# String behavior in electron orbitals?

It feels like this should be a incredibly easy question to find an answer to, but I’m coming up zeroes in my search - if I’ve overlooked a parallel question please feel free to close this.

Conceptually, the idea of atomic electrons existing as delocalized standing waves fully described by the orbital seems like a good fit (From Electron as a standing wave and its stability)

Certainly it feels more intuitively satisfying than trying to imagine a physically real point particle existing in a cloud of probability space, or a billiard ball being guided by some non-physical pilot wave. Ultimately if they give the same predictions and are indistinguishable in the laboratory, I get that this falls more under philosophy than physics and the first answer is likely don’t worry about it, just calculate.

Here’s my problem: Strings appear to be treated as very real, very physical things. Whereas I had years of being told to stop thinking about particle spin as a tiny sphere actually spinning.... well, in string theory it sounds like we’re waving our hands and saying “OK, OK, so there’s actually something physically spinning there after all. Sorry.” See: https://motls.blogspot.com/2014/08/do-stringy-electrons-spin-faster-than.html

My question: What is the behavior of strings in electron orbitals like?

Do they spread out into standing wave closed loops, occupying the entire orbital in a physically real way? Are we to abandon the idea of standing waves in fields and in stead think of tiny, physically real open strings zipping around orbital space as they physically rotate as well?

The conceptual grinding of the gears I’m having here is that it seems like string theory is a gigantic step backward, whereas atomic behavior seemed so intuitively elegant in field descriptions, and now we’re essentially back to billiard balls.

If someone could explain whatever the common understanding of string behavior in atomic orbitals is I’d appreciate it.

• The map between the particle states and the string is far less concrete than you seem to think, see e.g. my answer here – ACuriousMind Nov 24 '18 at 22:39
• ACuriousMind That helps a bit... but accepting that not all vibrational modes map to physically real observables doesn’t appear to help with the very real orbitals picture; if a physical rotation of a string is invoked for spin, I still have to ask what the standing waves of field theory map to in the stringy picture. – JPattarini Nov 25 '18 at 0:09
• The AdS/CFT correspondence seems to provide a non-perturbative definition of string theory (arxiv.org/abs/gr-qc/0602037, page 16), at least for asymptotically-AdS spacetime. This definition doesn't refer to strings at all. So maybe "strings" are to string theory like "virtual particles" are to QFT (which seems at least roughly consistent with the post that @ACuriousMind cited). This doesn't answer the question, but maybe it helps manage expectations. – Chiral Anomaly Nov 25 '18 at 0:27
• No one is invoking the "physical rotation" of a string for spin (unless you give a reference for that). The orbitals in an atom are not "standing waves of a field". In quantum field theory, the field is the operator, not the state, You seem to have taken all too literally some popularizations of quantum mechanics. It's hard to tell how to answer this question since it seems to rely on already questionable assumptions not mentioned in it. – ACuriousMind Nov 25 '18 at 0:46
• ACuriousMind I was basing the spin interpretation on Lubos’ blog post here motls.blogspot.com/2014/08/… – JPattarini Nov 25 '18 at 1:01

It seems we need to review how atoms are described in three kinds of quantum theory: nonrelativistic quantum mechanics, relativistic quantum field theory, and string theory.

These are all quantum theories, so they all utilize some kind of quantum framework, in which there are "observables" that can take possible values, with probabilities derived from wavefunctions, quantum states, path integrals, etc.

It's no secret that there is a lot of angst and contention regarding the right way to think about quantum objects. The most neutral yet accurate way to talk about it, would be to restrict oneself to statements about the observables - what they are, what their possible values are, and how the probabilities vary.

But if I am required to offer a word-picture, I would describe a quantum object as being subject to the uncertainty principle. This concept is most familiar when applied to the position and momentum of a point particle, but it also applies to other conjugate variables, like the position and velocity of a point on a moving string, or the amplitude and rate of change of a point in a field.

I'm not saying that this is the ultimate way to view reality, just that it is an accurate qualitative way to think about the meaning of quantum mechanics. (I am assuming that you understand the basics of quantum mechanics, like the Born rule and how operators represent observables.) We can attempt to talk about other conceptions, but if you go too far beyond this, you're going beyond quantum mechanics per se, to some other set of ideas.

Electron orbitals are a concept from the nonrelativistic quantum mechanics of electrons in atoms and molecules. They are elements of a (possibly many-body) wavefunction that obeys a Schrodinger equation. This wavefunction describes one or more electrons, which my qualitative word-picture describes as point particles subject to an uncertainty principle. That's what an orbital does, it gives you probabilities regarding the observable properties of a point particle.

In quantum field theory, we now have particle observables (like particle number) and field observables (like field intensity). According to my qualitative formula, the fundamental objects here are fields, boson fields and fermion fields. However, when the uncertainty principle is applied to a field, one obtains quanta of energy which behave like the quantum particles of nonrelativistic quantum mechanics.

If you can understand the energy levels of a quantum harmonic oscillator, you may understand how this works for boson fields. One considers the Fourier modes of the boson field (like the plane waves of Maxwell's classical equations) as independent harmonic oscillators. Apply the uncertainty principle to each one, and you will find that each field mode can contain zero, one, two... "quanta of energy", which behave like quantum particles in a pure momentum state. By superposing these, one can then build up any desired collection of localized wavefunctions, thus imitating a state from n-particle quantum mechanics. However, in theory it can all be interpreted as a superposition of states of the field.

Fermion fields will be more of a pain to convey. Each bosonic oscillator has an infinite number of energy levels. But a "fermionic oscillator" must only have two, in order to implement the exclusion principle. If we are to think of the fermion field as a field, i.e. an entity which has a value at each point in space, we need to think of its values as "Grassmann numbers", a special kind of "number" with unusual algebraic properties. This concept is employed in fermionic path integrals. So if you can swallow the notion of a Grassmann-valued field, then we can extend the qualitative word-picture and say that fermions too are quantum particles arising from a field subject to the uncertainty principle.

But a lot of physicists would say that Grassmann numbers aren't numbers, they're just formal objects, and they would focus just on the observables, and the algebraic properties of the relevant operators, like their commutation relations. For their physical intuition, they must use some other approach when thinking about fermions.

Now we are talking about quantum electrodynamics (QED). We have a boson field (for photons) and fermion fields (for the electron, and perhaps for the nucleons if we don't treat them classically). Here another complication rears its head, which is that bound states like atoms are not treated very well in a relativistic quantum field theory like QED. We don't have the simplicity of the nonrelativistic framework, in which we have a wavefunction evolving according to a universal time.

Instead, the fundamental object of relativistic quantum field theory is the S-matrix, the scattering matrix, describing transition probabilities between asymptotic states - the probability to go from a state in the infinite past, to another state in the infinite future. One usually imagines a scattering process in which particles start far apart, come close and interact, and then the products of the interaction move apart again. Rather than the Schrodinger equation, the fundamental method of calculation here is Heisenberg's operator picture or Feynman's sum over histories.

Anyway, since I am tiring of exposition, I will just say a few more things. It is a little complicated to represent bound states - like atoms - in the S-matrix framework. Because particle number can vary, and because there isn't a fixed universal time coordinate, they need to be "built" out of Feynman diagrams or other QFT constructs somehow. The basic one is the Bethe-Salpeter equation and it has been applied to very simple atoms.

In your question you say that orbitals are "standing waves in fields" but that's not really true. In the nonrelativistic picture, the orbital is a standing wave in a wavefunction, and a single-particle wavefunction is like a field in that it has a value at each point in space. But it's not a field. It's a "probability amplitude wave", and it's probably just a part of an entangled many-body wavefunction anyway.

Meanwhile, when we get to the actual field model of orbitals, in quantum field theory, it's going to be some complicated superposition of quantum field histories, in which the boson field quanta approximate the Coulomb potential of the nucleus, and the fermion field quanta approximate the orbitals.

And only now do we arrive at string theory. In the case of quantum field theory, Feynman diagrams offer a framework for calculation which looks like a sum over particle histories; you don't even need to concern yourself with the field picture. (Except that you do need the full framework of fields, for "nonperturbative" phenomena.) In the case of string theory, we have something like the Feynman diagrams, the topological diagrams which show strings splitting and joining; but the fundamental theory, analogous to the field picture of quantum field theory, is still quite obscure. There is a thing called "string field theory", and it has its uses, but hardly anyone would think that that is the fundamental formulation of string theory.

String theory, like quantum field theory, is an S-matrix theory. And I think the understanding of bound states is even more primitive in string theory, than in quantum field theory, partly because of mathematical complexity, partly because of the lack of an independent spacetime background that can anchor one's search for bound states within the S-matrix.

So, while ultimately strings are probably just a kind of excitation of some fundamental geometry or "pregeometry", ironically, the clearest picture we have of string theory is still the perturbative picture, analogous to the Feynman diagrams of quantum field theory, the picture in which what the theory is about, is vibrating interacting strings, strings that can split and join, and which are subject to an uncertainty principle.

This means, first of all, that if you want to understand an electron orbital in terms of string theory, you should first understand it in terms of a point particle, and then just imagine that the particle is actually a very tiny string. Of course, this is hardly any change at all.

You ask specifically about spin. Lubos presents an analysis for a closed bosonic string, in terms of the waves that move (let us say) clockwise and counterclockwise (usually one talks of right movers and left movers). This is indeed how the quantum states of a string are built up. However, for the specific case of fermions, like electrons, you need to have a Grassmann field on the string in order to get half-integer spin. For a purely bosonic string, the only property that a point on the string has is its position vector (and its velocity vector, if we want to consider that as well). The position vectors of the points on the string behave like bosonic fields. But each point on a superstring has set of Grassmann coordinates as well, so that it can produce fermionic states.

So for a genuinely fermionic object, Lubos's analysis is only suggestive. The string in question will be spatially extended, so some of the analysis will carry over. But the fermionic variables on the string must also contribute to the calculation of the angular momentum, through some kind of Grassmann integral, but I'd have to resort to a string theory textbook to figure out the details.

And this is where I'll stop. The question of how string theory would describe electron orbitals is actually of some interest, but I believe the key issues are quite other than those in your question, e.g. how to adapt the Bethe-Salpeter equation to string theory. (Another interesting but technical topic, interesting for me, is how massive Dirac fermions like electrons, produced by yukawa interactions with the Higgs, are built up from the stringy Weyl fermions. But one would first try to answer the Bethe-Salpeter question for a more straightforward case.)

As I've said, I think you are mixing up wavefunctions and fields. In the end, yes, particle wavefunctions are obtained from quantum fields, but you seem to be thinking of wavefunctions as like classical fields, because you can calculate a charge density from the wavefunction. You know, in the vicinity of an atomic nucleus, it may actually be possible to decompose the electron fermion field into modes that correspond to the orbitals. So there may be a sense in which an occupied orbital is a "standing wave in a field". But it would be a quantum excitation of a Grassmann-valued field, and not a standing wave in a charged classical field.

• Can't thank you enough for taking the time to write this. It's a great balance of the Sisyphean effort we non-mathematicians often ask for in this forum, which is to take that which is fully described by mathematics and deform it into the written word without completely losing the essence in the process. It also provides some great hooks for further reading. My sincere thanks. – JPattarini Dec 3 '18 at 13:19
• It was quite some work figuring out how to answer the question, but we got there... You could structure a string theory textbook around the theme of "getting to atoms". – Mitchell Porter Dec 4 '18 at 23:19