Problem understanding Wigner's 1939 paper

  • Context: Graduate 
  • Thread starter Thread starter facenian
  • Start date Start date
  • Tags Tags
    Paper
Click For Summary

Discussion Overview

The discussion revolves around the understanding of invariance of subspaces with respect to linear operators, specifically in the context of Wigner's 1939 paper. Participants explore the implications of invariance under linear operators and their inverses, particularly in relation to Lorentz transformations.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether a subspace A, invariant with respect to a linear operator, is also invariant with respect to the inverse operator.
  • Another participant acknowledges a misunderstanding and clarifies that if the subspace is invariant under all Lorentz transformations, it must also be invariant for the inverse transformation.
  • A participant asserts that if the linear operator is invertible and maps A onto A, then invariance holds.
  • It is noted that in the case of infinite dimensional spaces, the definition of invariance may not necessarily hold without additional assumptions.
  • Another participant agrees that more assumptions are needed in infinite dimensions and corrects their earlier statement accordingly.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which invariance holds, particularly in infinite dimensional spaces. There is no consensus on the implications of these conditions.

Contextual Notes

Participants highlight the need for additional assumptions in infinite dimensional spaces to support claims about invariance, indicating that the discussion is limited by these considerations.

facenian
Messages
433
Reaction score
25
if subspace A is invariant with respect to a linear operator, is it true that A is invariant with respect to the inverse operator? if not I think threre is a mistake in Wigners paper in "B. Some immediate simplifications" page 13
 
Physics news on Phys.org
I'm sorry it's my mistake, I didn't read well. At the beginning of the paragraph he says that the subspace is invariant under all lorentz transformation then it must be invariant for [itex]D(L^{-1})[/itex] too
 
facenian said:
if subspace A is invariant with respect to a linear operator, is it true that A is invariant with respect to the inverse operator?
Yes, if the linear operator is invertible and maps A onto A.
 
Last edited:
A. Neumaier said:
Yes, if the linear operator is invertible (and hence maps A onto A).
If invariant is defined as ##T(A) \subseteq A##, then I think this isn't necessarily true in an infinite dimensional space.
 
Samy_A said:
If invariant is defined as ##T(A) \subseteq A##, then I think this isn't necessarily true in an infinite dimensional space.
Indeed. In infinite dimensions you need to assume more. I corrected my statement accordingly.
 

Similar threads

Graduate Velocity dependence of operators in Inonu-Wigner contraction
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad If T is diagonalizable then is restriction operator diagonalizable?
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Wigner-Eckart theorem and reduced matrix element
  • · Replies 3 ·
Replies
3
Views
5K
Undergrad "Spatial Width," "Coherence Length," and the Measurement Problem
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Does Decoherence get rid of all Quantumness?
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Dot product of two vector operators in unusual coordinates
  • · Replies 14 ·
Replies
14
Views
3K
Graduate Wave Function Collapse and Thermodynamic Irreversible Processes
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad The collapse of a quantum state as a joint probability construction
  • · Replies 4 ·
Replies
4
Views
3K
Graduate What are some good pedagogical QM papers?
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Can anyone give me the details of creating a cat state in Circuit QED?
  • · Replies 16 ·
Replies
16
Views
4K