Since many people seem to have very odd ideas about this, let's address this from a much simpler point of view.
Let's suppose you have a friend who only knows math at the level of arithmetic of positive integers. You try to tell him about the existence of negative numbers, and he tells you,
That's stupid, there's obviously no such thing as "negative" numbers, how can I possibly measure something so stupid? Can you have negative one apple? No, you can't. I can owe you positive one apple, but there's clearly no such thing as negative apples.
How can you start to argue that there is such a thing as negative numbers?
A very powerful first step is mathematical consistency. You can list all of the abstract properties you believe to characterize everything about positive integer arithmetic:
- For all a,b,c, a(b+c) = ab+ac
- For all a,b, a+b=b+a, ab=ba
- There exists a number, called 0, such that, for all a, a+0=0+a=a, a0=0a=0
- There exists a number, called 1, such that, for all a, 1a=a1=a
(note that, in sharp contrast to the case of real numbers, the first property can be proved with induction, and need not be an axiom. Similarly, other listed properties can be proved from other ones designated to be more basic if one wishes, which can not be done in the case of the reals.)
So, once you both agree that these axioms characterize the positive integers completely, you can show that these hypothetical negative numbers, based on their formal properties, are consistent with the above axioms. What does this show?
The positive integers, with the addition of the negative integers, can do at least as much as the positive integers by themselves.
(STOP At this point, pause to realize how powerful this constraint is!! How many other ways could one generalize arithmetic, at this level, to something else that is consistent with the properties you want? Zero. There is absolutely no other way to do it. This is incredibly suggestive, and you should keep this in mind for the rest of the cartoon argument, and see how every argument that follows is secretly an aspect of this one!)
Your friend responds:
Sure, you can write down toy models like that, and they may be consistent, but they don't correspond to reality.
Now, what else do you need to demonstrate to your friend to convince him of the validity of the negative numbers?
You find something else they can do that you can't do with the positive numbers alone. Simply, you can state that every positive-integer-valued algebraic equation does not have a solution:
x + 1 = 0
does not have a solution.
But, it is a trivial fact that extending to the negative numbers allows you to solve such equations. Then, all that's left to convince your friend of the validity of negative numbers is to show that this is equivalent to solving an ("a priori") different problem which only involved arithmetic of positive integers:
x+1=0 <=> y + 1 = 1
So, y = 0, and y=x+1 is equivalent to the other problem.
To be complete, we also have to consider problems that are "unique" to the negatives, such as (-1)(-1) = 1, but in the realm of integers, these are trivial matters that are reducible to the above. Even in the case of reals, given the other things we've shown, these consequences are almost "guaranteed" to work out intuitively obviously.
Now, assuming your friend is a reasonable, logical, person, he must now believe in the validity of negative numbers.
What have we shown?
- Consistency, both with previous models and with itself
- The ability to solve new problems
- The reduction of some problems in the new language to problems in the old language
Now, to decide if this a good model for a particular system, you must look at the subset of problems that did not have a solution before, and see if the new properties characterize that system. In this case, that's trivial, because the properties of negative numbers are so obvious. In the case of applying more complicated things to describe the details of physical situations, it's less obvious, because the structure of the theory, and the experiments, is not so simple.
How does this apply to string theory? What must we show to convince a reasonable person of its validity? Following the above argument, I claim:
- String theory reproduces (by construction) general relativity
- String theory reproduces (by construction) quantum mechanics (and by the above, quantum field theory)
So string theory is at least as good as the rest of the foundations of physics. Stop again to marvel at how powerful this statement is! Realistically, how many ways are there to consistently and non-trivially write a theory that reduces to GR and QFT? Maybe more than one, but surely not many!
Now the question is--what new do we learn? What additional constraints do we get out of string theory? What problems in GR and QFT can be usefully written as equivalent problems in string theory? What problems can string theory solve that are totally outside of the realm of GR and QFT?
Only the last of these is beyond the reach of current experiments. The "natural" realm where string theory dominates the behavior of an experiment is is at very high energies, or equivalently, very short distances. Simple calculations show that these naive regions are well outside of direct detection by current experiments. (Note that in the above example of negative numbers, the validity of the "theory" strictly in the corresponding realm didn't need to be directly addressed to make a very convincing argument; pause to reflect on why!)
However, theoretical "problems" with the previous theories, such as black hole information loss, can be solved with string theory. Though these can't be experimentally verified, it's very suggestive that they admit the expected solution in addition to reproducing the right theories in the right limits.
There are two major successes of string theory that satisfy the other two requirements.
AdS/CFT allows us to solve purely field theory problems in terms of string theory. In other words, we have solved a problem in the new language that we could already solve in the old language. A bonus here is that it allows us to solve the problem precisely in a domain where the old language was difficult to deal with.
String theory also constrains, and specifies, the spectrum and properties of particles at low energies. In principle (and in toy calculations), it tells us all of the couplings, generations of particles, species of particles, etc. We don't yet know a description in string theory that gives us exactly the Standard Model, but the fact that it does constrain the low-energy phenomenology is a pretty powerful statement.
Really, all that's left to consider to convince a very skeptical reader is that one of the following things is true:
- It is possible for string theory to reproduce the Standard Model (e.g., it admits solutions with the correct gauge groups, chiral fermions, etc.)
- It is not possible for string theory to reproduce the Standard Model (e.g., there is no way to write down chiral theories, it does not admit the correct gauge groups, etc. This is the case in, e.g., Kaluza-Klein models.)
I claim, and it is generally believed (for very good reasons), that the first of these is true. There is no formal, complete, mathematical proof that this is the case, but there is absolutely no hint of anything going wrong, and we can get models very similar to the standard model. Additionally, one can show that all of the basic features of the Standard Model, such as chiral fermions, the right number of generations, etc, are consistent with string theory.
We can also ask, what would it mean if string theory was wrong? Really, this would signal that,
The theory was mathematically inconsistent (there is no reason to believe this)
At a fundamental level, either quantum mechanics or relativity failed in some fairly pathological way, such as a violation of Lorentz invariance, or unitarity. This would indicate that a theory of everything would look radically different than anything written down so far; this is a very precarious claim--consider what would happen in the example of arithmetic in the above if there were something "wrong" with addition.
The theory is consistent, and a generalization of GR and QFT, but is somehow not a generalization in the right "limit" in some sense. This happens in, e.g., Kaluza-Klein theory, where chiral fermions can't be properly written down. In that case, a solution is also suggested by a sufficiently careful analysis (and is one potential way to get to string theory).
Of these three possibilities, the first two are extremely unlikely. The third is more likely, but given that it is known that all the basic features can show up, it would seem very strange if we could almost reproduce what we want, but not quite. This would be like, in the arithmetic example, being able to reproduce all the properties we want, except for 1+ (-1) = 0.
If you're careful, you can phrase my argument in a more formal way, in terms of what it precisely means to have a consistent generalization, in the sense of formal symbolic logic, if you like, and see what must "fail" in order for the contrapositive of the argument to be true. (That is, (stuff) => strings are true, so ~strings => ~(stuff), and then unpack the possibilities for what ~(stuff) could mean in terms of its components!)
This post imported from StackExchange Physics at 2014-03-30 15:55 (UCT), posted by SE-user Mr X
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