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JustWannaKnow
  • Member for 4 years, 11 months
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Rigorous construction of fermionic field theory?
Pedro, very nice! Always good to hear from experts. So, Arai's book seems like a nice place to start while I'm learning the backgrounds for the algebraic approach right?
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Rigorous construction of fermionic field theory?
Pedro, okay. I don't know how to start a chat tho. This book does not seem well known. Yet, it seems pretty good to me. I'd like to know your opinion (you can take its discussion on the quantization of the Dirac field as a parameter for instance).
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Rigorous construction of fermionic field theory?
Pedro, let me take the oportunity to ask: what's your opinion on Arai's book "Analysis on Fock Spaces and Mathematical Theory of Quantum Field"? It has less algebraic approach and seems more suitable to me.
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Rigorous construction of fermionic field theory?
Pedro, if you could include some brief discussion to guide me, that would be great! Your comments are always really helpful. In the meantime I'll take a look at the reference you mentioned! Thanks again!
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Rigorous construction of fermionic field theory?
Pedro, thanks again for the comments. Well, I should apologize for my ignorance, but I've never have contact with any of those things, so it is really hard to me to follow your reasoning. On the other hand, this is the crucial part of the problem, since it provides the bridge between the general picture and the Dirac field quantization, the example I'm interested in. I don't think I can get where I want without undertanding your latest comments. Maybe you could suggest some references to me? It would be of huge help! And again,thanks for your patience.
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Rigorous construction of fermionic field theory?
sorry to revive the discussion after all this time, but I've been studying your answer and I'd like to ask whether it is possible for you to be a little more explicit about that the space $\mathfrak{k}$ should be. You mean $\mathfrak{k} = \operatorname{ker}(H_{D}-E)$, where $H_{D}$ is the Dirac operator $H_{D} := i\gamma^{\mu}\partial_{\mu}-m$ on $L^{2}(\mathbb{R}^{4};\mathbb{C}^{4})$?
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Frontiers of QM and QFT
@Buzz thanks for your comments. I appreciate this interaction! In terms of mathematics, it seems to me that there is some differences between QM and a "simple" free field because the latter puts space and time variables in equal foot. QFT demands operators to be indexed by points rather than elements of $\mathscr{H}$. But, at least in the case of non-interacting systems, this seems to accomplished just by setting a "misguided" notation; it is impressive to me, but I'm trying to understand all this better.
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Frontiers of QM and QFT
@MichaelEngelhardt this is true, indeed. However, as far as I know (please, correct me if I'm wrong) you don't explicitly define the Fock space, but you know there is an underlying Fock space which you recover from the vacuum state. Although this goes, in some sense, in the opposite direction as the scheme I posted, the question remains: the QFT formalism seems to arise as a matter of notation of a generic scheme that does not differentiate what is QM of what is QFT.
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Invariance of Lorentz measure
@idrinehart, thanks for your answer! It seems a very nice approach and I can get the idea behind it but, to be sincere, I have almost no background on differential geometry and it is hard to me to follow your answer 100%; I can appreciate the elegance of the approach, however. I was thinking more like some sort of "change of variables"; I mean, $\Omega_{m}(\Lambda E)$ is the integral over $j_{m}(\Lambda E)$ so we maybe we can redefine ${\bf{x}}$ or something like that to get back to $j_{m}(E)$. It is not clear to me yet, tho.
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Rigorous construction of fermionic field theory?
Pedro, by the way, you probably know this but there is a book by E. de Faria and W. de Mello called "Mathematical Aspects of Quantum Field Theory" where the quantization of Dirac fields is briefly discussed. But, there, the authors introduce Grassmann variables instead and I don't understand where the quantization is actually discussed. Anyways, I thought the quantization using Grassmann variables was used only when one is trying to quantize it via path integrals. If possible, could you comment on that?
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QFT and mathematical rigor
@Pedro, do you know if this is what is usually assumed to "define" Hilbert spaces in physics books? I've heard several times "this Hilbert space has an indefinite inner product" or even "A Hilbert space is a vector space with a positive-definite inner product" but I never understood this assertions. So actually the difference is between Hilbert spaces and Kreub spaces?
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QFT and mathematical rigor
@paulgarrett the "g" instead of "d" was a typo. Thanks for pointing it out!
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