Quantum reference frames and triality Lee Smolin∗ Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2J 2Y5, Canada and Department of Physics and Astronomy, University of Waterloo and Department of Philosophy, University of Toronto July 14, 2020 Abstract In a background independent theory without boundary, physical observables may be defined with respect to dynamical reference systems. However, I argue here that there may be a symmetry that exchanges the degrees of freedom of the physical frame of reference with the other degrees of freedom which are measured relative to that frame. This symmetry expresses the fact that the choice of frame of reference is arbi- trary, but the same laws apply to all, including observer and observed.. It is then suggested that, in a canonical description, this leads to an extension of the Born duality, which exchanges coordinate and momentum variables to a triality that mixes both with the temporal reference frame. This can also be expressed by extending 2n dimensional symplectic geometry to a d = 2n+1 dimensional geometry arXiv:2007.05957v1 [hep-th] 12 Jul 2020 with a cubic invariant. The choice of a temporal reference frame breaks the triality of the cubic invariant to the duality represented by the canonical two form. We discover that a very elegant way to display this structure which encompasses both classical and quantum mechanics, is in terms of matrix models based on a cubic action. There we see explicitly in either case how a spontaneous symmetry breaking leads to the emergence of a temporal reference frame. Note:: This article draws partly on [16] and in part supersedes [15]. ∗[email protected] 1 Contents 1 Introduction:trialityanddynamicalreferenceframes 2 2 Examples of extending dualities to trialities: classical theories 5 3 Cubic matrix theory as a template for a relational dynamics 8 3.1 Solutionsandtheemergenceofdynamics . ...... 10 3.2 Classicalorquantummechanics? . ..... 10 3.3 Breaking a symmetry to introduce a physical reference frame......... 10 3.4 Introducingadditionaldimensions.. ........ 12 3.5 Brokendualityandtheprincipleofinertia . ......... 13 3.6 Recoveryofthefreerelativisticparticle . .......... 14 3.7 CompactificationtoChern-Simonstheory . ....... 15 4 Conclusions 16 1 Introduction: triality and dynamical reference frames One of the most fundamental principles activating our search for a common completion of general relativity and quantum mechanics is background independence. This principle asserts that the laws of physics not depend on any, fixed, non-dynamical background structures[1, 2, 3, 4]. Background independence underlies all our modern understanding of the fundamental forces as its avatars in field theory are diffeomorphism invariance and local gauge invariance Another consequence of background independence is that the common completion of quantum mechanics and general relativity must be a cosmological theory. In [4] Mangabeira Unger and I make a detailed analysis of how a theory of the whole universe as a closed system must differ from a theory of a portion of the universe. This is also argued in [1, 2, 3]. In particular a theory of the whole universe contains its observers and frames of ref- erence as dynamical subsystems. As Einstein taught us, any complete description of a physical observable contains, implicitly at least, a description of how a frame of refer- ence, which gives a precise meaning to that observable, is established in each concrete application to a physical system. Most importantly, any theory that ascribes times to physical events, must contain a dynamical description of an internal clock. Similarly, any use of location or velocity must begin as a measurement of relative position-between the system under study and a physical system that is defined as a frame of reference. It is very important to understand that fundamentally, nothing-except a certain very minimal level of complexity, distinguishes the physical subsystems of the universe that we choose to play the roles of frame of reference and/or observer, from other subsystems, which we use the frame of reference to describe. Hence, any description that employs one 2 subsystem as the frame of reference, whose clock readings and rulers define physically meaningful quantities, breaks a symmetry. That symmetry mixes up observer and observed, as it expresses the obviously true fact that both the observer or frame of reference and the physical systems whose description they anchor are physical subsystems of the universe. Consequently, so long as each has sufficient complexity, it must be possible to give a description in which the two systems trade places-and what was the physical system is now the frame of reference, and vice versa. This implies that there must be two novel structural elements: first, a symmetrical de- scription of the system in which the degrees of freedom usually described are mixed up with degrees of freedom which, when frozen, create a reference system. Second, this sym- metry must be spontaneously broken, in a way that freezes and decouples those degrees of freedom that will serve as the reference systems. It follows directly from this that if there is already a symmetry amongst a number of the particles of the system, say a permutation symmetry, PN or a continuous symmetry SO(N), including the frame of reference might increase this to an SN+1 or SO(N + 1). In the canonical formalism this might extend an sp(n) symmetry of symplectic geometry to an SO(d +1,d) symmetry. We should also note the subtle behaviour of degrees of freedom attributed to physical reference frames. To serve their purpose, they must be very weakly coupled to the other physical degrees of freedom, hence they must be close to a limit in which some decou- pling parameter, λ is taken to zero. But one cannot take the limit in which λ = 0 lest they decouple completely. Often one is interested then in only the linear, response of the measuring instruments to the physical degrees of freedom they give access to. This paper continues and brings together, several previous lines of research, on the basis of the new perspective offered by the investigations into dynamical and quantum frames of reference[5]-[9][10]. These include the proposal to extend dualities to trialities put forward in [15]. A model for this extension is found in the study of a special kind of matrix model, cubic in the matrices[11]-[16]. These are models which have a diverse set of avators, which may emerge from them, including topological field theories, gauge theories and theories of gravity[16]. In this paper, we study first the most elementary cases, in which the reference degree of freedom is only a time parameter, so the gauge symmetry which enforces its redun- dency is time reparametrization. This is spontaneously broken, freezing the choice of time variables, which may then be identified as the reading of an internal clock. We next locate the symmetry whose breaking gives rise to the separation of a frame of reference which includes a clock, as an augmentation of the fundamental Born duality to a triality. The extra, third degree of freedom is a gauge parameter that under symmetry breaking to distinguish it, it becomes an operator that measures the time rate of change of other operators.. We will see in detail how this works in both classical and quantum mechanics. In the classical case this transformation amongst frames of reference is found to be 3 an element of a group of transformations- such as the Galilean or Poincare groups. But the quantum case is believed to be considerably more intricate, because the observer and system may be entangled or in superposition of states in which the frame of reference has definite values. Here, I propose a general solution to this cluster of problems based on the following schema (or principles1.) 1) Both classical and quantum dynamics are built on dualities. In classical mechanics this is the Born duality between momentum and position[27]. In quantum mechanics we also have a form of Born duality; and we have as well the duality between bra and ket: or in other words between preparation and measurement. 2) In both cases the duality pivots on a fixed, static point, which represents a fixed frame of reference. These are the clock, (the choice of which is also represented in the choice of symplectic structure (in classical physics), and the hermitian inner product of Hilbert space. This is no surprise, as we know the symplectic structure and the Hilbert space inner product are closely tied to each other. 3) In order to realize the symmetry that trades physical system and frame of reference, we must enlarge the duality to a triality, as suggested in [15], whose third element is precisely the symplectic structure or Hilbert space norm that gives meaning to the basic observations (which are always of relative quantities.). The classical triality is d xa ↔ p ↔ (1) b dt The quantum version is < Ψ|↔ |Φ >↔ † (2) I also describe below, the same idea expressed in terms of an extension of symplectic geometry, in which a 2n dimensional phase space, Γ is augmented by the addition of an explicit clock variable, to a d =2n +1 dimensional space, τ. The quadratic and antisym- metric Poisson structure, Ω, on Γ is augmented to a triple product, Φ, antisymmetric, on τ. Φ is represented by a three-form on τ, and we can see explicitly how its symmetry is broken by the choice of a time variable, reducing the three form, Φ to the symplectic two-form Ω. In this paper I present a model for how physical reference frames emerge from a spon- taneous breaking of this triality symmetry in both classical and quantum physics. The model is a particular matrix model[22]-[26], by which is meant a model whose degrees of freedom are described as large matrices.