
and
explicitly before the conclusion is stated.
Hypothesis 1: is self-adjoint on
.
This was established in Section 2 as a consequence of QM1 Theorem 5.2
and the standard characterization of the domain of the Laplacian on
.
Hypothesis 2: is symmetric with
.
Symmetry follows from the real-valuedness of , as recorded above.
The domain inclusion
holds because for ,
the product is in by the relative bound
Thus multiplication by is a well-defined map from
to .
Hypothesis 3: is relatively bounded with respect to
with relative bound .
This is precisely the admissibility result of Section 3, which
establishes that all scalar-conformal potentials
satisfy the relative bound with .
All three hypotheses are satisfied.
The Kato--Rellich theorem therefore applies and yields the following
result.
Theorem. Self-adjointness of the scalar-conformal Hamiltonian.
Let be an admissible scalar-conformal potential.
Then the scalar-conformal Hamiltonian
is self-adjoint on the domain
Equation. Hamiltonian domain.
where is understood in the distributional sense.
Proof.
The three hypotheses of the Kato--Rellich theorem have been verified
above: is self-adjoint on ,
is symmetric with
and is relatively bounded with respect to with
relative bound .
The Kato--Rellich theorem therefore applies with
and yields self-adjointness of
on
Remark.
A notable consequence of the self-adjointness theorem is that the
domain of the full Hamiltonian is the same as the domain
of the kinetic operator alone: both are
The addition of the potential does not shrink or enlarge the
operator domain.
This is a direct consequence of the relative bound : a potential
with is controlled enough by the kinetic operator that it does
not require additional regularity of the closure state beyond that
imposed by itself.
In physical terms, the scalar capacity modulation does not introduce
new short-distance singularities that would restrict the class of
admissible closure states beyond those already required to have
square-integrable second derivatives.
The self-adjointness theorem establishes that is self-adjoint
on the Sobolev domain .
An equally important result for the NUVO program is that is
essentially self-adjoint on the Schwartz domain
, which is the natural initial domain for
transport closure states.
Essential self-adjointness on means that
restricted to smooth, rapidly decreasing states has a unique
self-adjoint extension, and that extension is the operator of the
self-adjointness theorem.
No additional boundary conditions or extension choices are required.
Corollary. Essential self-adjointness on the Schwartz domain.
For , the Hamiltonian is essentially
self-adjoint on
Its unique self-adjoint closure is the operator of the self-adjointness
theorem, with domain .
Proof.
The kinetic operator is essentially self-adjoint on
by QM1 Theorem 5.2.
For potentials , the second part of the Kato--Rellich
theorem---which states that essential self-adjointness on a core is
preserved under an admissible perturbation---applies with core
Since is essentially self-adjoint on
and is admissible with relative bound , the full
Hamiltonian
is essentially self-adjoint on , with
closure equal to the self-adjoint operator on
established in the self-adjointness theorem.
Remark.
The physical significance of essential self-adjointness on
within the NUVO program is the same as
that recorded in QM1 for the free momentum operators: essential
self-adjointness on means that the
transport closure system selects a unique Hamiltonian operator from
the initial domain of smooth, rapidly decreasing states, without any
freedom of boundary condition.
The scalar capacity geometry uniquely determines the Hamiltonian, and
the Hamiltonian uniquely determines the dynamics via Stone's theorem
in Section 5.
The derivation chain
is therefore free of choices at every stage.
With established as a self-adjoint operator on ,
the spectral theorem of QM1 applies immediately, yielding a unique
projection-valued measure and a spectral decomposition
of every closure state in .
The present subsection records the qualitative spectral structure for
the admissible class of scalar-conformal Hamiltonians, which is needed
for the interpretation of the conservation laws in Section 7 and for
the scattering analysis of QM10.
Proposition. Spectral structure of the scalar-conformal Hamiltonian.
For an admissible scalar-conformal Hamiltonian
with , the following spectral properties hold.
The spectral theorem of QM1 applies to , yielding a unique
projection-valued measure on
such that
and a spectral decomposition
in the -norm for every .
The spectrum decomposes as
where is a discrete, possibly empty,
set of negative eigenvalues with normalizable eigenfunctions in
, and is the essential
spectrum determined by the asymptotic behavior of .
The discrete eigenstates
satisfying
are identified with the bound holonomic closure modes of the
Q-series; they form a complete orthonormal family within the
discrete-spectrum subspace.
Proof.
Part 1 follows directly from the spectral theorem, since is
self-adjoint by the self-adjointness theorem above.
Part 2 is a standard consequence of the Weyl essential spectrum theorem:
for with
the case of all physically relevant scalar-conformal potentials, the
essential spectrum of coincides with that of , which
is
Discrete eigenvalues below zero are possible and correspond to bound
states.
Part 3 follows from the identification of the bound holonomic closure
modes of the Q-series with the discrete eigenstates of the hydrogenic
Hamiltonian, established in the Q-series and recovered via the
hydrogenic recovery result of Section 5.
Remark.
The spectral structure described in the proposition is the precise form
of the discrete-continuous decomposition established in QM1, now applied
to the physical Hamiltonian of the scalar-conformal transport
closure system.
The discrete spectrum corresponds to
the bound holonomic closure modes---spatially localized, normalizable
states in ---that were identified in the Q-series and
constitute the QB-series finite-dimensional pre-Hilbert space
.
The continuous spectrum
corresponds to the unbound, scattering-type transport configurations
that are the subject of QM10.
With the spectral theorem now applied to the full , the
decomposition of arbitrary closure states into bound and scattering
components is available for all subsequent work in the QM-series.
Remark.
For the hydrogenic Hamiltonian
with Coulomb potential
the discrete spectrum is
with
recovering the energy levels established in Q4.
The continuous spectrum is
corresponding to ionized configurations in which the closure transport
is no longer holonomically bound.
These are consistent with the general spectral structure and provide an
internal consistency check on the self-adjoint extension established in
this section.
The self-adjoint Hamiltonian established in the
self-adjointness theorem is the input to the present section.
Stone's theorem converts this single operator-theoretic fact into a
complete dynamical framework: a unique unitary time-evolution group,
a well-posed initial value problem, and the Schrödinger equation
as its generator equation.
The logical structure is clean and irreversible in the sense that each
step has a unique output: the self-adjoint generator uniquely determines
the unitary group, the unitary group uniquely determines the evolution,
and the evolution uniquely determines the solution for each initial datum
in .
No dynamical postulate enters at any stage.
Stone's theorem is the functional-analytic result that establishes the
equivalence between self-adjoint operators and strongly continuous unitary
groups on a Hilbert space.
It is stated here in full so that its hypotheses can be identified
explicitly and verified against the operators at hand.
Theorem. Stone's theorem.
Let be a self-adjoint operator on with domain
.
Then the family of operators defined via the
spectral functional calculus by
Equation. Stone time-evolution family.
where is the projection-valued measure of , constitutes a
strongly continuous one-parameter unitary group on :
it satisfies
,
for all ,
for all ,
for each , the map
is continuous in the -norm.
Moreover, is recovered as the infinitesimal generator:
for each , the strong derivative
exists in and equals
Conversely, every strongly continuous one-parameter unitary group on
arises in this way from a unique self-adjoint generator.
The proof is a foundational result of functional analysis, cited for
orientation.
The converse direction---every strongly continuous unitary group has a
unique self-adjoint generator---is the part of Stone's theorem responsible
for the uniqueness claims recorded below.
Remark.
The definition
uses the spectral functional calculus for self-adjoint operators, which
applies to any bounded Borel-measurable function of the spectral variable.
Since
is bounded for each fixed , it has modulus one for all
, the operator is a well-defined bounded
operator on all of for each .
This is in contrast to itself, which is unbounded: the
time-evolution operator is always bounded even when the generator is not.
In the NUVO framework, this means that the full Hilbert space
is the appropriate domain for the dynamics: maps
to for all , while acts only on the dense subspace
Stone's theorem is applied to the scalar-conformal Hamiltonian by
identifying the self-adjoint generator with
The factor is required by dimensional consistency: the
generator of time evolution must have dimensions of inverse time, while
has dimensions of energy, so the combination
carries the correct dimensions.
Verification of the Stone's theorem hypothesis.
By the self-adjointness theorem, is self-adjoint on
Since is a fixed positive constant, the operator
is self-adjoint on the same domain
self-adjointness is preserved under multiplication by a positive real scalar.
The hypothesis of Stone's theorem is therefore satisfied with
Theorem. Time-evolution operator.
Let be the scalar-conformal Hamiltonian of the
self-adjointness theorem.
Then the family of bounded operators
Equation. Time-evolution operator.
where is the projection-valued measure of
established in the spectral-structure proposition, constitutes a
strongly continuous one-parameter unitary group on satisfying
properties 1--4 of Stone's theorem.
This group is the unique strongly continuous unitary group on
whose self-adjoint generator is .
Proof.
Apply Stone's theorem with
Self-adjointness of on is verified above.
Stone's theorem therefore yields a strongly continuous one-parameter
unitary group
on satisfying properties 1--4.
The spectral integral representation in the time-evolution operator
equation follows from the functional calculus definition
with
and the projection-valued measure of the spectral-structure
proposition.
Uniqueness follows from the converse direction of Stone's theorem: any
two strongly continuous unitary groups on with the same
self-adjoint generator must coincide.
Remark.
The spectral integral form of the time-evolution operator makes the
connection to the spectral decomposition of QM1 transparent.
For a closure state expanded in the spectral basis of
as
the time evolution acts by multiplying each spectral coefficient by a
phase factor:
Each spectral component acquires an independent oscillatory phase at
a rate determined by its eigenvalue.
The modulus of each coefficient and is preserved by
consistent with the Parseval identity and the closure conservation
established in QM1.
The Schrödinger equation is now derived as the generator equation
of the unitary group .
This derivation is the culmination of the logical chain that began with
the scalar capacity modulation in Section 3: the geometry determines
the potential, the potential determines the Hamiltonian, the Hamiltonian
determines the unitary group, and the unitary group determines the
equation of motion.
Theorem. The Schrödinger equation.
Let be the scalar-conformal Hamiltonian of the
self-adjointness theorem and let be the unitary group
of the time-evolution theorem.
For each initial datum
define the curve
Equation. Schrödinger solution.
Then:
for all .
The map is strongly differentiable in .
satisfies the time-dependent Schrödinger equation
Equation. Schrödinger equation.
strongly in for all .
Proof.
Part 1.
By Stone's theorem, the unitary group preserves the domain
of its generator.
Since the generator of is
with domain
and since unitary operators preserve the spectral structure of the
generator, the domain is invariant under
for all :
for all .
Part 2.
By Stone's theorem applied to
the map
is strongly differentiable in and its strong derivative
satisfies
Equation. Generator equation.
where in the last step part 1 is used to confirm that
is well-defined in .
Part 3.
Multiply both sides of the generator equation by :
This is the Schrödinger equation, which holds strongly in
for all .
Remark.
The Schrödinger equation
is the time-dependent Schrödinger equation of quantum mechanics.
In the standard formulation of quantum mechanics it is introduced as a
fundamental postulate: a law of nature asserted without derivation.
In the present framework it is a theorem, derived in three logically
ordered steps from the scalar--conformal geometry.
The derivation chain is:
No step in this chain introduces a dynamical postulate.
The derivation above establishes that
is a solution of the Schrödinger equation.
The converse direction of Stone's theorem establishes that it is the
only solution.
Corollary. Uniqueness of the Schrödinger evolution.
For each
the curve
is the unique strongly differentiable solution of the Schrödinger equation
in satisfying
Proof.
Suppose is a second strongly differentiable
solution of the Schrödinger equation with
and
for all .
Then defines a strongly continuous map satisfying
the generator equation for , which by the converse
direction of Stone's theorem implies that the group generated by
must coincide with .
In particular,
for all .
Remark.
The uniqueness corollary is the Hilbert-space expression of the
deterministic character of the scalar--conformal transport closure system.
The Q-series established that the coupled evolution is a
deterministic system: given the initial closure density and transport
phase, the subsequent evolution is uniquely determined by the transport
closure law.
The uniqueness corollary establishes the same conclusion at the level
of the Schrödinger equation on : given the initial state
no other evolution is consistent with the Schrödinger dynamics generated
by .
The statistical character of quantum measurements, inherited from the
Born frequency law of QB6, arises not from indeterminism in the dynamics
but from the coherence-gated interaction structure: the evolution itself
is deterministic and unique.
The Schrödinger equation of the Schrödinger theorem is a general result,
valid for all admissible scalar-conformal Hamiltonians
.
The Q-series Schrödinger-type compatibility result recalled earlier is
recovered as a special case.
Proposition. Recovery of the Q-series result.
The time-dependent Schrödinger equation of the Schrödinger theorem,
specialized to the hydrogenic Hamiltonian
with the Coulomb potential of the hydrogenic-specialization proposition,
extends and completes the Schrödinger-type compatibility result of the
Q-series in the following respects.
is self-adjoint on
and essentially self-adjoint on by the
self-adjointness theorem and its essential self-adjointness corollary.
The Q-series did not establish self-adjointness.
The unitary time-evolution group
is the unique strongly continuous unitary group on
generated by .
The Q-series did not establish uniqueness of the evolution.
The equation
holds as a strong equation on
for all
and all .
The Q-series established only a compatibility condition within the
finite-dimensional span of stationary modes.
The stationary closure modes of the Q-series are the
discrete eigenstates of in , with
eigenvalues
Under the full Schrödinger evolution,
so stationary modes remain stationary with time-dependent phase
factors, consistent with the Q-series result.
Proof.
Parts 1 and 2 follow directly from the self-adjointness theorem and the
time-evolution theorem applied to , whose admissibility
was established in the admissibility proposition.
Part 3 follows from the Schrödinger theorem applied to
.
For part 4: if
in , then by the spectral functional calculus,
since the exponential of a constant multiple of the identity acts as
scalar multiplication on eigenstates.
The eigenvalues
are recovered from the Q-series hydrogenic spectrum via the
hydrogenic-spectrum check.
Remark.
The recovery proposition is the explicit statement of the promotion
from compatibility to theorem that was identified as Gap (iii).
The Q-series result---that the hydrogenic transport closure system is
consistent with a Schrödinger-type equation---is now understood as
the restriction of the Schrödinger theorem to the hydrogenic sector.
Every aspect of the Q-series result that was tentative, including
self-adjointness not verified, uniqueness not established, and scope
limited to stationary modes, is now established as a rigorous consequence
of the general theorem.
The Q-series is thereby retroactively confirmed as correct within its
stated scope, and the present paper provides the general framework
within which it is embedded.
The unitary group established in the time-evolution theorem
carries with it, by definition, a set of norm-preservation and
inner-product-preservation properties that are recorded in the present
section.
These properties are not new results in the sense of requiring derivation
beyond what Stone's theorem already provides; rather, they are the
physical and structural consequences of unitarity that must be stated
explicitly in order to close the logical arc of the present paper.
Three items are addressed.
First, unitarity of is recorded as a theorem and its
norm-preservation consequence is stated.
Second, the relationship between this operator-level norm preservation
and the geometric norm preservation of QM1 Corollary 3.4 is analyzed;
their logical independence and mutual consistency constitute a non-trivial
internal check on the NUVO framework.
Third, the preservation of the full inner product---not merely the
norm---is established, and its consequences for the orthogonality
structure of the transport closure states are recorded.
Unitarity of is an immediate consequence of Stone's theorem
and requires no additional argument: property (iii) of Stone's theorem
states that
which is precisely the definition of unitarity.
The present theorem records this fact and draws the norm-preservation
consequence that is of primary physical importance.
Theorem. Unitarity and norm preservation.
The time-evolution operator of the time-evolution theorem
is unitary for every :
Equation. Unitarity.
Consequently, for any and all ,
Equation. Norm preservation from unitarity.
In particular, if
then
for all , where
is the Schrödinger evolution of the Schrödinger theorem.
Proof.
Unitarity is part (iii) of Stone's theorem, which established
and
by the group law.
For norm preservation:
where the second step uses the definition of the adjoint and the third
uses unitarity.
Taking square roots gives norm preservation.
The special case
follows immediately.
Remark.
The unitarity of can also be seen directly from the spectral
integral representation of the time-evolution operator:
Since
for every spectral value , the spectral functional calculus gives
and hence
Thus unitarity is the spectral expression of the fact that Schrödinger
evolution changes only the phases of the spectral components of a
closure state, never their magnitudes.
QM1 established norm preservation from a different starting point.
There, the conservation of total closure followed from the divergence
form of the continuity equation for the closure density:
Integrating over all space and imposing the appropriate decay or
boundary conditions yielded
Since the complex state encoding satisfies
this is precisely the statement that
The present section obtains the same conclusion from the operator
framework:
The two results are logically independent.
QM1 norm preservation is geometric: it follows from the transport
continuity equation before any Hamiltonian or Schrödinger dynamics has
been constructed.
QM4 norm preservation is operator-theoretic: it follows from the
unitarity of the time-evolution group constructed by Stone's theorem
from the self-adjoint Hamiltonian.
Remark.
The agreement between the geometric norm preservation of QM1 and the
operator-level norm preservation of the present section is an internal
consistency check on the NUVO program.
At the geometric level, norm preservation states that total closure is
conserved under admissible transport.
At the operator level, norm preservation states that Schrödinger evolution
is unitary on .
Both statements describe the same physical system from different
representational levels.
The agreement confirms that the scalar--conformal action produces both
the correct transport equations in the Q-series and the correct operator
algebra in the QB- and QM-series.
In subsequent papers, both routes remain available:
the geometric route for direct transport arguments, and the
operator-theoretic route for spectral and algebraic arguments.
Unitarity preserves not only the norm of each individual state but the
full inner product between any two states evolving under the same dynamics.
This is a stronger statement than norm preservation and has direct
consequences for the orthogonality structure of the transport closure
states.
Corollary. Inner product preservation.
For any
and all ,
Equation. Inner product preservation.
Proof.
Using the definition of the adjoint and unitarity:
The consequences of inner product preservation for the transport closure
state structure are recorded in the following remarks.
Remark.
Inner product preservation implies in particular that if
at , then
for all .
Orthogonality is preserved by the Schrödinger evolution.
In the NUVO framework this means that distinct holonomic closure classes,
which were shown to be mutually orthogonal under the QB3 inner product
and whose orthogonality was extended to in QM1, remain
orthogonal for all time under the Schrödinger dynamics.
The time evolution does not mix orthogonal closure classes; it transports
them unitarily within the Hilbert space.
Remark.
Inner product preservation also preserves transition amplitudes.
For any reference state and any evolving state
, the overlap
is constant when both states evolve under the same unitary group:
Thus the relative coherence between two co-evolving closure states is
preserved.
This is the dynamical extension of the holonomic coherence structure
established in QB3.
If instead is held fixed while only evolves, then
may vary in time.
This variation is the source of time-dependent transition amplitudes
and spectral oscillations in later papers of the QM-series.
Remark.
For an energy eigenstate
the Schrödinger evolution gives
The closure density is therefore time-independent:
Thus stationary closure modes remain stationary under the Schrödinger
evolution, with only a global phase factor changing in time.
The Born frequency law inherited from QB6 is consequently time-stable
for stationary states: the predicted interaction-event frequencies
associated with a stationary eigenmode do not change under time
evolution.
Remark.
For a superposition of energy eigenstates,
the magnitudes are preserved, but relative phases evolve as
The norm and total closure are conserved, but interference terms between
different spectral components can oscillate in time.
This is the dynamical source of beat phenomena, transition amplitudes,
and time-dependent expectation values in the later QM-series.
The preservation of the inner product guarantees that these oscillations
redistribute coherence within the state without changing the total
closure content.
The Schrödinger dynamics established in Section 5 and the unitarity results of Section 6 together provide the dynamical framework within which symmetry and conservation can be analyzed at the level of quantum expectation values.
The present section constructs the Noether bridge identified as Gap (iv) in Section 2: the derivation connecting the geometric transport symmetry of the M-series, which operates at the level of the scalar--conformal action functional, to the conservation of expectation values of quantum-mechanical observables under Schrödinger evolution.
The structure of the section is as follows.
A general conservation theorem is established first: if a self-adjoint observable commutes with the Hamiltonian on a dense domain, then the expectation value is conserved for all Schrödinger evolutions initialized in that domain.
This general theorem is then applied to the three fundamental symmetries of the scalar--conformal transport system: time translation, yielding energy conservation; spatial translation, yielding momentum conservation for free transport and the quantum Newton law for general potentials; and spatial rotation, yielding angular momentum conservation for rotationally symmetric potentials.
The closing remark records the Noether bridge explicitly, closing Gap (iv).
The classical Noether theorem, at the level of the scalar--conformal action, associates a conserved current to each continuous symmetry of the transport closure system.
The quantum-mechanical analogue operates at the level of expectation values: a symmetry of the Hamiltonian, expressed as a commutation relation with a self-adjoint generator, implies that the expectation value of that generator is time-independent under Schrödinger evolution.
The following theorem establishes this correspondence in full generality.
Theorem. Conservation law from commutation.
Let be a self-adjoint operator on with domain , and suppose that
is dense in .
Suppose further that
Equation. Commutation condition.
Then for any Schrödinger evolution
with
the expectation value
Equation. Expectation value.
is well-defined for all and satisfies:
Equation. Conservation law.
Proof.
Since
and the Schrödinger theorem establishes that
for all , it remains to verify that
for all so that the expectation value is well-defined.
This follows from the fact that preserves any domain that is invariant under ; since commutes with on the dense domain, preserves as well, a standard consequence of the spectral theory of commuting operators.
To establish the conservation law, differentiate
with respect to .
Since both
and
are strongly differentiable in , the product rule for inner products gives:
Substituting the Schrödinger equation
gives:
Pulling out the scalar factors and using conjugate linearity in the first argument:
Since is self-adjoint,
so the expression becomes
By the commutation condition and the fact that
one has
Remark.
The identity
established in the proof of the conservation theorem holds for any self-adjoint , not only those commuting with , and is the Schrödinger-picture form of the Heisenberg equation of motion for the observable .
In the NUVO framework it is not a separate postulate but a direct consequence of the Schrödinger dynamics and the self-adjointness of .
The conservation theorem is the special case
which yields
The general identity
will be used directly in Section 8 to derive the Ehrenfest equations.
The first application of the conservation theorem is the conservation of energy under Schrödinger evolution.
By Stone's theorem, the Hamiltonian is the generator of the time-translation group : it is the operator that controls the response of the system to infinitesimal time translations.
The invariance of the static scalar-conformal transport system under time translation---guaranteed by the time-independence of the scalar capacity field in the static sector---is reflected at the operator level by the trivial commutation
Corollary. Energy conservation.
For any Schrödinger evolution
with static Hamiltonian and
one has:
Equation. Energy conservation.
The expectation value is constant for all .
Proof.
Apply the conservation theorem with .
The commutation condition
holds trivially for all
Remark.
The conservation of is the quantum-expectation-value counterpart of the classical energy conservation that follows, via the classical Noether theorem, from the time-translation invariance of the M-series scalar--conformal action in the static sector.
In the NUVO framework, both results have the same geometric origin---the time-independence of the scalar capacity field ---but are derived at different levels: the classical result from the action variational structure, the present result from the Schrödinger dynamics and the commutator
For an eigenstate of with eigenvalue , the expectation value
is time-independent and constant for all , consistent with the identification of the discrete spectrum with the bound holonomic closure modes of the Q-series.
The momentum operators
are the generators of spatial translations in the transport closure system.
Their commutation with is determined by the spatial uniformity or non-uniformity of the potential .
Proposition. Momentum commutator and conservation.
On the dense domain
the commutator of
with is:
Equation. Hamiltonian-momentum commutator.
for all .
Consequently:
Free transport or :
on , and by the conservation theorem, is conserved for all .
Non-uniform potential:
in general, and the rate of change of mean momentum is:
Equation. Momentum rate.
Proof.
Compute
on .
First term.
The kinetic operator is
Since
and
because both are times commuting partial derivatives, one has
Second term.
For , compute:
Substituting
gives
Therefore
which is the Hamiltonian-momentum commutator equation.
Part 1 follows immediately: if
then
and the conservation theorem yields conservation of .
Part 2 follows from the general Heisenberg identity:
Substituting the Hamiltonian-momentum commutator gives:
Remark.
The momentum-rate equation is the quantum-mechanical analogue of Newton's second law: the rate of change of mean momentum equals the negative mean gradient of the potential, which is the mean force.
In the scalar--conformal NUVO framework, this is not an independent physical law but a direct consequence of the commutator structure of the transport generators with the Hamiltonian, which is in turn a consequence of the geometry of the scalar capacity modulation.
The force
is determined entirely by the gradient of the scalar capacity modulation field.
This result will appear again as the second Ehrenfest equation in Section 8.
The angular momentum operators are the generators of spatial rotations in the transport closure system.
Their full algebraic structure is developed in QM5; the present subsection establishes their commutation with for rotationally symmetric potentials, which is the content of angular momentum conservation needed for the QM-series.
Proposition. Angular momentum commutator and conservation.
Define the angular momentum operators by:
Equation. Angular momentum operator definition.
where is the Levi-Civita symbol.
Then on :
for all .
For a rotationally symmetric potential ,
for all .
Consequently, for
with ,
and by the conservation theorem,
is conserved for all .
Proof.
Part 1.
Write
Since
for all , one has
The second term vanishes since
For the first term,
Using the canonical commutation relation,
so
Therefore
Since is antisymmetric in and is symmetric, because the momentum operators commute, the contraction vanishes:
Hence
Part 2.
For , the operator depends only on
The angular momentum operators generate rotations, and is rotation-invariant.
Formally, for :
Expanding,
Since
one has
Thus the contribution is proportional to
Contracting with gives zero:
because is antisymmetric and is symmetric in .
Hence
Part 3.
By parts 1 and 2,
The conservation theorem with then yields
Remark.
The angular momentum proposition establishes angular momentum conservation as a theorem of the scalar--conformal dynamics for rotationally symmetric potentials.
The operators defined above are introduced here in their minimal form, as products of position and momentum operators, sufficient for the commutation calculation.
Their full algebraic structure---the commutation algebra
the joint spectrum of and , the identification of spherical harmonics as stationary angular closure eigenstates, and the integer holonomy quantization of orbital quantum numbers---is developed in QM5 using the Schrödinger dynamics and the operator framework of the present paper as its foundation.
The three conservation laws established in the energy conservation corollary, the momentum conservation proposition, and the angular momentum conservation proposition are special cases of a single general pattern, which constitutes the Noether bridge between the geometric transport symmetry of the M-series and the conservation of quantum-mechanical expectation values.
Remark. The Noether bridge.
The M-series established the scalar--conformal action to be invariant under three continuous symmetries in the static sector: time translation, invariance of under ; spatial translation, invariance of under for uniform ; and spatial rotation, invariance of under for .
At the level of the classical transport closure system, each symmetry generates a conserved Noether current.
The present section has established the quantum-mechanical counterpart: each symmetry corresponds to a transport generator that commutes with on a dense domain, and the associated expectation value is conserved under Schrödinger evolution by the conservation theorem.
The correspondence is:
This is the Noether bridge: Gap (iv) of Section 2 is explicitly closed.
The geometric transport symmetry established in the M-series, the self-adjoint operator structure derived from the scalar--conformal geometry in the QB-series and QM1, and the Schrödinger dynamics established in the present paper all combine to yield the quantum conservation laws without any classical-mechanics input.
Every element of the derivation is internal to the NUVO program.
The conservation laws of Section 7 addressed observables that commute
with the Hamiltonian.
The Ehrenfest theorem addresses the complementary case: observables that
do not commute with ---specifically, position and momentum---and
derives the equations governing the time evolution of their expectation
values.
The result is that these expectation values satisfy equations formally
identical to the classical Hamilton equations of motion, with the
potential gradient replacing the classical force.
This is an exact result of the Schrödinger dynamics, not an
approximation, and it holds for all normalized closure states and
all times.
In the NUVO framework it expresses the fact that the scalar--conformal
transport geometry generates both the full quantum dynamics and the
classical transport trajectories as a structural consequence, with the
latter emerging as the evolution of mean transport quantities under the
former.
For a Schrödinger evolution
the expectation values of position and momentum are defined as
Equation. Mean position.
and
Equation. Mean momentum.
These are the mean position and mean momentum of the closure state
at time .
In the NUVO framework, is the first spatial
moment of the normalized closure density
identifying it as the centroid of the closure distribution.
The mean momentum is the mean
transport-phase gradient, weighted by the closure density, representing
the average transport velocity of the closure configuration.
That the expectation values above are well-defined for all follows
from the Schrödinger theorem, which ensures
together with the growth conditions on and for
states in .
For initial data
both expectation values are finite and smooth in .
The rate of change of each expectation value is controlled by the
general identity established in the conservation-law section:
Equation. Heisenberg identity for expectation values.
for any self-adjoint with
The Ehrenfest theorem is the result of applying this identity to
and
in turn and computing the resulting commutators explicitly.
The two Ehrenfest equations are established as a single theorem, with
each proved in turn from the general Heisenberg identity and the
commutator calculations carried out explicitly.
Theorem. Ehrenfest theorem.
Let
be the Schrödinger evolution with
Then the expectation values of position and momentum satisfy:
Equation. First Ehrenfest equation.
and
Equation. Second Ehrenfest equation.
for all and all .
Proof.
First Ehrenfest equation: the position equation.
Apply the Heisenberg identity with
Compute
on .
Potential term.
Since both and are multiplication operators,
Therefore
Kinetic term.
Write
Using
one obtains
Summing over gives
and hence
Therefore
Substituting into the Heisenberg identity gives
which is the first Ehrenfest equation.
Second Ehrenfest equation: the momentum equation.
Apply the Heisenberg identity with
The commutator was computed in the momentum-conservation proposition:
Therefore
which is the second Ehrenfest equation.
Remark.
The second Ehrenfest equation is the precise quantum-mechanical
counterpart of Newton's second law in the scalar--conformal setting.
The quantity
is the restoring force per unit closure content generated by the
gradient of the scalar capacity modulation field.
The equation
states that the rate of change of mean momentum equals the mean
restoring force.
This is not a postulate imported from classical mechanics but a theorem
derived from the commutator of with , which
is in turn a consequence of the scalar--conformal transport geometry.
The derivation chain is:
The Ehrenfest equations are formally identical in structure to the
classical Hamilton equations:
Equation. Classical Hamilton equations.
The structural identity between the Ehrenfest equations and the
classical Hamilton equations is exact: the Ehrenfest equations are not
an approximation to the classical equations but the exact quantum
equations for the expectation values, which happen to take the same
form.
The distinction between the quantum and classical descriptions lies not
in the form of the equations but in their objects: the Ehrenfest equations
govern expectation values
and
which are spatial averages of the closure density and transport phase
gradient, while the classical equations govern the individual trajectory
coordinates of a point particle.
For a general closure state , the expectation-value trajectory
is the centroid trajectory of the closure distribution.
Individual fluctuations of around this centroid---encoded in the
higher moments of ---evolve according to the full
Schrödinger dynamics and are not described by the Ehrenfest equations
alone.
In particular, the spatial spread of the closure
distribution can grow in time even when the centroid follows a classical
trajectory, a feature that will be studied in the context of coherent
states in QM6.
Remark.
Within the NUVO framework, the Ehrenfest theorem carries the following
precise interpretation.
The scalar--conformal transport geometry generates a deterministic
evolution for the full closure state via the Schrödinger
equation.
This evolution encodes both the mean transport trajectory---the centroid
dynamics governed by the Ehrenfest equations---and the fluctuation
structure around that trajectory, encoded in the higher moments of
.
The mean transport trajectory follows the classical scalar--conformal
transport law exactly, without approximation.
This is not a separate postulate or a statement about a classical limit:
it is a theorem derived from the Schrödinger dynamics and the canonical
commutation relation of QM1.
The sense in which the classical transport trajectory is fully recovered
by the quantum dynamics---beyond the expectation-value level---requires
the notion of coherent states: minimum-uncertainty states for which
the expectation-value trajectory and the full quantum trajectory coincide
as closely as the uncertainty relations of QM3 permit.
For the harmonic oscillator Hamiltonian of QM6, coherent states evolve
as exact classical trajectories with no spreading of the closure
distribution, providing the sharpest possible realization of the
classical correspondence within the NUVO framework.
The present theorem establishes the exact result; QM6 develops the
optimal approximation.
Remark.
A technical point deserves explicit statement.
The Ehrenfest equations involve
the expectation value of the potential gradient, which is in general
not equal to the gradient of the potential evaluated at the mean
position:
Equation. Expectation of gradient versus gradient at expectation.
in general, with equality only when is linear in or when
the closure state is sufficiently sharply localized that the nonlinearity
of is negligible over the spatial extent of .
The departure from equality is controlled by higher moments of the
closure distribution and the nonlinearity of .
In the NUVO framework, this departure is a measurable structural feature
of the closure state: it quantifies the degree to which the closure
distribution is sensitive to the curvature of the scalar capacity
modulation field.
The Ehrenfest theorem as stated above is exact; the approximation
is an additional step that is valid only in the semiclassical regime
and is not assumed here.
The present section maintains the interpretive discipline established
in the Q-series and continued through the QB-series and QM1.
Four items are addressed: the logical status of the Schrödinger
equation within the NUVO framework, the derived character of the
Hamiltonian, the preservation of determinism through the dynamical
framework, and the scope of the present construction relative to the
remainder of the QM-series.
In the standard formulation of quantum mechanics, the time-dependent
Schrödinger equation
is introduced as a fundamental postulate: a dynamical law asserted as
a primitive constituent of the theory, with no derivation from more
basic principles.
Its justification is empirical---it produces predictions that agree
with experiment---and its status as a postulate places it outside the
scope of derivation within the standard framework.
In the present framework the same equation is a theorem.
The derivation proceeds in four steps, each of which introduces no
new postulate and relies only on results established in the prior series
and the present paper.
Step 1: The scalar capacity modulation correspondence, Section 3.
A spatial modulation of the baseline scalar capacity
generates a real-valued potential
in the exchange-sector transport closure law at first order.
This correspondence is derived from the exchange-sector action of the
M-series and does not require any assumption about the form of the
potential.
Step 2: Admissibility and the Kato class, Section 3.
The scalar-conformal potential arising from a localized modulation
belongs to the admissible class of admissible potentials,
satisfying the relative bound
with .
Step 3: Self-adjointness of via Kato--Rellich, Section 4.
With the kinetic operator self-adjoint on
and the potential admissible with relative bound ,
the Kato--Rellich theorem yields self-adjointness of
on .
No extension choices or boundary conditions are required; the domain
is uniquely determined.
Step 4: Stone's theorem and the generator equation, Section 5.
A self-adjoint operator on a Hilbert space uniquely determines a
strongly continuous one-parameter unitary group via Stone's theorem.
Applied to , this yields the unitary group
and the Schrödinger equation
is the generator equation of this group, not an independent assertion.
The logical status of the Schrödinger equation in the NUVO framework
is therefore that of a theorem derived from the scalar--conformal
geometry, the functional analysis of self-adjoint operators, and the
algebraic structure of unitary groups.
No dynamical postulate is introduced at any step.
The equation is not less reliable for being derived rather than
postulated; on the contrary, its derivation makes explicit the geometric
and algebraic conditions under which it holds and the precise class
of Hamiltonians to which it applies.
In the standard quantum-mechanical formalism, the Hamiltonian
is a choice: given a physical system, one writes down a Hamiltonian
that models its energy structure, guided by classical analogies,
experimental data, or symmetry arguments.
The Hamiltonian is therefore a primitive input to the theory rather than
a consequence of it.
In the present framework, the Hamiltonian is derived from the geometry.
The kinetic operator
arises from the free exchange-sector transport closure system with
uniform scalar capacity : it is the transport
energy of a closure state in the baseline geometric configuration.
The potential operator arises from the spatial modulation
of the scalar capacity field: it is the first-order
energy correction generated by the departure of the scalar capacity
from its baseline value.
The full Hamiltonian
is therefore not a choice but a structural consequence of the
scalar--conformal geometry of the transport closure system.
This has a direct implication for the admissible class of potentials.
The admissible class consists of those potentials for
which the Kato--Rellich theorem applies: potentials relatively bounded
with respect to with relative bound strictly less than one.
In the geometric picture, this class corresponds precisely to those
scalar capacity modulations that are localized and
controlled enough that the departure from the baseline geometry does
not dominate the free transport kinetic structure.
Potentials outside ---those with relative bound
or those not relatively bounded at all---correspond to scalar capacity
modulations so singular that the first-order expansion of the transport
closure law breaks down and the transport closure system in that
background is not well-defined at the level of the present analysis.
The admissible class is therefore not a technical convenience but a
statement about which geometric configurations support well-defined
transport closure at first order.
Remark.
The derivation chain
is unique at each step: the scalar-conformal potential correspondence
is determined by the M-series exchange-sector action and the first-order
expansion, the Kato--Rellich theorem yields self-adjointness on a
uniquely determined domain, and Stone's theorem yields a unique unitary
group.
The entire dynamical framework---Hamiltonian, time evolution,
conservation laws, Ehrenfest equations---is therefore uniquely determined
by the scalar capacity field .
There is no freedom of choice at any stage.
In subsequent papers of the QM-series, each physical sector, including
the harmonic oscillator, hydrogen atom, and barrier potentials, corresponds
to a specific scalar capacity configuration, and the Hamiltonian for that
sector is determined by the configuration without additional input.
The Q-series established that the exchange-sector transport closure
system---the coupled deterministic evolution of ---is
fully determined by its initial conditions.
Given , the subsequent evolution is uniquely
determined by the transport closure law.
This deterministic character of the underlying transport geometry is
reflected at the level of the Schrödinger dynamics.
The uniqueness of Schrödinger evolution establishes that for each
the Schrödinger evolution
is the unique strong solution of
with the given initial datum.
No other evolution is consistent with the dynamics generated by .
The determinism of the transport closure system is thus preserved in
the Hilbert-space formulation: the closure state at any future time
is uniquely determined by the closure state at the initial time.
The statistical character of quantum-mechanical predictions---the Born
frequency law of QB6, the measurement correspondence of QB7---is not
a consequence of indeterminism in the dynamics.
It arises from the coherence-gated interaction structure of the transport
system: the deterministic evolution of generates deterministic
closure density
at each time, and the Born frequency law assigns event frequencies to
coherence-gated interaction channels based on this density.
The statistical layer is logically separate from and does not alter
the deterministic character of the dynamical layer.
Remark.
The logical separation between the deterministic Schrödinger dynamics
and the statistical Born frequency law is a distinctive structural feature
of the NUVO framework.
In the standard formulation, the Schrödinger equation and the Born rule
are two independent postulates, and their logical relationship is a
matter of ongoing interpretive debate.
In the NUVO framework, the deterministic Schrödinger dynamics is a
theorem derived from the transport geometry, and the Born frequency law
is a theorem derived from the coherence-gated interaction structure of
QB6.
Both are results, and their logical independence from each other is
explicit: neither derivation depends on the other.
Their numerical consistency---both produce the same predictions for
measurement frequencies---is a non-trivial structural feature of the
scalar--conformal framework rather than an assumption.
The present paper establishes the dynamical foundation of the QM-series
in a form that is both general and rigorous.
It is equally important to record precisely what it does not establish,
so that the logical dependencies of subsequent papers are transparent
and no result is used before it has been derived.
The paper does not treat time-dependent potentials.
The scalar capacity modulation is taken to be
time-independent throughout Sections 3--8, producing a static Hamiltonian
that generates time-translation invariance and energy
conservation.
A time-dependent modulation would produce a
time-dependent potential , generating a time-dependent Hamiltonian
for which Stone's theorem does not directly apply and the
energy conservation established in the static sector fails.
The time-dependent sector---which is the natural setting for the
electromagnetic interaction and the minimal coupling of charges to
fields---is the domain of the RQM-series, where the full covariant
treatment of time-dependent backgrounds is undertaken.
The paper does not develop the full angular momentum algebra.
The angular momentum operators
are introduced and their commutation with rotationally symmetric
Hamiltonians is established, but the commutation algebra
the joint spectrum of and , the identification
of spherical harmonics as stationary angular closure eigenstates, and the
integer holonomy quantization of orbital quantum numbers are all developed
in QM5, which uses the Schrödinger dynamics and conservation structure
of the present paper as its foundational input.
The paper does not treat the harmonic oscillator or coherent states.
The harmonic potential
arising from a quadratically growing scalar capacity modulation, falls
outside the class of the main admissibility argument
and requires a separate self-adjointness treatment via the explicit
Hermite-function eigenbasis.
The algebraic ladder-operator structure of the harmonic oscillator, the
energy spectrum
and the identification of coherent states as minimum-uncertainty transport
configurations are all developed in QM6.
The paper does not treat multi-particle Schrödinger dynamics.
The single-particle Hamiltonian established here acts on the
single-particle Hilbert space
The multi-particle generalization requires replacing with
the -particle configuration space
constructing the multi-particle Hamiltonian as a sum of single-particle
kinetic terms and pair-interaction potentials, and imposing the exchange
symmetry constraints, symmetric for bosonic and antisymmetric for
fermionic closure structures, that arise from the topological identity
of indistinguishable loop configurations.
This construction, together with the derivation of the Pauli exclusion
principle as a structural consequence of antisymmetry, is undertaken
in QM7.
The paper does not treat spin dynamics.
The single-particle Hamiltonian of the present paper acts on
scalar-valued closure states in
and does not include a spin degree of freedom.
The spin- Hamiltonian requires the state space to be
extended to
with the spin-space component of constructed from the Pauli
matrices arising from the double-cover holonomy structure of QM8.
The magnetic moment coupling and the spin-orbit interaction are part
of the spin Hamiltonian developed in QM8, which uses the operator and
dynamical framework of the present paper throughout.
In each case, the present paper provides the essential dynamical
infrastructure---the Schrödinger equation, the unitary group, the
conservation structure, and the Ehrenfest theorem---on which the
subsequent development depends.
No paper in QM5 through QM11 bypasses the framework established here.
The present paper has established the complete dynamical framework for
the scalar--conformal NUVO transport closure system on the Hilbert space
of QM1, closing four structural gaps identified in the prior
series and constructing the Schrödinger dynamics, conservation laws,
and classical correspondence as theorems derived entirely from the
scalar--conformal geometry.
The principal results are as follows.
The scalar-conformal potential correspondence.
A spatial modulation
of the scalar capacity field generates, at first order in
, a real-valued multiplicative potential
in the exchange-sector transport closure equation.
This is Gap (i) closed.
The structural coupling constant is fixed by
consistency with the hydrogenic sector, which recovers the Coulomb
potential
as a special case, providing an internal consistency check on the
correspondence.
Scalar-conformal potentials arising from localized modulations
belong to the admissible class of admissible potentials,
satisfying the relative bound required by the Kato--Rellich theorem
with .
Self-adjointness of the Hamiltonian.
The scalar-conformal Hamiltonian
is self-adjoint on the second-order Sobolev domain
for all admissible potentials , by the
Kato--Rellich theorem.
This is Gap (ii) closed.
The domain is unchanged from that of the free kinetic operator
: the potential does not introduce additional regularity
requirements.
Essential self-adjointness on the Schwartz domain
follows as a corollary, confirming that
the scalar capacity geometry uniquely determines the Hamiltonian
without freedom of boundary conditions or extension choices.
The unitary time-evolution group and the Schrödinger equation.
Stone's theorem, applied to the self-adjoint operator
on , yields a unique strongly
continuous one-parameter unitary group
on .
This is Gap (iii) closed.
For each
the curve
is the unique strong solution of the time-dependent Schrödinger equation
with initial datum .
The Schrödinger equation is thereby established as a theorem derived
from the scalar--conformal geometry via the correspondence
with no dynamical postulate introduced at any stage.
The Q-series Schrödinger-type compatibility result for the
hydrogenic sector is recovered as a corollary of the general theorem.
Unitarity and norm preservation.
The time-evolution operator satisfies
for all , yielding norm preservation
and full inner product preservation
for all .
These results are consistent with and logically independent of the
geometric norm preservation of QM1.
Their agreement constitutes a non-trivial internal consistency check on
the NUVO framework.
Orthogonality of distinct holonomic closure modes, established in QB3
and extended to in QM1, is preserved for all time under the
Schrödinger evolution, and the Born frequency law of QB6 is time-stable
for stationary states.
Conservation laws from transport symmetry.
The general conservation theorem establishes that
for any self-adjoint observable , yielding conservation of
whenever
This is Gap (iv) closed.
Applied to the three fundamental symmetries of the static
scalar--conformal transport system: time-translation invariance yields
spatial-translation invariance yields
for free transport and
for non-uniform potentials; rotational invariance of yields
The Noether bridge is thereby recorded explicitly: the geometric
transport symmetry of the M-series is connected to the conservation of
quantum-mechanical expectation values by the Schrödinger dynamics of the
present paper, without classical mechanics input.
The Ehrenfest theorem.
The expectation values of position and momentum satisfy
and
equations structurally identical to the classical Hamilton equations.
These are exact results holding for all normalized closure states and
all times, derived from the Schrödinger dynamics and the canonical
commutation relation of QM1.
The mean transport trajectory of any closure state follows the classical
scalar--conformal transport law exactly, with departures from classical
behavior encoded in the higher moments of the closure distribution.
The present paper closes the four structural gaps that prevented the
QM-series from proceeding beyond QM3 on a rigorous foundation.
With the results now in place, the scalar--conformal transport geometry
supports a complete derivation chain:
with conservation laws and the Ehrenfest theorem following as corollaries
of the dynamics.
Every element of this chain is a theorem derived from prior NUVO results.
No dynamical postulate, no Hamiltonian chosen by analogy with classical
mechanics, and no conservation law imported from classical Noether theory
enters the framework.
The scalar--conformal geometry determines the dynamics completely and
uniquely.
The dynamical infrastructure established in the present paper propagates
forward through every subsequent paper in the QM-series without exception.
QM5 uses the conservation structure and the commutation algebra of the
angular momentum operators as the starting point for the full rotational
transport analysis.
QM6 uses the Schrödinger equation with a quadratic scalar capacity
modulation to establish the harmonic oscillator spectrum and to identify
coherent states as the closure configurations that most closely realize
the classical transport trajectory.
QM7 extends the single-particle Hamiltonian framework of the present
paper to the -particle configuration space
with multi-particle Hamiltonians constructed as sums of single-particle
kinetic terms and pair-interaction scalar-conformal potentials.
QM8 extends the state space to
and equips the Hamiltonian with a spin-space component constructed from
the Pauli matrices arising from double-cover holonomy.
QM9 constructs entangled states as non-factorizable elements of the
multi-particle state space of QM7, using inner product preservation to
track correlations.
QM10 uses the continuous-spectrum structure of and the
time-evolution group to construct the S-matrix and analyze
scattering and tunneling.
QM11 reformulates the transport generators as Lorentz-covariant objects
using the SR-series kinematics, and the covariant Schrödinger dynamics
that results prepares the transition to the Klein-Gordon and Dirac
equations of the RQM-series.
The Noether bridge closes a conceptual gap that has been present in the
NUVO program since the M-series established the geometric transport
symmetry structure.
The M-series showed that the scalar--conformal action is invariant under
time translation, spatial translation, and spatial rotation in the static
sector, and derived the associated Noether currents at the classical
geometric level.
The present paper shows that these same symmetries, when the transport
closure system is represented on the Hilbert space and
governed by the Schrödinger dynamics, imply the conservation of the
corresponding quantum-mechanical expectation values via the commutation
structure of the conservation theorem.
The derivation chain from the scalar--conformal geometry to all three
standard conservation laws of quantum mechanics is now complete, without
any step that appeals to classical mechanics, external symmetry
principles, or independent postulate.
The conservation of angular momentum established in the angular momentum
conservation proposition is the point of departure for QM5.
That paper develops the complete rotational transport algebra from the
operators
introduced in the present paper.
The commutation algebra
is derived from the canonical commutation relation of QM1 using the
Schrödinger dynamics of the present paper to identify the physical
significance of the algebraic structure.
The joint spectrum of and is then derived
from the integer holonomy quantization condition on rotationally closed
transport paths, producing the angular quantum numbers and
as structural outputs rather than assumed labels.
The spherical harmonics emerge as the
stationary angular closure eigenstates in the rotationally symmetric
sector, reconnecting to the hydrogenic structure of the Q-series and
completing the angular part of the hydrogen atom solution on the basis
of the dynamical framework established in the present paper and the
Hilbert space framework of QM1.