Timeline for Rigorous construction of fermionic field theory?
Current License: CC BY-SA 4.0
32 events
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| Nov 12, 2022 at 0:41 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
typo corrected
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| Nov 11, 2022 at 7:46 | comment | added | Pedro Lauridsen Ribeiro | In due time: the different scalar multiplication for $\bar{\mathfrak{k}}$ affects the reconstruction of the fermionic creation and annihilation operators from the Dirac field operators. I'm a bit short on time to recall the precise details, so I'll omit that part for now to keep the answer clear for the time being and add the missing pieces later. | |
| Nov 11, 2022 at 7:42 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Fixed error in the construction of Dirac field operators
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| Nov 11, 2022 at 6:23 | comment | added | Pedro Lauridsen Ribeiro | Hi Alan, yes, you're actually correct; with the Dirac fields defined as above, indeed we get $$\{\psi(f),\psi^*(g)\}=\frac{1}{2}(\{b(f),b^*(g)\}+\{c^*(f),c(g)\})=\frac{1}{2}(\{b(f),b^*(g)\}+\{c(g),c^*(f)\})=\frac{1}{2}(\langle f,g\rangle+\langle g,f\rangle)\mathbf{1}=\text{Re}\langle f,g\rangle\mathbf{1}\ .$$ I'll make appropriate amends to the answer. Thanks for the warning. | |
| Nov 9, 2022 at 13:51 | comment | added | Alan Garbarz | Pedro, could it be that in the Dirac construction, as it is written, the anticommutator $\left\{\psi(f),\psi^*(g)\right\}=\text{Re}\langle f , g \rangle \textbf{1}$? Instead, in order to get the inner product on th RHS of the anticommutator, may be one can take the conjugate (pre)Hilbert space $\overline{\mathfrak{k}}$ in the second slot, I mean $\mathfrak{h}=\mathfrak{k} \oplus \overline{\mathfrak{k}}$ ? Also this would make $f \mapsto \psi(f)$ anti-linear. | |
| Jul 6, 2021 at 0:10 | comment | added | Pedro Lauridsen Ribeiro | If you choose to follow this path, I'd suggest you to read Arai's book in tandem with some standard QFT textbook for physicists, so you can understand better the latter from a mathematical viewpoint and, at the same time, keep track of why certain things are defined the way they are in the former. I emphasize, however, that the algebraic approach is not only more efficient as I've just argued, it's also more robust when you move to interacting fields. The whole "canonical" framework as e.g. developed in Arai's book tends to break down in several places when interactions come to play. | |
| Jul 5, 2021 at 23:04 | comment | added | JustWannaKnow | 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? | |
| Jul 5, 2021 at 21:44 | comment | added | Pedro Lauridsen Ribeiro | OK, I've had a quick look at Arai's book. The book is certainly more pedestrian in the sense that it delivers more background material and tries to be closer to traditional QFT textbooks for physicists. The more algebraic approach I follow in my answer is equivalent but more, say, efficient - it's quite shorter and easier to generalize to other space-times (you can do it for sections of any vector bundle). Particularly regarding the one-particle space $\mathfrak{k}$, I've deliberately kept Fourier analysis to a minimum since this technique is not available in curved space-times. | |
| Jul 3, 2021 at 0:45 | comment | added | JustWannaKnow | 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). | |
| Jul 3, 2021 at 0:32 | comment | added | Pedro Lauridsen Ribeiro | Perhaps we should move this discussion to chat if we see fit to continue it in the future. | |
| Jul 3, 2021 at 0:31 | comment | added | Pedro Lauridsen Ribeiro | I don't know this book, I'll have a look at it. | |
| Jul 2, 2021 at 23:41 | comment | added | JustWannaKnow | 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. | |
| Jul 2, 2021 at 22:16 | comment | added | Pedro Lauridsen Ribeiro | The formulae for the Wightman 2-point function and the other Green functions for the Dirac field are discussed with proper care in the comprehensive book by N. N. Bogolyubov, A. A. Logunov, A. I. Oksak and I. T. Todorov, "General Principles of Quantum Field Theory" (Kluwer, 1990), see Section 8.4, pp. 340-358 therein. I'll try to include a brief discussion on this into my answer in the next few days. | |
| Jul 2, 2021 at 22:12 | comment | added | JustWannaKnow | 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! | |
| Jul 2, 2021 at 21:37 | comment | added | JustWannaKnow | 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. | |
| Jul 2, 2021 at 6:24 | comment | added | Pedro Lauridsen Ribeiro | Using the Fourier transform, one can rewrite $\omega(\overline{f},g)$ in terms of initial data for positive-energy solutions of the Dirac equation, as I mentioned in my answer above. The advantage of the form I've chosen in the two comments above is that it's explicitly Poincaré-covariant. | |
| Jul 2, 2021 at 6:16 | comment | added | Pedro Lauridsen Ribeiro | $S^R$ and $S^A$, on their turn, can be obtained by respectively applying the "conjugate" Dirac operator $i\gamma^\mu\partial_\mu+m$ to respectively $\Delta^R\otimes\delta$ and $\Delta^A\otimes\delta$, where $\Delta^R$ and $\Delta^A$ are respectively the retarded and advanced fundamental solutions of the Klein-Gordon operator $\Box+m^2$ with the same mass and $\delta$ is the four-dimensional Kronecker delta. | |
| Jul 2, 2021 at 6:09 | comment | added | Pedro Lauridsen Ribeiro | Not quite. One way to define $\mathfrak{k}$ is e.g. to take the completion of $\mathscr{D}(\mathbb{R}^4;\mathbb{C}^4)$ w.r.t. the scalar product $$\langle f,g\rangle=\langle 1,\psi(f)\psi^*(g)1\rangle=\langle 1,b(f)b^*(g)1\rangle=\omega(\overline{f},g)\ ,$$ where $\omega$ is the Wightman 2-point distribution of the Dirac field. One way to define the latter is as the positive-frequency part of the causal commutator $S=S^R-S^A$, which is the difference between the retarded and advanced fundamental solutions $S^R,S^A$ of the Dirac operator. | |
| Jun 30, 2021 at 21:56 | comment | added | JustWannaKnow | 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 $L^{2}(\mathbb{R}^{4};\mathbb{C}^{4})$? | |
| May 27, 2021 at 16:51 | comment | added | Pedro Lauridsen Ribeiro | Yes, I do know the book and his authors. The families of fermionic creation and annihilation operators generate each a Grassmann algebra - this is implicit in the CAR's. The (non-zero) anti-commutation relation relating both families to each other is the one involving Planck's constant $\hbar$, which is tacitly set to one in the above discussion - that's the only place where the CAR algebra differs from a Grassmann algebra, the one which makes it " quantum". As $\hbar\rightarrow 0$, both kinds of algebras coalesce - that's why we need the latter when quantizing fermions with path integrals. | |
| May 27, 2021 at 14:43 | comment | added | JustWannaKnow | 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? | |
| May 26, 2021 at 22:19 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Added formula
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| May 26, 2021 at 22:12 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Added formula
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| May 26, 2021 at 22:04 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Added explanation
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| May 26, 2021 at 22:03 | comment | added | JustWannaKnow | I will take a look at both references! Thanks! | |
| May 26, 2021 at 21:27 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Added correction
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| May 26, 2021 at 21:22 | comment | added | Pedro Lauridsen Ribeiro | You can look at Volume 1 of Bratteli-Robinson for an introduction to C*-algebras. Another nice book on the subject is G. Murphy's C${}^*\!$-Algebras and Operator Theory, but BR1 is more than enough to follow Volume 2. | |
| May 26, 2021 at 21:15 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Typos corrected, added formula
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| May 26, 2021 at 21:14 | comment | added | JustWannaKnow | Pedro, thanks for the answer. Although I'm not much familiar with $C^{*}$-algebras, I got the idea and appreciate the details and quality of your exposure. I know Bratelli & Robinson's book, so I might need to learn a little bit of $C^{*}$-algebra to fully understand your reasoning. In any case, your answer will be an important guide for me. Also, this discussion led me to think about another topic, and I should post another question soon. | |
| May 26, 2021 at 21:11 | vote | accept | JustWannaKnow | ||
| May 26, 2021 at 21:09 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Typos corrected, added formula
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| May 26, 2021 at 20:56 | history | answered | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |