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# A confusing answer to a simple question: What is quantum theory about?

Posted Mar28-12 at 07:05 AM by Demystifier

Quantum theory is called "quantum" because it is supposed to be about quanta - discrete spectrum of certain quantities which in classical theory have a continuous spectrum. But actually, quantum theory does not need to be about quanta at all, because many quantum quantities actually have a continuous spectrum.

If you think it's confusing, then you've seen nothing. Quantum mechanics is (or at least used to be) about Schrodinger equation, which is a quantum theory of electron. But Schrodinger equation is not really correct, because Schrodinger equation is not relativistic and does not describe the electron spin. Actually, Schrodinger derived this nonrelativistic equation from his relativistic equation, but relativistic Schrodinger equation of electron is not really called Schrodinger equation (but Klein-Gordon equation), does not really describe electron either (because it does not include spin) and is not really consistent (because it does not conserve probability). The consistent relativistic electron equation is Dirac equation, which includes spin and conserves probability. But even this equation is not fully correct, because one needs to second quantize it, after which it turns out that conservation of probability is no longer conservation of probability but conservation of charge. And similarly, it turns out that the Klein-Gordon equation also needs to be second quantized, so that its non-conservation of probability becomes irrelevant for a similar reason as conservation of probability for the Dirac equation. Thus the Dirac equation is not important because it conserves probability, but because it derives spin from linearization of the Klein-Gordon equation. But to derive spin you don't really need to linearize Klein-Gordon equation, because you can also get spin from linearization of the non-relativistic Schrodinger equation (which is simply called Schrodinger equation, and, by the way, also needs to be second quantized), leading to the Pauli equation who actually did not obtain that equation by linearizing the Schrodinger equation. However, the derivation of spin from linearization is not really a true derivation of spin, because the true derivation of spin comes from irreducible representations of the rotation group. More precisely, not of the proper rotation group SO(3), but of its simply connected covering group SU(2).

If you think that's it, you are deeply wrong. Actually it is misleading to say that all these equations above need to be second quantized, because there is only one quantization, but applied to different degrees of freedom. So what we called second quantization, and was really second quantization of particles, is actually first quantization of fields. So fundamental objects are fields, not particles. Or maybe not, because we measure particles, not fields. But not always, because sometimes we really measure fields. Actually only bosonic fields, because fermionic fields cannot be measured even in principle.

Can we at least say that quantum theory is not really about Schrodinger equation? No, because even relativistic quantum field theory, including spin and everything else we seem to need, still can be represented by a Schrodinger equation. But this Schrodinger equation is actually a generalized functional Schrodinger equation, and was not discovered by Schrodinger. But still, not all quantum field theories can be described by such a generalized Schrodinger equation, because quantum gravity is an exception, requiring Wheeler-DeWitt equation instead of the generalized Schrodinger equation. Wheeler-DeWitt equation is a generalized Klein-Gordon equation (but nobody calls it so). Yet, while Klein-Gordon equation is manifestly relativistic covariant, Wheeler-DeWitt equation is not, even though it is still relativistic. Actually, you don't really need Wheeler-DeWitt equation to do quantum gravity; there is also a manifestly relativistic-covariant way to do it. But that relativistic covariant way is not general-relativistic covariant, which a quantum theory of gravity should be.

Further complication comes from use of quantum field theory in condensed-matter physics, where it suggests that fields are not fundamental at all, not even bosonic ones, because the field description is appropriate only at large distances. Particles are more fundamental in condensed-matter physics, so quantum field theory is better called second quantization in condensed-matter community. But the second-quantized particles in condensed-matter physics are actually pseudo-particles (e.g., phonons), not the fundamental particles. In such an approach to condensed-matter physics, the fundamental particles are atoms and molecules, they are not quantized (despite the fact that they are actually quantum objects), and we know that they are not fundamental at all, because they consist of fundamental quarks and electrons, which are fundamentally described by another quantum field theory, which, by being fermionic field theory, describes fermionic fields which cannot be measured even in principle.

Now you might think that we really need a more fundamental theory of everything to clean up all that mess. So what our best candidate for the theory of everything - string theory - has to say about all this? First, it says that particles are not really particles but little strings. At first sight it does not change much, but at a second one it changes a lot, and at a third one it changes even more. One surprising result is that first quantization of strings is enough, so one does not need second quantization of strings, called also string-field theory. Actually string-field theory exists as well, but it is not consistent, and almost nobody uses it. But still, first-quantized theory of strings is also a quantum field theory - more precisely conformal quantum field theory in 2 dimensions. But it does not mean that strings live in 2 dimensions, because they really live in 10 dimensions. But the 2-dimensional strings (actually 1-dimensional if we don't count time) that live in 10 dimensions are not the end of the story, because the theory contains also branes - objects having more than 2 stringy dimensions. These branes are not really fundamental, because they are only special configurations of classical 10-dimensional fields, while these 10-dimensional fields themselves are not fundamental. Despite of being non-fundamental, these classical 10-dimensional fields are actually quantum 10-dimensional fields, but nobody knows how to quantize them because, as I said, string-field theory is not consistent (and not even needed).

I said that branes are not fundamental, but actually they are. More precisely, they are not fundamental in perturbative string theory, but they are fundamental in non-perturbative string theory. Therefore non-perturbative string theory is not really a string theory, but it is still called string theory. This non-perturbative string theory is more fundamental than perturbative string theory, but there is one little technical problem with it: nobody knows what this non-perturbative string theory is. At least we have a cool name for it - M-theory, but nobody knows what even "M" stands for. (Some good candidates are Membrane-theory, Matrix-theory, Mystery-theory and Witten-theory (where W is reversed M).) Whatever that M-theory might be, at least we know that in this theory strings and branes are equally fundamental (or better to say, equally non-fundamental) and that the theory lives in at least 11 dimensions, which may be actually 12 dimensions, or perhaps the number of dimensions is not fundamental at all.
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4.  As to what quantum theory is about, I would have said it 's about choosing an algebra of observables L and then constructing a mapping from L to scalars which admits a probability interpretation. E.g., if A is an element of L, then the mapping is denoted by $$\def\<{\langle} \def\>{\rangle} A ~\to~ \$$ The distinction between classical and quantum then comes from whether or not things like ##\## factorize into ##\\##. Then comes tensor products of this basic construction... :-) Posted Mar30-12 at 10:47 PM by strangerep Updated Mar30-12 at 10:53 PM by strangerep