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Countable basis for free electromagnetic fields

Ivan Fernandez-Corbaton

TL;DR

This work establishes a countable, physically meaningful basis for the free Maxwell-field Hilbert space $\mathbb{M}$ by constructing basis vectors $\ket{n j m \lambda}$ as simultaneous eigenstates of four commuting operators, yielding a separable Hilbert space isomorphic to $\ell^2$ and a polychromatic, single-photon energy spectrum $E_n=n\hbar c_0 k_0$. It unifies two prior bases (Mack–Todorov and Kastrup–Mayer) through the conformal-group structure, resolves dimensional inconsistencies by introducing a fixed length scale $k_0^{-1}$ (leading to $N=\hbar(P_0/k_0+k_0K_0)$ and $R=\exp(-i\pi(P_0/k_0+k_0K_0)/2)$), and analyzes the resulting basis properties: orthonormality, completeness, energy quantization, and smooth, wavelet-like time behavior with distinct incoming/outgoing components. The framework provides dimensionally consistent expressions for plane-wave and multipolar expansions and yields a practical, discretized approach to polychromatic light–matter interactions, potentially simplifying $S$/$T$-matrix calculations by replacing double frequency integrals with discrete sums. This countable basis thus strengthens both the theoretical foundation and computational toolbox for polychromatic Maxwell-field descriptions in scattering theory.

Abstract

Polychromatic electromagnetic fields are typically expanded as integrals over monochromatic fields, such as plane waves, multipolar fields, or Bessel beams. However, monochromatic fields do not belong to the Hilbert space of free Maxwell fields, since their norms diverge. Moreover, the continuous frequency integrals involved in such expansions complicate the treatment of light--matter interactions via the scattering operator. Here, we identify and study a polychromatic basis for free Maxwell fields whose basis vectors belong to the Hilbert space. These vectors are defined as simultaneous eigenstates of four commuting operators with integer eigenvalues. As a consequence, the basis set is countable, and the Hilbert space is separable and isomorphic to $\ell^2$, the Hilbert space of square-summable sequences. Each basis vector represents a polychromatic single-photon wave with quantized energy and a wavelet--like temporal dependence. Three versions of this basis are defined: Regular, incoming, and outgoing. The fields of the regular basis are smooth in both space and time. The incoming and outgoing fields are likewise smooth, except at the spatial origin. These results support and motivate the use of countable bases for both the theoretical description and the practical computation of light--matter interactions.

Countable basis for free electromagnetic fields

TL;DR

This work establishes a countable, physically meaningful basis for the free Maxwell-field Hilbert space by constructing basis vectors as simultaneous eigenstates of four commuting operators, yielding a separable Hilbert space isomorphic to and a polychromatic, single-photon energy spectrum . It unifies two prior bases (Mack–Todorov and Kastrup–Mayer) through the conformal-group structure, resolves dimensional inconsistencies by introducing a fixed length scale (leading to and ), and analyzes the resulting basis properties: orthonormality, completeness, energy quantization, and smooth, wavelet-like time behavior with distinct incoming/outgoing components. The framework provides dimensionally consistent expressions for plane-wave and multipolar expansions and yields a practical, discretized approach to polychromatic light–matter interactions, potentially simplifying /-matrix calculations by replacing double frequency integrals with discrete sums. This countable basis thus strengthens both the theoretical foundation and computational toolbox for polychromatic Maxwell-field descriptions in scattering theory.

Abstract

Polychromatic electromagnetic fields are typically expanded as integrals over monochromatic fields, such as plane waves, multipolar fields, or Bessel beams. However, monochromatic fields do not belong to the Hilbert space of free Maxwell fields, since their norms diverge. Moreover, the continuous frequency integrals involved in such expansions complicate the treatment of light--matter interactions via the scattering operator. Here, we identify and study a polychromatic basis for free Maxwell fields whose basis vectors belong to the Hilbert space. These vectors are defined as simultaneous eigenstates of four commuting operators with integer eigenvalues. As a consequence, the basis set is countable, and the Hilbert space is separable and isomorphic to , the Hilbert space of square-summable sequences. Each basis vector represents a polychromatic single-photon wave with quantized energy and a wavelet--like temporal dependence. Three versions of this basis are defined: Regular, incoming, and outgoing. The fields of the regular basis are smooth in both space and time. The incoming and outgoing fields are likewise smooth, except at the spatial origin. These results support and motivate the use of countable bases for both the theoretical description and the practical computation of light--matter interactions.
Paper Structure (13 sections, 45 equations, 3 figures, 1 table)

This paper contains 13 sections, 45 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction and summary
  2. The electromagnetic Hilbert space and the conformal group
  3. Two countable basis for $\mathbb{M}$ are equivalent
  4. New expressions for the countable basis
  5. Dimensional discrepancies in the existing expressions
  6. Resolution of the discrepancies
  7. Properties of the $\ket{njm\lambda}$ in $\mathbb{M}$
  8. Zero static fields, smoothness, and incoming/outgoing wavelets
  9. Conclusion and outlook
  10. Expansions in plane waves and multipolar fields
  11. Conventions in previous expressions
  12. Orthonormality, completeness, and energy of $\ket{njm\lambda}$
  13. Smoothness of the $\ket{njm\lambda}$ fields

Figures (3)

  • Figure 1: Left panels: Multipolar expansion functions $c_{nj}(k)$ of some members of the countable basis $\ket{njm\lambda}$ for the $j=1,2$ and $3$ multipolar orders, and $k_0=1\per m$. Right panels (shared legend): Spectral densities ($\text{d}k$) of the same basis vectors. Top right: Photon density. Bottom right: Energy density.
  • Figure 2: The derivatives of any order of a smooth function are continuous. The graph shows three smooth functions, equal to zero outside the following specified intervals. Step: $\exp(-1/x)$ for $x>0$; bump: $\exp\left(1/(x^2-1)\right)$ for $|x|<1$; and flat-top: $\exp\left(1-1/(1-u^2)\right)$ for $u=\text{max}(0,|x|-1)<1$. The inset shows a zoomed view of the upper-right corner of the flat-top on logarithmic axes.
  • Figure 3: Faint lines: Time dependence of the $l=j$ component of the electric field of $\ket{njm\lambda}$ in Eq. (\ref{['eq:Ftext']}) at spatial radius $r=5$, denoted here by $c_{nj}(\mathrm{c_0} t,r=5)$. Towards negative times, the faint lines are covered over by more intense lines which graph the incoming part of $c_{nj}(\mathrm{c_0} t,r=5)$. Clearly, the incoming and outgoing parts of $c_{nj}(\mathrm{c_0} t,r=5)$ dominate the negative and positive times, respectively. The wavelet centered around $\mathrm{c_0} t=-5$ is almost entirely of incoming character, while the wavelet centered around $\mathrm{c_0} t=5$ is almost entirely of outgoing character. This matches the expected behavior of an incoming spherical wave propagating from spatial infinity towards the origin and then continuing as an outgoing spherical wave propagating outwards from the origin towards spatial infinity. The value $k_0=1\per m$ has been assumed.