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arXiv:2505.20983 · International Journal of Theoretical Physics

Construction of Metaplectic Representations of $$SL_2(\mathbb {Z}_{2^n})$$ and Twisted Magnetic Translations

Emmanuel G. Floratos, Kimon Manolas, I. Tsohantjis

Published Oct 27, 2025• 0 citations

Abstract

Unitary metaplectic representations of the group $$\varvec{SL}_{\textbf{2}}(\mathbb {Z}_{\textbf{2}^{\varvec{n}}})$$ are necessary to describe the time evolution of $$\textbf{2}^{\varvec{n}}$$ -dimensional quantum systems, such as systems involving n qubits. It is shown that in order for the metaplectic property to be fulfilled, an increase in the dimensionality of the involved n-qubit Hilbert spaces, from $$\textbf{2}^{\varvec{n}}$$ to $$\textbf{2}^{\textbf{2}{\varvec{n}}}$$ , is necessary. Thus we construct the general matrix form of such representations based on the magnetic translations of the diagonal subgroup $$\varvec{HW}_{\textbf{2}{\varvec{n}}} \otimes \varvec{HW}_{\textbf{2}{\varvec{n}}}$$ . Comparisson with other approaches on this problem of the literature are discussed.

DOI: 10.1007/s10773-025-06103-5

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Construction of Metaplectic Representations of

SL_2(\mathbb {Z}_{2^n})

and Twisted Magnetic Translations

Unitary metaplectic representations of the group

\varvec{SL}_{\textbf{2}}(\mathbb {Z}_{\textbf{2}^{\varvec{n}}})

are necessary to describe the time evolution of

\textbf{2}^{\varvec{n}}

-dimensional quantum systems, such as systems involving n qubits. It is shown that in order for the metaplectic property to be fulfilled, an increase in the dimensionality of the involved n-qubit Hilbert spaces, from

\textbf{2}^{\varvec{n}}

to

\textbf{2}^{\textbf{2}{\varvec{n}}}

, is necessary. Thus we construct the general matrix form of such representations based on the magnetic translations of the diagonal subgroup

\varvec{HW}_{\textbf{2}{\varvec{n}}} \otimes \varvec{HW}_{\textbf{2}{\varvec{n}}}

. Comparisson with other approaches on this problem of the literature are discussed.

The symplectic group Sp(n) acts on phase space while the unitary representation of its double cover, Mp(n), the metaplectic group, acts on functions defined on configuration space. We will construct an extension Mp(n) of Mp(n) acting on square integrable functions on phase space. This is performed using previous results of ours involving explicit expressions of the twisted Weyl symbols of metaplectic operators and Bopp pseudodifferential operators, which are phase space extensions of the usual Weyl operators.

Gaussian unitary transformations are generated by quadratic Hamiltonians, i.e., Hamiltonians containing quadratic terms in creations and annihilation operators, and are heavily used in many areas of quantum physics, ranging from quantum optics and condensed matter theory to quantum information and quantum field theory in curved spacetime. They are known to form a representation of the metaplectic and spin group for bosons and fermions, respectively. These groups are the double covers of the symplectic and special orthogonal group, respectively, and our goal is to analyze the behavior of the sign ambiguity that one needs to deal with when moving between these groups and their double cover. We relate this sign ambiguity to expectation values of the form

\langle 0|\exp{(-i\hat{H})}|0\rangle

, where

|0\rangle

is a Gaussian state and

\hat{H}

an arbitrary quadratic Hamiltonian. We provide closed formulas for

\langle 0|\exp{(-i\hat{H})}|0\rangle

and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.

We construct explicitly the quantization of classical linear maps of SL(2,{\\Bbb R}) on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of SL(2,{\\Bbb Z}) to the subgroup SL(2,{\\Bbb Z})/\\Gamma_l, \\Gamma_l being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, O_{l}(2)\\subset SL(2,l), for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.