From Mechanical Momentum to Matter-Wave Phase

01How moving matter acquires a spatial phase

Fringes formed after particles pass through an aperture require two descriptions: the momentum carried by localized matter and the phase propagated by the surrounding field. ASD starts from a moving spatial structure. Its accompanying field develops through the joint adjustment of the core and surrounding field. The matter-wave problem asks why a phase scale in that field should correspond to the mechanical momentum measured in an experiment.

Mechanical momentum describes translation of the complete material structure. Let $m_0$ be its rest mass, $\mathbf v$ the core's translational velocity, $c$ the characteristic propagation speed and $\gamma=(1-v^2/c^2)^{-1/2}$. This page takes the following energy–momentum relations from upstream ASD dynamics, where they are constructed from the spatial response to motion:

Here $E$ is total energy, $\mathbf P$ is mechanical momentum of the complete structure and $P=|\mathbf P|$. These relations specify how velocity enters energy and momentum, but do not yet set an accompanying-field wavevector. A nearly monochromatic field has phase $\mathbf k\cdot\mathbf r-\omega t$: $\mathbf k$ specifies the direction and rate of spatial phase change, $\omega$ is angular frequency and the wavelength along propagation is $\lambda=2\pi/|\mathbf k|$. Connecting these two descriptions is the substantive matter-wave task.

Eliminating velocity and choosing the positive-energy branch gives manuscript equation (5.2.13):

The rationalized kinetic-energy expression is exact for the adopted energy–momentum relation. Its denominator approaches $2m_0c^2$ in the low-speed limit. Expanding further gives $E=m_0c^2+P^2/(2m_0)-P^4/(8m_0^3c^2)+O(P^6/(m_0^5c^4))$. This Taylor expansion is a separate low-momentum check, not the exact relation labelled (5.2.13) in the final manuscript.

02Calculating momentum from the complete field structure

The connection first requires a definition of field momentum. Spatial translation symmetry gives a conserved translation generator, the Noether momentum. This is calculated from the same action for the translating structure, rather than added as a second momentum assigned to a particle.

Chapter 5 illustrates this with a real field $q$. Here $q$ is a field amplitude, not the transfer wavevector used below for scattering. Let $Z$ be a constant kinetic weight, $B_{ij}$ a real symmetric dimensionless propagation matrix, and $W(q)$ the local potential-energy density. For the constant-coefficient action density of manuscript equation (5.2.9),

choose the sign convention in which a forward-propagating wave has positive momentum. The momentum density $g_i$ and energy flux $S_i$ are then

Here $\mathbf S$ denotes energy flux, distinct from the driving source in the accompanying-field equation. In this model $\mathbf S=c^2\mathbf g$ additionally requires isotropy, $B_{ij}=\delta_{ij}$. Higher spatial derivatives, mixed velocity terms or multiple coupled fields change the specific Noether currents; a complete ASD configuration therefore requires its own action calculation.

For conserved total energy, a local energy continuity equation and vanishing surface terms, the energy center obeys

The second relation follows by integrating the energy continuity equation by parts. Only when the action also supplies the isotropic flux–momentum relation above does it imply $\mathbf P=E\dot{\mathbf R}_E/c^2$. The connection between energy-center motion and mechanical momentum thus has explicit dynamical conditions. Experimental momentum includes the core, accompanying field and necessary near field. Assigning phase momentum to a bare core and then adding the same field contribution again would double-count it.

03How mode action connects energy, momentum and phase

A monochromatic periodic mode has not only frequency and wavevector but also a mode action $\mathcal I$, fixed by amplitude and energy normalization. With units of energy times time, it gives the action associated with one unit of angular phase. Chapter 5 uses the matching relations

The left-hand relations connect mode energy and momentum to phase generators. A classical linear mode can change its amplitude and hence its action; Fourier decomposition does not itself fix each excitation's action to $\hbar$. Matching one complete matter excitation to $\mathcal I=\hbar$ gives $E=\hbar\omega$ and $\mathbf P=\hbar\mathbf k$. The wavelength definition then gives $\lambda=h/P$, with $h=2\pi\hbar$.

A period must also be distinguished from angular frequency. The rest carrier has period $\tau=2\pi/\omega_0$. After matching $E_0=\hbar\omega_0=m_0c^2$, one complete period gives $E_0\tau=h$, while action per unit angular phase is $E_0/\omega_0=\hbar$. For a nonlinear periodic configuration, its orbit action $\oint p_A\,dq_A$ must be calculated before testing single-excitation matching; $q_A,p_A$ are its canonical coordinates and momenta.

The final manuscript adopts single-excitation action locking to connect the dressed structure to its phase: $E=\hbar\omega$, $\mathbf P=\hbar\mathbf k$, hence $\lambda=h/P$. These are usable matching relations of the present effective construction. The remaining task is to derive and maintain this locking from one nonlinear moving core and its surrounding fields, including their normalization; it is not to derive the wavelength algebra again. The phase–momentum locking problem records that microscopic closure task.

04Why phase, envelope and core have different velocities

A phase surface, a packet envelope and a material core are different objects. Along the wavevector, phase velocity is $v_{\rm ph}=\omega/|\mathbf k|$. Group velocity is the wavevector derivative of dispersion and describes the propagation of a narrow-band envelope peak. For a low-speed effective envelope with constant $B$,

Here $\mathbf p=\hbar\mathbf k$ is envelope phase momentum. Its identification with complete mechanical momentum $\mathbf P$ still requires the previous section's matching. In the isotropic free limit, $B=I$, giving $\mathbf p=m_0\mathbf v_g$. If $B$ depends on momentum or velocity, differentiation must also include $\partial B/\partial\mathbf p$; the constant-coefficient result cannot be extended unchanged to arbitrary speeds.

For the complete isotropic gapped dispersion $\omega^2=\omega_0^2+c^2k^2$, differentiating with respect to $\mathbf k$ and applying the adopted single-excitation matching gives

The final equality uses $E=\gamma m_0c^2$ and $\mathbf P=\gamma m_0\mathbf v$ for the same complete dressed excitation. Thus core translation and group motion agree within this matched isotropic free model. Computing a nonlinear moving family that independently supplies the assumed matching remains a separate microscopic task. Phase velocity may exceed $c$; the earliest information front is governed by the original second-order equation.

The manuscript's two-dimensional co-moving example makes the issue concrete. For motion at speed $v$ along $x$ and positive constant coefficients $b_x,b_y$, set $\xi=x-vt$ and write the co-moving amplitude as $\phi=e^{iq_v\xi}\chi$. Cancellation of the first spatial derivative requires

Here $q_v$ is a phase wavenumber and $\chi$ is the response amplitude after extracting that phase. This step connects translation and phase within the specified equation. The source-frequency offset and boundary conditions still determine whether the response propagates or decays. Interpreting the phase wavelength as an experimental matter wave requires both core–group-velocity locking and phase–total-momentum matching. Adjusting $b_x$ to fit fringes cannot replace those conditions.

05How packet weight propagates and is exchanged

A phase relation must be accompanied by a description of envelope redistribution. For the effective wave equation, take real $V$ and real symmetric $B_{ij}$. Multiply the equation by $\Psi^*$ and subtract its conjugate: the potential terms cancel and the gradient terms form a divergence, giving

$\Psi$ is the complex envelope, $n=|\Psi|^2$ its weight density, $\mathbf j$ its flux and $S_{\rm Sch}$ the effective source. Integration over a region $\mathcal V$ gives manuscript equation (5.2.8):

Weight can leave through the boundary or be exchanged with the core through the source. Total envelope weight is conserved only without a source and net boundary flux. If a complete dressed state has already absorbed the core into the initial state or effective Hamiltonian, source-free propagation may be used; the same core source must not then be inserted a second time.

For relative distributions one can normalize globally, $\psi=\Psi/(\int|\Psi|^2d^3r)^{1/2}$. This neither supplies a single-excitation energy condition nor makes sourced evolution automatically unitary. A time-dependent normalization adds a corresponding term to the normalized envelope equation. The result here is field-weight balance; its conversion into discrete records is addressed under detection and statistics.

06Why neutral composites retain a matter wave

Zero net charge does not remove an atom's or molecule's total energy, inertial response or overall phase. Electrical neutrality cancels the charge monopole; internal polarization, multipole fields and total core–field energy remain properties of the composite. Neutrality alone therefore cannot assign its matter wave exclusively to one open propagation channel.

Within an already adopted multicomponent low-speed envelope model, the manuscript demonstrates center-of-mass separation with two constituents. For masses and positions $m_1,m_2$ and $\mathbf r_1,\mathbf r_2$, define

$M$ is the total mass parameter in this low-speed model, $\mu$ the reduced mass, while $\mathbf R$ and $\boldsymbol\rho$ describe translation and internal relative motion. The chain rule gives $\nabla_1=(m_1/M)\nabla_R+\nabla_\rho$ and $\nabla_2=(m_2/M)\nabla_R-\nabla_\rho$. Substitution into the kinetic terms cancels the mixed derivatives:

If the internal potential depends only on relative position and the external field varies slowly across the object, the Hamiltonian separates into center-of-mass kinetic energy, an internal Hamiltonian $H_{\rm int}$, an external potential $V_{\rm ext}(\mathbf R)$ and an inhomogeneity correction $H_{\rm tidal}$. For an approximately frozen internal state $H_{\rm int}\chi_n=E_n\chi_n$, insert $\Psi_{\rm tot}=\psi_{\rm CM}(\mathbf R,t)\chi_n(\boldsymbol\rho)e^{-iE_nt/\hbar}$ and project to obtain

Here $P_{\rm CM}$ is center-of-mass mechanical momentum. The key conditions are a stable internal state, an external variation scale larger than the object and momentum transfer that does not resolve internal structure. With $q$ the transferred momentum divided by $\hbar$ and $a_c$ the object size, the last condition is $qa_c\ll1$. It does not require the center-of-mass wavelength to exceed object size: translational wavenumber may be large while the apparatus cannot excite or resolve the interior.

C₆₀ interference is an experimental requirement for such collective wave behavior. This separation follows within a specified multicomponent envelope; it is not an independent construction of general multiparticle states from a three-dimensional ASD local field. Including binding energy in inertial mass also requires matching $M$ to $M_{\rm phys}=E_{\rm rest}/c^2$ with consistent higher-order kinetic and interaction corrections.