Scalar and Vector Coherent State Representations of Compact and Non-compact Symplectic Groups
Bibliographic record
Abstract
When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation $T$ of a real algebraic Lie group $G$. This requires defining an inner product on the Hilbert space $\\mathbb{H}$ that carries the representation $T$. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of $T$, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet ($\\mathfrak{F}_{\\mathbb{H}_D}$, $\\mathbb{H}_D$, $\\mathfrak{F}^{\\mathfrak{H}_D}$) of Bargmann spaces. Coherent state wave functions -- the elements of $\\mathfrak{F}_{\\mathbb{H}_D}$ and of $\\mathfrak{F}^{\\mathbb{H}_D}$ -- are used to define the inner product on $\\mathbb{H}_D$ in a way that simplifies the calculation of the overlaps. This inner product and the group action $\\Gamma$ of $G$ on $\\mathfrak{F}^{\\mathbb{H}_D}$ are used to formulate expressions for the matrix elements of $T$ with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for $\\Gamma$ relies on matrix factorizations in the complexification of $G$ even though the representation $T$ does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and $\\Gamma$ action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in $\\mathbb{H}$. The scalar and vector coherent state methods will both be applied to Sp($n$) and Sp($n,\\mathbb{R}$).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".