Gaussian 09 Reference

[Jonathon Burnside]

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The Wheeler-DeWitt equation of vacuum geometrodynamics is turned into a Schrdinger equation by imposing the normal Gaussian coordinate conditions with Lagrange multipliers and then restoring the coordinate invariance of the action by parametrization. This procedure corresponds to coupling the gravitational field to a reference fluid. The source appearing in the Einstein law of gravitation has the structure of a heat-conducting dust. When one imposes only the Gaussian time condition but not the Gaussian frame conditions, the heat flow vanishes and the dust becomes incoherent. The canonical description of the fluid uses the Gaussian coordinates and their conjugate momenta as the fluid variables. The energy density and the momentum density of the fluid turn out to be homogeneous linear functions of such momenta. This feature guarantees that the Dirac constraint quantization of the gravitational field coupled to the Gaussian reference fluid leads to a functional Schrdinger equation in Gaussian time. Such an equation possesses the standard positive-definite conserved norm.

For a heat-conducting fluid, the states depend on the metric induced on a given hypersurface; for an incoherent dust, they depend only on geometry. It seems natural to interpret the integrand of the norm integral as the probability density for the metric (or the geometry) to have a definite value on a hypersurface specified by the Gaussian clock. Such an interpretation fails because the reference fluid is realistic only if its energy-momentum tensor satisfies the familiar energy conditions. In the canonical theory, the energy conditions become additional constraints on the induced metric and its conjugate momentum. For a heat-conducting dust, the total system of constraints is not first class and cannot be implemented in quantum theory. As a result, the Gaussian coordinates are not determined by physical properties of a realistic material system and the probability density for the metric loses thereby its operational significance. For an incoherent dust, the energy conditions and the dynamical constraints are first class and can be carried over into quantum theory. However, because the geometry operator considered as a multiplication operator does not commute with the energy conditions, the integrand of the norm integral still does not yield the probability density. The interpretation of the Schrdinger geometrodynamics remains viable, but it requires a rather complicated procedure for identifying the fundamental observables. All our considerations admit generalization to other coordinate conditions and other covariant field theories.

I am interested in constructive quantum field theory where Gaussian measures on duals of nuclear spaces (specifically, the space of tempered distribution $\mathcalS'(\mathbbR^n)$) play a key role. I would like to learn more about these measures so I can read some of the CQFT literature.

There is an older post (Measure theory in nuclear spaces) that asks for a similar reference and I have gone through most of the references suggested there. I have read Gelfand's book, and though the exposition is great, I find it presents the material in a way that is no longer standard today (please correct me if I am wrong here). I have also read the section suggested in B. Simon's book, but it was not much more than a very quick introduction.

To summarize the above, I am looking for a good reference that goes into as much detail as necessary so that one can begin reading the CQFT literature right after. If this does not exist, can someone suggest what topics are needed in Gaussian measure theory for my goal?

Let me stick to the original question and distributional setting. To learn the theory of probability measures on spaces of distributions like $\mathscrS'(\mathbbR^d)$,with a view to applications to constructive QFT, theGlimm-Jaffe and Gel'fand-Vilenkin books are okay, but not that great. They miss the most fundamental theorem needed: the Lvy Continuity Theorem (LCT). Namely, it is the characterization of weak convergence of probability measures using the pointwise convergence of the generating function of Schwinger functions, e.g., when one removes a UV cutoff.

There is an important fact about probability measures on spaces of distributions which I don't think is covered in the article by Bierm et al., nor the GJ and GV books: if moments exist, they are automatically jointly continuous, see:

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Background: The aim of this study was to develop new and useful criteria for partitioning reference values into subgroups applicable to gaussian distributions and to distributions that can be transformed to gaussian distributions.

Methods: The proposed criteria relate to percentages of the subgroups outside each of the reference limits of the combined distribution. Critical values suggested as partitioning criteria for these percentages were derived from analytical bias quality specifications for using common reference intervals throughout a geographic area. As alternative partitioning criteria to the actual percentages, these were transformed mathematically to critical distances between the reference limits of the subgroup distributions, to be applied to each pair of reference limits, the upper and the lower, at a time. The new criteria were tested using data on various plasma proteins collected from approximately 500 reference individuals, and the outcomes were compared with those given by the currently widely applied and recommended partitioning model of Harris and Boyd, the "Harris-Boyd model".

Results: We suggest 4.1% as the critical minimum percentage outside that would justify partitioning into subgroups, and 3.2% as the critical maximum percentage outside that would justify combining them. Percentages between these two values should be classified as marginal, implying that nonstatistical considerations are required to make the final decision on partitioning. The correlation between the critical percentages and the critical distances was mathematically precise in the new model, whereas this correlation is rather approximate in the Harris-Boyd model because focus on the difference between means in this model makes high precision hard to achieve. The application examples suggested that the new model is more radical than the Harris-Boyd model.

Conclusions: New percentage and distance criteria, to be used for partitioning gaussian-distributed data, have been developed. The distance criteria, applied separately to both reference limit pairs of the subgroup distributions, seemed more reliable and correlated more accurately with the critical percentages than the distance criteria of the Harris-Boyd model. As opposed to the Harris-Boyd model, the new model is easily adjustable to new critical values of the percentages, should they need to be changed in the future.

ERA5 data is produced and archived as spectral coefficients or on a reduced Gaussian grid, which has quasi-uniform spacing over the globe. Reduced Gaussian grids have a series of evenly spaced data points along each parallel (latitude), and parallels spaced at quasi-regular intervals. Near the poles there are only a few points along a parallel, but close to the equator there are many more data points along a parallel.

For a list of spectral, Gaussian and equivalent lat/lon grids see the Open IFS FAQ, section "OpenIFS questions: general and runtime", and there "What does the 'T' mean in 'T511', 'T1279' etc?" and "How do I know the grid from the 'T' number?"

All gridded data is made available in Decimal Degrees, lat/lon, with latitude values in the range [-90;+90] referenced to the equator and longitude values in the range [0;360] referenced to the Greenwich Prime Meridian. Some software applications automatically display these longitudes in the range [-180;+180].

When you download ERA5 data you have the option to interpolate the data to a custom grid and horizontal resolution (eg. 'grid':'0.5/0.5'). The default interpolation method is bilinear for continuous parameters (e.g. Temperature) and nearest neighbour for discrete parameters (eg. Vegetation).

Many software applications by default visualise regularly spaced data as a continuous tiled surface, as in (b). If you use this visualisation, think of coordinates as referencing the centroids of the tiles.

Some software applications do not recognise the spatial reference information embedded in the data file and may require you to manually assign a spatial reference. In this case use a 6367.47km sphere for all data if possible. This GRIB1 sphere does not have an EPSG code.

The users thereof use the information at their sole risk and liability. For the avoidance of all doubt , the European Commission and the European Centre for Medium - Range Weather Forecasts have no liability in respect of this document, which is merely representing the author's view.

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