Monday, January 23, 2012

Why "semiclassical gravity" isn't self-consistent

Sabine Hossenfelder recently discussed various gedanken experiments showing that "semiclassical gravity" can't be a consistent description of Nature:
Backreaction on Eppley-Hannah's thought experiment

Backreaction on this and Page-Geilker's thought experiment
While she admits that "semiclassical gravity" can't be right for various theoretical reasons, she still irrationally criticizes the two thought experiments above. In this inconsistent treatment of hers, she seems to misunderstand that the very purpose of gedanken experiments is to give us those "theoretical reasons" to know that Nature can't work in certain ways.

Now, I must tell you what we mean by "semiclassical gravity". Quite generally, the adjective "semiclassical" in physics means that certain parts of the physical system are being treated in the framework of quantum mechanics; others are treated using classical physics. If the two parts influence each other (in both directions), this is really an inconsistent approach, as I will discuss, and it doesn't make sense to develop the "semiclassical approach" too accurately.

In particular, the term "semiclassical approximation" is often applied to electrons in an external potential. The potential, e.g. the electrostatic potential induced by the atomic nuclei, is assumed to be a source of classical external forces. This is justifiable as an approximation because the nuclei are much heavier, and therefore "more classical", than the electrons and the same comment applies to the field they exert.

However, such a treatment automatically denies the existence of virtual photons etc. so it can't possibly be right at the quantum (loop) level. Only the leading quantum effects influencing the electron, those proportional to \(\hbar^1\), may be considered in this treatment. That's why the first quantum corrections to classical physics, e.g. one-loop diagrams in various quantum field theories, are often called "semiclassical" as well; some people view "semiclassical" and "one-loop" to be synonyms. The WKB approximation is a typical example of a semiclassical treatment in non-relativistic quantum mechanics.

In the context of gravity, people use the term "semiclassical gravity" either as a legitimate approximation that is aware of its limitations; it's the approximation that was used e.g. by Stephen Hawking to derive the thermal radiation emitted by black holes. Alternatively, some people use the term "semiclassical gravity" as a proposed "hybrid" quantum-classical picture of physics. Reasons why this idea is wrong will occupy the rest of this blog entry.




As the Wikipedia page clarifies, the "semiclassical gravity" is a hypothetical theory describing the real world in which matter such as protons and electrons obeys the rules of quantum mechanics but gravity doesn't. Instead, the gravitational field is a classical field and it evolves according to a "classical projection" of quantum Einstein's equations:
\[ R_{\mu\nu}-\frac{1}{2} R g_{\mu\nu} = \frac{8\pi G}{c^4} \left\langle \hat T_{\mu\nu} \right\rangle_{\ket\psi} \] What you see are almost ordinary equations of classical general relativity. However, because the energy and momentum density is composed of "quantum matter", one must extract a classical value of the full stress-energy tensor – the latter is an operator – and the seemingly natural choice is the expectation value.

If you take quantum mechanics seriously and you use a proper "positivist" interpretation of it, you immediately see that the equation above can't be right. Why? It's simple. The metric tensor is an "objective variable". However, the right hand side depends on the wave function \(\ket\psi\) which describes the state of knowledge of an observer. It's therefore subjective. Even the expectation values are subjective. An equation claiming that an objective variable such as the Einstein tensor is equal to a subjective variable – such as a function of the state vector – is as inconsistent as the claim that the surname of the president of your country coincides with each citizen's wife's first name.

The only way to keep the mixed equation alive is to assume that the wave function is an objective real wave, too. Consequently, you must assume that it collapses at some objectively determinable point and the collapse is an objective process, too. That's a big problem. Indeed, because the expectation value of the stress-energy tensor sources the gravitational field, you will be able to measure the time when the collapse occurred, at least in principle. At that moment, the gravitational field abruptly changes.

This assumption already contradicts the internal logic of quantum mechanics. The collapse of a wave function is otherwise a completely immaterial process. It's the process in which new facts are taken into account and the original probabilities are "abruptly" replaced (in your head) by conditional probabilities with the new known facts' being added among the conditions.

Even if you believe that the wave function is a tangible or classical wave, you must agree that quantum mechanics and anything that is empirically equivalent to it – anything that hasn't been ruled out empirically – unambiguously predicts that the precise moment of a collapse is unobservable. As TRF has discussed many times, Wigner's friend closed in a box together with Schrödinger's cat may "feel" that certain quantities already have sharp outcomes. But a more accurate observer outside the box treats Wigner's friend as another quantum object that evolves into linear superpositions as well. Such a treatment is necessary to correctly calculate various extreme interference processes that Wigner's friend himself may participate in in the future. Consequently, Wigner's friend and the external observer inevitably disagree when the "collapse" occurred. That's a sign of its being a subjective process.

But, as Hossenfelder correctly points out, the problem of the collapse's becoming an objective process is even more pathological and leads to more tangible problems if you try to design a classical gravitational field that reacts to the expectation values. Why? Well, the collapse changes the expectation value of the stress-energy tensor "instantaneously". It follows that if you allow the wave function \(\ket\psi\) to "physically collapse" at a certain moment, the expectation value \(\langle \hat T_{\mu\nu}\rangle\) calculated from the wave function will discontinuously change, too: a part of the Schrödinger's cat's mass disappears from a place once you determine whether it is dead or alive. It will violate the local conservation law, i.e.
\[ \nabla_\mu \left\langle \hat T^{\mu\nu} \right\rangle_{\ket\psi} \neq 0. \] That's too bad because the left hand side is locally covariantly conserved. The equation
\[ \nabla_\mu \left( R^{\mu\nu}-\frac{1}{2} R g^{\mu\nu} \right) = 0 \] is an identity, a tautology: one can prove it just by cancelling all the terms (various products of the metric tensor's components and their derivatives) regardless of the values of \(g_{\mu\nu}(x^\alpha)\). So the Einstein tensor can't possibly be equal to the expectation value of the stress-energy tensor: they have different covariant divergences! Consequently, the defining equation of "semiclassical gravity" has no solutions if you allow the wave function \(\ket\psi\) to "physically collapse" and that's needed to agree with the empirical facts if you want to claim that the wave function is an "objective object".

In the case that you think that this is an insurmountable paradox of any physics, I must tell you that proper quantum gravity has no problem. At long distances, the proper quantum equations of gravity are
\[ \hat R_{\mu\nu}-\frac{1}{2} \hat R \hat g_{\mu\nu} = \frac{8\pi G}{c^4} \hat T_{\mu\nu} \] There are no expectation values here. The Einstein equations hold as operator equations at the full quantum level. Only the expectation value of the Einstein tensor are linked to the expectation values of the stress-energy tensor. Both of them behave discontinuously in the moment of the "collapse" – more correctly, in the moment when you learn about the result of a measurement. And they are allowed to behave discontinuously. If you calculate the expectation value of both sides of the equations above, both of them will depend on \(\ket\psi\) which changes abruptly which is why there won't be any contradiction. However, the covariant divergence of both sides always vanishes at the operator level.

Gravity can't be classical in a quantum world. Classical and quantum objects can't be combined. If the fate of an object – e.g. the gravitational field (and the Earth whose journey is affected by it) – depends on some quantum objects and variables, it must be described by quantum, probabilistic variables as well. There's no way to escape this simple argument.

Hossenfelder tries to claim that the gedanken experiments showing that "semiclassical gravity" isn't a consistent theory can't be promoted to real experiments. The same meme is promoted by a self-described philosopher or, equivalently (using physics terminology), crackpot, named James Mattingly (who didn't even manage to fix the grammar in his title). Well, I disagree with that. It could be hard to design an experiment that would actually see some interference that depends on the quantum character of the gravitational field. But the equation of "semiclassical gravity" may be experimentally falsified, anyway.

When the Solar System was created, the Earth appeared at a pretty random direction relatively to the Sun (close to the ecliptic, as determined by i.e. perpendicular to the initial angular momentum of the Solar System). But it's fair to estimate that the expectation value of the position of the Earth was zero, near the Sun. Nevertheless, the Moon seems to orbit an Earth of the right mass that has a well-defined location, instead of a mixture of Earths that are distributed along the whole orbit.

If you wanted to claim that it is because the wave function of the Earth has already "objectively collapsed", you will find other things that contradict the experiments, too. Such an equation would predict lots of nonlocal phenomena and related violations of special relativity. But as I have explained above, the equations of "semiclassical gravity" won't have any solutions at all because the expectation value of the stress-energy tensor isn't locally conserved if you allow the state vector to collapse. So if you allow me to think, and physicists must be allowed to think, the equation of "semiclassical gravity" is empirically excluded. I don't really need any expensive observations at all. That's often the case. Most random ideas someone proposes as theories of physics are stillborn.

These basic questions about quantum gravity have been settled for a very long time. It is very obvious that e.g. Richard Feynman understood them in the early 1960s and probably earlier than that. After all, he constructed the simplest Feynman rules for the quantized general relativity which directly reproduces the low-energy limit of the rules in string theory.

Some people doubt that there are gravitons. But these doubts are silly. First of all, Einstein's equations – which have been verified in quite some detail – predict gravitational waves. We haven't seen the gravitational waves "quite directly" but the Scandinavian folks have already awarded a Nobel prize, namely the 1993 physics Nobel prize, for an experimental detection proving their existence. A pulsar was seen to change its frequency. The change accurately agreed with the prediction of general relativity that says that the accelerated motion of massive bodies produces gravitational waves which carry a calculable amount of energy away. With the calculable energy loss, the objects become increasingly bound and the frequency of the orbiting is therefore going up.

So let me assume that you understand that the gravitational waves exist and have been "almost directly" observed. Now, are they composed of gravitons in the same way as electromagnetic waves are composed of photons? They have to be. Imagine you have a monochromatic gravitational wave. Various observables in this wave are changing with frequency \(f=\omega/2\pi\) and periodicity \(t=1/f=2\pi/\omega\). But if you accept that the whole Universe, including its gravitational fields and bodies affected by them, is described by the same apparatus of quantum mechanics, you may derive other inevitable consequences of these assumptions.

In particular, every wave function may be decomposed into a linear superposition of energy eigenstates which evolve as
\[ \ket{n(t)} = \exp(E_n t/i\hbar) \ket{n(0)} \] If you take a general superposition of such states \(\ket{n}\) and if you assume that all expectation values and other things are periodic with period \(2\pi/ \omega\), then you may easily see that all the energy differences \(E_m-E_n\) between states \(\ket{m}\) and \(\ket{n}\) included in the superposition (with nonzero coefficients) have to be multiples of \(E=\hbar\omega\). That's the only way how the observables may be periodic: after the period \(\Delta t =2\pi/\omega\), the phase factors differ by the multiplicative factor of
\[\begin{align} \exp(E_m \Delta t/i\hbar) / \exp(E_n \Delta t/i\hbar) =\dots\\ \dots = \exp\left[\frac{(E_m-E_n)\Delta t}{i\hbar}\right] = \exp(2\pi N) \end{align} \] for \(E_m-E_n=-N\hbar\omega\) and they only return to the same "relative phase" if \(N\) is an integer. It follows that the energy isn't changing continuously; in a monochromatic wave or any periodic system in Nature, the energy has to respect the spacing \(\Delta E = \hbar\omega=hf\). This relationship between the energy and time doesn't apply to photons only; it applies to all systems in Nature. We just use the terms "photons" and "gravitons" for the changes of the frequency \(f\) electromagnetic and gravitational field (respectively) that increase the total energy by the minimum allowed step, namely by \(\Delta E = \hbar\omega=hf\).

At the one-loop level, there's nothing wrong about quantum gravity where the gravitational field is quantized in the fully analogous way as the electromagnetic field in QED. For example, linearized equations of general relativity are enough to determine the number of physical polarizations of a graviton (two in \(d=4\)). Quantum gravity is still conceptually harder because there are no rigid God-given slices of spacetime on which we could define our wave functions etc. But when we do the right calculations that may be generalized from QED, they work. The problems only occur when we try to calculate multiloop corrections because we encounter the non-renormalizable divergences hiding in quantized Einstein's equations. Only at this point, we realize that string theory is paramount.

At any rate, the goal of this blog entry was to convince at least some TRF readers that the idea that one could completely deny quantum mechanics in the gravitational sector (or any other sector) is one of the ideas that may be classified as bad physics. The people who are trying to advocate such theories never respect any high standards – because no consistent equations of this kind can even be written down – and their unusual constructions are symptoms of their broader misunderstanding of modern physics, including special relativity, locality, and the probabilistic character of quantum mechanics.

5 comments:

  1. Semiclassical gravity, is the same name used for "quantum field theory in curved-spacetime" ??? And what about the Unruh effect ??? That was based on semiclassical gravity approach???
    Thanks!

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  2. Semiclassical gravity is the same word used to "quantum field theory in curved spacetime" ???
    Thanks!

    ReplyDelete
  3. Yup, semiclassical gravity is used as a synonym of QFT on curved spacetimes. Well, it's QFT on curved spacetimes together with the classical Einstein's equations describing the evolution of the background spacetime geometry.


    Yes, Unruh's effect was first calculated in this framework. But it holds more generally. The thermal radiation that an accelerating observer experiences also includes gravitational waves at the same temperature...

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  4. Thank you for the replay! Well I have another question (actually a clarification). I have a professor who tell me that, semiclassical gravity may have inconsistencies has you point it out... but he told me that, QFT in curved backgrounds is a well established stuff. It is like to put QFT on a flat Minkowski space, but now instead of Minkowski space we use another background (a curved background), and study the theory just as in the Minkowski space case, he told me that, for example, if a QFT is renormalizable in Minkowski space, also it will be renormalizabled in the curved background (provide some assumptions). I mean, you choose some FIXED space-time background and put your favority QFT on it and calculate process like the way you do in the usual QFT on Minkowski space. So what do you think about this approach? (Note that this approach does not intent to describe the evolution of the spacetime geometry, since it is fixed, althout curve, from the begining)

    Thanks!

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  5. What you write is perfectly valid. If a QFT is renormalizable in flat space, then it will be renormalizable in curved spaces, too - I don't want to go into exceptions which are subtleties - because the renormalizability etc. is determined by the short-distance structure of the theory.


    However, a QFT in a predetermined curved spacetime isn't a framework to describe gravity because gravitational fields, by definition, must depend on the distribution of masses i.e. they must depend on the actual evolution of your quantum field theory. A predetermined spacetime doesn't do it so this spacetime doesn't really "respond" to what the masses are doing in your theory. Consequently, QFT in curved spacetime is only OK to study the motion of a small enough number of excitations whose total energy/mass/momentum is negligible relatively to the masses that source the surrounding gravitational field.

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