论文标题
离散间隔的量子riemannian几何形状
Quantum Riemannian geometry of the discrete interval and q-deformation
论文作者
论文摘要
我们在有限的晶格间隔$ \ bullet- \ bullet- \ cdots- \ bullet $的情况下解决量子Riemannian几何形状，并带有$ n $ nodes（type $ a_n $的dynkin图），并发现它们一定是$ q $ q $ q $ q $ q = e^= e^e^e^{\imathπ{\imathπ\ forn+1} $。这来自固有的几何形状，而不是假设图片中的任何量子组。具体来说，我们发现了一种新颖的“边界效应”，为了承认量子 - levi civita连接，任何边缘的“度量重量”被迫更大地指向边界的批量，而$（i+1）_q/（i+1）_q/（i+1）_q/（i+1）_q/（i）_q $在node $ i $ i $ i $ i $，$ i $（i）$ qug- $ q $ q $ q $ q $ q $ q。基督徒符号也是Q构成的。限制$ q \ to 1 $同样迫使自然数$ \ bbb n $的量子riemannian几何形状使理性度量倍数$（i+1）/i $在增加$ i $的方向上。在这两种情况下，都有一个独特的Ricci-Scalar平面度量，直到归一化。量子场理论和量子重力的元素以$ n = 3 $的形式展出，并且以$ \ bbb n $的几何形状的连续限制。标量 - 平台度量的拉普拉斯式变为x} {d^2 \ vy d x^2} $的通风方程运算符$ {1 \ airy方程式运算符。通过共形因子$ e^{ψ（i）} $缩放此度量，从而使RICCI标量曲率与$ {e^{ - ψ} \ vose x} {d^2ψ\ vo d x^2} $成比例成比例。
We solve for quantum Riemannian geometries on the finite lattice interval $\bullet-\bullet-\cdots-\bullet$ with $n$ nodes (the Dynkin graph of type $A_n$) and find that they are necessarily $q$-deformed with $q=e^{\imathπ\over n+1}$. This comes out of the intrinsic geometry and not by assuming any quantum group in the picture. Specifically, we discover a novel `boundary effect' whereby, in order to admit a quantum-Levi Civita connection, the `metric weight' at any edge is forced to be greater pointing towards the bulk compared to towards the boundary, with ratio given by $(i+1)_q/(i)_q$ at node $i$, where $(i)_q$ is a $q$-integer. The Christoffel symbols are also q-deformed. The limit $q\to 1$ likewise forces the quantum Riemannian geometry of the natural numbers $\Bbb N$ to have rational metric multiples $(i+1)/i$ in the direction of increasing $i$. In both cases, there is a unique Ricci-scalar flat metric up to normalisation. Elements of quantum field theory and quantum gravity are exhibited for $n=3$ and for the continuum limit of the geometry of $\Bbb N$. The Laplacian for the scalar-flat metric becomes the Airy equation operator ${1\over x}{d^2\over d x^2}$ in so far as a limit exists. Scaling this metric by a conformal factor $e^{ψ(i)}$ gives a limiting Ricci scalar curvature proportional to ${e^{-ψ}\over x}{d^2 ψ\over d x^2}$.
