Timeline for Frontiers of QM and QFT
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 17, 2021 at 0:03 | comment | added | Buzz | You can see the issue with the Fock state by looking at the Lee Model, which a toy model that has an exactly solvable spectrum, yet which still requires renormalization. If you try to set up the Fock space at the beginning, you will find that it does not really correspond to the state space of the theory, because the renormalization changes this fundamentally. | |
| Jun 16, 2021 at 18:30 | comment | added | Michael Engelhardt | Just to throw in an example that may help in sorting things: In light-front field theory, one argues that the vacuum is trivial (although this is a very subtle point). One then casts the theory in terms of a Fock space. It is still an interacting theory, the Hamiltonian is not quadratic yet! But one does have a Fock space representation, on the basis of which one can develop truncation schemes to approximately solve further, etc. This is why I write about "partially" solving. In light-front field theory, one has "partially solved" by making the argument that the vacuum is trivial. | |
| Jun 16, 2021 at 15:23 | comment | added | Michael Engelhardt | I think there are two separate things going on here. One is that QM and QFT are both quantum theories. The other is the idea that one can organize them both in terms of a Fock space. As far as the former is concerned, sure, QFT is just QM of infinitely many degrees of freedom (not that that's a trivial transition, cf. multivariable calculus vs. functional analysis). But a Fock space construction is something more specific that implies you've at least partially solved your theory. The interesting part of QFT is the solving, not the organizing into a Fock space at the end. | |
| Jun 16, 2021 at 12:39 | comment | added | JustWannaKnow | @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. | |
| Jun 16, 2021 at 12:32 | comment | added | JustWannaKnow | @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. | |
| Jun 16, 2021 at 7:41 | comment | added | Michael Engelhardt | To formulate a QFT, we don't begin with a Fock space. Constructing a Fock space implies that you already know what the vacuum is. In that sense, the Fock space is a way of presenting the (at least partial) solution of a QFT rather than formulating the QFT. | |
| Jun 16, 2021 at 3:17 | comment | added | Buzz | There is really nothing fundamental about making a connection between quantum mechanics and free quantum field theory. The relationship between the two is not subtle, and it can be expressed rigorously in many different fashions. It is only when there are nontrivial interactions that quantum field theory becomes a fundamentally different "kind" of thing from quantum mechanics, so you are never going to get the key insights into what makes quantum field theory interesting in its own right by looking at free theories. And how to treat the interacting cases with rigor is, generally, unknown. | |
| Jun 16, 2021 at 2:07 | history | asked | JustWannaKnow | CC BY-SA 4.0 |