Confusion (1) from Weinberg's QFT.(unitary representation)

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Discussion Overview

The discussion revolves around a question raised by a participant regarding a specific equation from Weinberg's Quantum Field Theory (QFT) concerning the unitary operator and normalization factors. The scope includes theoretical aspects of quantum mechanics, particularly focusing on the implications of unitary transformations and normalization in the context of continuous momentum spectra.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the necessity of a normalization factor \( N(p) \) in the equation \( \Psi_{p,\sigma} \equiv N(p)U(L(p))\Psi_{k,\sigma} \) if \( U \) is indeed a unitary operator.
  • Another participant suggests that \( N(p) \) can be chosen as 1, but this leads to complications in maintaining orthonormality in the scalar products of the states.
  • A participant argues that if \( N(p) = 1 \), it implies that \( U \) transforms normalized vectors into unnormalized ones, raising doubts about the unitarity of \( U \).
  • Concerns are raised about the normalization of states \( \Psi_{p,\sigma} \) and \( \Psi_{k,\sigma} \), with some participants noting that the latter are defined to be orthonormal while the former are not.
  • Discussion includes the implications of the infinite norm of the momentum basis vectors \( \Psi_{k,\sigma} \), suggesting they do not belong to the Hilbert space, which complicates the argument for unitarity.
  • Some participants highlight that the definitions used by Weinberg for different sets of states, despite using similar notation, lead to confusion regarding their orthonormal properties.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the normalization factor and the nature of the states involved. There is no consensus on whether the unitary operator \( U \) can be considered truly unitary given the issues raised about normalization and orthonormality.

Contextual Notes

Participants note that the discussion involves continuous spectra of momentum and that the normalization constants are crucial in defining the properties of the states. The implications of using \( N(p) = 1 \) lead to contradictions, and the definitions of the states in Weinberg's text are seen as inconsistent, contributing to the confusion.

  • #31
A. Neumaier said:
This describes ortho_normality_. Orthogonality is the waker statement assuming p and p' being different and concluding that the inner product vanishes - this is independent of the scaling of the metric. But for orthonormality, the precise factor matters, and different factors define different concepts of orthonormality.
[/QUOTE]
Ok, that's more or less satisfactory to me, thanks.
 
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  • #32
Fredrik said:
You keep coming back to this, but this is a result about inner product spaces, and we're not dealing with an inner product space.

Unfortunately I don't know rigged Hilbert spaces well enough to explain what you should be saying instead, at least not without making a bigger effort than I have time for right now.
Sorry about that. What you said makes sense, but some other people don't seem to insist that this is the crux of the problem, and I have to mention it to make my point sometimes.
 
  • #33
If what I said is the crux of the problem or not depends on if we're trying to explain why your original argument doesn't work, or how things work in the space we're actually dealing with. I've been focusing on the former. Someone who focuses on the latter will of course emphasize other things. I don't think any of them disagrees with me about what's wrong with the argument.
 

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