Monday, July 21, 2008

Bimetric pseudoscience and ghosts

This month, at least two preprints about "bimetric" theories of gravity have been submitted to the arXiv: one of them came from the Imperial College (IC) and the other one was written at the Perimeter Institute (PI).

Here, I want to explain why this kind of writing is ridiculously bad and lethally flawed if understood as science. I claim that the authors of these papers - and dozens of previous papers about the same subject - must be unaware of virtually all elementary facts about physics to be explained below and they should have been failed in their field theory courses.

Two geometries, two gravities, two Newtons? It may sound attractive :-) but it is physically unacceptable.

The IC paper talks about "two metric tensors" - as an example of the Variable Speed of Light misapproach - but there is really one metric tensor only (supplemented by a scalar with unusual interactions): the only thing that is "doubled" are the frames. You can define different frames - different conventions what you mean by a metric tensor and by distances.

Normally, frames (e.g. Einstein frame and string frame in string theory) are related to each other by scaling. In the IC case, they differ by additive terms - such as "∂_μ φ ∂_ν φ" - which means that they generally induce different causal structures but there is still one independent light spin-2 field so the adjective "bimetric" is a misnomer.

It is also wrong to call it a "Variable Speed of Light" theory because the unique metric tensor still defines a unique and fixed value of "c". Whether the IC paper defines a consistent theory depends on details but it is surely not an attractive or a well-motivated theory.

The IP paper is even more seriously flawed because it literally wants to have two metric tensors, "g" and "h", that can be used to measure distances on a manifold. We will explain that it is a physically inconsistent setup because
  1. there can only be one general covariance to remove ghosts
  2. if one introduces new non-local symmetries to get rid of the remaining ghosts, the second metric tensor becomes massive and disappears from long-distance physics
  3. consistent theories of quantum gravity automatically imply that there can only be one independent metric tensor
Ghosts in theories including fields with spin

So let me begin with an explanation what ghosts are. In physics, this concept has a somewhat more specific and abstract meaning than e.g. ghosts from the Ghostbusters movie. ;-) Physicists use the term for two related but different objects:
  1. bad ghosts: states in the Hilbert space whose (squared) norm is negative i.e. that have a negative probability to exist
  2. good ghosts: Faddeev-Popov ghost fields that are very convenient to deal with local (gauge) symmetries
The relationship between bad ghosts and good ghosts is that the refinement of a theory that employs good ghosts replaces the original Hilbert space with the physical Hilbert space of cohomologies of the BRST operator, Q, and if the theory is consistent, this (reduced) physical Hilbert space is free of bad ghosts. That's necessary for a theory to have a physical meaning because negative probabilities can't occur in reality. For example, no one has ever won -30% of elections, as long as he participated at least in one. :-)

Below, I will talk about bad ghosts only. In the first place, we should ask:

Why there should be any bad ghosts?

Well, the reason is simple. Consider a (non-gravitational) theory with a field whose spin is one or higher (spin-0 and spin-1/2 fields won't generate any ghosts: recall that the psi-dagger-psi Dirac inner product, measuring the probability density, is positively definite), for example QCD with spin-one Yang-Mills fields. When you draw the "t=0" slice and try to quantize it - which you eventually have to do because our world is a quantum world - you will have operators A_μ whose Fourier modes give you creation and annihilation operators for particles with certain momenta.

What is the norm of the one-particle states created by these operators? You may see that the inner product of the state created by A_μ with the state created by A_ν is actually proportional to g_μν, the metric tensor. This fact is pretty much required by the Lorentz invariance - the metric tensor is the only "universal" tensor with these indices - but you can derive it explicitly from the action, too. (A priori, you might think that the inner product can also depend on the timelike vector associated with the t=0 slice, but the set of 1-particle states is actually Lorentz-covariant and the inner product doesn't depend on the slicing, after all.)

Now, the metric tensor g_μν is indefinite: when you write it in a diagonal basis, some of its entries must be negative (because the space and time have opposite signature). Most of the entries of vectors and tensors are the purely spatial ones, and these have to generate the positively definite one-particle states in the Hilbert space leading to positive probabilities (regardless of your sign convention for the metric).

The negative-normed entries correspond to quanta - particles - that would lead to negative probabilities. Whenever you would create a final state with such a particle, its probability would be negative (a negative multiple of the squared complex |amplitude|), leading to logically unacceptable predictions - an inconsistent theory. The only way how such a theory could possibly survive is that such ghosts are never produced if you don't have them in the initial state.

Decoupling the ghosts

So if you want to save your theory, you must assume that there are no ghosts to start with. More importantly, the theory must also imply that if there are no ghosts to start with, they are never produced by the evolution. The probability amplitude (for a scattering etc.) including one bad ghost or more ghosts (in the final state) and many ordinary particles (in the final and initial state) must be zero. We say that the ghosts must be decoupled.

(In a theory treated with good, Faddeev-Popov ghosts, it is no longer true that the probability with all kinds of ghosts in the final state must vanish. Instead, the amplitudes with BRST-exact states must vanish, and they automatically do if the BRST axioms such as Q^2=0 are satisfied. Let us follow the old treatment without the good ghosts below.)

Now, it is a very nontrivial requirement that the ghosts must be decoupled from an arbitrary combination of ordinary particles. When you have a "generic" theory where ghosts (which are just "some" parts of your tensorial fields, after all) interact with the normal matter, you may be sure that most of the probability amplitudes will be nonzero. How can you possibly make all of them zero?

The answer is a "gauge invariance". You must "pay" gauge invariance for each potential ghost, and because it is so hard to find a lot of gauge invariance in your theory, you normally have enough gauge invariance to kill the timelike components only. That's why the timelike components must be those associated with the negative-normed one-particle states (regardless of your metric sign conventions).

At every point of spacetime, there must exist a parameter of gauge invariance for each component of bad ghost fields. This is enough. Why? Because the scattering amplitudes are encoded in the correlators of various operators - i.e. in Green's functions - and the operator corresponding to a bad ghost may be expressed, up to a momentum-dependent rescaling, as "∂_μ j^μ" which vanishes because the corresponding current "j" is conserved. It is conserved because of Noether's theorem applied to the global part of the corresponding gauge symmetry.

You can see that the very condition that a quantum version of your theory exists severely constrains how the theory can look like. In classical physics, you might think that any kind of field theory is just fine. You might write any field equations controlling any fields you like. However, most of them would lead to quantum theories that predict negative probabilities and they're not allowed. It follows that fields with spin can't be added arbitrarily: you need a gauge symmetry for each time-like (negative-normed) component of such fields!

That's great. So let us look whether we have enough gauge symmetry to kill the ghosts. Yes, we do. For example, in QCD whose gauge group is SU(3), there are eight Lorentz vectors "A_μ": the 1...8 index is the adjoint index of the 8-dimensional SU(3) Lie algebra. But we also have 8 currents "j_μ" and 8 corresponding parameters of SU(3) transformations at every point. If you think about it, this counting obviously works for every gauge group in every Yang-Mills theory.

Modification for spin-3/2 and spin-2

We will begin with the most difficult case, the spin-3/2 fields.

If the spin is 3/2 (i.e. 1.5), you deal with Rarita-Schwinger fields that effectively carry one spinor index and one vector index. The spinor index would generate a positively definite (=ghost-free) Hilbert space, like the Weyl or Dirac field, but the vector index can go timelike again. Once again, you are threatened by ghosts and negative probabilities. And once again, you need a local symmetry - and the corresponding current - that saves your theory.

How many parameters such a local symmetry should have? Well, if the vector index is "0" (time), you obtain ghosts. But the spinor index can still be anything: the ghosts have many components. There is a whole spinor of ghosts and the corresponding gauge symmetry to kill these ghosts must therefore be a spinor, too. Now, gauge symmetries (and conservation laws) whose parameters carry a spin are heavily constrained by the Coleman-Mandula theorem and its modern refinements. If you wanted a conservation law for too difficult an object (with a high spin), pretty much every momentum of every particle would have to be conserved and the interactions would vanish: too bad even for a remotely realistic theory.

For spin 1/2 conserved quantities, there exists one possibility, and it is called supersymmetry: the conserved charges are called supercharges and they transform as a spinor or spinors. If you want a consistent, ghost-free quantum theory with spin-3/2 fields, it must respect (local) supersymmetry! By pure algebra, you can argue that the anticommutator of two supersymmetries includes a (local) spacetime translation, so the coordinate reparametrizations must automatically be a part of the story. With spin-3/2 fields and nonzero interactions, you must inevitably deal with supergravity, a theory that includes both gravitinos and gravitons.

The spin 1/2 of the supercharge is already pretty high and the conservation law for the supercharges - which is equivalent to the symmetry called supersymmetry - is very constraining. The maximum number of supercharges that a physical theory can possibly have is 32 real components which is equivalent to one real chiral spinor in 11 dimensions or N=8 (complex chiral) Weyl spinors in four dimensions. This "N=8 supergravity" is extremely constrained: its low-energy interactions are fully determined by the required supersymmetry. The less supersymmetry you require, the more free and unconstrained your model building becomes. For N smaller or equal to 4, you can get non-gravitational yet supersymmetric theories but because they're non-gravitational, the supersymmetry must only be global, not local!

Now the main point: spin two

But spinors and supersymmetry might be too complex for some readers, so let's return to an integer spin, namely spin-2. Nevertheless, I will still have to assume that the reader knows why gravity must be carried by spin-2 particles i.e. that the arrogant and mostly anonymous imbeciles from Not Even Wrong and similar dumping grounds of the Internet no longer visit this blog. ;-)

Imagine that you deal with a spin-2 field h_μν - such as the metric tensor - that contains something like the normal kinetic term in the Lagrangian. Again, the Fourier modes of this quantum field will give you creation and annihilation operators. The inner product of the states created by h_μν and by h_μ'ν' will be proportional to something like g_μμ' g_νν'. Or another product of two metric tensors with these four indices or one of their linear combinations. If exactly two of these four indices are timelike (i.e. 0), you may obtain a negative result. For example, "g_00 g_33" is negative.

It follows that the h_01, h_02, h_03 components of the tensor field create ghosts. There are D-1=3 of them and you need roughly D-1 of the gauge parameters, too. Essentially a whole spacetime vector. If you carefully analyze the index structure, you will find out that the conserved currents must be components of a tensor that you may call the stress-energy tensor T_μν. This tensor must be conserved and because its divergence "∂^μ T_μν" must be related to the bad components of the tensor "h" you started with, you may see that the corresponding transformation of "h" must coincide with the standard transformation rule of the metric tensor under the coordinate transformations.

In fact, coordinate reparametrizations are the only local transformations of fields whose infinitesimal parameters naturally transform as vectors.

In other words, the only way how a spin-2 field can be nontrivially interacting with the world, while generating no bad ghosts, is to identify the field with the metric tensor in a generally covariant theory. Because gauge invariance may be viewed as a "fundamental concept", you may derive general relativity from this alternative starting point. See also Gravity from spin-2 gauge invariance.

Fine: so can you have two metric tensors?

You would need "two independent reparameterization invariances". That's not possible in a physically acceptable and interesting theory. When you consider one manifold, there only exists one set of coordinates, after all. So there are only D functions that determine the relationship between two choices of the coordinate system: D parameters of a local symmetry. (Let me not discuss the difference between D-1 and D, it's too subtle.)

In fact, there exists a loophole - a method to have two or more metric tensors; see Arkani-Hamed, Georgi, Schwartz (AGS) based on generalized deconstruction (a phenomenologist's toy model for extra dimensions). However, when you follow their analysis of the physics that follows from their approach, you will find out that only one metric tensor - the "real one" - remains massless. The other one acquires a mass and disappears from the long-distance physics. In fact, their approach is the most natural method to analyze massive spin-2 fields.

Morally speaking, the other spin-2 field in the AGS picture has a similar character as the massive Kaluza-Klein modes of the graviton (or even the massive spin-2 fields in string-theoretical tower of states): in all cases, there are very manifest reasons why the new spin-2 field cannot be massless. You shouldn't view the AGS theory as a very low-energy theory because the theory breaks down as an effective theory above the energy scale (cutoff) that goes to zero (=the theory breaks down everywhere) if the mass of the other graviton itself goes to zero, even though the cutoff goes to zero more slowly than the mass (it is a kind of weighted geometric average of the mass of the massive graviton and the Planck scale).

So if you want to consider theories similar to one envisioned by the IP paper and others - where you can literally measure distances on a large manifold by two independent metric tensors - you are inevitably led to an inconsistent theory with ghosts. A good student should be able to comprehend - and even re-discover - all these facts in a few hours as a part of one homework exercise. Some people who promote various wrong theories in the media or on the Internet couldn't get them at least since 2001 when the wrong bimetric theories first appeared in the literature.

One metric tensor in string theory

Every good field theorist knows why these bimetric papers, and many similar papers, are just wrong. You don't really need to know string theory to make such analyses. Nevertheless, string theory gives us a particularly clear picture why there can't be several massless spin-2 tensors.

You may often hear the crackpots saying that string theory makes no predictions about the reality. Well, it makes tons of predictions. For example, it implies that virtually every paper that the critics' friends have ever written must be completely wrong. It predicts that none of their predictions can ever materialize. Isn't this a powerful prediction, too?

In string theory, you don't have to work hard to derive similar qualitative facts - e.g. that the number of gravitons equals one and not two: they automatically follow from it, from its own technical tools that we're going to look at. I would be skeptical about any theory that doesn't naturally imply any of these general facts about quantum fields. Of course, no theory besides string theory does such things which is one of dozens of reasons why I am skeptical about any non-stringy way to go beyond quantum field theory, to say the very least.

Why is there one metric tensor in any background of string theory, including every single vacuum from the proverbial landscape of 10^{500} solutions? Well, it just seems to be true in every vacuum we have ever seen - perturbatively or nonperturbatively. However, perturbative string theory makes the answer exceptionally transparent and let's look at it.

We will begin with bosonic string theory. The ground state of one string is a tachyon. And you may add the alpha-oscillator excitations. Already the first excitation brings you to the massless level. Two excitations already create a massive particle. So if you're only interested in the massless spectrum, you can only add one excitation in the open string case - generating a spin-1 field - or two excitations (left-moving and right-moving) in the closed string case - generating a spin-2 field.

The open strings can carry additional labels - the Chan-Paton factors (colors associated with the endpoints) - so you can have many components of a spin-1 field. In fact, the Chan-Paton factors become two indices and that's how you generate the whole "square matrix" of the gauge fields, transforming in the adjoint representation.

However, the closed strings don't carry any additional labels similar to the Chan-Paton factors. For example, closed strings don't have any endpoints or other special places that could support additional degrees of freedom. Consequently, the spin-2 massless graviton is inevitably unique. Nothing changes about this conclusion even if you compactify some of the dimensions. Any excitation of the compactified degrees of freedom increases the squared mass. So if you don't want to surpass the massless level, you cannot combine the compactified excitations with the excitations of the noncompact dimensions (that define the "ordinary" spin).

An analogous counting applies to the NS-NS sector of the superstring. Its ground state is a tachyon, too. (This particular tachyon becomes unphysical due to the GSO projections.) The only difference is that the squared mass is 1/2 of the bosonic one and you can work with half-integer-moded fermionic oscillators, too.

The vertex operator - a conformal field theory operator whose correlators are used to calculate the scattering amplitudes for a particular particle in string theory - corresponding to the graviton is proportional to "∂ X_μ(z) ∂* X_ν(z*)". By looking at this object, you can also see that there is only "one type" of the graviton that you can construct. There is only one worldsheet, with one set of coordinates, z and z*, and its embedding into the conventional spacetime is only described by one set of the fields X_μ(z,z*).

There is just no way to ever get two gravitons (or two metric tensors) with indices along the large spacetime coordinates. The only, uninteresting exception is when you imagine that the two metric tensors belong to two "subtheories" that don't interact with one another at all. In this case, you can even have two independent coordinate reparametrizations. Operationally speaking, the other "segment" of the Universe besides your own doesn't really exist. Andrei Linde used this vision - that there can exist completely decoupled worlds on top of ours - to motivate some ideas about the multiverse. The IP paper cited Linde's paper but Linde has clearly nothing to do with the crackpot enterprise of writing theories with several interacting metric tensors!

By the way, string theory has its own, completely new technical way to prove that the ghosts decouple: you don't have to talk about the currents "j" in spacetime. In fact, you can prove that the decoupled unphysical components of fields - such as ghost gluons and ghost gravitons - have vertex operators that can be written as a total derivative of another operator with respect to one or two worldsheet coordinates. Consequently, the integrated vertex operator (over the effectively compact worldsheet) vanishes, and so does the scattering amplitude.

In the BRST treatment including the good ghosts, the arguments would be slightly more sophisticated but they would be equally powerful to derive the conclusions we need.

Related facts about the stringy spectrum

Similar arguments lead to many other, related general conclusions or general predictions of string theory, if you wish. There can't exist massless fields with spin higher than 2 - not even 5/2! ;-) The fields with spin-3/2 must be gravitinos and their number is related to the number of supersymmetries.

The spin-1 fields in string theory may come in many flavors. I have already mentioned the gauge fields produced by open strings with Chan-Paton factors (colors attached to the end points). But there are many other ways how string theory produces spin-1 fields. These pictures may be shown to transform into each other under various dualities and transitions.

But let's look at one more classical way how spin-1 fields may emerge from perturbative string theory: the Kaluza-Klein way. You only need a field whose one vector index is parallel to the large dimensions, to get spin-1: this index can arise from the right-moving side of a closed string. The other index, taken from the left-moving side, may be parallel to a compact dimension. That's equivalent to getting a gauge field, A_μ, from components of a higher-dimensional metric tensor, g_μ5, where 5 is a compact direction. That's nothing else than the Kaluza-Klein theory.

Instead of 5, you may also use e.g. the index 25 on the bosonic, left-moving side of the heterotic string. That also generates a Yang-Mills field. But it turns out that the gauge group in the heterotic string is neither a power of the U(1) nor the isometry of an ordinary manifold you could think about in a classical Kaluza-Klein theory. Instead, it is one of the two 496-dimensional groups, SO(32) or E8 x E8. This larger group is the extended, quantum, or "stringy" isometry of a particular chiral 16-dimensional torus used for the compactification of the 16 "redundant" dimensions on the bosonic side. Still, you can view the emergence of the gauge group in the heterotic string as a generalization of the Kaluza-Klein theory.

There are other ways how the spin-1 fields may come into existence - for example, through lower-dimensional D-branes (related to the open string's Chan-Paton factors discussed above by T-duality) or through F-theoretical singularities (related to and generalizing D7-branes). String theory includes and links all physically consistent ways how fields - including higher-spin fields - can be interacting with each other and how you can generalize this picture.

Whether we already know "everything" about this world is a different question. Of course, we don't yet know everything. But we already know something and the insights of string theory collected so far, much like the laws of mathematics themselves, are a crucial and inseparable part of it. Unfortunately, the physicists who don't know this crucial part are usually ignorant about many other, much more elementary parts of the modern physics cannon, too.

And that's the memo.

1 comment:

  1. A good student should be able to comprehend - and even re-discover - all these facts in a few hours as a part of one homework exercise