Is String Theory a Field or Quantum Mechanical Theory of the String rather than a Particle?
I should know this having studied this for a term, but we jumped into the deep end, without really covering the basics of the theory.
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1$\begingroup$ Yes, it is a field theory of a non-point-particle. $\endgroup$ – Chris Gerig Jun 13 '12 at 6:30
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$\begingroup$ From what I gather in en.wikipedia.org/wiki/… , string theory includes/explains qft . $\endgroup$ – anna v Jun 13 '12 at 8:20
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$\begingroup$ It is not a field theory--- it does not have local fields at space-time points. $\endgroup$ – Ron Maimon Jun 14 '12 at 0:47
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$\begingroup$ What do you call string fields? $\endgroup$ – Ernesto Ulloa Jun 19 '12 at 17:55
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1$\begingroup$ @ErnestoUlloa: String fields are nonlocal, they are not defined at space-time points. $\endgroup$ – Ron Maimon Jun 19 '12 at 21:01
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According to the definition, a field assigns a value (classically; or a distribution quantum mechanically) to every point in the space(-time). So field theory deals with point-like excitations in the space(-time). String theory, thus, is not quite a field theory, since it's excitations are defined on extended objects. To better understand the difference, I would look at http://en.wikipedia.org/wiki/String_field_theory. Also, another important difference to notice is that people consider a few fundamental particles interacting with each other when they do qft; however, in string theory there are an infinite number of fundamental excitations in the theory, leading to an infinite tower of fundamental particles.
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$\begingroup$ I deleted some inappropriate comments here. Those involved are free to continue the discussion in Physics Chat, but not here. $\endgroup$ – David Z♦ Jun 20 '12 at 21:13
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String theory isn't a quantum field theory. See What is the stress-energy distribution of a string in target space? and Statistics and macrolocality in string theory. See Do we need a quantum deformation of the diffeomorphism group in string theory? for a contrary opinion.
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In string field theory a string field creates string excitations from the vacuum that interact. Interactions are treated using perturbation theory. The theory uses string vertex operators and string propagators. SFT is definitely a quantum field theory, but not a point particle QFT. It is used mostly in the study of unstable branes, topological string theories and non-commutative geometry. The principal versions of SFT are Light-Cone SFT, Covariant BRST SFT & Witten’s SFT. In principle string theory should be formulated as a quantum field theory of strings, but due to technical reasons related to the difficulties inherent to the above string field theory formulations, or simply by the incomplete knowledge of the underlying theory, that most calculations in the string theory literature are done in the context of first quantized formulation or in low energy effective classical actions. So it can be said that string theory & M-theory are, in principle, quantum field theories for extended objects, even if calculations are not generally done in this formalism
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$\begingroup$ It is not a field theory, because the string field is nonlocal. The formulation is also problematic. $\endgroup$ – Ron Maimon Jun 19 '12 at 21:02
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$\begingroup$ Ron, again you are confused, because the question was not if string theory was local or not, it was if string theory is a quantum field theory or not. The original question was not about locality. $\endgroup$ – Ernesto Ulloa Jun 20 '12 at 12:42
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$\begingroup$ Ron, maybe you we have to think about the term "Field" in a more general term that what mathematicians call field. Locality is a concept that is rooted to the very postulates that define most of our concepts of real analysis. A mathematician will almost always define a field as a local function of space. There is not a calculus based in functions that are non local functions of x,y,z. You have to extend your notion of what is a field. My earlier comment was a little bit rude. I apologize. $\endgroup$ – Ernesto Ulloa Jun 20 '12 at 12:50
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$\begingroup$ You never have to apologize to me for rudeness! The term "field" requires that there are operators which commute which attach to each space-time point. This microcausality is the absolute irreducible minimum requirement for calling something a field theory. The string fields do not make microcausal anything, and the string theory does not have local fields. It's S-matrix theory, not field theory. $\endgroup$ – Ron Maimon Jun 20 '12 at 16:58
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1$\begingroup$ I am not confusing anything. Not every theory with fields is a QFT. If your fields live in loop space (the space of all curves on a surface) and not on the space itself, then they aren't attached to points. The "divergences" is important--- a field theory has independent degrees of freedom at every point and this is what gives rise to the divergences. It is also why string theory can't be formulated using local fields, only using loop fields. Loop fields are not local fields, and this is important to say, so that no string field theory is a field theory proper. I think we are arguing semantics $\endgroup$ – Ron Maimon Jun 20 '12 at 17:48
