Quantum mechanics is our most successful physical theory. Created to account for atomic phenomena, it has a vast range of applications extending well beyond the atomic realm, from predicting the abundances of the light elements created a few minutes after the Big Bang to understanding the properties of semiconductor materials that are the basis of advanced information technologies.
Quantum mechanics is also successful in its exquisitely accurate predictions of fundamental parameters, such as the value of the ‘magnetic moment’ of the electron, a property linked to its electric charge and magnetic behaviour, which is predicted to an accuracy of one part in ten trillion.
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Our best current understanding of the fundamental constituents of matter, the standard model of particle physics, is a quantum theory. And the statement that every physical system is, fundamentally, a quantum system has no known counter evidence.
In the case of gravity, however, nearly a century of effort has not resulted in a stable consensus even about the most promising grounds on which to build a theory of quantum gravity.
Why is quantum gravity proving more challenging than other quantum theories? The reasons lie partly in the lack of definitive observational phenomenology to guide us and partly in the character of gravity, which makes quantum gravity different from all other physical theories.
In this article I diagnose, in Albert Einstein’s words, “where the shoe pinches” in quantum gravity and describe one way forward based on physicist Richard Feynman’s alternative vision for quantum mechanics.
Quantum foundations
A good place to start is Werner Heisenberg’s famous 1925 breakthrough on the Danish island of Heligoland. In retrospect, the truly fresh and startling concept introduced by Heisenberg is that of transitions of the atom from one quantum state to another, transitions that do not occur in the familiar three dimensions of physical space. That breakthrough set up a conflict between the new rules of quantum prediction and the familiar, historically successful conceptual framework of physical goings-on occurring in real 3D space.
The question of where atomic transitions occur, if not in 3D space, was not raised explicitly by Heisenberg in 1925 but was answered soon afterwards. The whereabouts of these transitions was identified and formalized mathematically by Paul Dirac and others as being the ‘Hilbert space’ of the quantum system, which comprises all conceivable quantum states of the system. In mathematics, the Hilbert space takes the form of a ‘vector space’: think of the quantum state as an arrow pointing in a particular direction in a space with many, many dimensions.
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The transitions in Hilbert space from one quantum-state vector to another affect and are affected by what occurs in physical 3D space by means of special interactions between the quantum system and instruments that exist in 3D space. Such interactions are known as measurements, and the standard quantum rules make predictions about the outcomes of measurements registered by the instruments in 3D space.
To make these predictions, the goings-on in Hilbert space must be coordinated with the goings-on in 3D space, and this coordination is achieved using a concept of synchronized time. As time passes in Hilbert space, the vector arrow of the state of the quantum system changes. Its motion in Hilbert space is governed by a law of quantum physics, the Schrödinger equation, and is a type of rotation: the angle through which the quantum-state vector rotates in Hilbert space is proportional to the time that passes.
Time also passes for the measuring instruments, and physicists, in physical 3D space. The situation is like a round in the long-running British comedy radio show, I’m Sorry I Haven’t a Clue, in which each celebrity contestant has to sing along with a recording of a song, then continue to sing while the recording is muted. When the recording is unmuted again, the contestant wins points if they are still singing in synch with the recording. Seriously, it’s funnier than it sounds.
In this analogy, the song playing is like the quantum state evolving in Hilbert space, and the singer is like the measuring instrument in 3D space. The period during which the singer cannot hear the recording is like the period during which the measuring instrument is not making a measurement. The moment of unmuting is like the moment of measurement.
In the game, the singer can get out of synch with the recording, but in quantum mechanics, the synchronization is always perfect. The time that has passed in 3D space for the measuring instrument is always in perfect synch with the time that has passed in Hilbert space, marked by the angle the state vector has rotated through.
This perfect synchronization is so fundamental to quantum mechanics that, when working in the theory, the same symbol t is used for time in 3D physical space and for time in Hilbert space.
By means of this concept of synchronized time, Heisenberg and his colleagues in Copenhagen established a scientifically successful détente between the two notions of the whereabouts of physical goings-on. Although the evolution of the state of the quantum system occurs in Hilbert space, scientific predictions are about instances in which the instrument measures something, which occur in physical 3D space.
The past 100 years have proven that a quantum physicist can become extremely successful by accepting and working with the Copenhagen group’s strange ‘duality of location’. When the quantum system is gravity, however, the Copenhagen view cannot hold.
Gravity’s exceptionalism
Gravity is not like other physical systems. Our best theory of gravity is the general theory of relativity, which was published by Einstein in 1915. In general relativity, the physical entity that is the subject of the theory is space-time. Space-time is a physical 4D fabric with geometrical structure that bends, warps, ripples, carries energy and has its own laws of motion, which are as precise and experimentally successful as Newton’s laws of mechanics.
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In general relativity, space-time graduates from being a fixed stage on which what happens happens, to being both the stage and a dynamical actor in reality’s play in its own right. Gravity’s physical status is different from those of other systems. If particles, for example, are absent from a region of space-time, there can still be physical goings-on in that region, including warping or rippling of space-time itself. But if space-time is absent, there cannot be any particles, any electromagnetic radiation or anything else — because there’s nowhere and no-when for them to be.
Nothing is external to space-time: every physical thing that exists in general relativity either is space-time or is in space-time. The Copenhagen requirement for a physical measuring instrument in physical space external to the quantum system is incompatible with the quantum system being space-time.
Also, the character of physical time in general relativity stymies the synchronization that the Copenhagen détente requires. In general relativity, physical time passes individually for each particle, body or measuring instrument along its own unique, individual path, or ‘worldline’, in space-time. These individual physical worldline times cannot even be synchronized with each other and so cannot be synchronized with the time that passes in Hilbert space, as demanded by the Copenhagen rules. For quantum gravity, Heisenberg’s détente collapses, and an alternative approach is needed.
Feynman’s alternative
In 1985, Feynman gave a series of public lectures about quantum electrodynamics1, the theory unifying quantum mechanics and electromagnetism, for which he shared the 1965 Nobel Prize in Physics. In the lectures, he set out an alternative approach to quantum theory, in which there is no strange duality of location — quantum states evolving in Hilbert space were simply left out.
Instead, Feynman based his explanations of quantum phenomena on events and histories in space-time, concepts that are also fundamental in general relativity. Starting in the 1980s, physicists James Hartle and Rafael Sorkin, independently, sought to build on Feynman’s approach and develop it into an alternative foundation for a quantum theory suitable for describing gravity. Here’s how the Feynmanian approach works.
An event is something that can happen. It’s useful to have concrete examples in mind, so think of rain on a particular date in Bengaluru, India. An event has a space-time location, and it either happens or it doesn’t. Beforehand, it is uncertain whether a given event will or will not occur, but afterwards, there is no uncertainty: it either rained or it did not.
