Orthogonal Fermionic Wave Packets

• Orthogonal Fermionic Wave Packets are orthonormal superpositions of finite-energy modes that form a discrete basis for mesoscopic transport.
• They reorganize continuous energy spectra into deterministic, time-resolved electron emissions with spacing Δt=h/eV, underpinning Landauer conductance.
• These packets extend to Wilson–Wannier and Clifford algebra frameworks, linking transport physics to phase-space localization and abstract orthogonalization.

Orthogonal fermionic wave packets are orthonormal single-fermion modes built as localized superpositions rather than as delocalized plane waves. In mesoscopic transport, they furnish an exact time-domain basis in which the occupied bias window is reorganized into a discrete train of charge-carrying excitations, and the Landauer conductance emerges from a deterministic sequence of fermionic events rather than from momentum-space kinematics (Biborski, 17 Jun 2026). Related constructions appear in interacting-fermion theory through the Wilson–Wannier basis, which localizes fermionic states simultaneously in real space and momentum space (Ossadnik, 2013), and in abstract noncommutative orthogonalization through fermionic quantum procedures on Clifford systems, which model orthogonalization of fermionic modes algebraically rather than as explicit spatial wave packets (Lakos, 2015).

1. Finite-window packet construction in transport

In the transport formulation, the system is a one-dimensional ballistic channel connected to a source and drain. At zero temperature, the source is biased by $V$, so the occupied transport window is the finite energy interval $E\in[0,eV]$, measured relative to the drain Fermi level, which is set to zero. The basic fermionic modes are second-quantized energy operators $c_E^\dagger,c_E$ satisfying

$$\{c_E,c_{E'}^\dagger\}=\delta(E-E'),\qquad \{c_E,c_{E'}\}=\{c_E^\dagger,c_{E'}^\dagger\}=0.$$

The key step is to replace the continuum of occupied energy modes in $[0,eV]$ by a discrete set of time-labeled packet operators (Biborski, 17 Jun 2026).

The packet creation operator for the $n$-th packet emitted at time $t_n$ is defined by a finite-window Fourier transform,

$$\hat A_n^\dagger=\frac{1}{\sqrt{eV}}\int_0^{eV} dE\, e^{iEt_n/\hbar}c_E^\dagger.$$

This is the zero-temperature limit of a more general source-weighted definition. The corresponding one-particle state is $|\phi_n\rangle=\hat A_n^\dagger|0\rangle$, with energy-space amplitude

$C_n(E)=\frac{1}{\sqrt{eV}}e^{iEt_n/\hbar}.$

The defining feature of this construction is that each packet is a coherent superposition of all energies inside the bias window. The packet basis is therefore not a truncation of the continuum but a reorganization of it. In this representation, transport is described in terms of localized fermionic wave packets rather than infinite plane waves.

2. Orthogonality, Pauli exclusion, and the temporal lattice

The orthogonality structure follows directly from the packet anticommutator,

$E\in[0,eV]$0

with $E\in[0,eV]$1. The overlap vanishes exactly when

$E\in[0,eV]$2

so the fundamental temporal spacing is

$E\in[0,eV]$3

For this choice, the canonical algebra becomes

$E\in[0,eV]$4

which makes the packets a discrete orthonormal fermionic basis (Biborski, 17 Jun 2026).

The physical interpretation given in the transport construction is that the finite bandwidth $E\in[0,eV]$5 sets the inverse time resolution, while the Pauli exclusion principle enforces one fermion packet per time bin as the maximal occupation compatible with orthogonality within the finite energy window. This produces a temporal lattice of emission events. In the zero-temperature single-channel limit, the current is thus reorganized as a deterministic train carrying one electron charge per time bin.

A common misunderstanding is to read this temporal discretization as an approximation or sampling artifact. In the finite-window construction it is neither. The spacing $E\in[0,eV]$6 is the exact orthogonality condition generated by the Fourier dual of the transport window together with fermionic anticommutation.

3. Real-space wave functions and arbitrary dispersion

The time-labeled packets admit a real-space representation through the field operator of a right-moving channel, written in energy representation with group velocity $E\in[0,eV]$7 and inverse dispersion relation $E\in[0,eV]$8. The spatial wave function of the $E\in[0,eV]$9-th packet is obtained by projection,

$c_E^\dagger,c_E$0

The resulting packet is the paper’s generalized Wannier packet. Its energy integral carries a $c_E^\dagger,c_E$1 factor, identified as the Jacobian needed to preserve canonical anticommutation relations under the change from energy to position space (Biborski, 17 Jun 2026).

The spatial overlap is exactly orthonormal,

$c_E^\dagger,c_E$2

The proof uses the $c_E^\dagger,c_E$3-integration to generate a delta function in momentum and then uses monotonicity of $c_E^\dagger,c_E$4 to convert it back to an energy delta. The construction is exact for arbitrary energy dispersion $c_E^\dagger,c_E$5, provided the relevant branch is one-to-one over the transport window.

The packet shape depends on the dispersion. For linear dispersion, packets are rigid and sinc-shaped. For parabolic dispersion, packets spread and chirp. The orthogonality and conductance result, however, remain intact because the proof relies on the finite window and the canonical algebra rather than on linearization. This suggests that packet geometry and transport quantization are logically distinct components of the construction.

4. Exact Fourier duality with the plane-wave picture

Because transport is restricted to the finite interval $c_E^\dagger,c_E$6, the energy-mode operator can be reconstructed exactly from the packet basis as a discrete Fourier series,

$c_E^\dagger,c_E$7

The derivation uses Poisson summation to obtain an energy-space comb,

$c_E^\dagger,c_E$8

and since both $c_E^\dagger,c_E$9 and $$\{c_E,c_{E'}^\dagger\}=\delta(E-E'),\qquad \{c_E,c_{E'}\}=\{c_E^\dagger,c_{E'}^\dagger\}=0.$$0 lie in $$\{c_E,c_{E'}^\dagger\}=\delta(E-E'),\qquad \{c_E,c_{E'}\}=\{c_E^\dagger,c_{E'}^\dagger\}=0.$$1, only the $$\{c_E,c_{E'}^\dagger\}=\delta(E-E'),\qquad \{c_E,c_{E'}\}=\{c_E^\dagger,c_{E'}^\dagger\}=0.$$2 term survives. This is the mechanism by which cross terms disappear and the packet train reproduces a steady current (Biborski, 17 Jun 2026).

The significance of this step is conceptual as much as technical. The packet basis is not a heuristic replacement for scattering states but the exact Fourier dual of the finite-bandwidth transport window. The standard Landauer–Büttiker description uses stationary plane-wave scattering states and transmission probabilities in energy space. The packet construction instead resolves transport as explicit time evolution, with finite arrival times and transparent Pauli-induced temporal ordering, while remaining exactly equivalent to the stationary picture when the packet basis spans the full energy window.

In the limit of infinitesimally narrow energy windows, the packets delocalize and the basis reduces back to plane waves. The two descriptions are therefore mathematically dual rather than competing formulations.

5. Many-body packet states, Landauer current, and numerical decomposition

The many-body state for the zero-temperature source is a fully occupied train of orthogonal packets,

$$\{c_E,c_{E'}^\dagger\}=\delta(E-E'),\qquad \{c_E,c_{E'}\}=\{c_E^\dagger,c_{E'}^\dagger\}=0.$$3

Because the packet states are orthogonal fermionic modes,

$$\{c_E,c_{E'}^\dagger\}=\delta(E-E'),\qquad \{c_E,c_{E'}\}=\{c_E^\dagger,c_{E'}^\dagger\}=0.$$4

The mean current therefore reduces to a sum of individual-packet contributions,

$$\{c_E,c_{E'}^\dagger\}=\delta(E-E'),\qquad \{c_E,c_{E'}\}=\{c_E^\dagger,c_{E'}^\dagger\}=0.$$5

For parabolic dispersion, $$\{c_E,c_{E'}^\dagger\}=\delta(E-E'),\qquad \{c_E,c_{E'}\}=\{c_E^\dagger,c_{E'}^\dagger\}=0.$$6, and the expression becomes the standard probability-current formula summed over packets. Using the temporal comb and the exact cancellation of velocity factors yields

$$\{c_E,c_{E'}^\dagger\}=\delta(E-E'),\qquad \{c_E,c_{E'}\}=\{c_E^\dagger,c_{E'}^\dagger\}=0.$$7

so the conductance quantum is $$\{c_E,c_{E'}^\dagger\}=\delta(E-E'),\qquad \{c_E,c_{E'}\}=\{c_E^\dagger,c_{E'}^\dagger\}=0.$$8 (Biborski, 17 Jun 2026).

Within this formulation, the steady direct current is not imposed by hand. It emerges when the packet train is fully occupied and the observation time averages over many packets. The train is described as a discrete sampling of the transport window at the Nyquist rate set by the bias,

$$\{c_E,c_{E'}^\dagger\}=\delta(E-E'),\qquad \{c_E,c_{E'}\}=\{c_E^\dagger,c_{E'}^\dagger\}=0.$$9

As the number of packets increases, the current fluctuations shrink and the packet current converges to the constant Landauer value; the reported numerical deviation scales roughly like $[0,eV]$0, where $[0,eV]$1 is the number of packets used in the train.

The same framework is generalized by partitioning the total energy axis into disjoint windows $[0,eV]$2. Each window has its own spacing $[0,eV]$3 and packet operators $[0,eV]$4 satisfying

$[0,eV]$5

This yields an orthogonal decomposition of Fock space,

$[0,eV]$6

At $[0,eV]$7, the decomposition is exact and invariant under refinement of the energy bins; at finite temperature, it becomes a controlled coarse-graining of the Fermi–Dirac distribution, with packet occupations weighted by the thermal factor. Packet propagation can then be carried out independently in each sub-band, typically using Crank–Nicolson time evolution with complex absorbing potentials at the boundaries, after which observables are assembled by summing packet contributions. The paper identifies this structure as highly parallelizable and well suited to high performance computing. It also uses single-packet dynamics to extract resonance lifetimes in double-barrier structures, transmission coefficients $[0,eV]$8, and transient NESS formation (Biborski, 17 Jun 2026).

6. Related bases, algebraic analogues, and limitations

A distinct but related notion of orthogonal fermionic wave packets appears in the Wilson–Wannier basis for interacting fermions. There, the objective is not a time-resolved transport train but an orthogonal, phase-space localized wave packet basis adapted to low-energy states with short-range order. Starting from translated and modulated packets

$[0,eV]$9

one obtains an overcomplete tight frame when $n$0, and an orthonormal Wilson–Wannier basis when $n$1 by combining packets at $n$2 with definite parity. The resulting states are localized near $n$3, have momentum content concentrated near $n$4, and satisfy

$n$5

The basis is designed for systems in which important correlations are short-ranged in real space but tied to low-energy regions near the Fermi surface; the recommended regime is $n$6, with $n$7 the relevant pair size (Ossadnik, 2013).

In that setting, the orthogonal wave packet basis is used to analyze particle-particle pairing, particle-hole pairing, and more general short-range order. The transformed kinetic energy becomes local hopping on a coarse-grained $n$8-lattice, the interaction becomes short-ranged in the coarse coordinate, and in the half-filled one-dimensional Hubbard chain the renormalized Hamiltonian can be reduced to an effective Heisenberg chain. This suggests a different organizing principle from the transport construction: phase-space localization tailored to correlation length rather than temporal ordering tied to a bias window.

A further conceptual analogue is provided by fermionic quantum orthogonalization in filtered real algebras. In that framework, orthogonalization is formulated not for explicit Hilbert-space wave packets but for tuples of algebra elements organized into Clifford systems. The generalized Gram–Schmidt procedure is defined through the polarization map

$n$9

with recursive orthogonalization

$t_n$0

The framework also includes symmetric procedures, such as connection-based iterative schemes of the form

$t_n$1

and is explicitly described as an abstract algebraic analogue of orthogonalizing fermionic modes or states rather than a ready-made algorithm for physical wave packets (Lakos, 2015).

This distinction is important for scope. The Clifford-algebraic formalism does not define fermionic wave packets as functions on space, does not discuss localization or Fourier transforms, and does not give a Hilbert-space construction of packet orthogonality. Its relevance is structural: because Clifford systems encode fermionic anticommutation, orthogonalizing a tuple into a Clifford system is analogous to constructing orthogonal fermionic generators. The Wilson–Wannier basis, by contrast, does define explicit phase-space localized states, but its simplest treatments use leading-order gradient expansions, often keep only packet states near the Fermi points, and are developed in detail only in one dimension. Across these variants, “orthogonal fermionic wave packets” therefore names a family of constructions with a shared orthogonality principle but different physical targets: transport events, phase-space coarse-grained quasiparticles, or abstract fermionic mode generators.