Real-Space First-Quantised Representation

• Real-space first-quantised representation is a framework that expresses quantum systems directly in position space, bypassing second quantisation and enabling detailed analysis of spacetime entanglement.
• It employs grid discretizations and finite-difference approximations to simulate multi-particle interactions efficiently, supporting quantum simulations in chemistry, field theory, and condensed matter.
• The approach underpins advanced numerical algorithms such as adaptive grids, block-encoding, and worldline path integrals, reducing computational complexity and clarifying topological responses.

A real-space first-quantised representation refers to the formulation and manipulation of quantum systems (particles, fields, or materials) directly in terms of position-space wavefunctions, with all quantum degrees of freedom represented as operator-valued quantities acting on real-space or grid-based states. Unlike second quantisation—where fields are elevated to operator status and Fock space is used—this paradigm treats particles as primary and operates directly in coordinate (and, if relevant, time) space. Such representations underpin a wide range of contemporary theoretical, computational, and algorithmic advances, spanning relativistic quantum simulation, quantum chemistry, quantum field theory, topological matter, and quantum information processing.

1. Fundamental Formalism: Position, Time, and Operator Structure

In real-space first-quantised frameworks, both position and time are promoted to operator status. For relativistic electrons, the central object is the parametrized Dirac wavefunction $\psi(x, \tau)$, where $x^{\mu}$ ($\mu=0,1,2,3$) denotes spacetime coordinates and $\tau$ is an invariant evolution parameter. The governing equation is the parametrized Dirac equation: $$\left(\frac{\hbar}{ic}\right)\frac{\partial\psi}{\partial\tau} + \gamma^\mu \left[\frac{\hbar}{i}\frac{\partial}{\partial x^\mu} - \frac{e}{c}A_\mu\right]\psi = 0,$$ which, in natural units ($c = \hbar = 1$), becomes

$$i^{-1} \frac{\partial\psi}{\partial\tau} + \gamma^\mu (i^{-1} \partial_\mu - e A_\mu) \psi = 0.$$

Position and time operators satisfy $[x^\mu, p^\nu] = i g^{\mu\nu}$, with $p_\mu = (1/i) \partial/\partial x^\mu$ and allow superpositions with indefinite rest mass, bypassing the usual restriction to fixed-mass sectors. This enhanced flexibility enables descriptions of spacetime entanglement and localization beyond the Newton–Wigner paradigm (Bennett, 2014).

Grid-based representations use discrete coordinate eigenkets $|x_n\rangle$ (with $x_n = n a$) as basis states, and finite-difference approximations for derivative operators (e.g., momentum and Laplacians) rest on closed-form commutator-derived matrix elements, ensuring systematic convergence as the grid is refined and the representation order is increased (Ninno et al., 2017). Such discretizations underlie digital quantum simulation and efficient tensor representations on both classical and quantum hardware.

2. Multiparticle States, Entanglement, and Bound-State Equations

Many-particle systems in the first-quantised real-space approach are formulated by extending the Hilbert space to tensor products (or, for fermions, their antisymmetrized subspaces) of single-particle real-space wavefunctions. Explicitly, two-particle states obey

$$(1/i) \frac{\partial \Psi}{\partial\tau} + [\pi(x) \otimes I_4 + I_4 \otimes \pi(y)] \Psi = 0,$$

with generalized momentum operators $$\pi^\mu(z) = (1/i)\partial/\partial z_\mu - e A_\mu(z)$$ for $z \in \{x, y\}$ and suitable antisymmetrization for indistinguishable fermions (Bennett, 2014).

The formalism enables natural construction of spacetime-entangled states, including entanglement in time, and yields propagators and Møller operators for the construction and evolution of many-body states. By concatenation (integration over $\tau$ or other invariant parameters), the standard inhomogeneous Bethe–Salpeter equation for relativistic bound states is recovered: $$\Psi(x', y', m) = \Psi_0(x', y', m) + \iint d^4x'' d^4y'' \, \Gamma^0(x'-x'', y'-y'', m) V(x'', y'') \Psi(x'', y'', m),$$ where $\Gamma^0$ is a two-particle influence function (propagator) (Bennett, 2014).

Such an approach seamlessly connects first-quantised, real-space representations to precise calculations of physical spectra and scattering processes, supporting the analysis of correlated dynamics without explicit field quantisation or second-quantised formalism.

3. Quantum Simulation and Algorithmic Strategies

First-quantised, real-space simulation methods are critical in quantum algorithm design, particularly for quantum chemistry and many-body physics. The state of an $\eta$-particle system is encoded as a vector in a high-dimensional Hilbert space corresponding to an $\eta D$-dimensional spatial grid, with each register denoting a coordinate bin. The Hamiltonian is discretised on this grid: $H = T + V,$ where $T$ is approximated by high-order finite-difference formulas and $V$ is diagonal in the grid basis. Singularities in $V$ (e.g., Coulomb potential) are regularised by cutoffs. Time evolution is performed via a truncated Taylor series expansion, decomposing $H$ into a linear combination of unitaries (LCU). This results in scaling for the simulation of pairwise interactions as $\tilde{O}(\eta^2)$ in fixed-grid regimes, a substantial improvement over earlier methods (Kivlichan et al., 2016).

Efficient resource scaling depends on grid spacing, regularity assumptions, and the smoothness of the wavefunction; in favorable scenarios, exponential quantum speedups are retained, although worst-case performance can be sensitive to highly localized or singular states. The approach is specifically suited to position-basis quantum computing, avoiding the computational bottlenecks and memory overhead of second-quantised (Fock space) methods.

4. Quantum Chemistry and Reduced Complexity via Real-Space Representation

Recasting many-electron quantum chemistry problems in a real-space first-quantised framework can yield dramatic reductions in computational scaling for Hartree–Fock, Møller–Plesset, and coupled cluster methods. For example, the Hartree–Fock exchange energy becomes

$$E_\mathrm{HF} = \iint \frac{|\gamma(r_1, r_2)|^2}{|r_1 - r_2|} dr_1 dr_2,$$

with density matrix evaluation and grid quadrature reducing the scaling to $O(N^3)$ for the entire HF procedure (where $N$ is system size) (Mardirossian et al., 2017). Similar reductions occur for MP2 and certain coupled cluster diagrams, leveraging fast Fourier transforms and efficient grid convolution routines.

This does not introduce additional approximations—rather, computational savings are realized by transferring complex orbital contractions to efficient real-space operations. The representation enables exact treatments of interaction energies and response properties, bridging the gap between wavefunction-based and density-functional approaches for large-scale electronic structure calculations.

5. Quantum Field Theory, Path Integrals, and Topological Invariants in Real Space

The first-quantised real-space representation underpins alternative formulations of quantum field theory via the worldline (particle path integral) formalism. Here, amplitudes and effective actions are written as functional integrals over space–time trajectories, incorporating spin and gauge degrees of freedom via extended supersymmetry, auxiliary Grassmann or color fields, and operator-valued geometric representations. For example, the worldline propagator for a particle in a background field is given by

$$D^{x'x}[A] = \int_0^\infty dT \, e^{-m^2 T} \int_{x(0)=x}^{x(T)=x'} \mathcal{D}x(\tau) \, e^{-\int_0^T d\tau \left[\frac{1}{4}{\dot{x}}^2(\tau) + i e \dot{x}(\tau) \cdot A(x(\tau))\right]},$$

which unifies the calculation of Feynman amplitudes, including spin factors, color (via worldline fields), and non-commutative extensions (Edwards et al., 2019).

Topological invariants (e.g., winding number, first and second Chern numbers, Berry phase, polarization) are re-expressed in real-space via projector and commutator formulas: $$c_1 = (2\pi/i) \ \text{tr}\, P [X_\mu, P][X_\nu, P], \qquad c_2 = \frac{1}{2!(2\pi/i)^2} \text{Tr} \, \epsilon^{\mu\nu\rho\sigma} P \Pi_\mu P \Pi_\nu P \Pi_\rho P \Pi_\sigma P,$$ making these invariants robust to disorder and open boundary conditions (Shiina et al., 20 Feb 2025, Lin et al., 2021, Onaya et al., 21 Mar 2025). Such representations facilitate the computation and interpretation of topological responses and multipole moments even in the absence of translational invariance.

6. Numerical and Computational Implementations

Real-space, first-quantised algorithms admit a variety of adaptive and efficiently implementable discretizations:

• Adaptive, molecule-centered grids (e.g., Voronoi and Becke–Lebedev schemes) concentrate sampling in high-electron-density regions, minimizing the qubit and gate overhead by avoiding oversampling low-density domains and precisely resolving wavefunction cusps near nuclei. The discrete Laplacian and kinetic operators are constructed via finite-volume integration, with geometric weights reflecting Voronoi tessellation (Feniou et al., 28 Jul 2025).
• Transcorrelated (similarity-transformed) Hamiltonians are used to regularize Coulomb singularities, yielding cusp-free wavefunctions and reducing the grid resolution needed for chemical accuracy. The transformed Hamiltonian

$$\tilde{H} = e^{-\tau} H\, e^{\tau} = H + [H, \tau] + \frac{1}{2} [[H, \tau], \tau]$$

is non-Hermitian, but isospectral, and is handled via generalized quantum eigenvalue estimation (QEVE) protocols (Feniou et al., 28 Jul 2025).

• Block-encoding techniques are essential for mapping discretized Hamiltonians (including nonlocal pseudopotentials, as in the GTH scheme) to quantum circuits suitable for quantum phase estimation. Non-cubic cell geometries require coherent arithmetic for generalized reciprocal lattice representations, norm computation via Gramian matrices, and optimized state preparation (Berry et al., 2023).
• Finite-difference and translation-operator constructions for momentum and kinetic energy allow relativistic kinetic operators to be approximated perturbatively and encoded efficiently in variational circuits, compatible with both periodic and Dirichlet boundary conditions (Joo et al., 26 Sep 2025).

7. Broader Theoretical and Phenomenological Implications

Real-space first-quantised frameworks provide deep insight into the role of quantum symmetry, configuration space structure, and their classical limits. Noncommutative geometries, intrinsic in systems governed by extended Heisenberg–Weyl algebras, emerge naturally as representations of quantum symmetry; the contraction to classical spaces occurs via well-defined limiting procedures, recovering familiar Newtonian configuration or phase spaces (Chew et al., 2016). This links quantum mechanics, quantum gravity, and quantum field theory, offering routes to model quantum spacetime and suggests possible resolutions to foundational issues such as the cosmological constant problem by positing distinct quantization scales for matter and geometry (Lake, 2020).

In addition, the real-space approach is robust in inhomogeneous and disordered environments, facilitating both numerical studies and potential experimental probes (e.g., via cold atoms, photonics, or condensed matter systems), and supporting physically transparent calculation of entanglement, multipole moments, and other observables relevant to quantum information theory, topological phases, and materials science.

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The real-space first-quantised representation unifies the treatment of particles, fields, and correlated quantum systems directly in the position basis, leveraging covariant operators, grid discretizations, and projectors to support transparent physics, advanced simulation algorithms, and the computation of observable and topological quantities with high accuracy and broad applicability.