## Saturday, June 06, 2015

### It's childish to vigorously fight against one sign convention

A vitriolic blog regularly attacking modern physics has chosen a new target. If you Google search for
The West Coast Metric is the Wrong One,
you will find a rant where the well-known crackpot presents his "modest proposal" that
the HEP community should just admit that the West Coast convention was a mistake, and rewrite all the textbooks (Weinberg doesn’t have to…).
LOL. Crackpots (including this one) are usually repetitive and boring but occasionally, their silliness becomes creative and makes you smile.

Well, the West Coast Metric cannot be "the wrong one" or a "mistake" because it is just a damn convention and it is spectacularly obvious that it is exactly as internally consistent as the opposite one, the East Coast Metric. One may get used to by one convention or another, internalize it, and it becomes more convenient for him than the other.

But once he tries to rationalize it and pretend that the life cannot exist on the other coast, he becomes an irrational loon who is constantly deluding himself. I assure all the readers that I've spent at least a year on both coasts and the life is possible on both.

OK, what are we talking about here? Einstein's special theory of relativity has unified the space and time – and similarly the momentum and energy (and many other things). Let's talk about the space and time. The 3 spatial coordinates and 1 temporal coordinates were realized to be "almost the same thing" except that there is a sign difference between the spatial and temporal directions.

There exists an "invariant distance" between the origin $$(t,x,y,z)=(0,0,0,0)$$ and the point $$(t,x,y,z)$$ that has the same value from the viewpoint of all inertial systems (reference frames). This distance may be written as$ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2.$ It looks like the hypotenuse according to the Pythagorean theorem except that there is a relative minus sign between the terms. The spatial terms contribute with the opposite sign than the temporal term. Otherwise, you can't say which of them should be negative and which of them should be positive because the whole normalization of $$ds^2$$ is undetermined.

After all, you might divide $$ds^2$$ not just by $$-1$$ but also by $$c^2$$ or any other coefficient – and it would still be invariant. First, let us eliminate the the problem with the powers of $$c$$. We may always use units with the numerical value of the speed of light $$c=1$$ – and adult physicists who work with special relativity all the time i.e. both relativists and particle physicists are doing so all the time.

So let me correct the previous formula to$ds^2 = dt^2 - dx^2 - dy^2 - dz^2.$ and stop worrying about the powers of $$c$$ in the definition of all quantities – which may open a whole new can of irrelevant worms. However, even when the $$c$$ problem is killed, we still face an ambiguity concerning the overall sign.

Three ways to deal with the relative sign

The first convention to deal with this negative relative sign was actually neither the West Coast Metric nor the East Coast Metric. It was the $$({+}{+}{+}{+})$$ metric. How could it be the correct one? Well, someone defined$\eq{ x^1 &= x\\ x^2 &= y\\ x^3 &= z\\ x^4 &= it\\ }$ Well, the original form would be $$x^4=ict$$. You have four coordinates and a Lorentz invariant may be written simply as$\sum_{i=1}^4 (x^i)^4$ All the signs are plus. (Or all of them may be minus, if you wish.) The price you pay for the uniform sign is the non-uniform reality condition. While $$x^1,x^2,x^3$$ are real, $$x^4$$ is pure imaginary. (These two rules may be interchanged, too, but I don't want to promote fifty new possible conventions.)

If you said that this convention is a mistake, bad, ugly, or there must be something wrong with it, well, let me mention that this was the treatment of the invariants introduced by a man called Albert Einstein.

This real-imaginary mixed metric convention is as consistent as those we use today. Its generalization to general relativity would be a bit awkward, however, because in general relativity, the coordinates $$x^\mu$$ describing a curve space are neither universally spacelike nor timelike. They can be both and the character is influenced by the values of the metric tensor.

So in general relativity, it becomes unnatural to impose different reality conditions on different components $$x^\mu$$. That's one reason why both "coast" conventions are better for modern physics, especially if general curved metrics ever enter your calculations.

East Coast vs West Coast

Hermann Minkowski, the teacher who called student Einstein a "lazy dog", introduced the concept of the spacetime in 1908, three years after special relativity was discovered. If we modernize his formalism by using the $$c=1$$ units, his invariant was$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2.$ The overall sign is the opposite one than the convention I started with. But this overall sign flip may be explained by the fact that $$ds^2$$ in one convention simply means $$-ds^2$$ in the other, and vice versa. In other words, the invariant may be written as$ds^2 = \sum_{\mu,\nu=1}^4 g_{\mu\nu} dx^\mu dx^\nu$ and we may say that the metric tensor i.e. the coefficients $$g_{\mu\nu}$$ in one "language" means exactly the same thing as $$-g_{\mu\nu}$$ in the other "language". It should be obvious to an intelligent schoolkid that if one language is internally consistent, so must be the other. It's just like to create a new language out of English by replacing "the" with "het". The new language will be as consistent as English. In fact, it's called Dutch.

To make the story short – and still describe the history much more correctly than the crackpot – Minkowski's "mostly plus" convention was adopted by Pauli and when the U.S. became the world center of physics, it was used by relativists on the East Coast – think about Pennsylvania (where you have relativists like Ashtekar) as well as Princeton (Einstein) and Harvard (Schwinger used it, too).

On the other hand, there is the West Coast Metric or "mostly minus" convention. I will discuss it later.

Minkowski's "mostly plus" convention presents the spacetime as an extension of the usual 3D Euclidean spacetime we know very well – with its $$ds^2 = dx^2+dy^2+dz^2$$ Pythagorean theorem. This interpretation of the spacetime is sometimes used to defend the convention:
With the East Coast convention, the treatment of spatial coordinates is just like in the non-relativistic case. In the West Coast convention, as far as space goes, you have decided to work with a negative definite metric, which is a quite misguided thing to do for obvious reasons.
But this claim is sloppy, emotional, and from a science viewpoint, it is irrational. There are no "obvious reasons" why a negative definite metric should be "misguided". When all conventions are settled, certain quantities simply are positive and certain quantities are negative and stupid complaints are the last thing you can do against these facts. For the 3D Pythagorean theorem, we were using the "all plus" convention because we only needed one sign of $$ds^2$$. But we could have used the "all minus" convention in the 3D Euclidean space, too.

Moreover, there is no reason why the 4D spacetime "must" be presented as an extension of the "all plus" 3D space and copy all the old conventions. The only invariant fact is that some dimensions in the 4D spacetime have the opposite sign than others. We can't determine which of the dimensions are the "plus ones". It is a totally unphysical question. In relativity, we need to work with an indefinite $$ds^2$$, a quantity that may have both signs. Because both signs may appear, it's clearly just a matter of convention – both options are possible – to decide which kind of $$ds^2$$ (spacelike or timelike) should be positive and which should be negative.

The West Coast Metric

The East Coast Metric says that spacelike intervals are the positive ones. By saying that their sign is positive, we tend to say that these are the "normal" intervals. In the East Coast Metric, the spacelike intervals are implicitly said to be the "normal" ones.

However, $$ds^2$$ measures the squared proper length of a line interval in the spacetime. And the only lines and line intervals in the spacetime that are physically important - that "really exist" out there – are world lines of moving objects. And because of the relativistic causality, the world lines may only be timelike (or light-like, for massless particles)!

So if you will calculate $$ds^2$$ for some "truly physically important line interval", you will obtain the sign that you get from the timelike, not spacelike, intervals. You will always get the same sign and it makes sense to say that the universal sign of these "truly physical" line intervals is the positive one. So the positive sign should be associated with the timelike intervals and the metric is $$({+}{-}{-}{-})$$, mostly minus.

This is called the "West Coast Convention" not because of Feynman as the crackpot claimed but mostly because Bjorken and Drell who wrote an early influential textbook of quantum field theory. They worked at Stanford's SLAC and their convention was adopted not just by most other people in California but by most particle physicists, too.

One can try to rationalize the convention by many observations. The intervals $$ds^2\gt 0$$ for the timelike intervals, those that are associated with trajectories of massive particles. Similarly, the invariant$p^2 = p^\mu p_\mu = g_{\mu\nu} p^\mu p^\nu$ (with the Einstein sum rule) happens to be positive for particles we may actually observe, $$p^2=m^2\gt 0$$. The negative sign of $$m^2$$ would be connected with tachyons, superluminal particles that are prohibited in a stable spacetime.

But just like the "arguments" in favor of the East Coast Metric, these "arguments" in favor of the West Coast Metric don't mean that you "must" use one or the other. An intelligent schoolkid should be able to see that both conventions will be "simpler" than the competitor in some contexts. It has to be so simply because $$ds^2$$ of both signs may appear because special relativity deals with the indefinite metric. So if you make some of the things positive, the other ones will be negative, and vice versa! To say that "one of the two options clearly wins" means to be completely biased and only look at 50% of the "arguments".

My cultural background

As an undergraduate in Prague, I was made prepared for both conventions – I understood the cultural character of the convention much earlier than that, of course. The relativists (course of general relativity) would prefer the East Coast Convention, I believe. But I am sure that the quantum field theory courses preferred the West Coast Metric.

I am much more certain about the latter because I have done numerous calculations where one simply has to be careful about the relative signs and where the results (scattering amplitudes etc.) involve lots of contractions of the Lorentz vector indices. You must be careful about the number of factors of $$i$$, too. The QFT course – I mean especially those by Dr Jiří Hořejší – probably had many more "mechanical exercises" with well-defined calculations so the convention was more important there, and I was trained to use the "mostly minus" convention.

So I have always considered myself culturally a West Coast guy – even though I spent 90% of my decade in the U.S. on the East Coast. But in contrast with what the crackpot bloggers wants you to believe, you won't get excommunicated from the East Coast if you use the West Coast convention and vice versa. Many people on both coasts do prefer the other coast's convention. All sensible physicists realize that it's silly to "fight" about these choices and the individual who does is a crackpot, indeed (for this reason and for tons of other reasons).

String theory, a unification of the QFT+GR cultures

A few historical comments above make you understand that the "mostly plus" metric, the East Coast Metric, is mostly used by the (general) relativists (who largely ignore quantum mechanics), while the "mostly minus", the West Coast or "Bjorken Drell" Metric, is mostly used by particle physicists who work with quantum mechanics all the time (and neglect gravity in most of their careers).

There may be some "intrinsic" reasons why this correlation between the sign convention and the subdiscipline of theoretical physics has developed. But at the end, I think that it's mostly due to sociological historical coincidences. Both relativists and particle physicists know that $$p^2$$ is a rather natural quantity and it may be nicer if it is positive for ordinary particles that exist. Both of them know that the 4D spacetime may be obtained by adding a direction with a different sign on top of the 3D Euclidean space, and so on.

They use their convention because that's how they were trained to think.

String theory is the only consistent unifying theory of quantum field theory and general relativity. The two subdisciplines that are being unified prefer opposite conventions. Because string theorists need to build on both parts of physics, you could think that there may be some schizophrenia about the convention in string theory. And indeed, that's the case. There's some schizophrenia. Or lots of tolerance towards both choices and a great flexibility of string theorists who can switch from one convention to the other very quickly whenever it matters.

Well, in some cases, the flexibility is so impressive that it should be called schizophrenia, after all. For example, one-half of Michael Dine's recent textbook uses one convention while the other one-half uses the other! Michael could have undoubtedly unified the convention but he chose not to. (Polchinski uses the "mostly plus" metric and although I was mostly trained as a "mostly minus" guy, I had absolutely no problem to adapt to Polchinski's textbook. One simply has to be careful about various signs if he wants the accurate results. Most of the important or qualitative results that we describe "verbally" don't depend on the convention, anyway. They are much deeper than that. But anyone who thinks that one of the conventions is "deeply flawed" is clearly misunderstanding the total basics.)

You should appreciate the example of Dine's book. If even one author doesn't find it important to preserve one particular convention throughout one book, you may believe me that it's even harder and less justified to try to impose some unity on all physicists. The crackpot's ludicrous demand to "translate" all the books to one convention would mean to rewrite hundreds of books and tens of thousands of papers (maybe you would be ordered to recycle the books on your bookshelves and buy new ones, too). You know, the crackpot has never read or written a scientific paper but physicists often do, so if you want to unify the conventions, you can't ignore these tens of thousands of papers.

No one will rewrite the tens of thousands of papers and most particle physicists just won't voluntarily switch to the East Coast Metric because they were trained to think and talk in the West Coast Metric and it's not trivial to "retrain yourself". They would be making mistakes. And they taught themselves to think that the arguments in favor of the West Coast Metric are "a bit stronger", anyway. So why should they switch?

The costs of the transition to the "unified metric" would probably greatly exceed the benefits.

At the end, professional physicists will agree that the real losers are not those who prefer the opposite convention; but rather those such as the aforementioned notorious crackpot who don't understand that a convention is just a convention. The crackpot shows his childish stupidity really clearly. There are lots of sentences saying that "it obviously has to be this way and not this way" even though it's always totally obvious that there can't exist any rational reason why it should be in one way or another way – both options are just conventions and they are equally consistent. The crackpot is like a kid in the kindergarten who screams that "18 is clearly a better number than 20" and beats the other kid if he doesn't recant that "20 is a better number". "Eighteen is clearly the best number because it's the only one that is three times six." "No, only 20 is four times five, so for the obvious reason, it's the best number." Kids, there is clearly no "better number" among the two!

You may see that certain mentally weak people face huge problems even when they learn things that are simple as a sign convention. The relationship between the two sign conventions is a trivial example of a "duality" – one in which the equivalence of the two languages is manifest. For an equivalence to be called a "duality", the relationship must be surprising to an intelligent person. So the two sign conventions aren't really "dual" according to the normal understanding of "dualities". But it's clear that if someone doesn't understand as trivial things as switching signs to a different convention, he can't ever possibly understand as complex things as string theory and the dualities in it. Modern physics in general and string theory in particular really does require you to become able to quickly figure out which features, signs, and properties matter and which of them are artifacts of conventions or descriptions. And count how many truly inequivalent choices and parameters are there, and so on. If you can't easily assure yourself that physics may be translated from one sign convention to another, you can't possibly understand modern physics because you're just way too stupid for that.

Some people on the obnoxious crackpot blog are sensible, others aren't. But the weird comments go beyond the choice of one sign convention. Noboru Nakanishi wrote:
The use of the words, “East Coast Metric” and “West Coast Metric”, is unwelcome; high-energy physicists are not necessarily Americans.
Holy cow. These terms are terms in the U.S. English, anyway. Americans have the right to speak in the U.S. English and even if they didn't have the right, they would still do so. And almost no Czech (to mention a specific example of another language) has used the Czech translations of these terms more than 10 times in his life. Moreover, in other nations, there exist no clear correlations between the convention and easy-to-describe geographic locations (or similar labels) so it would be hard to invent new names of the conventions that would be related e.g. to the Japanese culture or the Japanese geography.

More importantly, English is just the most important language in (not just) theoretical physics and the U.S. is the most important country that does research in (not just) theoretical physics – and it's been the case at least for 70 years. So even physicists from other countries simply have to pay some attention to the events that take place in the U.S. physics. Mr Nakanishi's attempt to "discourage" the usage of terms that refer to the American culture (or, in this case, geography) is an example of political correctness run amok – also because the relevant American culture is largely the civilized mankind's global culture, too.