Relational Physics

Prerequisites: Canonical Quantum Mechanics, General Relativity. Full path (28 pages)

Best explanations

Concrete Space is blue and birds fly through it
Abstract Why Gauge?

Why is it interesting?
Intuitive
Concrete
Abstract

Why is it interesting?

It is usually stated that the fundamental insight of general relativity is that there is no gravitational field at all. The gravitational force is rather an illusion due to the curvature of spacetime.

However, we can also make a completely contrary argument, which is what Einstein realised after several decades. Instead of the statement above, we can take as the fundamental insight of general relativity that there is no spacetime, only the gravitational field! This line of thought is described nicely in the book "Quantum Gravity" by Rovelli and is based on the "hole argument". Se also this essay.

Similar statements can be made for other theories, like gauge theories. This is discussed, for example, in the section "Towards a Pointless Theory" in the book "Some elementary gauge theory concepts".

These ideas are summarized under the notion "relational physics".

The basic idea is that the only thing that matters are the points where worldlines of different objects meets. Everything else is not observable and hence has no meaning. In this sense there is no spacetime and no "inner spaces". The only things that matter are relative relations between objects. Everything else can not be observed, because an observation necessarily implies that objects meet.

'There is no law except the law that there is no law… Ultimate mutability is the central feature of physics.' -John Wheeler

The requirement of general covariance “takes away from space and time the last remnant of physical reality Einstein

Einstein believed that the hole argument implies that the only meaningful definition of location and time is through matter. A point in spacetime is meaningless in itself, because the label which one gives to such a point is undetermined. Spacetime points only acquire their physical significance because matter is moving through them. In his words: "All our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meeting of two or more of these points."[7] He considered this the deepest insight of general relativity. According to this insight, the physical content of any theory is exhausted by the catalog of the spacetime coincidences it licenses. John Stachel called this principle, the point-coincidence argument.[1] Generally what is invariant under active diffeomorphisms, and hence gauge invariant, are the coincidences between the value the gravitational field and the value the matter field have at the same 'place' because the gravitational field and the matter field get dragged across together with each other under an active diffeomorphism. From these coincidences one can form a notion of matter being located with respect to the gravitational field. As Carlo Rovelli puts it: "No more fields on spacetime: just fields on fields."[4] This is the true meaning[clarification needed] of the saying "The stage disappears and becomes one of the actors"; space-time as a 'container' over which physics takes place has no objective physical meaning and instead the gravitational interaction is represented as just one of the fields forming the world. Einstein referred to his resolution as "beyond my wildest expectations."

https://en.wikipedia.org/wiki/Hole_argument

Traveler, there are no paths; Paths are made by walking. Antonio Machado

Intuitive

Think about how you would tell a friend where you are. You never give coordinates relative to empty space. You say "at the corner of the bakery" or "two metres left of the tree". Every position you can actually check is a relation between things: this object next to that object, this clock reading when that light flash arrived.

Relational physics takes this everyday habit seriously and asks whether physics needs anything more. Newton thought it did. In his picture there is an absolute space, a fixed stage, and objects move around on it. Leibniz disagreed: for him there is only the set of relations between objects, and talk about "where the whole universe is" has no content.

General relativity pushes strongly toward Leibniz. The equations of the theory look the same in any coordinate system, and this freedom is so large that a point of spacetime, taken on its own, cannot be identified at all. What can be identified is a meeting: two particles colliding, a light signal reaching a detector, the hand of a clock pointing at 12 when a train arrives. Such coincidences are the same in every description, and every measurement we make is built from them.

The same question comes up elsewhere. In gauge theories, the phase of a charged field at a single point is not measurable, only the phase difference along a path between two points. In the relational interpretation of quantum mechanics, proposed by Carlo Rovelli, the state of a system is taken to describe the system relative to another system that interacts with it, and different observers can assign different states to the same system.

The common thread is a rule for which questions make sense: only relations between physical things are observable, and a theory should be formulated so that everything else drops out.

Concrete

The central problem is that of the points of space. The debate goes back to Leibniz and Newton. For Leibniz the constituent of all things, both material and spiritual, is provided by monads, which are windowless (we would say they have no internal structure), and the only things that matter are their mutual relationships. Here we seem to recognize the first “definition” given by Bourbaki at the beginning of his discussion of set theory: A set is composed of elements capable of having certain properties and having certain relations among themselves or with elements of other sets. From this ontological point of view the elements, or points, are pre-existent and the problem is to organize them, to give them a structure. From the physical point of view, one postulates with Newton the existence of an absolute space, in which the phenomena occur: positions are predetermined, destined to be inhabited by the accidents of matter. In Mach’s philosophy, on the contrary, space is determined by matter ; the most advanced mathematical form is certainly furnished by the Einstein gravitational equations: $R_{\mu\nu}-\frac{1}{2}Rg_{\mu \nu} = 8 \pi \kappa T_{\mu\nu}.$ The right-hand side, $T_{\mu\nu}$ , is determined by the matter that happens to be present, or, in more modern form, it is a function of the nongravitational fields, while the left-hand side is a function of only the gravitational field $g_{\mu \nu}$ , identified with the metric tensor that defines the geometry. In contrast to Newton’s views, the space is no longer a mere receptacle, but an actor in physics, as the bending of light rays in a gravitational field shows. For Mach and Einstein, a point then only appears as a label making it possible to identify an event.

Probably the right mathematical tool to make the ideas of relational physics precise is category theory.

A great recent summary of the relational interpretation of quantum mechanics is "Space is blue and birds fly through it" Carlo Rovelli

Abstract

See the section "Towards a Pointless Theory" in Some "Elementary Gauge Theory Concepts" by Sheung Tsun Tsou, Chan Hong-Mo and Section 2.2.5 and Section 2.3.2 Rovelli's Quantum Gravity book

Why Gauge? by Carlo Rovelli
C. Rovelli, “Relational quantum mechanics,” Int. J. Theor. Phys. 35 (8), 1637-1678 (1996).
M. Smerlak and C. Rovelli, “Relational EPR,” Found. Phys. 37, 427-445 (2007).
N. D. Mermin, “What is quantum mechanics trying to tell us?,” Am. J. Phys. 66 (9), 753-767 (1998).

Relational Physics

Why is it interesting?

Intuitive

Concrete

Abstract

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