## March 10, 2006

### Avatars of Nonlocality?

Blogging about physics is a bit of a tightrope walk. On the one hand, I don’t want to say something manifestly foolish in this very-public forum. If I’m going to comment on some topic, I need to put in the (sometimes considerable) effort to make sure I’m saying something sensible. On the other hand, if I put in enough effort, the result is sometimes better-presented as a paper on the arXiv than as a blog post.

That’s the story of one half-written blog post that’s been sitting on my computer for a few weeks. I’d been somewhat confused by the recent paper of Adams, Arkani-Hamed, Dubovsky, Nicolis and Rattazzi. Finally, after a discussion with Ben Grinstein, who was visiting this past week, I think I understand the source of my confusion and, at least in some examples, how it is resolved. Together with Ira Rothstein, we’re writing up a short paper, but in the meantime, I’m going to break my usual rule and explain what’s going on in a blog post as well.

Adams et al study the leading higher-dimension operators which, generically, occur in effective field theory. These appear in the effective lagrangian in the form $\frac{c_i}{M^{d_i-4}} O_i$ where $M$ is the cutoff on the effective field theory, $d_i$ is the engineering dimension of the Lorentz-invariant operator, $O_i$. $c_i$ is a dimensionless coupling constant, about which we typically assume that not much can be said, except that we expect it to be $\sim \mathcal{O}(1)$.

The claim of Adams et al is that more, in fact, can be said about the $c_i$. They claim that the coefficients of the leading irrelevant operators are such that the classical stress-energy tensor computed including the contribution of these terms satisfies the dominant energy condition. Only effective theories satisfying this constraint can be completed to local quantum field theories (with or without gravity) in the UV.

They don’t use the phrase “dominant energy condition.” Instead, they talk about superluminal propagation of signals in the background of a nontrivial classical solution. These are, however, really the same condition.

Propagation in a nontrivial translationally-invariant background, given by summing a geometric series in 4-point forward scattering amplitudes.

They then go on to argue that the same positivity constraints on the coefficients $c_i$ follow from looking at the dispersion relations satisfied by the tree-level amplitude for forward scattering. The connection between these two points of view is illustrated in the diagram at left. The propagator in a nontrivial background, which determines the speed of signal propagation, is determined by summing a geometric series of forward-scattering diagrams off the background. (Here, we’re specializing to a translationally-invariant background, so that the momentum is conserved.)

The thing that has confused me is why it is legitimate to treat these leading irrelevant operators as part of the classical action. In the effective field theory, the contribution of nonrenormalizable interactions to scattering occurs at the same order in the derivative expansion as loops involving renormalizable interactions. In a generic effective field theory, it would be incorrect, to this order, to ignore the loop corrections to the forward scattering amplitude. Equally, it would be incorrect to include the nonrenormalizable operator as a part of the classical action, for the purpose of computing the speed of signal propagation.

Of course, in very particular theories, this may not be true. Consider Maxwell theory,

(1)
$L = -\frac{1}{4e^2} F_{\mu\nu}F^{\mu\nu} + \frac{c_1}{\Lambda^4} (F_{\mu\nu}F^{\mu\nu})^2 + \frac{c_2}{\Lambda^4} (F_{\mu\nu}\tilde{F}^{\mu\nu})^2 +\dots$

In this theory, the renormalizable part of the theory is free. The leading nontrivial scattering effects come from treating $c_1,c_2$ at tree level.

Another example is the effective Lagrangian for the Goldstone boson of a spontaneously-broken $U(1)$ global symmetry,

(2)
$L= \frac{1}{2} (\partial \pi)^2 + \frac{2 c}{f^4} (\partial\pi)^4 + \dots$

(They also consider a slightly more general Lagrangian, without a $\pi\to -\pi$ symmetry, which contains a $(\partial\pi)^2 □ \pi$ interaction. More on that below.)

In both of these theories, the leading interactions come from the nonrenormalizable terms we’ve highlighted. Studying such examples is interesting, but may not give an accurate picture of the state of affairs in a general effective field theory.

That was pretty much the state of my understanding when I started talking to Ben. It became clear that a good model to test these ideas was the interacting version of (2), the chiral Lagrangian for the spontaneous breaking of $SU(2)\times SU(2)\to SU(2)$.

(3)
\array{\arrayopts{\colalign{right left}} L =& \frac{f^2}{4} Tr (\partial_\mu\Sigma^\dagger\partial^\mu\Sigma) + \frac{m^2 f^2}{4} Tr(\Sigma+\Sigma^\dagger-2) \\ &+ \frac{l_1}{2} Tr(\partial_\mu\Sigma^\dagger\partial^\mu\Sigma)^2 + \frac{l_2}{2} Tr(\partial_\mu\Sigma^\dagger\partial_\nu\Sigma\partial^\mu\Sigma^\dagger\partial^\nu\Sigma)+\dots }

where \pi = \frac{1}{2}\left(\array{\arrayopts{\colalign{center}} \pi_0 & \pi_1-i\pi_2\\ \pi_1+i\pi_2& -\pi_0}\right),\qquad \Sigma=e^{2i\pi/f}

and $l_1,l_2$ are the leading 4-derivative couplings, whose sign Adams et al would like to bound.

Note that we’ve turned on a nonzero pion mass. This is actually necessary to be able to apply dispersion relations to the forward scattering amplitude. As soon as you turn on interactions, there are 2-particle cuts (in both the $s$- and $u$-channels), which extend all the way down to zero when the particles are massless. At the end, we can try to extrapolate our answers down to zero pion mass (which is where a constant pion gradient, the configuration of interest to Adams et al, is a solution to the equations of motion).

The forward scattering amplitude for, say, $2\pi_0\to 2\pi_0$, is an even function of $\tilde{s}= s-2m^2$.

(4)
\array{\arrayopts{\colalign{right left}} A_{2\pi_0\to 2\pi_0}(t=0) =& \frac{m^2}{f^2} + \frac{1}{96\pi^2 f^4} \left[(8C-27) m^4 + 4 (C-3) \tilde{s}^2\right. \\ & +3(3m^4-4m^2\tilde{s}+2\tilde{s}^2)J(2m^2-\tilde{s}) \\ &\left. +3(3m^4+4m^2\tilde{s}+2\tilde{s}^2)J(2m^2+\tilde{s})\right] }

where $J(y)= 2+x\log\left(\frac{x-1}{x+1}\right),\qquad x=\sqrt{1-4m^2/y}$ and $C = 96\pi^2 \left(l_1(\mu) + l_2(\mu) - \frac{1}{32\pi^2}\log(m^2/\mu^2) \right)$ is an RG-invariant combination of the couplings. To obtain this forward scattering amplitude to order-$s^2$, we had to sum a set of graphs in which the 4-derivative terms in (2) enter at tree-level, while the other terms appear to 1-loop.

The first thing to notice, of course, is that there’s no meaning to trying to bound the sign of the couplings, $l_i$, themselves. They are additively renormalized, and can have either sign, depending on what scale you evaluate them. What we can bound is the RG-invariant quantity, $C$. Applying standard dispersion-relation arguments, one finds

(5)
$C\gt (9\pi -36)/32$

Repeating the same analysis for $A_{\pi_+\pi_0\to\pi_+\pi_0}(t=0)$, one obtains

(6)
$C' \gt (39\pi-92)/48$

where $C' = 48\pi^2\left(l_2(\mu) - \frac{1}{48\pi^2}\log(m^2/\mu^2)\right)$ is RG-invariant. From the sign of the $\beta$-functions, we see that the running couplings increase as we flow to the IR. The “bare” couplings, $l_i(\mu=\Lambda_{\chi SB})$, could perfectly well start out negative, and still satisfy the bounds on $C,C'$ by flowing to suitable positive values, $l_i(\mu= m)$, in the IR. (Note that $l_1(\mu=m)$ can be negative, and indeed is likely negative in the real world.)

Note that, in order to get these results, we had to include loop-contributions from the two-derivative terms in the action. In other words, the computation is not a classical one. We may, in the end, obtain a bound on some RG-invariant quantities. But we do not expect (and do not obtain) a constraint on the signs of the “bare” couplings in the classical action.

The same remarks, obviously, hold in more general theories. The leading higher-derivative terms in the action should not be thought-of as part of the classical action. Rather, they contribute to physical processes at the same order as loop corrections. This applies, whether one is computing scattering amplitudes, or studying propagation is a nontrivial background.

As another example of this, consider, instead of Maxwell Theory (1), QED with $F^4$ interactions. The contributions to light-by-light scattering (equivalently, the speed of light in a constant elecromagnetic field background) include not just the “classical” $F^4$ terms, but also electron loops. Even at weak coupling, the latter dominate over the former, for $m_e\ll \Lambda$.

One of the most interesting applications of their work is an attempt to kill the Dvali-Gabadadze-Porrati model, a plausible-sounding braneworld model. In the DGP model, there is both Einstein gravity on the brane and in the 5D bulk.

In a certain decoupling limit, the theory reduces to a single 4D scalar, with Lagrangian $L = 3 (\partial \pi)^2 - \frac{(\partial\pi)^2 □ \pi}{\Lambda^3} + \mathcal{O} (\partial^m(\partial^2\pi)^n)$

where $\Lambda\sim M_5^2/M_4$ is held fixed, as we send the 4D and 5D Planck masses to $\infty$.

This theory looks nontrivial, but all the interaction terms vanish by the equations of motion. So they do not contribute to on-shell scattering. The S-matrix is trivial, and one does not obtain any constraint on the theory by considering this limit. (That’s not to say that the DGP model makes sense; just that you can’t kill it this way.)

#### Update (3/22/2006):

I have had a longer update to this post sitting on my computer, waiting till I and my colleagues, Ben and Ira, could come to some consensus about our incipient paper (or non-paper, depending on the day of the week). But, since Nima is annoyed that I haven’t corrected the last paragraph of this post yet, let me go ahead and do that.

The $□ \pi$ factor in the above interaction vanishes when applied to an external line in any scattering amplitude. However, as they compute in their paper, there are nontrivial contributions to scattering, where the $□$ hits only internal lines in the graph. Normally, such interaction terms can be shifted away by field redefinitions, but — in this case — not in a way compatible with the shift symmetry, $\partial_\mu\pi\to \partial_\mu\pi+c_\mu$, which they wish to maintain. At tree-level, they get a nonvanishing 4-point scattering amplitude, $A(s,t) = \frac{1}{\Lambda^6}(s^3+t^3+u^3)+\dots$ This tree-level amplitude is nontrivial, but vanishes in the forward direction, not just to $O(s^2)$, which already causes problems for the dispersion relation, but to all orders $s^3+t^3+u^3 = 3 s t u + (s+t+u)(s^2+t^2 +u^2 - s t - t u - u s) = - 3 s t (s+t)$ (A little thought shows that all tree diagrams will have this kind of structure.)

Having said this much, there is still a loophole in this argument. To apply dispersion relations, we need to turn on a small mass, so as to separate the cuts which, in the massless limit, extend all the way down to $s=0$. When you do this, you regenerates a nonzero contribution to the forward scattering amplitude $A(s, t=0) \propto \frac{m^2}{\Lambda^6}(s-2m^2)^2 +\dots$ Presumably, everything now goes through. We can compute the dispersion relation. The coefficient of $s^2$ is positive, as it must be, but vanishes in the $m^2\to 0$ limit, which still violates unitarity.

It sure looks, then that the DGP model is ruled out.

#### Update (4/28/2006):

I’ve put off updating the physics of this discussion until our paper was ready. Now that it’s done, I feel I can bring this post up-to-date. The classical superluminality constraints for the Lagrangian (3) (at $m=0$) can be derived by expanding about the background $\pi = c^\mu x_\mu \tau^3$. The result is
(7)
$l_1 + l_2 \gt 0,\quad l_2 \gt 0$

These look just like the constraints (5),(6) above, except that the latter:

• involve the renormalized couplings, $l_i(\mu=m)$,
• have a finite shift on the right-hand side of the inequality.

Bollocks!” you say, “I can absorb a finite shift like that in a change in renormalization scheme for these couplings.” Up to an astute choice of such a renormalization scheme, all I seem to have done is to replace the classical couplings, by their renormalized counterparts.

But that’s because I’ve been holding out on you. There’s actually a third constraint

(8)
$C+C' \gtrsim 0.911166$

following from dispersion relations for $\pi^+\pi^+$ scattering, that isn’t reproduced by the classical superluminality analysis. The true shape of the allowed region is shown in the figure below.

The region allowed by the dispersion relation analysis is above all the lines. In Gasser & Leutwyler’s notation, $\overline{l}_1=C-C'$ and $\overline{l_2}=C'$.

The extra disallowed region is depicted in burnt-orange. Some wags have suggested that the original disallowed region should have been drawn in crimson, instead of pale blue.

#### Update (5/12/2006):

For more, see this followup post.
Posted by distler at March 10, 2006 6:36 AM

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### Re: Avatars of Nonlocality?

Dear Jacques,

first of all, I was equally surprised that our friends did not want to talk about the dominant energy conditions. Most of the authors simply consider energy conditions to be a “shit”, and my explanations why DEC is equivalent to the condition did not change it.

As a result of a debate, I wrote the article

Energy conditions

Whether or not you can use effective actions as classical actions depends on the the comparison of the cutoff scale vs. the scale of energies of the processes you study. If these two scales are close to each other, you don’t really include any loops to the amplitudes because the cutoff is so low, which means that the effective action becomes the classical action - only tree level graphs matter.

This was an attempt to formulate the problem in terms of the Wilsonian effective action. Using the language of the 1PI effective action, the relation between the classical action and the effective action would probably be more straightforward.

All the best
Lubos

Posted by: Lubos Motl on March 11, 2006 10:00 AM | Permalink | Reply to this

### Re: Avatars of Nonlocality?

The thing that one treats classically (for the purpose, say, of computing superluminal propagation of signals) is the 1PI effective action. For length-scales long compared with the Compton wavelength of the particles running around the loops, this coincides with the Wilsonian effective action (evolved down to that scale).

In the chiral ($m\to0$) limit of the pion effective Lagrangian, there isn’t (as far as I can tell) a regime where you can safely ignore pion loop effects.

Posted by: Jacques Distler on March 12, 2006 9:35 AM | Permalink | PGP Sig | Reply to this

### Dominant energy condition

I looked over the proof of the dominant energy condition theorem and what is really relevant over nonvacuum backgrounds is the relative stress-energy tensor which is defined to be the difference between the “actual” stress-energy tensor and the stress-energy tensor of the background solution and it is really the relative stress-energy tensor which has to satisfy the dominant energy condition. The proof also assumes that the background metric is fixed, so I am wondering if GR, which leads to a dynamical metric might lead to a loophole. But at any rate, all the models that Arkani-Hamed et al looked at were over a fixed metric (in fact, a flat metric).

Brandon Carter pointed out in Energy dominance and the Hawking Ellis vacuum conservation theorem that Randall-Sundrum models with negative tension orbifold branes violate the dominant energy condition. Does that mean that superluminality is present in RS models? Classical AdS orbifolds admit solutions where time ends (or is reflected in) a spacelike orbifold brane.

Posted by: Jason on March 23, 2006 9:45 AM | Permalink | Reply to this

### Re: Dominant energy condition

There’s a dominant energy condition theorem? By that do you mean Nima et al’s paper? Otherwise, I thought there was a result (Epstein-Glaser-Jaffe, maybe) that there always existed a state in a QFT that violated all the pointwise energy conditions.

Posted by: Aaron Bergman on March 23, 2006 10:08 AM | Permalink | Reply to this

### Re: Dominant energy condition

There alway exists a normalizable state of the QFT, such that the expectation-value of the stress tensor in that state violates whatever pointwise energy condition you choose.

That’s not what’s relevant here. We are interested in translationally-invariant states (so that there’s a conserved energy and a conserved momentum, whose dispersion relation, $\omega=\omega(\vec{k})$, we wish to compute).

Posted by: Jacques Distler on March 23, 2006 10:36 AM | Permalink | PGP Sig | Reply to this

### Re: Dominant energy condition

I know that (we discussed it after all). That’s why I was asking if this “DEC theorem” referred to the paper or something else.

Posted by: Aaron Bergman on March 23, 2006 12:50 PM | Permalink | Reply to this

### Re: Dominant energy condition

Sorry for the confusion. By the DEC theorem, I didn’t mean a theorem stating that the DEC condition has to hold. I meant the theorem that assuming DEC holds, changes in the stress-energy tensor can’t propagate superluminally.

Posted by: Jason on March 24, 2006 10:13 AM | Permalink | Reply to this

### Re: Avatars of Nonlocality?

Hi Jacques,

That is an interesting paper, I am pretty happy with the message elaborated in section 3, that some assumptions about the UV completion translate readily to statements about measurable physics. Conversly, some measurments in low energy can tell us something completely crazy is going on…

Section 2 confuses me for the same reason, there the statement is that EFTs which violate the DEC by irrelevant operators are inconsistent, without any reference to any UV completion. That statement is much stronger and more surprising, I would think, based on usual reasons of decoupling, that anytime one try to probe such inconsistency the EFT description would break down. For example it is well-known that R^2 terms give rise to ghosts, but those appear just beyond the cutoff. I would think similar statement must apply in this case, but maybe things are more subtle and decoupling somehow does not hold…

Anyhow, thought provoking paper…

Posted by: Moshe on March 11, 2006 10:48 AM | Permalink | Reply to this

### Re: Avatars of Nonlocality?

So can we think of your bounds on C and C’ as a way of making explicit the discussion on page 24 of the Adams et. al. paper of identifying a “matching contribution” and a “running” contribution and getting a non-trivial constraint, or is there a flaw in their discussion? I found it confusing. Thanks for clearing up what the correct constraint is.

Posted by: Anon. on March 12, 2006 2:52 PM | Permalink | Reply to this

### Superluminal Rarita-Schwinger?

What do you think of the example of a Rarita-Scwinger spinor minimally coupled to the electromagnetic field as shown in the following paper?

Propagation and Quantization of Rarita-Schwinger Waves in an External Electromagnetic Potential

by Giorgio Velo and Daniel Zwanziger?

This particular example isn’t mentioned in the recent paper by Arkani-Hamed et al.

Posted by: Jason on March 20, 2006 3:17 PM | Permalink | Reply to this

### Re: Superluminal Rarita-Schwinger?

They consider a Rarita-Schwinger field coupled to an electromagnetic background (such things occur in extended supergravity models, where an R-symmetry is gauged).

The model has ordinary $U(1)$ gauge invariance, and the gravitino shift-symmetry, $\psi^\mu\to\psi^\mu +\gamma^\mu \chi$.

They gauge-fix the latter symmetry, and — it appears — their gauge-fixing procedure breaks down if the background magnetic field is too strong.

Perhaps I’ve missed something, but none of what I got from a quick scan of their paper seemed particularly surprising.

Posted by: Jacques Distler on March 20, 2006 5:44 PM | Permalink | PGP Sig | Reply to this

### Re: Superluminal Rarita-Schwinger?

Isn’t the gravitino shift symmetry only present for a massless Rarita-Schwinger field?

I don’t think they gauge-fixed a nonexistant gauge symmetry (for the case of a massive field). What they mentioned is the form of the action changes upon a field redefinition and that they chose a particular field redefinition which gives rise to a certain action.

What I’m really thinking about is the consequences for effective field theories. In QCD, we have charged spin 3/2 baryon resonances and QCD definitely isn’t superluminal. This immediately tells us that from an effective field theory point of view, the couplings of these baryons to the electromagnetic field can’t possibly be a minimal coupling.

Similarly, there might be some restrictions on supergravity with a gauged R-symmetry. (I’m not sure why it has to be an extended SUSY; even with N=1 SUSY, the gravitino has a nonzero axial R-charge)

Posted by: Jason on March 21, 2006 10:51 AM | Permalink | Reply to this

### Dumping gauged R-symmetries into the swampland?

It’s interesting that it isn’t easy to come up with a gauged R-symmetry from string theory (Disclaimer: I’m not a string theorist and so I don’t know what I am talking about :) ). First of all, since the SUSY generators are charged under the R-symmetry and the commutator of two SUSY generators generates translations, gauging the R-symmetry requires GR, and hence supergravity; but then we now have charged gravitinos coupled to a R gauge background.

But let’s try to come up with a concrete example. 11d M-theory (and SUGRA) doesn’t have any R-symmetries. Of all the 5 (perturbative) superstring theories in 10d, only type IIB has a $SO(2)$ R-symmetry, but unfortunately (or fortunately), it’s not gauged. The same comment applies to type IIB SUGRA.

But what if we try to compactify 10d string theory/SUGRA or 11d M-theory/SUGRA? We wish to preserve at least a N=1 SUSY in 4d, so let’s compactify over a Calabi-Yau 3-fold or a $G_2$ holonomy manifold (which we will call X). If we can find an isometry of X which acts as a continuous rotation about a fixed point on X, this will act as a 4D R-symmetry. Moreover, this R-symmetry will be gauged thanks to general relativity. But a compact X doesn’t have any continuous group of isometries (Metric symmetries and spin asymmetries of Ricci-flat Riemannian manifolds by Brett McInnes).

But what if we drop the condition of compactness? Let’s say we happen to live on a p-brane where $3\leq p \leq 7$? Then we have a continuous KK tower of gravitinos and gravitons and minimal coupling is not a good approximation at any distance scale. No matter how weak ($F \sim \Lambda^2$ with an arbitrarily low $\Lambda$) the background R gauge field strength is, it still doesn’t decouple from the KK excitations.

What if we break SUSY entirely, but keep the gauged R-symmetry? There certainly are many compact manifolds with a continuous isometry group with a fixed point; spheres for example. And we still have Kaluza-Klein Rarita-Schwinger excitations with a nontrivial R-charge. But these excitations have a compactification scale mass and there are also many other KK excitations with compactification scale masses and we can’t disregard the coupling of the RS field to the other KK degrees of freedom.

The way this whole thing works seems like magic to me, I have to admit.

On other somewhat related issue; GR admits solutions with closed timelike loops (acausality), but when we try to embed GR within superstring theory, there seems to be some additional magic at play. Chronology Protection in String Theory

Posted by: Jason on March 24, 2006 11:31 AM | Permalink | Reply to this

### Re: Dumping gauged R-symmetries into the swampland?

AdS/CFT (or, more prosaically, Freund-Rubin compactification of M-theory on AdS7$\times S^4$, IIB on AdS5$\times S^5$ and M-theory on AdS4$\times S^7$) provides examples of maximally-supersymmetric ($SO(5)$, $SO(6)$ and $SO(8)$)gauged supergravity in 7,5,4 dimensions.

You might say that this is a bit of a cheat, as the radius of the sphere is not parametrically smaller than the radius of the AdS space.So we are not completely justified in treating it as a 7-, 5- or 4-dimensional theory.

Posted by: Jacques Distler on March 24, 2006 2:16 PM | Permalink | PGP Sig | Reply to this

### Re: Avatars of Nonlocality?

Hi Jacques,

This post is a great illustration of what I dislike about blogs and more specifically trackbacks. As I explained to you when you were visiting Harvard last week, your first point about the RG running is standard effective field theory (with an abbreviated discussion in our paper because it is fairly common knowledge–read Georgi’s book). I of course don’t object to your writing a paper to clarify these points to yourself or others. But this is minor. More importantly, as I also explained to you both in email and in person, what you write about the DGP model is totally wrong. The theory is an interacting one, as a 1-line computation of the tree-level S-matrix reveals (which was of course done in the paper). Our arguments certainly do apply to DGP–indeed this is the most interesting application of what we are talking about. You told me that you would correct this discussion on your blog. A week has gone by, you have posted on many other things, but you haven’t fixed any of it.

Now, in general I don’t care about what is said on blogs, as I believe they largely fulfill the primate desire to look and see what the other monkeys are doing, and I think they are a big waste of time. But I do object to having a trackback, linked from my paper, to a post about it that claims that one of the central claims is wrong, when a 45 second computation, even done for the reader’s convenience in the paper itself, refutes the argument.

If you were to write a paper saying DGP is a free theory in disguise, you would probably have to do the 45 second computation of the tree S-matrix, find to your astonishment that its non-vanishing, and not write the paper after all. But on a blog, you can just throw it out there, fancy equation script and all–and this is a big drawback to physics blogging. Of course we all make tonnes of trivial little mistakes as we understand new things–but there is a good reason why these stay in our heads and notebooks. Communicating new ideas is hard enough without adding more noise to the system. So count me with the people who say trackbacks should be abolished. Any serious comment about correct or incorrect physics can be made in the usual way–by writing a paper about it.

Best,

Nima

Posted by: Nima Arkani-Hamed on March 21, 2006 11:25 PM | Permalink | Reply to this

### Re: Avatars of Nonlocality?

A week has gone by, you have posted on many other things, but you haven’t fixed any of it.

A week is a very short period, on the timescale of writing papers. And, indeed, it’s a short period on the timescale of progress in physics.

It does, however, seem like an eternity on the timescale of the blogosphere.

As I said, I fully intended to correct the above statement about the DGP model, along with clarifying several other points in this post. I’ve gone ahead and done the former. The latter will have to wait.

But on a blog, you can just throw it out there, fancy equation script and all…

And you can write in to point out the error. The useful thing about this medium is that it has some self-correcting mechanisms built in.

There are plenty of wrong papers in the literature which never get corrected, or if there is a correction somewhere, not one that is easy to find (see the paper of Pham and Truong that you cite).

They, too, constitute a kind of noise, but one that is much, much, harder to filter out.

Posted by: Jacques Distler on March 22, 2006 9:53 AM | Permalink | PGP Sig | Reply to this

### Re: Avatars of Nonlocality?

Hmm but without the trackback it had already taken if not a week a couple dats for a generic seach engine as technorati or google to index this entry about Nima’s paper, and even then he should be aware of it only if explicitly searching for his own papers. So probably without trackbacks (nor comments) the page had had an uncorrected comment for a long time.

Posted by: Alejandro Rivero on March 23, 2006 6:09 AM | Permalink | Reply to this

### 1PIs, effective actions and time-ordering

I have another question.

What do the 1PI’s give in the presence of a nonvacuum background? The same question applies to the n-point functions given by summing over the Feynman diagrams (or from the Schwinger-Dyson equations or from the generating functional). In ordinary subluminal quantum field theory, these n-point functions (which can be gotten from the 1PI’s and vice versa) are the time-ordered functions. Time ordering is perfectly fine in a subluminal theory where spacelike separated fields commute/anticommute.

In the models mentioned by Arkani-Hamed et al (and Velo and Zwanziger), the perturbative theory about the vacuum isn’t superluminal and so, time-ordering makes sense perturbatively and the standard interpretation of 1PI’s makes sense. But when expanding about a nonvacuum background, spacelike separated fields no longer commute/anticommute and in fact, if we choose a background with no global causal structure, it’s not even possible to define any time-ordering. This certainly raises questions on what the effective action means as we can always perform a Taylor expansion of the effective action about a nonzero background.

Posted by: Jason on March 22, 2006 9:45 AM | Permalink | Reply to this

### Re: 1PIs, effective actions and time-ordering

Presumably, we should restrict ourselves to backgrounds that satisfy the quantum equations of motion (are stationary points of the 1PI effective action). In good theories, propagation in any such background will be subluminal

In bad theories, it’s not clear what is supposed to make sense.

Posted by: Jacques Distler on March 23, 2006 1:41 AM | Permalink | PGP Sig | Reply to this

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