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arXiv:math-ph/0310065

Geometric phase and modulus relations for SU(n) matrix elements in the defining representation

Alonso Botero

Published Oct 29, 2003• 0 citations
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Abstract

A set of relations between the modulus and phase is derived for amplitudes of the form $\mels{\hatu(x)}$ where $\hat{U}(x) \in SU(n)$ in the fundamental representation and $x$ denotes the coordinates on the group manifold. An illustration is given for the case $n=2$ 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 ${\cal R} = {\mathbb C}P^n$. The connection between the two is discussed.

DOI
10.48550/arxiv.math-ph/0310065

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A set of relations between the modulus and phase is derived for amplitudes of the form \mels\hatu(x)\mels{\hatu(x)}\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.
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.
Following a recent group theoretical quantization of the symplectic space S={(phi in R mod 2pi, p>0)} in terms of irreducible unitary representations of the group SO(1,2) the present paper proposes an application of those results to the old problem of quantizing modulus and phase in interference phenomena: The self-adjoint Lie algebra generators K_1, K_2 and K_3 of that group correspond to the classical observables p cos(phi), -p sin(phi) and p > 0 the Poisson brackets of which obey that Lie algebra, too. For the irreducible unitary representations of the positive series the modulus operator K_3 has the positive discrete spectrum {n+k, n=0,1,2,...; k > 0}. Self-adjoint operators for cos(phi) and sin(phi) can then be defined as (K_3^{-1}K_1 + K_1 K_3^{-1})/2 and - (K_3^{-1} K_2 + K_2 K_3^{-1})/2 which have the theoretically desired properties for k >0.32. Some matrix elements with respect to number eigenstates and with respect to coherent states are calculated. One conclusion is that group theoretical quantization may be tested by quantum optical experiments.
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Grossmann and Royer have recently shown that the Wigner functions are closely related to the set of all translations and inversions of phase space. This allows the phase space representation of quantum mechanics to be constructed directly on the group of phase space translations and inversions. Starting from this observation, we have derived analytical expressions for the matrix elements of the translation and inversion operators, in the harmonic oscillator representation, without introducing coordinate or momentum wavefunctions.
An elementary derivation of the system of phase factors defining the ray representations (those faithful to within a phase factor) of the Galilei group is presented. The proof employs the group elements themselves. In particular, the operation of conjugation (which corresponds to coordinate transformation) is used extensively to effect the desired result, i.e., in the notation of Levy-Leblond, U(b′,a′,v′,R′)U(b,a,v,R)=±exp [i(12m)(a′·R′v−v′·R′a+bv′·R′v)]×U(b′+b,a′+R′a+bv′,v′+R′v,R′R).