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Scale anomaly as the origin of time

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Abstract

We explore the problem of time in quantum gravity in a point-particle analogue model of scale-invariant gravity. If quantized after reduction to true degrees of freedom, it leads to a time-independent Schrödinger equation. As with the Wheeler–DeWitt equation, time disappears, and a frozen formalism that gives a static wavefunction on the space of possible shapes of the system is obtained. However, if one follows the Dirac procedure and quantizes by imposing constraints, the potential that ensures scale invariance gives rise to a conformal anomaly, and the scale invariance is broken. A behaviour closely analogous to renormalization-group (RG) flow results. The wavefunction acquires a dependence on the scale parameter of the RG flow. We interpret this as time evolution and obtain a novel solution of the problem of time in quantum gravity. We apply the general procedure to the three-body problem, showing how to fix a natural initial value condition, introducing the notion of complexity. We recover a time-dependent Schrödinger equation with a repulsive cosmological force in the ‘late-time’ physics and we analyse the role of the scale invariant Planck constant. We suggest that several mechanisms presented in this model could be exploited in more general contexts.

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Notes

  1. Particle models are also known to provide useful insights to the problem of time [9]. Conformal quantum mechanics, moreover, has also recently attracted attention in the study of the AdS/CFT correspondence [10].

  2. The idea of considering RG as the origin of time in Shape Dynamics is due to S. Gryb. See also [11].

  3. Its predictions can also be compared with the model explored in [12, 13] with Newtonian (degree \(-1\)) potential in which the single global degree of freedom that breaks scale invariance is traded classically for an internal time, after which the remaining dynamical degrees of freedom are all scale invariant. There are in fact two ways to achieve scale invariance, one (which we study here) stronger than the other (studied in [12, 13] ).

  4. The \(SO(3)\) gauge symmetry of \(N\)-body dynamics is probably responsible for a lot of interesting features of this model, in particular at the quantum level, as shown by the wealth of results obtained by Littlejohn and Reinsch [17], who took seriously the fibre-bundle structure of the configuration space.

  5. Saari [18] calls \(V_{\scriptstyle \mathrm {shape}}\) the configuration measure.

  6. We shall show that the functions \(\varphi _n\) are countable, so we adopt this notation already.

  7. One of the authors has proposed, in [19], that a notion of time might be hidden in the frozen wavefunction resulting from the reduction before quantization approach. This might be the case if the wavefunction of the universe is concentrated on ‘time capsules’, i.e., configurations which suggest a history. In the reduction-before-quantization approach, the only allowed solution of (17) is one with \(\lambda _n=0\), corresponding to zero total energy and a frozen wavefunction. Rewriting (17) in the form \(\hat{H}_{\scriptstyle \mathrm {shape}} \, \varphi _n = - \Delta _{S^{3N-4}}\varphi _n + (2/ \hbar _{\scriptstyle \mathrm{si}}^2) V_{\scriptstyle \mathrm {shape}} \varphi _n = (\lambda _n /\hbar _{\scriptstyle \mathrm{si}}^2)\; \varphi _n \), we see that \(2/\hbar _{\scriptstyle \mathrm{si}}^2\) plays the role of a dimensionless coupling constant. It might perhaps be possible to adjust its value to ensure the existence of the zero eigenvalue. This would be a novel first-principles derivation of the strength of quantum effects.

  8. For a pedagogical introduction, see [27].

  9. This is just a consequence of the fact that the eigenvalue zero in the strong regime is an accumulation point in the spectrum of every self-adjoint extension of each radial Hamiltonian without being an eigenvalue associated with a proper eigenfunction.

  10. The label \(m\) does not affect the choice of \(\theta _n\) in the strong regime because it corresponds to a complete turn around the cylinder (2). Moreover in the weak regime the label \(m\) is not present at all.

  11. An advantage of this proposal is that the usual internal time scenarios generically have great difficulty in meeting the requirement of monotonicity whereas in our framework time, being associated to RG flow, is monotonic by definition.

  12. Regularizing the potential is a complementary approach to that of self-adjoint extensions because, like it, it amounts to fixing a boundary condition near the origin. But the self-adjoint extensions approach is non perturbative and regulator-independent, so we keep both languages at hand to supplement each other and guard ourselves against artefacts of the regularization choice.

  13. We will not discuss here the major interpretational issue—which as yet has no definitive answer—that every quantum theory of the Universe must face: what is the meaning of these probabilities?

  14. See also the model proposed recently in [32] and related papers.

  15. We do not have the “multiple choice problem” of usual semiclassical approaches because any function \(f\) of \(R\) and \(D, f(R,D),\) is singled out by the anomaly. This is a significant advantage of the shape-dynamic approach.

  16. The assumption that \(\eta \) is complex could be unjustified because is taken to be a solution of a real equation (see [33]).

  17. Similar equations are standard in the semiclassical approach. See e.g. [9].

  18. D. Kendall firstly studied the concept of shape sphere. Anderson [9] used it extensively.

  19. Strictly speaking, the scale-invariant Hamiltonian is not self-adjoint due to the three singularities at the two-body collisions, and a \(U(3)\) family of self-adjoint extensions would be needed. But these singularities are not as bad as the \(1/R^2\) singularity at the origin, because they go like \(1/R\). Moreover, these singularities reproduce the Newtonian gravitational interactions among particles, and there are good reasons just to take the Friedrichs extension for all of them, which are the extensions that allow us to best model the hydrogen atom.

  20. We noted earlier, in footnote 7, that in the case of quantization after reduction the spectrum, if it exists at all, consists solely of a zero eigenvalue. If \(h_{\scriptstyle \mathrm {si}}\) can be ‘tweaked’ to ensure existence of such an eigenvalue, the theory will still be incomplete if the eigenvalue is degenerate and corresponds to a superposition of zero-eigenvalue wavefunctions.

  21. There is no corresponding notion of a most complex state: there is an Alpha, but no Omega [8, 13, 38].

  22. The caveat “in its usual representation” relates to the possibility noted in footnote 3 of achieving scale invariance by trading a global degree of freedom for an internal time.

  23. Admittedly this model is still affected by the quantum-mechanical measurement problem. The wavefunction in fact spreads over macroscopically distinguishable configurations in shape space.

  24. See Sect. 7.3 for a discussion which generalize this proposal to N body problems.

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Acknowledgments

We would like to thank S. Gryb for his initial input that proved crucial for the beginning of this project. We thank also J. Louko, P. Hoehn, E. Anderson and G. Canevari for useful comments and discussions during the preparation of this paper. M.L. thanks St. Hugh’s College for hospitality when working on his Master Thesis in a joint exchange programme with Collegio Ghislieri; he also thanks the Institute for Advanced Studies of Pavia for partial funding. This work was supported by a grant from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi Time and Foundations 2010 to the Foundational Questions Institute. It was also made possible in part through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.

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Authors and Affiliations

  1. College Farm, South Newington, Banbury, Oxon, OX15 4JG, UK

    Julian Barbour

  2. Department of Physics, Imperial College London, Prince Consort Rd, London, SW7 2AZ, UK

    Matteo Lostaglio

  3. School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK

    Flavio Mercati

  4. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON,  N2L 2Y5, Canada

    Flavio Mercati

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  1. Julian Barbour
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Correspondence to Flavio Mercati.

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M. Lostaglio: This work has been submitted in partial fulfillment of the Master Degree in Physics at the University of Pavia. J. Barbour: Visiting Professor in Physics at the University of Oxford.

Appendix

Appendix

1.1 Jacobi coordinates

A simple algorithm [44] for creating Jacobi coordinates requires specification of a complete directed binary tree, whose leaves represent the \(N\) particles, and the \(N-1\) internal vertices are associated with the Jacobi coordinates. The algorithm works this way: given such an arbitrary tree, one assigns to each “parent” vertex four numbers: the three coordinates of the centre of mass of the two “child” vertices and the sum of their masses, which are taken of course from the leaves (Fig. 4).

Fig. 4
figure 4

The binary tree algorithm for the three-body system: the two Jacobi coordinates associated with the internal nodes are \({\varvec{\rho }}^1 =\mathbf{r}^1 -\mathbf{r}^2,\, {\varvec{\rho }}^2 = (m_1 \mathbf{r}^1 + m_2 \mathbf{r}^2)/(m_1+m_2) - (m_1 \mathbf{r}^1 + m_2 \mathbf{r}^2 +m_3 \mathbf{r}^3)/(m_1+m_2+m_3)\)

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Then one defines the Jacobi coordinates as a function on the internal nodes. This function associates to each node the difference of the coordinates of its two “child” nodes: \({\varvec{\rho }}^\mathrm{parent} =\mathbf{r}^\mathrm{left} -\mathbf{r}^\mathrm{right}\). When applied to the root node, this function gives the \(N\)-th Jacobi coordinates, that is, the coordinates of the centre of mass of the system, which has now been decoupled and can be discarded, so that one can work with just with the first \(3N-3\) coordinates. The great advantage of the coordinates, besides the centre-of-mass decoupling, is that they leave the kinetic metric diagonal [44]. Its eigenvalues in this coordinate system are the Jacobi effective masses \(\mu _{\scriptstyle \mathrm {\,J}}\), which are just the masses of the nodes associated with each Jacobi coordinate. This algorithm represents a relational way to move to the centre-of-mass reference frame.

1.2 Diagonalization of the inertia tensor

We can solve the angular momentum constraint with a gauge-fixing, that is, with a choice of orientation of our axes. A global gauge-fixing, though, has to satisfy certain requirements to be good. The main requirement is that the constraint surface defined by the gauge fixing must never be parallel to the vector field generated by the constraint it is supposed to gauge fix. This translates into the requirement that the Poisson bracket between the gauge fixing and the constraint must be invertible, which means it must be everywhere (weakly) non-vanishing.

A popular choice of axes among \(N\)-body specialists is the one defined by the principal axes of the moment-of-inertia tensor [17]. It is attractive for its simplicity and symmetry, and more than anything else for its relational character, but it fails to be global for the reason mentioned above: it becomes degenerate at certain points of configuration space.

The moment-of-inertia tensor is defined as

$$\begin{aligned} I^{ab} = \sum _{I=1}^N m_I \, \left( r^I_a - r^\mathrm{cm}_a\right) \left( r^I_b - r^\mathrm{cm}_b\right) - \delta ^{ab} \sum _{I=1}^N m_I \, || \mathbf{r}^I - \mathbf{r}^\mathrm{cm} ||^2. \end{aligned}$$
(69)

Its eigenvectors are the three principal axes of inertia of the \(N\)-body configuration. The matrix \(I^{ab}\) is symmetric, and therefore can be diagonalized with a rotation that puts the three principal axes of inertia respectively on the \(x,y\) and \(z\) axes. The three constraints that impose such a condition are the ones that set the off-diagonal elements to zero:

$$\begin{aligned} I^{12} = I^{23} = I^{13} = 0, \end{aligned}$$
(70)

This condition, though, does not always uniquely fix the axes for two reasons:

  1. 1.

    When the system is in a symmetric configuration (spherically or axisymmetric) there are identical eigenvalues (two if axisymmetric, three if spherically symmetric), and the matrix is already (block) diagonal.

  2. 2.

    Even in the non-symmetric case, the solution to Eq. (70) is not unique: there are three solutions, corresponding to the residual symmetry under exchange of the axes. But this is harmless as one can resolve the ambiguity by requiring the eigenvalues to be arranged in order of magnitude: \(I^{11} < I^{22} < I^{33}\).

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Barbour, J., Lostaglio, M. & Mercati, F. Scale anomaly as the origin of time. Gen Relativ Gravit 45, 911–938 (2013). https://doi.org/10.1007/s10714-013-1516-y

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