# A Classical fields: EM fields vs. Gravitational fields

#### SherlockHolmes

"Quantum field theory naturally began with the study of electromagnetic interactions, as the electromagnetic field was the only known classical field as of the 1920s".

Why wasn't Newtonian gravitation regarded as a classical field theory in 1920?

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#### dextercioby

Homework Helper
You are absolutely correct, and this is a point in which this free encyclopedia is wrong. Physics should be read from textbooks. Encyclopedia articles cannot cover subtle points or details.

Edit: The reply above was written by assuming the OP meant GR, not Newtonian gravity.

#### PeterDonis

Mentor
Why wasn't Newtonian gravitation regarded as a classical field theory in 1920?
Because it had been supplanted by GR, and it was not clear in 1920 whether or not GR was a "classical field theory" of the same sort as electromagnetism. We now know that it can in fact be viewed that way, but that was not clear in 1920.

You are absolutely correct,
Not quite. See above.

#### PeterDonis

Mentor
Why wasn't Newtonian gravitation regarded as a classical field theory in 1920?
Also, even if we leave out GR and only consider SR, Newtonian gravity, considered as a field theory, is not compatible with SR (because the "field" has to propagate at infinite speed), whereas electromagnetism is (because EM fields propagate at the speed of light). So it would not have occurred to anybody in 1920 to try to make a quantum field theory based on Newtonian gravity, since the whole point of quantum field theory was to make QM compatible with special relativity.

Encyclopedia articles cannot cover subtle points or details.
That's correct, but I think adding back the subtle points and details makes the Wikipedia article's claim seem reasonable as a basic overview of the subject.

#### dextercioby

Homework Helper
Because it had been supplanted by GR, and it was not clear in 1920 whether or not GR was a "classical field theory" of the same sort as electromagnetism. We now know that it can in fact be viewed that way, but that was not clear in 1920.

Not quite. See above.
Oh. I am sorry. I swear I thought the OP meant GR as a classical field theory, not Newtonian gravity. My bad.

#### SherlockHolmes

OK.

So if we consider Newtonian gravitation for a second, would it be true (leaving GR aside for now) that that is a classical field theory?

What was making it hard for them to see GR as a field theory back then? what was it that changed that?

Thanks!

#### PeterDonis

Mentor
if we consider Newtonian gravitation for a second, would it be true (leaving GR aside for now) that that is a classical field theory?
Only if you allow a "classical field theory" to have fields that propagate at infinite speed, violating special relativity. But the physicists who were working on quantum field theory did not allow that; as I have said, their purpose was to make QM consistent with SR, so Newtonian gravitation was not something they would have considered a "classical field theory" that could be quantized this way.

#### SherlockHolmes

This is interesting, so what is there in the Newtonian inverse square law that leads to the infinite propagation speed? Why does Maxwell's field formulation have a concrete propagation speed but Newton's does not?

#### PeterDonis

Mentor
What was making it hard for them to see GR as a field theory back then? what was it that changed that?
What made it hard was the fact that GR was interpreted as describing the geometry of spacetime, which was not considered a "field". It is true that Einstein, and a few others, used the term "field" (or its German equivalent) to describe the metric of spacetime, but this usage did not lead to any development of GR as a field theory like electromagnetism at that time. It was only in the 1960s and 1970s that it was discovered that you could, in fact, construct GR as the classical field theory of a massless, spin-2 field, similar to the way Maxwell's electrodynamics can be constructed as the classical field theory of a massless, spin-1 field.

#### SherlockHolmes

I'm going to peruse this, it discusses the stuff you spoke about:

My math is quite rusty and so it'll be a struggle.

#### PeterDonis

Mentor
I'm going to peruse this
This is a doctoral thesis, and on a rather advanced topic, so it might not be the best introduction to the subject. A textbook on QFT would be better.

#### vanhees71

Gold Member
OK.

So if we consider Newtonian gravitation for a second, would it be true (leaving GR aside for now) that that is a classical field theory?

What was making it hard for them to see GR as a field theory back then? what was it that changed that?

Thanks!
I would not call a theory with "action at a distance" "field theory". For me the field idea is due to the intuitive insight (afaik first by Faraday) that "actions at a distance" are in some way "unnatural" and that in fact the instantaneous forces of Newtonian Gravity and older theories on electric an magnetic forces (Ampere, Neumann) are to be substituted by interactions as "mediated" by fields and the forces are a local phenomenon, i.e., the force on a particle is due to the presence of a field (like the electromagnetic field as described by Maxwell) whose changes propagate at a finite speed. The final understanding came with the works of Heaviside, Poincare, Lorentz, FitzGerald and of course finally with Einstein and Minkowski.

#### martinbn

I would not call a theory with "action at a distance" "field theory". For me the field idea is due to the intuitive insight (afaik first by Faraday) that "actions at a distance" are in some way "unnatural" and that in fact the instantaneous forces of Newtonian Gravity and older theories on electric an magnetic forces (Ampere, Neumann) are to be substituted by interactions as "mediated" by fields and the forces are a local phenomenon, i.e., the force on a particle is due to the presence of a field (like the electromagnetic field as described by Maxwell) whose changes propagate at a finite speed. The final understanding came with the works of Heaviside, Poincare, Lorentz, FitzGerald and of course finally with Einstein and Minkowski.
What about a theory where the field has infinite speed of propagation. Would you call that a field theory?

#### Nugatory

Mentor
so what is there in the Newtonian inverse square law that leads to the infinite propagation speed?
The inverse square law says that the force on a test mass points towards the gravitating body, no matter where it is right now. It’s the “right now” that implies an infinite speed of propagation. When we move the gravitating object the force calculated from the inverse square changes everywhere in the universe, no matter how far away.
Why does Maxwell's field formulation have a concrete propagation speed but Newton's does not?
Maxwell’s laws contain $\frac{\partial}{\partial{t}}$ terms allowing for changes over time, and they describe the forces at a point using only values that are known at that point. The electric force at a point isn’t determined by the distance to the charged body, it is determined by the strength of the electric field at that point.

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#### vanhees71

Gold Member
What about a theory where the field has infinite speed of propagation. Would you call that a field theory?
No, that would be action at a distance!

#### martinbn

No, that would be action at a distance!
Yes, but through a field. It's just that the field propagates instantaneously. For instance the heat equation has infinite speed of propagation, and some would call the temperature a field.

#### vanhees71

Gold Member
Of course, it's an approximation. The heat equation in its usual form is not relativistic!

#### martinbn

Of course, it's an approximation. The heat equation in its usual form is not relativistic!
So, when you said "field theory" you meant "relativistic field theory"?

#### king vitamin

Gold Member
In principle I don't see a problem with defining a "Newtonian action"
$$\mathcal{S} = \int dt \left[ \mathrm{KE} - \frac{ 1 }{ 8 \pi G } \int d^d x \, \partial_{ i } \phi ( x ) \partial^{ i } \phi ( x ) + \int d^d x \, \rho ( x ) \phi ( x ) , \right],$$
where $\rho(x)$ is a mass density (perhaps a sum over delta functions), and KE is the kinetic term for the matter (perhaps a sum over $m_i \dot{x}_i(t)/2$). But it's sort of a funky field theory, since the Newtonian gravitational field is non-dynamical (there's no "kinetic" term for $\phi(x)$). This means that the field can change instantaneously in time without a kinetic energy "penalty," which is something undesirable in most field theories (whether they are relativistic or not).

#### atyy

https://www.edge.org/response-detail/26729
"... As soon as Einstein announced his elegant theory, other physicists began trying to reconcile general relativity with quantum mechanics. Quantum mechanics is the physical theory that governs matter at its smallest and most fundamental scales. The last century has also seen tremendous advances in the application of quantum mechanics to study elementary particles, solid state physics, the physics of light, and the fundamental physics of information processing. Pretty much as soon as the print on Einstein's papers had dried, physicists began trying to make a quantum theory of gravity. They failed.

The first theories of quantum gravity failed because scientists did not understand quantum mechanics very well. It was not until a decade after Einstein's results that Erwin Schroedinger and Werner Heisenberg provided a precise mathematical formulation of quantum mechanics. By the beginning of the 1930s, Paul Dirac had formulated a version of quantum mechanics that incorporated Einstein's earlier—and by definition less general—theory of special relativity. ..."

I wonder what work Seth Lloyd is referring to when he says "As soon as Einstein announced his elegant theory, other physicists began trying to reconcile general relativity with quantum mechanics."

#### PeterDonis

Mentor
I wonder what work Seth Lloyd is referring to when he says "As soon as Einstein announced his elegant theory, other physicists began trying to reconcile general relativity with quantum mechanics."
That certainly doesn't seem right to me. Einstein published the EFE in November 1915. The first work on quantum gravity that I'm aware of is in the late 1950s. What's more, in 1915 the only "quantum mechanics" around was Planck's 1900 paper on black-body radiation, Einstein's 1905 paper on the photoelectric effect, and the original Bohr-Sommerfeld model of the atom, and physicists were still struggling how to even do experiments that probed the simplest quantum phenomena. So I don't know what he's talking about.

#### king vitamin

Gold Member
I would guess that people attempted the Bohr-Sommerfeld quantization scheme for general relativity (though Sommerfeld's paper was 1916). After all, it did successfully account for special relativistic corrections to atomic spectra. But I wouldn't be surprised if you can't find many papers on the subject, since I wouldn't expect the method to get very far.

The earliest "quantum gravity" result I can think of is Fierz and Pauli's observation that a QFT of spin-2 particles leads to linearized GR (1939).

#### atyy

I emailed Seth Lloyd, and he pointed me to The Early History of Quantum Gravity (1916–1940) by John Stachel. The article is not free, but here are excerpts.

In his first paper on gravitational radiation [1916], Einstein argued that quantum effects must modify the general theory of relativity:
Due to the intra-atomic movement of electrons, atoms would have to radiate not only electromagnetic but also gravitational energy, if only in tiny amounts. As this is hardly true in nature, it appears that quantum theory would have to modify not only Maxwellian electrodynamics, but also the new theory of gravitation (p.209).

The intimate connection between space, time and gravitation in general relativity suggested the need for a quantum-based modification of general relativity to Oskar Klein [5]. In 1927 ...
Then follows a footnote:
For example, Einstein light-deflection by a collection of moving material bodies in a thermal equilibrium state would destroy the equilibrium of the radiation, which allows one to suspect a sort of gravitational Compton effect. One would expect a corresponding result in the case of statistical equilibrium between gravitational waves and light waves according to the general theory of relativity, due to the non-occurence of Planck's constant in the relevant equations.

Others were less cautious in approaching the problem of a quantum theory of gravity. In their first paper on quantum electrodynamics, Heisenberg and Pauli asserted [1929]:
Quantization of the gravitational field, which appears to be necessary for physical reasons [in a footnote, they refer to the works of Einstein and Klein cited above}, may be carried out without any new difficulties by means of a formalism fully analogous to that applied here (p.3).

#### PeterDonis

Mentor
I emailed Seth Lloyd, and he pointed me to The Early History of Quantum Gravity (1916–1940) by John Stachel. The article is not free, but here are excerpts.
Hm, very interesting!

Due to the intra-atomic movement of electrons, atoms would have to radiate not only electromagnetic but also gravitational energy, if only in tiny amounts. As this is hardly true in nature
I'm not sure we can actually rule out GW emission by electrons in atoms on observational grounds, since it would be much, much too weak for us to detect. So "as this is hardly true in nature" seems to be typical Einstein, similar to his attitude when someone asked him what he would think if (IIRC) Eddington's eclipse expedition didn't end up observing bending of light by the Sun (quoting from my best memory from previous reading): "then I would feel sorry for nature; the theory is correct".