Tuesday, May 15, 2007

Some elementary myths about quantum gravity

A freshman talks how excited he is about Carlip's lectures at angryphysicist.wordpress.com. There is some confusion in the text that comes from various sources. Because I keep on hearing some of these opinions from physicists who are often much older than freshmen, including new grad students, it may be a good idea to mention them explicitly.

Myth: The right theory of quantum gravity may be completely non-local

Well, in reality, the correct theory must be very accurately local because the local effective theories have been verified experimentally. Locality means that we can't send signals faster than light and different regions of space can't influence each other immediately: it is an important part of the relativistic description of reality.

Quantum gravity is almost certainly somewhat non-local when processes like black hole evaporation are taken into account: this feature seems to be necessary to preserve the information and almost certainly follows from string theory. But a correct theory must always explain why these non-local processes have a tiny impact on the usual observable phenomena otherwise the theory is dead.

Myth: Quantization of gravity implies that spacetime is discrete

This myth, much like many other myths, arises because their authors use certain words whose meaning they don't understand well and they only use the methods of comparative literature - a manipulation with fuzzy words instead of sharp equations - to derive their conclusions. In this particular case, it's important to realize that the word "quantization" is really used in two inequivalent albeit related ways.

Quantization of a physical theory is a method to construct a quantum theory whose classical limit agrees with a given classical theory: the classical observables are promoted to operators and a Hilbert space is found. Such a process implies that the spectrum of some observables becomes discrete but it is not true that the spectrum of all observables is discrete. For example, the spectrum of both position and momentum remains continuous in quantum mechanics unlike e.g. the energy of the harmonic oscillator.

Discreteness of allowed values - another meaning of "quantization" - can sometimes follow from the rules of quantum mechanics but sometimes it doesn't. More concretely, it is extremely unlikely that quantum gravity in 4 dimensions or higher has any useful description in terms of discretized geometry because such a discreteness would violate certain important symmetries of the laws of physics - and more generally, it is simply an unjustified, extremely constraining assumption. It is a random guess that is unlikely to be true much like when you conjecture that Elvis Presley will be found alive on the Moon. Theories without any continuous description form an unlikely, infinite-codimension subclass of possible theories.

Myth: Quantization of gravity makes no sense at all

If we apply the standard rules of quantization to general relativity, we end up with a theory that works pretty well, until we study it very accurately. At the crudest qualitative level, there is no real difference between electrodynamics and general relativity. Both of them imply that waves are composed of particles and their interactions are calculable by Feynman diagrams or related devices. Photons have spin one while gravitons have spin two. But it's a difference in technical details.

Problems with naively quantized gravity only emerge if we study it at a multi-loop level and discover infinitely many types of divergences, or if we look at situations where the causal structure differs from the causal structure of empty space. In perturbation theory, gravitons are thought of as infinitesimal perturbations of the background whose causal structure is respected. On the other hand, one can see that this fact doesn't mean that semiclassical gravity can't be used for other curved backgrounds.

Myth: Gravity may be classical

It is impossible to couple a classical system with a quantum system consistently. According to quantum mechanics, the positions of objects are quantum observables that can have different probability amplitudes for having different values. Because these positions influence the right-hand side of Einstein's equations, it follows that the metric tensor is an operator with a probabilistic interpretation, too. It can't be otherwise. If something depends on a "random generator", everything else that is influenced by it depends on this random generator, too.

Myth: Only expectation values of operators follow their equations

The angry physicist writes something like the expectation values of Einstein's equations, claiming that maybe, no other laws can be valid. This is a misunderstanding how quantum mechanics works. Quantum mechanics implies a linear equation for the time evolution of the wave function, if I use the Schrödinger picture. This picture is equivalent to precise laws for the evolution of all & whole operators in the Heisenberg picture.

If only some expectation values followed the laws and others would not, one could show that the laws of physics either don't determine some probabilities at all - in which case they wouldn't really determine almost anything, if you think about it - or they would be equivalent to a non-linear Schrödinger equation because the expectation value, for example, is bilinear in "psi". It is very likely that any attempt to formulate such new laws for "psi" or the equivalent laws for the operators must be (very) non-local if it is (very) inequivalent to a standard linear Schrödinger equation.

Myth: Gravity waves could have continuous energy

The existence of classical gravitational waves, predicted by general relativity, has been indirectly verified by observations of pulsars and awarded by a Nobel prize in the 1990s. Some could still argue that the energy of a gravitational wave doesn't have to be a multiple of the graviton energy because the gravitational field is different from the electromagnetic field.

That's not really possible. A monochromatic gravitational wave is a classical solution that is periodic in time. Because such a classical wave must be allowed to have a somewhat fuzzy value of the ADM energy, we see that the different quantum microstates contributing to the classical configuration must have energies that differ by a multiple of "E=hf". If the differences were different, the quantum time-dependent phase "exp(Et/i hbar)" wouldn't be synchronized for different microstates and some observables in the classical wave would thus be aperiodic.

To summarize, "E=hf" is a universal law for the quantum of energy carried by any kind of a wave in a quantum theory. In other words, gravitons must exist.

Myth: Quantum stress-energy tensor isn't conserved

I don't exactly know why but the angry physicist argues that the covariant divergence of "T_{mn}" is not zero in the quantum theory. Well, the quantum equations should follow the classical ones with a hat. The only new subtlety are ordering ambiguities that can be rephrased as higher-derivative operators generated by quantum loops. But surely one wants to construct a theory whose effective description reproduces the normal continuity equation. This continuity equation follows from Einstein's equations and Einstein's equations follow from the defining formulae of the theory - such as the action in the path integral. There can't be any contradiction here.

Myth: The only task is to add nice hats

In the semiclassical approximation where only the terms proportional to the first power of Planck's constant are considered, quantum gravity is analogous to quantum electrodynamics and other quantum field theories. It produces a Fock space and can be used to calculate interactions in the same approximation. However, when we study the theory more accurately, this picture is spoiled by new divergences and related problems.

Nature doesn't guarantee and cannot guarantee that the right description of the theory will always involve the metric tensor. Actually, it is almost certain that above 3 spacetime dimensions, such a pure-metric quantum theory of gravity is impossible. Some people want to mimic the cargo cult and invent nicer shapes of their headphones - the hats above the operators in this case - but they are not ready to modify their fundamental assumption that the metric tensor is everything there is.

The known facts make it extremely unlikely that the metric remains the correct degree of freedom at the Planck scale, in the heavily quantum regime. The known facts include the non-renormalizable UV divergences of gravity - analogous to those in Fermi's theory where a more UV complete theory was later found (gauge theory). They also depend on the decades of failures to make any sense out of multi-loop pure gravity in 4 dimensions or higher, and the complete consistency of the analogous calculations that has been observed in string theory. I think that only a bigot could fail to learn a lesson here.

Myth: In the context of singularities, the only goal of quantum gravity is to make things look finite

This myth is shared by most of the loop quantum gravity and arguably by some string theorists, too. The real problem with a singularity is not that it looks infinite. The real problem with a singularity is that it cripples the predictive power of your theory. When things are infinite, you can't safely subtract them from each other. You don't know how much infinity minus infinity is. That's why you don't know what will happen which is the bad thing.

You can regulate a theory with singularities in many ways but generically, the result will depend on the details how you regulate them. This means that you have made no progress whatsoever. The dependence of your physical predictions on the cutoff is just another way of reformulating the unpredictability resulting from the singularity. It is, once again, just a superficial method to paint an ugly face by pink color but it doesn't change the essence of the problem.

The only way to really solve the problem is to replace the singular theory by a theory that only requires us to determine a finite number of parameters to make the predictions that the theory is supposed to make. Any other procedure to make things smooth or discrete is a physically worthless sleight of hand. This conclusion, of course, applies to bizarre statements such as the statement that loop quantum gravity solves some ultraviolet problems of quantized general relativity. It doesn't. It only looks at these problems through too strong eyeglasses.

Myth: The Hilbert space of black hole microstates is universal

This is really Steve Carlip's myth but it naturally fits into this text. In string theory, one can calculate the entropy of huge classes of black holes and other black things with charges, angular momenta, and diverse topologies in various dimensions. The calculation typically reduces to the Cardy's formula: the microscopic machine to get the right exponential degeneracy of states boils down to the same method of counting of states in a conformal field theory.

But the nature of this conformal field theory may depend on the context. In various backgtrounds, it has a different spectrum of operators, it has different degrees of freedom. Steve Carlip doesn't like it and he would prefer if all black holes had the same degrees of freedom. Well, I think it's obvious that his conjecture that the degrees of freedom are universal has been falsified. There are different CFTs that describe black hole microstates in different backgrounds. Even the local spacetime description has different degrees of freedom because the metric is coupled to different kinds of matter fields in different contexts.

The only statistical-physics thing shared by all the black holes is the leading Bekenstein-Hawking estimate for the entropy. It is shared because all the black holes live in theories that share the metric tensor and the Einstein-Hilbert term in the action. All other terms and degrees of freedom in the action depend on the context which is really why the details of the entropy counting depend on the situation, too.

Myth: The problem of time means that everyone must work with non-local observables all the time

General relativity in a generic background has no gauge-invariant local observables. Local observables should be associated with points but there is no way to select "the" point: coordinates don't have a direct physical meaning in general relativity. We can only identify a point by its position relatively to other point which requires us to deal with an extended region of space: such an approach is non-local.

But it doesn't really mean that the majority of physics must be reduced to a proper way to define and manipulate with non-local gauge-invariant quantities. It is a myth spread primarily in the algebraic quantum field theory community that one is never allowed to work with observables that are not gauge-invariant. In reality, it is only the final physical predictions that must be gauge-invariant. But the intermediate calculations don't have to be gauge-invariant.

The main advantage of gauge-dependent degrees of freedom is that we can use them, after all. This simplifies many calculations. Additional auxiliary degrees of freedom and symmetries make many theories more elegant and they make the calculations more efficient. There is nothing wrong with unphysical degrees of freedom as long as the final results are physical. Insisting that gauge-dependent degrees of freedom are banned is an irrational denial of a successful technology that grew increasingly more important in the 20th century.

Myth: We don’t know how to renormalize wave functions, and thus cannot really know how to get probabilities

Probabilities are always obtained as squared absolute values of probability amplitudes and there is never a problem with the normalization of the thing that is properly called a wave function - the state vector. It can simply be defined to be normalized to one. What we call "wave function renormalization" in quantum field theory is actually a renormalization of the field operators, not the actual wave function. The word "wave function" is only used for these operators because they may be thought as arising from single-particle wave functions by the second quantization.

Myth: "We don’t know if quantum gravity is generally covariant" is a meaningful sentence

As argued above, gauge symmetries are a useful technical tool to formulate many theories in an elegant way and make calculations more straightforward. Nevertheless, gauge symmetries aren't a part of the real physics. The question whether a theory has a gauge symmetry is not a physical question. There are ways to calculate physical quantities such as the S-matrix without a gauge symmetry - e.g. string theory in the light cone gauge - but it doesn't mean that there aren't other approaches. Moreover, I think that every theory that deserves to be called "quantum gravity" must respect general covariance in any context in which the metric tensor is a good degree of freedom. If one formulates or tries to formulate a theory with the metric tensor, we thus know, by definition, that quantum gravity is generally covariant.

If someone looks for a theory where it's not, he is really solving a very different problem and he can't use the usual justifications of general relativity to promote his ideas. People should try to be careful before they declare their assorted ideas to be relevant for quantum gravity. Gravity is associated with concepts such as Newton's law, general covariance, and gravitons. If neither of these objects is present, the idea is not connected to gravity.

Myth: The Hamiltonian is 0 in any Hamiltonian formulation of classical general relativity

The Hamiltonian is only zero in a compact space. However, whenever one can define asymptotic quantities such as the ADM energy, the Hamiltonian has additional boundary terms that make it nonzero and identical to this ADM energy. The vanishing of the Hamiltonian in a certain situation is not a problem of a theory of quantum gravity; instead, it is a proof that we have asked a wrong question. If we want to fix the problem, the solution is not to look for a different dynamical theory but rather for a better question.

Myth: A major task for quantum gravity is to find a nice field redefinition

Field redefinitions should be viewed as technicalities. A correct field redefinition can be useful to find a solution to a problem. But it can't allow one to find the right theory if it were impossible without the field redefinition. Every ambiguity in certain coordinates or degrees of freedom can be translated to an analogous ambiguity formulated in terms of the redefined degrees of freedom. Every inconsistency or other problems may be translated as well.

Myth: Ordering ambiguities are an independent problem of a local quantum theory

If one wonders what's the right ordering of operators replacing a classical expression, such a question is already included in our ignorance about the higher-dimension operators. Various possibilities to change the ordering are equivalent to a subset of possible higher-derivative terms that can be added to the action or other defining equations of the theory.

But the list of the higher-derivative terms in the action can be more extensive. If we define a theory using an action, all terms that are consistent with certain symmetries and general consistency criteria must be allowed whether or not they can be simply written using some form of deformation quantization or non-commutative geometry.

Simplicity defined as compactness of formulae may be a good strategy to guess the right laws of physics but it is not a fully rational argument. When the theory is understood more completely, the argument about compactness of formulae is replaced by more solid arguments. For example, the reason why we neglect curvature-squared terms in general relativity is not to make the equations short which is what Einstein essentially thought. Instead, the rational reason is that we can show that the curvature-squared terms only influence physics at very short (ultramicroscopic) distances, not the long (astronomical) ones.

Similar principles such as the renormalization group often lead to the conclusion that the right equations describing reality indeed look simple but in a fully rational approach, the argument of simplicity must always be replaced by a more solid one.

Myth: Perhaps we could abandon this notion of the graviton and *gasp* move forward?

The existence of gravitational waves has been proven by the pulsars so that even the Swedes are satisfied. And the existence of quanta of energy carried by these waves is essentially proven at the beginning of the text. When quantum gravity is defined at a technical level, the scattering matrix for gravitons is not only the most important set of observables we have but, in some sense, the only one.

Also, any simple attempt to show a contradiction about the existence of gravitons is a result of sloppy thinking. Gravitons neither violate laws of thermodynamics nor they create infinite recursion, and the first somewhat technical analysis one can make shows that they are philosophically analogous, with almost all details, to photons.

Myth: We should quantize the curvature instead

When the metric tensor is quantized in the semiclassical framework, the Riemann tensor automatically becomes an operator, too. There is no way how to promote the Riemann tensor to an operator without making the same thing with the metric as long as we want to avoid a dramatic increase of the number of degrees of freedom. Such an increase would prevent us from calling our theory a theory of quantum gravity.

Myth: Gravity must be treated as geometry, not a field

The metric tensor is usually associated with a rather special gauge symmetry but the difference between general covariance and e.g. Yang-Mills symmetry is a purely technical one. At a general level, the metric tensor is just another field. Whether or not we like to use the word "geometry" is a matter of taste: this question has no testable consequences.

On the other hand, it is completely irrational to try to promote the metric tensor to a "higher" type of a field that should be treated with more dignity. All degrees of freedom that exist in a given theory must be treated using the same, proper, general methods of physics. Any attempt to divide fields to different groups that are treated completely differently is a quasi-religious silliness that is irrational and biased almost by definition and that contradicts the unity of the laws of physics.

Myth: But a geometric approach is better, isn't it?

In physics, the primary way of dividing theories is into correct theories and wrong theories. A general attempt to divide ideas and tools into geometric ones and non-geometric ones is typically ill-defined - it depends on the definition of "geometry" which is a matter of historical and social coincidences in mathematics rather than a matter of well-defined differences. Our understanding what geometry is has been evolving for centuries. More importantly, the approach that is labeled "more geometric", whether or not the reasons behind this terminology are rational or not, doesn't have to be "more correct".

Physics of string theory can be defined to be the right "generalized geometry". At this level, it is just an empty word.

The basic dynamics of general relativity admits a geometric interpretation whether or not we like to use the word "geometry" more often than "fields" or less often. Arguing how often certain words and dogmas should be repeated doesn't belong to physics.

Myth: Something's wrong with the weak-field expansions because they're against the philosophy of GR

There exist some important differences between physics and philosophy. In philosophy, one may decide to believe in dogmas or promote the thinking of a philosopher one likes.

Physicists prefer to believe things that can be demonstrated to be true by observations, experiments, with a help of solid calculations that can make our reasoning and decisions more indirect than ever before while they remain solid. Whether something agrees with some philosophy is a very different question than the question whether it is true. It is the latter question - whether it is true - that matters in physics. All other types of arguments to debunk ideas are at most secondary.

I have no concrete idea what kind of philosophy those weak-field haters have in mind. I doubt it is a good philosophy - it must be some misarrangement of the wheels and gears in their brains - but I am sure that it is very bad physics. A proper physics analysis implies that one can get the right answers - qualitatively and quantitatively - by weak-field expansions from the right theory. These right answers include, among other things, virtually all experimental tests of general relativity that have ever been made.

Einstein himself relied on the weak-field expansions intensely. That's how he derived the Newton's potential - even though he may have been able to find the exact Schwarzschild solution, too. And Einstein has also derived the existence of gravitational waves from the weak-field expansions, even though he used to love Mach's principle that disagreed with the existence of gravity waves.

Perturbative expansions are among the paramount tools of physics and, indeed, all of science. Whoever denies their critical importance shows that he's not really interested in the true answers to well-posed physical questions. Don't get me wrong: I think that the full non-linear equations of general relativity are prettier if printed on a T-shirt than some particular calculation in the weakly curved regime. Well, it's because detailed calculations are almost always uglier than the fundamental laws. But the real importance of Einstein's equations as well as other fundamental laws is to allow us to make calculations in concrete situations - and the weak-field situations dominate.

The fact that a weak-field calculation looks less elegant than the equations you started with doesn't allow you to say that there's something wrong with this calculation. Only simpletons could say that something is wrong - or not even wrong - because of these irrational reasons. And they, in fact, do. As Einstein has said, only two things are infinite - human stupidity and the Universe - and we're not sure about the latter.

The role of these expansions grows even more important in any quantum theory of fields.

The weak expansions give us important and correct qualitative answers to important qualitative questions and they may be used as a systematic method to obtain extremely accurate - and, in many cases, exact - quantitative answers. Whether someone likes their philosophy is much less important than the fact that they're critical to find the true answers.

Myth: All the components should be first quantized in isolation

This is what not only the angry physicist but whole communities of people think. They fail to realize that their statement is based on the assumption that the different components can exist and make sense in isolation. In science, statements can often be wrong. String theory actually does show that this assumption is incorrect - the different low-energy fields and particles can't really be separated from each other: they're inseparable manifestations of more fundamental laws. But even if we didn't have string theory, the assumption would be unjustified.

The whole history of physics is flooded with examples of concepts that were thought to be independent but turned out to be closely related and inseparable. This unification is not just a fact about physics but it is arguably one of its most fascinating features. The interconnectedness of the concepts may be one of the most faithful criteria to measure the depth of our understanding.

Motion of apples and planets can't be separated. Electricity, magnetism, and light have been unified for 150 years. Relativity has showed that magnetism must exist if electricity does. Mass can't be separated from momentum, momentum can't be separated from energy, and mass is really identical to energy. Quantum mechanics showed that the energy is always associated with a frequency. W-bosons and photons must also be unified and quarks can't exist without leptons (or vice versa) because anomalies wouldn't cancel. I could give hundreds of other examples.

The purest three-dimensional AdS gravity you can get is probably inseparable from the monster group.

The opinion that some phenomena or components of the laws of physics are disconnected from each other is an artifact of an incomplete understanding of these laws. Quantum gravity is clearly another example: probably the best one. In the Planckian quantum gravity regime, all kinds of matter, fields, particles, geometry, and effects influence each other as much as they can - this situation in fact maximizes the amount of interactions among any objects - and any attempt to separate the phenomena into decoupled clusters in the context of quantum gravity is bound to be wrong.

It is wrong as a description of the real world around us because we kind of know that electromagnetism and other forces don't get turned off at the Planck scale; but it should also be expected that such an attempt to divide in this heavily interacting regime won't lead to any mathematically solid or interesting theoretical insights. The messy and entirely unimpressive history of the attempts to quantize gravity in isolation beyond the semiclassical level seems to confirm this prediction spectacularly well. This shouldn't come as a surprise: quantum gravity is obviously inherently linked to unification of forces and matter and this statement would be quite obviously true even if we didn't know string theory.

One can try to think about all possible ideas inspired by physics but such ideas don't lead to new and deeper descriptions of reality or profound mathematical insights too often. If they do, we can usually see something special about these ideas that justifies that these ideas were worth our attention.

And that's the memo.

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