The task of quantizing general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when applied to so fundamental a structure as the space-time geometry itself. This paper seeks to clarify the foundations of quantum mechanics. It presents a reformulation of quantum theory in a form believed suitable for application to general relativity. The aim is not to deny or contradict the conventional formulation of quantum theory, which has demonstrated its usefulness in an overwhelming variety of problems, but rather to supply a new, more general and complete formulation, from which the conventional interpretation can be deduced. The relationship of this new formulation to the older formulation is therefore that of a metatheory to a theory, that is, it is an underlying theory in which the nature and consistency, as well as the realm of applicability, of the older theory can be investigated and clarified.

"relative State" Formulation of Quantum Mechanics

The existence of a fundamental scale, a lower bound to any output of a position measurement, seems to be a model-independent feature of quantum gravity. In fact, different approaches to this theory lead to this result. The key ingredients for the appearance of this minimum length are quantum mechanics, special relativity and general relativity. As a consequence, classical notions such as causality or distance between events cannot be expected to be applicable at this scale. They must be replaced by some other, yet unknown, structure.

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Quantum Gravity and Minimum Length

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A First Course In General Relativity

A large body of theoretical work has been developed in attempts to understand and predict phenomena of the sort described in the previous chapter. In most cases the cause of the instability can be formulated in a way that gives some physical understanding of the processes involved. However, this is usually opposed by some stabilizing process, such as the damping action of viscosity, and the physical argument provides no precise information about when the instability will occur. It also provides little indication of the detailed structure of the motion resulting from the instability.

The Theory of Hydrodynamic Stability

Der speziellen Relativitatstheorie liegt folgendes Postulat zugrunde, welchem auch durch die Galilei-Newtonsche Mechanik Genuge geleistet wird: Wird ein Koordinatensystem K so gewahlt, das in bezug auf dasselbe die physikalischen Gesetze in ihrer einfachsten Form gelten, so gelten dieselben Gesetze auch in Bezug auf jedes andere Koordinatensystem K′, das relativ zu K in gleichformiger Translationsbewegung begriffen ist. Dieses Postulat nennen wir „spezielles Relativitatsprinzip“. Durch das Wort „speziell“ soll angedeutet werden, das das Prinzip auf den Fall beschrankt ist, das K′ eine gleichformige Translationsbewegung gegen K ausfuhrt, das sich aber die Gleichwertigkeit von K′ und K nicht auf den Fall ungleichformiger Bewegung von K′ gegen K erstreckt.

Die Grundlage der allgemeinen Relativitätstheorie

Fundamental Principles of General Relativity Theories

In relativistic theory of irreversible thermodynamical processes near equilibrium, generally a series of assumptions is made having, in particular, the consequence that the temperature vector is a Killing vector. We show that, in contrast to usual approaches, in equilibrium (i) the temperature vector can also be a conformal Killing vector, (ii) as an implication of the Killing property of the temperature vector, most assumptions made can be derived, without restricting the matter configuration to a perfect fluid, (iii) for non-vanishing rotation of the fluid, the heat-flow is unequal to zero, (iv) for vanishing acceleration of the fluid the Friedmann radiation cosmos is the only physically significant solution of Einstein's equations and (v) the equilibrium conditions are of the Cattaneo form such that a causal propagation of temperature can be expected.

Thermodynamical equilibrium and spacetime geometry

Starting out with the balance equations for energy-momentum, spin, particle and entropy density, an approach is considered which represents a framework for special- and general-relativistic continuum thermodynamics. A general entropy density 4-vector, containing particle, energy-momentum, and spin density contributions, is introduced. This makes possible, firstly, to test special entropy density 4-vectors used by other authors with respect to their generality and validity and, secondly, to determine entropy supply and entropy production. Using this entropy density 4-vector, material-independent equilibrium conditions are discussed. While in literature, generally thermodynamic equilibrium is determined by introducing a variety of conditions by hand, the present approach proceeds as follows: For a comparatively wide class of space–time geometries, the necessary equilibrium conditions of vanishing entropy supply and vanishing entropy production are exploited. Because these necessary equilibrium conditions do not determine the equilibrium, supplementary conditions are added systematically motivated by the requirement that also all parts of the necessary conditions have to be fixed in equilibrium.

Entropy identity and equilibrium conditions in relativistic thermodynamics

According to Poincare, only the “epistemological sum of geometry and physics is measurable”. Of course, there are requirements of measurement to be imposed on geometry because otherwise the theory resting on this geometry cannot be physically interpreted. In particular, the Weyl-Cartan space problem must be solved, i.e., it must be guaranteed that the comparison of distances is compatible with the Levi-Civita transport. In the present paper, we discuss these requirements of measurement and show that in the (purely affine) Einstein-Schrodinger unified field theory the solution of the Weyl-Cartan space problem simultaneously determines the matter via Einstein's equations. Here the affine field Γikl represents Poincare's sum, and the solution of the space problem means its splitting in a metrical space and in matter fields, where the latter are given by the torsion tensor Γi[kl].

The Weyl-Cartan Space Problem in Purely Affine Theory

We discuss the meaning and prove the accordance of general relativity, wave mechanics, and the quantization of Einstein's gravitation equations themselves. Firstly, we have the problem of the influence of gravitational fields on the de Broglie waves, which influence is in accordance with Eeinstein's weak principle of equivalence and the limitation of measurements given by Heisenberg's uncertainty relations. Secondly, the quantization of the gravitational fields is a “quantization of geometry.” However, classical and quantum gravitation have the same physical meaning according to limitations of measurements given by Einstein's strong principle of equivalence and the Heisenberg uncertainties for the mechanics of test bodies.

Horst-Heino von Borzeszkowski

Papers

Fundamental Principles of General Relativity Theories

Thermodynamical equilibrium and spacetime geometry

Entropy identity and equilibrium conditions in relativistic thermodynamics

The Weyl-Cartan Space Problem in Purely Affine Theory

Quantum theory and Einstein's general relativity