Typos found and reported so far will be listed here, and periodically updated. Readers should take into account this list if reporting typos or if reporting, suggesting or discussing corrections.
The ebook version of the book contains a substantial number of typos along the lines of font changes that were neither solicited by the Author nor passed by the Author before being made. Due to their large number, these will only be comprehensively dealt with at a rather later date. Particularly egregious or confusing examples among these excepted.
Comment 1. The Book reviews the Problem of Time from its posing by Wheeler and DeWitt in the late 60s (and of smaller pieces of it from somewhat earlier, most notably under other names in the works of Dirac) through to the date of the Book (2017). It is a rather comprehensive review of these, around 1 order of magnitude more so than previous acclaimed Reviews of this subject from the early 1990s by Kuchar and by Isham.
Some of this extra ink is used to lay down how time is manifested in each of the observationally accepted paradigms of Physics. I.e. Newtonian Physics, Special Relativity, Ordinary Quantum Theory, Special-Relativistic Quantum Theory and General Relativity. This is crucial because time is a substantially different concept in General Relativity than in the other paradigms (for all that there are more subtle differences in time concepts between the others as well). A systematic and conceptual account of this had, in fact, never been detailed before in one piece. Part I of the Book serves to precisely fill this gap, and argue from it to the pieces of the Problem of Time that, successively, Dirac, Kuchar, and Isham found.
Kuchar and Isham did not know what mathematical structure they were moving toward a formulation of in their carefully cataloguing the greatly nonlinearly interfering facets of the Problem of Time as manifested in hundreds of papers by other authors (and themselves) that tried to address the Problem of Time, only to find themselves blocked out after considering some facets by other facets. What I found in 2010-2017 is that most of these facets could be reconceptualized as underlying Background Independence aspects. Earlier work from 1982 to around 2005 was initially helpful in this regard. Though it quite quickly became apparent that Dirac’s work was going to be central to understanding what the Problem of Time is. This represents a shift from conceptualizing about kinematical preliminaries to a core of constrained dynamics.
Comment 2. In 2018 to 2020, I focused in particular on the classical manifestation of the Problem of Time. This is meaningful because, while the Problem of Time has often been addressed as between General Relativity and Quantum Theory, it is, at a deeper level between Background Independent Physics and Background Dependent Physics. And this clash persists at the classical level. Every aspect underlying every facet is nontrivially realized at the classical level. Moreover, it turns out to be rather more straightforward to mathematically identify the classical version of the aspects. And moreover, these do form a recognizable whole, whose large amounts of linear interferences were already well-understood from elsewhere in Mathematics. Indeed, carrying out Kuchar and Isham’s reverse engineering from facets to aspects to a coherent whole, at the local and classical level, turns out to be a recovery of a well-defined large chunk of Lie Theory! The short version of this stunning result was published in 2021 in Geometry, Integrability and Quantization, Volume 22 pp 43-63 as “Lie Theory Suffices to Resolve the Local Classical Problem of Time”. The full version has just now been published in 2023 in Volume 3 of Time and Science, ed. R. Lestienne and P.A. Harris (World Scientific) as “Problem of Time: Lie Theory Suffices to Resolve It”.
Comment 3. This space shall serve to point the Reader to preprints superceded by each of these articles. As well as to longer accounts in multiple parts which shall be cast into monograph form over the next few years.
Comment 4. Dirac worked on constrained dynamical systems, with an eye to being general enough to include General Relativity, in around 1949 to 1964. Unfortunately, Lie Theory was largely dormant from Lie’s era (1880s-1890s) until the 1960s. Exceptions to this shall be highlighted further down this comments page, but the main point is that Lie Theory only became a mainstream, rapidly expanding and widely applied subject in the 1960s. Because of this unfortunate timing, Dirac and the revivers of Lie Theory missed each other rather than joining forces. Nor did subsequent authors spot how Dirac’s approach to constrained dynamics and Lie Theory could in fact strengthen each other.
Comment 5. This local classical solution of the Problem of Time is moreover categorically poseable. For instance, it can be posed for other brackets algebras. This includes for quantum operator level brackets algebras. More simply, it can be posed for flat geometry, resulting in a new derivation of there being two competing top geometries in flat space: projective geometry and conformal geometry. This by itself is already indicatory of unexpected fertility.
Comment 6. Does this mean that the Problem of Time is also solved at the quantum level? No. Because “posed” does not mean “solved”. The difference between “posed” and “solved” turns out to be this. Certain cohomology groups need to be suitable, and the analogue of Teitelboim’s Refoliation Invariance solution of the Foliation-Dependence Problem facet of the Problem of Time needs to work out. So each category one is interested in requires specific cohomological calculation, identification of what analogue of Differential Geometry’s foliations it has, and of whether this analogue has an analogue of the Refoliation Invariance property that Teitelboim showed the Dirac constraint algebroid of General Relativity to possess. In other words, there are two concrete calculations that need to be carried out for each category of interest. Conceptual and philosophical consideration of the Problem of Time has thereby been replaced with a precise prescription for two “local” (only mildly topological) mathematical calculations.
Comment 7. It is highly natural for the local classical Problem of Time to be resolved by Lie Theory. For General Relativity’s Metric Geometry is mounted upon the Differential-Geometric framework. And Lie Theory is then the simplest way of conceiving of many physically-significant features at this level of Mathematics. This includes the famous application of Killing vector ideas to General Relativity. A theory of observables, for instance as developed along the lines of Lie’s integral theory of geometrical invariants. And the theory of foliations upon slicing up spacetime into a series of spatial slices.
Comment 8. What are the main known implications of the result that the local classical Problem of Time is solved by Lie Theory?
Let us split this up into three parts. Firstly, a quite general expectation was that the Problem of Time was a series of conceptual problems that were preventing Quantization of Gravity. So perhaps if one understood Kuchar and Isham’s facets, then one would have new mathematical structure which would offer new possibilities for such Quantization. However, an already well-known mathematical structure drops out: Lie Theory. This resolves these particular conceptual problems: un-identified mathematics has been determined to be an already well-known, basic and establishedly widely-applicable branch of Mathematics.
So, suddenly two or three orders of magnitude more people know what the Problem of Time is: those with substantial prior experience in Lie Theory. Or they would know, were they to take interest in this newly-identified application of Lie Theory. It likewise becomes possible to explain what the Problem of Time is in quite some detail to fourth-year University students in Mathematics or in Theoretical Physics. This is quite some leap from it being a very advanced, highly obscure field of research only considered by a small number of specialist researchers, and of generalists: each capable of writing reviews in many different areas of the Theory of STEM.
But there is no new mathematical framework, at least to start off with. Indeed, instead of a Quantum Gravity theory dropping out of this work, a theory of Background Independence has dropped out. The first main port of call is in fact that the Principles of Dynamics can be reformulated in Background-Independent form. This is a subsequent large upgrade to Dirac’s passing from the standard Principles of Dynamics to its extension by a specific prescription for systematically handling constrained dynamical systems. This first port of call is indeed the form that the Author’s next monograph on the Problem of Time shall take. Its working title is: “Principles of Dynamics: from the Standard Formulations to the Lie-Dirac Reformulation that Locally Solves the Problem of Time”.
We have also hit upon two specific technical questions to ask of each sufficiently Background-Independent program for Quantum Gravity. For understanding that the Problem of Time is just Lie Theory at the local classical level, and that the way this works out categorically poseable, and solved in the case of General Relativity by a brackets algebra obstruction being absent (cohomology) and a foliation-type result working out (Teitelboim’s result), gives us two technical criteria to look at category by category.
On the one hand, the first of these technical criteria is however of an already well-known kind. For at the Quantum level, the main kind of brackets algebra obstructions are known as anomalies. So the idea that conceptual thinking about time would somehow give us a Quantum Gravity free from this major technical problem has turned out to be wishful thinking. This is very siginifcant, since may Quantum Gravity experts have agreed since the 1980s that anomalies are a particularly severe restriction on possible theories of Quantum Gravity. And now those hoping that the Wheeler, DeWitt, Dirac, Kuchar and Isham line of thinking would somehow get around this have to face the music: it does not.
On the other hand, the second of these technical criteria has not often been considered. A careful study of the Problem of Time has thus revealed that, while nothing can be done about brackets algebra obstructions, anomalies included, a second difficult technical problem accompanies them. Classical General Relativity manages to work out because of the extrinsic geometry (higher-d version of Gauss’ Theorema Egregium), foliations, Refoliation Invariance tower. Which is automatially solved by the form taken by the Dirac algebroid formed from General Relativity’s constraint equations. This is furthermore a deep aspect of Background Independence. And so one needs to ask what the analogues of extrinsic geometry, Theorema Egregium type results, foliations and Refoliation Invariance are in each category of interest. Such as for plausible operator algebra categories for formulating Quantum Gravity theories. This is certainly an item of progress, in the sense of a further technical restriction on what can work has been identified. It is also a subsequent impediment to progress: Quantizing Gravity (in a Background-Independent manner) has become harder as a result of its identification.
Secondly, Lie Theory itself has been strengthened. This is because Dirac’s approach does not presuppose consistency, and is consequently a selection principle. Lie’s Algorithm missed this key feature (which is the powerhouse behind Dirac’s constrained dynamics). Thus Dirac’s work is not merely a subcase of Lie Theory, but also contains an improvement. This is how the above new result for the foundations of Flat Geometry arises. It can also be expected that many other areas of STEM to which Lie Theory is usefully applicable will have new results of this kind.
The Dirac Algorithm, and by extension the Lie-Dirac generalization of the Lie Algorithm is capable of finding foundational results systematically. Within the local classical Problem of Time, this gives that the Relativity must be locally Lorentzian, locally Galilean, locally Carrollian, or the General Relativity Initial Value Problem’s constant-mean-curvature (CMC) slicing. And that either the DeWitt supermetric coefficient of 3 + 1 split GR must be present or the gravity must be ‘strong’. But this would have permitted humanity to hit upon GR in the absence of Einstein’s (in this case universalistic) genius. In fact, Einstein missed the third logical possibility of Carrollian Relativity (the CMC case is a “Dirac surprise” rather than an a priori logical possibility to the universalist mindset).
Within Flat Geometry, this gives that if there are quadratic generators, then they must be either special-projective or special-conformal. But not both at once, since the two mutually brackets-obstruct each other. And this could have been used to derive flat space’s Projective Geometry and (for technical reasons 3-d or higher) Conformal Geometry if these had not already been worked out. Systematic Projective Geometry started with Desargues in the 17th Century. Conformal Geometry is much more of a 19th and 20th Century subject, without any clear one founder.
So now let us go after other major applications of Lie Theory and see what we get!
A) The symmetries of differential equations are an obvious target.
B) So are other brackets’ analogues. Notable other brackets including the anticommutator of fermionic and then supersymmetric theory, the Nijenhuis bracket (a prototype of quentum operator algebra brackets) and the Nambu bracket (used in a so far minor approach to M-theory). I have already covered these three cases to some extent. The Problem of Time book already documents that Supergravity is qualitatively different from General Relativity as regards Problem of Time facets. And I put preprints on arXiv on the Nijenhuis-Dirac and Nambu-Dirac analogues of the Lie-Dirac Algorithm. In essence, 70 years’ worth of developing brackets algebra variants of Lie brackets can be injected ‘overnight’ into the recent Problem of Time/Lie-Dirac advance. And the above three mentions are a first step for each of these programs. Perhaps somebody else could work out if any of these have brackets obstructions, and what the analogue of the above tower gives in each case… Were nobody else to bestir themselves thus, the Author would eventually get round to writing such matters up.
Thirdly, understanding that the local classical Problem of Time is solved by Lie Theory has revolutionized understanding of the Global Problem of Time. The Kuchar 1991 to the Problem of Time Book 2017 era had order 10 subaspects to its global Problem of Time. Having identified the local version as Lie Theory, however, we have the benefit of the past 70 years of systematic study of Lie Theory, so we can now more precisely formulate order 100 subaspects of this! I made about 1/4 of these publicly visible between here: https://conceptsofshape.space/2020/12/07/global-problem-of-time-sextet-1-introduction-and-notions-of-globality/ , and here: https://conceptsofshape.space/2020/12/21/global-problem-of-time-sextet-o-relational-preliminaries-generator-provision-and-stratification/ .
Working on this requires a substantial amount of preparation. As well as understanding both local Lie Theory, the abovementioned paradigms of Physics and the Problem of Time, one now also needs to be a generalist Applied Topologist, or to be part of a large enough research group to have this skill-set. It is also well worth emphasizing that one and the same Topology subject arises no matter which area of STEM one sudies the theory of in sufficient depth. So a large investment in topological skills comes with a large number of benefits.
Indeed, the way I am progressing toward a cataloguing of Global Problem of Time aspects – a Kuchar-Isham facet cataloguing counterpart for the 2020s – is to write out a tower of Topology books. More precisely, I am splitting Mathematics for the Theory of all STEM into Pre-Topology and Topology, as an upgrade of the current widespread practise of splitting basic Mathematics into Pre-Calculus and Calculus. With the initial summit of the Pre-Topology series being the above-mentioned Lie-Dirac reformulation of the Principles of Dynamics. And longer-term summits for the Topology Series being a suitable presentation of Global Lie Theory feeding into a Global Principles of Dynamics and a Global Quantum Theory of Background Independence.
Comment 9. Apart from the Lie-Dirac replacement for Lie’s Algorithm, are there any other differences between the Problem of Time’s local Lie Theory and the Lie Theory taught to beginners at grad school?
Yes. We need not only Lie algebras but also Lie algebroids, because the Dirac algebroid of GR is a such (or, better, with one axiom more, a Poisson algebroid). In the Quantum Gravity literature, we have Martin Bojowald to thank for this mathematical observation. These have not structure constants but structure functions. It is through having particular examples of such that the Dirac algebroid manages to locally encode all possible foliations at once. Also most grad school introductory courses on Lie Theory will not cover Lie Theory as applied to foliations. Most do not mention observables either, and some do not even mention Killing’s equation (Lie derivative of metric = 0) whose solutions are the Killing vectors. And yet each of these topics can be well outlined in around 3 lectures. So one is just talking about a 12-lecture supplement to a typical introductory-level grad school course on Lie Theory. A thus-supplemented first course on Lie Theory will become available shortly, as the monograph preceding my Principles of Dynamics. I will also very soon cite here a series of research papers to make do as regards such a supplement in the meanwhile.
Comment 10: So who worked on Lie Theory between Lie’s day and the 1960s? And who produced key work in reviving Lie Theory then?
A short answer to the first question is that Cartan did the lion’s share. See his Collected Works (1955) for the material!
As to the second question, key work that the Problem of Time makes use of on Lie Theory from the 1960s is Gerstenhaber’s and Rinehart’s. Though also the Nijenhuis-type generalization was under development the. Some of this goes as far back as the mid 1950s, with some being joint with Schouten, Frolicher or Richardson. Also the first modern textbook on Lie Theory was Jacobson (1962). Lie Theory was of course also explosively applied to Particle Physics in the late 1960s, thus entering into mainstream Theoretical Physics very fast indeed.
Let me then expand the first answer to mention Chevalley and Harish-Chandra as other pioneers.
Comment 11. The Author wishes to emphasize, following Chris Isham (a mentor from 2007 to 2015), that Quantum Theory is inherently global. As such, it makes limited sense to consider a local quantum Problem of Time. Next, following Dirac’s maxim that classica systems are easier than their quantum counterparts, it makes good sense to attempt the classical Global Problem of Time before the quantum Global Problem of Time. The Author will consequently I) just write a Part of a monograph on quantum Problem of Time considerations at the local (Pre-Topology) level. II) while targetting the quantum Global Problem of Time with a tower of monographs passing through the classical Global Problem of Time. It is fair to say that I) is a very quick fix compared to II).
Concepts of Shape 