This book examines the foundational consistency of quantum mechanics incorporated within relativistic frameworks. Quantum physics remains a perplexing formalism that, although very successful in explaining physical phenomena, poses many philosophical and interpretational questions. Several of the subtleties of quantum physics become more manifest when quantum processes are described using relativistic dynamics. For instance, the successful connection of spin to quantum statistics is a consequence of the consistent incorporation of special relativity into the quantum formalism. There should be similar profound explanations awaiting discovery as gravitating phenomena are successfully incorporated into quantum formulations.

The common theme of this manuscript is the examination of the incorporation of relativistic behaviors upon the foundations of quantum physics. The approach is to keep all formulations as close to observed phenomena as possible, rather than to present a set of speculative models whose primary motivations are internal aesthetics. In the search for the most elegant models of physical phenomena, one must recognize that at its core, physics is an experimental science. The dimensional analysis of fundamental units, taught at the very beginning of introductory physics classes, demonstrates that phenomenology lies at the foundations of physics. Fundamental ideas such as correspondence, the principle of relativity, and complementarity provide direct contact with the physics used to guide this exploration. This manuscript is an elaboration and expansion on previously published work, but also contains some new material.

A synopsis of Big Bang cosmology

One of the fundamental principles guiding the expected behaviors of the cosmology is a generalization of ideas of Copernicus, known as the cosmological principle. This principle presumes that no non-rotating observer at rest to the CMB radiation is more special than any other. Since the observed universe has large-scale uniformity, the cosmological principle imposes an overall homogeneity and isotropy to the universe. In addition, the dynamics of most of the aggregate features in the universe can be described assuming that the energy-momentum content of the cosmology is consistent with being an ideal fluid.

The Friedmann-Lemaitre equations 8.6, which describe the dynamics of an ideal fluid cosmology, are spatially scale invariant (if the cosmological constant is negligible), but not temporally scale invariant. The form of those equations that govern a spatially flat (k = 0) expansion satisfies spatial scale invariance (at least to a very good approximation), due to the fact that the energy densities that drive the dynamics are intensive thermodynamic variables. However, there is apparently a beginning time t0 ≈ 13.7(±0.2) billion years ago, which represents the earliest backwards-looking extrapolation of the standard model expansion called the Big Bang. The physics during these earliest moments is an active field of research. Thus, the cosmological principle does not refer to the temporal evolution of the universe.

Fundamentals of quantum mechanics

Quantum mechanics remains one of the most successful, yet enigmatic, formulations of physics. Uncertainty and measurement constraints are incorporated at the core of this fundamental description of micro-physical dynamics. Quantum mechanics successfully describes the observed structures in chemistry and materials science. In particular, the impenetrability of matter due to Pauli exclusion is a consequence of incorporating special relativity into quantum mechanics. Microscopic causality, or the requirement that communications cannot propagate at greater than the speed of light, relates a particle's spin to its quantum statistics but does not exclude the space-like correlations associated with a coherent quantum state.

This section will examine some of the foundations of quantum physics. An emphasis will be placed upon how the microscopic fundamentals of quantum mechanics relate to the geometric fundamentals of gravitational mechanics.

Quantum formalism

There are several interpretations of quantum mechanics offering models of the underlying (often hidden) dynamics that generate the observed quantum phenomenology (for example, the Copenhagen interpretation, or the many-worlds interpretation). Since any interpretation consistent with quantum physics cannot be proved or disproved by experiment, only those aspects of quantum formalism directly connected to experimental observables will be developed, with minimal interpretation imposed. To establish the conventions utilized in what follows, a brief overview of the formalism that describes those calculable observables modeled by quantum physics will be given.

The fundamental “players” in the cosmological arena are microscopic particles and the interactions by which they exchange well-defined quantum numbers. Many of the critical properties of micro-physics can be determined by their behaviors in nearly flat space-time, as described by Minkowski. There are several requirements that a successful model of fundamental processes should fulfill. Among these characteristics are:

1. • Describes quantum phenomenology: Quantum mechanics successfully describes the subtle behaviors of matter and energies undergoing microscopic exchanges. Quantum behaviors inherently have aspects beyond measurement.

2. • Conservation properties: Most particles have internal quantum numbers, like charge, lepton number, baryon number, etc., that are carried undiminished throughout complicated interactions with other particles. In addition, they carry properties like mass and spin, whose kinematic transformations under space-time transformations are well-defined, and which satisfy composite conservation laws (energy, momentum, angular momentum) for sufficiently isolated homogeneous and isotropic systems.

3. • Unitarity: Despite being unable to follow quantum coherent particle properties while that coherence is maintained, the evolution of those properties is described in a manner that ensures conservation of probability. This means that these properties do not just “pop” into or out of existence between the detections and interrelations of the particles.

4. • Cluster decomposability and classical correspondence: Classical physics is quite successful in describing much of common phenomenology. Classical models exploit those characteristics of a system that can be isolated, studied, and parameterized independent of observation.

5. […]

The incorporation of quantum mechanics into gravitational dynamics introduces perplexing issues into modern physics. In contrast to other interactions like electromagnetism, the classical trajectory of a gravitating system is independent of the mass coupling to the gravitational field. As previously discussed, this allows the gravitation of arbitrary test particles to be described in terms of local geometry only, the basis of general relativity. Thus, the geometrodynamics of classical general relativity are most directly expressed using localized geodesics. However, quantum dynamics incorporate measurement constraints that disallow complete localization of physical systems. A coherent quantum system is not represented by a path or a classical trajectory; rather, it self-interferes throughout regions. This complicates the use of classical formulations in describing inherently quantum processes.

In addition, the equations of general relativity are complex and non-linear in the interrelations between sources and geometry, which makes solutions of even classical systems complicated. The key to describing complex systems is to determine the most useful set of parameters and coordinates that give concise predictive explanations of those systems. This chapter will develop tools for examining quantum behaviors in gravitating systems.

Quantum coherence and gravity

The behaviors of quantum objects in Minkowski space-time are well understood, despite a lack of consensus on the various interpretations (Copenhagen, many worlds, etc.) of the underlying fundamentals of the quantum world, or concerns of the completeness of quantum theory. There have also been tests of systems modeled by equations that involve both Newton's gravitational constant GN and Planck's constant ħ, as described in Section 2.3.1.

One of the principles of modern physics that is most adhered to is the expectation that models constructed to describe the phenomena of the physical universe should not depend upon any absolute frame of reference. The discovery of the CMB radiation perhaps demonstrates a counter example to this supposition, due to the preferred frame at rest relative to the energy content of the universe during its initial phase of expansion, as will be discussed in the next chapter. However, for most phenomena, the co-variance of the laws modeling those phenomena is consistent with the expectation of independence of the fundamental physics from the particular frame of reference utilized by the observer. This principle is embodied in the concept of complementarity [23] in the description of black holes. In its most direct expression, complementarity simply states that no observer should ever witness a violation of a law of nature. In particular, one expects that for a freely falling observer, there should be no local effects of gravitation as espoused by the principles of equivalence and relativity.

In this treatment, a horizon will always be a light-like surface that globally separates causally disconnected regions of space-time. Since light itself is characterized by both classical and quantum properties, geometries with horizons offer insights into the subtle relationships between general relativity and quantum physics. Generally, horizons can be only globally (not locally) defined, which means that local experiments performed by freely falling, inertial observers cannot detect the presence of such horizons.

This book is the result of decades of efforts by the author to understand physics at its most basic foundations, and to construct models that can address some of the unanswered questions about gravity, quantum mechanics, and the spectrum of fundamental particles. It is felt that foundational explorations should examine the conceptual and philosophical basis of a discipline. As such, less emphasis was placed upon the mathematics of complex calculations, while more emphasis was placed upon the consistencies and critiques of basic premises, in the preparation of this book. As a scientist, there is often a tendency to be drawn towards mathematical and formal pursuits, simply because of the beauty and elegance of mathematics. Such approaches sometimes bypass difficult conceptual approaches and thought experiments, or develop speculative formulations just because of their elegance. This book attempts to establish a balance towards the exploration of basic concepts and puzzles.

The prior book on black holes by Lenny Susskind and myself serves as an introduction to quantum physics and relativity for static geometries, as well as information in horizon physics. However, some of my more recent explorations indicate that qualitative modifications in descriptions of the physics occur once dynamics has been incorporated. This then calls into question any intuitions from static geometries that have been used to imply that those very geometries cannot be static. This book incorporates dynamic, spatially coherent geometries, as well as expanding upon the well-established foundations of the prior work.

Groups and special relativity

Fundamentals of group theory

Properties of groups

A group is a set of elements that have the following properties:

1. • Contains the identity transformation 1;

2. • If E and E′ are elements in the group, then there exists a group operation (generically called “multiplication”) that always produces an element of the group, E″ = E′ · E (closure);

3. • For every transformation element E, there exists in the group an inverse element E-1, where E-1 • E = 1;

4. • The group operation is associative, E′ · (E′ · E) = (E″ · E′) · E.

A particular type of group satisfies an additional property. For an abelian group, the group operation yields the same answer regardless of the order, E′ · E = E · E′. Often, a subset of the elements within a group satisfies all four group properties. This subset is referred to as a subgroup.

Generally, two different groups are isomorphic if there is a one-to-one relationships between the elements of the groups with regards to group operations. More generally, if a set of elements in one group are in direct relationship with one element in another, the groups are homomorphic.

A particular class of groups is quite useful for describing transformations in quantum physics. Invertible N × N matrices (i.e., matrices with non-vanishing determinants) form the set of linear groups.