We can do quantum field theory just fine on the curved spacetime background of general relativity.
Then he adds this caveat:
What we have so far been unable to do in a convincing manner is turn gravity itself into a quantum field theory.
These carefully selected quotes are from a recent post by Toth on Quora where he is a regular contributor. His area of expertise is in cosmology, including the study of black holes. On another post he explains how the 2 theories are mathematically ‘incompatible’ (my term, not his):
The equation is Einstein’s field equation for gravitation, the equation that is, in many ways, the embodiment of general relativity:
Rμν−12Rgμν=8πGTμν.
The left-hand side of this equation represents a quantity formed from the spacetime metric, which determines the “deformation of spacetime”. The right-hand side of this equation is a quantity that is formed from the energy, momentum, angular momentum and internal stresses and pressure of matter.
He then goes on to explain that, while the RHS of the equation can be reformulated in QM nomenclature, the LHS can’t. There is a way out of this, which is to ‘average’ the QM side of the equation to get it into units compatible with the classical side, and this is called ‘semi-classical gravity’. But, again, in his own words:
…it is hideously inelegant, essentially an ad-hoc averaging of the equation that is really, really ugly and is not derived from any basic principle that we know.
Anyway, the point of this mini-exposition is that there is a mathematical conflict, if not an incompatibility, inherent in Einstein’s equation itself. One side of the equation can be expressed quantum mechanically and the other side can’t. What’s more, the resolution is to ‘bastardise’ the QM side to make it compatible with the classical side.
You may be wondering what all this has to do with the title of this post. The fundamental constant at the heart of general relativity is, of course, G, the same constant that Newton used in his famous formula:
On the other hand, the fundamental constant used in QM is Planck’s constant, h, most famously used by Einstein to explain the photo-electric effect. It was this paper (not his paper on relativity) that garnered Einstein his Nobel prize. It’s best known by Planck’s equation:
E = hf
Where E is energy and f is the frequency of the photon. You may or may not know that Planck determined h empirically by studying hot body radiation, where he used it to resolve a particularly difficult thermodynamics problem. From Planck’s perspective, h was a mathematical invention and had no bearing on reality.
G was also determined empirically, by Cavendish in 1798 (well after Newton) and, of course, is used to mathematically track the course of the planets and the stars. There is no obvious or logical connection between these 2 constants based on their empirical origins.
There is a third constant I will bring into this discussion, which is c, the constant speed of light, which also involves Einstein, via his famous equation:
E = mc2
Now, having set the stage, I will invoke the subject of this post. If one uses Planck units, also known as ‘natural units’, one can see how these 3 constants are interrelated.
I will introduce another Quora contributor, Jeremiah Johnson (a self-described ‘physics theorist’) to explain:
The way we can arrive at these units of Planck Length and Planck Time is through the mathematical application of non-dimensionalization. What this does is take known constants and find what value each fundamental unit should be set to so they all equal one. (See below.)
Toth (whom I referenced earlier) makes the salient point that many people believe that the Planck units represent the physical smallest component of spacetime, and are therefore evidence, if not proof, that the Universe is inherently granular. But, as Toth points out, spacetime could still be continuous (or non-discrete) and the Planck units represent the limits of what we can know rather than the limits of what exists. I’ve written about the issue of ‘discreteness’ of the Universe before and concluded that it’s not (which, of course, doesn’t mean I’m right).
Planck units in ‘free space’ are the Universe’s ‘natural units’. They are literally the smallest units we can theoretically measure, therefore lending themselves to being the metrics of the Universe.
The Planck length is
1ℓP=1.61622837∗10−35m
And Planck time is
1tP=5.3911613∗10−44s
If you divide one by the other you get:
1ℓP/1tP=299,792,458m/s
Which of course, is the speed of light. As Johnson quips: “Isn’t that cool?”
Now, Max Planck derived these ‘natural units’ himself by looking at 5 equations and adjusting the scale of the units so as they would not only be consistent across the equations, but would non-dimensionalise the constants so they all equal 1 (as Johnson described above).
In fact, the definition of the Plank units (except charge) includes both G and
The point is that I was able to derive G from h using Planck units. The Universe lends itself to portraying a consistency across metrics and natural phenomena based on units derived from constants that represent the extremes of scale, h and G. The constant, c, is also part of the derivation, and is essential to the dimension of time. It’s not such a mystery when one realises that the ‘units’ are derived from empirically determined constants
Addendum: For a comprehensive, yet easy-to-read, historical account, I’d recommend John D. Barrow’s book, The Constants of Nature; From Alpha to Omega.
Addendum (13 April 2023): According to Toth, the Planck units don't provide a limit on anything:
...the Planck scale is not an inherent limit of anything. It is simply the set of “natural” units that characterize Nature.
So the Planck scale is not a physical limit or a limit on what can be observed; rather, it’s a limitation of the theory that we use to describe the quantum world.
Read his erudite exposition on the subject here:
https://www.quora.com/Why-is-the-Planck-length-the-smallest-measurable-length-Why-cant-it-be-smaller/answer/Viktor-T-Toth-1
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