⚛️ quantum physics

Position operators in terms of converging finite-dimensional matrices: Exploring their interplay with geometry, transport, and gauge theory

This paper proposes a convergent finite-dimensional matrix representation (CRM) for the position operator to resolve the divergence issues inherent in traditional representations, thereby clarifying its distinct conceptual nature from spin matrices, its relationship with the Berry connection and Bloch space, and its implications for transport theory and gauge invariance.

Original authors: B. Q. Song, J. D. H. Smith, J. Wang

Published 2026-02-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: B. Q. Song, J. D. H. Smith, J. Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The "Infinite" Position

Imagine you are trying to measure the exact location of an electron in a crystal (a solid material like silicon or graphene). In the world of quantum mechanics, we usually describe things using matrices (grids of numbers) or wave functions (smooth curves).

For a long time, physicists have been stuck on a specific problem: The Position Matrix.

When physicists try to write down the position of an electron as a grid of numbers (a matrix), the numbers on the diagonal (the main line from top-left to bottom-right) blow up to infinity. It's like trying to weigh a feather on a scale that breaks immediately because the number is too big.

This "divergence" (infinity) causes a mess. It makes it impossible to calculate things like electric currents or how electrons move when hit by light. It's like trying to build a house on a foundation that is constantly sinking into a black hole.

The Old Way vs. The New Way

The Old Way (The Divergent Matrix):
Think of the old method as trying to describe a continuous ocean using a single, giant bucket. You try to scoop up the whole ocean at once. Because the ocean is infinite, your bucket overflows, and you get a mess. The math says the position operator is $i\partial_p$ (a derivative), but when you force this into a matrix, it breaks.

The New Way (The Convergent Matrix):
The authors of this paper say: "Stop trying to scoop the whole ocean at once. Let's use a different tool."

They propose a Convergent r-Matrix (CRM). Instead of one giant, broken grid, they use a series of smaller, manageable grids that fit together perfectly. The numbers in these grids are finite and well-behaved.

The Core Idea: Changing the "Room"

To understand how they fixed it, imagine you are trying to describe a dance.

The Old Room (Bloch Space): Imagine a dance floor that is infinitely large. The dancers (electrons) can be anywhere. If you try to count their positions, the numbers get too big. The floor is too big to measure.
The New Room (Quotient Space): The authors realize that while the dance floor is infinite, the pattern of the dance repeats. They create a smaller, finite room that captures the essence of the dance without the infinite size.

They use a mathematical "projection" (like a shadow) to map the infinite dance floor onto this smaller, finite room. In this new room, the position numbers stop being infinite and start being normal, calculable numbers.

The "Ribbon" Analogy

The paper introduces a cool concept called a "Ribbon."

Standard Matrices (like Spin): Imagine a stiff ruler. If you rotate the ruler, the numbers on it change in a predictable, rigid way. This is how we usually handle things like "spin" (a type of quantum rotation).
Position Matrices (Differential Operators): Imagine a stretchy rubber band or a ribbon. If you twist or rotate a ribbon, it doesn't just change numbers; it stretches and bends. The math for position is like this ribbon. It has a "memory" of how it was twisted.

The authors realized that for a long time, physicists were treating the "ribbon" (position) like a "stiff ruler" (spin). That's why the math was breaking. You can't treat a stretchy ribbon like a rigid ruler. You have to account for the stretching (the derivative) when you rotate it.

Why Does This Matter? (The "Transport" Problem)

Why should you care if a math formula is infinite or not? Because it affects how we understand electricity and light.

The Current Problem: When light hits a crystal, it can knock electrons loose, creating a current. Physicists have different formulas to calculate this current (Shift Current, Adiabatic Current, etc.).
The Confusion: Because the position math was broken (infinite), these formulas were often guesses or approximations. Some said the current was a "single particle" effect; others said it was a "many-particle" effect. They couldn't agree because the foundation was shaky.
The Solution: With the new Convergent Matrix, the authors show that these different currents are actually connected. They prove that what looks like a single-particle jump at high speeds is actually part of a larger, coordinated dance of many particles at low speeds.

It's like realizing that a single drop of water (high speed) and a flowing river (low speed) are made of the same stuff, just viewed differently. The new math unifies them.

The "Gauge" Secret

The paper also talks about Gauge Invariance. Think of this as a rule that says: "The physics shouldn't change just because you change your coordinate system (like switching from miles to kilometers)."

The authors found that for position, this rule is very fragile. If you miss even one electron in your calculation, the whole "gauge" breaks, and the math becomes nonsense. This explains why some transport theories are so sensitive to particle numbers. The new matrix helps us see exactly why this happens and how to fix it.

Summary: The Takeaway

The Problem: The old way of writing down the position of an electron in a crystal resulted in "infinity," making calculations impossible and confusing.
The Fix: The authors created a new way to write the position matrix using finite, converging numbers. They did this by changing the mathematical "room" the electrons live in, mapping the infinite to the finite.
The Insight: Position isn't a rigid ruler; it's a stretchy ribbon. You have to treat it differently than other quantum properties like spin.
The Result: This new math unifies different theories of how electricity flows in crystals. It helps us understand that electrons in a crystal act as a coordinated team, not just independent individuals, and provides a solid foundation for designing future electronics and understanding topological materials.

In short: They fixed the broken ruler so we can finally measure the quantum world accurately.

1. Problem Statement

The paper addresses a fundamental inconsistency in the matrix representation of the position operator ( $\hat{r}$ ) in solid-state physics, particularly within the context of Bloch bases and transport phenomena.

The Divergence Issue: In wave mechanics, $\hat{r}$ is represented as $i\partial_p$ . However, when transformed into a matrix form (e.g., under a Bloch basis $|\psi_{n,k}\rangle$ ), the diagonal elements diverge. This "Divergent r-matrix" (DRM) arises because the position operator acts on an unbounded continuous spectrum, leading to infinite norms and ill-defined expectation values $\langle \hat{r} \rangle$ .
Theoretical Gap: Standard quantum mechanics relies on the commutation relation $[\hat{r}, p] = i\hbar$ $[\overset{r}{^}, p] = i ℏ$ (Weyl algebra). However, it is a known mathematical fact that Weyl algebras admit no finite-dimensional matrix representations. Despite this, the DRM is widely used in literature (e.g., for calculating Berry curvature and shift currents) by heuristically substituting $\hat{r} \to i\partial_k$ $\overset{r}{^} \to i \partial_{k}$ . This substitution leads to logical gaps:
- The Bloch space ( $H_B$ ) is shown to be incomplete for $\hat{r}$ .
- The distinction between a "matrix operator" (like spin) and a "differential operator" (like position) is blurred.
- Observable extraction (e.g., adiabatic current) relies on ambiguous procedures that often require ad-hoc renormalization or cutoffs.

2. Methodology

The authors propose a rigorous mathematical framework to resolve the divergence by redefining the spaces and algebras involved.

Extension to N-th Weyl Algebra ( $A_N$ ): Instead of using the standard 1st Weyl algebra ( $A_1$ , where $\hat{r} \to i\partial_k$ ), the authors introduce the N-th Weyl algebra. They define the position operator as a sum over $N$ variables:
$\hat{r} = \frac{1}{N} \sum_{m=1}^N i \frac{\partial}{\partial k_m}$
This algebraic modification allows for a finite-dimensional treatment.
Space Decomposition (Isomorphism): The authors introduce a projection map $\Pi$ $Π$ that decomposes the infinite-dimensional Bloch space ( $H_B$ $H_{B}$ ) into a tensor product of two finite-dimensional quotient spaces:
$H_B \cong V \otimes E$
- $V$ : A quotient space associated with band indices (dimension $N$ , the number of bands).
- $E$ : A quotient space associated with crystal momentum $k$ (dimension $N$ , related to the discretization of the Brillouin zone).
- This mapping replaces the continuous coordinate integration with discrete inner products, effectively "folding" the infinite space into a well-behaved finite-dimensional manifold.
Ribbon Band Formalism: To handle the transformation properties of differential operators, the authors introduce the concept of "ribbon bands" (maps from the Brillouin zone manifold $K$ $K$ to the vector space $V$ $V$ ). This distinguishes between:
- Matrix Operators: Transform via similarity transformations ( $U O U^\dagger$ ).
- Differential Operators: Transform via inhomogeneous rules involving derivatives of the transformation matrix ( $U O U^\dagger + U i\partial_k U^\dagger$ ).

3. Key Contributions

A. Derivation of the Convergent r-Matrix (CRM)

The primary result is the derivation of the Convergent r-Matrix (CRM). Unlike the DRM, the CRM is finite in both diagonal and off-diagonal elements.

The matrix elements are defined on the quotient space $V$ rather than the divergent Bloch space.
The CRM is not a representation of the Weyl algebra (it does not satisfy $[\hat{r}, k] = i$ ), but rather a procedure for encoding the operator $\hat{r}$ into finite dimensions.
The CRM resolves the divergence by separating the "mass center" of the crystal (a constant term $\bar{R}$ ) from the relative position shifts, ensuring that physical observables (which depend on differences) are well-defined.

B. Clarification of Geometric Quantities

The paper rigorously establishes that geometric quantities like the Berry connection and Berry curvature are defined on the quotient space $V$ (spanned by periodic functions $u_{n,k}$ ), not on the full Bloch space $H_B$ .

This clarifies why the Berry connection is often calculated using $u_{n,k}$ rather than $\psi_{n,k}$ : the latter space is incomplete for $\hat{r}$ , while the former is isomorphic to the finite-dimensional space where the CRM lives.

C. Distinction Between Matrix and Differential Operators

The authors formalize the difference between matrix operators (e.g., spin) and differential operators (e.g., position):

Hermiticity: While $\hat{r} = \hat{r}^\dagger$ holds for the operator, the matrix relation $r_{m,n} = r^*_{n,m}$ does not imply the operator is associative in the same way. The paper proves $r_{m,n} = r^*_{n,m} \nLeftrightarrow \hat{r} = \hat{r}^\dagger$ in the context of basis-free notation.
Transformation Rules: Differential operators obey a non-homogeneous transformation rule (Leibniz rule), which introduces an extra term under gauge/ribbon transformations. This explains why standard "basis-free" bra-ket notation often fails for position operators.

D. Unified Gauge Theory and Transport

The paper redefines Gauge Transformation ( $T_G$ ) and Gauge Invariance in the context of transport:

Gauge transformations are shown to be induced by Ribbon Transformations ( $T_R$ ).
Observables: The paper argues that for differential operators, observables cannot be extracted simply by taking diagonal elements (as with spin). Instead, they must be extracted via functionals that integrate over the Brillouin zone (e.g., $\oint r_{n,n}(k) dk$ ) to ensure $U(1)$ gauge symmetry.
This resolves the ambiguity in defining currents (shift current vs. adiabatic current) by showing they correspond to different limits of a unified theory based on the CRM.

4. Key Results

Non-Existence of DRM Representation: It is mathematically proven that no matrix representation of the Weyl algebra exists; thus, the DRM is fundamentally flawed as a representation, not just a divergent approximation.
Incompleteness of Bloch Space: The Bloch space $H_B$ is proven to be incomplete for the position operator $\hat{r}$ because the dimension of the space spanned by position eigenstates (uncountably infinite) exceeds that of the Bloch basis (countably infinite).
CRM vs. DRM: The CRM is distinct from the DRM. Taking the limit $N \to \infty$ in the CRM does not recover the DRM; they are fundamentally different mathematical objects.
Berry Curvature Validity: The celebrated Berry curvature formula for adiabatic currents is shown to be conditionally true, relying on the integration over a closed manifold (the Brillouin zone) to cancel non-invariant terms. Locally, the formula is not gauge invariant for differential operators.
N-Particle Nature of Transport: The analysis suggests that adiabatic currents are inherently N-particle phenomena (where N is the number of bands/unit cells), rather than single-particle effects, due to the necessity of a closed topology for gauge invariance.

5. Significance and Impact

Foundational Correction: The paper corrects a long-standing logical gap in condensed matter physics regarding the matrix representation of position. It moves the field from heuristic, divergent calculations to a rigorous, convergent framework.
Unified Transport Theory: By resolving the divergence, the authors provide a pathway to unify disparate transport mechanisms (shift current, injection current, adiabatic current) under a single theoretical umbrella based on the CRM and gauge symmetry principles.
Clarification of Geometry: It rigorously links the geometric interpretation of electronic states (Berry phase, curvature) to the specific quotient space $V$ , clarifying the role of periodic functions $u_{n,k}$ versus full Bloch waves.
Methodological Shift: The introduction of "ribbon bands" and the distinction between matrix and differential operators offers a new language for handling gauge theories and transport in topological materials, potentially impacting the interpretation of ultrafast experiments and topological responses in THz regimes.

In summary, this work replaces the ill-defined, divergent matrix representation of position with a convergent, finite-dimensional matrix (CRM) derived from the N-th Weyl algebra. This resolves mathematical inconsistencies, clarifies the geometric nature of transport phenomena, and establishes a rigorous foundation for extracting physical observables in crystals.