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October 8, 2018 9:29 Contemporary Physics Palmer˙Sciama˙Rev˙2f Contemporary Physics Vol. 00, No. 00, Month 0000, 1–29 Lorenz, G¨odel and Penrose: New Perspectives on Determinism and Causality in Fundamental Physics T.N.Palmer University of Oxford, Clarendon Laboratory, Parks Rd, Oxford (v3.0 released March 2014) Despite being known for his pioneering work on chaotic unpredictability, the key discovery at the core of me- teorologist Ed Lorenz’s work is the link between space-time calculus and state-space fractal geometry. Indeed, properties of Lorenz’s fractal invariant set relate space-time calculus to deep areas of mathematics such as G¨odel’s Incompleteness Theorem. These properties, combined with some recent developments in theoretical and observational cosmology, motivate what is referred to as the ‘cosmological invariant set postulate’: that the universe U can be considered a deterministic dynamical system evolving on a causal measure-zero fractal invariant set IU in its state space. Symbolic representations of IU are constructed explicitly based on permu- tation representations of quaternions. The resulting ‘invariant set theory’ provides some new perspectives on determinism and causality in fundamental physics. For example, whilst the cosmological invariant set appears to have a rich enough structure to allow a description of quantum probability, its measure-zero character en- sures it is sparse enough to prevent invariant set theory being constrained by the Bell inequality (consistent with a partial violation of the so-called measurement independence postulate). The primacy of geometry as embodied in the proposed theory extends the principles underpinning general relativity. As a result, the physi- cal basis for contemporary programmes which apply standard field quantisation to some putative gravitational lagrangian is questioned. Consistent with Penrose’s suggestion of a deterministic but non-computable theory of fundamental physics, a ‘gravitational theory of the quantum’ is proposed based on the geometry of IU , with potential observational consequences for the dark universe. 1. Introduction There were three great revolutions in 20th Century theoretical physics: relativity theory, quan- tum theory and chaos theory (Fig 1). Each has had a profound impact on the development of science, and yet their domains of impact remain quite distinct. Despite over a half century of intense research, there is still no consensus on how to combine quantum theory and general relativity theory into a supposed ‘quantum theory of gravity’, nor even a consensus about what such a notion means physically (left edge of triangle). Moreover, the unpredictability of non- linear chaotic systems is generally considered quite unrelated to the indeterminism of quantum measurement (right edge of triangle). Finally, whilst an essential characteristic of chaos is the arXiv:1309.2396v2 [quant-ph] 25 Mar 2014 existence of positive Lyapunov exponents (measuring exponential divergence of neighbouring state-space trajectories in time) this is not a relativistically invariant characteristic; under a logarithmic transformation of time, a positive exponent can be transformed to a zero exponent (bottom edge of triangle). The purpose of this paper is to develop new ideas which may lead to some unification of these three revolutions. These ideas evolve around Einstein’s great insight that geometry provides the ultimate expression of the laws of physics. Motivated by develop- ments in cosmology on the one hand and nonlinear dynamics on the other, here we attempt to extend this geometric insight from space-time to state-space. The key idea which motivates the discussion in this paper is the fractal state-space geometry associated with certain classes of nonlinear dynamical system exemplified by the Lorenz equations [29] (see Section 2). The key cosmological developments which justify this discussion are those of quasi-cyclic cosmologies, Email: [email protected] ISSN: 0010-7514 print/ISSN 1366-5812 online c 0000 Taylor & Francis DOI: 10.1080/0010751YYxxxxxxxx http://www.informaworld.com October 8, 2018 9:29 Contemporary Physics Palmer˙Sciama˙Rev˙2f 2 T.N.Palmer Quantum Theory Iu Relativity Chaos Theory Theory Figure 1. It is proposed that the three great revolutions of 20th Century physics can be unified if the universe is considered a deterministic dynamical system evolving on a fractal invariant set IU in state space. the discovery of a positive cosmological constant and the black-hole no-hair theorem. In Section 2 it is shown how properties of Lorenzian fractal geometry have links to one of the most famous theorem in 20th Century mathematics - the G¨odel incompleteness theorem - and indeed also to some of the number-theoretic tools used in Wiles’ proof of Fermat’s Last Theorem. If this geometry relates to deep concepts from 20th Century mathematics, could it also relate to deep but still troublesome concepts from 20th Century physics; concepts such as state superposition, incompatible observables and quantum non-locality? Above all, could a focus on state-space geometry provide some new perspectives on the problem of ‘quantum gravity’? In Section 3, the strong formal similarities between the Schr¨odinger equation and the classical Liouville equation for conservation of state-space probability are discussed. Not least, both equations are linear. Just as the linearity of the Liouville equation is no constraint to how nonlinear and chaotic may be the underpinning deterministic evolution equations, is it possible that the linearity of the Schr¨odinger equation hides some underpinning deterministic dynamic that is also profoundly nonlinear? The standard view is that it is not possible if we require these dynamics to be locally causal; this is the Bell Theorem, a focus for discussion in this paper. By introducing a causal concept called ‘the cosmological invariant set postulate’ [36], an im- portant conclusion of this paper is that this standard acausal interpretation of the Bell Theorem is incorrect. The cosmological invariant set postulate, that the universe U is evolving causally and deterministically on some measure-zero fractal invariant set IU in its state space, is intro- duced in Section 4. As discussed in Section 5, this postulate is a potential game changer as far as the Bell Theorem is concerned, raising the possibility that the experimentally verified existence of nonclassical correlations can be explained deterministically without resorting to any violation of local causality in space-time (the technical role of the invariant set postulate in the Bell Theorem is that it provides a non-conspiratorial cosmological basis for a partial violation of the so-called measurement independence postulate [23] [24]). In Section 6 these ideas are de- veloped more explicitly. In particular, a symbolic representation of IU is constructed; from this it is shown how key properties of qubit quantum physics are emergent. ‘Invariant set theory’ provides the means to explain the three key differences between the Schr¨odinger equation and the classical Liouville equation: the existence of the square root of minus one and of Planck’s constant (Section 6.1), and of the complex Hilbert Space (Section 6.2). In particular the sym- bolic construction of IU makes use of a novel permutation/negation representation of complex numbers and it is outlined how the complex Hilbert Space of quantum theory can be viewed as the singular limit [6] of symbolic bit strings associated with the construction. In Section 7, two October 8, 2018 9:29 Contemporary Physics Palmer˙Sciama˙Rev˙2f Contemporary Physics 3 features which emerge from this causal deterministic construction are discussed: the notion of incompatible observables and violation of the Bell inequalities. In both cases, the emergence of these features hinges on an application of simple (but profound) number-theoretic properties of the cosine function. As discussed, in Section 7, these features also allows a new perspective on the iconic two-slit experiment. This paper is based on the author’s 9th Dennis Sciama Memorial Lecture given in 2013 1. The author was a student of Dennis Sciama in the 1970s, working on the gravitational energy- momentum problem in general relativity theory [33] [34]. During this period, he was strongly influenced by the work of Roger Penrose, one of the protagonists in the title of this paper. As Penrose wrote in the 1970s [39]: Despite impressive progress . towards the intended goal of a satisfactory quantum theory of gravity, there remain fundamental problems whose solutions do not appear to be yet in sight. [I]t has been argued that Einsteins equations should perhaps be replaced by something more compatible with conventional quantum theory. There is also the alternative possibility, which has occasionally been aired, that some of the basic principles of quantum mechanics may need to be called into question. Consistent with this ‘alternative possibility’, it is suggested that quantum theory is not fun- damental, but rather should be considered the (singular) limit of a causal deterministic theory of gravity which is geometric not only in space-time but also in state space. That is to say, it is proposed that invariant set theory be considered the basis for a causal ‘gravitational the- ory of the quantum’. As a result, in Section 8 it is claimed that contemporary programmes, which seek a ‘quantum theory of gravity’ by applying standard field quantisation to some puta- tive gravitational lagrangian, are

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