Infinities
How this infinite series comes up in string theory is puzzling, and I've not read or heard a good explanation for it. In the only string theory textbook I have, it is called in various places 'a surprising interpretation', 'an inspired guess' and an equality with a question mark. In other words, I believe the author has no clue as to how to make a rigorous statement. The interpretation is consistent with the prediction that the dimension of spacetime is 26, but I do not know how fundamental it really is.
Since I have been puzzling about analytic continuation for years, it is no surprise that my confusion has bubbled out of me and reached you. Though a specific incident escapes me. For the past few weeks I have delved into books on functional analysis and the theory of the spectrum of operators. I have found that the theory of infinite-dimensional operators is much richer than its finite dimensional counterpart. Not to be confused with the dimension of spacetime. The dimension here refers to the number of degrees of freedom. An operators maps a function to another function, and there are usually an infinite number of functions it can act on. There is a general belief that infinite-dimensional operators in quantum mechanics accurately describe the world, and that finite-dimensional operators provide the necessary caricature, without which little progress can be made.
The latest paper I am writing is about holography: how certain information in the d-dimensional bulk is encoded in the (d-1)-dimensional boundary. Here I revert back to the dimension of space. I am trying to prove a one-to-one correspondence between a topological invariant that is defined by bulk wavefunctions and the energy spectrum of boundary wavefunctions. The proof requires that I develop a level of mathematical rigor which I have never felt to be necessary, but my recent immersion into subjects that I have previously disdained have altered my stance. I want to transcend arguments and say something with absolute certainty. This is very difficult, but the work I have put in has honed a healthier attitude toward mathematics. Before I was largely content in expanses of ignorance, today I am pained by the weak arguments I took for granted, and I seek redress when I can.
This paper feels like a culmination, it draws on insights I've gained on every paper I've written in Princeton. I have been quietly working on this project for a month, mostly in Singapore, without speaking a word to anybody. It was with some trepidation I presented my results to Andrei yesterday, for fear that he believes my month-long effort to be trivial. If he does, I have already convinced myself to publish without him, the paper is nearly complete. He does not, which gave me a thrill of validation.
Since I have been puzzling about analytic continuation for years, it is no surprise that my confusion has bubbled out of me and reached you. Though a specific incident escapes me. For the past few weeks I have delved into books on functional analysis and the theory of the spectrum of operators. I have found that the theory of infinite-dimensional operators is much richer than its finite dimensional counterpart. Not to be confused with the dimension of spacetime. The dimension here refers to the number of degrees of freedom. An operators maps a function to another function, and there are usually an infinite number of functions it can act on. There is a general belief that infinite-dimensional operators in quantum mechanics accurately describe the world, and that finite-dimensional operators provide the necessary caricature, without which little progress can be made.
The latest paper I am writing is about holography: how certain information in the d-dimensional bulk is encoded in the (d-1)-dimensional boundary. Here I revert back to the dimension of space. I am trying to prove a one-to-one correspondence between a topological invariant that is defined by bulk wavefunctions and the energy spectrum of boundary wavefunctions. The proof requires that I develop a level of mathematical rigor which I have never felt to be necessary, but my recent immersion into subjects that I have previously disdained have altered my stance. I want to transcend arguments and say something with absolute certainty. This is very difficult, but the work I have put in has honed a healthier attitude toward mathematics. Before I was largely content in expanses of ignorance, today I am pained by the weak arguments I took for granted, and I seek redress when I can.
This paper feels like a culmination, it draws on insights I've gained on every paper I've written in Princeton. I have been quietly working on this project for a month, mostly in Singapore, without speaking a word to anybody. It was with some trepidation I presented my results to Andrei yesterday, for fear that he believes my month-long effort to be trivial. If he does, I have already convinced myself to publish without him, the paper is nearly complete. He does not, which gave me a thrill of validation.
posted by Dr. Broke at 2:34 PM
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