Modeling Time’s Arrow

Modeling Time’s Arrow Quantum gravity, the initial low entropy state of the Universe, and the problem of time are interlocking puzzles. In this article, we address the origin of the arrow of time from a cosmological perspective motivated by a novel approach to quantum gravitation. Our proposal is based on a quantum counterpart of the equivalence principle, a general covariance of the dynamical phase space. We discuss how the nonlinear dynamics of such a system provides a natural description for cosmological evolution in the early Universe. We also underscore connections between the proposed non-perturbative quantum gravity model and fundamental questions in non-equilibrium statistical physics. The second law of thermodynamics, perhaps the deepest truth in all of science, tells us that the entropy is a non-decreasing function of time. The arrow of time implies that the Universe initially inhabited a low entropy state and the subsequent cosmological evolution took the Universe away from this state. A theory of quantum gravity must explain the singular nature of the initial conditions for the Universe. Such a theory should then, in turn, shed light on time, its microscopic and directional nature, its arrow. The quest for a theory of quantum gravity is fundamentally an attempt to reconcile two disparate notions of time. On the one hand, Einstein’s theory of general relativity teaches us that time is ultimately an illusion. On the other hand, quantum theory tells us that time evolution is an essential part of Nature. The failure to resolve the conflict between these competing notions is at the heart of our inability to properly describe the earliest moment of our Universe, the Big Bang. The puzzle of the origin of the Universe intimately connects to a second fundamental issue: should the initial conditions be treated separately from or in conjunction with the basic framework of the description of the dynamics? The discord between the general relativistic and quantum theoretic points of view has very deep roots. In this essay we will trace these roots to their origins. Then, drawing inspiration from the profound lessons learned from relativity and quantum theory, we propose a radical yet conservative solution to the problem of time. In quantum mechanics, time manifests as the fundamental evolution parameter of the underlying unitary group. We have a state |ψ⟩|ψ〉, and we evolve it as e−iℏHt|ψ⟩e-iℏHt|ψ〉. The Hamiltonian operator of a given system generates translations of the initial state in time. Unlike other conjugate quantities in the theory such as momentum and position, the relation between time and energy, which is the observable associated to the Hamiltonian, is distinguished. Time is not an observable in quantum theory in the sense that generally there is no associated “clock” operator. In the Schrödinger equation time simply enters as a parameter. This conception of time as a Newtonian construct that is global or absolute in a post-Newtonian theory persists even when we promote quantum mechanics to relativistic quantum field theory. In contrast, time in general relativity is local as well as dynamical. Suppose we promote general relativity to a quantum theory of gravity in a naïve fashion. In the path integral, the metric of spacetime is one more dynamical variable. It fluctuates quantum mechanically. So notions such as whether two events are spacelike separated become increasingly fuzzy as the fluctuations amplify. Indeed, Lorentzian metrics exist for almost all pairs of points on a spacetime manifold such that the metric distance is not spacelike. Clearly the notion of time, even locally, becomes problematic in quantum gravitational regimes. The commutation relation [O(x),O(y)]=0[O(x),O(y)]=0 when x and y are spacelike separated, but this is ambiguous once the metric is allowed to fluctuate. The failure of microcausality means that the intuitions and techniques of quantum field theory must be dramatically revised in any putative theory of quantum gravity. Crafting a theory of quantum gravity that resolves the problem of time is a monumental undertaking. From the previous discussion, we see that the standard conceptions of time in quantum theory and in classical general relativity are in extreme tension: time in the quantum theory is an absolute evolution parameter along the real line whereas in general relativity there can be no such one parameter evolution. A global timelike Killing direction may not even exist. If the vacuum energy density is the cosmological constant Λ, we may inhabit such a spacetime, de Sitter space. In any attempt to reconcile the identity of time in gravitation and canonical quantum theory, one is also immediately struck by the remarkable difference in the most commonly used formulations of the two theories. Whereas general relativity is articulated in a geometric language, quantum mechanics is most commonly thought of in algebraic terms within an operatorial, complex Hilbert space formalism. A principal obstacle to overcome rests with the rôle of time being intrinsically tied to the underlying structure of quantum theory, a foundation which—as recapitulated below—is rigidly fixed. Yet when quantum theory is examined in its less familiar geometric form, it mimics general relativity in essential aspects. In fact, these parallels provide a natural way to graft gravity into the theory at the root quantum level. Particularly, the geometric formulation illuminates the intrinsically statistical nature and rigidity of time in quantum theory and points to a very specific way to make time more elastic as is the case in general relativity. Thus, by loosening the standard quantum framework minutely, we can surprisingly deduce profound implications for quantum gravity, such as a resolution of the problems of time and of its arrow. In subsequent sections, we will lay out a framework for general quantum relativity in which the geometry of the quantum is identified with quantum gravity and time is given a dynamical statistical interpretation. Some bonuses stem from this point of view. We achieve a new conceptual understanding of the origin of the Universe, the unification of initial conditions, and a new dynamical framework in which to explore these and other issues. Here to reach a broader audience, conceptual rendition takes precedence over mathematical formalism the details of which are available in the literature given below. In Section 2, we examine the problem of time’s arrow. In Section 3, we briefly recall the geometric formulation of standard quantum mechanics, which naturally leads to a generalized background independent quantum theory of gravity and matter, the latter of which is embodied by M(atrix) theory. In Section 4, we apply this prescription to a theory of quantum gravity in its cosmological setting: the initial low entropy state of the Universe and cosmological evolution away from this state. Specifically we will discuss key properties of the new space of quantum states—a nonlinear Grassmannian—features which notably embody an initial cosmological state with zero entropy and provides a description of cosmological evolution when viewed as of a far from equilibrium dissipative system. We also will elaborate on a more general connection between quantum gravity, the concept of holography and some fundamental results in non-equilibrium statistical physics. Finally, Section 5 offers some concluding remarks.