Abstract
The basic physical principles and ideas of geometro-stochastic (GS) quantum theory were described in Secs. 1.3–1.5, and their implementation within a general relativistic context will be carried out in the next and subsequent chapters. However, as a basic testing ground for these principles and ideas in general, and of the central concept of GS propagation in particular, we shall choose in this chapter the more familiar, as well as experimentally more extensively investigated, territory of nonrelativistic quantum theory. In this realm the concept of sharp localization gives rise to no foundational inconsistencies at the theoretical level, and the experimental confirmation of the conventional theory is indubitable and conclusive. Thus, we shall demonstrate that in the nonrelativistic context the proposed GS framework merges in the sharp-point limit (cf. Sec. 3.5) into a framework that is equivalent to conventional nonrelativistic quantum mechanics in the presence of an external Newtonian gravitational field. In this manner, we shall reach the assurance that the GS framework gives rise to no inconsistencies with that part of quantum theory that has already received experimental confirmation under the empirical limitations stipulated by the imposition of the nonrelativistic regime. Hence, the motivation for the investigations in this chapter is, in this last respect, analogous to the motivation for considering the linearized gravity approximation of the classical theory of general relativity (CGR): the existence of such a weak-gravity limit, in which classical general relativistic solutions merge into their Newtonian counterparts, has provided, at the inception1 of general relativity, the necessary assurance that the CGR framework does not give rise to any conflict with the wealth of observational data supporting Newtonian theory, despite the total dissimilarities in the mathematical structures of these two frameworks for describing gravitational phenomena.
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Notes to Chapter 4
As authoritatively recounted by Pais (1982), investigations of the agreement in weak gravitational fields between Newtonian theory and gravitational models that eventually were to become part of the CGR framework can be traced to a series of papers published by Einstein already in the ten-year period preceding his seminal 1916 paper. A central role in these early considerations was played by attempts to derive quantitative estimates of the perihelion precession [M,W] of the planet Mercury by supplying corrections to Newtonian theory based on the idea that gravitational interaction is transmitted with the finite speed of light, rather than instantaneously. The possibility of such an explanation for Mercury’s perihelion precession can be traced to P. Gerber (1898), to whom E. Mach accredits that “from the perihelion motion of Mercury, forty-one seconds in a century, [he] finds the velocity of propagation [of gravitational interaction] to be the same as that of light” (Mach, 1907, p. 535).
As reviewed by Havas (1964), spacetime formulations of Newtonian mechanics were provided already by P. Frank (1909) and H. Weyl (1923) before Cartan’s work. A condensed comparison between the Newton-Cartan and the Einstein classical theory of gravity can be found in Box 12.3 on p. 297 of [M].
This result was first derived by Trautman (1963), who also imposed the conditions βiRj k/m = βkRj ilm, which turned out to be redundant (Künzle, 1972, p. 352).
Whereas a and e i assume values in the tangent spaces of M, this is not the case with X. However, as mentioned in Note 3, M can be embedded in a 5-dimensional Bargmann manifold M 5, where there are charts for which the role of % is taken over by vectors e 5= d/∂x 5 that belong to the 5-dimensional tangent spaces of M 5.
From a purely mathematical point of view, any basis in the present 11-dimensional Lie algebra could be chosen. The indexing as well as the choice of bases is, however, conditioned in the present context by physical considerations, which assign a physical meaning to the Lie algebra basis elements in (2.2), as will become clear in the next section — cf. (3.11)-(3.14).
This is the map which takes the pair (u,ψ) ∈ P × F into the equivalence class ψ ∈ P×G F containing it — cf. Proposition 5.4 on p. 55 of (Kobayashi and Nomizu, 1963). Note that this reference deals with finite-dimensional manifolds and bundles, but that this result, as well as other results and definitions in it, remain valid in the case of infinite-dimensional manifolds and bundles (cf. Chapter VII in [C]), such as the quantum bundles in this monograph. Note also that in this reference, as well as in some other references (cf., e.g., Sec. 3.3 in [I]), the very definition of an associated bundle with typical fibre F and structure group G is that of a G-product P ×G F, where P is a principal bundle with the same structure group G.
The term “soldering map” is not usually used for the mapping in (3.2a), but it proved convenient in adapting the more specialized concept of “soldered bundle” in (Kobayashi, 1956, 1957; Drechsler, 1977–1990) to the quantum regime, by viewing the infinite-dimensional quantum bundles studied in this monograph as “soldered” to the (tangent spaces of the) base manifold by such maps. In the previous papers dealing with quantum bundles (Prugovečki, 1987–89; Prugovečki and Warlow, 1989) the soldering maps were associated with sections s of the principal bundle P, rather than with the individual elements in it. The same pattern of construction could be followed in the present case by soldering the quantum frames directly to the Bargmann frames of the B M bundle in accordance with (3.7), and subsequently defining the wave functions ψ in (3.2b), by means of (3.8b), as components of fibre coordinates. Such a method of construction will be actually presented from Chapter 6 onwards, once the quantum frame bundles become sufficiently familiar to be regarded as the fundamental principal bundles with which various other quantum bundles are associated, rather than as some auxiliary mathematical objects. We give here, as well as in Sec. 5.1, an alternative formulation for pedagogical reasons, and also because it is more in keeping with the definition in standard physics literature [C,I] of associated bundles as G-products P ×G F, where P is a principal bundle consisting of classical rather than quantum frames. However, the former point of view actually becomes mandatory in the GS quantization of non-Abelian gauge fields, where the use of classical frame bundles becomes insufficient, and Grassmannian gauge degrees of freedom of quantum origin have to be introduced.
In all the cases treated in Chapter 2 the vector bundles had finite-dimensional fibres, so that the limits in (2.4.11), as well as (2.5.20), could be unambiguously defined as in vector calculus, i.e., by taking the corresponding limits for vector components in an arbitrary vector basis. In the present case the fibres are infinite-dimensional Hilbert spaces, so that a number of inequivalent topologies are of importance, of which the weak topology and the strong (or norm) topology [PQ] play the most significant roles. In a particular context, the choice of topology is conditioned by fundamental existence theorems. In the case of (3.10), the central theorem is a theorem which bears no specific name in the literature (in [PQ], p. 288, it is denoted as Theorem 3.1), but which is very closely related to the well-known Stone’s theorem. This central theorem states that if A is a self-adjoint operator, then the vector-valued function exp(iAt)ψ will be differentiable in the strong sense at t = 0 (and therefore also at all other t ∈ R 1) if and only if the vector ψ belongs to the domain of definition of A — which in case of unbounded A is only dense in the given Hilbert space. This theorem therefore ensures the existence of the strong limit in (3.10) under the subsequently discussed conditions.
A core of a self-adjoint operator A is a domain of essential self-adjointness, i.e., a linear space to which the restriction of A has a unique self-adjoint extension — which, of course, is A itself (cf. [PQ], p. 366, for a more general definition that applies to any closed operator). The most suitable cores for the covariant derivative operators in (3.11) are obtained by taking the linear spans of all elements of quantum frames belonging to some section of the quantum frame bundle QM, whose domain of definition contains the points where the covariant derivative is being computed — cf. Theorem A.2 in (Prugovečki and Warlow, 1989b).
As mentioned in Note 12 to Chapter 3, the case of an arbitrary choice of spin and mass can be treated by a suitable choice of SQM system of covariance.
This formulation and verification of (4.22) was explicitly carried out at the formal level by De Bièvre (1989a), who constructed a quantum Newton-Cartan bundle with L 2(R 3) as typical fibre, and (implicitly) used quantum frames Фx;k(Χq 0,q) built from the plane waves Фk(q) = (2π)−3/2exp(ik.q) — cf. Eqs. (3.6b,c) on p. 735 of (De Bièvre, 1989a). Such a construction reflects the fact that in QM textbooks, a plane wave Фk(x) is routinely interpreted as providing the “probability amplitudes” for observing a quantum point particle of 3-momentum k at the spatial location x in relation to a given inertial frame (which is conventionally always envisaged as being macroscopic and behaving in a totally classical manner). Of course, plane waves do not belong to L 2(R 3), but the resulting framework can be made mathematically more rigorous by using typical fibres which are rigged or equipped Hilbert spaces (Antoine, 1969; Prugovečki, 1973). However, the long-standing difficulties with the precise mathematical meaning of Feynman path integrals still remain even under those circumstances. Moreover, physical difficulties stemming from the uncertainty principle are also present, since the elements of such bundles, by their very definition, represent sharply localized states — and yet some of those elements are given in the form of plane waves which purportedly correspond to sharp values of 3-momentum. On the other hand, all these difficulties can be removed by regarding plane waves as representing sharp-point limits of quantum states giving rise to POV measures associated with systems of covariance in L 2(R 3) — namely of states represented by wave functions belonging to L 2(R 3), such as the ones in (3.6.7).
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© 1992 Springer Science+Business Media Dordrecht
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Prugovečki, E. (1992). Nonrelativistic Newton-Cartan Quantum Geometries. In: Quantum Geometry. Fundamental Theories of Physics, vol 48. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7971-1_4
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DOI: https://doi.org/10.1007/978-94-015-7971-1_4
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