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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2503.18039 |
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| _version_ | 1866909652183154688 |
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| author | Haba, Z. |
| author_facet | Haba, Z. |
| contents | We consider a path integral representation
of the time evolution $\exp(-\frac{i}{\hbar}tH)$ for Lagrangians of the variable $A$ which can be represented in the form (quadratic in $Q$) ${\cal L}(A)=\frac{1}{2}Q(A){\cal M}Q(A)+\partial_μL^μ$. We show that
$\exp(-\frac{i}{\hbar}tH)\exp(\frac{i}{\hbar}\int d{\bf x}L^{0}) =\exp(\frac{i}{\hbar}\int d{\bf x}L^{0})$ up to an $A$-independent factor. We discuss examples of the states $\exp(\frac{i}{\hbar}\int d{\bf x}L^{0})$ in quantum mechanics and in quantum field theory (the Chern-Simons states in Yang-Mills theory, Kodama states in quantum gravity). We show the relevance of these states for a determination of the dynamics in terms of stochastic perturbations of self-duality equations. The solution of the Schrödinger equation can be expressed by the solution of the self-duality equation in the leading order of $\hbar$ expansion. We discuss applications to gauge theory on a Lorentzian manifold and gauge theories of gravity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_18039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chern-Simons "ground state" from the path integral Haba, Z. High Energy Physics - Theory General Relativity and Quantum Cosmology We consider a path integral representation of the time evolution $\exp(-\frac{i}{\hbar}tH)$ for Lagrangians of the variable $A$ which can be represented in the form (quadratic in $Q$) ${\cal L}(A)=\frac{1}{2}Q(A){\cal M}Q(A)+\partial_μL^μ$. We show that $\exp(-\frac{i}{\hbar}tH)\exp(\frac{i}{\hbar}\int d{\bf x}L^{0}) =\exp(\frac{i}{\hbar}\int d{\bf x}L^{0})$ up to an $A$-independent factor. We discuss examples of the states $\exp(\frac{i}{\hbar}\int d{\bf x}L^{0})$ in quantum mechanics and in quantum field theory (the Chern-Simons states in Yang-Mills theory, Kodama states in quantum gravity). We show the relevance of these states for a determination of the dynamics in terms of stochastic perturbations of self-duality equations. The solution of the Schrödinger equation can be expressed by the solution of the self-duality equation in the leading order of $\hbar$ expansion. We discuss applications to gauge theory on a Lorentzian manifold and gauge theories of gravity. |
| title | Chern-Simons "ground state" from the path integral |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2503.18039 |