## Gradient expansions and quantum field theory

It is more than two years that I am working on quantum field theory in the strong coupling limit and I am generally very satisfied with the acceptance by the community about my views. Of course, these are new ideas and may take some time to be accepted. So, I keep on working on them trying to clarify them at best so that people can have a clear understanding of their strengths and weaknesses. One of the ways we researchers have to know how our colleagues consider our views is peer-review. This system is indeed crucial to any serious scientific endeavor and, indeed, I am proud of my achievements only when my peers agree about their value. But peer-review is also useful to my work to know what are the main objections to it. It can happen that sometime these objections are deeply wrong and may be worthwhile to discuss them at length also to have an idea on how such a prejudice arose.

We should know that when a mathematical theory enters into the description of nature, whatever mathematical method one uses to exploit it is always correct. So, natural laws in physics are described by differential equations and  whatever method you know to solve them is good provided is also mathematically legal. You should consider mathematics for physicists as a severe judge that grants no appeal. You are right or wrong depending on the correctness of your computation. But in physics there is something more and these are assumptions we start with. You can do the beautiful mathematics in the world but if you started with a wrong concept about how nature works your computations are simply rubbish.

One of the criticisms I have received on trying to get my papers published is that one cannot do a gradient expansion because this breaks Lorentz/Poincare’ invariance. This is completely wrong from a mathematical standpoint. As an exercise  you can consider the wave equation in two dimensions as

$\partial^2_{tt}u-\partial^2_{xx}u=0$

and consider the case where the spatial part is not so important. This can be easily obtained by rescaling time as $t\rightarrow\sqrt{\lambda}t$ and taking the limit $\lambda\rightarrow\infty$. One gets the solution series

$u=u_0+\frac{1}{\lambda}u_1+\frac{1}{\lambda^2}u_2+\ldots$

solving the equations

$\partial^2_{tt}u_0=0$

$\partial^2_{tt}u_1=\partial^2_{xx}u_0$

$\partial^2_{tt}u_2=\partial^2_{xx}u_1$

and so on. All this is perfectly legal from a mathematical standpoint and I get a true solution of the wave equation. But, as you can see, I have broken Lorentz invariance, a symmetry of this equation. So, mathematics says yes while physics seems to say no. The answer is quite simple and is known since a long time: The computation is right but Lorentz invariance is no more manifest. This is due to the fact that I have separated time and space. But if I am able to resum all the terms of the expansion series I will get the right answer

$u=f_1(x-t)+f_2(x+t)$

that is Lorentz invariant. So, both physics and mathematics give the same answer and is a resounding yes, it works and it works so well that we are left with a kind of strong coupling expansion.

So, what should do a smart referee with such a doubt, admitting that a smart referee does not know such mundane facts of physics and mathematics? It should realize that here one is facing a really interesting problem of physics: Could we formulate a gradient expansion in such a way to have Lorentz invariance manifest? I have not an answer yet to this question but I grant to you that is a matter I would like to publish a paper about  somewhere. This is an interesting mathematical problem as well. We know that people met a similar problem at the start of the deep understanding of QED due to Feynman, Schwinger, Tomonaga and Dyson. I think that an answer to this question would have the same scientific value.

### 6 Responses to Gradient expansions and quantum field theory

1. carlbrannen says:

Expansion in the gradient, in Lorentz invariant way, wouldn’t that imply the use of the Klein-Gordon equation? My intuition runs towards choosing a parameter that when nonzero gives a violation of Lorentz invariance, and at the end, letting that parameter go to zero, sort of like a method of Lagrange multipliers.

This gets down to the fundamental question, is it the equations which make up the soul of physics, or is it the philosophy? I think it is the first, since it is the equations that give numbers that can be compared to experimental results. You get the equations from the philosophy, but any mathematician can tell you that there are an infinite number of philosophies that still give the same set of equations.

What you are doing is compatible with the equations of relativity, but not with the philosophy. Of course there was a similar problem with early quantum mechanics.

2. mfrasca says:

Hi Carl,

Klein-Gordon or wave equation is the same thing. The only difference is a mass term that is zero for the latter. Really, there is no Lorentz violation at all because the solution series and the exact solution are identical.

Some time ago, a colleague of mine suggested to me to add a further dimension and work with this. Then, you do a Wick rotation on the remaining part and you are done. This is also the way Parisi-Wu method applies. I have not exploited further this idea but seems promising. I hope to get some time to exploit it better.

Once you have the correct equations, it does not matter philosophy. What really matters are the mathematical methods you know to cope with them. And if these methods are correct your solutions are too.

Indeed, you are right. People working in QED met this serious difficulty. Perturbation theory gave gauge and Lorentz violating terms and infinities were all around. The crucial work of Feynman, Schwinger, Tomonaga and Dyson solved all these difficulties giving us a powerful approach in quantum field theory. Renormalization criterion is foundational to any quantum field theory that pretends to describe reality.

Marco

3. carlbrannen says:

There’s been discussion of Lorentz invariance on various blogs recently. Having a night’s sleep, another thing comes to mind.

When people talk about “manifest Lorentz invariance” as opposed to equations that exhibit LI but not explicitly, it seems to me that nearly always the non manifest LI appears pretty much the way yours does, that is, space and time are treated completely differently.

But there is another way of getting to Lorentz invariance. Consider the modified Klein Gordon wave equation where the time 2nd deriviative has its sign changed. Then you’re working in the Euclidean metric instead of Lorentz. Sort of a Wick rotation.

But the operators are related. 3-d gradient is the average of the D’alembertian and the 4-d Euclidean gradient. So could you obtain three different series, one LI, one Euclidean relativistic, and one 3-d gradient, and obtain a relationship between the series?

4. mfrasca says:

Carl,

You are right about the question of manifest Lorentz invariance. Lorentz invariance may be hidden in some computations due to a different way to treat space and time components. What really matters in the end is that a Lorentz invariant result is obtained. It is just a matter of taste to be able to do computations manifestly Lorentz invariant and, anyhow, a nice way to manage physical questions.

The idea to work with another dimension and treating the 4D part as Euclidean after a Wick rotation is what my colleague proposed to me. He also said to work with Parisi-Wu technique.

I hope to get some time to work this scenario out. Currently I am involved with Ricci flow machinery and have no much time for other.

Marco

5. carlbrannen says:

Well it’s nice that someone else sees the obvious things.

The (I think) unnatural usefulness of Wick rotations is part of how I ended up in the outskirts of physics. This is related to another relativity puzzle, “why is it that proper time is coincidentally the only thing that everyone agrees about, but on the other hand, proper time is not a fundamental part of the geometry of spacetime?”

So my path to crackpottery began with finding a modified geometry that would allow relativistic quantum field theory to be written dispersion free, that is, all particles would travel at the same speed c, all the time. This makes the wave equations simpler as it eliminates mass.

It turns out that this idea is not so uncommon, it’s called Euclidean relativity, but no two people practice quite the same version of it. (In mine, you have to do some resummations to get the usual momentum and energy.)

Anyway, that’s the roundabout reason why my recent papers are written from a quantum information theory viewpoint; I don’t have to argue about which version of relativity to assume. The disadvantage is that I lose out on all the nice Fourier transforms. But I end up with discrete Fourier transforms on the particles.

6. Daniel de França MTd2 says:

Hi Carl,

Besides Euclidean relativity, I’d reccomend you Wesson gravity, for the idea that all particles run in the speed of light, but descended from a higher dimension, usualy 5D:
http://arxiv.org/find/gr-qc/1/au:+Wesson_P/0/1/0/all/0/1
http://astro.uwaterloo.ca/~wesson/

I think twistors could help you address your first question. Roughly, it maps the point in space time to the celestial sphere (2D). I sorts of “glues” the referencial, thus it might reference you concerns about thhe proper time (3D), to a space-like slice of the universe. I is a 5D theory.

Note that these 2 theories usualy use similar signatures (+—+).

Cheers,

Daniel.