Against Spacetime

Abstract

The notion of “location” physics really needs is exclusively the one of “detection at a given detector” and the time for each such detection is most primitively assessed as the readout of some specific material clock. The redundant abstraction of a macroscopic spacetime organizing all our particle detections is unproblematic and extremely useful in the classical-mechanics regime. But I here observe that in some of the contexts where quantum mechanics is most significant, such as quantum tunneling through a barrier, the spacetime abstraction proves to be cumbersome. And I argue that in quantum-gravity research we might limit our opportunities for discovery if we insist on the availability of a spacetime picture.

Appendices

Appendix A: More on Covariant Quantum Mechanics

Within the manifestly-covariant formulation of special-relativistic quantum mechanics, which matured significantly over the last decade [6–8], the spatial coordinates and the time coordinate play the same type of role. And there is no “evolution”, since dynamics is codified within Dirac’s quantization as a constraint, just in the same sense familiar for the covariant formulation of classical mechanics (see, e.g., Chap. 4 of Ref. [23]).

Spatial, ${\hat{X}}$, and time, ${\hat{T}}$, coordinates are well-defined operators on a “kinematical Hilbert space”, which is just an ordinary Hilbert space of normalizable wave functions [8, 9], where they act multiplicatively: ${\hat{X}} \Psi (x,t)=x \Psi (x,t)$, ${\hat{T}} \Psi (x,t)=t \Psi (x,t)$. And one has a standard description on this kinematical Hilbert space of their conjugate momenta [8, 9]:

$$[{\hat{P}}_0,{\hat{T}}] = i \,,~~[{\hat{P}} , {\hat{X}}] = -i \,,~~[{\hat{P}},{\hat{T}}] = [{\hat{P}}_0 , {\hat{X}}] = 0$$

Observable properties of the theory are however formulated on the “physical Hilbert space”, obtained from the kinematical Hilbert space by enforcing the constraint of vanishing covariant-Hamiltonian, which in the case of a free special-relativistic particle takes the form

$$({\hat{P}}_0^2-{\hat{P}}^2) \Psi _{physical} = 0$$

The observables of the theory, the “Dirac observables”, must commute with the constraint, and this is where one sees the root of the localization problem for special-relativistic quantum mechanics, which in different fashion had already been noticed by Newton and Wigner [3]: both ${\hat{T}}$ and ${\hat{X}}$ are not good observables, even within the free-particle theory, since they evidently do not commute with ${\hat{P}}_0^2-{\hat{P}}^2$.

Appendix B: More on Quantum Tunneling

The study of quantum tunneling has a very long history. However, the quality of related experimental results has improved significantly over the last two decades [12, 13], particularly starting with the measurements reported in Ref. [14], where a two-photon interferometer was used to measure the time delay for a single photon (i.e. one photon at a time) to tunnel across a well-measured barrier.

And also the understanding of quantum tunneling, and particularly of the tunneling time, has improved significantly in recent times. Previously there was much controversy particulary revolving around relativistic issues. Under appropriate conditions [12–14] a particle prepared at time $t_i$ in a quantum state with peak of the probability distribution located at a certain position to one side of a barrier is then found on the other side of the barrier at time $t_f$ with distribution peaked at a distance $L$ from the initial position, with $L$ bigger than $c(t_f-t_i)$. It is by now well established that this apparently “superluminal” behavior is not in conflict with Einstein’s relativity. Key for this emerging understanding is appreciating that in such measurement setups at first there is a large peaked distribution approaching the barrier from one side, and then a different (and much smaller, transmitted) peaked distribution is measured on the other side of the barrier. Contrary to our classical-limit-based intuition, as a result of quantum-mechanical effects (such as interference) the peak observed after the barrier is not some simple fraction of the peak that was approaching the barrier. This sort of travel times of distribution peaks, are not travel times of any signal, and indeed it is well known that for smooth, frequency-band limited, distributions the precursor tail of the distribution allows one to infer by analytic continuation [12, 14, 15] the structure of the peak. Even for free propagation, by the time the peak reaches a detector it carries no “new information” [12] with respect to the information already contained in the precursor tail. An example of “new information” is present in modified distributions containing “abrupt” signals [12, 15], and indeed it is found that when these new-information features are sufficiently sharp they never propagate superluminally [15].

It is important for the thesis advocated in these notes that, as these theoretical issues get clarified, and theoretical results get in better agreement with experiments, we are also getting more an more robust evidence of the fact that in quantifying the analysis of quantum tunneling we do not have the luxury of referring to some objective spacetime picture. In particular, there is no single “time of spacetime” but rather several possibilities for a “time of a specific clock” [12].

Appendix C: Relative Locality and Curved Momentum Space

As mentioned in the main text of these notes the main sources of interest in relative locality originate from results obtained in studies of certain types of Lie-algebra spacetime noncommutativity [19, 20] and of “group field theory” [21], where the generators of translation-symmetry transformations are not described as linear sums of single-particle momenta. In this appendix I want to highlight a particularly powerful formulation of relative locality that emerges in these theories by formally taking the limit [17, 18] of both $\hbar \rightarrow 0$ and $G_{N} \rightarrow 0$ but keeping their ratio $\hbar /G_{N}$ fixed. In this regime both quantum mechanics and gravity are switched off, but the mentioned nonlinearities for the composition of momenta are found to survive [17, 18] and to take the form of a manifestation of a nontrivial geometry for momentum space.

The implications for the phase space associated with each particle in this regime are found to be rather striking [17, 18]: this phase space is the cotangent bundle over momentum space, which one may denote by $\Gamma ^{RL} = \mathcal{T}^{*} (\mathcal{P})$. So this regime is, at least in this respect, dual to the standard classical-gravity regime, where the single-particle phase space is the cotangent bundle of the spacetime $\mathcal M$, which one may denote by $\Gamma ^{GR} = \mathcal{T}^{*}(\mathcal{M} )$. And just like in the general-relativistic formulation of classical gravity momenta of particles at different points of spacetime, $x$ and $y$, can only be compared by parallel-transporting along some path from $x$ to $y$, using the spacetime connection, one finds on $\Gamma ^{RL}$ an analogous problem for the comparison of spacetime coordinates on the worldlines of two particles, $A$ and $B$, with different momenta. These coordinates $x^\mu _A$ and $x^\mu _B$ live in different spaces and they can be compared only in terms of a parallel transport on momentum space. All this is formalized in Refs. [17, 18] where the relative-locality features mentioned in the main text of these notes are shown to admit a fully geometric description (in terms of the geometry of momentum space).