Momentum Space Correlation Functions in 2D Galilean Conformal Algebra

Abstract: Galilean Conformal Algebra (GCA) arises as a controlled nonrelativistic limit of the relativistic conformal algebra. In this paper, we initiate the study of momentum space correlation functions in two-dimensional GCA. We derive and solve momentum space Ward identities to obtain two-point and three-point functions. However, relating them to position space correlation functions presents a challenge as Fourier transforms of the latter do not exist. This is resolved by analytically continuing the boost eigenvalues to imaginary values. In this regime, the Fourier transform of the position space two-point and three-point functions exist and match exactly with the momentum space two-point and three-point function obtained by solving the Ward identities.

• The paper introduces analytic continuation of boost eigenvalues to regularize Fourier transforms of divergent 2D GCA correlators.
• It employs momentum-space Ward identities and nonrelativistic limits to explicitly derive two- and three-point functions for scalar primaries.
• The findings establish a robust framework for nonrelativistic holography and momentum-space bootstrap applications in flat-space contexts.

Momentum Space Correlation Functions in 2D Galilean Conformal Algebra

Introduction and Motivation

The Galilean Conformal Algebra (GCA) emerges from a nonrelativistic contraction of the relativistic conformal algebra, paralleling the relationship between Galilean and Poincaré algebras. In two dimensions (2D), this procedure yields an infinite-dimensional algebra, directly related to the asymptotic symmetry group BMS$_3$ of three-dimensional flat spacetime—a structure extensively studied in the context of flat-space holography. While previous literature has focused on position-space correlators in GCA-invariant field theories, momentum-space techniques have proven pivotal for conformal field theories (CFTs), especially in making analytic properties and factorization channels explicit [Bzowski et al., (Bzowski et al., 2013); Gillioz, (Gillioz, 2019)].

This work initiates a systematic study of momentum-space correlation functions in 2D GCA, targeting scalar primaries. The key technical challenge addressed is the exponential growth of 2D GCA position-space correlators, which precludes the existence of their Fourier transforms for real momenta. To overcome this, the authors analytically continue boost eigenvalues to imaginary values, enabling tempered distributions and well-defined Fourier transforms that match the momentum-space correlators obtained from the Ward identities.

Position-Space GCA and Representation Theory

GCA arises from contracting the relativistic conformal algebra via the Wigner-Inönü procedure. The resulting generators in 2D—$L_m$ and $M_n$—span an infinite-dimensional algebra akin to the Virasoro algebra's contraction. Scalar primary fields are characterized by dilatation eigenvalue $\Delta$ and boost eigenvalue $\xi$. The Ward identities in position space fix the functional forms of two-point and three-point correlators:

• Two-point: $$G^{(2)}(x,\tau) = C^{(2)}\, \delta_{\Delta_1,\Delta_2}\, \delta_{\xi_1,\xi_2}\, \tau^{-2\Delta_1} \exp\left(\frac{2\xi_1 x}{\tau}\right)$$
• Three-point: Exponentially dependent on coordinates, with tensor structures set by the algebra [Bagchi & Mandal, (0903.4524)].

For real boost eigenvalues, these correlators are not tempered and diverge at large separations. By continuing $\xi \to -i\xi$, exponential growth is tamed, Fourier transforms exist, and analytic continuation provides a self-consistent framework for relating position and momentum space.

Momentum-Space Ward Identity Formalism

The main analytical tool consists of momentum-space Ward identities derived from the algebraic action of generators. For a scalar primary $\widetilde{O}(E,k)$, the boost and dilatation generators act via differential operators in $(E, k)$-space. For the two-point function $G^{(2)}(k, E)$, translation invariance reduces it to functions of energy and momentum differences.

The Ward identities read: $$(-k \partial_E + 2\xi) G^{(2)}(k,E) = 0,\quad (-E\partial_E - k\partial_k + 2\Delta - 2) G^{(2)}(k,E) = 0$$ Solving these yields the general form: $$G^{(2)}(k, E) = C^{(2)} k^{2\Delta-2} \exp\left(\frac{2\xi E}{k}\right)$$ with selection rules $\Delta_1 = \Delta_2$ and $\xi_1 = \xi_2$.

The same structure arises from (i) direct solution of Ward identities, (ii) analytic continuation followed by Fourier transformation of the position-space correlator, and (iii) nonrelativistic limit of relativistic CFT momentum-space correlators [Setare & Kamali, (Setare et al., 2010)]. This triple redundancy provides a strong consistency check.

Three-Point Functions: Solutions and Matching

The momentum-space three-point correlators pose higher complexity. The position-space structure, once boost eigenvalues are analytically continued, can be Fourier transformed, yielding three-point functions matching explicit solutions to the Ward identities. The latter are solved via change of variables and homogeneity properties stemming from the dilatation generator, reducing the problem dimensionally and allowing explicit expressions for all functional dependencies.

Both momentum-space and Fourier-transformed position-space three-point functions satisfy the full set of GCA Ward identities—boosts, dilatation, and special conformal constraints—confirming their equivalence up to overall normalization. The momentum-space form is characterized by delta-function constraints on energy-momentum conservation and explicit dependence on scaling dimensions and boost labels.

Analytic Continuation and Temperedness

A pivotal technical advance is the demonstration that analytic continuation of boost eigenvalues regularizes otherwise divergent spatial integrals in Fourier space. For $\xi \to -i\xi$, the exponential position-space correlators acquire oscillatory behavior, ensuring their Fourier transforms exist as tempered distributions. This regime corresponds physically to non-trivial imaginary boost parameters and may be interpreted as probing non-local sectors or analytic continuations of the operator spectrum.

Practical and Theoretical Implications

Momentum-space analyses facilitate the extraction of singularity structure, causality constraints, and factorization properties critical for scattering amplitude analogs, cosmological bootstrap programs, and holographic correspondence in flat spacetime [Bzowski et al., (Bzowski et al., 2019)]. The tractable nature of the momentum-space GCA three-point function, versus the complicated triple-K integrals of relativistic CFTs, suggests that nonrelativistic limits could streamline the study of correlators and amplitudes in related holographic models.

The methodology opens pathways for generalization to:

• Operators with spin and conserved current sectors.
• Momentum-space bootstrap and higher-point (four-point) correlators.
• Systematic study of boundary/bulk dualities in flat space holography, leveraging the isomorphism BMS$_3 \cong$ GCA$_2$.

The analytic structures uncovered here are expected to underpin more advanced studies of bulk scattering in three dimensions, correlation functions in nonrelativistic quantum field theories, and possibly in effective field theories relevant for condensed matter systems and non-equilibrium statistical mechanics.

Conclusion

This work rigorously constructs momentum-space correlation functions for 2D Galilean Conformal Algebra, resolving foundational challenges related to exponential divergence in position space via analytic continuation of boost eigenvalues. The equivalence between direct Ward identity solutions, Fourier-transformed correlators, and nonrelativistic limits of CFT expressions is exhaustively demonstrated for two- and three-point functions. The results provide robust foundations for momentum-space bootstrap approaches, analytic studies of nonrelativistic holography, and future generalizations integrating spin, conserved currents, and higher-point functions. The analytic techniques and correspondence elucidated are poised to inform both practical computations and deeper theoretical development in non-Lorentzian conformal systems (2512.21733).