Reconciling QM with string theory

[Sophrosyne]

I am trying to read about and understand string theory. But in trying to understand how it reconciles with the world of quantum field theory and quantum mechanics, I am getting a little confused. How does the string move through and propagate through the quantum field?

Does string theory, for example, propose that there is only one field, the String field, in which the strings vibrate? And the different particles are just different vibrations of the string of this fundamental String field?

 

[Demystifier]

That's a very good question, the explicit answer to which is not easy to find in the string literature.

The first thing one needs to know is that "string theory" is not one theory. There are at least 3 versions of the theory, namely
1) perturbative string theory
2) string field theory
3) M theory
Each of these versions provides a different answer to your question.

1) Perturbative string theory is the best understood version of string theory. According to this theory, there are no fields at the fundamental level. There are only strings, which scatter according to the "first quantization" rules similar to that in the book Bjorken and Drell "1", Relativistic Quantum Mechanics. But once you calculate the scattering amplitude and make a low energy limit, you observe that the same scattering amplitude can be obtained by a certain field theory. This field theory is only an effective theory valid at low energies, not a fundamental theory. Fields are emergent in perturbative string theory, not fundamental. The problem with perturbative string theory is that it is almost certainly incomplete. Therefore any general conclusions such as the ones I've just made should not be taken too seriously.

2) String field theory is an attempt to formulate string theory as a "second quantized" theory, similar to that in the book Bjorken and Drell "2", Relativistic Quantum Fields. According to this theory, fundamental objects are certain string fields. If ordinary field is thought of as a function of the form $\Phi({\bf x},t)$, the string field is a functional of the form $\Phi[{\bf X}(\sigma);t)$ where $\sigma$ is a real parameter along the string. At large distances one can use an approximation ${\bf X}(\sigma)\approx {\bf x}$, which is why at low energies one can approximate string field theory with ordinary field theory. String field theory is well understood for bosonic strings, but this theory does not describe fermions so is not realistic. Unfortunately, string field theory is not well understood for more realistic superstrings. Most string theorists believe that string field theory is not the correct approach to describe string theory at the fundamental level.

3) M theory is widely believed to be the most fundamental formulation of string theory. In principle, it should give the best answer to your question. Unfortunately, M theory is not very well understood. It is not even known what the fundamental objects in this theory are supposed to be. There is even no agreement what the "M" stands for. Therefore, at this moment, M theory does not provide a clear unambiguous answer to your question.

 

[Urs Schreiber]

The royal road to understanding how perturbative string theory is related to perturbative quantum field theory is to first understand the latter in the worldline formalism. This makes the idea of perturbative string theory completely clear, and in fact intellectually compelling:

 

[John G]

https://en.wikipedia.org/wiki/D-brane

The zero-mass states in the open-string particle spectrum for a system of N coincident D-branes yields a set of interacting quantum fields which is exactly a U(N) gauge theory. (The string theory does contain other interactions, but they are only detectable at very high energies.) Gauge theories were not invented starting with bosonic or fermionic strings; they originated from a different area of physics, and have become quite useful in their own right. If nothing else, the relation between D-brane geometry and gauge theory offers a useful pedagogical tool for explaining gauge interactions, even if string theory fails to be the "theory of everything".

 

[Urs Schreiber]

The zero-mass states in the open-string particle spectrum for a system of N coincident D-branes yields a set of interacting quantum fields which is exactly a U(N) gauge theory.

Not sure why you just recalled this, but this gives me occasion to highlight that this pillar of string theory lore (underlying absolutely everything, from geometric engineering of gauge theories, via intersecting D-brane model building to AdS/CFT duality) had remained without real proof until maybe recently (quoting references from here):

The idea that on N coincident D-branes there is gauge enhancement to U(N)-gauge field theory is due to

• Edward Witten, section 3 of Bound States Of Strings And p-Branes, Nucl.Phys.B460:335-350, 1996 (arXiv:hep-th/9510135)

There, this is called an “obvious guess” (first line on p. 8). Subsequently, most authors cite this obvious guess as a fact; for instance the review

• Robert Myers, section 3 of Nonabelian Phenomena on D-branes, Class.Quant.Grav. 20 (2003) (arXiv:hep-th/0303072)

By actual computation in open string field theory “convincing evidence” (see bottom of p. 22) was found, numerically, in

• Erasmo Coletti, Ilya Sigalov, Washington Taylor, Abelian and nonabelian vector field effective actions from string field theory, JHEP 0309 (2003) 050 (arXiv:hep-th/0306041)

Similar numerical derivation, as well as exact derivation at zero momentum, is in

• Nathan Berkovits, Martin Schnabl, Yang-Mills Action from Open Superstring Field Theory, JHEP 0309 (2003) 022 (arXiv:hep-th/0307019)

The first full derivation seems to be due to

• Taejin Lee, Covariant open bosonic string field theory on multiple D-branes in the proper-time gauge, Journal of the Korean Physical Society December 2017, Volume 71, Issue 12, pp 886–903 (arXiv:1609.01473)

which is surveyed in

• Taejin Lee, Deformation of the Cubic Open String Field Theory, Phys. Lett. B 768 (2017) 248 (arXiv:1701.06154)

That on D0-branes this reproduces the BFSS matrix model and on D(-1)-branes the IKKT matrix model is shown in

• Taejin Lee, Covariant Open String Field Theory on Multiple Dp-Branes (arXiv:1703.06402)

Derivation via rational parameterized stable homotopy theory applied to the brane bouquet is in

• Vincent Braunack-Mayer, Hisham Sati, Urs Schreiber, Gauge enhancement of Super M-Branes (arXiv:1806.01115)

 

[John G]

Not sure why you just recalled this, but this gives me occasion to highlight that this pillar of string theory lore (underlying absolutely everything, from geometric engineering of gauge theories, via intersecting D-brane model building to AdS/CFT duality) had remained without real proof until maybe recently (quoting references from here)...

You relating the worldsheet to Feynman path integral wordlines reminded me of years ago when you mentioned on Woit's blog that there's a deep relation between the worldsheet and path integrals. Tony Smith was in that conversation and U(N) gauge theory is the string theory-like part of Smith's model. I think this is one area where you, Smith, Witten, Polchinski, etc. have had some seriously good intuition even without a lot to go on empirically. It is I think the best response to give to those who think string theory can't be related to known physics (Standard Model and relativity). U(N) gauge theory doesn't even need supersymmetry so even those tired of waiting for superpartners should I would think stay interested in string theory. For Polchinsky via Tony, I've been thinking recently about this part.

http://www.tony5m17h.net/MonsterStringCell.pdf

Polchinski says "... there will also be r^2 massless scalars from the components normal to the D-brane. ... the collectives coordinates ... X^u ... for the embedding of n D-branes in spacetime are now enlarged to nxn matrices. This 'noncummutative geometry' ...[may be]... an important hint about the nature of spacetime. ..."

Tony relates Cl(8) to elementary cellular automata and I used to work with cellular automata at IBM. I've always though thought I should understand Tony's relating better than I do so I've been trying to put it into a formal Rodrigo Obando-like rule space.

http://wpmedia.wolfram.com/uploads/sites/13/2018/02/24-1-2.pdf

Obando in the above paper talks about primitives in a basis vector-like way but you can get the 14 Cl(8) Pertti Lounesto 4-vector primitive idempotent terms to show up on matrix diagonals of an Obando rule space (position-momentum spacetime for Tony). There's however kind of two different rule space basis vector orderings to use. There's one very close to what Tony has mentioned that seems related most to Clifford duals (gradings 0 vs 8,1 vs 7, 2 vs 6, 3 vs 5, and 4 self dual) and this shows the matrix diagonals (using either Tony's order or the closely related order). Then there's an order related most to getting group theory Cartan subalgebras and deriving rather than showing the Pertti Lounesto primitive idempotent terms for the matrix diagonals.

I'm still trying to understand it better but unfortunately I'm much better at the rule space part than the Clifford algebra/group theory part (and Tony is perhaps a bit different than others in this area). I came here and looked at the recent topics after looking up some old Carl Brannen and Marni Sheppeard conversations about Primitive Idempotents. I probably need to get hold of Tony, but he's not responding to the email I used with him ten years ago or so.

 

[Urs Schreiber]

It is I think the best response

This is a very common perspective, that string theory is at least a conceptual dictionary of the space of quantum field theories, giving geometric interpretation to surprising relations between different quantum field theories. Sometimes this is referred to as "geometric engineering of quantum field theories".