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?

  • $\begingroup$ We wonder if you are refering to Quantum Field Theories in general, with emphasis on the viewpoint of representations of the Poincare group (a Lie Group, which cerainly admits a Lie algebra) or if you are specifically interested in the gauge group aspects of the Standard model. $\endgroup$ – Nikolaj-K Jan 21 '12 at 14:18
  • $\begingroup$ Thanks for responding. My wife and I are helping our son make a report to his high-school science club. QFT and gauge theory are valid topics. We'd like to summarize the main points of the SM as a QFT gauge theory and include references to Hilbert space (as a quantum physical system: states, observables/operators, transformations/dynamics), then show how Lie theory (ala boson realizations and fermion realizations in Iachello chapters 7-8) relates to the Hilbert space formulation of the SM. Otherwise we would outline how Hilbert space and Lie algebra cover different aspects of the SM. $\endgroup$ – user7234 Jan 21 '12 at 14:52
  • $\begingroup$ High-school? QFT and Lie Algebras? I don't understand. How deep will such a science club project go? Are you and your wife mathematicans? If yes, then much Hilbert space stuff follows from using compact Lie Groups (for example as gauge groups like $SU(3)$) alone, and as far as QFTs are concerned, some features are summarized here, although this is quite far away from "applications like the Standard Model". $\endgroup$ – Nikolaj-K Jan 21 '12 at 15:06
  • $\begingroup$ There is a book of Arnold Neumaier on Lie algebras in CM and QM: mat.univie.ac.at/~neum/ms/QML.pdf $\endgroup$ – Vladimir Kalitvianski Jan 21 '12 at 18:06
  • $\begingroup$ Thanks for the reference to the Neumaier book. If anyone else knows of a book or website where they use a combination of Hilbert space and Lie theory in a discussion of the Standard Model, please post the title or link. Thanks. $\endgroup$ – user7234 Jan 22 '12 at 13:29

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.


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.

  • $\begingroup$ Thanks. I'll try to put it in my own words, according to what I understand from the Reality book by Roger Penrose (sections 20.5 and 26.6): There are physical systems (classical, QM and QFT), consisting of states, observables and dynamics. In QM states are organized in a Hilbert space H, observables are operators on H, and dynamics is/are defined in terms of a Lagrangian and an action S (an integral of the Lagrangian). In QFT, could you define the physical system? How is the Lagrangian defined as a function(al) of fields and derivatives of the fields? Are actions and Lie theory involved? $\endgroup$ – user7234 Apr 10 '12 at 22:04
  • $\begingroup$ About QM you are right on the money. About QFT, there is such a diversity of possible approaches, it is hard to comment on your statement. The approach that Dr. Peter Morgan, quite an expert, likes is that the observable are abstract operators, they do not act on any particular space at all. The states are functions on the whole algebra of these operators: a state is a way of assigning a value to each observable. The dynamics is a one-parameter group operation on the algebra of operators, which can also be made an operation on the states, of course. Lie Groups come in as symmetry gro $\endgroup$ – joseph f. johnson Feb 12 '13 at 15:24
  • $\begingroup$ ups of the whole algebra of operators. Weinberg does not like this approach. Another approach is that the states are operator-valued functions on space--time, where the operators in fact are concrete operators on a far-out Hilbert space. The Lie groups enter in as groups of symmetries on this space. But Lie algebras are not much used in either approach, in particular, the Lie algebra structure of the observables is not used. The Lie Groups act on the configuration space: space-time plus spin plus colour plus.... the configuration variables. Again, the dynamics is a one-parameter group $\endgroup$ – joseph f. johnson Feb 12 '13 at 15:28
  • $\begingroup$ QFT is just a mess....I never bothered to learn it...learn Stat Mech instead. $\endgroup$ – joseph f. johnson Feb 12 '13 at 15:29
  • $\begingroup$ And, yes, the dynamics can be given by an action principle and a Lagrangian (except in the abstract operator algebra approach). The Lagrangian is a functional that acts on the operator-valued functions on space-time, it involves them, various ordered products of these operators, and some derivatives, too. The formulas look like, well, at least have a family resemblance to, the formulas from classical mechanics or quantum mechanics. $\endgroup$ – joseph f. johnson Feb 12 '13 at 15:32

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.


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