尺度不變重力

Abstract

從Einstein的幾何重力理論出發，Weyl將Einstein的「運動的相對論」觀念推廣到「量值的相對論」，以具有局部尺度對稱的理論來描述重力。在掌握這些尺度不變重力理論的作用量之後，我們試圖討論這些理論的黑洞物理與宇宙學。首先在黑洞方面，我們找到了這些理論的球對稱黑洞解，並在Weyl重力理論、它的無質量的特例，以及含有對稱破缺項的Weyl理論下，分別證明了這些黑洞的形狀並不會受到Weyl規範場與純量測度場的電荷所影響。然後，由於量子世界中，存在一個絕對的尺度標準─Planck常數，使目前的宇宙沒有尺度對稱，因此我們在應用Weyl重力論來研究宇宙學之前，得先引入量子修正項來把尺度對稱破壞掉。為了避免結果曖昧不明，我們應用Vilkovisky與DeWitt (VD) 的方法來計算系統的量子等效位能，因為這套方法所算出的等效位能形式，和背景場的變換無關，也和規範條件的選取無關。最後我們探討具有這個VD等效位能的Weyl理論下的宇宙學，我們發現這個理論的宇宙暴漲模型，其相變是以Hawking-Moss瞬子解來描述，而這個理論在暴漲階段之後的宇宙學性質，則和標準熱大爆炸模型相差極微。除了以上成果之外，我們也釐清，一般誤認在規範場論下VD等效位能的曖昧性，其實只是所有場論中初始場變數的選取問題而已。

A natural generalization from the idea of "relativity of motion" in Einstein's geometrical theory of gravity is Weyl's "relativity of magnitude" realized by local scale symmetry of space-time. Given the actions of scale invariant gravity, we study the black hole physics and cosmology for these theories. First, we found the spherically symmetric black hole solutions and proved that there is no classical hair of the Weyl's gauge fields and the scalar measuring field in Weyl gravity, its massless limit and the ones with symmetry-breaking potential. Second, by noting that scale symmetry is not appreciated by the quantum nature because of the existence of the absolute Planck scale, we were led to introduce quantum corrections to break the scale symmetry dynamically before the study of the cosmology for Weyl gravity. To get rid of ambiguities, we calculated the one-loop effective potential for Weyl gravity by Vilkovisky-DeWitt (VD) method, which guarantees the field re-parameterization covariance and gauge independence of the results. In the last part of this thesis, a cosmology based on Weyl gravity with one-loop VD effective potential is studied in detail. It is found that the model admits an inflation via Hawking-Moss bubbling. However, the deviation from the standard hot big-bang model is vanishingly small after inflation. We also found that the metric dependence of VD effective potential relates to initial field parameterization rather than gauge symmetry as it was thought of as.