By Claudio Simeone
During this publication, homogeneous cosmological types whose Hamilton-Jacobi equation is separable are deparametrized through turning their motion sensible into that of a regular gauge method. Canonical gauges imposed at the gauge process are used to outline an international section time by way of the canonical variables of the minisuperspaces. The method sincerely exhibits how the geometry of the constraint floor restricts the alternative of time. the implications that this has for direction imperative quantization are mentioned, and the transition amplitude is acquired for relativistic isotropic versions, relativistic anisotropic types (Kantowski-Sachs and Taub) and isotropic string cosmologies. an entire bankruptcy concerning the software of the deparametrization software to the standard canonical quantization scheme can also be incorporated.
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During this booklet, homogeneous cosmological versions whose Hamilton-Jacobi equation is separable are deparametrized by means of turning their motion practical into that of a standard gauge procedure. Canonical gauges imposed at the gauge procedure are used to outline an international part time when it comes to the canonical variables of the minisuperspaces.
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Extra resources for Deparametrization and Path Integral Quantization of Cosmological Models
For k, f > 2, the horizontal dual of a k by C. rectangle is a — 1 by C + 1 rectangle, and the vertical dual is a k + 1 by C — 1 rectangle Also, (Rh )" (R'') h R. , joining the left-hand side of R to the right-hand side. We write 11(R) for the event that R has such a crossing Similarly, we write 17(R) for the event that I? has an open. VC/ heat crossing, defined analogously. 52 Bond ingeolaboa O n Z -- the flail Theorem M I MI I M S11W 111111 1111111111111 Figure 1 A rectangle 1? (solid litand its horizontal dual Rh (dashed lin R h is the Nettie& dual of B As a minimal open houzontal crossing of a rectangle I?
Inequality (8) holds Cot ever) i. For t = 2, this observation is sometimes known as the 'square-root trick; for t = this is the ''nth-toot trick' If p : Q" R is any function, then we may think of measure on Q" Thus, For E C Q n we have II as a signed Fr(x). = and the integral of a function h : Q" — R with respect to p is h ip= h(3)/1(3) = (HO(Q") EQ. In pat ticultu, wining l f for the diameter istic function of a set E C we have It^ dt = p(E) In this notation. 13 (Pp > A Li dIPp(9) whenever A. B C Q" me up-sets Inequality (9) yields a more high-blow formulation of Harr is's Lenarta Although the terminology is self-explanatory, we note that a function It : Q" R is (monotone) increasing if h(r) G h(g) whenever < y Lemma 4.
E if there are i — 1 values 1 < t with z s = z, then we test whether = 1 Note that i < A — 1: initially, z Bas i c cancrids and results z IS incident v1/4-ith at most A untested bonds. ' with z, and we cannot choose zs if there is only one such bond remaining Since each variable X,, is tested at most once, each test succeeds with conditional probability p, and the sequence has exactly the distribution of By the construction of I', if e iyi z i was the bowl chosen at step t and e, E L b then the test at step t succeeded, so one of the X, is it follows by induction on equal to 1, and z i is open.
Deparametrization and Path Integral Quantization of Cosmological Models by Claudio Simeone