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The non-critical string theory describes the relativistic string without enforcing the critical dimension. Although this allows the construction of a string theory in 4 spacetime dimensions, such a theory usually does not describe a Lorentz invariant background. However, there are recent developments which make possible Lorentz invariant quantization of string theory in 4-dimensional Minkowski space-time.[citation needed]
There are several applications of the non-critical string. Through the AdS/CFT correspondence it provides a holographic description of gauge theories which are asymptotically free.[citation needed][1] It may then have applications to the study of the QCD, the theory of strong interactions between quarks.[1] Another area of much research is two-dimensional string theory which provides simple toy models of string theory. There also exists a duality to the 3-dimensional Ising model.[citation needed]
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Julia Galef: Think Rationally via Bayes' Rule
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String Efficiency Part - 1 | Thakar Ki Pathshala
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What’s the difference between a scientific law and theory? - Matt Anticole
Transcription
I'd like to introduce you to a particularly powerful paradigm for thinking called Bayes' Rule. Back in the Second World War the then governor of California, Earl Warren, believed that Japanese Americans constituted a grave threat to our national security. And as he was testifying as much to Congress, someone brought up the fact that, you know, we haven't seen any signs of subterfuge from the Japanese American community. And Warren responded that, "Ah, this makes me even more suspicious. This is an even more ominous sign because that indicates that they're probably planning some major secret timed to attack á la Pearl Harbor. And this convinces me even more that the Japanese Americans are a threat." So this pattern of reasoning is what sustains most conspiracy theories. You see signs of a cover up -- well, that just proves that I was right all along about the cover up. You don't see signs of a cover up, well that just proves that the cover up runs even deeper than we previously suspected. Bayes' Rule is provably the best way to think about evidence. In other words, Bayes' Rule is a formalization of how to change your mind when you learn new information about the world or have new experiences. And I don't think that the math behind -- the math of Bayes' Rule is crucial to getting benefit out of it in your own reasoning or decision making. In fact, there are plenty of people who use Bayes' Rule on a daily basis in their jobs -- statisticians and scientists for example. But then when they leave the lab and go home, they think like non-Bayesians just like the rest of us. So what's really important is internalizing the intuitions behind Bayes' Rule and some of the general reasoning principles that fall out of the math. And being able to use those principles in your own reasoning. After you've been steeped in Bayes' Rule for a little while, it starts to produce some fundamental changes to your thinking. For example, you become much more aware that your beliefs are grayscale, they're not black and white. That you have levels of confidence in your beliefs about how the world works that are less than one hundred percent but greater than zero percent. And even more importantly, as you go through the world and encounter new ideas and new evidence, that level of confidence fluctuates as you encounter evidence for and against your beliefs. Also I think that many people, certainly including myself, have this default way of approaching the world in which we have our preexisting beliefs and we go through the world and we pretty much stick to our beliefs unless we encounter evidence that's so overwhelmingly inconsistent with our beliefs about the world that it forces us to change our minds and, you know, adopt a new theory of how the world works. And sometimes even then we don't do it. So the implicit question that I'm asking myself that people ask themselves as they go through the world is when I see new evidence, can this be explained with my theory. And if yes, then we stop there. But, after you've got some familiarity with Bayes' Rule what you start doing is instead of stopping after asking yourself can this evidence be explained with my own pet theory, you also ask well, would it be explained better with some other theory or maybe just as well with some other theory. Is this actually evidence for my theory.
The critical dimension and central charge
In order for a string theory to be consistent, the worldsheet theory must be conformally invariant. The obstruction to conformal symmetry is known as the Weyl anomaly and is proportional to the central charge of the worldsheet theory. In order to preserve conformal symmetry the Weyl anomaly, and thus the central charge, must vanish. For the bosonic string this can be accomplished by a worldsheet theory consisting of 26 free bosons. Since each boson is interpreted as a flat spacetime dimension, the critical dimension of the bosonic string is 26. A similar logic for the superstring results in 10 free bosons (and 10 free fermions as required by worldsheet supersymmetry). The bosons are again interpreted as spacetime dimensions and so the critical dimension for the superstring is 10. A string theory which is formulated in the critical dimension is called a critical string.
The non-critical string is not formulated with the critical dimension, but nonetheless has vanishing Weyl anomaly. A worldsheet theory with the correct central charge can be constructed by introducing a non-trivial target space, commonly by giving an expectation value to the dilaton which varies linearly along some spacetime direction. (From the point of view of the worldsheet CFT, this corresponds to having a background charge.) For this reason non-critical string theory is sometimes called the linear dilaton theory. Since the dilaton is related to the string coupling constant, this theory contains a region where the coupling is weak (and so perturbation theory is valid) and another region where the theory is strongly coupled. For dilaton varying along a spacelike direction, the dimension of the theory is less than the critical dimension and so the theory is termed subcritical. For dilaton varying along a timelike direction, the dimension is greater than the critical dimension and the theory is termed supercritical. The dilaton can also vary along a lightlike direction, in which case the dimension is equal to the critical dimension and the theory is a critical string theory.
Two-dimensional string theory
Perhaps the most studied example of non-critical string theory is that with two-dimensional target space. While clearly not of phenomenological interest, string theories in two dimensions serve as important toy models. They allow one to probe interesting concepts which would be computationally intractable in a more realistic scenario.
These models often have fully non-perturbative descriptions in the form of the quantum mechanics of large matrices. Such a description known as the c=1 matrix model captures the dynamics of bosonic string theory in two dimensions. Of much recent interest are matrix models of the two-dimensional Type 0 string theories. These "matrix models" are understood as describing the dynamics of open strings lying on D-branes in these theories. Degrees of freedom associated with closed strings, and spacetime itself, appear as emergent phenomena, providing an important example of open string tachyon condensation in string theory.
See also
- String theory, for general information about critical superstrings
- Weyl anomaly
- Central charge
- Liouville gravity
References
- ^ a b Kiritsis, Elias (26 Jan 2009). "Dissecting the string theory dual of QCD". Fortschritte der Physik. 57 (5–7): 369–417. arXiv:0901.1772. Bibcode:2009ForPh..57..396K. doi:10.1002/prop.200900011. S2CID 2236596.
- Polchinski, Joseph (1998). String Theory, Cambridge University Press. A modern textbook.
- Vol. 1: An introduction to the bosonic string. ISBN 0-521-63303-6.
- Vol. 2: Superstring theory and beyond. ISBN 0-521-63304-4.
- Polyakov, A.M. (1981). "Quantum geometry of bosonic strings". Physics Letters B. 103 (3): 207–210. Bibcode:1981PhLB..103..207P. doi:10.1016/0370-2693(81)90743-7. ISSN 0370-2693.
- Polyakov, A.M. (1981). "Quantum geometry of fermionic strings". Physics Letters B. 103 (3): 211–213. Bibcode:1981PhLB..103..211P. doi:10.1016/0370-2693(81)90744-9. ISSN 0370-2693.
- Curtright, Thomas L.; Thorn, Charles B. (1982-05-10). "Conformally Invariant Quantization of the Liouville Theory". Physical Review Letters. 48 (19): 1309–1313. Bibcode:1982PhRvL..48.1309C. doi:10.1103/physrevlett.48.1309. ISSN 0031-9007. [Erratum-ibid. 48 (1982) 1768].
- Gervais, Jean-Loup; Neveu, André (1982). "Dual string spectrum in Polyakov's quantization (II). Mode separation". Nuclear Physics B. 209 (1): 125–145. Bibcode:1982NuPhB.209..125G. doi:10.1016/0550-3213(82)90105-5. ISSN 0550-3213.
This page was last edited on 30 July 2021, at 19:55