Dimensions of Hilbert Spaces confusion

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Discussion Overview

The discussion revolves around the dimensions of Hilbert spaces in quantum physics, particularly focusing on the nature of separability and the implications of different cardinalities, including finite, countably infinite, and uncountably infinite dimensions. Participants explore examples from quantum electrodynamics (QED) and quantum field theory (QFT), raising questions about the conditions under which Hilbert spaces are considered non-separable.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants propose that Hilbert spaces can be finite, countably infinite, or uncountably infinite, with examples in quantum physics illustrating these variations.
  • There is a question regarding which specific Hilbert spaces in QED are non-separable, with references to the continuum of directions in 3-space as a potential example.
  • One participant mentions that the dimension of the Hilbert space of the universe should be at least the cardinality of the continuum, citing external sources.
  • Another participant explains that the Fock space can be finite dimensional if the number of modes is finite, but becomes infinite dimensional and non-separable if the allowed modes are continuous.
  • There is a discussion about the confusion surrounding the separability of state spaces, particularly in non-relativistic quantum mechanics, with some asserting that the state space for a single particle is non-separable due to an uncountable basis.
  • One participant acknowledges a misunderstanding and clarifies that for a single particle with one degree of freedom, the Hilbert space is infinite dimensional and separable, while a system with an infinite number of degrees of freedom leads to a non-separable space.
  • There is a debate about the nature of the cardinality of uncountably infinite bases, with references to the continuum hypothesis and the uncertainty surrounding the exact cardinality.
  • Participants discuss the construction of the Hilbert space in standard QFT, noting that it is built from a separable one-particle Hilbert space and questioning whether it is a direct sum or direct product.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which Hilbert spaces are separable or non-separable, and there is no consensus on the specific examples or definitions being discussed. The discussion remains unresolved regarding the implications of different cardinalities and the nature of the Hilbert spaces in question.

Contextual Notes

There are limitations in the discussion related to the assumptions made about the cardinalities and definitions of Hilbert spaces, as well as the mathematical subtleties involved in the transition from finite to infinite degrees of freedom.

  • #31
samalkhaiat, thanks for the excellent summary, but I think you miss some important points.

The Gauß law is identical to the generator of gauge transformations only modulo integration by parts, i.e. surface terms; of course they vanish on the compact 3-torus.

In QCD on compact spaces it is natural that physical states are color-neutral states. And I don't see why there should be a problem at all if the theory predicts strict color-neutrality.

So there may be obstacles on R3 but not on T3.
 
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  • #32
tom.stoer said:
samalkhaiat, thanks for the excellent summary, but I think you miss some important points.

Well, I missed a lot of important points. I already said I would.

The Gauß law is ... i.e. surface terms; of course they vanish on the compact 3-torus.

Do they? Can you show me how you can get rid of the “Schwinger” type terms in the commutator algebra that come from the anomalous divergence of the source current? As far as I know, with careful calculations one ends up with
[ i G_{ a } ( x ) , G_{ b } ( y ) ] = f_{ a b c } G_{ c } ( x ) \delta^{ 3 } ( x – y ) - i S_{ a b } ( A ; x , y ) \ \ (1)
It is known fact that whenever a dynamical current, i.e. a source current in a gauge theory, possesses a anomalous divergence, then also the equal-time commutators for its components must contain anomalous terms. This is because a divergence can be represented by commutator with the translation generators P^{ \mu } and the commutator must be anomalous since the divergence is. Since P^{ 0 } = H contains J^{ \mu }_{ a }, one expects that the equal-time commutator [ J^{ \mu }_{ a } , J^{ 0 }_{ b } ] is anomalous. This commutator is one source of anomaly in (1). The other coming from the commutator of ( D_{ j } \delta / \delta A_{ j } )_{ a } with J^{ 0 }_{ b }. This commutator induces infinitesimal gauge transformation on any explicit dependence of J^{ 0 }_{ b } on the gauge field.

In QCD on compact spaces it is natural that physical states are color-neutral states. And I don't see why there should be a problem at all if the theory predicts strict color-neutrality.

That does not help me to determine how much spin do the quarks and gluons contribute to the proton.
 
  • #33
samalkhaiat said:
See also the following review

[2] Lavelle, M. and McMullan, D. Phys. Rep. 279, 1(1997).
It's so cool that you mention these authors! :cool: Their papers are really interesting. (I've been working with some of their other papers that deal with IR issues and gauge invariance in QED.)

For the benefit of other readers, that paper is freely available here.
 
  • #34
samalkhaiat said:
Can you show me how you can get rid of the “Schwinger” type terms in the commutator algebra that come from the anomalous divergence of the source current? As far as I know, with careful calculations one ends up with
[ i G_{ a } ( x ) , G_{ b } ( y ) ] = f_{ a b c } G_{ c } ( x ) \delta^{ 3 } ( x – y ) - i S_{ a b } ( A ; x , y ) \ \ (1)
With gauge-invariant regularization these Schwinger terms do vanish for gauge currents. (I don't know whether this works for chiral gauge theories)

samalkhaiat said:
That does not help me to determine how much spin do the quarks and gluons contribute to the proton.
Of course it doesn't. I don't see why the local color gauge algebra shall be able to do that. For the color-neutral axial current things may change, but that's not a gauge current.
 
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  • #35
strangerep said:
It's so cool that you mention these authors! :cool: Their papers are really interesting. (I've been working with some of their other papers that deal with IR issues and gauge invariance in QED.)

For the benefit of other readers, that paper is freely available here.

They understand the role played by Poincare’ generators in gauge field theories. And that is good enough for me. Plus, one of them is a good teacher.
 
  • #36
tom.stoer said:
With gauge-invariant regularization these Schwinger terms do vanish for gauge currents. (I don't know whether this works for chiral gauge theories)

I think, I said that the anomalies come from the source current not the gauge field current.


I don't see why the local color gauge algebra shall be able to do that.

That is a very long story. See any paper by E. Leader on the “proton angular momentum controversy”. Or read page 49-51 in the above mentioned review by Lavelle & McMullan.
 

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