A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang-Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase-space of the system under consideration. The ''matter fields'' are local generalizations of states and observables: they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase-space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group theoretical and differential geometrical interpretation. A novel background-quantum split symmetry plays a central role. (orig.)
A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang–Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase space of the system under consideration. The "matter fields" are local generalizations of states and observables; they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group-theoretical and differential-geometrical interpretation. A novel background-quantum split symmetry plays a central role.
Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.
In this thesis we study the roles played by symplectic geometry in quantum mechanics,in particular quantum dynamics and quantum information theory treated as two separate parts. The common ground for both parts is the geometrical formulation of quantum mechanics. In Chapter 2, we review the associated complex projective Hilbert space of quantum pure states, with symplectic and Riemannian structures and their roles in quantum dynamics and kinematics. In Chapter 3, we motivate the idea of information-theoretic constraint on the differentiable manifold of probability distributions through the maximum uncertainty principle and the linear Schrodinger equation by introduction of the wave function. In Chapter 4, we review both regular and singular symplectic reduction of a symplectic manifold, which is acted upon properly and symplectically by a compact Lie group. Chapter 5 contains the author's original contributions to the first part of the thesis. In this chapter, by using the same information-theoretic discussion of the Chapter 3 we propose a non-relativistic, spin-less, non-linear quantum dynamical equation,with the Fisher information metric replaced by the Jensen-Shannon distance information. Furthermore, we show that the non-linear Schrodinger equation is in fact a Hamiltonian dynamics, namely it preserves the symplectic structure of the complex Hilbert space. The projected dynamics on the corresponding projective Hilbert space is derived and its properties are highlighted in further details. Chapter 6 contains the author's primary contributions to the second part of the thesis. In particular, by using the singular symplectic reduction of the Chapter 4 we explicitly construct the space of entanglement types of three-qubit pure states with a specific (shifted) spectra of single-particle reduced density matrices. Moreover, we obtain the image of the symplectic quotient under the induced Hilbert map, by using local unitary invariant polynomials. Then the symplectic structure on the principal stratum of the symplectic quotient is derived. Finally, it is discussed that other lower dimensional strata are relative equilibria on the original manifold and their stability properties are investigated under compact subgroups of the local unitary transformations.
This work reformulates quantum mechanics on the real plane
(R2), establishing an algebraic isomorphism to the complex field
(C) and demonstrating that the complex structure inherent to the Schrödinger equation can be fully captured by coupled real-valued dynamics. By constructing a field
(R2,+,⋅) isomorphic to
C, the real counterpart of the Schrödinger equation is derived, yielding a system of coupled real partial differential equations. This system reveals a symplectic structure where the symplectic matrix
J geometrically represents the imaginary unit
i, acting as a phase-space rotation operator. Time-independent solutions decouple into identical real Helmholtz equations, preserving spectral equivalence. The reformulation provides geometric insights into foundational aspects: the geometric representation of
i, the emergence of probability conservation from real-component interactions, and the encoding of quantum phase relationships through the multiplicative structure of the
R2 field. This perspective reinforces connections to classical Hamiltonian mechanics and geometric quantization, suggesting that complex numbers are structural, emerging from field operations on the real plane, rather than fundamental. Furthermore, quantum computation elements like qubits and gates are shown to have equivalent representations as real symplectic and special orthogonal matrices, maintaining unitary equivalence. We introduce a framework for quantum computation within a real vector space and provide a generalized protocol for generating entangled Bell states. These results offer a new perspective on the necessity of complex numbers in quantum theory, providing a framework with potential computational advantages for hybrid quantum-classical systems and enhanced geometric interpretability.
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary — or, in general, projective unitary — representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.