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We are looking for a publication or website that explains the Standard Model in terms of Hilbert space and Lie algebra.

We are reading Debnath's Introduction to Hilbert Spaces and Applications and Iachello's Lie Algebras and Applications. Is there a book or website that combines the two approaches (Hilbert and Lie)? If they can't be combined, can you provide a link to articles that compare/contrast them?

Qmechanic
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user7234
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3 Answers3

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One answer to the question about books that explicitly link Hilbert space and Lie theory in a discussion of the SM is the three volume set by Eberhard Zeidler, Quantum Field Theory I: Basics in Mathematics and Physics. According to the table of contents on amazon, chapter 7 of Volume 1 has sections on Hilbert spaces, Lie algebras and Lie groups. I ordered the book through Interlibrary Loan and will post details later for anyone who's interested.

user7234
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I don't think there is any, for a reason: there is a paradigm shift from Quantum Mechanics to Quantum Field Theory. see https://physics.stackexchange.com/a/20387/6432 .

In Quantum Mechanics, all the dynamical variables are treated on an equal footing, and the number of particles is fixed. There is no annihilation or pair-creation. The Hilbert space is the space of states of, e.g, 27 electrons, neither more nor less. The observables are operators on that space. But in Quantum Field Theory, the number of particles is treated as an operator, and there are creation and annihilation operators, and one switches to treating the state of a field as a function on space-time whose values are field observables. The observables are no longer studied as operating on the space of such functions, and researchers tend to ignore Hilbert space aspects of it. See also Quantum field theory variants for a discussion of QFT, and What's the exact connection between bosonic Fock space and the quantum harmonic oscillator? for a short discussion of how Fock space replaces, in QFT, the usual one-particle Hilbert Space of QM, in order to allow the number of particles to change as some get annihilated.

Community
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joseph f. johnson
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First, a possibly unwelcome comment: You need more than Lie algebras to define the Standard Model's particle content and couplings. You need the representation theory of Lie groups, for a zillion different reasons, e.g.:

  1. The difference between $\mathbb{R}$ and $U(1)$ -- which have the same Lie algebra -- is related to the existence of charge quantization.
  2. The parity groups in the standard model -- generated by elements $C$, $P$, and $T$ which square to 1 -- play an import role in understanding the character of weak interactions, the difference between fermions and antifermions, and such.
  3. You need the representation theory of the Poincare group $ISO(3,1)$ which really isn't the same thing as $ISO(4)$, to understand the classification of the different kinds of spinors fields that exist in nature.

Then, in an attempt to be helpful: Chapter 2 of Volume 1 of Weinberg's The Quantum Theory of Fields describes the classification of single particle states in some some detail. Also, Baez & Huerta's article http://arxiv.org/abs/0904.1556 has some nice exposition in it.

user1504
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