Discussion Overview
The discussion revolves around the existence and examples of uncountable infinite dimensional Hilbert spaces, with a focus on their implications in physics, particularly in quantum field theory. Participants explore definitions, properties, and specific examples, while also addressing the concept of separability in Hilbert spaces.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant inquires about examples of uncountable infinite dimensional Hilbert spaces, referencing Banach spaces but noting they are not Hilbert spaces.
- Another participant provides an example from a textbook, describing a space of complex-valued functions defined on the real numbers that vanish almost everywhere, asserting it is non-separable.
- There is a discussion on the importance of separability in quantum physics, with one participant stating that quantum theories operate only with separable Hilbert spaces.
- Some participants argue that non-separable Hilbert spaces, like Fock space, play significant roles in physics, although there is uncertainty about their properties.
- One participant challenges the claim that Fock space is non-separable, presenting a detailed argument about the separability of tensor products of Hilbert spaces.
- Another participant disputes the logic of using induction to claim separability for infinite-dimensional spaces, citing counterexamples.
- There are claims regarding the equivalence of separable Hilbert spaces and the implications of Haag's theorem in quantum field theory, with some participants expressing confusion over the connections made.
- Disagreement arises over the relevance of certain examples and definitions, particularly concerning the construct of Fock space and its separability.
Areas of Agreement / Disagreement
Participants express multiple competing views regarding the properties of Fock space and the implications of separability in Hilbert spaces. The discussion remains unresolved, with no consensus on whether Fock space is separable or non-separable.
Contextual Notes
Participants reference various mathematical theorems and properties, but there are unresolved assumptions and dependencies on definitions that affect the clarity of the arguments presented.