# Can a free String Field Theory be described in terms of a “wavefunctional” of classical string field configurations?

A free scalar QFT can be understood as a wavefunctional that maps classical field configurations to complex numbers representing amplitudes.  An eigenstate of this basis is a classical field configuration that assigns a specific real number to each point in spacetime $$n$$.  Each of these points can then be thought of as a local oscillator, and the global field as an array of oscillators.  One can calculate the expectation value of the displacement or field strength $$q(n)$$ of such local oscillators in arbitrary global states (which for practical purposes are never actually eigenstates of this basis).

For example, when one thinks of a quasi-localized one particle state in a free QFT (see diagram), the wavefunctional of such a state is one in which the oscillators at and near the localization point have higher expectation values for their displacement relative to other sites.

In this sense it is apparent how and why the underlying degrees of freedom are fields and how particle states, coherent states, etc., are particular superpositions of configurations of fields.

Diagram taken from Linde, Helmut: "A New Way of Visualizing Quantum Fields", (2019), https://arxiv.org/abs/1907.11311

Does this reasoning generalize to a free scalar String Field Theory?

In furtherance of this, I am imagining a system where each of the local oscillators  - each $$n$$ in the diagram - is replaced by a space of all unique classical 1D string "shapes" or "curves" $$\sigma_i$$ which pass through $$n$$. Perhaps these are the curves with n as its midpoint, as its center of mass, or as some arbitrary point, given some fixed choice of coordinates along the string.

Then, in an eigenstate of the string field wavefunctional, one assigns a real number to each such $$\sigma_i$$ at each point $$n$$.  So classical string field values $$q(n[\sigma_i])$$ are assigned to each $$n[\sigma_i]$$

Note a field configuration is now an infinite set of values at each point $$n$$, one for each $$\sigma_i$$, and not merely a single value at each $$n$$.  In a general state of the string field, which is a superposition of these classical configurations, one has an expectation value for each $$q(n[\sigma_i])$$.  The underlying system is then like an array of "string oscillators" which are displaced about their origin.

I expect this procedure would have to be duplicated for each internal mode of the quantized string itself, treated as a separate field/array.  A point $$n$$ could be indexed to a reference background spacetime or to a brane worldvolume.

Is a straightforward analogy like this sensible?  Or are there string-theoretic reasons why in the generalization to strings, only a Fock representation is a viable form of second quantization?

I'm not expecting that this is in any way a practical or useful approach to string theory, just whether it is a faithful or misleading way to think of the general idea of a string field theory.

String field theory is vastly much more sophisticated than that.

Let me mention a few points to illustrate why the way you propose to visualize string fields theory is not a good one.

1. Take the example of closed bosonic string field theory. If you look at the highly complicated action you can learn that a string field has an infinite amount of "excitations" and those components have typically non zero ghost number. Not to mention that every excitation of a string field is an ordinary field. What is a "classical superposition of fields" whith excitations that are fields themselves and whose components have non-zero ghost numbers?

2. The rules of string theory are still valid when the underlying spacetime is noncommutative or when the worldsheet theory sits at a Gepner point or even when the notion of target spacetime topology is meaningless. What does a "classical superposition of string fields" mean when the spacetime is fundamentally noncommutative or non-geometric or lacks sensible notion of spacetime topology?

The best I can do is to recommend some good divulgative texts in the beautiful subject of string field theory (and related string issues).

String Field Theory

How and why strings generalize geometry

• Thanks for this answer. So it seems like in String Field theory one cannot identify analogs of the local field operators of QFT. How then does a QFT state appear as the low energy/long distance limit of a String Field Theory? Is it only through identifying states in the respective Fock representations? – Felix Wotter Jun 6 '20 at 0:42
• It was a real pleasure. How String Field theory reduces to ordinary QFTs? String fields create and destroy strings, first quantized string theory is the limit of SFT that assumes that a fixed number of strings were created once and for all. Then one study the spectrum of states of a string and discover that some string states are in one to one correspondence with the ordinary fields of ordinary QFT. – Ramiro Hum-Sah Jun 7 '20 at 4:08
• but in which representation of QFT can we see this 1 to 1 correspondence? It is established in (free, flat) QFT that the Fock and wavefunctional reps are unitarily equivalent. But you seem to be saying a wavefunctional SFT is not sensible and "String fields create and destroy strings" sounds like you are speaking of SFT in Fock terms. So, when we consider quantum gravity, are Fock reps privileged (perhaps due to the dynamical geometry)? This would be a fascinating turn, given the consensus in curved QFT is to downplay Fock (because of no preferred vacuum) in favor of local fields. – Felix Wotter Jun 7 '20 at 4:59
• This answer refutes itself. It states "the way you propose to visualize string field theory is not a good one," then cites articles that do just that. OP didn't ask "is string field theory as unsophisticated as QFT?" He asked if string theory had any functional or field configuration forms, reminiscent of QFT. The answer is "Yes in this sense." – alexchandel Jul 2 '20 at 19:55
• No, you are wrong. Taylor's paper does not write something reminiscent of a Monte-Carlo distribution for string field amplitudes. My answer emphasize how different is SFT from QFT because that answers the actual question. 1) The most basic questions: How does that Monte-Carlo simulation describe the creation of a string? That's basically what a string field does. Do you think plausible to simulate the metric spacetime fluctuations with a simple model of harmonic oscillators? (eq. 1.2). 2) String fields are not a system of coupled harmonic oscillators (requirement for Monte-Carlo visualization – Ramiro Hum-Sah Jul 3 '20 at 2:07