# If Particles are simply excitations in a field, how does string theory change this description

I am a computer scientist with a passing love of Quantum Physics, but perhaps because I am a lefty I prefer to visualize when possible what the quantum world looks like. So my question here is I read everywhere that thinking of photons/Electrons as particles is simply a poor view, they are excitations in their fields. Does String Theory change this view to there being actual 1-dimensional objects existing, or are strings 1-d excitations in a field. I am asking this to someone who accepts String Theory. No math, paint a picture.

We usually distinguish between first-quantized and second-quantized descriptions.

The first-quantized picture treats particles as well.. particles, while the second-quantized description treats them as excitations of fields.

Disclaimer: because OP asks to "draw a picture" without math, many things are clumsy about this answer. One example being the apparent failure to distinguish virtual and real particles. I do know that these are different, and I do know how they are different. Please keep in mind that I am trying to "draw a picture".

### First-quantized description

In first-quantized description, particles are ad hoc. We start by describing a single (relativistic) particle. We discover that we can add additional properties (massive/massless, spin, charge) to the particle which influence the equations that we use to describe the particle.

Later we discover that our theory is inadequate as it is unable to describe interactions between different elementary particles. E.g. we know that electrons interact by means of exchanging a photon. We can add interactions by hand by allowing particle worldlines to meet in the interaction nodes.

By combining ad hoc expressions for the interactions in the nodes and the theories of quantum relativistic particles, I can arrive at the Feynman diagrams, and reproduce the measurable predictions derived from these diagrams (cross sections, decay rates, etc).

### Second-quantized description

The first-quantized description has a major disadvantage: it is not a quantum-mechanical theory.

Indeed, any quantum-mechanical theory has to have a Hilbert space and self-adjoint operators acting on the Hilbert space. Instead, we have a Hilbert space attached to any quantum particle represented by the edge in the graph.

We could solve this by requiring that the overall Hilbert space of the graph is given by the tensor product of its edges. But this only gives a Hilbert space for any graph, not one Hilbert space to rule them all.

The second-quantized description solves this problem. Basically, we re-interpret the quantum-mechanical wave equations for wavefunctions of different types of particles as classical field equations. Then we quantize the field. Actually, the name second quantization comes from this peculiar feature: it seems like we are quantizing an already quantized theory of the particle. But actually, we are quantizing the theory of the field, and only once.

Particles arise as excitations of the field. The field can have multiple excitations corresponding to multiple particles. Moreover, the field can be in a superposition of states with different numbers of particles (thus, the total number of particles in the field is, just like every observable in quantum mechanics, fuzzy or undetermined).

Feynman graphs arise as terms in the perturbative series for quantum transition amplitudes between different states of fields (the IN- state and the OUT-state, which are given by superpositions of configurations of particles).

Another major advantage of this approach is that we have much less freedom in choosing possible interactions. These interactions are strongly suppressed by the requirements of Lorentz invariance, gauge invariance, renormalizability and unitarity.

Consequently, a particular model (Standard model) of interacting quantum fields has been found, corresponding to an incredibly accurate description of the real world.

### The deal with strings

Strings are first-quantized. String theory has initially been formulated in a first-quantized fashion, and there are reasons for that. Here goes.

First, when we draw a string's path through spacetime, the resulting figure is a surface called the worldsheet of the string (opposed to a curve called the particle's worldline). You can do a lot more with surfaces than you ever could with curves.

As an example, consider a Feynman diagram and its stringy analogue:

You can observe that the second picture doesn't have any "special points" in the interaction nodes. Interactions are made of... strings, just like strings themselves. In a theory of particles, we have to explicitly give expressions for interaction nodes. In string theory, these are given by the theory itself. String theory is already an interacting theory.

Also, a string can be interpreted as a particle with mass, spin and charge. Thus, strings already model different types of particles which we encounter in the first-quantized (and second-quantized) description for particles.

As for Hilbert spaces, string theory in the form described above is not a quantum-mechanical theory. It uses quantum mechanics heavily, as its mathematical description, the conformal field theory (CFT) on the string worldsheet, is a quantum-mechanical theory. But physical predictions are obtained in another way.

UPDATE: this claim of mine caused some confusion in the comments. @MeerAshwinkumar claims that there's a well-defined Hilbert space ${\cal H}$ of the string given by the cohomology of the BRST operator, and he is absolutely right. But here's what I meant: this Hilbert space does not describe the string as a quantum object, but rather its fluctuations (different modes). The string is given: it is classical. There is no state in ${\cal H}$ that corresponds to the superposition of "there is a string" and "there isn't a string". There's always a string, different states in ${\cal H}$ only determine the position of its center of mass and vibration modes.

### Conclusion

Strings are not excitations of something, because we use the first-quantized approach to handle them. There is compelling evidence that this approach is much more adequate for strings than it is for particles (we aren't required to specify interaction nodes, strings explain particle properties, etc).

### Second-quantized strings?

There are several approaches to a nonperturbative, "second-quantized" description for strings:

1. String field theory

As far as I know, these are still being investigated heavily.

• Ok, I spend 15 minutes of my time to write this detailed answer, and I receive a downvote with no explanation. Rage downvote? I'd like to see a reason behind it. – Prof. Legolasov Jan 14 '17 at 7:54
• @SolendonParadoxus Voted down, because the answer is very long but incorrect. There is a String Field Theory, which answers the question, and it is mentioned only under the question mark in the very end of the post. – Andrey Feldman Jan 14 '17 at 7:55
• @AndrewFeldman 1. My answer being long is not necessarily bad. It might mean that I put a lot more details in it than you had. 2. What is incorrect about it? SFT is a research topic, it is clear that OP wants to know about the common formulation of string theory. – Prof. Legolasov Jan 14 '17 at 7:57
• @MeerAshwinkumar I think Solonedon means that the strings themselves are not "quantum objects" in this first-quantized theory, just like the notion of "particle" in non-QFT QM doesn't really exist - we're looking at states of particles, but "a particle" doesn't really exist on that level. – ACuriousMind Jan 14 '17 at 15:44
• @MeerAshwinkumar please see my update. – Prof. Legolasov Jan 15 '17 at 7:12

Let me slightly extend my post in order to clarify the answer, which may be necessary given a totally wrong answer by Solenodon Paradoxus.

There are two perturbatively equivalent formulations of particle physics -- first-quantized and second-quantized. The latter usually called QFT. Non-uniqueness of vertex operators inserted in the nodes of the first-quantized theory graphs is dual to non-uniqueness of the interaction term in the Lagrangian of QFT, they are not more arbitrary or "ad hoc". This story is infinitely long, so let me just cite a few references. Rewriting of QFT amplitudes in the first-quantuzed language is given in great details in Field Theory Without Feynman Diagrams: One-Loop Effective Actions by Strassler. The close similarity to stringy case is obvious and mentioned by the author. Analogous possibility is spelled in Section VIII.C.5 in Fields by Siegel for the case of Yang-Mills theory. Very clear and rigorous discussion of the topic and comparison to stringy case can be found in the beginning of D'Hoker lectures in Quantum Fields and Strings: A course for Mathematicians. Volume 2. One should notice that there are also non-perturbative objects in QFT, which cannot be directly seen in the first-quantized theory -- instantons, monopoles, skyrmions, etc. The situation in String Theory is analogous -- D-branes aren't seen in the first-quantized perturbation theory.

String Theory can also be formulated in the second-quantized language, as well as in the first-quantized. The second-quantized theory, named String Field Theory, describes strings as excitations of a unified string field in which all the fields describing particles are packed. The field formulation of String Theory can be shown to be equivalent to the worldsheet (first-quantized) formulation in analogy to the theory of point particles. However, such a formulation appears to be very complicated and hard to deal with, so almost always the first-quantized formulation is used. Moreover, such a fundamental and fruitful phenomenon of String Theory as dualities is very hard to see (if possible at all) in a second-quantized formalism. Of course, the worldsheet formulation is a genuine quantum mechanical theory, namely 2D Conformal Field Theory. The topic is beautifully presented in, say, Polchinski's "String Theory".

If you are interested in String Field Theory, check this extensive review. There is also a list of recommended literature here.

• Can you clarify what you mean by "two perturbatively equivalent formulations of particle physics"? I'm asking because its not how I've seen it presented. – Mozibur Ullah Jan 15 '17 at 7:39
• Really? QFT is usually presented perturbatively, QM isn't; it's usually presented axiomatically; I don't quite see how a single particle theory can be equivalent to a many particle theory. – Mozibur Ullah Jan 15 '17 at 7:46
• No, I'm afraid that's not really relevant when the basics aren't in order. – Mozibur Ullah Jan 15 '17 at 7:52
• I don't think that you know what you're talking about; so why bother reading references by someone who doesn't know what he's talking about? What do you ** think**? – Mozibur Ullah Jan 15 '17 at 7:55
• Personally, I'm concerned why you're trying to present yourself as a professionally qualified physicist, when you're not; please stop it, faking jargon is not helpful. – Mozibur Ullah Jan 15 '17 at 8:02