Branching rules for conformal embeddings

After an introduction to the classic theory of the Jahn–Teller effect for octahedral complexes, an account is given of Lie groups and their relevance to the F+ center in CaO. The coincidence of the three-fold and two-fold vibrational modes (both of angular frequency ω) leads to a study of U5 and R5, the unitary and rotation groups in five dimensions. The language of second quantization is used to describe the weight spaces and branching rules. Pairs of annihilation and creation operators for phonons are coupled to zero angular momentum and used as the generators of the noncompact group O(2, 1). This facilitates the evaluation of matrix elements of V, the interaction that couples the oscillations of the octahedral complex to the electron in its interior. Glauber states are used near the strong Jahn–Teller limit, corresponding to [Formula: see text]. The possible extension of the analysis to incorporate the breathing mode is outlined. Correspondences with problems in nuclear physics are mentioned.

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Alternative branching rules for some nonconvex problems

Optimization Methods and Software ◽

10.1080/10556788.2014.885521 ◽

2014 ◽

Vol 30 (2) ◽

pp. 365-378 ◽

Cited By ~ 4

Author(s):

M. Locatelli

Keyword(s):

Nonconvex Problems ◽

Branching Rules

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Quantum fluctuations of the metric, three-geometries and conformal embeddings in four-manifolds

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887821502340 ◽

2021 ◽

Author(s):

Simon Davis

Keyword(s):

Path Integrals ◽

Invariance Group ◽

Two Dimensional ◽

Four Dimensions ◽

Physical Conditions ◽

Exponential Expansion ◽

Cotangent Bundles ◽

Four Manifolds ◽

Conformal Embeddings ◽

Natural Relation

In this paper, connections between the path integrals for four-dimensional quantum gravity and string theory are emphasized. It is shown that there is a natural relation between these two path integrals based on the theorems on embeddings of two-dimensional surfaces in four dimensions and four-dimensional manifolds in ten dimensions. The isometry groups of the three-geometries that are spatial hypersurfaces confomally embedded in the four-manifolds are required to be subgroups of [Formula: see text], which is the invariance group of the Pfaffian differential system satisfied by one form in the cotangent bundles on the four-manifolds. Based on this and other physical conditions, the three-geometries are restricted to be [Formula: see text], [Formula: see text] and [Formula: see text] with a boundary, which may be included in the quantum gravitational path integral over four-manifolds which are closed at initial times followed by an exponential expansion compatible with supersymmetry.

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Conformal embeddings and hidden symmetries of free fermion theories

Nuclear Physics B ◽

10.1016/0550-3213(87)90194-5 ◽

1987 ◽

Vol 290 ◽

pp. 392-416 ◽

Cited By ~ 10

Author(s):

Jürgen Fuchs

Keyword(s):

Free Fermion ◽

Hidden Symmetries ◽

Conformal Embeddings

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Tree Trimming: Four Non-Branching Rules for Priest’s Introduction to Non-Classical Logic

The Australasian Journal of Logic ◽

10.26686/ajl.v12i2.2066 ◽

2017 ◽

Vol 12 (2) ◽

Author(s):

Marilynn Johnson

Keyword(s):

Modal Logic ◽

Classical Logic ◽

Variable Domain ◽

Free Logic ◽

Intuitionist Logic ◽

Branching Rule ◽

Branching Rules ◽

Graham Priest

In An Introduction to Non-Classical Logic: From If to Is Graham Priest (2008) presents branching rules in Free Logic, Variable Domain Modal Logic, and Intuitionist Logic. I propose a simpler, non-branching rule to replace Priest’s rule for universal instantiation in Free Logic, a second, slightly modified version of this rule to replace Priest’s rule for universal instantiation in Variable Domain Modal Logic, and third and fourth rules, further modifying the second rule, to replace Priest’s branching universal and particular instantiation rules in Intuitionist Logic. In each of these logics the proposed rule leads to tableaux with fewer branches. In Intuitionist logic, the proposed rules allow for the resolution of a particular problem Priest grapples with throughout the chapter. In this paper, I demonstrate that the proposed rules can greatly simplify tableaux and argue that they should be used in place of the rules given by Priest.

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