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arXiv:2610.25057

Super Howe duality and rigidity of hybrid Fourier transforms

Jing-Song Huang, Yongzhi Luan, Yinuo Xue

quant-phmath.RTSubmitted Oct 18, 2026• Announced Oct 20, 2026• 0 citations
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Abstract

We characterize the hybrid Fourier transform on the Schwartz--Grassmann space $\caS(\bbR^m)\otimesΛ_{2n}$ by operator intertwining relations. Using the bosonic oscillator Howe decomposition and the fermionic Lefschetz decomposition, we describe the action of the hybrid Fourier transform on both tensor factors. We show that first-order covariance with respect to coordinate multiplication and differentiation determines the transform up to a scalar. For $m,n\ge1$, we prove that covariance under all even quadratic operators, together with any one nonzero mixed quadratic operator, determines the linear span of the hybrid Fourier transform and its inverse. This rigidity result is obtained by computing the ordinary commutants of the even and full quadratic oscillator actions of $\fosp(4n|2m)$. After a degree-scaling change of normalization, we make the hybrid Fourier transform unitary with respect to the occupation number inner product, and factor it into a bosonic oscillator rotation and (up to a scalar phase) an $SU(2)$ rotation in the fermionic Lefschetz representation. For $m=1$ and $n\ge2$, we show that full even quadratic covariance together with the parity-preserving cubic relation for $qQ_f$ determines the unitary transform up to a global phase and a fermion-parity phase. Finally we calculate the unitary conjugation formulas for Hubbard--Holstein Hamiltonian for a hybrid continuous-variable and discrete-variable quantum system.

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24 pages

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We characterize the hybrid Fourier transform on the Schwartz--Grassmann space \caS(\bbRm)⊗Λ2n\caS(\bbR^m)\otimesΛ_{2n}\caS(\bbRm)⊗Λ2n​ by operator intertwining relations. Using the bosonic oscillator Howe decomposition and the fermionic Lefschetz decomposition, we describe the action of the hybrid Fourier transform on both tensor factors. We show that first-order covariance with respect to coordinate multiplication and differentiation determines the transform up to a scalar. For m,n≥1m,n\ge1m,n≥1, we prove that covariance under all even quadratic operators, together with any one nonzero mixed quadratic operator, determines the linear span of the hybrid Fourier transform and its inverse. This rigidity result is obtained by computing the ordinary commutants of the even and full quadratic oscillator actions of \fosp(4n∣2m)\fosp(4n|2m)\fosp(4n∣2m). After a degree-scaling change of normalization, we make the hybrid Fourier transform unitary with respect to the occupation number inner product, and factor it into a bosonic oscillator rotation and (up to a scalar phase) an SU(2)SU(2)SU(2) rotation in the fermionic Lefschetz representation. For m=1m=1m=1 and n≥2n\ge2n≥2, we show that full even quadratic covariance together with the parity-preserving cubic relation for qQfqQ_fqQf​ determines the unitary transform up to a global phase and a fermion-parity phase. Finally we calculate the unitary conjugation formulas for Hubbard--Holstein Hamiltonian for a hybrid continuous-variable and discrete-variable quantum system.
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The position--momentum correspondence combines several mathematically distinct identifications that are often conflated. For the additive group (R,+)(\mathbb R,+)(R,+), every continuous character has the form x↦eikxx\mapsto \mathrm e^{ikx}x↦eikx. After choosing a coordinate p=αkp=αkp=αk on the dual group, one obtains eipx/α\mathrm e^{ipx/α}eipx/α, so Pontryagin duality alone does not identify a particular dual coordinate with physical momentum. For the corresponding unitary Fourier transform, multiplication in position space is transported to convolution in momentum space, while the diagonal distribution is transported to the specific composition with the addition map (p,q)↦p+q(p,q)\mapsto p+q(p,q)↦p+q; it is not transported to pointwise multiplication in momentum space. The family P^α=−iα d/dx\hat P_α=-iα\,d/dxP^α​=−iαd/dx has commutator [X^,P^α]=iαI[\hat X,\hat P_α]=iαI[X^,P^α​]=iαI on S(R)\mathcal S(\mathbb R)S(R), and the standard quantum-mechanical normalization [X^,P^]=iℏI[\hat X,\hat P]=i\hbar I[X^,P^]=iℏI therefore selects α=ℏα=\hbarα=ℏ within this family. Equivalently, the standard normalization of the translation generator gives P^=−iℏ d/dx\hat P=-i\hbar\,d/dxP^=−iℏd/dx. The resulting characters eipx/ℏ\mathrm e^{ipx/\hbar}eipx/ℏ have spatial period h/∣p∣h/|p|h/∣p∣ for p≠0p\neq0p=0. All distributional statements are formulated in the Schwartz rigging and no product or pullback of arbitrary tempered distributions is used.
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