Complex Dimensions in Fractal and Quantum Systems

• Complex Dimensions are complex-valued quantities that capture scaling laws and log-periodic oscillations in both fractal geometry and quantum physics.
• They arise as poles of meromorphically continued zeta functions in fractal settings and as complex scaling dimensions in operator renormalization.
• Their study offers practical insights into Minkowski measurability, heat-content asymptotics, and the dynamics of discrete scale invariance in diverse systems.

Complex dimensions are complex-valued quantities that encode scaling beyond a single real exponent. In fractal geometry, they are defined as the poles of meromorphic continuations of geometric, scaling, distance, or tube zeta functions; in quantum and field-theoretic settings, they appear as complex scaling or anomalous dimensions when continuous scale invariance is replaced by a discrete subgroup or when operator mixing becomes non-Hermitian. Across these settings, the real part governs the leading power law, while the imaginary part generates log-periodic oscillations, geometric spectra, or other discrete-scale effects (Dettmers et al., 2015, Moroz, 2010, Calcagni, 2017). A recurrent terminological ambiguity is that the same phrase denotes poles of zeta functions in fractal geometry and complex-valued scaling exponents in physics; the operational definitions differ, but the analytic role of the imaginary part is closely related.

1. Zeta-function definitions in fractal geometry

The classical fractal-geometric definition starts from a Dirichlet-type series. For a one-dimensional Cantor set, one may enumerate the intervals arising in its approximants and define the geometric zeta function

$\zeta(s)=\sum_{j=1}^\infty d_j^s,$

where $d_j$ is the length of the $j$th interval. In the middle-third case,

$$\zeta(s)=\sum_{n=0}^\infty 2^n(3^{-n})^s=\frac{1}{1-2\cdot 3^{-s}},$$

which extends meromorphically to $\mathbb C$ with simple poles

$$s_k=\frac{\ln 2+2\pi i k}{\ln 3},\qquad k\in\mathbb Z.$$

These poles are the complex dimensions (Sidorov, 2023).

For self-similar systems of similitudes $\Phi=\{\phi_1,\dots,\phi_m\}$ with scaling ratios $0<\lambda_\phi<1$, the scaling zeta function is

$$\zeta_\Phi(s)=\frac{1}{1-\sum_{\phi\in\Phi}\lambda_\phi^s}.$$

Its poles are the solutions of the Moran equation

$1=\sum_{\phi\in\Phi}\lambda_\phi^\omega,$

and the unique real solution $d_j$0 is the similarity dimension (Hoffer et al., 13 Aug 2025).

The same framework extends beyond Euclidean subsets. In a $d_j$1-regular Ahlfors metric measure space $d_j$2, Lapidus and Watson define the distance zeta function of a bounded set $d_j$3 by

$d_j$4

initially for $d_j$5 large. The upper box dimension $d_j$6 is the abscissa of absolute convergence, $d_j$7 is holomorphic on $d_j$8, and the visible complex dimensions in a domain $d_j$9 are the poles of the meromorphic continuation of $j$0 in $j$1 (Lapidus et al., 7 Apr 2025).

A related object is the tube zeta function

$j$2

linked to the distance zeta function by

$j$3

For $j$4, the two functions have the same poles, with the same multiplicities, in any common domain of meromorphic continuation (Lapidus et al., 7 Apr 2025).

2. Critical lines, measurability, and the lattice/nonlattice dichotomy

A central result of the theory developed by Lapidus and van Frankenhuijsen is that the structure of complex dimensions on the critical line controls Minkowski measurability. For a languid fractal string $j$5, the following are equivalent: $j$6 is the only pole of $j$7 on $j$8 and is simple; the counting function satisfies

$j$9

and the boundary of the associated bounded open set is Minkowski measurable. In that case,

$$\zeta(s)=\sum_{n=0}^\infty 2^n(3^{-n})^s=\frac{1}{1-2\cdot 3^{-s}},$$0

This makes complex dimensions a criterion for measurability, not merely a descriptive invariant (Dettmers et al., 2015).

For self-similar strings, the lattice/nonlattice dichotomy determines the pole pattern. In the lattice case, where the logarithms of the contraction ratios generate a discrete subgroup, the complex dimensions lie on finitely many vertical arithmetic progressions with oscillatory period $$\zeta(s)=\sum_{n=0}^\infty 2^n(3^{-n})^s=\frac{1}{1-2\cdot 3^{-s}},$$1. Hence there are infinitely many poles on $$\zeta(s)=\sum_{n=0}^\infty 2^n(3^{-n})^s=\frac{1}{1-2\cdot 3^{-s}},$$2, and the boundary is not Minkowski measurable. In the nonlattice case, $$\zeta(s)=\sum_{n=0}^\infty 2^n(3^{-n})^s=\frac{1}{1-2\cdot 3^{-s}},$$3 is the sole pole on $$\zeta(s)=\sum_{n=0}^\infty 2^n(3^{-n})^s=\frac{1}{1-2\cdot 3^{-s}},$$4, while the remaining poles lie to the left and accumulate toward the critical line; then the boundary is Minkowski measurable (Dettmers et al., 2015).

This dichotomy persists in higher-dimensional box-counting zeta functions. Under OSC or $$\zeta(s)=\sum_{n=0}^\infty 2^n(3^{-n})^s=\frac{1}{1-2\cdot 3^{-s}},$$5-disjointness, the renewal-equation approach yields a denominator of the form $$\zeta(s)=\sum_{n=0}^\infty 2^n(3^{-n})^s=\frac{1}{1-2\cdot 3^{-s}},$$6, so the pole set again reflects whether the logarithms of the contraction ratios are commensurable. In the lattice case one obtains infinite periodic pole families; in the nonlattice $$\zeta(s)=\sum_{n=0}^\infty 2^n(3^{-n})^s=\frac{1}{1-2\cdot 3^{-s}},$$7-disjoint case, the box-counting dimension is the only pole on its critical line and the set is box-counting measurable (Dettmers et al., 2015).

The nonlattice case nevertheless retains structure. The Lattice String Approximation algorithm approximates a nonlattice self-similar string by lattice strings whose periods are

$$\zeta(s)=\sum_{n=0}^\infty 2^n(3^{-n})^s=\frac{1}{1-2\cdot 3^{-s}},$$8

The associated stability theorem shows that zeros of the approximating lattice Dirichlet polynomial track the zeros of the original nonlattice polynomial in large regions of the complex plane. This yields a quasiperiodic pattern of complex dimensions rather than exact periodicity (Lapidus et al., 2020).

3. Extensions beyond disjoint self-similarity

The classical theory is often formulated under the Open Set Condition, but more recent work shows that complex dimensions remain meaningful when overlaps are present. Sidorov replaces the attractor by a push-down measure for an iterated function system

$$\zeta(s)=\sum_{n=0}^\infty 2^n(3^{-n})^s=\frac{1}{1-2\cdot 3^{-s}},$$9

with weights $\mathbb C$0, $\mathbb C$1, and defines a limiting zeta function from cylinder masses close to the expected size $\mathbb C$2. The resulting function $\mathbb C$3 is meromorphic on the half-plane $\mathbb C$4, holomorphic for $\mathbb C$5, and its set of poles coincides with the vertical strip $\mathbb C$6 (Sidorov, 2023).

This is a substantial departure from the pole lattices familiar from disjoint self-similar systems. In the overlapping setting, the pole set need not reduce to finitely many vertical lines; the strip $\mathbb C$7 can itself be the pole set. Sidorov also studies the boundary line $\mathbb C$8 through Fourier-type sums

$\mathbb C$9

whose limit $$s_k=\frac{\ln 2+2\pi i k}{\ln 3},\qquad k\in\mathbb Z.$$0 satisfies $$s_k=\frac{\ln 2+2\pi i k}{\ln 3},\qquad k\in\mathbb Z.$$1 for all real $$s_k=\frac{\ln 2+2\pi i k}{\ln 3},\qquad k\in\mathbb Z.$$2, is in $$s_k=\frac{\ln 2+2\pi i k}{\ln 3},\qquad k\in\mathbb Z.$$3, and is generally aperiodic (Sidorov, 2023).

An explicit example is the Bernoulli convolution with parameter $$s_k=\frac{\ln 2+2\pi i k}{\ln 3},\qquad k\in\mathbb Z.$$4, given by

$$s_k=\frac{\ln 2+2\pi i k}{\ln 3},\qquad k\in\mathbb Z.$$5

with equal weights $$s_k=\frac{\ln 2+2\pi i k}{\ln 3},\qquad k\in\mathbb Z.$$6. In that case,

$$s_k=\frac{\ln 2+2\pi i k}{\ln 3},\qquad k\in\mathbb Z.$$7

so the poles satisfy $$s_k=\frac{\ln 2+2\pi i k}{\ln 3},\qquad k\in\mathbb Z.$$8, equivalently

$$s_k=\frac{\ln 2+2\pi i k}{\ln 3},\qquad k\in\mathbb Z.$$9

The real part $\Phi=\{\phi_1,\dots,\phi_m\}$0 lies in the strip $\Phi=\{\phi_1,\dots,\phi_m\}$1, consistent with the general theorem (Sidorov, 2023).

A different generalization replaces Euclidean ambient space by Ahlfors regular metric measure spaces. There, the main Euclidean properties of the distance zeta function carry over: holomorphy on $\Phi=\{\phi_1,\dots,\phi_m\}$2, equality of the abscissa of convergence with $\Phi=\{\phi_1,\dots,\phi_m\}$3, and blow-up as $\Phi=\{\phi_1,\dots,\phi_m\}$4 when $\Phi=\{\phi_1,\dots,\phi_m\}$5, the Minkowski dimension exists, and the lower $\Phi=\{\phi_1,\dots,\phi_m\}$6-dimensional Minkowski content is positive. This prevents holomorphic continuation across the critical line under those hypotheses (Lapidus et al., 7 Apr 2025).

4. Oscillatory geometry, tube formulas, and heat content

Complex dimensions are not only singularities of zeta functions; they govern asymptotic expansions of geometric and analytic quantities. In self-similar domains with fractal boundary, the possible complex dimensions determined by the similitudes control the heat-content asymptotics. If $\Phi=\{\phi_1,\dots,\phi_m\}$7 is the total heat content for the Dirichlet problem on a bounded open region whose boundary is self-similar, then under admissibility conditions

$\Phi=\{\phi_1,\dots,\phi_m\}$8

where $\Phi=\{\phi_1,\dots,\phi_m\}$9 are residue coefficients and $0<\lambda_\phi<1$0 is the pole set of the scaling zeta function (Hoffer et al., 13 Aug 2025).

In this expansion, $0<\lambda_\phi<1$1 sets the power-law exponent and $0<\lambda_\phi<1$2 produces log-periodic oscillations. The lattice case yields poles on finitely many vertical lines and hence exact periodicity in $0<\lambda_\phi<1$3; the generic nonlattice case yields a quasiperiodic set dense in a strip. For generalized $0<\lambda_\phi<1$4-von Koch snowflakes, the relevant poles are the solutions of

$0<\lambda_\phi<1$5

where $0<\lambda_\phi<1$6. Each such pole contributes a term $0<\lambda_\phi<1$7 to the heat-content expansion (Hoffer et al., 13 Aug 2025).

A parallel bridge between geometry and complex dimensions is provided by generalized Steiner formulas and support measures. For a compact $0<\lambda_\phi<1$8, Hug–Last–Weil support measures $0<\lambda_\phi<1$9 lead to the shell functions

$$\zeta_\Phi(s)=\frac{1}{1-\sum_{\phi\in\Phi}\lambda_\phi^s}.$$0

and corresponding scaling exponents

$$\zeta_\Phi(s)=\frac{1}{1-\sum_{\phi\in\Phi}\lambda_\phi^s}.$$1

Radunović’s survey states that

$$\zeta_\Phi(s)=\frac{1}{1-\sum_{\phi\in\Phi}\lambda_\phi^s}.$$2

so the support-measure exponents refine but preserve the outer Minkowski dimension (Radunović, 5 Sep 2025).

The distance zeta function

$$\zeta_\Phi(s)=\frac{1}{1-\sum_{\phi\in\Phi}\lambda_\phi^s}.$$3

admits the exact decomposition

$$\zeta_\Phi(s)=\frac{1}{1-\sum_{\phi\in\Phi}\lambda_\phi^s}.$$4

Hence the poles of $$\zeta_\Phi(s)=\frac{1}{1-\sum_{\phi\in\Phi}\lambda_\phi^s}.$$5 coincide with the poles of those basic zeta functions whose abscissae of convergence attain the relevant scaling exponents. In the Sierpiński gasket example, one finds poles at

$$\zeta_\Phi(s)=\frac{1}{1-\sum_{\phi\in\Phi}\lambda_\phi^s}.$$6

recovering the classical nonreal complex dimensions and the associated fractal tube formula (Radunović, 5 Sep 2025).

5. Complex scaling dimensions in nonrelativistic quantum mechanics

In nonrelativistic quantum mechanics, complex dimensions arise from operator scaling rather than from zeta-function poles. For two particles in $$\zeta_\Phi(s)=\frac{1}{1-\sum_{\phi\in\Phi}\lambda_\phi^s}.$$7 spatial dimensions interacting through the inverse-square potential $$\zeta_\Phi(s)=\frac{1}{1-\sum_{\phi\in\Phi}\lambda_\phi^s}.$$8, the local $$\zeta_\Phi(s)=\frac{1}{1-\sum_{\phi\in\Phi}\lambda_\phi^s}.$$9-wave composite operator

$1=\sum_{\phi\in\Phi}\lambda_\phi^\omega,$0

has scaling dimensions obtained through the operator–state map: $1=\sum_{\phi\in\Phi}\lambda_\phi^\omega,$1 This follows from separating center-of-mass and relative motion in a harmonic trap and identifying the total trapped energy with the scaling dimension (Moroz, 2010).

The critical coupling is $1=\sum_{\phi\in\Phi}\lambda_\phi^\omega,$2. In the over-critical regime,

$1=\sum_{\phi\in\Phi}\lambda_\phi^\omega,$3

the square root becomes imaginary and one writes

$1=\sum_{\phi\in\Phi}\lambda_\phi^\omega,$4

The appearance of $1=\sum_{\phi\in\Phi}\lambda_\phi^\omega,$5 is directly tied to discrete scale invariance and to a geometric tower of bound states (Moroz, 2010).

The renormalized two-point function of the composite operator takes the closed form

$1=\sum_{\phi\in\Phi}\lambda_\phi^\omega,$6

where $1=\sum_{\phi\in\Phi}\lambda_\phi^\omega,$7 is a short-distance renormalization scale. Its poles yield the bound-state energies

$1=\sum_{\phi\in\Phi}\lambda_\phi^\omega,$8

with exact ratio

$1=\sum_{\phi\in\Phi}\lambda_\phi^\omega,$9

Thus the energies form an exact geometric progression accumulating at zero, and the operator $d_j$00 represents the entire tower (Moroz, 2010).

The physical mechanism is a quantum scale anomaly. Classically, $d_j$01 is scale invariant, but the singularity at $d_j$02 requires regularization and renormalization. The renormalization-group flow runs on a limit cycle rather than to a fixed point, breaking continuous scale invariance to the discrete subgroup $d_j$03 with $d_j$04. Higher angular-momentum sectors obey the analogous formula

$d_j$05

which becomes complex when $d_j$06 (Moroz, 2010).

6. Discrete scale invariance, complex spacetime dimensions, and nonunitary anomalous dimensions

A second physical usage of the term arises when a scale-invariant quantity obeys only a discrete dilation symmetry. If

$d_j$07

with fixed $d_j$08, then the Mellin transform leads to poles satisfying

$d_j$09

The corresponding inverse Mellin expansion produces a log-periodic Fourier series in $d_j$10, so the imaginary parts $d_j$11 act as complex dimensions in the sense of discrete scale invariance (Calcagni, 2017).

Calcagni, Rodríguez Fernández, and collaborators apply this structure to quantum gravity. In the ultraviolet, if spacetime geometry exhibits discrete scale invariance, then the Hausdorff and spectral dimensions acquire a complex part: $d_j$12 with $d_j$13 and $d_j$14 equal to an integer multiple of $d_j$15. The associated log oscillations can propagate into observables such as the primordial power spectrum, which at leading harmonic takes the form

$d_j$16

The cited summary reports that Planck 2015 and related analyses constrain the leading amplitude to $d_j$17 at $d_j$18 confidence level for frequencies $d_j$19–$d_j$20, with no strong detection but some modest improvement of fit in some cases (Calcagni, 2017).

A conceptually distinct route to complex dimensions appears in non-integer-dimensional gauge theory through evanescent operators. Jin and collaborators compute operator norms and one-loop renormalization matrices in Yang–Mills theory using on-shell form factors. In pure YM at canonical dimension $d_j$21 and length $d_j$22, the norm of a $d_j$23-type evanescent operator contains a factor $d_j$24, so it becomes negative for $d_j$25. In the same $d_j$26-even $d_j$27 evanescent subsector, the one-loop dilatation matrix has a complex-conjugate eigenvalue pair

$d_j$28

and a length-5 sector similarly yields

$d_j$29

The paper interprets these results as evidence that general gauge theories are non-unitary in non-integer spacetime dimensions (Jin et al., 2023).

The physical interpretation differs from the zeta-function setting, but the analytic signature is parallel. Complex anomalous dimensions imply power-law behavior modulated by logarithmic oscillations, while the indefinite Gram matrix and negative-norm states indicate violation of positivity. In the strict integer-dimensional limit, the evanescent sector decouples; away from that limit, the scaling spectrum becomes genuinely complex (Jin et al., 2023).

Taken together, these literatures show that complex dimensions are not a single formalism but a family of closely related analytic structures. In fractal geometry they are poles governing tube volumes, heat content, and measurability; in nonrelativistic quantum mechanics they are operator dimensions tied to RG limit cycles and geometric bound-state spectra; in quantum spacetime they encode discrete scale invariance; and in non-integer-dimensional gauge theory they signal nonunitary operator mixing. The common feature is that a nonzero imaginary part records broken continuous scaling in a way that is directly visible in oscillatory asymptotics.