*"My brain feels very Boltzmannian today",*or

*"The customer density in this store is too high for my metastable mental balance"*. I have a friend who calls Chinese take out

*"the canonical choice"*and another friend who, when asked whether he had made a decision, famously explained

*"I don't yet want my wave-function to collapse"*. My ex-boyfriend once called it

*"the physicist's Tourette-syndrome"*[2].

One of my favourite physics-speech words is self-consistent. Self-consistency is tightly related to nothing. You know, that "nothing" that causes your wife to conclude her whole life is a disaster, we're all going to die in a nuclear accident, her glasses vanished (again!), and btw that's all your fault (obviously). But if you ask her what's the matter. Well, nothing.

*"There's nothing I hate more than nothing*

*Nothing keeps me up at night*

*I toss and turn over nothing*

*Nothing could cause a great big fight*

*Hey -- what's the matter?*

*Don't tell me nothing."*

**1. Self-consistent**

Science is our attempt to understand the world we live in. We observe and try to find reliable rules upon which to build our expectations. We search for explanations that are useful to make predictions, a framework to understand our environment and shape our future according to our needs. If our observations disagree with our rules, or observations seemingly disagree with each other (I swear I left my glasses in the kitchen), we are irritated and try to find a mistake. Something being in contradiction with itself [3] is what I mean with not self-consistent (What's the matter? - Nothing!).

On a mathematical basis this is very straight forward. E.g. If you assume my mood is given by a real valued continuous function

*f*on the compact interval [

*now, then*] with

*f*(

*now*)

*f*(

*then*) smaller than 0, this isn't self-consistent with the expectation it can do so without having a zero [4]. For more details on my mood, see sidebar.

Self-consistency is a very powerful concept in theoretical physics: if one talks about a probability, that probability better should not be larger than one. If one starts with the axioms of quantum mechanics, it's not self-consistent to talk about a particle's definite position

*and*momentum. The speed of light being observer independent is not compatible with Galileo invariance and the standard addition law for velocities. Instead, self-consistency requires the addition law to be modified. This lead Einstein to develop Special Relativity.

A particularly nice example comes from multi-particle quantum mechanics, where an iterative approach can be used to find a 'self-consistent' solution for the electron distribution e.g. in a crystal or for an atom with many electrons (see self-consistent field method or Hartree-Fock method). A state of several charged particles will not be just a tensor product of the single particles, since the particles interact and influence each other. One starts with the tensor product as a 'guess' and applies the 'rules' of the theory. That is, by solving the SchrÃ¶dinger equation with the mean- field potential which effectively describes the interaction, a new set of single particle wave functions can be computed. This result will however in general not agree with the initial guess: it is not self-consistent. In this case, one repeats the procedure with using the result as an improved guess. Given that the differential equations behave nicely, this iterative procedure leads one to find a fixed point with the properties that the initial distribution agrees with the resulting one: it is self-consistent.

A similar requirement holds for quantum corrections. A theory that is subject to quantum corrections but whose initial formulation does not take into account the existence of such extra terms is strictly speaking not self-consistent (see also the interesting discussion to our recent post on Phenomenological Quantum Gravity).

There are some subtleties one needs to consider, most importantly that our knowledge is limited in various regards. Self-consistency might only hold under certain assumptions or in certain limiting regimes, like small velocities (relative to the speed of light), large distances (relative to the Planck length) or at energies below a certain threshold. Likewise, not being self-consistent might be the result of having applied a theory outside these limits (typically, using an expansion outside a radius of convergence). In some cases (gravitational backreaction), violations of self-consistency can be negligible.

However, one might argue if it is possible at all to arrive at such a disagreement then at least one of the assumptions was unnecessary to begin with, and could have been replaced by requiring self-consistency. Unfortunately, this is often more easily said than done -- physics is not mathematics. We rarely start with writing down a set of axioms which one could check for self-consistency. Instead, in many cases one starts with little more than a patchwork of hints, and an idea how to connect them. Self-consistency in this case is somewhat more subtle to check. My friends and I often kill each others ideas by working out nonsensical consequences. Here, at least as important as self-consistency is that a theory in physics also has to be consistent with observation.

**2. Consistent with Observation**

The classical Maxwell-Lorentz theory is self-consistent. However, it is in disagreement with the stability of the atom. According to the classical theory, an electron circling around the nucleus should radiate off energy. The solution to this problem was the development of quantum mechanics. The inconsistency in this case was one with observation. Without quantizing the orbits of the electron, atoms would not be stable, and we would not exist.

This requirement is specific to sciences that describe the real world out there. Such a theory can be 'wrong' (not consistent with observation) even though it is mathematically sound. Sometimes however, these two issues get confused. E.g. in a recent Discover issue, Seth Lloyd wrote:

*"The vast majority of scientific ideas are (a) wrong and (b) useless. The briefest acquaintance with the real world shows that there are some forms of knowledge that will never be made scientific [...] I would bet that 99.8 percent of ideas put forth by scientists are wrong and will never be included in the body of scientific fact. Over the years, I have refereed many papers claiming to invalidate the laws of quantum mechanics. I’ve even written one or two of them myself. All of these papers are wrong. That is actually how it should be: What makes scientific ideas scientific is not that they are right but that they are capable of being proved wrong."*

~Seth Lloyd, *You know too much*

The current issue now had a letter in reply to this article:

*"I was taken aback by Seth Lloyd's assertion that "99.8 percent of ideas put forth by scientists are [probably] wrong" and even more so by his statement that "of the 0.2 percent of ideas that turn out to be correct ... [t]he great majority of them are relatively useless." His thesis omits a basic trait of what we call science -- that it is a continuous fabric, weaving all provable knowledge together [...] we do science for a science sake, because a fundamental principle of science is that we never know when a discovery will be useful"*

~Eric Fisher, Springfield, IL.

Well, the majority of

*my*scientific ideas are definitely (a) wrong and (b) useless, but these usually don't end up in a peer review process. However, the reply letter apparently referred to the word 'correct' as 'provable knowledge', and to science as the 'weave' of all that knowledge. It might indeed be that the mathematical framework of a theory that is not consistent with observation turns out to be useful later but that doesn't change the fact that this idea is 'wrong' in the meaning that it does not describe nature. Peer review today seems to be mostly concerned with checking self-consistency, whereas being non-consistent with observation is ironically increasingly tolerated as a 'known problem'. Like, the CC being 120 orders of magnitude too large is a known problem. Oohm, actually the result is just infinity. But, hey, you've turned your integration contour the wrong way, the result is not infinity, but infinity + 2 Pi.

The requirement of consistency with observation was for me the main reason to chose theoretical physics over maths. The world of mathematics, so I found, is too large for me and I got lost in following runaway thoughts, or generalizing concepts just because it was possible. It is the connection to the real world, provided by our observations, that can guide physicists through these possibilities and lead the way. (And, speaking of observations and getting lost, I'd really like to know where my glasses are.)

**3. Self-contained**

Unlike maths, theoretical physics aims to describes the real world out there. This advantageous guiding principle can also be a weakness when it comes to the quantities we deal with. Mathematics deals with well defined quantities whose properties are examined. In physics one wants to describe nature, and the exact definitions of the quantities are in many cases subject of discussion as well. Consider how our understanding of space and time has changed over the last centuries!

In physics it has often happened that concepts of a theory's constituents only developed with the theory itself (e.g. the notion of a tensor or the Fock-space). As such it happens in physics that one can deal with quantities even though the framework does not itself define them. One might say in such a case the theory is incomplete, or not self-contained.

Due to this complication, I've known more than one mathematician who frowned upon approaches in theoretical physics as too vague, whereas physicists often find mathematical rigour too constraining, and instead prefer to rely on their intuition. Joe Polchinski expressed this as follows:

*"[A] chain of reasoning is only as strong as its weakest step. Rigor generally makes the strongest steps stronger still - to prove something it is necessary to understand the physics very well first - and so it is often not the critical point where the most effort should be applied. [A]nother problem with rigor [is]: it is hard to get it right. If one makes one error the whole thing breaks, whereas a good physical argument is more robust."*

~Joe Polchinski, *Guest Post at CV*

When it comes to formulating an idea, physicists often set different priorities than mathematicians. In some cases it might just not be necessary to define a quantity because one can sit down and measure it (e.g. the PDFs). Or, one can just leave a question open (will be studied in a forthcoming publication) and get a useful theory nevertheless. All of our present theories leave questions open. Despite this being possible, it is unsatisfactory, and the attempt to make a theory self-contained has lead to many insights throughout the history of science.

Newton's dynamics deals with forces, yet there is nothing in this framework that explains the origin of a force. It contains masses, yet does not explain the origin of masses. Maxwell's theory provides an origin of a force (electromagnetic). It has a source term (

*J*), yet it does not explain the dynamics of the source term. This system has to be closed, e.g. with minimal coupling to another field whose dynamics is known. The classical Maxwell-Lorentz theory does this, it is self-contained and self-consistent. However, as mentioned above, this theory is not consistent with observation. Today we know the sources for the electromagnetic field are fermions, they obey the Dirac equation and Fermi statistic. However, if you look at an atom close enough you'll notice that quantum electrodynamics alone also isn't able to describe it satisfactory...

Besides the existence of space and time per se, the number of space-time dimensions is one of these open questions that I find very interesting. It has most often been an additional assumption. An exception is string theory where self-consistency requires space-time to have a certain number of dimensions. However - if it also contains an explanation why we observe only three of them, nobody has yet found it. So again, we are left with open questions.

4.

**Simple and Natural**[5]

The last guiding principle that I want to mention is simplicity, or the question whether one can reduce a messy system of axioms and principles to something more simple. Is there a way to derive the parameters of the standard model from a single unified approach? Is there a way to derive the axioms of quantization? Is there a way to derive that our spacetime has dimension three, or Lorentzian signature?

In my opinion, simplicity is often overrated compared to the first three points I listed. We tend to perceive simplicity as elegance or beauty, concepts we strive to achieve, but these guidelines can turn out to be false friends. If you can find your glasses, look around and you'll notice that the world has many facettes that are neither elegant nor simple (like my husband impatiently waiting for me to finish). Even if you'd expect the underlying laws of nature to be simple, you'll still have to make the case that a certain observable reflects the elementary theory rather than being a potentially very involved consequence of a complex dynamical system, or an emergent feature. A typical example are the average distances of planets from the sun, a Sacred Mystery of the Cosmos that today nobody would try to derive from a theory of first principles (restrictions apply).

Also, we tend to find things simpler the more familiar we are with them, up to the level of completely forgetting about them (did you say something?). E.g. we are so used to starting with a Lagrangian that we tend to forget that its usefulness rests on the validity of the action principle. It is also quite interesting to note that researchers who are familiar with a field often find it 'simple' and 'natural'... I therefore support Tommaso's suggestions to renormalize simplicity to the generalized grandmother.

In this regard I also want to highlight the argument that one can allegedly derive all the parameters in the standard model 'simply' from today's existence of intelligent life. Notwithstanding the additional complication of 'intelligent', could somebody please simply explain 'existence' and 'life'?

**Bottomline**

Much like classical electrodynamics, Einstein's field equations too have a source term whose dynamics one needs to know. The system can be closed with an equation of state for each component. This theory is self-consistent [6], and it is consistent with all available observations. It reaches its limits if one asks for the microscopic description of the constituents. The transition from the macro- to the microscopic regime can be made for the sources of the gravitational field, but not also for the coupled gravitational field (oh, and then there's the CC, but this is a known problem).

Two theories that yield the same predictions for all observables I'd call equivalent (if you don't like that, accept it as my definition of equivalence.) But our observations are limited, and unlike the case of classical electrodynamics not being consistent with the stability of the atom, there is presently no observational evidence in disagreement with classical gravity.

For me this then raises the question:

- Is there more than one theory that is self-consistent, self-contained and consistent with all present observations?

In a recent comment, Moshe remarked:

*"To paraphrase Ted Jacobson, you don't quantize the metric for the same reason you don't go about quantizing ocean waves."*That sounds certainly reasonable, but if I look at water close enough I will find the spectral lines of the hydrogen atom and evidence for its constituents. And their quantization. To me, this just doesn't satisfactory solve the question what the microscopic structure of the 'medium', here space-time, is.

And what have we learned from all this...?

Let me go back to the start: If you ask a question and the answer is 'Nothing', you most likely asked the wrong question, or misunderstood the answer.

Ah... Stefan found my glasses (don't ask).

See also: Self-Consistency at The Reference Frame

[1]This habit is especially dominant -- and not entirely voluntarily -- among the not native English speakers, whose vocabulary naturally is most developed in the job related area.

[2] Unintentional cursing and uttering of obscenities, called Coprolalia, is actually only a specific feature of the Tourette syndrom.

[3] However, some years ago I was taught the word 'self-consistency' in psychology has a different meaning, it refers to a person accumulating knowledge from his/her own behaviour. A person whose thoughts and actions are in agreement and not in contradiction is called 'clear'. (At least in German. I couldn't find any reference to this online, and I'm not a psychologist, so better don't trust me on that.).

[4] See: Bolzano's theorem.

[5] "Woman on Window", by F.L. Campello. For more, see here.

[6] Note that this theory is self-consistent at arbitrary scales as long as you don't ask for the microscopic origin of the sources.

TAGS: PHYSICS, SCIENCE, MATHEMATICS