Robertson intelligent states

Trifonov, D. A.

doi:10.1088/0305-4470/30/17/006

Cited by 53 publications

(121 citation statements)

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“…As these certainly include correlated states, including those with position-momentum correlations, the claim of Born reciprocity to set position and momentum 'coordinates' on an equal footing, is once again demonstrated -for a given uncertainty measure σ , there is no natural distinction between uncorrelated and correlated states. Let us consider the 8 × 8 covariance matrix Σ in more detail (compare [17]). It is easily established using elementary algebra that Σ is a positive semi-definite matrix, Σ ≥ 0 .…”

Section: Discussionmentioning

confidence: 99%

“…7 In fact in this case of canonical observables, the sqeezed states are equivalent to the well-known Barut-Girardello group-like coherent states (see for example [17]). …”

Section: Discussionmentioning

confidence: 99%

“…'Pure' position and momentum states should in this context be thought of as singular (non-normalisable) limits of such |z , for inadmissible values of the ratio a/b [17]. However, from the viewpoint of quaplectic transformations which can rotate unitarily between X and P , the attribution of these quantities, regarded as having sharp classical values or not, is purely frame-dependent.…”

Section: Discussionmentioning

confidence: 99%

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Born Reciprocity and the Granularity of Spacetime

Jarvis

Morgan

2006

Found Phys Lett

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The Schrödinger-Robertson inequality for relativistic position and momentum operators X µ , Pν , µ, ν = 0, 1, 2, 3 , is interpreted in terms of Born reciprocity and 'non-commutative' relativistic positionmomentum space geometry. For states which saturate the Schrödinger-Robertson inequality, a typology of semiclassical limits is pointed out, characterised by the orbit structure within its unitary irreducible representations, of the full invariance group of Born reciprocity, the so-called 'quaplectic' group U (3, 1) ⊗s H(3, 1) (the semi-direct product of the unitary relativistic dyamical symmetry U (3, 1) with the Weyl -Heisenberg group H(3, 1) ). The example of the 'scalar' case, namely the relativistic oscillator, and associated multimode squeezed states, is treated in detail. In this case, it is suggested that the semiclassical limit corresponds to the separate emergence of space-time and matter, in the form of the stress-energy tensor, and the quadrupole tensor, which are in general reciprocally equivalent.

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Section: Discussionmentioning

confidence: 99%

“…7 In fact in this case of canonical observables, the sqeezed states are equivalent to the well-known Barut-Girardello group-like coherent states (see for example [17]). …”

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Born Reciprocity and the Granularity of Spacetime

Jarvis

Morgan

2006

Found Phys Lett

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“…For real w and v = u * the operator uI − (t) + u * I + (t) + wI 3 (t) is Hermitian, therefore [32] the states |z, u, u * , w = w * ; κ, t minimize the Robertson inequality [23,32] for the three observables I j ( I = (I 1 , I 2 , I 3 )) :…”

Section: Wave Functions and Algebra Related Coherent Statesmentioning

confidence: 99%

“…Then the second moments σ ij ( L) of L j in any state are simply related to those of I j (t) [25,32]: σ ij ( L) = λ in (t)σ nm ( I)λ jm (t).…”

Section: Wave Functions and Algebra Related Coherent Statesmentioning

confidence: 99%

Exact solutions for the general nonstationary oscillator with a singular perturbation

Trifonov¹

1999

J. Phys. A: Math. Gen.

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Three linearly independent Hermitian invariants for the nonstationary generalized singular oscillator (SO) are constructed and their complex linear combination is diagonalized. The constructed family of eigenstates contains as subsets all previously obtained solutions for the SO and includes all Robertson and Schrödinger intelligent states for the three invariants. It is shown that the constructed analogues of the SU (1, 1) group-related coherent states for the SO minimize the Robertson and Schrödinger uncertainty relations for the three invariants and for every pair of them simultaneously. The squeezing properties of the new states are briefly discussed.

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Quantization of the optical phase space in terms of the group

Kastrup¹

2003

Fortschritte der Physik

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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 \documentclass{article}\pagestyle{empty}\begin{document}$\mathcal{S}^2 = \{\varphi \bmod{2\pi}, I> 0\}$\end{document} has the global structure \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$S^1 \times \mathbb{R}^+$\end{document} (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 \documentclass{article}\pagestyle{empty}\begin{document}$\mathcal{S}^2$\end{document} has the conic structure of an orbifold \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathbb{R}^2/Z_2$\end{document}. 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 \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$Sp(2,\mathbb{R})$\end{document}. 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 \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathfrak{so}(1,2)$\end{document}. 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.

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Robertson intelligent states

Cited by 53 publications

References 53 publications

Born Reciprocity and the Granularity of Spacetime

Born Reciprocity and the Granularity of Spacetime

Exact solutions for the general nonstationary oscillator with a singular perturbation

Quantization of the optical phase space in terms of the group

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