On the Elementarity of Measurement in General Relativity: Toward a General Theory
The evolution of theoretical physics, from the earliest studies in the Greek era to the present period, is characterized by one essential feature — an aim at generalization in the underlying bases for natural phenomena. When one takes to its logical extreme the premise which asserts the existence of the generalization that is sought, the conclusion is reached that all of the fundamental processes are, in fact, manifestations of a unified theory of the universe. The approach that will be taken in this lecture assumes that such a theory does indeed exist. It will be argued that if a logically consistent generalization were to be built from present day theories, then it must necessarily reject a part of their underlying premises. Nevertheless, it will be assumed that a reasonable direction to be taken in the construction of a general theory may indeed be guided by the mathematically consistent features of the earlier developed formulations which have yielded predictions that agree with a large group of facts. This would ensure, from the outset, that the newly constructed general theory will agree, in particular limits, with the successful predictions of the standard formalisms. In this way, the investigation that will be discussed attempts to build on the scientific knowledge that has been acquired in the previously developed theories.
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- 1.The remarkable discovery in 1957 that (at least) one type of physical phenomenon — the weak interaction — does violate spatial reflection symmetry, appeared as one further lesson of nature that our investigations must be based, entirely, on objective analyses, rather than on the subjective feelings that are suggested by our immediate perceptions.Google Scholar
- 2.The algebraic properties of the quaternion were first discovered by W. R. Hamilton, more than 120 years ago. It seems that science has indeed been impeded from the lack of interest that has been shown (until very recently) in the use of quaternion analysis.Google Scholar
- 3.The physical reason for this procedure (called second quantization) follows from the non-deterministic requirement of the quantum theory that not all of the dynamical variables of a particle description can represent simultaneously measurable quantities (to arbitrary accuracy). The implication in a field description is that not all of the field components that are associated with the particle can represent simultaneously measurable quantities, within a sufficiently small region of space-time. This statement is represented mathematically by using an algebra of non-commuting field operators, rather than the ordinary functions of a ‘classical’ field theory.Google Scholar
- 4.One of the original motivations for the general theory that is discussed in this lecture is the failing of the quantum field theory to fulfil the minimum requirement of providing a demonstrably consistent theory. For further discussion of this comparison, see M. Sachs, British Journal for the Philosophy of Science 21 (1964) 213.Google Scholar
- 6.It should be noted at this point that the integrals in this theory which relate to the constants of the motion, have in the linear limit of the non-linear formalism, precisely the same forms as the constants which are predicted by the quantum theory. Thus, it is only an approximation in this general theory to say that observables, such as momentum and energy, are in one-to-one correspondence with a set of linear operators. On the other hand, the latter correspondence is a necessary one in the quantum theory, independent of the approximation that may be used to solve a particular problem.Google Scholar
- 7.The bilinear form ensures that the weighting function will be positive-definite.Google Scholar
- 9.This automatically removes the infinite self-energy difficulty that appears in classical electrodynamics as well as in the quantized version.Google Scholar
- 10.The description of electromagnetic experiments without the use of photons is commonly referred to as action-at-a-distance. So as not to be confused with the classical Newtonian concept of action-at-a-distance, the former should be called delayed-action- at-a-distance since the information between parts of an interacting system does propagate with finite speed.Google Scholar
- 11.One other experimental observation that is conventionally interpreted in terms of photons which are uninfluenced by charged matter is the spectral distribution of black- body radiation. Nevertheless, it has been shown (Bull. Amer. Phys. Soc., Ser. II. 10 (1965) 536; ‘Blackbody Radiation from a Self-Consistent Field Theory of Quantum Electrodynamics’, Nuovo Cimento 37 (1965) 977) that the observed Planck distribution will follow from a description of the interactions of an ideal gas of positronium atoms (each in its ground state of null energy-momentum) with the charged matter of the cavity walls and the detecting apparatus of the experiment. The reason is that the present field theory leads to a description of these interactions in the cavity in terms of distinguishable modes of vibration of a classical field, whose energies are proportional to integral multiples of the fixed driving frequency of the detecting apparatus. Further, the dynamical features of the ground state solution of the positronium atom lead to the result that they do not couple, as a unit, to charged matter. This leads to agreement with the experimental facts that the spectral distribution is independent of the material that constitutes the cavity walls. Still, the incoherent coupling of the interacting components of the positronium atoms, individually, to the matter of the cavity walls, leads to the establishment of thermodynamic equilibrium in the gas, at a constant temperature when the cavity walls are maintained at this temperature.Google Scholar
- 12.M. Sachs and S. L. Schwebel, ibid.Google Scholar
- 14.M. Sachs and S. L. Schwebel, ibid.Google Scholar
- 16.Note added in proof.) In a recent extension of this analysis (M. Sachs, Nuovo Cimento, in press) it is shown that the expression of the Einstein formalism, which is quadratic in the quaternion variables, can be ‘factored’ into a pair of coupled time- reversed field equations that transform singly as the quaternion variables and their conjugates, respectively.Google Scholar
- 17.It should be remarked that others have investigated formulations of Einstein’s equations in terms of variables that are equivalent to the quaternion form discussed here. Nevertheless, a new feature that has not been added until now, which lends itself to a unification of the matter, electromagnetic and gravitational fields, is the expression of the Maxwell equations in a first-rank spinor form. This leads to a situation that resolves one of the difficulties that occurs in the attempt to unify the vector-tensor forms of gravitation and electromagnetism. In particular, the standard formalism faces the difficulty of combining the (tensor) electromagnetic field intensity Fμv with the corresponding geometrical term that occurs in the gravitational theory — the affine connection field [μv, a]. The difficulty lies in the fact that the latter field does not transform like a tensor (or like any other covariant field). On the other hand, the corresponding terms in the spinor-quaternion formalism (spinors, quaternions and the spin- affine connection) do transform alike under the continuous coordinate transformations of general relativity.Google Scholar