A different perspective on the problem of time in quantum gravity

TL;DR

This paper proposes that the time parameter in quantum mechanics should be viewed as the local Minkowski time in curved spacetime, resolving the traditional 'problem of time' in quantum gravity.

Contribution

It offers a new perspective that aligns quantum mechanical time with local Minkowski coordinates, challenging the conventional view of an incompatible 'problem of time' in quantum gravity.

Findings

Reinterprets quantum mechanical time as local Minkowski time.

Argues against the notion of an inherent 'problem of time' in quantum gravity.

Provides a conceptual framework linking quantum mechanics and general relativity.

Abstract

The perspective is advanced that the time parameter in quantum mechanics corresponds to the time coordinate in a Minkowski flat spacetime local approximation to the actual dynamical curved spacetime of General Relativity, rather than to an external Newtonian reference frame. There is no incompatibility, as generally assumed in the extensively discussed "problem of time" in Quantum Gravity.

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Full text

1 Introduction

The “problem of time” (PoT) in the quantization of General Relativity (GR)[1, 2, 3, 4, 5, 6, 7, 8, 9, 10] arises in part from considering that time in the Time Dependent Schrödinger Equation (TDSE) is a parameter (an element of an external frame of reference as the absolute space and absolute time introduced by Newton), whereas in General Relativity (GR) time acquires a dynamical character due to the curvature of spacetime that reflects the presence of mass. It is thus stated that: “On the other hand, one faces the problem of time. Whereas time in quantum theory is absolute (the parameter in Schrödinger equation has been inherited from Newtonian mechanics), time as part of the space-time obeying Einstein’s equations is dynamical. A more fundamental theory is therefore needed to gain a coherent concept of time” [4]; and “The greater part of the PoT occurs because the “time” of GR and the “time” of Quantum Theory are mutually incompatible notions” [1]. This view is furthermore supported by Pauli’s objection to the existence of a time operator in Quantum Mechanics (QM) [11].

2 A different perspective

Based on recent theoretical and experimental developments, this outlook of the problem of time is modified if one takes into account the following:

a) The relativistic quantum equation as formulated by Dirac satisfies Lorentz invariance. This is achieved by integrating the time parameter of the time dependent Schrödinger equation (TDSE) with the three space coordinates into a four dimensional spacetime Minkowski frame of reference[12, 13]. As such, neither the time nor the space coordinates are represented by operators, as postulated in quantum mechanics (QM) to be associated to the observable properties of the system under study; nor time can be considered to be canonically conjugate to the Hamiltonian [14].

b) Closed systems (foremost the Universe, although finite macroscopic and atomic systems have been so considered for all practical purposes in the formulations (non relativistic first and then relativistic) of classical and quantum mechanics. Being closed, they are static, i.e., they do not evolve. Therefor in QM the basic equation is the time independent Schrödinger equation (TISE).

c) Time in the TDSE is the laboratory time transferred by the entanglement of the microscopic system with its macroscopic environment where clocks are found; thus the TDSE is a classical quantum equation where $t$ is part of a dynamical reference frame in the curved spacetime [15, 16, 17]. A similar situation is expected to follow in the case of the time independent Wheeler-deWitt equation (WdW) in the canonical quantization of GR [15].

d) QM and GR are at present  assumed to be universally valid in the cosmological development of the Universe, from the Big Bang to the progressive expansion and cooling, the appearence of fundamental particles, the aggregation into atoms and, as a consequence of decoherence, into the massive components of the present world and the remnant cosmic background radiation (CMBR), and possibly also dark matter and dark energy, all inmersed in a GR curved spacetime [8]. Although this is still an open problem, “Local agreement with SR (Special Relativity) is also required. A natural hypothesis here is Einstein’s that SR inertial frames are global idealizations of GR’s local inertial frames that are attached to freely falling particles. Furthemore, in parallel with the developmeent of SR, Einstein retained a notion of metric $g_{\mu\nu}$ on spacetime to account for observers in spacetime having the ability to measure lengths and times if equipped with standard rods and clocks, encode the distinction between time and space, as $g_{\mu\nu}$ reduces locally to GR’s $\eta_{\mu\nu}$ everywhere the other laws of Physics take their SR form”[1]. And also: “Any acceptable quantum gravity theory must allow us to recover the classical spacetime in the appropiate limit. Moreover, the spacetime geometrical notions should be intrisically tied to the behavior of matter that probes them” [18].

e) A composite closed system ( micoscopic system plus macroscopic environment -the laboratory-) is found in a curved spacetime representing the pervasive gravitational interaction. However, the minimal (less than 0.02%) correction to the hydrogen spectrum that arises from a Dirac equation extended to curved spacetime suggests that the laboratory is subject to a very weak curvature [19], so the Minkowski flat spacetime of SR is a good local approximation to the GR curved spacetime.

Furthermore, Dirac’s formulation of Relativistic Quantum Mechanics (RQM) allows the introduction of a self-adjoint ”time” operator for the microscopic system $T=\mathbf{\alpha.r}/c+\beta\tau_{0}$, in analogy to the Hamiltonian $H=c\mathbf{\alpha.p}+\beta m_{0}c^{2},$ where $\alpha=(\alpha_{x,}$ $\alpha_{y,}\alpha_{z})$ and $\beta$  are the Dirac matrices. It represents in principle an additional observable. This operator generates a unitary transformation that shifts momentum - whose spectrum is continuous and unbounded -, and ensuingly the energy in both positive and negative energy branches, thus circumventing Pauli’s objection. It also provides a time energy uncertainty relation directly related to the space momentum uncertainty, as envisionned originally by Bohr in the uncertainty of the time of passage of a wave packet at a certain point; and a formal basis for de Broglie’s daring assumption of associating a wave of frequency $\upsilon=mc^{2}/h$ to a particle of mass $m$ [20, 21, 22, 23].

3 Conclusion

In view of the above one can conclude that in relativistic classical and quantum mechanics, time is part of a Minkowski reference frame that locally approximates well the actual GR dynamic curved spacetime where the laboratory is locates. It is not Newtonian. Its origin is dynamical111Another source of confusion arises because “dynamical” is attached to two different aspects. One is the variation associated with  the impact of matter on the spacetime reference frame. The other is the dependence on the parameter $t$ of the observables of the system, that in QM is made explicit in the Heisenberg picture.. There is therefore no incompatibility. To quote Einstein: “Newton, forgive me; you found the only way which, in your age, was just about possible for a man of highest thought and creative power. The concepts, which you created, are even today still guiding our thinking in physics, although we now know that they will have to be replaced by others farther removed from the sphere of immediate experience, if we aim at a profounder understanding of relationships” [24]. This apology and recognition should be extended to the creators of the Hamilton Jacobi formulation of classical mechanics and to the creators of quantum mechanics.

To be pointed out is that not every link of the present perspective has been fully developed at present. The quantization of GR is still an open question in many aspects [1, 18, 25]. However, this point of view removes the question of whether time is to be identified before or after quantization, in favour of a timeless interpretation of quantum gravity [10], where time would emerge as the observable that conditions all the others, as proposed by Page and Wootters [26]. The proposed self-adjoint time operator that complements the Dirac formulation of RQM may play the role of that observable [27], in response to the objection of Unruh and Wald [28]. And perhaps it may also help in removing the ambiguity with respect to time in the foliation of spacetime in the canonical approach to Quantum Gravity.

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