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arXiv:quant-ph/0510234 · Physical Review A

Quantization of the canonically conjugate pair angle and orbital angular momentum

H.A. Kastrup

Published May 12, 2006• 86 citations
arXivHTMLPDF

Abstract

The question how to quantize a classical system where an angle $\ensuremath{\varphi}$ is one of the basic canonical variables has been controversial since the early days of quantum mechanics. The problem is that the angle is a multivalued or discontinuous variable on the corresponding phase space. The remedy is to replace $\ensuremath{\varphi}$ by the smooth periodic functions $\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}$ and $\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}$. In the case of the canonical pair $(\ensuremath{\varphi},{p}_{\ensuremath{\varphi}})$, where ${p}_{\ensuremath{\varphi}}$ is the orbital angular momentum (OAM), the phase space ${\mathcal{S}}_{\ensuremath{\varphi},{p}_{\ensuremath{\varphi}}}={\ensuremath{\varphi}∊\mathbb{R}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}\phantom{\rule{0.2em}{0ex}}2\ensuremath{\pi},{p}_{\ensuremath{\varphi}}∊\mathbb{R}}$ has the global topological structure ${S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}$ of a cylinder on which the Poisson brackets of the three functions $\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi},\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}$, and ${p}_{\ensuremath{\varphi}}$ obey the Lie algebra of the Euclidean group $\mathrm{E}(2)$ in the plane. This property provides the basis for the quantization of the system in terms of irreducible unitary representations of the group $\mathrm{E}(2)$ or of its covering groups. A crucial point is that, due to the fact that the subgroup $\mathrm{SO}(2)\ensuremath{\cong}{S}^{1}$ is multiply connected, these representations allow for fractional OAM $l=\ensuremath{\hbar}(n+\ensuremath{\delta}),n∊\mathbb{Z},\ensuremath{\delta}∊[0,1)$. Such $\ensuremath{\delta}\ensuremath{\ne}0$ have already been observed in cases like the Aharonov-Bohm and fractional quantum Hall effects, and they correspond to the quasimomenta of Bloch waves in ideal crystals. The proposal of the present paper is to look for fractional OAM in connection with the quantum optics of Laguerre-Gaussian laser modes in external magnetic fields. The quantum theory of the phase space ${\mathcal{S}}_{\ensuremath{\varphi},{p}_{\ensuremath{\varphi}}}$ in terms of unitary representations of $\mathrm{E}(2)$ allows for two types of ``coherent'' states, the properties of which are discussed in detail: nonholomorphic minimal-uncertainty states and holomorphic ones associated with Bargmann-Segal Hilbert spaces.

DOI
10.1103/physreva.73.052104

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The question how to quantize a classical system where an angle \ensuremathφ\ensuremath{\varphi}\ensuremathφ is one of the basic canonical variables has been controversial since the early days of quantum mechanics. The problem is that the angle is a multivalued or discontinuous variable on the corresponding phase space. The remedy is to replace \ensuremathφ\ensuremath{\varphi}\ensuremathφ by the smooth periodic functions cos\ensuremathφ\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}cos\ensuremathφ and sin\ensuremathφ\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}sin\ensuremathφ. In the case of the canonical pair (\ensuremathφ,p\ensuremathφ)(\ensuremath{\varphi},{p}_{\ensuremath{\varphi}})(\ensuremathφ,p\ensuremathφ​), where p\ensuremathφ{p}_{\ensuremath{\varphi}}p\ensuremathφ​ is the orbital angular momentum (OAM), the phase space S\ensuremathφ,p\ensuremathφ=\ensuremathφ∊Rmod2\ensuremathπ,p\ensuremathφ∊R{\mathcal{S}}_{\ensuremath{\varphi},{p}_{\ensuremath{\varphi}}}={\ensuremath{\varphi}∊\mathbb{R}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}\phantom{\rule{0.2em}{0ex}}2\ensuremath{\pi},{p}_{\ensuremath{\varphi}}∊\mathbb{R}}S\ensuremathφ,p\ensuremathφ​​=\ensuremathφ∊Rmod2\ensuremathπ,p\ensuremathφ​∊R has the global topological structure S1\ifmmode×\else\texttimes\fiR{S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}S1\ifmmode×\else\texttimes\fiR of a cylinder on which the Poisson brackets of the three functions cos\ensuremathφ,sin\ensuremathφ\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi},\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}cos\ensuremathφ,sin\ensuremathφ, and p\ensuremathφ{p}_{\ensuremath{\varphi}}p\ensuremathφ​ obey the Lie algebra of the Euclidean group E(2)\mathrm{E}(2)E(2) in the plane. This property provides the basis for the quantization of the system in terms of irreducible unitary representations of the group E(2)\mathrm{E}(2)E(2) or of its covering groups. A crucial point is that, due to the fact that the subgroup SO(2)\ensuremath≅S1\mathrm{SO}(2)\ensuremath{\cong}{S}^{1}SO(2)\ensuremath≅S1 is multiply connected, these representations allow for fractional OAM l=\ensuremathℏ(n+\ensuremathδ),n∊Z,\ensuremathδ∊[0,1)l=\ensuremath{\hbar}(n+\ensuremath{\delta}),n∊\mathbb{Z},\ensuremath{\delta}∊[0,1)l=\ensuremathℏ(n+\ensuremathδ),n∊Z,\ensuremathδ∊[0,1). Such \ensuremathδ\ensuremath≠0\ensuremath{\delta}\ensuremath{\ne}0\ensuremathδ\ensuremath=0 have already been observed in cases like the Aharonov-Bohm and fractional quantum Hall effects, and they correspond to the quasimomenta of Bloch waves in ideal crystals. The proposal of the present paper is to look for fractional OAM in connection with the quantum optics of Laguerre-Gaussian laser modes in external magnetic fields. The quantum theory of the phase space S\ensuremathφ,p\ensuremathφ{\mathcal{S}}_{\ensuremath{\varphi},{p}_{\ensuremath{\varphi}}}S\ensuremathφ,p\ensuremathφ​​ in terms of unitary representations of E(2)\mathrm{E}(2)E(2) allows for two types of ``coherent'' states, the properties of which are discussed in detail: nonholomorphic minimal-uncertainty states and holomorphic ones associated with Bargmann-Segal Hilbert spaces.
The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle $\ensuremath{\theta}$ and angular momentum ppp is solved. While a key element for the construction of Wigner functions for the planar phase space ${(q,p)\ensuremath{\in}{\mathbb{R}}^{2}}$ is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space ${(\ensuremath{\theta},p)\ensuremath{\in}{S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}}$ is the Euclidean group E(2)E(2)E(2) of the plane and its unitary representations. Here the angle $\ensuremath{\theta}$ is replaced by the pair $(cos\ensuremath{\theta},sin\ensuremath{\theta})$, which corresponds uniquely to the points on the unit circle. The main structural properties of the Wigner functions for the planar and the cylindrical phase spaces are strikingly similar. A crucial role is played by the sincsincsinc function, which provides the interpolation for the discontinuous quantized angular momenta in terms of the continuous classical ones, in accordance with the famous Whittaker cardinal function well known from interpolation and sampling theories. The quantum mechanical marginal distributions for the angle (continuous) and angular momentum (discontinuous) are, as usual, uniquely obtained by appropriate integrations of the $(\ensuremath{\theta},p)$ Wigner function. Among the examples discussed is an elementary system of simple cat states.
The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.
Recently a paper on the construction of consistent Wigner functions for cylindrical phase spaces ${\mathbb{S}}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}$, i.e., for the canonical pair angle and orbital angular momentum, was published [H. A. Kastrup, Phys. Rev. A 94, 062113 (2016)] in which the main properties of these functions are derived and discussed and their usefulness is illustrated with examples. The present paper is a continuation which compares the properties of the Wigner functions for cylindrical phase spaces with those of the well-known Wigner functions for planar phase spaces in more detail. Furthermore, the mutual (Weyl) correspondence between Hilbert space operators and their phase-space functions is discussed. The $\ensuremath{\star}$ product formalism is shown to be completely implementable. In addition, basic dynamical laws for Wigner and Moyal functions are derived as generalized Liouville and energy equations. They are very similar to those in the planar case but also show characteristic differences.
Abstract The problem of quantizing properly the canonical pair “angle and action variables”, φ and I, is almost as old as quantum mechanics itself and since decades an intensively debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantization in terms of the group SO(1,2): The crucial point is that the phase space has the global structure (a simple cone) and cannot be quantized in the conventional manner. As the group SO(1,2) acts transitively, effectively and Hamilton‐like on that space its irreducible unitary representations of the positive discrete series provide the appropriate quantum theoretical framework. The phase space has the conic structure of an orbifold . That structure is closely related to a Z2 gauge symmetry which corresponds to the center of a 2‐fold covering of SO(1,2), the symplectic group . The basic variables on the phase space are the functions h0 = I, h1 = I cos φ and h2 = ‐I sin φ the Poisson brackets of which obey the Lie algebra . In the quantum theory they are represented by the self‐adjoint Lie algebra generators K0, K1, and K2 of a unitary representation, where K0 has the spectrum {k + n, n = 0, 1, …; k > 0}. A crucial prediction is that the classical Pythagorean relation h12 + h22 = h02 can be violated in the quantum theory. For each representation one can define three different types of coherent states the complex phases of which may be “measured” by means of the operators K1 and K2 alone without introducing any new phase operators! The SO(1,2) structure of optical squeezing and interference properties as well as that of the harmonic oscillator are analyzed in detail. The additional coherent states can be used for the introduction of (Husimi type) “Q” distributions and (Sudarshan‐Glauber type) “P” representations of the density operator. The three operators K0, K1, and K2 are fundamental in the sense that one can construct composite position and momentum operators out of them! The new framework poses quite a number of fascinating experimental and theoretical challenges.
In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr-Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)-model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 -(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = -Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)-model of the HO the three hj correspond to the self-adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)-Hamiltonian . For an m-fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)-model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.