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Timeless Configuration Space and the Emergence of Classical Behavior

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Abstract

The inherent difficulty in talking about quantum decoherence in the context of quantum cosmology is that decoherence requires subsystems, and cosmology is the study of the whole Universe. Consistent histories gave a possible answer to this conundrum, by phrasing decoherence as loss of interference between alternative histories of closed systems. When one can apply Boolean logic to a set of histories, it is deemed ‘consistent’. However, the vast majority of the sets of histories that are merely consistent are blatantly nonclassical in other respects, and further constraints than just consistency need to be invoked. In this paper, I attempt to give an alternative answer to the issues faced by consistent histories, by exploring a timeless interpretation of quantum mechanics of closed systems. This is done solely in terms of path integrals in non-relativistic, timeless, configuration space. What prompts a fresh look at such foundational problems in this context is the advent of multiple gravitational models in which Lorentz symmetry is not fundamental, but only emergent. And what allows this approach to overcome previous barriers to a timeless, conditional probabilities interpretation of quantum mechanics is the new notion of records—made possible by an inherent asymmetry of configuration space. I outline and explore consequences of this approach for foundational issues of quantum mechanics, such as the natural emergence of the Born rule, conservation of probabilities, and the Sleeping Beauty paradox.

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Notes

  1. For the most up to date achievements of ‘environment-induced-decoherence’, see [2].

  2. Here \(*\) refers to an arbitrary smooth Riemannian metric on \(\mathcal {Q}\). For my purposes here it is enough to leave it at this level of abstraction. For more detail see [18].

  3. To be more careful, I should only ascribe to them the title of ‘subsets’, not submanifolds. However, under reasonable assumptions, as shown in the accompanying paper [10] they indeed form submanifolds.

  4. As much as possible, I want to avoid technicalities which will not be required here. Having said this, formally one would have had to define the so-called kinematical Hilbert space \(\mathcal {K}\) for the quantum states over \(\mathcal {Q}\) by using a Gelfand triple over \(\mathcal {Q}\) with measure \(d^dq^a=dq^1\cdots dq^d\), i.e. \(\mathcal {S}\subset \mathcal {K}\subset \mathcal {S}'\). This is not necessary in my case, because we will not require a Hilbert space.

  5. In [23], the conditions under which Briggs and Rost derive the time dependent Schroedinger equation from the time-independent one, corresponds to the system being deparametrizable. I.e. one can isolate degrees of freedom (the environment, in Briggs and Rosen) which are heavy enough to not suffer back reaction. It is a stronger condition, in that there \(H_o\) is supposed independent of t.

  6. Alternatively, we could have defined a “vacuum state” \(|\phi ^*\rangle \) existing over a trivial (one complex dimension) Hilbert space \(H_{\phi ^*}\) with the usual complex space inner product, over \(\phi ^*\), and then taking \({\hat{W}}(\phi ^*,\phi ):H_{\phi ^*}\rightarrow H_{\phi }\) as an operator such that \(|\psi (\phi )\rangle := {\hat{W}}(\phi ^*,\phi )|\phi ^*\rangle \).

  7. Note that in this case the action has dimensions of configuration space length, and thus \(\rho \) has the right dimensions, and also that the path parameter along each geodesic becomes proportional to the configuration space length.

  8. In this simple example, the action was not presumed to be given by the usual length functional on configuration space, as is (26). However, if it was, and the distance \(d((\pi ,0), (\pi ,1))\gg \hbar \), there would still be negligible interference.

  9. And quadratic in momenta so that we could write the path integral solely in configuration space.

  10. This is not true for general relativity in ADM form [35], which possesses refoliation symmetry. But it does hold for conformal geometrodynamic theories, such as shape dynamics, with an inner product of the form

    $$\begin{aligned} \langle u, v\rangle _g=\int _M \sqrt{g}d^3x \, \, \sqrt{C^{ab} C_{ab}}\,\,u_{ab}g^{ac}g^{bd} v_{cd} \end{aligned}$$

    for \(C_{ab}\) the Cotton tensor, and \(u,v\in T_g\mathcal {Q}\). And the argument here goes through without a hitch [36].

  11. The riddance of \(T^{(2)}\) of (37) in this expression can be seen as a Jacobian coming from fixing the variations along the transversal planes, at the base space point of the double-tangent bundle: \(\phi ^\perp _{tm}={\tilde{\phi }}_{tm}\).

  12. This is not true for orbifolds (which does not mean the formula doesn’t hold, it just means this proof needs to be amended). Also note that it is not necessary for this proof that all of the curves define the same orientation in the codim = 1 surface. I.e. it is not necessary that the define the same “direction of time”.

  13. Note that arc-length, for example, does increase along each path, but it is not a scalar function in configuration space.

  14. Of course, one should take into account that the Definition 2 of the record region itself, has some thickness, which does not require \(f_k(q)\rightarrow \delta (q-q_k)\). See Fig. 2.

  15. This reciprocal relation is heuristic: consider configuration space with Jacobi metric action, and a coordinate system parametrized by the outgoing geodesics, with the screens at constant distance. Then the geodesics are Eulerian, and their inverse density parametrizes the area element. To be more precise I know of no way other than to have a specific action functional. For a Jacobi system, with metric \(G^{ab}\), the normal to the constant S surfaces are given by \(j_a=\Delta \frac{\delta S}{\delta \phi ^a}=\Delta \nabla _a S\). Then reparametrization invariance implies that semi-classically one obtains the conservation equation \(G^{ab}\nabla _a j_b =0\). For the full proof that this implies conservation, see [18].

  16. Again, one shouldn’t count all of the instants in which one sees the coins for conservation of probabilities. Although in this case it is immaterial, because in all of the branches the behavior of the coin becomes static (wrt to some other subsystem, such as clock).

  17. In the work here, made implicitly through a choice of action functional, initial point, coarse-graining, etc.

  18. Projectors of a single history commute, but associated to different choices they might not: in QM there is no truth functional associated to the product of non-commuting projectors: PQ, even if Q and P are projectors (associated to truth functionals). One can only combine different frameworks when all projectors commute (in which case there is a larger set for which both subsets are fine-grainings). Once one has a certain splitting of the entire system under consideration into an ‘environment+system+apparatus’ tensorial product, one can use (generalizations of) the tri-decomposition uniqueness theorem, which states that for an orthogonal basis for each subsystem there is a unique choice of branches, but of course this begs the question of where the separation itself comes from, which is what consistent histories was supposed to evade.

  19. One should note that Fujiwara’s et al [46] formalization of the path integral through piece-wise classical paths is one of the more robust formal mathematical treatments of the path integral.

  20. Note that for \(E<V\) we have an imaginary length functional. Accordingly, tunneling emerges from paths that extremize the Euclidean action, or seen otherwise, as paths that are themselves imaginary. We do not want to expand on this subject here. For more information on this [47], and on how to obtain the correct tunneling amplitude from a real path integral, see [48].

  21. For Finsler metrics and infinite-dimensions there are caveats (the geodesics come arbitrarily close to any given endpoint in infinite-dimensions for example), but nothing that would bother us here.

  22. The formal treatment of the time-slicing approach used here in the context of piece-wise extremal curves was developed in [46].

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Acknowledgements

I would like to thank Lee Smolin, Flavio Mercati, Tim Koslowski, and Simon Saunders for comments, and Clement Delcamp for help with the figures. This research was supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

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Correspondence to Henrique Gomes.

Appendices

Appendix

The Jacobi Metric

The Jacobi version of the Maupertuis principle establishes some instances when dynamics can be viewed as geodesic motion in an associated Riemannian manifold. That is, if the action can be written as

$$\begin{aligned} S = \int (T - U)dt \end{aligned}$$

for a system defined in a Riemannian manifold, (Mg), with smooth potential U, the extremal trajectories of S with energy \(E = T + U\) coincide with the extremals (geodesics) of the length functional

$$\begin{aligned} L[\gamma ]=\int |\gamma |ds \end{aligned}$$
(53)

defined in (Mh), where h is the constructed Jacobi metric, conformally related to g by \(h = 2(E - V)g\), for \(E>V\), and the norm is calculated using this metric.Footnote 20 In general I will call a Jacobi metric, any metric in configuration space whose geodesics are extremal paths of the action.

1.1 Geodesically and Dynamically Connected Manifolds

For any finite-dimensional metrically complete Riemannian manifold, the Hopf-Rinow theorem guarantees that two points of the manifold will be connected by a geodesic.Footnote 21

However, even for finite-dimensional spaces, there is already here a very robust obstruction to this procedure: namely, non-smooth potentials U (such as a Dirac \(\delta \)-function potential, which are ubiquitous in physics). In that case, it can be shown that there might be no extremal paths connecting two given points a and b. However, Marsden has shown that by considering geodesic flows with corners, one can re-obtain a version of the Hopf-Rinow theorem for systems with a Jacobi metric. That is, one must use piece-wise geodesics. For questions on the generic behavior of “critical connectedness” in Hamiltonian systems, see [29].

DeWitt Expansion and Piecewise Extremal Paths

1.1 The DeWitt Semi-classical Expansion

Coarse-grainings and Zeno The simplest type of coarse-graining one could try to define would be through paths which never leave a certain region \(O\subset \mathcal {Q}\). However, as shown in [14], this definition will suffer from the quantum Zeno effect. This happens as follows: In standard particle quantum mechanics, upon calculating the path integral by the limit:

$$\begin{aligned} W(\mathbf {x}_i,t_f;\mathbf {x}_f,t_f)=\lim _{ n\rightarrow \infty } \int _R d^dx^1\cdots \int _R d^dx^{n-1} W(\mathbf {x}_k,t_k;\mathbf {x}_{k-1},t_{k-1}) \end{aligned}$$
(54)

where R is the space-time region, each integration acts as a projector of the state, by

$$\begin{aligned} P=\int _R d^d x|x\rangle \langle x| \end{aligned}$$

and in the limit, this constitutes a continuous projector of the state onto the region. By the quantum zeno effect, this implies that the restricted propagation is already unitary, or, equivalently, the particle is never allowed to leave R. Equivalently, Halliwell has shown that defining a set of paths by “never entering” a region, amounts to implementing reflecting boundary conditions on said region, which alter the path integral one would like to study.

Thus the issue to evade with our choice of coarse-graininings in configuration space is the formalization of a coarse-graining implicitly through reflecting boundary conditions. To do so, I will use a different approach, which is based on the techniques developed in [27] for the computation of the path integral based on integrating over deviation vector fields at geodesics. That work is too technical for a comprehensive treatment here. The important point for us, is only that, if an extremal path between \(\mathbf {x}_i\) and \(\mathbf {x}_f\) exists, call it \(\gamma _{\text{ cl }}\), one can replace the integration over paths by an integration over the space of deviation vector fields over \(\gamma \), with a measure given by the Jacobian matrix. The benefit is that this will give a constructive definition for coarse-grainings; not an implicit one as e.g. all paths that enter or don’t enter a region (which gives rise to the quantum zeno effect).

Here I will very briefly report on some results of Cecille deWitt in [27], concerning a higher order semi-classical expansion around extremal paths. In that paper, the problem of computing the path integral is re-expressed completely in terms of classical paths and variations around it. The rationale is to expand the action in higher variations around a segmented classical path, obtaining arbitrarily high orders of approximation to the full path integral. Although the case studied in [27] is for finite-dimensional manifolds, the work was extended to infinite-dimensional Banach spaces in [49].

The technical steps that make it possible are, very roughly,

  1. 1.

    The path integration can be replaced by an integral over the space of vector fields \(\mathbb {X}\) (along the stationary path) which vanish at the endpoints. This is still assumed for all possible vector fields, therefore it encompasses all the paths.

  2. 2.

    One segments the extremal path into many subsegments. Now let \(\beta (u,t)\) be a geodesic variation of \(\gamma _{\text{ cl }}(t)\), i.e. one which does not necessarily keep the endpoints fixed and is itself extremal, i.e. \(\frac{d}{du}_{|u=0}S[\beta (u)]=0\) and \(\beta (0)=\gamma _{\text{ cl }}\). A Jacobi vector field, \(\beta ^{\prime }{}(0)\), can be defined from initial and final data on each segment. Using broken Jacobi vector fields at each segment, for geodesic variations that vanish at one endpoint but not the other of each of these segments, one can find a measure on the space of deviation vector fields \(\mathcal {D}(u)\), which has cylindrical (i.e. composition) properties.

  3. 3.

    This measure involves an integration over Jacobi fields at each endpoint (not at midpoints of the segments). This is because the Jacobi matrix serves as an analogue of the Feynman propagator. It is a matrix relating initial and final vectors.

  4. 4.

    It is then shown that the infinite limit of the path integral obtained with this projected measure at each segment yields the standard path integral.

    $$\begin{aligned} W({\phi ^*}, \phi _f)=\Delta _\gamma ^{1/2}e^{iS[\gamma ]/\hbar }\sum _{n=0}^\infty (i\hbar )^n A_n \end{aligned}$$
    (55)

    where \(A_0=1\) and the next orders depend on further variations around the classical action (and their projection to the cylindrical measures at the endpoints through the Jacobi matrix).

Equation (55) may provide a way to maintain preferred coarse-grainings centered on extremal paths and yet obtain higher order of expansions in \(\hbar \) beyond the usual semi-classical approximation.Footnote 22

1.2 The Piecewise Semi-classical Approximation

There might be cases in which there are no extremal paths between two given configurations \({\phi ^*}\) and \(\phi _f\). This failure can be most easily visualized by connecting the least action principle to the existence of a corresponding metric and geodesics over the same space, in which case the manifold with the induced metric becomes smooth-geodesically incomplete, as discussed at length in [29], and briefly recounted in Appendix Appendix A. The easiest way in which we can envisage this property even in the finite-dimensional case are for non-smooth potentials (such as Dirac delta barriers, or mirrors) in configuration space whose associated symplectic flows have “corners” [29] (or here called vertices).

On the other hand, even for non-geodesically complete Riemannian manifolds, if two of its points are in the same connected component they can be connected by paths which piece-wise are geodesics. In the metric case, the space of all paths on \({\mathcal {M}}\) can be densely covered by the (inductive limit in N of the) space of piecewise geodesic segments:

$$\begin{aligned} \Gamma ^N_{ \text{ pc.geod }}({\phi ^*},\phi _f)= & {} \left\{ \prod _{i=0}^{N}\gamma _i~|~\gamma _i:[0,1]\rightarrow \mathcal {Q}~ \text{ is } \text{ geodesic, } \text{ and }\,\,\gamma _{i+1}(0)\right. \nonumber \\= & {} \left. \gamma _i(1), \forall i, \gamma _0(0)={\phi ^*}, \gamma _N(1)=\phi \right\} \end{aligned}$$
(56)

Let us assume that this is also the case here—mutatis mutandi for configuration space (endowed with a Lagrangian density) instead of a Riemannian manifold, and extremal paths wrt the action we have on configuration space instead of geodesics.

The piece-wise approximation used here can be roughly seen as the inverse procedure as the Feynman–Dirac standard one of obtaining the path integral, using the decomposition of unity at successive time intervals. This approach to the Feynman path integral is known as the “time slicing approximation with piece-wise classical paths”, and it is extensively used in the literature not only for a formal definition of the path integral (see [Fujiwara] [46] for a rigorous treatment), but also as a practical method of computation. In the simplest case, it consists in splitting the usual particle path integral time interval into regular \(N+1\) sub-intervals \(\delta t\), the amplitude kernel becomes

$$\begin{aligned} W((q_i,0), (q_f, \tau ))= \int \prod _{i=1}^N\langle (q_f,\tau )|(q_N,N\delta \tau )\rangle \cdots \langle (q_1,\delta \tau )|(q_i,0)\rangle \end{aligned}$$
(57)

One obtains the path integral by taking the infinite limit \(N\rightarrow \infty \). The proposal is to take the time intervals small enough so that one can always find a classical path between the intermediary points and approximate each segment semi-classically.

Here the proposal is precisely parallel to the particle quantum mechanics case. The main difference is that in the absence of time we take the Jacobi-length intervals of the action to be small, and since we have not rigorously proven the analogous result, we state it as a conjecture:

Conjecture 1

(The piece-wise expansion) Given any two configurations \({\phi ^*}\) and \(\phi _f\) (not necessarily connected by an extremal path, in an action of Jacobi type), for a given order \(\epsilon \), there is a number \(N(\epsilon )\) and a finite distance \(D(\epsilon )>0\) (in the Jacobi metric) such that

$$\begin{aligned} W({\phi ^*},\phi _f)= \int \prod _{j=1}^N{{\mathcal {D}}}\phi _j W({\phi ^*}, \phi _1)\cdots W(\phi _N, \phi _f) +\mathcal {O}(\epsilon )\end{aligned}$$
(58)

where extremal paths of length D / N exist between \(\phi _j\) and \(\phi _{j+1}\) (and thus \(W(\phi _j, \phi _{j+1})\) is given to any order in \(\hbar \) by Eq. (55), and to first order in \(\hbar \) by (11)).

If there is a unique extremal path between a given \(\phi _{j-1}\) and \(\phi _{j+1}\), one can use Eq. (30) to ensure that \(\phi _j\) is in that segment (thereby collapsing this subdivision). The approximation holds true for \(\epsilon \sim \hbar \) in the trivial case when \({\phi ^*}\) and \(\phi _f\) are classically connected, by Eq. (55), when \(N(\epsilon )=1\) and \(D(\epsilon )\) is the geodesic distance between the points. It also holds for \(\epsilon =0\) in the limit \(N\rightarrow \infty \) (as then we can approximate each segment by the Lagrangian, as in the infinite limit of (57)).

For instance, if we focus on the space of piecewise geodesics with two segments, between \(v_{i-1}\) and \(v_{i+1}\), one has a well-defined calculus of variations where the position of the intermediary vertex \(v_i\) is the variation parameter, and it makes sense to say that a certain \(v_i^*\) is extremal. A simple, and yet non-trivial example in 2D Euclidean geometry is to consider two points (without loss of generality), \(p_1=(x_1,0)\) and \(p_2=(x_2,0)\) and remove from \(\mathbb {R}^2\) an open line segment \(L=(0,]-a,a[)\) between \(p_3=(0,-a)\) and \(p_4=(0,a)\), so as to make the manifold geodesically incomplete. The variational principle yields the two piecewise straight segments \(\gamma _1=\overline{p_1p_3}\cdot \overline{p_3p_2} \) and \(\gamma _2=\overline{p_1p_4}\cdot \overline{p_4p_2}\).

1.3 Piecewise Extremal Coarse-Grainings

Using Definition 1 for EC’s, we define

Definition 3

(Piecewise extremal coarse-grainings (PECs)) A piecewise extremal coarse graining (PEC) is a coarse-graining, where each element of the coarse graining set, \(C_{\alpha }\), is composed of extremal “segments”:

$$\begin{aligned} \{C_\alpha =C_{\gamma ^\alpha _1}\cdot C_{\gamma ^\alpha _2} \dots C_{\gamma ^\alpha _{N_\alpha }}~|~\gamma ^\alpha _1\cdot \gamma ^\alpha _2\dots \gamma ^\alpha _{N_\alpha }\in \Gamma ^{N_\alpha }({\phi ^*},\phi _f), \alpha \in I\} \end{aligned}$$
(59)

where each bundle of paths is defined as in the previous Definition 1 as contained in its tubular neighborhood \(\nu _{\gamma ^\alpha _i}(\rho _\alpha )\). The intermediary points (the endpoints of each segment) will be called the vertices of the PEC.

Set selection and PECS Let me first define the order of a PEC as the largest number of segments of any element of the PEC. From (59):

$$\begin{aligned} \mathcal {O}(I):=\max _{\alpha \in I}{N_\alpha } \end{aligned}$$
(60)

A minimal-\(\epsilon \) PEC is one that has the minimal order and is still exhaustive to order \(\epsilon \):

$$\begin{aligned} \{C_\alpha \in \Gamma ^{N_\alpha }({\phi ^*},\phi _f), \alpha \in I_{{\text{ min }}}~|~\mathcal {O}(I_{{\text{ min }}})= \inf _{I\in \text{ PEC }({\phi ^*},\phi _f)}{\mathcal {O}(I)}\} \end{aligned}$$
(61)

where \(\text{ PEC }({\phi ^*},\phi _f)\) parametrizes the space of all possible piecewise extremal coarse-grainings between the two configurations. Apart from the formalities, a minimal PEC, if it exists, can be seen as just that minimal set of segments of extremal paths such that its corresponding PEC yields the approximate amplitude for the given process. In that way, I propose that the existence of a unique minimal PEC yields the preferred coarse-graining for any given transition amplitude between an initial and a final configuration.

Uniqueness of the minimal PEC should be related to a unique solution of the variational problem of the transition amplitude wrt the vertices of the minimal PEC. A non-trivial example of a minimal PEC is of course the semi-classical approximation when extremal paths (without segmentation) exist between \({\phi ^*}\) and \(\phi \), (and \(\phi \) is not a focusing point). In this case it is given implicitly by (11), i.e. each extremal path seeds one of the coarse-grained sets (and indeed variation on an intermediary vertex will project it back to the extremal curve). However, this is the only case in which we use minimal PECs quantitavely in this paper. A more quantitative treatment of this approach using the expansion (55) is possible but will not be explored here..

Using minimal PECs, I can expand the definition of the semi-classical record used in Definition 2.

Definition 4

(Semi-classical record (general)) Given an initial configuration \({\phi ^*}\), \(\phi \) and \(\{C_\alpha \}_{\alpha \in I}\) the minimal \(\epsilon \)-extremal coarse graining (see Definition 3 for PECs and Eq. (61) for the minimal criterion) between \(\phi ^*\) and \(\phi \), of radius \(\rho _{ \text{ max }}\), \(\phi \) holds a semi-classical record of a field configuration \(\phi _r\) to order \(\epsilon \), if the ball \(B_{\rho _{ \text{ min }}}(\phi _r)\) is contained in every \(C_\alpha \), i.e. \(B_{\rho _{ \text{ min }}}(\phi _r)\subset C_\alpha , \forall \alpha \in I\).

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Gomes, H. Timeless Configuration Space and the Emergence of Classical Behavior. Found Phys 48, 668–715 (2018). https://doi.org/10.1007/s10701-018-0172-1

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