Analytical properties of standard subspaces and reflection positivity in AQFT
| Project/Area Number |
22KF0082
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| Project/Area Number (Other) |
22F21312 (2022)
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| Research Category |
Grant-in-Aid for JSPS Fellows
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| Allocation Type | Multi-year Fund (2023)
Single-year Grants (2022) |
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| Section | 外国 |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | The University of Tokyo |
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Principal Investigator |
河東 泰之 東京大学, 大学院数理科学研究科, 教授 (90214684)
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| Co-Investigator(Kenkyū-buntansha) |
ADAMO MARIA STELLA 東京大学, 大学院数理科学研究科, 外国人特別研究員
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| Project Period (FY) |
2023-03-08 – 2024-03-31
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| Project Status |
Granted (Fiscal Year 2023)
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| Budget Amount *help |
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2023: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2022: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | 場の量子論 / 代数的場の量子論 / 反転正値性 / 作用素環 / 共形場理論 |
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| Outline of Research at the Start |
これまでハンケル作用素と，単位円及び上半平面上のハーディー空間の核作用素について研究してきた．ベータ帯状領域について同様の研究を目指すが，領域の境界が連結ではないことが問題となる．
2次元共形場理論でのカイラル成分のプライマリー場と群状の表現圏を組み合わせてワイトマン場をすでに構成した．このとき組み紐構造はスカラーとなり，非スカラーの場合に我々の結果をどう拡張するかが問題となる．非可換Lp空間の性質に基づき，そのようなLp空間上の代表汎関数の連続性を証明し，非有界KMS状態の摂動を研究することを目指している．
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| Outline of Annual Research Achievements |
For the beta-strip, we see that the one-parameter group of multiplication acts as a unitary on its lower boundary. However, its action is not unitary on the upper boundary since an exponential factor depending on beta appears. On the other hand, we find a realization of the Hardy space of the beta-strip as a graph of a translation operator R on the imaginary axis, which is invariant under the above non-unitary one-parameter group. Such multiplication on the lower boundary and the translation operator R on the imaginary axis verify canonical commutation relations, which produce a normal form result: for a Hilbert space equipped with a unitary one parameter group and a self-adjoint operator which satisfy a CCR produce a realization of the Hilbert space as L2-functions, the one-parameter group as translations and the self-adjoint operator as multiplication. Similarly, we show that for every positive self-adjoint operator on a Hilbert space, its graph is a standard subspace in a double copy of the original Hilbert space endowed with a suitable complex structure.
In the framework of AQFT, we consider full conformal field theories in 2-dimensions whose chiral components admit representation categories with enough irreducible automorphisms.
We construct the Wightman fields and their Hilbert structure for such theories using a suitable combination of primary fields for each chiral component. We also build the 2-dimensional Haag-Kastler net that corresponds to the Wightman fields. We worked out our construction when the two chiral components are given by the U(1)-current.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Reproducing kernels for Hardy spaces on the unit disk and the upper half-plane are used to investigate reflection positive representations for the integer and the real line group, respectively. For the beta strip, one obtains reproducing kernels using the biholomorphism between simply connected domains. Using a translation operator R on the imaginary axis, we can write the Cauchy-Szego kernel for the strip as a series expansion using a “periodization process” for the Cauchy-Szego kernel of the upper half-plane. Performing a similar procedure in the abstract case produces a Hermitian contraction operator on the Hardy space of the upper half-plane, opening the way to study more general reproducing kernel Hilbert spaces.
The unitary one-parameter group on the lower boundary and the translation operator R moving lower boundary to upper one suggest a setting to prove a one-sided version of the Beurling-Lax theorem for the beta-strip case.
We are working out the class of loop groups of the special unitary Lie group, in particular for the case n=2, which should fit our construction of Wightman fields and a class of 2-dimensional conformal field theories obtained as fixed points of the U(1)-current wrt the automorphism which sends J in -J.
We obtained continuity results for the case of non-commutative Lq spaces of a von Neumann algebra equipped with a trace state for the case q=2, and for a general q, under an auxiliary assumption of continuity, which replaces the continuity wrt the norm of the von Neumann algebra in the case q=2. We study continuity for the tracial weight case.
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| Strategy for Future Research Activity |
We consider a larger family of reproducing kernel Hilbert spaces on the beta-strip that we aim to investigate in terms of measures parametrized by a positive constant s. For specific values of the parameter, we should recover known cases. Namely, the case s=1 should give the Cauchy-Szego kernel for the Hardy space, while the case s=2 should correspond to the Bergman space.
We have been studying the link between positive self-adjoint operators on a Hilbert space and standard subspaces in the direct sum of the original Hilbert space with itself endowed with a suitable complex structure. Such results appear related to studying the complex structures in the fixed points of an L2-space over R, for which the fixed points of a Hardy space of the upper half-plane become a standard subspace.
We conjecture that our Wightman field construction for the U(1)-current case could be extended further, corresponding to a particular case of Moriwaki’s work. Similarly, we seek to find inequivalent extensions with the same modular invariant while we aim to generalize our construction to the non-scalar case to consider more examples. Such construction will prepare the ground to study perturbations of such fields, which we aim to investigate to find possibly new non-conformal nets from a conformal one.
For unbounded perturbations of KMS states, the state turns out to be representable if we assume it is defined everywhere in the Lp space. Such state is usually assumed continuous wrt a dual continuity, thus we aim to investigate such perturbations in the framework of representable functionals.
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Report
(1 results)
Research Products
(1 results)
