Holomorphic Plane Waves in Quantum Field Theory

• Holomorphic plane waves are complexified solutions of the Klein–Gordon equation that extend analytically in Minkowski, de Sitter, and anti–de Sitter spaces.
• They provide a coordinate-invariant and group-theoretic framework for spectral representations in Lorentzian quantum field theory by leveraging holomorphy in specific tube domains.
• This unified approach simplifies the analytic continuation between different spacetime geometries and underpins the construction of two-point functions and propagators.

Holomorphic plane waves constitute a distinguished class of solutions to the Klein–Gordon equation in complexified Minkowski, de Sitter (dS), and anti–de Sitter (AdS) spaces. These solutions extend holomorphically into specific domains of the corresponding complex manifolds and form the foundation for a coordinate-invariant, group-theoretic approach to harmonic analysis and spectral representations in Lorentzian Quantum Field Theory (QFT). Holomorphic plane waves are uniquely compatible with analyticity requirements dictated by the spectral condition (positive energy, thermal KMS, or global time generator positivity) and enable global expansions of two-point functions and propagators, subsuming conventional mode decompositions and facilitating analytic continuation between different space-time geometries (Moschella, 23 Mar 2024).

1. Algebraic Construction in Minkowski, de Sitter, and Anti–de Sitter Spaces

In Minkowski space $M_d^{(c)} \simeq \mathbb{C}^d$, holomorphic plane waves are constructed as

$$\psi_p^{(\pm)}(z) = \exp\left(\pm i\, p \cdot z\right), \quad p^0 = \omega_{\vec p} = \sqrt{|\vec p|^2 + m^2}$$

where $z = x + i y$ and $p \cdot z = p_\mu z^\mu$. These solve $(\Box + m^2)\psi = 0$ and admit holomorphic extension in all $z \in M_d^{(c)}$.

For de Sitter space, $$dS_d = \{x \in \mathbb{R}^{1,d} \mid x \cdot x = -R^2\}$$, and its complexification, the basis is given by

$$\psi_\lambda(z, \xi) = (z \cdot \xi)^\lambda = \exp\big[\lambda \ln(z \cdot \xi)\big]$$

with $$\xi \in C^+ = \{\xi \in \mathbb{R}^{1,d}: \xi^2 = 0,\, \xi^0 > 0\}$$, and parameter $\lambda$ subject to $m^2 = -\lambda(\lambda + d-1)$. For $m^2 \geq 0$, the principal series with $\lambda = -\frac{d-1}{2} \pm i\nu$, $\nu \in \mathbb{R}$, is used.

In anti-de Sitter space, $$AdS_d = \{x \in \mathbb{R}^{2,d-1} \mid x \cdot x = +R^2\}$$, the construction is analogous, with plane waves

$\psi_\ell(z, \xi) = (z \cdot \xi)^\ell$

defined for integer $\ell$ on the ambient null-cone $$C_{d} = \{\xi \mid \xi^2 = 0\} \subset \mathbb{R}^{2,d-1}$$, and $m^2 = \ell(\ell + d-1)$. These solutions also extend holomorphically into prescribed domains of $AdS_d^{(c)}$ (Moschella, 23 Mar 2024).

2. Analyticity Domains and the Spectral Condition

In all geometries, physically meaningful two-point functions $W(z_1, z_2)$ arise as boundary values of holomorphic functions defined on products of analyticity domains or "tubes," intimately connected to the appropriate spectral condition.

• Minkowski tubes: $$T_{\pm} = \{z = x + i y \in M_d^{(c)} \mid \pm y \in \overline{V}_+ \}$$, $V_+ = \{x^2 > 0, x^0 > 0\}$.
• de Sitter analytic tubes: $$\mathcal{T}_\pm = dS_d^{(c)} \cap T_\pm = \{z = x + i y \in dS_d^{(c)} \mid \pm y \in V_+ \}$$.
• AdS chiral tubes: $$\mathcal{Z}_\pm = \{z = x + i y \in AdS_d^{(c)} \mid y^2 > 0,\, \epsilon(z) = \pm 1\}$$ with $\epsilon(z) = \mathrm{sign}(y^0 x^d - x^0 y^d)$.

The spectral condition (positive frequency/energy, KMS state, $M_{0d}$ positivity) is equivalent to holomorphy in the corresponding tube product. For instance, the positive-frequency Wightman function in Minkowski spacetime is holomorphic for $z_1 \in T_-$ and $z_2 \in T_+$ (Moschella, 23 Mar 2024).

3. Covariance and Group-Theoretic Structure

Holomorphic plane waves transform covariantly under the respective isometry groups:

• Minkowski: $\psi_p(\Lambda z) = \psi_{\Lambda^{-1}p}(z)$ for $\Lambda \in \mathrm{ISO}(1,d-1)$.
• de Sitter: $$\psi_\lambda(g z, \xi) = \psi_\lambda(z, g^{-1} \xi)$$ for $g \in SO_0(1,d)$.
• AdS: $\psi_\ell(g z, \xi) = \psi_\ell(z, g^{-1} \xi)$ for $g \in SO(2,d-1)$.

Integration measures on the light-cone are invariant under the full symmetry group, ensuring the manifest invariance of all two-point function expansions. This group-theoretic foundation makes all constructions globally covariant, removing dependence on explicit coordinate charts or local patching procedures (Moschella, 23 Mar 2024).

4. Spectral Expansions of Two-Point Functions

Holomorphic plane waves provide the natural basis for spectral integral representations of two-point Wightman functions and propagators:

• Minkowski:

$$W_m(z_1, z_2) = \int \frac{d^{d-1}p}{2(2\pi)^{d-1}\omega_{\vec p}} \exp[-i p \cdot (z_1 - z_2)], \quad z_1 \in T_-,\, z_2 \in T_+$$

with the retarded propagator $R_m(x_1, x_2)$ built from boundary values.

• de Sitter:

$$W_\nu(z_1, z_2) = N_{d, \nu} \int_\gamma (z_1 \cdot \xi)^{-\frac{d-1}{2} - i\nu} (z_2 \cdot \xi)^{-\frac{d-1}{2} + i\nu} d\mu_\gamma(\xi)$$

where $$N_{d, \nu} = \frac{\Gamma(\frac{d-1}{2} + i\nu)\Gamma(\frac{d-1}{2} - i\nu) e^{\pi\nu}}{2^{d+1}\pi^d}$$, and with the maximally analytic form expressed in terms of Legendre functions of the first kind $P$ in the invariant variable $\zeta = z_1 \cdot z_2$.

• AdS:

$$W^d_\nu(z_1, z_2) = \tilde N_{d,\nu} (\zeta^2 - 1)^{-\frac{d-2}{4}} Q_{-\frac{1}{2} + \nu}^{\frac{d-2}{2}}(\zeta)$$

with normalization $$\tilde N_{d, \nu} = \frac{\Gamma(\frac{d-1}{2} + \nu)}{2 \pi^{\frac{d-1}{2}} \Gamma(\nu + 1)}$$, $m^2 = \nu^2 - \frac{(d-1)^2}{4}$, and $Q$ the Legendre function of the second kind. All expansions are performed over the appropriate complex projective null cone, with analyticity guaranteed by the integration domains (Moschella, 23 Mar 2024).

5. Analyticity, Spectral Condition, and Physical Consequences

Analyticity in tube domains is directly tied to the spectral condition through the properties of holomorphic plane waves and their decay. In Minkowski space, the support in positive/negative energy is equivalent, under the Fourier–Laplace transform, to holomorphy in $T_-\times T_+$. In de Sitter space, the KMS (thermal) property, associated with de Sitter invariance, is reflected in holomorphy in $\mathcal{T}_- \times \mathcal{T}_+$. In AdS, positivity of the global Hamiltonian maps to holomorphy in $\mathcal{Z}_- \times \mathcal{Z}_+$.

Locality and positivity are automatically implemented. Positivity is inherited from the unitarity of the group representation on the null cone. Locality, i.e., vanishing commutators at spacelike separation, corresponds to boundary values of holomorphic functions differing by "cuts" across the relevant causal hypersurfaces. The plane-wave formalism naturally encodes these structural QFT properties (Moschella, 23 Mar 2024).

6. Comparative Advantages and Unified Perspective

Holomorphic plane waves eliminate the need for coordinate-dependent mode expansions, providing a construction that is group-theoretic and globally valid. Analyticity is manifest, and the spectral condition is directly linked to the geometry of tube domains. The approach gives closed-form spectral/Källén–Lehmann expansions of both two-point and multi-point functions (e.g., bubble diagrams) via generalized Mehler–Fock transforms, facilitating powerful harmonic analysis methods such as Mellin–Barnes and Harish–Chandra techniques.

A unified viewpoint emerges: Minkowski, de Sitter, and anti–de Sitter QFTs correspond to different real sections of a single complex ambient manifold, with holomorphic plane waves as globally defined functions that project naturally onto the different geometries. This approach not only clarifies analytic continuation and structure across backgrounds but yields direct routes to demonstrating unitarity, covariance, locality, and spectral support (Moschella, 23 Mar 2024).