**From mathematical concepts to physical models and "Euclidean" almost commutative geometries to "Lorentzian" almost commutative spacetimes**

The set-up of the Dirac equation [1] was a prodigious step towards the unification of quantum and relativistic principles. Surprisingly enough, the differential operator engaged in the Dirac equation turned out to play a pivotal role in differential geometry [2]. Moreover, it lies at the heart of Connes’ theory of noncommutative geometry [3, 4], which extends such classical notions as differentiation, distance [3] and causality [5] to an abstract algebraic setting.Nowadays, noncommutative geometry centred around the concept of the Dirac operator provides a compelling framework for the study of fundamental interactions [6], yielding concrete testable predictions in the domain of elementary particles [7, 8] and gravitational physics [9, 10].

Building upon Connes’ ideas [11] we model a single massive Dirac fermion with the help of an almost commutative spacetime. The latter turns out to provide a geometric description of one of the peculiarities of the Dirac theory – the Zitterbewegung[12]. We show that the causal structure of the almost commutative spacetime at hand puts an explicit bound on the frequency of the ‘trembling motion’ of a single Dirac fermion propagating in a possibly curved spacetime and interacting with the electromagnetic and a Higgs-like scalar field. Next,we explain how the concept of quantum simulation [13] can be promoted to emulate almost commutative spacetimes, thus opening the door to a direct experimental test of Connes’ theory.

...

The original explanation of the mechanism behind Zitterbewegung given by Schrödinger[12], and refined by several authors [14, 16, 17],relates it to the interference between the positive and negative energy parts of the Dirac wave packet...

The tangibility of Zitterbewegung for actual fermions ... is generally questioned[18] (see however [19]).On the other hand, the realness of this effect has been confirmed in various Dirac-like systems [15, 20, 21] and was found responsible for the appearance of a minimal conductivity and a sub-Poissonian shot noise in graphene[22, 23].

There exists an alternative viewpoint on Zitterbewegung (known also under the name of ‘chiral oscillations’ [24, 25]) relating it to the spin components of the fermionic wave function[26–30]. Any Dirac spinor can be uniquely decomposed as a sum of two Weyl spinors ψ=ψ^{+}+ψ^{−}... The Weyl spinors ψ^{±}, being eigenstates of the chirality operatorγ^{5}a... have opposite chirality. By acting with the projector P^{±}ℏon Equation (1) we obtain iγ^{µ}∂_{µ}ψ^{±}= mcψ∓ ... which can be seen as two coupled equations for Weyl spinors ψ±, one acting as a source of the other [30]. Since Weyl spinors are massless, they move with the speed of light and one obtains a ‘zigzag picture’ of a massive fermion [29, Figure 25.1]. In the fermion’s rest frame, the period of oscillations between the two eigenstates of chirality precisely equals T_{ZB}= πℏ/(mc^{2}). [28].

The two points of view on Zitterbewegung are closely related in the Dirac wave packet formalism [25]. In particular, purely positive/negative energy solutions to the Dirac equation also do not exhibit chiral oscillations [24].

Hestenes contended [26–28] that the ‘chiral’ interpretation of Zitterbewegung is more natural, as the origin of the effect resides in the geometry of spacetime. In the present Letter we support this claim, although we argue that the very notion of geometry needs to be refined...

The basic objects of noncommutative geometry à la Connes [3] are spectral triples (A, H, D) consisting of a (dense subalgebra of a)C^{*}-algebra A, a Hilbert space H with a faithful representation of A and an unbounded self-adjoint operator D acting on H.The original framework was designed to describe spaces of Euclidean signature and has recently been extended to encompass the Lorentzian ones[31–36]. In the latter case, the main conceptual change consists in endowed the Hilbert space H with an indefinite inner product, turning it into a Krein space K [37] — a vector space equipped with an indefinite non-degenerate inner product — and in requiring D to be self-adjoint with respect to the indefinite inner product.

The noncommutative geometry of Zitterbewegung

Michał Eckstein, Nicolas Franco, Tomasz Miller (Submitted on 31 Oct 2016)

*"*Connes theory" could mean "theory of causal cones" in spectral noncommutative geometry≼Causality is one of the most fundamental principles underlying physical theories.Within Einstein’s theory it is defined as a partial order relation on the set of events: pq means that q lies in the future of p. However, noncommutative spaces (not only in the framework of spectral triples) typically admit only a global description and the very notion of an event does not make sense. This raises the question:what is the scene for causal relations and what is an operational meaning of a ‘noncommutative spacetime’?

In [5] we put forward the idea that a ‘noncommutative spacetime’ ought to be understood as the space of (pure) states of a, possibly noncommutative,C^{*}-algebra.The motivation behind this step is twofold: Firstly, if the algebra at hand is of the form A_{M}=C^{∞}_{c}(M) ..., then its pure states P(A_{M}) are in one-to-one correspondence with the events in M[5]. Secondly,C^{*}-algebras provide a unified framework for an operational formulation of both classical and quantum physics [39–41]. In this context, states on an abstractC^{*}- algebra of observables can be understood as the actual states of a given physical system. For instance, ifA_{F}= M_{n}(C) ..., then pure states in P(A_{F}) are precisely the n-qubits, whereas the mixed states S(A_{F}) correspond to the density matrices [41]. In [5] we have shown that sucha ‘noncommutative spacetime’ admits a sensible notion of a causal structure associated with a Lorentzian spectral triple. To this end, one has to identify a specific subset C of the algebra A named the ‘causal cone’...

The space of pure states of an almost commutative geometry has a particularly pellucid physical interpretation: Let A =A_{M}⊗A_{F}..., then P(A)≃M×F for some finite dimensional space F. In other words, all pure states on A are separable [43]. Hence, the space of pure states of an almost commutative spacetimeis the Cartesian product of the spacetime M and an ‘inner’ space of states of the model.

...in almost commutative spacetimes, Einstein’s causality in the{commutative}spacetime component is not violated. On the other hand, in [38] and [44] we discovered that the extended causal structure imposes highly non-trivial restrictions on the evolution also in the ‘inner’ space of the model. We shall now apply the mathematical results obtained in [44] to lift the veil on the nature of Zitterbewegung...

Connes first observed [11] that the (Euclidean version of the) above almost commutative model provides a geometric viewpoint on Zitterbewegung. However, Connes remark was only qualitative and focused on regarding the Higgs field as a gauge boson operating on the finite space {−,+} (see also [50]).We discovered that taking into account the Lorentzian aspects of this model reveals a deeper, quantitative, connection between geometry and the ‘trembling motion’ of fermions(cf. [44, Theorem 9]):Theorem 6. Let τ(γ) be the proper time along a causal curve in M. Two states (p,−),(q,+) ∈ P(A) are causally related with (p,−)≼(q,+) if and only if there exists a causal curve γ giving p≼q in M, such that τ(γ)≥πℏ/(2|µ|c^{2}).

Restoring the physical dimensions in the model (which is unambiguous as slashed D has the dimension L^{-1}, thus D_{F}must have so), we arrive at:

(p,−)≼(q,+) ⇔ p≼q and τ (γ) ≥πℏ/(2|µ|c^{2}). (6)

The number on the RHS of (6) is precisely half of the Zitterbewegung period of a Dirac fermion of mass |µ| = m.It is striking to realise that the possibility of Zitterbewegung is encoded in the geometry of a purely classical model. The bound on the frequency of the fermion’s quivering is ofkinematic origin– we have invoked {a fermionic} action... only to identify the relevant degrees of freedom. The apparent abrupt change of state implied by Theorem 6 becomes more transparent when one considers the subspace of mixed states of indefinite chirality. In the space M×[−1,+1] ⊂ S(A) the boundary of the causal cone becomes a continuous surface [44, Fig. 2] permitting a smooth evolution of the expectation value of chirality.

The advantage of the presented model is its general covariance, which guarantees that Theorem 6 applies in any globally hyperbolic spacetime M. But the framework of noncommutative geometry is even more flexible and allows one to accommodate, via the ‘fluctuations’ of the Dirac operator, other fields interacting with the fermion[8]...

**Emulating almost commutative spacetimes ?**

In an (analogue) quantum simulation some aspects of the dynamics of a complicated quantum system are mimicked in a simpler one, which is under control [13]. Given an almost commutative spectral triple... together with {a} fermionic action... one can extend this concept to probe almost commutative spacetimes in table-top experiments...

The quantum simulation of a single free Dirac fermion in a flat 2-dimensional spacetime has been successfully accomplished with cold atoms [15], trapped ions [20] and photonic systems [21]. Furthermore, a suitable experimental setup has been proposed using superconductors [54], semiconductors [55, 56] and graphene [57]. In the trapped-ion setting [20, 58], the mass of the simulated fermion can be introduced dynamically, what enables a simulation of a Dirac fermion coupled to a (real) scalar field. Moreover, a possibility of studying the impact of an external electromagnetic field on Zitterbewegung using the framework of [20] was suggested in [59].

In the presented almost commutative model... Zitterbewegung in the wave packet formalism is not a single-frequency oscillation [16]. Nevertheless,given concrete initial and final states of a simulator system one can unravel the consequences of{the Connes theory}drawing from the fact that the causal cone is completely characterised in [44] and suitable computational tools to handle mixed states were devised in [60]...

For a free electron the period of Zitterbewegung... is of the order of 10^{-20}s, which is well beyond the currently available experimental time resolution. Moreover, to model a genuine electron one would need to employ the quantum field theoretic description, which seems to exclude Zitterbewegung [18], at least in flat spacetimes [19]. The scheme presented in this Letter suggests, however, that the very foundations of quantum gauge theories might need to be refined. The principle of micro-causality — requiring the observables in space-like separated regions to commute — is at the core of all axiomatic approaches to quantum field theory [39, 61]. If the fields have additional degrees of freedom then their background geometry is that of an almost commutative spacetime. Consequently, when constructing a quantum theory of fields, one should take into account its inherent causal structure.

The concept of causality in the space of states is at the core of the presented model. When applied to other almost commutative spacetimes (see [38] for another gauge model), it might cast a new light on such perplexing phenomena as neutrino or quark mixing, which also involve a ‘motion’ in the fermion’s internal space. Finally, one can reach beyond the almost commutative setting and study the causal structure of genuinely noncommutative spacetimes [62]. This perspective suggests that,contrary to the common belief [63, 64], causal structure need not breakdown at the Planck scale, but the very notion of spacetime geometry needs to be refined.