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arXiv:math-ph/0203044 · Journal of Mathematical Physics

Geometric phase and modulus relations for probability amplitudes as functions on complex parameter spaces

Alonso Botero

Published Oct 24, 2003• 8 citations
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Abstract

We investigate general differential relations connecting the respective behaviors of the phase and modulus of probability amplitudes of the form 〈ψf|ψ〉, where |ψf〉 is a fixed state in Hilbert space and |ψ〉 is a variable state, treated as a section of a U(1) bundle over a complex subspace of the corresponding ray space R=CPn. Amplitude functions on such holomorphic line bundles, while not strictly holomorphic, nevertheless satisfy generalized Cauchy–Riemann conditions involving the U(1) Berry–Simon connection on the parameter space. These conditions entail invertible relations between the gradients of the phase and modulus, therefore allowing for the reconstruction of the phase from the modulus (or vice versa) and other conditions on the behavior of either polar component of the amplitude. As a special case, we consider amplitude functions valued on the space of pure states, the ray space R=CPn, where transition probabilities have a geometric interpretation in terms of geodesic distances as measured with the Fubini–Study metric. In conjunction with the generalized Cauchy–Riemann conditions, this geodesic interpretation leads to additional relations, in particular, a novel connection between the modulus of the amplitude and the phase gradient, somewhat reminiscent of the WKB formula. Finally, a connection with geometric phases is established.

DOI
10.1063/1.1612895

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We investigate general differential relations connecting the respective behaviors of the phase and modulus of probability amplitudes of the form 〈ψf|ψ〉, where |ψf〉 is a fixed state in Hilbert space and |ψ〉 is a variable state, treated as a section of a U(1) bundle over a complex subspace of the corresponding ray space R=CPn. Amplitude functions on such holomorphic line bundles, while not strictly holomorphic, nevertheless satisfy generalized Cauchy–Riemann conditions involving the U(1) Berry–Simon connection on the parameter space. These conditions entail invertible relations between the gradients of the phase and modulus, therefore allowing for the reconstruction of the phase from the modulus (or vice versa) and other conditions on the behavior of either polar component of the amplitude. As a special case, we consider amplitude functions valued on the space of pure states, the ray space R=CPn, where transition probabilities have a geometric interpretation in terms of geodesic distances as measured with the Fubini–Study metric. In conjunction with the generalized Cauchy–Riemann conditions, this geodesic interpretation leads to additional relations, in particular, a novel connection between the modulus of the amplitude and the phase gradient, somewhat reminiscent of the WKB formula. Finally, a connection with geometric phases is established.
A set of relations between the modulus and phase is derived for amplitudes of the form $\mels{\hatu(x)}$ where U^(x)∈SU(n)\hat{U}(x) \in SU(n)U^(x)∈SU(n) in the fundamental representation and xxx denotes the coordinates on the group manifold. An illustration is given for the case n=2n=2n=2 as well as a brief discussion of phase singularities and superoscillatory phase behavior for such amplitudes. The present results complement results obtained previously \cite{PMrel1} for amplitudes valued on the ray space R=CPn{\cal R} = {\mathbb C}P^nR=CPn. The connection between the two is discussed.
Internal Phase, Amplitude, and Probability in UPG extends the Unified Phase Geometry (UPG) program by adding a mathematically precise amplitude and probability layer to the existing three‑paper framework. In UPG, quantum phase geometry and Lorentzian spacetime geometry are treated as distinct but related structures: the phase manifold carries a positive Fubini–Study–type Kähler metric, while spacetime carries a Lorentzian metric or causal cone, with the two connected by a projection map and horizontal lifts. This paper introduces an internal U(1) phase degree of freedom modeled as a Hermitian line bundle over the phase manifold. The internal U(1) action is generated by a Hamiltonian vector field whose moment map gives the conserved internal action or charge. A key clarification is that the complex amplitude is not itself the Noether charge; instead, the squared Hermitian norm ∣A∣2 is the natural local action or probability density. The paper distinguishes coherent projection, which sums fiber contributions as complex amplitudes before taking the norm, from incoherent pushforward, which sums intensities. Under coherent projection, and assuming Hermitian norm, linear superposition, positivity, normalization, and additivity over exclusive alternatives, the projected probability density takes the standard Born form p=∣ψ∣2. Projection alone does not enforce the Born rule; it only restricts observables to internal‑phase‑invariant quantities. The paper also shows that gauge‑invariant interference requires a U(1) connection so that relative phase between fibers is defined via holonomy. This construction integrates cleanly with the relational action of UPG Paper #3 and preserves the central two‑geometry principle: the Fubini–Study metric measures quantum phase separation, while Lorentzian geometry governs spacetime causality. This work completes the foundational kinematic layer of UPG by providing a rigorous treatment of internal phase, amplitude, and probability within the two‑geometry framework.
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The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system with a complex projective Hilbert space as its phase space, thus equipped with a Riemannian metric in addition to a symplectic structure. This paper extends the geometric quantum theory to include aspects of the symplectic topology of the state space by identifying the Robertson-Schrödinger uncertainty relation for pure quantum states as the differential version of the energy identity in the theory of JJJ-holomorphic curves. We consider a family of maps from a Riemann surface into a finite-dimensional quantum state space by using the vector fields generated by two quantum observables,and show that the Fubini-Study metric tensor pulls back by such a map to the covariance tensor for the two observables. By calculating the map energy density in the pull-back metric, the uncertainty relation is represented as an equality that compares the map energy differential to the sum of the pull-back of the symplectic form and a positive definite term that vanishes when the map is holomorphic. Saturation of the Robertson-Schrödinger inequality occurs when the map is conformal and the off-diagonal covariance terms vanish. For compact Riemann surfaces where such a map can be globally defined, if the map is holomorphic, it is harmonic and its image is a minimal surface. In this case, the uncertainty product integral is a topological invariant that depends only on the homology class of the curve modulo its boundary.
We theoretically analyze the polar decomposition for quantum modular values under various pointer states approaches. We consider both the finite-dimensional discrete pointer state and continuous pointer state cases. In contrast to that, a weak value of an observable is usually divided into its real and imaginary parts; here, we show that separation from the modulus and phase is necessary to a modular value. We show that the modulus of the modular value is related to the pointer post-selection conditional probability, and the phase of the modular value is connected to the summation of a geometric phase and an intrinsic phase. We also discuss a relationship between the modulus and phase, and therein, the derivative of the phase is related to the derivative of the logarithm of the modulus via a Berry-Simon-like connection which is in the form of a weak value. As a consequence, the modulus-phase relation allows us to obtain these polar components whenever the connection is specified. One of the possible applications of our results is to evaluate the weak value (the Berry-Simon-like connection) when the polar modular value is experimentally obtained.
In a quantum system initially in the nnn-th eigenstate, an adiabatic evolution of the Hamiltonian ensures that the system remains in the corresponding instantaneous eigenstate while acquiring a phase factor. This phase has two components: one resulting from standard time evolution and another associated with the dependence of the eigenstate on the varying Hamiltonian, known as the Berry phase. In this work, we explore the concept of geometric amplitudes in the context of a Hermitian Hamiltonian. We introduce the notion of geometric amplitude and provide a novel derivation of this concept. Our study reveals that a system undergoing cyclic evolution under adiabatic conditions acquires an additional amplitude factor of purely geometric origin. To illustrate this idea, we apply it to a concrete case: a generalized inverted harmonic oscillator. Although a pseudo-inner product can be introduced to make resonance states formally normalizable, this procedure relies on a non-unitary metric operator and defines a modified Hilbert space; it does not restore normalizability nor self-adjointness in the standard L2(R)L^2(R)L2(R) framework.